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ELECTRON OPTICS
III. PROBLEMS OF THEORETICAL ELECTRON OPTICS
O. Scherzer, Darmstadt
Contents: 1) Paraxial rays, 2) Electron-optical aberrations, 3) Correction of electron-optical systems, 4) Conclusions.
In Busch’s article a survey was given of the basic principles of electron optics; in Brüche’s article the experimental part was illuminated. The present article is devoted to the question of what theoreticians might contribute to the further development of electron optics.
At first glance it becomes clear that electron optics provides abundant material for theoretical investigations. Questions concerning suitable, in each individual case, sources of electrons, the best material for a luminescent screen, and the most expedient manufacture of the object are connected with the atomic nature of matter; in the study of this nature a wide field of activity is offered to theoreticians. Moreover, both these and many other questions cannot be included in the theory of electron optics alone, since they are connected with other areas of physics and technology as well. The fundamental problem of electron optics consists in selecting such electric and magnetic fields as would determine the prescribed paths of electrons. The solution of this problem may be divided into three parts, which will be considered below. In doing so we shall have in mind, first of all, the electron microscope.
1. Paraxial Rays
The first step in solving this problem is Gaussian dioptrics. This concept is borrowed from light optics. By Gaussian dioptrics is meant the analysis and calculation of the course of rays close to the axis. In these calculations certain simplifications are introduced which ensure the linearity of the differential equations. The theory of Gaussian dioptrics was given essentially in Busch’s first papers[^1]. The consideration of Gaussian dioptrics, necessary in the design of any electron-optical instrument, does not at present constitute a difficult theoretical problem. This can be shown by the example of an immersion objective (Fig. 1, a). Here such a form of diaphragm is chosen that the potential throughout space can be represented by elementary analytic functions.
functions.* Let us imagine that a potential of 1 thousand V relative to the cathode is applied to the outer diaphragm. The question arises: how many volts must be applied to the inner diaphragm so that the entire system gives an image of the cathode on a distant fluorescent screen? In the scale of the drawing the screen must lie several meters to the right, which in practice means infinity.
For an approximate calculation let us imagine that both diaphragms act as thin lenses and that the ray paths between them are rectilinear (Fig. 1, б). The optical power of the lens is calculated by means of the formula given by Davisson and Calbick for a single diaphragm
\[ \frac{1}{f}=\frac{\Phi'_b-\Phi'_a}{4\Phi_0}, \]
where the curve of the change of potential along the optical axis can be represented schematically by the broken line shown by the dashed line in Fig. 1, в. In this formula \(f\) denotes the focal distance, \(\Phi_0\) the potential of the point of inflection, and \(\Phi'_a\) and \(\Phi'_b\) the derivatives of the potential in front of and behind the lens. If the calculation is carried out taking these simplifications into account, then we obtain the potential at the center of the aperture of the inner diaphragm equal to 150 V, as is also indicated in Fig. 1, в. The solid curve gives the exactly calculated change of potential, obtained by lengthy and tedious integrations of the differential equation. In this case very good agreement is found. On the basis of the dashed curve one can estimate the order of magnitude
Fig. 1. Approximate consideration of the Gaussian dioptric in an immersion objective
* In the case represented in Fig. 1, the potential is calculated as follows:
\[ \varphi(z,r)=203\left\{|z+a_2|\left(\frac{1}{m_2}+\operatorname{arc\,tg} m_2\right)-\right. \]
\[ \left.-(z-a_2)\left(\frac{1}{m_{-2}}+\operatorname{arc\,tg} m_{-2}\right)\right\} -193\left[(z+a_1)\left(\frac{1}{m_1}+\operatorname{arc\,tg} m_1\right)-\right. \]
\[ \left.-(z-a_1)\left(\frac{1}{m_{-1}}+\operatorname{arc\,tg} m_{-1}\right)\right]; \]
\[ a_1=\frac{3}{2};\qquad a_2=3 \]
\[ m_{\pm1}= \]
\[ =\sqrt{\frac{1}{2}\left[(z\pm a_1)^2+r^2-1\right] +\frac{1}{2}\sqrt{4(z\pm a_1)^2-\left[(z\pm a_1)^2+r^2-1\right]^2}}; \]
the corresponding expression is also obtained for \(m_{\pm2}\).
potential of the inner diaphragm. The exact value of the diaphragm potential, given by a rigorous calculation, is 32 V, as is indicated in Fig. 1a. This value can be established experimentally after all the apparatus has been made; therefore, in the calculation it is sufficient to estimate only the order of magnitude of the potential value. The latter can be done with a sufficient degree of accuracy by means of an approximate calculation, so that in practice there is no need to use an exact calculation, which requires several weeks for its execution. This situation—that the experimental route can yield results more quickly than exact theoretical calculations—may extend to all other applications of Gauss dioptrics, such as, for example, the mass spectrograph and Braun tubes. However, this still does not mean that such a state of affairs will last long. When the rapid development of electron optics has, in the main, been completed and the question arises of improving already existing electron-optical systems, Gauss dioptrics will have to be subjected once again to detailed consideration from the most varied points of view.
2. Electron-optical aberrations
The second step in the development of theoretical electron optics is the calculation of the aberrations produced by focusing fields. If the equations of Gauss dioptrics were strictly correct, then electron microscopes, in accordance with the small wavelength of electron rays (1 Å for electrons having a velocity corresponding to 150 V), would have to possess enormous resolving power. However, in the imaging of incandescent cathodes the relations are less favorable, since the electrons leave the object with a small velocity and, consequently, with a large wavelength. Thus the resolving power is limited by the wavelength measured at the object. Nevertheless, the fundamental limit of the resolving power is also extremely high here, since for the small velocity of 0.1 V, with which the majority of electrons emerge, the wavelength is no greater than
\[ \frac{1}{200\,000}\ \text{mm}. \]
In order to answer the question why this high resolving power is in fact unattainable, one must first of all bear in mind the low stability of objects, as well as the mutual repulsion of electrons. Until now, however, the actually attainable resolving power has been determined not by these disturbances but by the defects of the imaging system. Among these defects one must take into account, along with the unavoidable errors of adjustment, the aberrations known from light optics. By this are meant deviations of rays from the ideal directions that occur in a rigorous calculation. The formulae of Gauss dioptrics predict an ideal image. The calculation of aberrations shows how the known properties of electric and magnetic fields disturb the trajectory of the ray and thus lead to an indistinct image.
*
As in light optics, one must above all keep chromatic aberration in mind. The role of color in chromatic aberration is played here by the initial velocity of the electrons. Electrons of different velocities describe somewhat different paths and fall on the image plane not exactly at one point. In addition, as in light optics, third-order aberrations can occur here, such as: curvature of the image, curvature of the focal plane, astigmatism, coma, and spherical aberration; moreover, magnetic fields introduce certain peculiarities of their own. Second-order aberrations play a significant role only in instruments in which fields lacking axial symmetry are used (Braun tubes, spectrographs), and do not appear in the case of strict axial symmetry of the fields (electron microscope). Finally, in obtaining images of a heated cathode there also arise distortions that may be attributed to the small initial velocity of the electrons. The question of how to calculate the aberrations of a given lens is, in the main, already clear today. The field is specified by the course of variation of the electric potential and of the magnetic-field strength along the optical axis. Determination of the field in the case of an arbitrary arrangement of conductors encounters insurmountable difficulties. In calculating aberrations, as in all electron-optical problems, two methods may be used. The first proceeds, according to Busch’s work, from conceptions of the motion of electrons, being based, in this way, on Newton’s second law of dynamics for a material point. The second method consists in applying an analogy from light optics, the course of the electron rays being considered in the same way as the course of light rays in the corresponding optical problem. The answer to the question of which of these methods is simpler and more intuitive depends on the habits and tastes of the investigator. Of course, the results of both methods are the same. Glaser², using the second method, carried out the first systematic calculation of certain aberrations. Some of his interesting results have already been published. On the basis of these results one can establish that the calculation of aberrations in each case leads to the computation of a more or less complicated integral. With sufficient patience this integral can be evaluated and the value obtained compared with experiment. In this way one can verify the correctness of the calculation and the quality of the experiment.
3. Correction of electron-optical systems
The calculation of aberrations still does not provide a solution to the basic problem of electron optics. It is not enough to be able to calculate the aberrations introduced by a given field; it is also important to know what these fields must be like so that the aberrations introduced by them are as small as possible. Thus, it is necessary to find an electron-optical system with a minimum of aberrations. The first question that arises here is this: are these aberrations completely removable? The answer depends on what exactly is meant. Such aberrations as curvature of the image
PROBLEMS OF THEORETICAL ELECTRON OPTICS
distortions and coma can be eliminated; others, such as spherical and chromatic aberrations, cannot. One may require that the unavoidable aberrations be made as small as possible. There is still much work for theorists in this field. Since the magnitude of aberrations is expressed in the form of integrals, the problem of reducing them is solved by the method of the calculus of variations. The two independently variable functions are the values of the electrostatic potential and of the magnetic-field strength along the optical axis. After the values of the electric potential and of the magnetic-field strength on the optical axis satisfying the requirements of the posed problem have been found by means of variations, it is also necessary to find the shape and arrangement of conductors by means of which this field can be realized.
For this purpose the electric and magnetic fields are analytically extrapolated from the optical axis into space. Knowledge of the equation of the potential and of axial symmetry is sufficient to carry out this extrapolation uniquely. If these fields in space are known, it is not difficult to find the shape of the conductors that would create these fields. For example, in the case of an electrostatic field, in order to realize it it is sufficient to choose several suitable equipotential surfaces, make them of metal, and charge them to the corresponding potential. The complicated mathematical formulas, however, in some cases lead to such a form of the equipotential surfaces that their practical realization does not appear possible. In such cases one has to resort to changing the computed form of the conductors, so as, as far as possible, not to disturb the part of the field traversed by the electron rays.
All that has been said above can be illustrated by the example of a purely electric weak lens. Let us see how such a lens must be arranged so that it has the smallest possible spherical aberration. Weak lenses are practically useless, but they are of methodological interest and, above all, are simple. In the latter sense they are not typical representatives of electron-optical lenses, which usually lead to involved calculations. Consideration of Gaussian dioptrics gives the following data for the focal length \(f\) and the distance of the principal plane \(h\) of a weak lens:
\[ \frac{1}{f}=\frac{3}{16\Phi_0^3}\int_{-\infty}^{+\infty}\Phi'^2\,dz;\qquad h=\frac{3}{16\Phi_0^2}\int_{-\infty}^{+\infty}z^2\Phi'^2\,dz. \]
\(\Phi=\Phi(z)\) is the value of the potential measured along the optical axis. \(\Phi_0\) is the value of \(\Phi\) outside the lens.
Primes denote derivatives with respect to \(z\). If one also introduces the image length \(b\) and the radius of the lens aperture \(r_B\), then calculation of the magnitude of the spherical aberration gives:
\[ r_b=\frac{5br_B^3}{64\Phi_0^2}\int_{-\infty}^{+\infty}\Phi''^2\,dz. \]
The requirement is thus reduced to making \(r_b\) minimal for constant \(f\) and \(h\). The solution of the Euler equation relevant here is as follows:
\[ \Phi=\Phi_0(1+Ae^{Bz^2}). \]
The constants are determined with the aid of supplementary conditions. For calculating the field in space, the equation used is
\[ \varphi(z,r)=\frac{1}{2\pi}\int_{0}^{2\pi}\Phi(z+ir\sin\alpha)\,d\alpha, \]
which represents the above-mentioned analytic extrapolation. The condition \(\varphi(z,r)=\text{const}\) leads to the system of surfaces shown in Fig. 2. The solid lines denote surfaces having positive potential, the dashed lines negative potential. In this case one may be confident that the paths traversed by the electrons are indeed such that the image obtained on a screen placed at a large distance on the right-hand side is almost completely free from distortions produced by spherical aberration.
Fig. 2. Arrangement of conductors of a weak electric lens free from spherical aberration.
In summary, one may say that, despite great difficulties, theoretical electron optics has broad possibilities and is also of great practical interest.
REFERENCES
- Busch, Ann. d. Phys., 81, 974, 1926; 18, 583, 1927.
- Glaser, Z. Physik, 97, 177, 1935, and later works. On computations by other methods see O. Scherzer, Z. Physik, 101, 597, 1936.
- O. Scherzer, Z. Physik, 101, 23, 1936.