Electron Optics*

I. Fundamentals and Development of Electron Optics

G. Busch, Darmstadt

Geometrical electron optics considers the motion of electrons from an optical point of view. Such a mode of treatment has long been known; as early as 100 years ago Hamilton showed that the path of a material point in a force field determined by the potential \(\Phi\) is identical with the path of a light ray in an inhomogeneous optical medium with variable refractive index \(n\), the role of \(n\) being played by \(\sqrt{\Phi}\). This, however, by no means signifies that electron optics consists in the simple application of this Hamiltonian analogy. The field of electron optics is, on the one hand, more general, because it also includes the motion of electrons in magnetic fields, to which Hamilton’s analogy cannot be extended directly**; and, on the other hand, more restricted, for here not all electron paths are considered, but only those which give an image, analogous to the way this is done in geometrical light optics. An essential condition for this is the requirement that electrons issuing as a beam from one point should again come together in some (imaginary or real) focus***. Thus geometrical electron optics may be defined as the doctrine of the concentration and dispersion of electron beams and of the images obtained as a result. Electron optics, understood in this sense, began to develop from the moment when systems possessing such focusing properties (“electron lenses”) were indicated. The development of these systems followed a path entirely independent of light optics; only later were common paths worked out. This is explained by the fact that optical and electron lenses are constructed on entirely different principles. In optical lenses refracting surfaces are used, i.e., surfaces on which the refractive index undergoes a discontinuous change. In electron optics, constant electric or magnetic fields are used for the same purpose. In more

* Z. techn. Phys. 17, 584–604, 1936, translated by D. V. Zernov.
** On the extension of Hamilton’s analogy to the case of a magnetic field, see Glaser’s article, Z. techn. Phys., 17, 617, 1936.
*** In some cases the focus is a straight line, analogously to cylindrical lenses in light optics.

At a later stage in the development of electron optics, attempts were made to realize electron lenses by means of refracting surfaces. Such a refracting surface in electron optics, by Hamilton’s analogy, corresponds to a jump in electric potential, i.e. an electric double layer. This layer can, for example, be realized by means of two closely spaced, differently charged thin wire grids. In practice these experiments had no significance, owing to the impossibility of making the metallic surfaces required to delimit the double layer, which would at the same time satisfy the requirements of good permeability for electrons and complete homogeneity of the fields lying between them. Such lenses, made of wire mesh, act like glass lenses with a surface curved in accordance with the structure of the wire mesh. This path, however, proved unacceptable.

Let us now see what constant electric and magnetic fields can possess focusing properties. In the first place one may have in mind fields acting on electrons with a force directed immediately toward the axis of the beam, i.e. a radial electric field (a cylindrical condenser) or a circular magnetic field (the field of a straight current).

These systems, which were tested many times, proved unsuitable, since, apart from the disturbances created by the shadow of the wires placed in both systems along the axis of the beam, an important requirement is not fulfilled here: for complete concentration it is necessary that all rays of the beam join at one and the same point. This must occur for inner and outer rays alike, and, consequently, the outer rays must be deflected through a larger angle, i.e. be subjected to the action of a greater force, than the inner rays. This latter circumstance leads to the requirement that the deflecting force increase in proportion to the distance from the axis. However, in both of the systems mentioned above the deflecting force is proportional to \(\frac{1}{r}\). Therefore these systems cannot be regarded as a satisfactory solution of the problem of concentrating electron beams, and in particular of obtaining electron images. In reality, the solution of the problem is exceedingly simple; nature in this case comes to our aid. There exists a completely general rule: any constant* magnetic or electric field possessing axial symmetry exerts a concentrating action on paraxial (close to the axis) rays, and therefore this field may be used for focusing and for obtaining an electron image. Since this rule is of fundamental importance for electron optics, it is necessary to trace the line of reasoning that led to its derivation.

* Constant in the sense that not only the field intensities, but also their derivatives, are constant throughout the space penetrated by the electron rays, and in particular near the optical axis. In both examples cited above, this condition is not satisfied for the surface of the central electrode.

Let us consider two examples of a mechanism by means of which the focusing of electron rays can be accomplished. This mechanism is basically different for magnetic and electric fields. In the case of a magnetic field, a simple example may be a uniform longitudinal field (Fig. 1). In such a field the electrons describe helical lines, which arise because the longitudinal component of the velocity undergoes no change, while the radial component, directed perpendicular to the magnetic field, experiences the action of a force which gives the projection of the ray on a plane perpendicular to the magnetic field the form of a circle.

Fig. 1. Model of electron trajectories in a uniform longitudinal magnetic field

This transverse force, deflecting the path of the ray toward the axis, is proportional to the radial component of the velocity, i.e. to the initial angle of the electron path relative to the axis. The outer rays are thus deflected more strongly than the inner ones, and this produces good concentration of the beam.

A completely different mechanism of ray focusing occurs in an electric field. Here a uniform longitudinal field gives no radial force whatever. On the contrary, in a nonuniform field possessing axial symmetry, owing to the condition \(\Delta V = 0\) (for simplicity we restrict ourselves to a field free of space charges), a change of the longitudinal component along the axis \(E_z\) inevitably entails the appearance of a radial component

\[ E_r \simeq -\frac{r}{2}\frac{\partial E_z}{\partial z}. \]

Here too there is a radial force proportional to the distance from the axis, and consequently complete concentration also occurs. This is evident from a direct consideration of the following example. In Fig. 2 a simple electric lens is shown (i.e. a lens on both sides of which there are regions of equal constant potentials, or—speaking optically—of equal constant refractive indices). It consists of three diaphragms, of which the two outer ones are connected to each other, while the middle one is at a higher or lower potential. The figure shows equipotential surfaces; it is obvious that with increasing distance from the axis the inclination of the equipotential surfaces increases, and together with it the radial component of the electric-field strength increases. It is also seen from the drawing that this radial component inside the lens has a sign opposite to that which it has in the external space. Thus the separate parts of the lens exert a partially converging action,

partly scattering, so that the total action is represented by their difference.

Here Hamilton’s analogy may be applied. The equipotential surfaces shown in the drawing correspond to the surfaces \(n = \mathrm{const}\). With a positive potential of the middle diaphragm, \(n\) has a maximum in the middle. This system acts approximately as if \(n\) were constant between two equipotential

Fig. 2. Potential distribution in an electric lens

surfaces, and changed discontinuously on the surfaces themselves. Thus the equipotential surfaces act as refracting surfaces in optics, and, as is seen from the figure, the inner surfaces, curved inward, must act as scattering surfaces, while the outer ones, curved outward, as collecting ones.

To the considerations illustrated by these two examples, we shall return again in the general theory. The final aim is the calculation of electron paths for the case of paraxial rays. For this purpose one finds the equations of motion of the electrons and expands the field strengths entering into them near the axis in a series in powers, neglecting terms of higher order. In doing so it is necessary to take into account the fundamental equations of magnetic and electric fields free from space charges and vortices.* This leads to the fact that only terms with definite powers of \(r\) enter into the series, the coefficients of the series depending only on the value of the field strength—

* The theory also remains valid in the case of the presence of space charges, provided that the charge density \(\rho\) has axial symmetry and is constant throughout space. This is of importance for explaining gas concentration. In what follows, for simplicity, we shall assume \(\rho = 0\).

G. BUSH

...of the field intensity on the axis itself. The final result is the following differential equation for the electron trajectories:

\[ \frac{1}{2\eta}\ddot r=\Phi r''+\frac{1}{2}\Phi' r'=-r\left(\frac{1}{8}\eta H^2+\frac{1}{4}\Phi''\right) \]

(\(H=H(z)\) is the magnetic-field intensity at \(r=0\); \(\Phi=\Phi(z)\) is the electrostatic potential* at \(r=0\); \(\eta=\left|\frac{e}{m}\right|\). Primes denote derivatives with respect to \(z\), dots—derivatives with respect to \(t\).)

An essential property of this equation is the proportionality of the radial force \(m\ddot r\) to the distance \(r\) from the axis. From the fact that this equation is homogeneous and linear with respect to \(r\), there follows a proof of the general rule set forth above. Indeed, the solution of the equation for the trajectories of electrons emerging from some point on the \(z\)-axis must have the form:

\[ r=Cf(z), \]

so that the next point of their intersection with the \(z\)-axis, independently of the integration constant \(C\), must have one and the same value for all rays. Thus all the electrons of the beam, independently of the angle at which they fly out, must meet at one and the same point of the \(z\)-axis, i.e. at the focus. This proves the rule set forth above. It follows, therefore, from the linearity and homogeneity of the trajectory equations, and the latter, in turn, from the fundamental equations of electric and magnetic fields.

These focusing properties are thus determined by the properties of constant magnetic and electric fields; moreover, it is not without interest to note that a certain analogy to this case may also be observed in light optics. What is meant here is Exner’s gelatin-glycerin cylinder, in which, owing to diffusion, a radial fall of concentration is established and, together with it, a radial fall of the refractive index. The conditions following from the law of diffusion determine such a distribution of the refractive index that this cylinder exerts a good concentrating action on light rays. The considerations given above may serve, to a certain extent, as proof of the existence of electron optics. For practical applications, however, an exact calculation of the paths of electrons is necessary, i.e. a solution of the equation of the trajectory. In the general case the solution cannot be given in closed form, but this is possible for one important special case, the so-called short electron lens. By a short lens we mean such a lens whose effective length, i.e. the segment of the axis on which the field has appreciable values, is small in comparison with the distance between the initial point and the focus (Fig. 3). In this case the trajectory is curved only over a short segment

* The zero value of the potential \(\Phi\) is chosen here so that it corresponds to zero values of the electron velocities. Thus \(\Phi\) is nothing other than the velocity of the electrons expressed in volts.

$A—B$ inside the lens and may be regarded approximately as a broken straight line. We have here the same conditions as in thin lenses of optics.

Mathematically this leads to the fact that inside the lens one may regard $r$ as a constant quantity $(r = r_0)$ (this does not apply to the derivatives of $r$). On the right-hand side of the trajectory equation there therefore stands a function only of $z$, and the equation is integrated.

Fig. 3. Path of a ray in a “short” electron lens

The integration is carried out most simply in the case of a purely magnetic lens; then $\Phi$ is constant and equal to $\Phi_0$; $\dfrac{dr}{dz}$ drops out, and integration with respect to $z$ gives:

\[ -\left[\frac{dr}{dz}\right]^B_A \simeq r_0 \frac{\eta}{8\Phi_0}\int_A^B H^2(z)\,dz . \]

The left-hand side represents the difference between the angle at which the ray enters the lens and the angle at which it leaves it, i.e. the angle through which the ray is deflected in the lens. The equation thus shows that the angle of deflection 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 in light optics for a thin lens.

From the figure 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 the preceding equation becomes

\[ \frac{1}{a}+\frac{1}{b} = \frac{\eta}{8\Phi_0}\int_A^B H^2(z)\,dz = \frac{1}{f}, \]

i.e. the formula which, on the one hand, expresses the well-known lens law and, on the other, determines the focal length of the magnetic lens in definitive form.

In a similar way, and with the same results, a short electric lens can be considered. This latter is likewise analogous to the thin lens of light optics. The expression for the focal length of this lens can be derived from the same considerations,

The foundations of electron optics were developed in 1926–1927 as applied to the magnetic focusing of cathode rays. The possibility of realizing an electric lens as an independent focusing device had not yet been revealed at that time. This possibility was clarified only in 1931–1932, independently by Davisson and Calbick, on the one hand, and by Brüche and Johannson, on the other. But this was still not electron optics. True electron optics was created by experimental work carried out in 1931 by Knoll and Ruska with collaborators in the high-voltage laboratory of the Technische Hochschule in Berlin and, independently, by Brüche with collaborators in the laboratory of the AEG research institute. Both groups showed for the first time an image of an object by means of an electron lens, the former with a magnetic one and the latter with an electric one. By means of a combination of two lenses they constructed an electron microscope; especially large magnifications were achieved by Brüche with his magnetic microscope.

Both groups applied their electron microscopes to the study of the surfaces of cathodes emitting electrons. Brüche and his collaborators brought the methodology of these experiments to complete perfection. At the same time, the theory was also constructed. Here, first of all, the works of Scherzer and Glaser must be mentioned. All these works advanced the development of electron optics so far that by the beginning of 1934 it became possible for a general systematic survey to appear in the form of the excellent book by Brüche and Scherzer.