Phase Shift upon Reflection of Light by Thin Films

In recent years, films—especially metallic ones—whose thickness is measured in a few tens or hundreds of angstroms have been finding ever wider application. In connection with this, chiefly in connection with the use of such films in interference light filters and for the purpose of studying the microrelief of surfaces by methods of multiple-beam interferometry, the problem of studying the optical properties of thin films is becoming increasingly important.

The relatively few direct measurements so far of the refractive index of various substances in thin films, as well as the practice of manufacturing interference light filters, have shown that the optical properties of a substance deposited on an underlying surface in the form of a thin film differ substantially from the optical properties of the same substance in massive samples. Moreover, here we are dealing not at all with trivial effects caused by the presence of very closely spaced boundary surfaces, but with a real change in the effective optical constants of the film itself. As the thickness of the film decreases below a certain, characteristic value for a given substance, there is, generally speaking, a sharp increase in both the real and imaginary parts of the effective refractive index. In some cases there is even a change in the sign of the difference between the phase velocity of light in the film and in vacuum (which occurs for silver films at a thickness of \(30\ \text{\AA}\))[^1]. Both the degree and the character of the observed changes depend to a high degree on the thickness of the film. The suggestion has been made[^2] that in such films there also takes place a real change in the optical constants of the substance forming the film, caused by structural changes in it as the film thickness increases. Thus, for example, electron-diffraction measurements have shown that films of \(\mathrm{SiO_2}\), \(\mathrm{ZnS}\), and \(\mathrm{MgF_2}\), having a thickness of less than \(1000\ \text{\AA}\), are amorphous[^3].

However, recent investigations of the structure of thin metallic films of various nature and at various stages of their formation, carried out with the aid of an electron microscope and by other methods[^3,^4], if they do not completely refute this supposition, at any rate bring other causes to the fore. These investigations have clearly shown that in very thin layers the substance forming the film is not distributed uniformly over the underlying surface, but forms separate, mutually isolated particles, the size and shape of which are determined, apparently, by the character of the surface forces. As the amount of deposited material increases, the individual particles begin to form agglomerates, and the gradually granulated structure of the coating gives way to a porous structure, resembling in outward appearance a sponge. At the same time, a considerable part of the underlying surface remains uncovered by the film. Only at more substantial thicknesses of the deposited layer does there apparently occur its rearrangement and the formation of a homogeneous polycrystalline film. It is interesting to note that such a transformation of the structure of the film in the course of its growth is connected not only with a change in its optical properties (for example, the reflection coefficient and its dependence on wavelength[^3]), but also with a change in its electrical properties—the specific resistance of metallic films is \(10^3\)–\(10^7\) times greater than the specific resistance of the same metals in massive samples[^5].

Thus, at a small thickness of the deposited layer it does not constitute a uniform film and, from the optical point of view, should be regarded as a turbid medium (we note that the dimensions of the inhomogeneities are considerably smaller than the wavelength of light).

Consequently, there are no convincing grounds for assuming an actual change in the optical constants of the substance itself when it forms thin films. On the contrary, the observed changes in effective optical constants can apparently be accounted for by a change in the role and character of light scattering by the inhomogeneous, granular or porous structure of the films.

However, measurements in transmitted or reflected light cannot provide a criterion for distinguishing both effects: coherent scattering of light by structural inhomogeneities in the direction of the transmitted or reflected beam, which leads to phase shifts in these beams, turns out to be completely equivalent to a change in the effective optical constants of a homogeneous medium.

The question of the presence or absence of actual changes in the optical properties of the substance forming a thin film, depending on the thickness of this film, can be resolved only on the basis of investigating light scattering in thin films (such investigations, apparently, have not yet been carried out), and also by structural analysis of the films themselves. Nevertheless, clarification of the phase relationships in light reflected by thin films, and of their dependence on film thickness, as well as comparison of them with the predictions of a theory based on the assumption of film homogeneity and invariability of the optical constants of the substance, is of not only practical interest. An investigation of this kind was carried out by the authors of the paper under review^6.

Fig. 1. Scheme of the arrangement of coatings on the mirror of a Fabry–Perot interferometer in the study of silver layers.
a—in cross section, b—in plan.

I—MgF\(_2\) layer about 1000 Å thick, II—Ag layer about 300 Å thick, III—Ag layer of variable thickness.
The figures indicate the total thickness of the silver layer (II + III) in angstroms.

The authors investigated the dependence of the magnitude of the phase shift upon reflection both from the front side and from the reverse side (facing the substrate surface) of the film.

In the first case the layers under investigation were deposited on one of the mirrors of a Fabry–Perot interferometer.

For depositing the layers, the technique of evaporation in a high vacuum was used, and special measures were taken to ensure and control the uniformity of the coating over its entire extent. The mutual arrangement of the individual areas of the coating is shown in Fig. 1. First a comparatively thick layer of MgF\(_2\) (layer I) was deposited; its purpose was to increase the distance \(d\) (see Fig. 1) to a value convenient for measurements. Then a silver layer II was deposited, the thickness of which was chosen so that it would provide practically complete opacity of the layer. After that a second silver layer III was deposited, whose optical properties were to be measured. This arrangement of the layers was chosen in view of the necessity of excluding the influence of the layer thickness on the path difference of the interfering rays (the distance \(d\) does not depend on the thickness of layer III). The phase jump upon reflection from the area of layer II, superposed upon

onto layer III, also does not depend on the thickness of layer III. (For an opaque layer the phase jump \(\psi = 30.3^\circ\) and does not depend on the layer thickness.) Therefore, measuring the phase difference of the rays reflected from region II + III and from region II, as a function of the thickness of layer III, yielded the immediate possibility of determining the phase shift upon reflection from layer III as a function of its thickness.

The measurement results were compared with theoretical calculations carried out under the assumption of homogeneity of the layer and taking into account reflections from its lower boundary. It turned out that, for silver layers whose thickness exceeds 200 Å, fairly good agreement with theory is observed. (At thicknesses exceeding 400 Å, the phase jump practically coincides with the value corresponding to a completely opaque layer.) If, however, the thickness of the silver layer is less than 200 Å, then the agreement with theory breaks down, and at the same time irregular changes appear in the phase jump from specimen to specimen and as the specimen ages (from week to week).

This result is of serious importance for measurements of the microstructure of surfaces by the multiple-beam interference method, when the surface under investigation, in order to obtain an interferometer, is covered with a layer of silver.

By an analogous method the phase jump upon reflection from thin dielectric layers deposited on an almost transparent silver layer was investigated.

The method of depositing the films remained the same.

The scheme of the mutual arrangement of the individual regions of the coating is shown in Fig. 2. First, an opaque silver layer I, about 600 Å thick, was deposited; then a dielectric layer II of variable thickness; and, finally, a silver layer III, also about 600 Å thick. Measurements of the phase difference of the reflected rays 1 and 2 (Fig. 2) made it possible to determine the geometrical thickness of the dielectric layer, while measurements of the phase difference of the reflected rays 3 and 4 made it possible to determine the effective optical thickness of this layer. The study was carried out for films of ZnS, MgF\(_2\), cryolite, and CaF\(_2\). For MgF\(_2\) and cryolite the experimentally determined values over a wide range of layer thicknesses (approximately from 500 to 5000 Å) lie well on the theoretical curves. From this it may be concluded that, in the investigated interval of thicknesses, these substances form a homogeneous film. On the contrary, for ZnS\(_2\) the agreement of the experimental data with the theoretical curve is rather poor. An especially large number of measurements was made in this case for thicknesses in the interval from 500 to 1200 Å. They undoubtedly indicate the presence of significant systematic deviations from the theoretical curve for thicknesses less than 1000 Å. As for CaF\(_2\), the authors confine themselves to indicating the values of the effective refractive index: in the bulk specimen \(n = 1.43\), for layers thinner than 3000 Å \(n = 1.27\), and for layers 40,000 Å thick \(n = 1.23\).

Fig. 2. Scheme of the arrangement of coatings on the mirror of a Fabry–Perot interferometer in the study of dielectric films.

I — opaque silver layer about 600 Å thick, II — layer of the dielectric under investigation, III — silver layer.

Considering this as evidence of a high degree of porosity of the layers of this substance, the authors stipulate that the investigation of CaF₂ layers is continuing.

Fig. 3. Diagram of the interferometer for measuring the phase shift upon reflection from the reverse side of the film.
\(D\)—semi-transparent plate, \(C\)—compensator, \(M_1\)—mirror, \(F\)—film under investigation, \(M_2\)—supporting glass plate, \(C'\)—compensating glass plate identical with \(M_2\) (slightly rotated relative to the beam to avoid reflections).

The next series of measurements by the authors concerns the phase shift upon reflection from the reverse side of thin layers of silver and gold deposited on a glass plate. For this purpose a Michelson interferometer was used (Fig. 3). Figure 4 gives the results of the measurements. As is seen from the figure, the actual phase shift differs sharply from the pre—

Silver

Gold

Fig. 4. Calculated and measured values of the phase shift upon reflection from the reverse side of the layer. The phase shift \(\psi\) is represented by the angle between the abscissa axis and the radius vector drawn from the origin to the point of the curve corresponding to the given layer thickness. The length of the radius vector is proportional to the intensity of the reflected light. The numbers indicate the thickness of the layer in Å. Solid curve—calculations; dashed curve—measurements.

described by the theory, based on the assumption of homogeneity of the layer and invariance of the constants.

Attention is drawn not only to the increase in the discrepancies as the layer thickness decreases (at thicknesses \(> 400\) Å they are absent), but also to the mutually opposite character of the predicted and observed change of the phase shift. At a certain small layer thickness (about 60 Å for silver and 25 Å for gold) the phase shift changes sign, and with further decrease in the layer thickness the change in the magnitude of the phase shift occurs extremely rapidly. Let us recall that, according to Ishiguro’s data,^1 at a thickness of a silver film of 30 Å there also occurs a change in sign of the phase difference between the ray passing through the layer and the ray bypassing it. At certain layer thicknesses, for some films (gold, coalesced silver), the intensity of the reflected light becomes zero.

For comparison the authors give an analogous graph calculated for a hypothetical medium whose refractive index is equal to \(3 + 2.5 i\). This graph to some extent (at small thicknesses) imitates the course of the curve obtained by the authors for gold; however, the agreement of the curves is far from sufficient for a gold layer to be ascribed such a value of the effective refractive index.

In summary, it may be said that the measurements by the authors of the paper under review have shown that the phase relations upon reflection of light from thin films undergo very substantial changes as the film thickness changes, and that these changes are different for different substances. This circumstance must not be ignored when using thin films in interference light filters and for the purposes of multiple-beam interferometry.

In addition, the authors have shown that at film thicknesses of the order of 400 Å (for silver and gold), i.e., for practically opaque films, deviations from the predictions of the theory based on the assumption of a homogeneous layer are not observed. But precisely at thicknesses of this order the granulated or porous structure of films ceases to be observed. Thus, the authors’ measurements are convincing evidence in favor of the absence of real changes in the optical constants of a substance when it forms thin films.

V. Yur’ev

References

1. Kōzō Ishiguro, J. Opt. Soc. Am. 40, 789 (1950).
2. M. Perrot, Rev. d’Optique 28, 564 (1949); P. Rouard, ibid., p. 569.
3. See, for example, G. Hass and N. Scott, J. Opt. Soc. Am. 39, 179 (1949).
4. See, for example, H. Levinstein, J. Appl. Phys. 20, 306 (1949); G. W. Johnson, J. Appel. Phys. 21, 449 (1950).
5. F. T. S. Appellegard, Proc. Phys. Soc. 49, 118 (1937).
6. L. G. Schulz and F. J. Scheilner, J. Opt. Soc. Am. 40, 761 (1950).