NEW INSTRUMENTS AND METHODS OF MEASUREMENT

NEW INTERFEROMETRIC METHODS AND THEIR APPLICATION TO THE STUDY OF CRYSTALS*.

S. Tolansky.

NEW TECHNIQUES.

The present article gives a brief review of certain new techniques developed by the author recently in the field of interferometric optics. These techniques are especially suitable for studying the details of the topography of nearly plane surfaces; in particular, they have been applied in the investigation of crystal surfaces. The methods described below are very powerful: they make it possible to measure irregularities of molecular dimensions, using only light of visible wavelengths. These methods employ interference of low orders of multiply reflected rays; the multiply reflected rays are obtained in a thin layer enclosed between two highly reflecting metallic surfaces (usually silver). They may be roughly divided into observations with monochromatic light and observations with white light. We shall first consider observations made with monochromatic light; as the light source, the light of a mercury arc in vacuum, passed through a green filter, is usually used.

The application of interference methods to the study of surface form is very old. But in most cases, in practice, interference of only two rays was used. Similarly, methods with multiply reflected rays have long been used for investigating the quality of optically plane surfaces. Lines of equal thickness, formed by multiply reflected rays in a wedge, were used even in the classical determinations of the length of the meter by Fabry, Perot, and Benoît. If, however, one examines the early work attentively, one sees that it did not realize the conditions necessary for attaining the best resolution; nor were the broad possibilities afforded by the improved technique adequately appreciated. From what follows it will become clear how extensive these possibilities are.

*) Journ. Sci. Instruments 22, no. 9, 1945. Translated by V. M. Kharitonov.

In the experiments described in the present article, interference of a multiply reflected beam was obtained with the aid of semitransparent silver layers with high reflectivity, deposited on the surfaces for which we wished to obtain interference patterns. To achieve a product of reflection coefficients approaching 90% without noticeable absorption is a rather difficult experimental task, which, however, was completely solved. In exactly the same way as in the case of applying Airy’s well-known theory of multiple beams to the Fabry–Perot interferometer, when summing all the rays obtained from silvered surfaces, very sharp and distinct lines appear. For lines of equal inclination (parallel rays) summation is easy to obtain by concentrating the rays with a lens; for lines of equal thickness (a wedge) the conditions of summation, however, are quite different. It seems to us that until now it has not been clearly appreciated that the necessary condition is the requirement that the maximum lateral displacement of all rays participating in the interference, along the surface of any of the optical components, should not exceed a small fraction (say, \(1/50\)) of the distance between neighboring orders. This condition is best achieved if we use an almost point source of light placed in the focus of a good lens, and normal incidence of the beam, and reduce the distance between the reflecting surfaces to the smallest possible value, preferably, if possible, of the order of the wavelength of the light used. The higher the reflection coefficient of the silver, the more important the rays of higher orders of reflection are and the more stringent the optical conditions become. With the best possible coating one can clearly count up to 100 reflections, and under these conditions a very precise setup is required if we wish to achieve the best resolution.

Fig. 1. Typical example of the application of the new interference technique (Newton’s rings modified by silvering).

A typical example of the successful application of the technique described is shown in Fig. 1. It shows Newton’s rings obtained with

corresponding silvering of the glass surfaces and under optimal optical conditions. The light source was the green line of a mercury arc. The interference lines are remarkable for their sharpness. In order that the details should be visible, the lines were photographed on a low-contrast plate. Thus, the sharpness obtained is not artificial photographic sharpness. The irregular bends that occur in the lines indicate defects on the surface of the lens. If we take into account that the distance between neighboring orders is equal to 2700 Å and that, at such high resolution, we can distinguish deviations of a line on the order of \(1/135\) of the distance between them, then it is evident that we can detect a local deviation of only about 20 Å. This is a quantity on the order of molecular dimensions. Obviously, with first-class optical surfaces one can achieve very great accuracy in determining the diameters of rings. Newton’s rings are still often used in metrological measurements. The advantage of lines obtained by using a multiply reflected ray, as compared with ordinary two-beam lines, is so obvious that they should always be used instead of the latter.

Fig. 2. Illustration of interferometric phenomena with multiply reflected rays at oblique incidence: (a) without a polaroid, (b) and (c) with a polaroid turned in two mutually perpendicular directions.

Lines with multiply reflected rays are so sharp that new interferometric phenomena are then observed. At incidence other than normal, the rings split into doublets, as shown in Fig. 2, a. It can be shown that the lines of these two systems are formed by rays plane-polarized in mutually perpendicular directions, which is explained by the different change in the phase of polarization for different directions upon reflection from the silver layer. In Fig. 2, b and 2, c, the behavior of the rings is shown when a polaroid plate oriented in two mutually perpendicular directions is introduced. The angle of incidence for these photographs was \(60^\circ\). The doublet splitting gives us a new and very powerful method for measuring the difference in phase change at different angles of incidence. The values obtained for a silver layer \(550\) Å thick are shown in Fig. 3. Obviously, by using other

metals with high reflecting power, or by introducing various liquids between metallized surfaces, opens up a broad field for possible investigations.

We shall recall that the earlier classical methods for determining phase change were connected with measuring the ellipticity of polarization and required expensive and complicated optical instruments. The simplicity of the new method is not the least thing that makes it attractive.

Thus this method makes it possible to carry out new investigations of the optical properties of metals. Unfortunately, space does not permit us to consider further interesting results that are obtained with Newton’s rings; references to some of these works may be found in the brief bibliography appended at the end.

Fig. 3. Quantities obtained from the splitting of lines by measuring the phase difference at various angles of incidence for a silver film 550 Å thick. I — experimental curve, II — curve given by the classical electromagnetic theory, III — difference between curves I and II.

INVESTIGATION OF THE SURFACE OF NATURAL CRYSTALS

The technique just described of monochromatic multiply reflected rays has been successfully applied to the study of the topographical features of approximately plane crystalline surfaces. It is necessary to consider two cases, namely: (a) optically good, transparent crystals; (b) incompletely transparent or completely opaque crystals, including metallic surfaces, or any more or less plane surfaces. Since the silver layer is obtained by evaporating silver in a vacuum, it is obviously necessary that the surface under investigation not be destroyed in vacuum. For case (a) one may use both lines in transmitted light and in reflected light, whereas for case (b) only lines in reflected light may be used. Silver is evaporated onto the surface of the crystal and onto the surface of an optically plane quartz plate of the highest quality, accurate to \(1/60\) of a wavelength. These two surfaces are placed very close to one another, and the lines produced as a result of the interference of multiply reflected rays are observed. In investigations on mica it was shown that the deposited silver exactly po-

repeats the outlines of all the topographical features of the crystal surface. This is true up to the limit of observational accuracy (20 Å) and was verified in cases where the surface magnification did not exceed \(100\times\). Experiments at higher magnifications have not yet been carried out.

Typical results obtained by the method described are shown in Figs. 4 and 5, which relate to a highly polished (100) surface of a left-handed quartz crystal; the surface area is about one square centimeter. The bends of the lines are determined by the contours of various regions, while the particular observed pattern depends on the angle of inclination between the silvered surfaces. If this fact is taken into account, then two patterns that are quite different in external appearance will lead to the same results. In Fig. 4 are shown the lines obtained in transmitted light over the greater part of the surface; the optical conditions were such that they did not allow the entire surface to be covered.

Fig. 4. Investigation of a highly polished (100) surface of a levorotatory quartz crystal by means of a monochromatic multiply reflected beam (transmitted light).

Simultaneously the lines obtained for the yellow and green mercury lines are shown. In the photographs we see sharp, narrow, bright lines on a dark background. A characteristic feature of the success of the method is the narrowness of the line in comparison with the distance between neighboring orders, which makes it possible to record the finest details. (In all the figures of the present article there is no photographic increase of sharpness by the use of contrast materials. Low-contrast plates and paper were used both to obtain a natural picture and to preserve fine details.)

In Fig. 5 green lines in reflected light are shown, in the form of thin dark “absorption” lines on a broad light background. The distinctness of the lines is simply astonishing and even better than in a Fabry–Perot interferometer with the same silvering (in the latter, exact parallelism is never achieved, and all surface defects appear in the resulting pattern). From the pattern of Fig. 5 it is clear that this technique can also be successfully applied to opaque surfaces, in particular metallic ones, for studying, for example, the influence of polishing, etching, etc. By placing the cry-

of the crystal over various regions of the optical plane, it can be shown that the crystal is characterized by a whole series of remarkable topographic features; lack of space unfortunately does not allow us to consider this question in detail. It turns out, among other things, that on the surface of the crystal small elevations and depressions are found, only about 20 molecules in height (depth); our technique also makes it possible to detect local surface inclinations of only about \(1/100\) of an angular minute. (The method described below, using white light, makes it possible to detect inclinations of about \(1/300\) of an angular minute.) Interferometric methods thus give results at least 100 times better than those obtained with the most expensive precision goniometers, and, in addition, make it possible to carry out a separate study of very small areas of the surface.

Fig. 5. Study of the same crystal as in Fig. 4, but in reflected light.

STUDY OF CLEAVAGE SURFACES

In the preceding paragraph we considered the study of the surfaces of natural crystals; here we shall consider our method as applied to the study of cleavage surfaces of mica and selenite. The silvered cleavage surface was placed near the same silvered optically plane surface; illumination was carried out with the aid of a collimator in such a way that observation was conducted under optimal conditions. Fig. 6 shows a typical topography of the cleavage surface of mica over an area of about one square centimeter. A very large number of mica surfaces was investigated and practically

Fig. 6. Topography of the surface of mica (about 1 sq. cm).

for all, the picture proved analogous. The study of thin mica cleavages by means of interference of two rays is very old, but it is essential to note that interesting details appear only when we apply the improved technique of interference of multiply reflected rays. On mica surfaces studied in this way, elevations and depressions are usually observed, whose dimensions range from small fractions of the wavelength of light up to \(^{1}/_{200}\) mm. The most interesting features visible in Fig. 6 are: (a) smooth continuous lines and (b)—sharp “cleavage lines,” representing separate cleavage steps, the length of which may vary up to several millimeters, and in some extreme cases even centimeters.

Fig. 7. Graph of a section about 6 mm long in a direction more or less perpendicular to a typical group of cleavage lines of a muscovite specimen.

In Fig. 7 is shown a graph of a transverse section of the surface of a muscovite specimen over a length of about 6 mm, passing more or less perpendicularly to a group of typical cleavage lines. By exact measurements it was established that the jumps along the cleavage lines in most cases differ only by a small integer multiple from a magnitude almost equal to 20 Å (with an inaccuracy of less than 1 Å). Steps of 20, 40, 60, etc. Å were actually measured. Crystallographic studies by means of X-rays show that the constant of the molecular space lattice of mica in the direction perpendicular to the cleavage surface is exactly 20 Å. Consequently, using visible light, we can indeed obtain 20 Å just as in studying the space lattice. Twenty angstroms may be regarded as a magnitude for which one can vouch when carrying out optical investigations with the aid of multiply reflected rays.

A very important circumstance is that the observed value of the step height proved constant along the cleavage line, which shows that, during silvering, the silver layer exactly repeats the contours of the surface. Moreover, on the basis of the smoothness and continuity of the interference lines it can be shown that mica:

actually cleaves along the cleavage plane, at least with an accuracy of up to 30 Å, i.e., almost certainly within the limits of a single molecule; this fact may be supported by independent experiments involving the growth of ammonium iodide crystals on mica. We have examined in somewhat more detail only a little of what can be learned from studying photographs of mica.

Fig. 8 shows the outlines of the cleavage surface of a selenite crystal (an area of the order of one square centimeter). As in the case of mica, typical cleavage lines are clearly visible; however, it was found that, in contrast to mica, the adjacent surfaces adjoining the cleavage line are slightly inclined toward one another, with a slope toward the cleavage line. The observed magnitudes of the inclination vary within a small range, of the order of 0.011—0.063 minutes of arc, from which the resolving power of this method is evident. For comparison, Fig. 9 shows a typical three-dimensional diagram of the surface contour over a length of 8 mm (minor details on the diagram have been omitted). The black lines show the deviation of the surface from horizontality.

Fig. 8. Cleavage surface of a selenite crystal (about 1 sq. cm).

Observations with large dispersion show that the diffraction lines become ragged because of the presence of a secondary surface structure consisting of small, numerous, elongated areas shifted upward and downward by a small integer multiple of 15 Å, which is the value of the constant of the spatial molecular lattice of selenite obtained from X-ray observations.

OBSERVATIONS WITHOUT AN OPTICALLY PLANE SURFACE.

In all the experiments described above, the surface under investigation was placed near an optically plane surface of the highest quality. In the experiments described below, we dispensed with such a surface. Thin flakes of mica, of thickness of the order of \(1/50—1/150\) mm, were silvered on both sides and then subjected to investigation. Despite the already known complexity of the topography of each of the surfaces, the Fizeau lines obtained as a resul-

as a result of the interference of multiply reflected rays, prove to be surprisingly simple. On specimens similar to those shown in Fig. 10, considerable areas are observed, bounded by cleavage lines and colored quite uniformly (at this stage we

Fig. 9. Graph of the outline of the surface of a selenite crystal (Fig. 8) over a length of 8 mm (small details omitted).

may neglect the secondary system of faintly visible lines). The constant coloration in a multiply reflected ray is very sensitive to a change in thickness, especially if one uses light from a

Fig. 10. Typical examples of cleaved mica flakes.

mercury arc not passed through a light filter. Uniformly colored regions correspond to areas on which the mica sheets have one and the same thickness with an accuracy down to a small fraction of the wavelength.

waves. It follows from this that protrusions and depressions on one side correspond exactly to depressions and protrusions on the other, and the difference in thickness occurs only along the cleavage line. The surfaces are thus perfectly parallel; as a result, it is impossible to determine either the height or the direction of the steps and, consequently, impossible to interpret the observed picture (the resolution of this difficulty is given in the next section, which considers interference lines formed when observed in white light).

Nevertheless, the interference lines are still of considerable interest. Besides showing that the rear surface, with respect to deviations from flatness, follows the front one, this method of investigation has greater sensitivity, since the formula includes the relative refractive index of mica; as a result, measurements are obtained with 60% greater accuracy. Thus it is possible to detect the presence of local growth nuclei, individual inclusions, variation of chemical composition, double refraction, etc. A method of observation with the aid of a microscope was developed; the interference lines were observed with linear magnification up to \(100\times\); details and some results will be considered in subsequent paragraphs.

INTERFERENCE LINES IN WHITE LIGHT

In photographs 6 and 8, obtained on mica and on selenite, it is impossible to distinguish protrusions from depressions or to determine the actual direction of the steps. New interferometric methods, besides possessing other properties, also resolve these difficulties. It will hardly be an exaggeration to say that the new methods using the interference of multiply reflected rays, described below, are the most powerful in the field of precision optics employing interference of small orders. In the following exposition this assertion will be sufficiently substantiated. In the present method the interference lines are formed in white light and at the same time have numerous advantages in comparison with lines in monochromatic light. On the basis of theoretical considerations it is proposed to call these lines lines of equal chromatic order. In order to show how these lines are formed and to elucidate some of their properties, we shall consider Newton’s rings obtained on an arrangement with silvered surfaces, which gives the Fizeau lines shown in Fig. 1. The optical arrangement in this case is very simple. The image of a white light source is formed on a small circular aperture located at the focus of a good achromatic lens. A parallel beam of white light falls normally on the surfaces from which the interfering rays are reflected. The image of the surfaces is projec-

is then projected onto the narrow slit of a good spectrograph, and the lines obtained on the plate are photographed. The magnitude of the enlargement in projecting the image of the surface onto the spectrograph slit is of great importance and strongly affects the distinctness of the lines and even their appearance; however, we cannot consider this question in detail here.

The lines for the visible region that are obtained on the apparatus for obtaining Newton’s rings with silvered surfaces are shown in Fig. 11 (the scale in the photograph gives the wavelengths). These, to the highest degree unusual, interference lines possess the following properties: (1) each line is remarkably colored in a large number of colors; (2) all the lines are distinct to the highest degree, despite the fact that white light was used; the width of the lines is only a small fraction of the distance between neighboring orders; (3) the observed form of the lines can be predicted by means of a very simple theory; (4) local defects, visible (in enlarged form) also on our Fizeau lines, immediately show on the lines whether we are dealing with an elevation or a depression (in the present case the surface of the lens is an elevation); (5) the direction of convexity (concavity) of the lines immediately shows whether we are dealing with an elevation or a depression.

Fig. 11. Lines of equal chromatic order for the visible region of the spectrum, obtained by means of an apparatus for obtaining Newton’s rings with silvered surfaces.

It is easy to show that for each line the ratio \(t/\lambda\) is constant, where \(t\) is the distance between the silvered surfaces at the point under consideration (we recall that the image of these surfaces is projected onto the slit of the spectrograph). With the aid of these lines, which are very sharp despite the fact that we are not using a monochromatic light source, it is possible to resolve what remained uncertain in the preceding sections and to obtain much additional information. The possibilities in applying the method are suggested by two simple formulas, which are obtained as follows. If \(\Delta \lambda\) is the difference in wavelength between two neighboring lines horizontally, then \(t = \frac{1}{2}\Delta \lambda\). If a small change in thickness \(dt\) causes, for a given wavelength, a change of order by \(dn\), then \(dt = dn \cdot \frac{1}{2}\lambda\). Hence it is evident at first glance that po-

Since the quantity \(dn\), measured from the photograph, is very small, we can without difficulty estimate the magnitude of the defects as being of molecular dimensions.

When this method is used, the highest degree of accuracy can be attained. We shall now consider some of the possible applications. Lines of equal chromatic order can be used successfully for studying the topography of crystal surfaces. Figs. 12 and 13 show two examples of lines obtained from a sheet of mica pressed against an optically plane surface. In Fig. 12 a sharp ridge-like elevation of the surface is visible, together with other elevations and depressions, which can now easily be interpreted (the focusing is accurate only in the region \(4000\)—\(5000\) Å). A characteristic feature of our method is that exact measurements can be obtained directly from the print with the aid of only a single ruler. The actual ridges are magnified on the lines by approximately \(50\,000\times\). In Fig. 13 a cleavage line crossing the slit is shown. A simple glance is sufficient to establish the true direction of the step, and for a more precise determination one may use a simple count of the coincidences of the lines, applying the nonius formula. All doubts as to whether there is a protrusion or a depression, and as to the direction of the step, disappear completely, and the latter can be determined with great accuracy.

Fig. 12. Lines of equal chromatic order obtained from a sheet of mica pressed against an optically plane surface; a ridge on the surface of the mica is distinctly visible.

Fig. 13. Lines of equal chromatic order obtained with the aid of a sheet of mica pressed against an optically plane surface; a cleavage line crossing the slit is clearly visible.

We must pay special attention to the sharpness of the lines that is obtained, despite the fact that we use a white light source. This is very important in that we can have lines over a large region, and very bright ones, and this, in turn,

enables one, in measurements, to obtain an accuracy unattainable with any other type of interference fringes. In all methods using the interference of multiply reflected monochromatic rays (as, for example, the Fizeau fringes and the Fabry–Perot fringes), in which semitransparent silver layers are used, the thickness of the latter is limited by the brightness of the light source, and thus the sharpness of the fringes is limited. The brightness of monochromatic light sources is strictly limited by the requirements of monochromaticity, since for bright sources the line width is impermissibly large. As for fringes of equal chromatic order, very bright sources of white light may be used to obtain them, such as, for example, a Pantalit, a voltaic arc, and others; even with a very powerful voltaic arc, excellent fringes are obtained. Consequently, very dense coatings are permissible; as a result, the fringes obtained are very fine and sharp. Moreover, since we can use such bright light sources, then, as will be shown below, these fringes can be examined with the aid of powerful microscopes.

The fringes can be obtained from any crystalline surface both in transmitted light (for transparent materials) and in reflected light (for opaque substances, such as, for example, metals).

We shall now consider further applications of this method.

FURTHER APPLICATIONS.

Of particular interest are the fringes of equal chromatic order obtained on mica plates silvered on both sides. Two limiting cases are shown in Figs. 14 and 15. They were obtained respectively with the specimens shown in Figs. 10a and 10b. We shall not discuss the many details visible in the figures, and shall merely note that: (a) it is confirmed with great accuracy that, although the opposite surfaces are not plane, they are strictly “parallel”; (b) the change in thickness occurs in jumps, the magnitude of the jump being “quantized” in units of 20 Å; (c) owing to the double

Fig. 14. Fringes of equal chromatic order obtained from a mica specimen silvered on both sides and shown in Fig. 10a.

the birefringence lines are doubled; (d) in addition, much other information can be obtained.

The birefringence can be determined for the entire region of wavelengths from a single photograph, even if it is only 0.0001. We thus obtain a new method for determining this quantity. It is especially noteworthy that the magnitude of the splitting of the lines due to birefringence is completely independent of the thickness of the mica plate, which is entirely unexpected for an interference phenomenon. This fact follows already from the simplest theory and is fully confirmed by experiment.

Fig. 15. Lines of equal chromatic order obtained from an image of a mica specimen, silvered on both sides and shown in Fig. 10b.

METHOD OF APPLYING LINES OF EQUAL CHROMATIC ORDER.

Lines of equal chromatic order give a section of the surface along some line. In order to examine the whole surface, it is necessary to move the image on the slit perpendicular to the slit. This can be done quickly in visual observations by using a Hilger instrument with constant deviation, or some other spectrometer. It is often useful first to photograph the pattern obtained as a result of the interference of a multiply reflected monochromatic beam, covering a considerable area, after which one can choose the direction in which more detailed data on the section of the surface should be obtained by means of lines of equal chromatic order. We shall consider two examples of such a combined technique, in which small local features of mica flakes silvered on both sides were investigated. The corresponding photographs are given in Figs. 16—19. Fizeau lines for one such feature are shown in Fig. 16, and the corresponding lines of equal chromatic order in Fig. 17. The Fizeau lines were obtained with the aid of a microscope having a magnification of 80×. It was shown that this feature, in all probability, represents a nucleus of crystal growth. It has a pyramidal shape and consists of separate steps, 50% of which are me—

fewer than three molecules in height, and 80%—less than 8 molecules in height. Such information is obviously of great importance for the theory of crystal growth.

Another irregularity (less than 1 mm in cross section) consists of three groups of Fizeau lines of elliptical form, on each of which

Fig. 16. Fizeau lines for a small local feature.

Fig. 17. Lines of equal chromatic order for the same feature as in Fig. 16.

the doubling due to double refraction is clearly visible (Fig. 18). This picture cannot be interpreted until we obtain lines of equal chromatic order, approximately through the middle of the central region (Fig. 19), after which we obtain a complete idea of the shape of the surface. The final conclusion leads us to the inference that we are dealing here with a small foreign inclusion in the form of a thin petal. It is necessary to pay attention to the sharpness of both photographs on which this feature is represented (magnification 80×).

Let us also note here that the limit of accuracy is set by the magnification that can be used in examining Fizeau lines for multiply reflected rays. It was found that the width of the lines becomes apparent at a magnification of 200×. This phenomenon arises as a result of the necessity of increasing the angle of the light cone, as a consequence of which the very strict illumination conditions are violated. Independently of this, the provision of a suitable light source is in all cases a rather difficult matter.

On the other hand, lines of equal chromatic order do not have such a drawback. A layer consisting of two silvered surfaces may be regarded as a light filter through which pass only those waves whose lengths satisfy the relation \(n\lambda = 2\mu t\cos\theta\). In reality, it is advantageous when the surfaces from which the reflection of the interfering...

rays are parallel, since we then have optimal optical conditions for the superposition of the rays. Under such optimal conditions we can use maximum magnification. The lines were obtained already at a magnification of \(1000\times\). It is only necessary, in the case of interferometry, to regard the microscope as a very small diaphragm.

It should be pointed out that precisely in the case of reflected lines, lines in white light have a considerable advantage over lines in monochromatic light. For the latter, strict monochromatization is necessary upon reflection, since otherwise the sharpness of the lines disappears.

Fig. 18. Fizeau lines for another peculiarity.

Fig. 19. Lines of equal chromatic order for the same peculiarity as in Fig. 18.

All these difficulties disappear when lines of equal chromatic order are used.

In the present compressed and brief survey it was, of course, impossible either to consider the technique of the matter in detail or to give numerical results. Further details may be found in the author’s works listed below, as well as in works that have already been submitted and will soon appear in Proceedings of the Royal Society and in Philosophical Magazine.

REFERENCES TO THE AUTHOR’S WORKS:

Nature, 152, 722, 1943; 153, 195, 1944; Phil. Mag., 32, 120, 1944; 35, 179, 1944.
Proc. Roy. Soc. A, 184, 41, 1945; Proc. Roy. Soc. A, 184, 51, 1945.