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FROM CURRENT LITERATURE
INVESTIGATION OF SURFACE MICRORELIEF USING MULTIPLE-BEAM INTERFERENCE FRINGES OF EQUAL CHROMATIC ORDER
Among the various methods of using multiple-beam interferometry for investigating microrelief, one of the most precise is the use of the so-called fringes of equal chromatic order. A Fabry–Perot interferometer, formed by the surface under investigation (previously silvered) and by a reference plate with a semitransparent coating, is projected (in transmitted or reflected white light) onto the slit of a spectrograph. As a result, the spectrogram shows a set of bright (in transmitted light) or dark (in reflected light) fringes of “equal chromatic order,” the shape of which reproduces (as it were in section) the form of the air gap between the plates of the interferometer[^1]. In fact, the position of a fringe corresponds to the condition that the interference order be an integer for a given wavelength $\lambda$. On the other hand, the order of interference $n$ is related to the thickness of the air gap $t$ by the relation
\[ n = \frac{2t}{\lambda} + f, \tag{1} \]
where $f$ is the additional phase shift arising upon reflection of light from semitransparent coatings and upon passage of light through them[^1]. If at some point the thickness of the gap changes by $t$, then at the corresponding point of the spectrum the fringe is displaced into a position corresponding to a new value of $\lambda$ at the same integral value of $n$. If $f$ does not depend on $\lambda$, as is usually assumed, then $\Delta t = -\frac{1}{2} n\,\Delta \lambda$. On the other hand, the order of interference can be determined from the difference in wavelengths corresponding to two different integral orders of interference at one and the same thickness of the gap. Indeed:
\[ n\lambda = rt + f\lambda,\qquad (n+\Delta n)\lambda' = 2t + f\lambda', \]
whence
\[ (n-f)=\frac{\Delta n\cdot\lambda'}{\lambda-\lambda'}=\frac{2t}{\lambda}. \tag{2} \]
Usually, without sufficient grounds, it is assumed that $f=0$. In work[^2] it is shown that, generally speaking, for silver coatings $f$ is close to unity, as a result of which in most works devoted to the use of fringes
FROM CURRENT LITERATURE
of equal chromatic order, a systematic error is introduced (cf. \(^{1,3}\)).
Moreover, as it turns out, \(f\) depends appreciably on \(\lambda\), which is reflected in a systematic discrepancy in the values of \(t\) obtained by means of fringes of different order (in light of different wavelengths). Since \(f(\lambda)\) varies little with \(\lambda\), in a small spectral interval one may expand \(f(\lambda)\) in a series in \((\lambda-\lambda_0)\) and retain only the first term of the expansion:
\[ f(\lambda)=f_0+b(\lambda-\lambda_0). \]
Accordingly, instead of (2) we have:
\[ (n_0-f_0-b\lambda)=\frac{\Delta n\cdot \lambda}{\lambda_0-\lambda} =\frac{2t}{\lambda_0}-b\lambda. \tag{3} \]
Since relation (3) does not depend on \(n\), it can be used to determine \(b\) and \(t\). The results of measurements of \(f(\lambda)\) for a number of coatings in the wavelength interval \(4500\text{--}6300\,\text{\AA}\) are shown in Fig. 1. The measurements were made in reflected light when the lower coating was opaque and had, in white light, a reflection coefficient \(R>90\%\). The transparency of the upper coating is indicated in the figure.
As may be seen, for freshly prepared silver layers \(f\) is practically independent of \(\lambda\). As they age (in air), the curves of \(f(\lambda)\) begin to show a break: for short wavelengths \(b\ne0\), while for long wavelengths (beginning with some \(\lambda_0\)) \(b\ne0\) and \(f\) is practically independent of \(\lambda\) \((b\simeq 3.2\cdot10^{-5}\,\text{\AA}^{-1})\). As aging proceeds, the break point shifts toward shorter wavelengths in different ways for coatings of different transparency.
A calculation based on the use of the optical constants of massive silver for \(\lambda=5893\,\text{\AA}\) led the author to the value \(f=0.83\) (for the case of a pair of layers with \(R=0.82\)). The experimental values \(f_0\) for the corresponding case proved to be 0.95 and 0.86 for layers whose age was measured, respectively, by one hour and by one week. Thus, as a layer ages, its effective optical constants apparently tend toward the values corresponding to the massive specimen.
From the data presented, the author concludes that if in (2) one puts \(f=1\), then for fresh silver layers the systematic error will be equal to
\[ \frac{1-f}{\,n-f\,}, \]
which, for the typical value \(f\simeq0.93\) and \(n=8\), amounts to only about \(1\%\). We note that the influence of the magnitude of \(f\) and of its dependence on \(\lambda\) on the form and position of the interference fringes was considered in general form by Tol'den as early as 1949; however, as a rule, it was not taken into account. At the same time, as the author has shown, taking this circumstance into account makes it possible to increase substantially the accuracy of determination of the relief. In fact, the position of the fringe can be determined under favorable conditions on a spectrogram with an accuracy of the order of \(\pm 0.14\,\text{\AA}\), which corresponds to an error in the determination of \(t\) of the order of \(1\text{--}2\,\text{\AA}\).
Introducing the necessary refinements into the calculation formulas, the author used the described method for determining the microrelief of various surfaces \(^{4,5}\). First of all, the author turned to the study of the process of polishing a standard glass plate.
Let us recall that the question of the mechanism of glass polishing has not yet been clarified. Herschel suggested that simple grinding down of the existing irregularities takes place here. Thomson \(^{6}\) believed that the polishing particles, penetrating into the irregularities of the surface, cut them off. At the same time, a number of authors, beginning with Beilby \(^{7}\), adhere to the point of view that
polishing of glass proceeds analogously to the polishing of metal, i.e., by viscous flow of the protrusions into the hollows surrounding them. An investigation of polished surfaces of glass and diamond by means of multiple-beam interferometry^8 revealed that it is possible to obtain surfaces
Fig. 1. Dependence of \(f(\lambda)\) for a Fabry–Perot interferometer with silver coatings in reflected light.
\(a\)—an example of the experimental determination of \(f(\lambda)\) for one of the cases;
\(b\)—the dependence of \(f(\lambda)\) for layers with different reflectivity and of different age.
about one square millimeter in size, flat to an accuracy of up to \(0.004\lambda\), and smooth to this accuracy, except for the presence of a certain number of polishing scratches with depths from 10 to 40 Å. Investigations by means of X-rays^9 and an electron microscope^10 also confirmed the presence of surface irregularities of the order of \(\pm 10\) Å. All the authors named were inclined in favor of the theory of viscous flow, but were unable to obtain convincing evidence in its favor. Köhler,^4 using the technique
of bands of the same chromatic order, studied the irregularities observed in these bands, assuming that they are due to irregularities in the thickness of the interferometer spacer (i.e., irregularities of both surfaces forming it). As an example we present interferograms and the corresponding surface contours for cases of polishing with barnesite (BF) for 20 and 450 minutes (Fig. 2, a and b).
The author adheres to a direct interpretation of the pattern obtained by him and believes that the small irregularities of the interference bands accurately reproduce the height and shape of the irregularities of the glass surfaces (more precisely, the thicknesses of the air interlayer). Thus, in the author’s opinion, the surface of polished glass is, as it were, a multitude of round (or almost round) pointed cones of approximately the same height (10–100 Å), having at the base a diameter of less than 100 Å, closely pressed against one another and resting on a smooth base. Proceeding from this interpretation, the author determines experimentally the dependence of the distribution of cones by height on the duration of the polishing process, as well as the dependence of the mean cone height and the mean fluctuation in cone heights on the duration of polishing. It is found that as the surface is polished both the mean height of the cone and the mean fluctuation of the height decrease according to a parabolic law. If, after 12 minutes of polishing, the distribution of cones by heights covers the interval from zero to 120 Å with a maximum around 65 Å (cf. Fig. 2, a), then after 450 minutes of polishing the distribution extends practically only to 60 Å and has a maximum around 20 Å (cf. Fig. 2, b). At the same time the author determined interferometrically the polishing rate, i.e., the thickness of the layer removed from the surface per unit time. It turned out that in the initial stage (during the first hour) the polishing rate increases from \(\sim 700\) Å per minute to \(\sim 900\) Å per minute, after which there begins at first a rapid, and then an ever slower decrease in the polishing rate, so that after about 6 hours an almost unchanged value of \(\sim 400\text{–}500\) Å per minute is established. (These values were obtained under a quite definite polishing regime, and the author intends subsequently to determine their dependence on the regime.)
Considering the results he obtained, the author comes to the conclusion that in polishing glass no viscous flow takes place, and that the mechanism of polishing is simply the chipping away of the material by particles of the polishing substance embedded in the polishing disk. On the basis of these ideas he gives a qualitative explanation of all the features described above, including the time dependences of the polishing rate and of the shape of the polished surface.
With all the interest which the regularities revealed by the author present, the direct interpretation used by him calls for a very cautious attitude. First of all, let us recall\(^{1,11}\) that multiple-beam interference microscopy, as is known, is characterized by a high resolving power in depth, but by a low resolving power in the plane of view (unlike electron microscopy). Generally speaking, it is directly applicable to objects whose transverse cross-section exceeds \(\sim 1\ \mu\). In any case the resolving power in the plane of view cannot be greater than the resolving power of the microscope used. In the present case, however, the author believes that it reaches several angstroms. Obviously, diffraction phenomena, complicated by the conditions of multiple-beam interferometry\(^{11}\), must here lead to a complete distortion of the actual form of the object (let us recall that a diffraction theory of the multiple-beam interference microscope still does not exist).
Fig. 2. Irregularities observed in bands of equal chromatic order, and their interpretation.
a — glass surfaces polished for 20 minutes; b — glass surfaces polished for 450 minutes.
Visible labels in the figure:
- \(5790.7\,\text{Å}\), \(5769.7\,\text{Å}\), \(4916.0\,\text{Å}\); \(n=5\), \(n=6\), \(A\)
- \(5790.7\,\text{Å}\), \(4358.3\,\text{Å}\); \(n=7\), \(A\)
- Vertical axis: \(\Delta t\,(\text{Å})\)
- Horizontal axis: Distance (mm)
- \(a)\), \(b)\)
- [[unclear: label inside shaded region, likely identifying cracks/scratches]]
In support of his interpretation, the author gives the following arguments.
The surface of the interferometer is projected with 50-fold magnification onto the slit of a spectrograph of size \(10 \times 0.01\) mm, i.e. each time an area of size \(200 \times 0.2\,\mu\) is examined. (Obviously, the transverse size of this area is smaller than the size of the diffraction spot.) Further, the author found that for two regions separated from one another by a distance smaller than the average distance between two peaks (irregularities), i.e. \(\sim 100\,\text{Å}\), no correlation is observed between the irregularities detected. Finally, a smooth displacement of the interferometer image perpendicular to the spectrograph slit leads to a smooth change in the height of the peaks: from zero to a maximum at distances smaller than the average distance between peaks, which, in the author’s opinion, indicates the approximate roundness of the shape of their cross section.
In view of the foregoing, what is involved here is evidently not the form of the inhomogeneities themselves, but the form of the diffraction spots corresponding to them, and therefore the author’s interpretation requires serious theoretical analysis.
Moreover, let us recall that, to obtain multiple-beam interference, the surface is silvered, the coating thickness reaching \(300\)—\(400\,\text{Å}\). It is well known \(^{1,12}\) that in this process the vertical, but not the horizontal, relief of the surface is preserved (on average!) with high accuracy. The preservation of peaks only a few tens of angstroms thick at such a height is more than doubtful. In addition, it is well known \(^{12}\) that the surface of a silver coating is not smooth, but granular, with grain sizes from several tens of angstroms to \(\sim 1\,\mu\). Therefore, if the author’s interpretation is correct, it should be attributed not to the surface of polished glass, but to the surface of the silver layer. Since it is known \(^{12}\) that the structure of the layer depends strongly on the structure of the underlying surface, the results obtained by the author can apparently have only indirect significance for elucidating the polishing process, but they are of serious interest for the study of thin metallic coatings.
The last of the works reviewed \(^{5}\) is devoted to the investigation of cleavage planes of topaz crystals. By carrying out, with the aid of equal-chromatic-order bands, a large number of measurements of the height of steps observed on mica surfaces, the authors determined the height of the unit cell of the topaz crystal lattice in the direction perpendicular to the cleavage plane. In this way they obtained the value \(8.81 \pm 0.07\,\text{Å}\), in good agreement with the value obtained on the basis of X-ray structural analysis—\(8.78\,\text{Å}\).
Such high accuracy was achieved to a considerable extent at the expense of abundant statistics. The error of individual measurements of the step height is on average \(\pm 1.2\,\text{Å}\), in accordance with the estimate of the method’s capabilities given above (cf. \(^{1}\)).
The authors note interesting features of the cleavage surface in the case of topaz. Long cleavage lines are often curved. In this case the cleavage lines extending for several millimeters have, in a few hundred unit cells, the usual height and are readily observed in an ordinary microscope. The height of the step varies noticeably along the cleavage line. By contrast, short (several tenths of a millimeter) cleavage lines are not detectable with an ordinary microscope and have a constant step height along their length, amounting to several tens of unit cells. It is striking that the steps are not perpendicular to the cleavage plane.
G. R.
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