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Electron Optics and Its Applications
D. Zernov, Moscow
I. Foundations of Electron Optics
1. Introduction
Electron optics is a comparatively young branch of applied physics, which in the last several years has grown into an independent, extensive field of this science. Electron optics received its name owing to the well-known analogy between electron and optical phenomena. This analogy was already noticed by the first investigators working in this field, who knew that a body placed in the path of cathode rays emitted by the cathode of a Crookes tube casts on a fluorescent screen a sharply outlined shadow of the same shape as would be produced by light rays issuing from the cathode. This fact serves as clear proof of the rectilinear propagation of cathode rays. Thus the behavior of a stream of electrons in space free from the action of electric and magnetic fields proves to be analogous to the behavior of a light ray in an optically homogeneous medium.
At the present time the analogy between light and electron phenomena can be carried considerably further. It is known that both a light flux and a flux of electrons reveal both a wave and a quantum nature. Proceeding from the concepts of modern wave mechanics, we assume that an electron, in its motion, is associated with a phase wave, whose wavelength can be calculated from the formula:
\[ \lambda=\frac{h}{mv}, \]
where \(h\) is Planck’s constant, \(m\) is the mass, and \(v\) is the velocity of the moving corpuscle (electron). The wave nature of electrons is brilliantly confirmed by the well-known experiments of Davisson and Germer, Thomson, and others. These experiments, as is known, made it possible to observe the phenomenon of electron diffraction upon reflection from a surface.
crystal, or when an electron beam passes through a thin crystalline film, entirely analogous to the diffraction of short-wavelength (X-ray) radiation obtained under the same conditions.
Of especially great interest for practical purposes is geometrical electron optics, which deals with the question of electron trajectories in an electric and magnetic force field. In this case the motion of electrons is considered from the optical point of view. The basis for this is the analogy, established as early as the beginning of the nineteenth century by W. Hamilton, between the laws of propagation of a light ray in an optically inhomogeneous medium and the motion of a material particle moving in a conservative force field.
The impetus for the development of geometrical electron optics was the purely practical problem of obtaining narrow electron beams in Braun tubes, used for oscillography and similar purposes. Thus, geometrical electron optics set itself, above all, the goal of finding methods for concentrating electron rays issuing from one common point into some common focus. Electron-optical systems that made it possible to focus electron rays received the name “electron lenses.” As will be seen below, the role of such an electron lens may be played by any electric and magnetic field possessing axial symmetry.
2. Electron-Optical Focusing Systems
In the earliest designs of electron-optical focusing devices for concentrating an electron beam, constant electric or magnetic fields were used, acting on the electrons with a force directed directly toward the axis of the beam. This property is possessed by the radial electrostatic field of a cylindrical capacitor and by the circular magnetic field of a direct current. Thus Budde and Rogowski, for the concentration of an electron beam, attempted to use the device shown in Fig. 1. Along the
Fig. 1. Concentration of an electron beam in the field of a cylindrical capacitor
axis of the beam there is a thin metal rod, charged positively with respect to the cylinder surrounding it. In the radial field of such a cylindrical capacitor the electrons acquire an acceleration directed toward the axis of the beam and are drawn in toward the axis.
However, this system proved to be of little use for the following reason. It is obvious that for complete concentration of both the internal (close to the axis) and the external rays it is necessary that, when passing through the field of the condenser, the external rays be deflected through a larger angle than the internal ones. The latter condition leads to the requirement that the deflecting force increase proportionally to the distance from the axis. In the system described above, however, the deflecting force not only does not increase but, on the contrary, decreases with increasing distance from the axis, varying inversely proportionally to \(r\). Thus the internal rays, which experience the action of a larger radial force, are deflected more strongly than the external ones. As a result, rays inclined to the axis in different ways come together at different points. In optical language, such an electron lens has an entirely inadmissible amount of spherical aberration. The same may also be said of the field of a direct current (located along the axis of the beam), where the radial force is likewise proportional to
\[ \frac{1}{r}. \]
Thus the two systems considered cannot be regarded as a satisfactory solution of the problem of focusing electron rays. A necessary condition for electric or magnetic fields to be considered as an electron lens is, as indicated above, the proportionality of the radial force with which these fields act on moving electrons to the distance of the electrons from the axis of the beam. Let us see what electric and magnetic fields satisfy this requirement.
By the theoretical work of Busch and a number of other authors it was proved that a sufficiently narrow beam of electrons is focused by any constant electric or magnetic field possessing axial symmetry. Such a field may be created by any system of conductors possessing axial symmetry. It is assumed here that the vectors of the electric and magnetic field strengths and their derivatives are continuous and single-valued functions of the coordinates throughout the entire space penetrated by the electron rays (in the two examples considered above, this requirement is not fulfilled for the surface of the central electrode).
Before proceeding directly to the proof of this fundamental proposition of electron optics, let us consider the mechanism of focusing electron rays by electric and magnetic fields in several very simple examples. The simplest example of the focusing of electron rays by a magnetic field is the case of a uniform longitudinal magnetic field whose direction coincides with the direction of the axis of the beam.
Let us consider an electron \(e\) (Fig. 2), emerging from some point situated on the axis of the beam, with initial velocity \(v_0\), making with the axis of the beam (the \(z\)-axis) and, consequently, with the direction of the magnetic-field strength vector \(\mathbf{H}\), a certain pro-
arbitrary angle \(\alpha\). The magnitude of the Lorentz force acting on an electron moving in a magnetic field is then determined from the relation
\[ F = evH \sin \alpha = ev_{xy}H, \]
where \(v_{xy}\) is the projection of the electron’s initial velocity onto the plane \(xy\).
Fig. 2
This force is directed perpendicular to the plane passing through the vectors \(v\) and \(H\), and, consequently, is perpendicular to the \(z\)-axis, as well as to the velocity components in the direction of the \(z\)-axis and in the \(xy\)-plane (\(v_z\) and \(v_{xy}\)). Since the component of the force in the direction of the \(z\)-axis is, evidently, equal to zero at every instant of time, the motion of the electron in this direction will be uniform, with velocity \(v_z = v_0 \cos \alpha\). In the plane perpendicular to the \(z\)-axis, the motion will be rotational, with constant velocity \(v_{xy} = v_0 \sin \alpha\) (since the force always remains perpendicular to the velocity component in this plane), under the action of the centripetal force \(F = ev_{xy}H\). The radius of the circular orbit described by the projection of the electron onto the plane \(xy\) may be found on the basis of the known law of dynamics
\[ \frac{mv_{xy}^{2}}{R} = ev_{xy}H, \]
whence
\[ R = \frac{mv_0}{eH}\cdot \sin \alpha \tag{1} \]
Thus the radius of this circle depends on the magnitude of the initial angle of the electron trajectory with the \(z\)-axis (the angle of emission).
Let us now determine the angular velocity of rotation of the electron in the \(xy\)-plane. Setting \(v_{xy} = \omega R\), we have
\[ \omega = \frac{eH}{m}. \tag{2} \]
Expressions (1) and (2) show that electrons emerging from a certain point located on the beam axis describe, in the \(xy\)-plane, circles of different radii (depending on the magnitude and direction of the initial velocity), but with the same angular velocity, and again intersect the \(z\)-axis after equal intervals of time (Fig. 3). Suppose that the initial velocities of all electrons are equal in magnitude, and restrict ourselves to considering a sufficiently narrow beam, so that \(\cos \alpha\) may be taken as equal to unity. Then, taking into account that the motion of all elec-
trons in the direction of the \(z\)-axis occurs with the constant velocity \(v_z = v_0 \cos \alpha \simeq v_0\), we arrive at the conclusion that all electrons which have emerged from some point of the \(z\)-axis will again intersect this axis at one and the same point. The trajectories of the electrons in space will then be helices, whose pitch \(d\) is equal to the period of revolution of the electron around the circumference, \(T\), multiplied by the velocity of motion along the \(z\)-axis \((v_z)\)
\[ d = T \cdot v_z = \frac{2\pi m v_z}{eH}. \]
In Fig. 4 a model is presented of the trajectories of electrons in a homogeneous magnetic field. Electrons emerging from a common point, describing helical lines, again gather at a common focus located on the optical axis of the system (the axis of the beam). Thus the action of the magnetic field on electron rays proves to be, to a certain degree, analogous to the action of an ordinary glass converging lens on light rays.
Fig. 3
Fig. 4. Model of the trajectories of electrons in a homogeneous magnetic field
The mechanism of focusing electron rays by an electric field is essentially different from the mechanism of focusing by a magnetic field discussed above. Here a homogeneous longitudinal field creates no radial force. Applying Hamilton’s analogy, one can show that an electron ray in an electrostatic field behaves in exactly the same way as a light ray in an optically inhomogeneous medium, with the role of the refractive index \(n\) in this case being played by \(\sqrt{V}\), where \(V\) is the potential of the electrostatic field. The only difference is that in optics we usually deal with a discontinuous change of the refractive index (when a ray passes from one medium into another), whereas the potential of an electrostatic field varies continuously from point to point. Here too, however, a surface may be conceived on which the refractive index undergoes a discontinuous change. Such a surface may be provided by an electric double layer. Let us imagine two parallel plates, situated at an infinitely small distance from one another, one of which has-
has potential \(V_1\), and the other \(V_2\). Such two plates divide space into two parts, each of which has a constant value of the potential; moreover, in passing from one side of such a double electric layer to the other, the potential undergoes a discontinuous change. Let us suppose that the surface \(MN\) (Fig. 5) separates two regions having constant, but different, values of the potential. Let the potential above \(MN\) be \(V_1\), and below it \(V_2\), with \(V_2 > V_1\). Now let us consider an electron moving in the direction \(SA\) with velocity \(v_1\). As it passes through the double layer, a force acts on it normal to the surface \(MN\), in the direction \(AO'\). As a result, after the electron has passed through the surface \(MN\), its velocity changes and acquires the value \(v_2\). Since the force causing the change of velocity is normal to the surface \(MN\), only the normal component of the velocity \(v_n\) is changed, while the tangential component of the velocity \(v_\tau\) remains unchanged. From the drawing (Fig. 5) it follows that \(v_\tau = v_1 \sin i = v_2 \sin r\), whence
Fig. 5. Refraction of an electron beam at the surface of a double layer
\[ \frac{\sin i}{\sin r} = \frac{v_2}{v_1} = n. \tag{3} \]
This relation is the well-known law of geometrical optics. The ratio \(\frac{v_2}{v_1}\) thus plays the role of the refractive index of the second medium relative to the first.
The expression obtained for the refractive index may be given a somewhat different form. The work of the field in the transition of the electron from the first medium into the second is equal to \(e(V_2 - V_1)\). On the basis of the law of conservation of energy we may write
\[ \frac{mv_2^2}{2} = \frac{mv_1^2}{2} + e(V_2 - V_1), \]
whence, calculating \(\frac{v_2}{v_1}\), we have
\[ n = \frac{v_2}{v_1} = \sqrt{\,1 + \frac{e(V_2 - V_1)}{\dfrac{mv_1^2}{2}}\,}. \]
Assuming that the initial velocity of the electron \(v_1\) corresponds to the potential \(V_1\), we have
\[ \frac{mv_1^2}{2}=eV_1, \]
and, consequently, the refractive index proves to be equal to
\[ n=\sqrt{\frac{V_2}{V_1}}. \tag{4} \]
The last relation may also be regarded as a special case of the general law established by Hamilton’s analogy. If \(V_2 < V_1\) and if \(e(V_2 - V_1)\) exceeds, in absolute value, the part of the kinetic energy of the electron corresponding to the normal component of its velocity,
\[ \frac{m(v_1\cos i)^2}{2}, \]
then the electron will not pass through the double layer at all, but will be reflected from its surface; moreover, the normal component of its velocity remains unchanged in magnitude, but reverses its direction. The direction of the reflected electron then forms, evidently, the same angle \(i\) with the normal to the surface as does the direction of the incident electron. This latter case is analogous to the well-known optical phenomenon of total internal reflection. Thus the behavior of an electron beam at the surface of a double electric layer proves to be entirely analogous to the behavior of a light beam at the boundary between two media possessing different refractive indices.
Knoll and Ruska made use of this circumstance, constructing a number of electron-optical systems in which the electric double layer was realized by means of two closely spaced, differently charged wire meshes. By giving these meshes one form or another, it is possible to produce any electron-optical system analogous to an optical system possessing the corresponding form of refracting surfaces. In Fig. 6 various electron-optical systems of Knoll are presented.
The laws known from optics for reflection from plane and spherical mirrors, as well as the laws of refraction in prisms and lenses, can naturally be extended also to Knoll’s electron-optical systems. In particular, the refraction of electron beams by Knoll electron lenses must evidently obey the well-known law
\[ \frac{1}{a}+\frac{1}{b}=\frac{1}{f}, \]
where the focal length \(f\) is determined by the known relation
\[ \frac{1}{f}=(n-1)\left(\frac{1}{r_1}+\frac{1}{r_2}\right), \]
where \(n\) is the refractive index of the electric double layer, whose expression was given above, and \(r_1\) and \(r_2\) are the radii of curvature of the refracting surfaces (grids).
| Shape | Systems with acceleration of motion | Shape | Systems with deceleration |
|---|---|---|---|
| Plane | [diagram of electron rays reflected/refracted at a plane surface] | ||
| Concave mirror | [diagram of electron rays reflected by a concave mirror] | ||
| Prism Refraction toward the base |
[diagram of prism with refraction toward the base, marked \(O\), \(\lambda\), \(+\)] | Prism with refraction upward | [diagram of prism with refraction upward, marked \(O\), \(\lambda\)] |
| Converging lens | [diagram of converging lens, marked \(O\), \(\lambda\), \(+\)] | Diverging lens | [diagram of diverging lens, marked \(O\), \(\lambda\), \(-\)] |
| Diverging lens | [diagram of diverging lens, marked \(O\), \(+\)] | Converging lens | [diagram of converging lens, marked \(O\)] |
Fig. 6. Knoll’s electron-optical systems
These systems made it possible to verify experimentally the laws of geometrical electron optics. In practice, however, they were of no importance, owing to the impossibility of fabricating the metal grids required for realizing the electric double layer—grids which would simultaneously satisfy the requirement of good permeability for electrons and of the greatest possible homogeneity of the electric field in the space between them. On the flight of electrons
inevitably, the nonuniformity of the field, which occurs at the edges of the mesh cells, will have an effect. This nonuniformity will be the greater the sparser the mesh is. On the other hand, if the mesh is too fine, a large part of the electrons will be stopped by it, which will lead to a considerable decrease in the intensity of the electron beam. At present—
Fig. 7. Potential field of an electric lens
the focusing of an electron beam by an electrostatic field is carried out differently. Axially symmetric electrostatic fields are used as electron lenses; these fields can be created by electrodes possessing axial symmetry. An example of such an electron-optical system may be—
pose the electron lens shown in Fig. 7. It consists of three diaphragms, of which the two outer ones are connected to each other, while the inner one is at a higher or lower potential. The equipotential surfaces shown in Fig. 7 correspond to surfaces of constant refractive index \((n=\mathrm{const}.)\). To clarify the action of such a system, one may assume that \(n\) is constant between two neighboring equipotential surfaces, and changes discontinuously at the surfaces themselves. Then the whole system may be regarded as a system of lenses possessing different refractive indices, the equipotential surfaces acting as refracting surfaces in optics. The natural curvature of the equipotential surfaces of an axially symmetric electrostatic field provides the focusing action of the entire system.
In reality, the refractive index inside such a lens will, of course, vary continuously from point to point, since the potential of the electrostatic field is a continuous function of the coordinates. However, in optics too, lenses are known that possess a continuously varying refractive index. An example of such a lens is Exner’s gelatin-glycerin cylinder, in which, owing to the diffusion of glycerin, a radial concentration drop is established and, together with it, a radial variation of the refractive index. Such a cylinder exerts a good focusing action on light rays.
3. Determination of the optical constants of an electron lens. Equation of the trajectory of an electron in an axially symmetric field
For a complete characterization of the focusing properties of any optical system, it is necessary, as is known, to know the position of its principal planes and the magnitude of its focal lengths. In the case of an electron lens, the position of its principal points can be found by calculating the trajectories of electrons in a given force field. Indeed, if the trajectories are known of two electrons entering the electron lens from two opposite sides, so that both initially move parallel to the optical axis of the system (the axis of symmetry), then the points at which these trajectories, on leaving the lens, intersect the optical axis determine the positions of the principal foci, while the planes perpendicular to the optical axis and passing through the points of intersection of the initial and final directions of motion of each electron will be the principal planes of the system. The focal lengths are then determined as the distances between the focus and the corresponding principal plane. In Fig. 8 two such “fundamental trajectories” and the principal points of an electrostatic electron lens (the field of two cylinders), found by the method described above, are shown. Thus, the determination of the focusing properties of any electron lens reduces above all to the determination
trajectories of electrons in the given field of the lens. The equation of the trajectory of an electron moving in an axially symmetric electrostatic or magnetic field can be found from the laws of classical dynamics. The equation of motion of the electron
Fig. 8. Determination of the principal points of an electron lens
in the general case of the presence of both an electric and a magnetic field may be represented in the form
\[ m\frac{d^{2}s}{dt^{2}}=-e\{\mathbf{E}+[\mathbf{v}.\mathbf{H}]\}. \tag{5} \]
or, in coordinate form,
\[ m\ddot{x}=-e\{E_x+(\dot{y}H_z-\dot{z}H_y)\}, \tag{6} \]
\[ m\ddot{y}=-e\{E_y+(\dot{z}H_x-\dot{x}H_z)\}, \tag{7} \]
\[ m\ddot{z}=-e\{E_z+(\dot{x}H_y-\dot{y}H_x)\}. \tag{8} \]
(Dots denote derivatives with respect to \(t\).) The equation of the trajectory will be obtained if time is eliminated from these equations.
Let us note that from equations (6), (7), and (8) one can easily obtain the energy equation. Multiplying equation (6) by \(\dot{x}\), (7) by \(\dot{y}\), and (8) by \(\dot{z}\), and adding them, we find
\[ m\ddot{x}\dot{x}+m\ddot{y}\dot{y}+m\ddot{z}\dot{z} =-e\{\dot{x}E_x+\dot{y}E_y+\dot{z}E_z\}. \]
Taking into account that \(E=-\operatorname{grad} V\), we have
\[ m\ddot{x}\dot{x}+m\ddot{y}\dot{y}+m\ddot{z}\dot{z} = e\left\{\dot{x}\frac{\partial V}{\partial x} +\dot{y}\frac{\partial V}{\partial y} +\dot{z}\frac{\partial V}{\partial z}\right\} \]
or
\[ \frac{d}{dt}\left\{\frac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)\right\} = e\frac{dV}{dt}, \]
whence
\[ \frac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)=eV+\mathrm{const}. \]
Assuming that the electron velocity becomes zero at the point having zero potential, we finally have
\[ \frac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)=eV. \tag{9} \]
Thus the kinetic energy of the electron depends only on the magnitude of the potential of the electrostatic field.
We shall take as a basis the first two equations of motion (6) and (7) and the energy equation (9). Owing to axial symmetry, the components of the vectors \(\mathbf{E}\) and \(\mathbf{H}\) along the axes \(x\) and \(y\) may be expressed through their components in the direction of the radius vector \(r\):
\[ E_x=\frac{x}{r}E_r \quad \text{and} \quad E_y=\frac{y}{r}E_r. \]
Similarly,
\[ H_x=\frac{x}{r}H_r \quad \text{and} \quad H_y=\frac{y}{r}H_r. \]
Thus we arrive at the following system of equations:
\[ \left\{ \begin{aligned} m\ddot{x} &=-e\frac{x}{r}E_r-e\dot{y}H_z+e\frac{y}{r}\dot{z}H_r,\\ m\ddot{y} &=-e\frac{y}{r}E_r+e\dot{x}H_z-e\frac{x}{r}\dot{z}H_r,\\ \frac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)&=eV. \end{aligned} \right. \tag{10} \]
The subsequent computations are simplified if we introduce the complex variables
\[ w=x+iy \quad \text{and} \quad \overline{w}=x-iy. \tag{11} \]
Making the change of variables in equations (10), we obtain the equation of motion in the form
\[ m\ddot{w}=-e\frac{w}{r}E_r+ie\dot{w}H_z-ie\frac{w}{r}\dot{z}H_r, \tag{12} \]
and the energy equation in the form
\[ \frac{m}{2}\left(\dot{z}^{2}+\dot{w}\cdot\dot{\overline{w}}\right)=eV. \tag{13} \]
In order to eliminate time from these equations, we pass from derivatives with respect to \(t\) to derivatives with respect to \(z\):
\[ \frac{d}{dt}=\dot{z}\frac{d}{dz}. \]
Then equations (12) and (13) assume the following form (primes denote derivatives with respect to \(z\)):
\[ m\dot{z}\frac{d\dot{z}\,w'}{dz} = -e\frac{w}{r}E_r + ie\dot{z}\left(w'H_z-\frac{w}{r}H_r\right), \tag{14} \]
$$ \frac{m}{2}\dot z^{2}\left(1+w'\cdot \overline{w'}\right)=eV . \tag{15} $$
Determining $\dot z$ from equation (15) and substituting into equation (14), we obtain an equation independent of time
$$ \sqrt{\frac{2V}{1+w'\cdot \overline{w'}}}\cdot \frac{d}{dz}\left(w'\sqrt{\frac{2V}{1+w'\cdot \overline{w'}}}\right)= $$
$$ =-\frac{\overline w}{r}E_r +i\sqrt{\frac{e}{m}}\cdot \sqrt{\frac{2V}{1+w'\cdot \overline{w'}}} \left(w'H_z-\frac{w}{r}H_r\right). \tag{16} $$
The last equation is a differential equation for the trajectory of an electron in an axially symmetric field. The components of the vectors $\mathbf E$ and $\mathbf H$ entering equation (16) can be determined from the potential of the electrostatic field $V$ and the vector potential of the magnetic field $\mathbf A$ on the basis of the known relations
$$ \mathbf E=-\operatorname{grad} V \quad \text{and} \quad \mathbf H=\operatorname{rot}\mathbf A . \tag{17} $$
Thus, if the potential of the electrostatic field and the vector potential of the magnetic field were known functions of the coordinates $r$ and $z$, or, equivalently, functions of $w$, $\overline w$ and $z$ (since $r^{2}=w\cdot \overline w$), then equation (16) would give, in principle, the possibility of finding $w$, and consequently also the coordinates $x$ and $y$ as functions of $z$, i.e. of finding the equation of the electron trajectory.
4. Calculation of the electric and magnetic fields of an electron lens
It follows from the foregoing that, in order to compute the trajectory of an electron in the field of a lens, it is first of all necessary to know the distribution of the potential of the electrostatic field and of the vector potential of the magnetic field in the part of space penetrated by the electron rays.
The distribution of the potential of the electrostatic field can be found either if the distribution of charges is known [the volume density $\rho(x,y,z)$], in which case
$$ V=\iiint \frac{\rho(x,y,z)}{r}\,dx\,dy\,dz, \tag{18} $$
or if the shape and potentials of the conductors forming the field are known. The latter case is considerably more interesting in practice. In this case the distribution of the potential can be found by solving Poisson’s equation
$$ \Delta V=-4\pi\rho . \tag{19} $$
If the space charge may be regarded as equal to zero (as is usually the case in electron lenses), then the potential is determined as the solution of Laplace’s equation
\[ \Delta V = 0. \tag{20} \]
In integrating these equations it is necessary to take into account the boundary conditions: the potential assumes prescribed values on the surfaces of the conductors.
In the majority of practical cases, however, it proves impossible to obtain a simple analytic solution of these equations. Therefore, in order to find the distribution of the potential in space one must proceed in another way. It is easy to show that knowledge of the distribution of the potential on the optical axis of an electron lens (the axis of symmetry) is sufficient for the distribution of the potential to be determined, to any desired degree of accuracy, throughout the space surrounding the axis.
To prove this, let us use Laplace’s equation (20). Introducing cylindrical coordinates \(r, z\) and \(\psi\), and taking into account that the field possesses axial symmetry, i.e. that
\[ E_\psi = -\frac{1}{r}\frac{\partial V}{\partial \psi}=0, \]
we have
\[ \frac{\partial^2 V}{\partial r^2}+\frac{1}{r}\frac{\partial V}{\partial r}+\frac{\partial^2 V}{\partial z^2}=0. \tag{21} \]
Let \(V(r,z)\) be a solution of equation (21). Represent \(V(r,z)\) in the form of an infinite series
\[ V(r,z)=V_0(z)+r^2V_2(z)+r^4V_4(z)+\ldots \tag{22} \]
[terms with odd powers of \(r\) are absent, since, owing to axial symmetry, a change in the sign of \(r\) must not affect the value of \(V(r,z)\)].
Substituting (22) into (21), collecting the coefficients of equal powers of \(r\), and equating them to zero, we determine the coefficients of the series (22). In this way we find
\[ V(r,z)=V_0(z)-V_0''(z)\left(\frac{r}{2}\right)^2+\frac{1}{2^2}V_0^{(4)}(z)\left(\frac{r}{2}\right)^4-\ldots \]
or
\[ V(r,z)=\sum_{n=0}^{\infty}\frac{(-1)^n V_0^{(2n)}}{(n!)^2}\left(\frac{r}{2}\right)^{2n}. \tag{23} \]
Putting \(r=0\) in this expression, we find
\[ V(0,z)=V_0(z), \]
i.e. \(V_0(z)\) represents the distribution of the potential on the optical axis. Thus the distribution of the potential in space can be found by means of (23), if the distribution of the potential on the optical axis of the lens is known.
The vector potential of the magnetic field \(\mathbf A\) is determined by the formula
\[ \mathbf A=-4\pi \iiint \frac{\mathbf j}{r}\,dx\,dy\,dz, \tag{24} \]
where \(\mathbf j\) is the current-density vector. In the case where the field possesses axial symmetry, an expression analogous to expression (23) for the potential of an axially symmetric electrostatic field may be obtained for the vector potential.
Since the currents forming an axially symmetric magnetic field have no component in the direction of the \(z\)-axis, and since the vector potential is composed, as is evident from formula (24), of vectors parallel to the current vectors, it is obvious that the component of the vector \(\mathbf A\) in the direction of the \(z\)-axis \((A_z)\) is in this case equal to zero. From symmetry considerations it also follows that the component of \(\mathbf A\) in the direction of the radius vector \(r\,(A_r)\) is equal to zero. Thus, for an axially symmetric magnetic field,
\[ A_z=A_r=0 \quad \text{and} \quad A_\psi=A, \]
i.e., the vector lines of the vector potential have the form of concentric circles whose common axis is the optical axis of the system.
As is known, in a space free of current, to the vector \(\mathbf H\) one can assign a single-valued scalar potential \(\varphi\), satisfying Laplace’s equation
\[ \Delta\varphi=0, \]
or, for an axially symmetric field,
\[ \frac{\partial^2\varphi}{\partial r^2}+\frac{1}{r}\frac{\partial\varphi}{\partial r}+\frac{\partial^2\varphi}{\partial z^2}=0. \tag{25} \]
The solution of this equation, as was already shown above, can be represented in the form of an infinite series
\[ \varphi(r,z)=\varphi_0(z)-\varphi_0''(z)\left(\frac r2\right)^2+\frac{1}{2^2}\varphi_0^{(4)}(z)\left(\frac r2\right)^4-\cdots \]
or
\[ \varphi(r,z)=\sum_{n=0}^{\infty}\frac{(-1)^n\varphi_0^{(2n)}(z)}{(n!)^2}\left(\frac r2\right)^{2n}, \tag{26} \]
where \(\varphi_0(z)=\varphi(0,z)\).
Further, since \(\mathbf H=-\operatorname{grad}\varphi\) and, on the other hand, \(\mathbf H=\operatorname{rot}\mathbf A\), then
\[ \operatorname{rot}\mathbf A=-\operatorname{grad}\varphi . \tag{27} \]
or, in cylindrical coordinates,
\[ \begin{gathered} \frac{1}{r}\frac{\partial A_z}{\partial \psi}-\frac{\partial A_\psi}{\partial z}=-\frac{\partial \varphi}{\partial r},\\ \frac{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r}=-\frac{1}{r}\frac{\partial \varphi}{\partial \psi},\\ \frac{1}{r}\frac{\partial (rA_\psi)}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \psi}=-\frac{\partial \varphi}{\partial z}; \end{gathered} \tag{27'} \]
since, owing to axial symmetry, \(A_z=A_r=0\) and \(A_\psi=A\), it follows directly from equations \((27')\) that
\[ \frac{\partial A}{\partial z}=\frac{\partial \varphi}{\partial r}. \]
From equality (26),
\[ \frac{\partial \varphi}{\partial r} = z \sum_{n=0}^{\infty} \frac{(-1)^n\varphi_0^{(2n+2)}}{n!(n+1)!} \left(\frac{r}{2}\right)^{2n+1}. \]
Carrying out the integration with respect to \(z\), we find
\[ A= -\sum_{n=0}^{\infty} \frac{(-1)^n\varphi_0^{(2n+1)}}{n!(n+1)!} \left(\frac{r}{2}\right)^{2n+1}. \tag{28} \]
Let us denote the magnetic-field intensity vector measured on the optical axis of the system by \(H_0\). Then
\[ H_0=H(0,z)=-\operatorname{grad}\varphi_0=-\varphi_0'. \]
Replacing in expression (28) the derivatives of the scalar potential of the magnetic field \(\varphi_0'\) by the measured value of the intensity \(H_0\) on the optical axis, we obtain the final expression for the vector potential of an axially symmetric magnetic field in the form
\[ A= \sum_{n=0}^{\infty} \frac{(-1)^n H_0^{(2n)}}{n!(n+1)!} \left(\frac{r}{2}\right)^{2n+1}. \tag{29} \]
Thus, the values of the vector potential of the magnetic field in the space surrounding the axis can be determined with any degree of accuracy if the distribution of the magnetic field on the optical axis of the electron lens is known.
Knowing the distribution of the potential of the electrostatic field \(V(r,z)\) and the vector potential of the magnetic field \(A(r,z)\), one can determine the quantities \(E_r\), \(H_r\), and \(H_z\) entering equation (16) from the formulas
\[ E_r=-\frac{\partial V}{\partial r}; \qquad H_r=\operatorname{rot}_r A=-\frac{\partial A}{\partial z}; \qquad H_z=\operatorname{rot}_z A=\frac{1}{r}\frac{\partial (rA)}{\partial r}. \tag{30} \]
5. Experimental methods for determining the distribution of the electric and magnetic fields on the optical axis of an electron lens
It was shown above that, in order to calculate the electric and magnetic fields of an electron lens, it is necessary to know the distribution of the potential of the electrostatic field and of the magnetic-field intensity vector on the optical axis of the system. In most practical cases, however, this distribution cannot be expressed in a simple analytical form. In these cases the field distribution is found experimentally.
Fig. 9. Diagram for measuring the potential distribution in an electrolytic bath
For determining the distribution of the potential of an electrostatic field, the following method is usually used. An enlarged model of the electrodes is immersed in an electrolytic bath, and the potential distribution is measured by means of a probe connected to a sensitive bridge (Fig. 9). The potential distribution in the electrolytic bath is then obtained in the same way as the distribution of the potential of the electrostatic field produced by the given system of electrodes. Indeed, if we denote by $\mathbf{j}$ the current density and by $\lambda$ the specific conductivity of the electrolyte, then
\[ \mathbf{j}=\lambda \mathbf{E}=-\lambda \operatorname{grad} V . \]
Since the vector $\mathbf{j}$ in the electrolyte has no sources, $\operatorname{div}\mathbf{j}=0$, and consequently,
\[ -\lambda \operatorname{div}\operatorname{grad} V=-\lambda \Delta V=0, \]
whence
\[ \Delta V=0. \]
Thus the potential distribution in the electrolyte satisfies the same Laplace equation as does the potential, free from space charges, of the electrostatic field. If the boundary conditions are the same as in the case of the electrostatic field, then the potential distribution will have the same form. It is necessary, however, to bear in mind the influence of the walls of the electrolytic bath and to take special measures to eliminate it.
The distribution of the magnetic field on the optical axis of an electron lens can most simply be found in the following way. A small flat loop (or coil) is placed at the point of the magnetic field under investigation so that the plane of the loop is perpendicular to the direction of the lines of force. Then the loop is rapidly removed beyond the limits of the magnetic field. The induced e.m.f. thereby produced is measured with a ballistic galvanometer. This
e.m.f., is obviously proportional to the magnetic-field strength at the point under investigation, to the area of the loop, and to the number of turns of the loop. Thus, knowing the magnitude of the induced e.m.f. and the dimensions of the loop, one can calculate the magnitude of the magnetic-field strength at the point under investigation.
6. Paraxial electrons
In calculating focusing electron-optical systems, one may, in the first approximation, confine oneself—as is usually done also in light optics—to computing the trajectories of electrons very close to the optical axis of the lens. Such paraxial electrons are characterized by the fact that their distances from the axis, as well as the angles of inclination of the trajectory to the axis, are quantities so small that, in the first approximation, second and higher powers of these quantities may be neglected. Consequently, in this case, in equation (16) and in expressions (23) and (29) one may neglect all terms containing the quantities \(r\), \(w\), \(\overline{w}\), \(w'\), and \(\overline{w'}\) in powers higher than the first, as well as their products. The equation of the trajectory (16), under these assumptions, after replacing the quantities \(V\), \(E_r\), \(H_r\), and \(H_z\) occurring there on the basis of formulas (23), (29), and (30), takes the following form:
\[ \sqrt{V_0}\,\frac{d}{dz}\left(w'\sqrt{V_0}\right) = -\frac{w}{4}V_0'' + i\sqrt{\frac{eV_0}{8m}}\left(2w'H_0+wH_0'\right). \tag{31} \]
The imaginary term on the right-hand side of the last equation shows that the magnetic field entails a rotation of the plane of the electron trajectory. This rotation, superposed on the plane motion, is known as “Larmor precession.”
For practical purposes it is more convenient to use a differential equation of the trajectory free of imaginary quantities. Such an equation can be obtained by transforming equation (31), introducing a new coordinate \(u\), rotating with the “Larmor frequency” \(\omega_L\),
\[ u=w\cdot e^{-i\int \omega_L\,dt}, \tag{32} \]
where
\[ \omega_L=\frac{eH_0}{2m}. \]
We transform the integrand in equality (32) as follows:
\[ \omega_L\,dt=\frac{eH_0}{2m}\,dt=\frac{eH_0}{2m}\,\frac{dz}{\dot z}. \tag{33} \]
From the energy equation (15) it follows directly that for paraxial electrons
\[ \dot z=\sqrt{\frac{2eV_0}{m}}. \]
Substituting the obtained value of \(\dot z\) into (33), we find
\[ \omega_L\,dt=\sqrt{\frac{e}{8mV_0}}\cdot H_0\,dz, \tag{34} \]
Thus, we introduce new complex variables
\[ u=w\cdot e^{-i\chi}\quad \text{and}\quad \bar u=\bar w\cdot e^{i\chi}, \]
where
\[ \left. \begin{aligned} \chi&=\sqrt{\frac{e}{8m}}\int_{z_a}^{z}\frac{H_0}{\sqrt{V_0}}\,dz . \end{aligned} \right\} \tag{35} \]
Then
\[ \left. \begin{aligned} w&=ue^{i\chi},\\ w'&=\left(u'+\frac{i}{4}u\sqrt{\frac{2e}{mV_0}}\cdot H_0\right)e^{i\chi},\\ w''&=\left[u''-\frac{euH_0^2}{8mV_0} +i\sqrt{\frac{2e}{mV_0}}\left(\frac{u'}{2}H_0+\frac{u}{4}H'_0-\frac{u}{8}\frac{V'_0}{V_0}H_0\right)\right]e^{i\chi}. \end{aligned} \right\} \tag{36} \]
Substituting the obtained values of \(w\), \(w'\), and \(w''\) into equation (31), we arrive at an equation free of imaginary quantities,
\[ V_0u''+\frac{1}{2}V'_0u'+\frac{21}{4}V''_0u+\frac{eH_0^2}{8m}u=0. \tag{37} \]
Equation (37) may serve as the basis for calculating the trajectories of paraxial electrons in axially symmetric electrostatic and magnetic fields. This equation is linear and homogeneous with respect to \(u\); consequently, its solution can be represented in the form
\[ u=C_\alpha r_\alpha(z)+C_\beta r_\beta(z). \tag{38} \]
where \(r_\alpha(z)\) and \(r_\beta(z)\) are two particular independent integrals of equation (37); \(C_\alpha\) and \(C_\beta\) are constants of integration determined by the initial conditions.
An analogous expression can be obtained for the conjugate of the complex quantity \(u\)
\[ \overline{u}=\overline{C}_{\alpha}r_{\alpha}(z)+\overline{C}_{\beta}r_{\beta}(z). \tag{38'} \]
Knowing \(u\), one can, by separating the real and imaginary parts, easily find the equation of the trajectory in Cartesian coordinates on the basis of the equality
\[ x+iy=w=ue^{i\chi}. \]
However, owing to the axial symmetry, in determining the trajectory it is considerably more convenient in practice to use cylindrical coordinates.
Let us introduce cylindrical coordinates \(z,\ r,\ \psi\) and represent the variables \(w\) and \(\overline{w}\) in the form
\[ w=re^{i\psi};\quad \overline{w}=re^{-i\psi}, \]
but, on the basis of (35),
\[ ue^{i\chi}=re^{i\psi};\quad \overline{u}e^{-i\chi}=re^{-i\psi}, \]
whence
\[ u=re^{i(\psi-\chi)};\quad \overline{u}=re^{-i(\psi-\chi)}. \]
Denote \(\psi-\chi=\theta\),
then
\[ u=re^{i\theta};\quad \overline{u}=re^{-i\theta}. \tag{39} \]
Thus, if the quantity \(u\) has been found, then by separating the real and imaginary parts the equation of the trajectory in cylindrical coordinates can be found. The coordinate \(r\) is then determined as the modulus of the complex quantity \(u\); its argument \(\theta\), in sum with the quantity \(\chi\), determines the angular coordinate \(\psi\),
\[ \psi=\sqrt{\frac{e}{8m}}\int_{z_a}^{z}\frac{H_0}{\sqrt{V_0}}\,dz+\theta. \tag{40} \]
Let us now consider equality (38). Suppose that in some plane \(z=z_a\) (the object plane) one of the functions, for example \(r_{\alpha}\), vanishes, while the other, \(r_{\beta}\), takes the value 1. Further suppose that in this plane \(r'_{\alpha}(z_a)=1\) and \(r'_{\beta}(z_a)=0\). Determine the constants \(C_{\alpha}\) and \(C_{\beta}\) from the initial conditions that in the plane \(z=z_a\) the quantities \(r,\ r',\ \psi\) and \(\psi'\) take the prescribed values \(r_a,\ r'_a,\ \psi_a\) and \(\psi'_a\), and for simplicity set \(\psi_a=0\). Then, on the basis of (38), (39), and (40), we find
\[ C_{\beta}=r_a;\quad C_{\alpha}=r'_a+ir_a\theta'_a =r'_a+ ir_a\left(\psi'_a-\sqrt{\frac{e}{8mV_a}}\cdot H_a\right). \tag{41} \]
Let us consider another plane \(z=z_b\). Suppose that in this plane \(r_\alpha\) again vanishes, i.e. \(r_\alpha(z_b)=0\). Then in this plane \(u_b=r_a r_\beta(z_b)\). Since the quantity \(u_b\) in the present case has a real value, it is obvious that in the plane \(z=z_b\), \(\theta_b=0\) and \(u_b=r_b\). Consequently, in the plane \(z=z_b\),
\[ r_b=r_a\cdot r_\beta(z_b) \quad\text{and}\quad \psi_b=\sqrt{\frac{e}{8m}}\int_{z_a}^{z_b}\frac{H_0}{\sqrt{V_0}}\,dz. \]
Thus we arrive at the following conclusion: all electrons emerging from some point of the plane \(z=z_a\), situated at a distance \(r_a\) from the axis, will again be gathered in the plane \(z=z_b\) into one point, situated at a distance \(r_b\) from the axis, independently of the initial angle of inclination of the trajectory to the \(z\)-axis. In the plane \(z=z_b\) we thus obtain an image of the plane \(z=z_a\) magnified in the ratio \(r_\beta(z_b):1\). In the presence of a magnetic field the entire image is rotated through the angle \(\psi_b\). The above reasoning proves the fundamental proposition of geometrical electron optics: every axially symmetric electrostatic or magnetic field exerts a focusing action on electron rays. In other words, every axially symmetric electrostatic or magnetic field may be regarded as an “electron lens.”
In a number of practical cases one may assume that at the initial point (in the object plane \(z=z_a\)) the magnetic-field strength and the angular velocity of rotation of the electron vanish, i.e. \(\psi'_a=H_a=0\). In this case the quantity \(u\), as is seen from (38) and (41), acquires a purely real value and, consequently, becomes equal to the distance \(r\) of the electron from the \(z\)-axis. Under this assumption equation (37) may be written in the form
\[ V_0 r''+\frac{1}{2}V'_0 r' +\frac{1}{4}V''_0 r +\frac{eH_0^{\,2}}{8m}\,r=0. \tag{42} \]
The angular displacement of the electron in this case will obviously be directly determined by the formula
\[ \psi=\sqrt{\frac{e}{8m}}\int_{z_a}^{z}\frac{H_0}{\sqrt{V_0}}\,dz. \tag{43} \]
In the case of a purely electric lens the last term in equation (42) vanishes, and the equation of the trajectory is represented in the form
\[ V_0 r''+\frac{1}{2}V'_0 r' +\frac{1}{4}V''_0 r=0. \tag{42′} \]
The equation (42′) is linear and homogeneous with respect to \(V_0\). Thus, if all the potentials are increased proportionally by some constant factor, then equation (42′) will be satisfied for the same \(r\), and, consequently, the form of the trajectory will not change. Hence it is clear that, in the case of a purely electric lens, its optical constants will depend only on the ratio of the focusing voltages, and not on their absolute values. This latter circumstance is in complete agreement with experiment.
It is self-evident that in purely electric lenses rotation of the electron is absent \((\psi=0)\), and the trajectory has a plane form.
7. Approximate Calculation of the Trajectories of Paraxial Electrons
The determination of the optical constants of an electron lens in the general case reduces to computing the trajectories of paraxial electrons by integrating the differential equation (37). Provided that the distribution of the electrostatic and magnetic fields on the optical axis of the lens is known, this problem can be solved. However, since the field distribution is usually not given in analytic form, the integration of equation (37) must be carried out by an approximate method. There are several methods for the approximate calculation of the trajectories of paraxial electrons; here we shall restrict ourselves to describing only one of them, as applied to the particular case represented by equations (42) and (43). The method described consists in the following. The experimentally found distribution of the potential of the electrostatic field and of the magnetic-field strength on the optical axis of the system is plotted as a graph showing the dependence of \(V_0(z)\) and \(H_0(z)\) on \(z\). Next, by graphical or numerical differentiation, the values \(V'_0(z)\) and \(V''_0(z)\) are found and are likewise plotted on the graph. The \(z\)-axis is divided into \(n\) intervals, so small that the quantities \(V_0\), \(V'_0\), \(V''_0\), and \(H_0\) may be regarded as constant within each interval.
Let us consider the \(i\)-th interval and put
\[ \frac{1}{2}\frac{V'_0(z)}{V_0(z)}=-2A \quad \text{and} \quad \frac{1}{4}\frac{V''_0}{V_0}+\frac{eH_0^2}{8mV_0}=-B. \]
The quantities \(A\) and \(B\) are constant within the \(i\)-th interval. Then equation (42) may be represented in the form
\[ \frac{d^2 r}{dz^2}-2A\frac{dr}{dz}-Br=0. \tag{44} \]
The solution of this equation has the form
\[ r=\frac{(r'_i-r_i m_2)e^{m_1 z}}{2\sqrt{A^2+B}} -\frac{(r'_i-r_i m_1)e^{m_2 z}}{2\sqrt{A^2+B}}, \tag{45} \]
where \(r_i\) and \(r'_i\) are the values of \(r\) and \(\dfrac{dr}{dz}\) at the beginning of the \(i\)-th interval, \(m_1=A+\sqrt{A^2+B}\) and \(m_2=A-\sqrt{A^2+B}\). From (45) we find
\[ \frac{dr}{dz} = \frac{(m_1 r'_i+r_i B)e^{m_1 z}}{2\sqrt{A^2+B}} - \frac{(m_2 r'_i+r_i B)e^{m_2 z}}{2\sqrt{A^2+B}}. \tag{46} \]
The values of \(r\) and \(\dfrac{dr}{dz}\) for the end of the \(i\)-th interval are taken as \(r_{i+1}\) and \(r'_{i+1}\) for the \(i+1\)-st interval, etc. Continuing in this way, one can determine the values of \(r\) along the entire trajectory of the electron in the field of the lens.
The angular displacement of the electron can be determined by the same method. Indeed, put, for the \(i\)-th interval,
\[ \sqrt{\frac{e}{8mV_0}}\,H_0=C, \]
where \(C\) is constant within the \(i\)-th interval. Then, on the basis of (43), we have
\[ \psi=\psi_i+\int_{z_i}^{z} C\,dz \quad \text{or} \quad \psi=\psi_i+C(z-z_i), \tag{47} \]
where \(\psi_i\) is the angle of rotation at the beginning of the \(i\)-th interval. The value of \(\psi\) for the end of the \(i\)-th interval is then taken as \(\psi_{i+1}\) for the \(i+1\)-st interval, etc. In this way the trajectory of a paraxial electron can be completely determined step by step throughout all space.
If the intermediate values of \(\psi\) are not of interest, then one can directly determine the final value of the angle of rotation of the image by graphical integration of expression (43).
In the general case the problem can be solved in an analogous manner. By approximate integration, two independent integrals \(r_\alpha(z)\) and \(r_\beta(z)\) of equation (37) can be found. Further, from the initial conditions one can determine the constants \(C_\alpha\) and \(C_\beta\), after which the complex quantity \(u\) can be represented as a function of \(z\). Knowing \(u\), one can, by separating the real and imaginary parts, find the equation of the trajectory in Cartesian or cylindrical coordinates, using the relation
\[ x+iy=re^{i\psi}=ue^{i\chi}, \]
where
\[ \chi=\sqrt{\frac{e}{8m}} \int_{z_0}^{z} \frac{H_0}{\sqrt{V_0}}\, dz . \]
8. Thin Lenses
As is clear from the preceding, the determination of the optical constants of an electron lens in the general case leads to very lengthy and laborious calculations. The problem is greatly simplified for one particular case, important from the practical point of view, of the so-called thin electron lens. By a thin electron lens one usually means a lens whose effective length, i.e. the segment of the axis over which the field has appreciable values, is small in comparison with the distance to the initial point and to the focus. In this case the trajectory is curved only over the short segment \(AB\) (Fig. 10) inside the lens and may be regarded as a straight line with a break. Further, in the case of a thin lens one may consider the distance of the electron from the axis \(r\) inside the lens to be a constant quantity \((r=r_0)\). Under these assumptions equation (37) can be easily integrated.
Fig. 10. Path of rays in a thin magnetic lens
Let us consider, first of all, the case of a thin purely magnetic lens. In this case the potential of the electrostatic field is constant throughout space and its derivatives vanish. Equation (37) therefore assumes the following form (obviously, in the present case \(u=r\)):
\[ -\frac{d^2 r}{dz^2}=r_0 \frac{eH_0^2}{8mV_0}. \]
Integration of this equation with respect to \(z\) gives
\[ -\left[\frac{dr}{dz}\right]_{A}^{B} = r_0 \frac{e}{8mV_0} \int_{A}^{B} H_0^2(z)\, dz . \]
The left-hand side of the last equality represents the difference between the angle at which the electron ray enters the lens and the angle at which the ray emerges from it, i.e. the angle through which the ray is deflected in the lens. The equality shows that this angle is pro-
is proportional to the distance from the axis of the point at which the ray passes through the lens \((r_0)\), as is the case for a thin lens in light optics. From the drawing it follows that
\[ -\left[\frac{dr}{dz}\right]^B_A = -\left[\frac{dr}{dz}\right]^B + \left[\frac{dr}{dz}\right]^A = \frac{r_0}{b} + \frac{r_0}{a}. \]
Thus, after cancellation by \(r_0\), we have
\[ \frac{1}{a}+\frac{1}{b} = \frac{e}{8mV_0}\int_A^B H_2^0(z)\,dz. \]
Here \(a\) is the distance to the object and \(b\) is the distance to the image. Comparing the last expression with the well-known formula of a thin optical lens
\[ \frac{1}{a}+\frac{1}{b}=\frac{1}{f} \]
for the focal length of a thin magnetic lens, we find
\[ f= \frac{8V_0}{\dfrac{e}{m}\int_A^B H_2^0(z)\,dz}. \tag{48} \]
In a similar way a thin electric lens may be considered. In this case equation (37) assumes the following form:
\[ \frac{d^2 r}{dz^2} + \frac{V_0'}{2V_0}\frac{dr}{dz} + \frac{V_0''}{4V_0}r_0 = 0. \tag{49} \]
Put
\[ \frac{\sqrt{V_0}}{r_0}\frac{dr}{dz}=P. \]
Then equation (49) takes the form
\[ \frac{dP}{dz} + \frac{V_0''}{4\sqrt{V_0}} = 0. \]
Assuming that the electrostatic field is confined within a narrow zone \(A \le z \le B\), and integrating the last equation over these limits, we find:
\[ P_B-P_A+\int_A^B \frac{V_0''}{4\sqrt{V_0}}\,dz=0. \tag{50} \]
Let us consider an electron moving in the object space \(z \le A\) parallel to the optical axis, at a distance \(r_0\) from it (Fig. 11).
Then, for \(z=A\), \(\dfrac{dr}{dz}=0\), and consequently, \(P_A=0\). For \(z=B\) the electron leaves the lens still at a distance \(r_0\) from the axis, but with an inclination to the axis equal to
\[ \frac{dr}{dz}=-r'(B); \]
hence
\[ P_B=-\frac{\sqrt{V_0(B)}}{r_0}\cdot r'(B). \]
Denoting the focal distance in image space by \(f_2\), from the drawing in Fig. (11) we have
Fig. 11. Ray path in a thin electric lens
\[ f_2=\frac{r_0}{r'(B)}; \]
thus
\[ P_B=-\frac{\sqrt{V_0(B)}}{f_2}. \]
Substituting these values of \(P_B\) and \(P_A\) into equation (50), we find
\[ \frac{1}{f_2}=\frac{1}{4\sqrt{V_0(B)}}\int_A^B \frac{V_0''}{\sqrt{V_0}}\cdot dz. \]
Whence, for the value of the focal distance in image space of a thin electric lens, we find
\[ f_2= \frac{4\sqrt{V_0(B)}}{ \displaystyle \int_A^B \frac{d^2V_0}{dz^2}\cdot \frac{dz}{\sqrt{V_0(z)}} }. \tag{51} \]
In an analogous way, the focal distance of a thin electric lens in object space can be found:
\[ f_1= -\frac{4\sqrt{V_0(A)}}{ \displaystyle \int_A^B \frac{d^2V_0}{dz^2}\cdot \frac{dz}{\sqrt{V_0(z)}} }. \tag{52} \]
Thus the calculation of the focal distances of a thin electric and magnetic lens is reduced to simple quadratures.
9. Various Types of Electron Lenses
a) Electric lenses
The simplest electric lens may be a circular aperture in one of the plates of a plane capacitor (a diaphragm). The equipotential surfaces emerging from this aperture form an electron lens, the distribution of the potential on whose optical axis is shown in Fig. 12a. In the general case such a diaphragm may be the boundary between two homogeneous electrostatic fields possessing different gradients (Fig. 12, b and c). Thus, the characteristic feature of a lens of this type is that it consists of a diaphragm bordering, at least on one side, on a homogeneous electrostatic field.
Fig. 12. Field distribution on the axis of various electric lenses
By combining two or several diaphragms (or another type of electrodes possessing axial symmetry), a lens may be obtained on both sides of which there lie regions of constant potentials. If these potentials are equal to one another (Fig. 12, f and g), we obtain a lens equivalent to a single glass lens in light optics (the so-called unipotential lens). The potentials on the two sides of the lens may also be unequal (Fig. 12, d and e). In this case the electron lens corresponds to optical lenses separating an “immersion liquid” from air. Therefore such electron lenses are usually called immersion (or bipotential) lenses.
A special case of an immersion lens, very important for practical purposes, is an accelerating field adjoining a converging lens (Fig. 12, h). In accordance with the analogous device in light optics, this system is usually called an immer-
sion objective. This name also implies a very close position of the object.
Let us consider these four basic types of electric electron lenses in somewhat greater detail. Figure 13 shows the potential field of an electron lens of the 1st type (diaphragm) for the case when the field adjoins the diaphragm only on one side (a),
Fig. 13. Potential field of a diaphragm
and for the cases when the diaphragm borders on the field on both sides (b, c, and d). The form of the field can be represented more clearly in the form of the spatial model shown in Fig. 14. It is evident that the electrons passing through the aperture of the diaphragm (Fig. 15) will experience the action of a radial force, directed—
...either toward the axis of the beam (if the diaphragm has a negative potential), or away from the axis (if the diaphragm has a positive potential). Accordingly, such an electron lens will in the first case act as a converging lens, and in the second as a diverging one.
Fig. 14. Model of the potential field of the diaphragm
Fig. 15. Trajectories of electrons in the field of the diaphragm
The focal distance of such a lens can be readily calculated if this lens is regarded as thin. We shall assume the potential inside the lens to be everywhere the same and equal to the potential of the diaphragm \(V_0\). If the lens is concentrated on the segment \(ab\), then, applying formula (51) for electrons moving from the left of the diaphragm parallel to the axis, we find
\[ f=\frac{4\sqrt{V_0}}{\displaystyle \int_a^b \frac{V''}{\sqrt{V_0}}\,dz} =\frac{4V_0}{V'_b - V'_a}, \]
but \(V'_a=-E_a\) and \(V'_b=-E_b\).
Thus the focal distance of the diaphragm proves to be equal to
\[ f=-\frac{4V_0}{E_a-E_b}, \tag{53} \]
where \(V_0\) is the potential of the diaphragm, \(E_a\) is the field gradient in front of the lens, and \(E_b\) is the field gradient behind it. Depending on the sign of \(E_a-E_b\), such a lens will be converging or diverging.
It is necessary, however, to note that equality (53) does not give the actual position of the point of intersection with the beam axis of parallel electron rays in the case when the diaphragm borders on the field on both sides. In this case the trajectory of the electron upon leaving the lens will be curved by the field. However, knowing the value of \(f\), one can determine the angle at which the electron leaves the lens, and then calculate the trajectory itself.
The type of electron lens considered is of comparatively little interest from the practical point of view. Much more
interesting in this sense are the immersion and the solitary (unipotential) lenses and the immersion objective.
An immersion lens can be obtained as the result of a combination of two diaphragms, on both sides of which the potential has constant values. Combining two immersion lenses, we obtain a solitary lens, on both sides of which lie regions of equal potentials, namely the potentials of the outer diaphragms. A solitary lens is distinguished by the property that, like a glass lens in light optics, it can be moved along the direction of propagation of the rays without introducing changes into the surrounding medium, i.e., without affecting the magnitude of the potential of the surrounding space.
Here it is necessary to note the following: in light optics it is customary to assume that a ray passing through the center of a lens does not change its direction. The situation is otherwise in the case of an immersion lens. For a thin immersion lens the central ray entering it at an angle \(i\) will undergo refraction, where
\[ \frac{\sin i}{\sin r}=\sqrt{\frac{V_a}{V_b}}. \]
Thus, in calculating the magnification given by an immersion lens, instead of the well-known relation
\[ m=\frac{b}{a} \]
one must use the formula
\[ m=\frac{b}{a}\cdot\sqrt{\frac{V_a}{V_b}}. \]
In Fig. 16 various variants of a solitary lens are presented. If these figures are cut along a plane passing through their middle and perpendicular to the axis of symmetry, then possible variants of an immersion lens are obtained.
Fig. 16. Various variants of a solitary (unipotential) lens
In Fig. 17 a solitary lens is presented, the potential field of which is shown in Fig. 18. A spatial model of the field of the same lens is presented in Fig. 19. A very widespread type of immersion lens is the field of two cylinders. An immersion lens may be formed by two diaphragms possessing different potentials.
Considering the potential field of a solitary lens (Fig. 18), it is not difficult to see that in the case where, for example, the inner dia-
the diaphragm has a positive potential relative to the outer parts, then an electron passing through the field of the lens will experience the action of a radial force directed toward the axis only in the outer parts of the field of this lens, where the equipotential surfaces are curved outward. In the inner parts of the lens the radial component of the electric force will be directed in the opposite direction.
Fig. 17. External appearance of a single lens (after Johannson and Scherzer)
Fig. 18. Potential field of a single lens
Fig. 19. Spatial model of the potential field of a single lens
Thus the outer parts of the field of this lens will have a converging action, whereas the inner parts will have a diverging action. Nevertheless, despite this, the lens as a whole will always be converging. This is explained by the fact that the electron in this case, evidently, passes
the inner parts of the lens at a greater speed than the outer ones and, consequently, for a shorter time are subjected to the action of the scattering forces than of the converging ones. The same thing will occur if the inner diaphragm is charged negatively relative to the outer ones, with only the difference that now the electron will pass more rapidly through the outer, scattering parts of the lens and more slowly through the inner—converging—parts. Thus a unipotential lens is always a converging lens. The same applies also to immersion (bipotential) lenses.
An immersion objective, usually used as a system making it possible to obtain an enlarged image of an electron-emitting surface, consists for the most part of a cathode, at the surface of which the potential field begins, and one or two accelerating electrodes, arranged one after another at a small distance from the cathode surface. To obtain a sharp image it proves necessary to create at the cathode surface a field whose equipotential surfaces would have a curvature directed toward the cathode. This is achieved either by giving the cathode a concave shape, or, in the case of a plane cathode,
Fig. 20. Various designs of an immersion objective
Fig. 21. Potential field of an immersion objective
by using two electrodes placed in front of the cathode and creating a field of the required form. Plane equipotential surfaces at the cathode are incapable of causing the focusing of electrons emitted by the cathode, since the motion of electrons in a homogeneous electrostatic field will, as is known, take place along parabolas.
In Fig. 20 various designs of an immersion objective are compared. Fig. 20, a and b correspond to the case of a concave cathode; Fig. 20, c and d to the case of a plane cathode. Fig. 21 shows the potential field of an immersion objective, measured with the aid of an electrolytic tank, and the course of the electron rays emerging from a certain point of the cathode surface.
The calculation of the optical constants of an immersion objective can be carried out approximately by means of formula (53), if it is assumed that both diaphragms (Fig. 21) act as thin lenses and that the paths of the rays between them are rectilinear. Denoting the initial tangent of the angle of inclination of the potential curve (Fig. 22) by \(V_0'\), the tangent of the angle of inclination at the point of inflection by \(V_1'\), the value of the potential in the region of the first diaphragm by \(V_1\) and of the second by \(V_2\), for the optical power of the first lens we find
\[ \frac{1}{f_1}=\frac{V_1'-V_0'}{4V_1}. \]
The optical power of the second lens will be equal to
\[ \frac{1}{f_2}=-\frac{V_1'}{4V_2}. \]
The formulae obtained make it possible to estimate, as a guide, the order of magnitude of the optical constants of an immersion objective, or the order of magnitude of the focusing voltages.
Fig. 22. Distribution of the potential on the axis of an immersion objective
Fig. 23. Various types of magnetic lenses
A more rigorous calculation of the focusing properties of an immersion objective, as of any other electric lens, can be carried out by computing the trajectories of electrons in a given electrostatic field. In doing so it is necessary to take into account the initial velocities of the electrons with which they leave the surface of the cathode. In addition, when calculating electron trajectories in the field of an immersion objective, corrections have to be introduced for the change of the field near the cathode surface under the influence of space charge.
b) Magnetic lenses. Various types of magnetic electron lenses are shown in Fig. 23.
The simplest magnetic lens is a turn traversed by current a. From a single turn one may pass to a cylindrical coil, which creates a longitudinal, uniform magnetic field b. In practice, short multilayer coils c are usually used as magnetic lenses; in order to reduce the scattering of the magnetic field, they are often enclosed in an iron jacket, leaving an air gap only in that part where the magnetic lens d is to be located. In this case the lines of force extend only over a very small space, and such a magnetic lens may be regarded as thin. To eliminate, in certain cases, the undesirable rotation of the electron image produced by the magnetic field, two focusing coils are used, traversed by currents in opposite directions.
The types of electric and magnetic electron lenses considered are the basic elements of any electron-optical system. Combinations of these lenses make it possible either to concentrate electrons emerging from some source into a beam of very small cross-section, or to obtain an enlarged image of an object emitting electrons. Both kinds of combinations are widely used in practice in various electron-optical instruments.
10. Electron-Optical Aberrations
In electron optics, as in light optics, one may speak of aberrations, which are a consequence of the deviation of the actual path of rays in the field of a lens from the ideal path calculated by means of equation (37). The presence of these aberrations leads to indistinctness of the image obtained.
Among the aberrations observed in electron optics, chromatic aberration is foremost; in the present case the role of color is played by the initial velocity of the electrons. Owing to the difference in the initial velocities with which the electrons leave the object, rays emerging from one point will describe different trajectories and will be focused at different points. Chromatic aberration sometimes reaches a very considerable magnitude. As an example one may cite the graph (Fig. 24), showing the dependence of the distance between the cathode and its image, obtained by means of an immersion objective, on the magnitude of the initial velocity of the electrons, expressed in equivalent volts. As is seen from the curve, the chromatic aberration in this case is very considerable. However, the magnitude of the chromatic aberration can be reduced by a corresponding choice of the form of the field near the—
Fig. 24. Change in the position of the cathode image as a function of the initial velocity of the electrons (after Maloff and Epstein)
surfaces of the cathode. Thus, for example, if the equipotential surfaces adjacent to the cathode have a sufficiently large positive potential relative to it, then the electrons leaving the cathode at its very surface acquire considerable velocities, and since the magnitude of the initial velocity usually does not exceed several tenths of a volt, the difference in the velocities of the electrons at the surface of the cathode will be small and the chromatic aberration small.
In addition to chromatic aberration, electron lenses also possess monochromatic aberrations, known from light optics. These may include spherical aberration, coma, astigmatism, etc. The magnitude of these aberrations may be calculated on the basis of the more rigorous equation (16).
For this purpose we introduce into equation (16) the variables \(u\) and \(\bar u\), defined by the relations
\[ u = w \cdot e^{-i\chi} \quad \text{and} \quad \bar u = \bar w e^{-i\chi}, \tag{54} \]
where
\[ \chi = \sqrt{\frac{e}{8m}} \int_{z_a}^{z} \frac{H_0}{\sqrt{V_0}}\, dz . \]
Next, we set
\[ \left. \frac{V}{1 + w'\bar w'} = V_0(1 - 2\omega), \right\} \]
where
\[ \left. \omega = \frac{1}{2}\left(1 - \frac{1}{w'\bar w'}\frac{V}{V_0}\right). \right\} \tag{55} \]
Substituting the values \(w, \bar w, w'\) and \(\bar w'\), expressed in terms of the new variables \(u, \bar u, u'\) and \(\bar u'\), into equation (16) and replacing \(E_r, H_r\) and \(H_z\) on the basis of equalities (23), (29), and (30), after regrouping the terms we arrive at the following equation:
\[ V_0 u'' + \frac{1}{2} V_0' u' + \frac{1}{4} V_0'' u + \frac{eH_0^2}{8m}u = B_0, \tag{56} \]
where
\[ \left. \begin{aligned} B_0 ={}& 2\omega\left[ V_0 u'' - \frac{eu}{8m} H_0^2 + \frac{u'}{2}V_0' + i\sqrt{\frac{eV_0}{2m}} \left(u'H_0 + \frac{u}{2}H_0'\right) \right] \\ &+ V_0\omega'\left( u' + \frac{i}{4}u\sqrt{\frac{2e}{mV_0}}\,H_0 \right) \\ &+ i\left(\sqrt{1-2\omega}-1\right)\sqrt{\frac{2V_0}{2m}} \left( u'H_0 + \frac{i}{4}u\sqrt{\frac{2e}{mV_0}}\,H_0^2 + \frac{u}{2}H_0' \right) \\ &- \sum_{n=1}^{\infty} \frac{(-1)^n (u\bar u)^n}{4^n (n!)^2} \left[ \frac{u V_0^{(2n+2)}}{4(n+1)} \right. \\ &\left. \left. \qquad - i\sqrt{\frac{2V_0}{2m}(1-2\omega)} \left\{ \left(u' + \frac{i}{4}u\sqrt{\frac{2e}{mV_0}}\,H_0\right)H_0^{(2n)} + \frac{uH_0^{(2n+1)}}{2(n+1)} \right\} \right] \end{aligned} \right\} \tag{57} \]
The left-hand side of equation (56) coincides with the left-hand side of equation (37). Thus, the first approximation in calculating the trajectories of electrons is the assumption that \(B_0=0\) (for paraxial electrons). The quantity \(B_0\) determines the deviations from the ideal path of the rays and, at the same time, the magnitude of the aberrations. The second approximation consists in retaining, on the right-hand side of equation (56), the terms with the lowest powers of the small quantities \(u\), \(\bar u\), \(u'\), and \(\bar u'\), and assigning to them the values obtained as a first approximation in solving equation (37). \(B_0\) thereby becomes a known function of \(z\). Equation (56) then takes the form of an equation linear with respect to the unknown variable \(u\), with a right-hand side, and can be integrated. Since the lowest power in which the small quantities \(u\), \(\bar u\), etc., enter the right-hand side of equation (56) is the third, we obtain in this case “aberrations of the third order.” Further approximations lead to the determination of aberrations of higher order.
Neglecting, on the right-hand side of equation (56), all terms smaller than the third order of smallness, and substituting \(u''\) expressed from equation (37), we have
\[ \begin{aligned} B_0={}&u'\left[ V_0\omega' + i\sqrt{\frac{eV_0}{2m}} \left(\omega H_0-\frac{u\bar u}{4}H_0''\right) \right]+ \\[3pt] &+u\left[ \frac{u\bar u}{32}V_0^{(4)} -\frac{\omega}{2}V_0'' +\frac{eu\bar u}{16m}H_0H_0'' -\frac{e\omega}{4m}H_0^2+ \right. \\[3pt] &\left.\qquad\qquad +i\sqrt{\frac{eV_0}{8m}} \left(\omega H'+\omega'H_0-\frac{u\bar u}{8}H_0'''\right) \right]. \end{aligned} \tag{58} \]
The quantity \(\omega\) in this case, according to (54), (55), (23), and (29), will be equal to
\[ \omega = -\frac{u\bar u V_0''}{8V_0} +\frac{u'\bar u'}{2} +\frac{eu\bar u H_0^2}{16mV_0} +\frac{i}{8}(u\bar u'-\bar u u') \sqrt{\frac{2e}{mV_0}}\,H_0 . \tag{58a} \]
Assigning to the quantities \(u\), \(\bar u\), etc. the approximate values (38) and (38′), we obtain \(B_0\) in the form of a known function of \(z\). On the right-hand side of equation (56) we set the variable quantity \(u\) equal to the sum of the approximate solution \(u^*\) and the aberration \(\Delta u\), i.e.
\[ u=u^*+\Delta u . \]
Since the approximate solution \(u^*\) makes the left-hand side of equation (56) vanish, the aberration \(\Delta u\) will satisfy the same equation
\[ V_0\Delta u''+\frac{1}{2}V_0'\Delta u' +\frac{1}{4}V_0''\Delta u+\frac{eH_0^2}{8m}\Delta u=B_0 . \]
The solution of this equation will be represented in the form
\[ \Delta \bar{u} = \frac{r_\alpha}{\sqrt{V_a}} \int_{z_a}^{z}\frac{B_\varphi r_\beta}{\sqrt{V_0}}\,dz - \frac{r_\beta}{\sqrt{V_a}} \int_{z_a}^{z}\frac{B_\varphi r_\alpha}{\sqrt{V_0}}\,dz, \tag{59} \]
where \(r_\alpha\) and \(r_\beta\) still constitute the fundamental system of solutions of equation (37). The index \(a\) denotes quantities measured in the plane of the object.
Knowing \(\Delta u\), one can easily determine the magnitude of the aberrations in the directions of the \(x\)- and \(y\)-axes,
\[ \Delta x+i\Delta y=\Delta w=\Delta u\cdot e^{i\chi}, \]
where
\[ \chi=\sqrt{\frac{e}{8m}}\int_{z_a}^{z}\frac{H_0}{\sqrt{V_0}}\,dz. \]
Thus, if the distribution of the field on the optical axis is known, the magnitude of the aberrations can be calculated. However, knowledge of the magnitude of the aberrations alone is insufficient. The principal task of electron optics is to calculate the field of a lens for a minimum of aberrations. Since the magnitude of the aberrations is expressed in the form of integrals, the problem of finding the field that ensures a minimum of aberrations is solved by the methods of the calculus of variations. In this way one can first of all find the distribution of the field on the optical axis that satisfies the requirements of the problem posed. Next, the field distribution obtained is extrapolated into the space surrounding the axis. Knowing the basic equations of the electromagnetic field, this extrapolation can be carried out uniquely. Finally, from the form of the field found, the form of the conductors producing this field is also determined. It must be noted that the theoretical investigation of the aberrations of electron lenses leads to very complicated calculations and at the present time is hardly used in practice. However, the aberrations of electron lenses have to be taken into account in a whole series of existing electron-optical instruments and measures must be taken to eliminate them. In this connection, the only practical method for determining the magnitude of the aberrations is the experimental one. Nevertheless, the further improvement of electron-optical instruments requires a detailed development of this question from both the practical and the theoretical points of view. In any case, theorists still have a great deal of work ahead in this field.
II. ELECTRON-OPTICAL INSTRUMENTS AND THEIR APPLICATIONS
The development of electron optics has led to the improvement of previously known electron-optical instruments and to the creation of a number of new ones. Among the instruments known for a comparatively long time, but brought to a high degree of technical perfection only with the development of electron optics, are, for example, the Braun tube,
used for oscillography and television. The increased requirements now imposed on the degree of concentration and the power of the electron beam in instruments of this type could be met only after the focusing action of electric and magnetic fields possessing axial symmetry had been studied. The electron-optical system which, in such instruments, performs the functions of producing and concentrating electron beams received the name “electron projector,” or “electron gun.”
Later, with the development of electron optics, it became possible to construct electron-optical instruments analogous to optical ones, such as, for example, the electron microscope and the spectrograph. Electron optics also found a broad field of application in a number of special devices, such as the electronic image converter, secondary-electron multipliers, and so on. Below we shall consider the most characteristic applications of electron optics, which are, first of all, the electron projector and the electron microscope.
1. Electron projector
The purpose of the electron projector, as has already been indicated above, is to produce and concentrate an electron beam; in particular cases the projector must also allow the possibility of controlling the intensity of the electron current in the beam.
The source of electrons in electron projectors is usually an oxide cathode, made either in the form of a heated wire to the middle of which a metal rod is welded, the end of the rod being coated with oxide (Fig. 25), or in the form of a nickel cap heated from within by a tungsten spiral, the end of which is likewise coated with oxide, thus constituting the emitting surface (Fig. 26).
Fig. 25. Oxide cathode with direct heating
Fig. 26. Oxide cathode with indirect heating
In the present case the problem is to focus the electrons emitted by the cathode in order to obtain, on a distant screen, a spot of very small dimensions. To achieve this purpose, the focusing of the electrons in the electron projector is usually carried out in two stages. Directly at the surface of the cathode there is placed the first electron lens (an immersion objective), which collects the electrons emitted by a comparatively large surface of the cathode into almost a single point. Next the electrons emerging from this point enter the field of the second (main) electron lens and are focused on the screen of the tube.
Fig. 27. Zworykin projector
As the first lens there is usually used an electrostatic axially symmetric field; the second lens may be either electrostatic or magnetic.
A typical example of an electron projector with purely electrostatic focusing of the electron beam is the projector proposed by Zworykin for television cathode-ray tubes (Fig. 27). The source of electrons in this system is a heated oxide cathode \(A\). Electrode \(B\) (the Wehnelt cylinder) surrounds the cathode and has a circular aperture located exactly opposite the emitting surface of the cathode. Farther on there is a cylindrical electrode \(C\) (the first anode), furnished with a series of diaphragms cutting off the peripheral parts of the beam; and, finally, a second anode \(D\), made either in the form of an independent cylindrical electrode or in the form of a conducting layer deposited on the inner surface of the glass bulb.
The electrons emitted by the cathode enter the accelerating field in the space cathode—Wehnelt cylinder—first anode. This field forms an electron lens (immersion objective), giving an image of the emitting surface of the cathode. The role of electrode \(B\), besides its participation in producing the electron lens, consists in controlling the intensity of the electron current of the beam. This electrode creates a negative potential near a certain part of the emitting surface of the cathode, preventing electrons from leaving this region. The controlling action of electrode \(B\) is clear from Fig. 28, which shows the distribution of potential in the cathode—anode space at zero and at \(-30\ \mathrm{V}\) on the control electrode. When the potential of the control electrode is zero, the entire surface of the cathode emits; at \(-30\ \mathrm{V}\) only its central part emits, while the negative potential near the outer edges of the emitting surface prevents electrons from leaving this part of the cathode.
Calculation of the electron trajectories in the field of the first electron lens (in the cathode—first anode region) shows that if all elec-
electrons left the cathode with zero initial velocity, then, with the corresponding form of the field in this space, all the electrons would intersect the optical axis of the system approximately at one point.
Fig. 28. Distribution of the potential in the cathode–first-anode space of an electron projector: A) at 0 V on the Wehnelt cylinder, B) at −30 V on the Wehnelt cylinder
Fig. 29. Trajectories of electrons in the field of the first lens
point. However, the presence of radial components of the initial velocities of the electrons leads to the fact that these trajectories have the form shown in Fig. 29. The trajectories of electrons emerging from point \(A\) and possessing diametrically opposite radial-
components of the initial velocity are represented by curves 1 and 2. At point \(A_1\) all the electrons that have emerged from point \(A\) and possessed various values of the radial component of the initial velocity are collected. The same applies to points \(B\) and \(B_1\) and to curves 3 and 4.
The electronic image of the emitting surface of the cathode is thus obtained in the plane \(A_1B_1\). In the segment \(ab\), rays emerging from different points of the emitting surface of the cathode intersect. The section \(ab\) may then be regarded as a certain new source of electrons and, by means of the appropriate electron optics, be projected onto the screen of the tube.
In order to reduce the dimensions of the final cross section of the electron beam (the image), it is necessary to strive to make the dimensions of the area of the first intersection of the electron rays \(ab\) (the object) as small as possible. A theoretical analysis of the first intersection of the electron rays leads to the following relation between the magnitude of the current \(I\) that has passed through the aperture of a diaphragm placed in the plane of the first intersection of the electron rays (the center of the diaphragm aperture coincides with the optical axis), and the total emission current \(I_s\):
\[ \frac{I}{I_s}=1-\varepsilon^{-ar^2V}, \]
where
\[ a=\frac{e}{kT}\frac{F^2}{z^2}. \]
Here \(\varepsilon\) is the base of natural logarithms, \(e\) is the electron charge, \(k\) is Boltzmann’s constant, \(T\) is the temperature of the cathode in degrees Kelvin, \(z\) is the distance from the cathode to the plane of the first intersection of the electron rays \((ab)\), \(r\) is the radius of the diaphragm aperture, \(V\) is the total voltage applied to the system of electrodes forming the field of the first lens, and \(F\) is a coefficient depending on the distribution of the field in the region of the first lens.
It is clear from the formula given that, in order to concentrate as large a part of the electron current as possible within as small an area as possible, i.e., in order that, for a minimal \(r\), the ratio \(\frac{I}{I_s}\) should have a maximal value, one should apply the greatest possible potentials to the first anode, use as low a cathode temperature as possible, and also obtain such a distribution of the potential in the cathode–first-anode region as would ensure a maximal value of the quantity \(F\). In this way it is possible to ensure that the dimensions of the area of the first intersection of the electron rays will be considerably smaller than the dimensions of the emitting surface of the cathode.
In Zworykin’s electron projector, the focusing of the rays emerging from the region of the first intersection is accomplished by means of
by the action of an electron lens formed by an electrostatic field produced by the potential difference between the first and second anodes (the field of two cylinders). Owing to the presence of this lens, an enlarged image of the region of the first intersection of the electron rays is obtained on the tube screen. The magnification obtained in this way may be approximately calculated from the formula
\[ m=\frac{b}{a}\sqrt{\frac{V_1}{V_2}}, \]
where \(a\) is the distance from the object to the lens, \(b\) is the distance from the image to the lens, \(V_1\) is the potential of the first anode, and \(V_2\) is the potential of the second anode.
Fig. 30 shows the path of the electron rays in the electron projector of Zworykin’s system and the corresponding optical analogy. This electron-optical system makes it possible to obtain an electron current in the beam of the order of several hundred microamperes with a diameter of the final cross-section of the beam of about \(0.5\) mm. The voltages used for focusing are \(1\)—\(2\) kV on the first and \(5\)—\(10\) kV on the second anode.
Fig. 30. Path of rays in Zworykin’s projector and optical analogy
The focusing system described is, of course, not the only possible one. Any system of electrodes producing an electrostatic field with axial symmetry may be used here.
Electron projectors with magnetic focusing differ from the system described above in most cases only in that the final focusing of the electron rays is carried out by means of a magnetic lens. Any axially symmetrical magnetic field may serve as the magnetic lens. Such a field can easily be obtained by means of a short coil slipped over the neck of the tube. The system of electrodes forming the first intersection of the electron rays may in this case be exactly the same as in projectors with purely electrostatic focusing.
An example of an electron projector with magnetic focusing is the system currently used in America (by RCA) in powerful projection television tubes. Such a projector consists of a heated oxide cathode and a number of diaphragms with gradually increasing potentials, whose purpose is to obtain the most advantageous distribution of potential in the region of the first intersection of the electron rays (Fig. 31). Directly at the point of intersection there is placed a limiting diaphragm with a very narrow aperture (\(\varnothing = 0.1\) mm),
cutting off the peripheral part of the beam. The potential distribution is chosen in such a way that 90% of the total emission current of the cathode, whose working-surface diameter is 1 mm, is concentrated in this aperture. The aperture in the limiting diaphragm is further regarded as an independent source of electrons and is imaged on the tube screen by means of a magnetic lens (Fig. 32). Such a projector, with a voltage on the limiting diaphragm equal to 10 kV, makes it possible to obtain an electron beam several tenths (0.2–0.3) of a millimeter in diameter, with a beam current of the order of 1.5–2 mA.
Fig. 31. System of electrodes forming the first lens in the projector for projection electron-beam television tubes
Electron projectors have a wide field of application in all electronic devices where it is required to have a narrow beam of electrons. These include cathode-ray tubes used for oscillography and television, transmitting television tubes, electronic commutators, etc. A description of these devices is beyond the scope of the present article.
Fig. 32. Electron-beam projection television tube with magnetic focusing (after Law)
2. Electron Microscope
The most characteristic example from the field of applications of electron optics is the electron microscope. This instrument, like the microscope of light optics, serves to obtain a greatly magnified image of small objects. The electron microscope can be used both for the microscopic investigation of the electron emission of cathodes and for obtaining magnified images of objects that do not possess electron emission of their own.
Depending on which lenses—electric or magnetic—are used for focusing electron beams, electric and magnetic electron microscopes are distinguished. A typical example of an electric electron microscope is the “immersion objective” considered above (Fig. 21), consisting
Fig. 33. Magnetic electron microscope: a — external view, b — principal diagram, c — optical analogy
of two diaphragms located near the emitting surface of the cathode. The field formed by these diaphragms exerts a focusing action on the electrons emitted by the cathode, and, with a suitable choice of voltages, it is possible to obtain on the fluorescent screen of the tube a magnified image of the emitting surface of the cathode. Such microscopes can, naturally, be used only for studying objects that possess their own electron emission (chiefly thermionic). Usually they are used without an “eyepiece,” since the magnification they provide is sufficiently large even without one.
A magnetic microscope designed for observing objects that do not possess their own electron emission is shown in Fig. 33. Electrons emitted by the cathode are accelerated by an electrostatic field and, by means of a short magnetic coil that plays the role of a condenser lens, are directed onto the object under investigation. The object is placed either on a thin metal mesh or on a thin collodion film sufficiently permeable to electrons. The next coil, located behind the object, acts as the objective and gives an enlarged electron image of the object on an intermediate screen. A diaphragm situated in this plane cuts out part of this image, which, in still more enlarged form, is projected by means of a third coil onto the fluorescent screen of the tube. Thus this instrument corresponds to the projection microscope of light optics. In the newest design of a magnetic microscope of this type (by Siemens), the objective gives an intermediate image of the object enlarged in the ratio \(80:1\). Then, by means of a projection coil, the intermediate image is enlarged in the ratio \(350:1\). The image obtained on the screen of the tube is therefore enlarged in the ratio \(\sim 30\,000:1\). This image is further enlarged optically by approximately the ratio \(3:1\), so that the magnification ultimately achieved by this instrument is of the order of \(100\,000:1\) (!). The velocity of the electrons irradiating the object is then \(100\,000\ \mathrm{V}\).
For investigating the structure and emission properties of cathodes, electron microscopes possessing comparatively low resolving power may be used. Such a microscope (magnetic), used for the investigation of photocathodes, is shown in Fig. 34. The photocathode \(K\) is illuminated by means of a quartz mercury lamp \(Q\), the light from which passes through the quartz window \(F\). The image of the cathode is projected by means of the magnetic lens \(M\) and the coil \(A\) onto the fluorescent screen \(S\). The window \(G\) serves for direct observation of the photocathode.
For observations of an object with the aid of an electron microscope, it is naturally necessary that this object, for one reason or another, emit electrons. The following list may be given of the conditions under which this requirement can be fulfilled:
-
Electrons are emitted by the object as a result of: a) thermionic emission, b) photoelectric emission, c) secondary emission under the action of bombardment by a primary beam of electrons or ions.
-
Electrons are emitted by a thin active layer deposited on the object under investigation as a result of: a) thermionic emission, b) photoelectric emission, c) secondary emission.
-
The object is irradiated by a beam of fast electrons. In this case the elastically reflected electrons, possessing approximately the initial value of the velocity, form an image of the object (Fig. 35).
- The object is penetrated by fast electrons. The electrons passing through the object almost do not change their velocity and produce its image (Fig. 33).
In the first two cases, the electrons arise on the object itself and leave it with an insignificant velocity. In the last two cases, the electrons are emitted by a special source (“illuminator”) and reach the surface of the object with high velocities. Obviously, the latter two methods make it possible to observe a variety of objects that do not possess their own electron emission.
Fig. 34. Electron microscope for the study of photocathodes (after Mahl)
Fig. 35. Apparatus for obtaining electron images by means of elastically reflected electrons
The development of electron microscopy is of great interest for the following reasons. First of all, electron optics, as Ruska observed, makes it possible “to see things in another light,” which in itself is already important in the comprehensive investigation of one or another object. Further, electron optics makes it possible considerably to extend the limits of the resolving power of microscopes, since the wavelength of an electron is several orders of magnitude smaller than the wavelength of visible light. Finally, electron microscopes make it possible to observe objects that are at high temperature, thereby revealing features of their crystalline structure, whereas, when incandescent bodies are observed with the aid of a light microscope, owing to the identical temperature of all parts of the object, the latter appears completely homogeneous.
Electron-optical investigations of emitting surfaces have made it possible to study the activation process of thoriated, oxide, and other cathodes. As an example, one may cite here photographs of the surface of thoriated tungsten (Fig. 36). In the first photograph (a) the centers from which thorium passes onto the metal surface are clearly visible. With further heating, thorium spreads almost uniformly over the entire surface of the cathode (b). When heated to an even higher temperature, as a result of the evaporation of thorium, the surface is partially deactivated. At the same time the crystalline structure of the tungsten becomes apparent and new centers appear for the emergence of thorium onto the surface (c).
Electron-optical investigations of the emission of pure metallic surfaces and of surfaces covered with adsorbed (monomolecular) films of gases and especially of alkali and alkaline-earth metals have made it possible to obtain a picture of the crystalline structure of the cathode material. These investigations show that different faces of crystals may have unequal work functions, differing by tenths of a volt. The picture of the crystalline structure of the cathode becomes especially clearly visible when the cathode surface is covered with a thin film of an alkali or alkaline-earth metal. In this case there exists a certain coating thickness that is optimal in this sense.
Of particular interest is the possibility of investigating changes in the structure of metals at critical temperatures. Thus, for example, it proves possible to observe the process of recrystallization of iron at \(900^\circ\)C (the transition of \(\alpha\)-iron into \(\gamma\)-iron; see Fig. 37).
Finally, of extremely great interest is the possibility of using the electron microscope for observing microorganisms and other microscopic objects that do not possess their own electron emission and that are poorly, or not at all, amenable to observation with light microscopes because of their extremely small dimensions. For this purpose an electron microscope of the type shown in Fig. 33 may be used. As examples of images obtained with the aid of such a microscope, one may cite photographs of thrombocytes (Fig. 38) and of the virus of the mosaic disease of tobacco (Fig. 39). Thus, at the present stage of its development the electron microscope is a very valuable instrument for a whole series of investigations. As it is further improved, the range of its applications will undoubtedly expand rapidly.
3. Electron image converter and other electron-optical instruments
A special case of the electron microscope is the instrument that has received the name of electron image converter. The purpose of this instrument is the conversion of an optical image into an electronic one. The basic scheme of the instrument is as follows:
To D. Zernov’s article
Fig. 36. Electron image of thorized tungsten (after Brüche and Mahl). Magnification 25:1.
a — after heating for 1 min. at a temperature of 2800° K; b — after heating for 2 min. at a temperature of 2350° K; c — after heating for 2 min. at a temperature of 2750° K.
Fig. 37. Recrystallization of iron at the critical temperature.
Magnification 30:1.
Fig. 38. Electron image of thrombocytes (blood platelets).
Magnification 10,000:1.
Fig. 39. Electron image of the tobacco mosaic virus.
Magnification 20,000:1.
(fig. 40). By means of lens \(L_1\), the object is projected onto the semi-transparent photoelectric cathode \(K\). The photoelectrons emitted by the cathode in quantities corresponding to the illuminations of its individual regions are focused, by means of the electron lens \(L_2\), on the fluorescent screen \(E\). Thus a visible image of the object appears on screen \(E\).
Such an instrument may be used to amplify the brightness of images, since the brightness of the image on the fluorescent screen can be regulated by changing the accelerating electron field; moreover, in principle it is possible to resort to amplification of the primary electron current emitted by the photocathode by means of secondary emission, placing activated grids or thin films in the path of the electron beams.
Fig. 40. Diagram of an electronic image converter
In addition, this instrument may be used to convert invisible images, projected onto the photocathode in infrared or ultraviolet light, into visible ones. The latter is very important for a number of practical purposes, such as infrared and ultraviolet microscopy, observation of distant objects in infrared light (electron telescope), etc. In our Union, such instruments have found application for visual quality control of motion-picture film projected onto the photocathode of the converter in infrared light. Finally, the conversion of visible images into electronic ones is used in special types of transmitting television tubes (Farnsworth dissector, super-iconoscope).
The field of practical applications of electron optics is not limited to the most characteristic examples described above. Among the instruments in whose design the laws of geometrical electron optics must also be taken into account are electron multipliers, spectrographs, monochromators, etc. The essence of the action of an electron multiplier consists in the fact that electrons emitted by some source are successively directed onto a series of electrodes possessing gradually increasing potentials, releasing secondary electrons on them. The schematic diagram of the operation of an electron multiplier is shown in Fig. 41. However, in such a simple form this instrument would not provide any amplification, since the primary electrons emitted by the source would, for the most part, be directed immediately to the collector. In order to force the electrons to be successively concentrated on all intermediate elect-
rays (emitters), it proves necessary to focus them by means of specially selected electric or magnetic fields.
The operation of instruments intended for investigating the distribution of velocities in a stream of charged particles (spectrographs) is based on the dependence of the shape of a particle’s trajectory in an electromagnetic field on its initial velocity. In mass spectrographs use is made of the dependence of the shape of the trajectory on the magnitude of the ratio of the particle’s charge to its mass \(\left(\frac{e}{m}\right)\). In this case it is obviously necessary to resort to the simultaneous action on the particle of both an electric and a magnetic field, since the trajectory of a particle in a purely electrostatic field, as is clear from the preceding discussion, does not depend on the magnitude of \(\frac{e}{m}\). Finally, the influence of electric and magnetic fields on the shape of the trajectories of particles possessing different values of velocity can be used to isolate a stream of charged particles possessing one definite value of velocity (monochromator).
Fig. 41. Diagram of the operation of an electron multiplier
From all that has been set forth above, it is clear what enormous significance electron optics has for the most varied practical purposes. The development of electron optics up to the present time can by no means be considered complete. Much work still lies ahead for both theoreticians and practitioners in the field of further improvement of the results already achieved, as well as in the direction of revealing new possibilities which, in all probability, are still far from exhausted.
Literature
E. Brüche u. O. Scherzer, Geometrische Elektronenoptik, Berlin, 1934.
I. G. Maloff a. D. V. Epstein, Electron Optics in Television, New York, 1938.
H. Busch u. E. Brüche, Beiträge zur Elektronenoptik, Leipzig, 1937.
R. R. Low, Television, II, 328—350, 1937.
B. v. Borries u. E. Ruska, Z. techn. Phys., 19, No. 11, 402, 1938;
B. v. Borries u. E. Ruska, Z. Ver. Ing., 82, 937, 1938; Electronics, November, No. 11, 30—33, 1938.
W. Schaffernicht, Z. techn. Phys., 17, 596—604, 1936; see also Uspekhi Fizicheskikh Nauk, 17, 491, 1937.
V. K. Zworykin, Z. techn. Phys., 17, 170—183, 1936; see also Uspekhi Fizicheskikh Nauk, 16, 814, 1936.