MULTIPLE-BEAM INTERFEROMETRY AND INTERFERENCE FILTERS. I

G. V. Rozenberg

CONTENTS

Part I. Thin films, their optical properties and applications

1. Metallic films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
   a. The relation between the optical properties of a film and its structure . . . . . . . . . . . 5
   b. Optical properties of silver films . . . . . . . . . . . . . . . . . . . . . . 12

2. Dielectric films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
   a. Methods for calculating thin films . . . . . . . . . . . . . . . . . . . . . 25
   b. Single-layer dielectric films on a transparent dielectric. Antireflection of optics . . . . . . . . 29

3. Multilayer dielectric coatings . . . . . . . . . . . . . . . . . . . . . . . . 35
   a. Achromatic antireflection coatings . . . . . . . . . . . . . . . . . . . . 36
   b. Coatings with a large reflection coefficient . . . . . . . . . . . . . . . 37
   c. Dielectric coatings on metal . . . . . . . . . . . . . . . . . . . . . . . 40
   d. Polarization devices . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Part II. Interference devices and methods. Light filters

1. The Fabry–Perot interferometer. 5. Study of the microrelief of surfaces and determination of optical constants. 6. Interference light filters. 7. Interference-polarization light filters.

At the basis of any interferometric device lies the division of an electromagnetic wave into two or more components, which are then recombined and form an interference pattern. The methods of splitting beams are very diverse. In optics, the most widespread have been semitransparent, partially reflecting surfaces that divide a light beam into a transmitted and a reflected one. Such semitransparent surfaces are most often produced by depositing, on the surface of a transparent dielectric (for example, glass or quartz), very thin inhomogeneous films that provide the desired values of the transmission and reflection coefficients of the surface. The intensity-

the development of techniques for depositing thin metallic and dielectric layers, which has taken place over the last two decades, has made it possible not only substantially to improve the optical properties of such beam-splitting devices and to ensure their reproducibility, but also to create multilayer coatings possessing a number of most important advantages. At the same time, new possibilities of interference technique, which had until then remained outside the field of view, were revealed. Interferometry, which for a long time had seemed one of the most established and developed branches of optics, thereby received a new stimulus for development, and at the present time we are witnessing the creation of a whole series of very remarkable devices and methods that have substantially broadened the field of its application. The present review is devoted to the description of these devices and methods.

Our attention will be directed mainly precisely to those devices and methods whose development is connected, to one degree or another, with progress in the technology of making thin films and which can be grouped, somewhat conventionally, under the general heading of multiple-beam interferometry. Therefore a number of important questions of interferometry, deserving independent consideration, remain outside the scope of our review. These include practically all two-beam interferometry, spectroscopic and metrological applications of interference technique, interference microscopy, and also interference methods for studying hydrodynamic processes and the kinetics of chemical reactions. At the same time we have considered it expedient to include within the proposed review a section devoted to interference-polarization light filters. On the one hand, the principle of operation of these light filters, from a certain point of view, has much in common with the principle of operation of interference light filters (multiple-beam interferometers). On the other hand, the important question of nonabsorbing light filters has as yet been almost unilluminated in the domestic review literature.

The list of literature cited does not claim to be exhaustive. We have nevertheless tried to make it sufficiently complete so that the reader can, without difficulty, find his bearings in the literature on one or another special question. Additional bibliographic information may be found in 2, 160, 183, where, however, references to the works of Soviet scientists are almost entirely absent.

Part I. THIN FILMS, THEIR OPTICAL PROPERTIES AND APPLICATIONS

Thin films, i.e., layers of matter whose thickness is measured in a few tens or hundreds of angstroms, are, in essence, a special aggregate state. The physical properties of a substance deposited on a substrate surface...

properties in the form of a thin film, generally speaking, differ substantially from the properties of the same substance in bulk form and depend both on the thickness of the film and on the conditions of its formation. Taking these special features into account is absolutely necessary for the rational use of such films, in particular for the purposes of interferometry. Therefore, a modern survey of interferometric devices must inevitably be preceded by a presentation of data on thin films. Such a presentation is all the more necessary because these questions have not yet received adequate treatment in the Russian monographic literature. However, the properties of thin films will interest us only insofar as they are directly related to interference devices; moreover, here too, for reasons that will be clarified below, we shall essentially refrain from a detailed consideration of the theoretical questions of the optics of single-layer and multilayer coatings. The reader will find a detailed account of these questions, as well as of questions concerning the technology of film production, the determination of their thickness and properties, and the physicochemical problems associated with them, in special monographs (for example,¹˒² *).

1. METALLIC FILMS

a. Relation of the optical properties of a film to its structure

The optical properties of thin metallic films have been the object of repeated and very thorough investigations. However, the very first studies showed that substantial discrepancies arise among the data of different authors. The reason for these discrepancies was seen in the fact that the optical properties of thin metallic films depend to a great extent both on their thickness and on the technology by which they are deposited. At the same time, the peculiar dependence of the optical properties on the film thickness left no doubt that we are dealing here not with trivial interference effects caused by the presence of two interfaces situated very close to one another, but with real changes in the effective optical constants of the film itself. At present the nature of such changes is clear in its general outlines. On the one hand, it is beyond doubt that

) Let us note that in recent years a number of Soviet scientists have carried out an intensive and fruitful study of the optical properties of photocathodes and, in particular, of the interference phenomena observed in them. See, for example, N. D. Morgulis, DAN 52, 681 (1946); N. D. Morgulis, P. G. Borzyak and B. I. Dyatlovitskaya, DAN 56, 925 (1947); Izv. AN SSSR, ser. phys. 12, 126 (1948); N. S. Khlebnikov and N. S. Zaitsev, ZhTF 9, 44 (1931); D. M. Khorosh, ZhTF 17, 341 (1947); M. I. Menshikov, ZhTF 17, 579 (1947); P. G. Borzyak, DAN 56, 815 (1947); ZhTF 20, 928 (1950); N. D. Morgulis and N. E. Karhanina, Scientific Notes of Kiev University 9, issue 2, Proceedings of the Faculty of Physics* No. 5 (1950), and others.

in very thin layers, as their thickness increases, a restructuring of the structure of the substance itself forming the layer takes place^3,4. Thus, for example, X-ray structural studies and electron-diffraction studies^5,6,7 have shown that an increase in film thickness is accompanied by a rearrangement of the crystal lattice (in aluminum films the face-centered tetragonal cell is rearranged, as growth proceeds, into the face-centered cubic cell characteristic of massive aluminum^8). In very thin layers some (far from all) metals (for example, rhodium) apparently have an amorphous structure^9. In addition to the restructuring of the structure of the substance, the optical properties of a film must also change with changes in its thickness because, for films in the thickness range under consideration (\(0—1000\) Å), the electron mean free path is comparable with the thickness of the film (or greater than it) and changes as the latter changes. Thus, with the thickness of the film, its specific electrical conductivity changes, and consequently so do its optical properties^10,11,12,13. It has also been pointed out that, owing to the looseness of the film material itself (see below), the role of bound electrons must change as well, which will also entail a change in the dispersion curve^14. However, the principal cause of the change in the optical properties of a film is, evidently, that in the process of increasing its thickness the aggregate state of the substance forming the film changes substantially.

Faraday^15 already drew attention to the similarity of the colors of semitransparent gold films and its colloidal solutions, and suggested that the films have a granular structure. The granular character of the structure was also indicated by the phenomenon of light scattering in very thin films^13,16; moreover, strongly scattering films usually also possess strong absorption, which even provided grounds for dividing thin films into two categories: strongly scattering and strongly absorbing, and weakly scattering and weakly absorbing^16; both types of films have the same thickness, but are obtained under somewhat different conditions. Changes in specific resistance with increasing film thickness^17,18,19 also testify to the granular nature of the structure. The specific resistance of films only a few angstroms thick is practically infinitely large. Noticeable conductivity (a specific resistance \(10^3—10^7\) times greater than that of the massive metal)^7,17,19 appears only beginning with thicknesses on the order of \(50—100\) Å (film transparency \(40—80\%\)); moreover, the magnitude of the resistance depends substantially both on the conditions of preparation of the film (for example, the temperature of the underlying surface) and on the time elapsed since its preparation (aging)^17.

The granularity of thin metallic films and its dependence on the conditions of formation of the film and on the state of the underlying

surfaces have repeatedly been observed also directly in microscopic investigations ^20,21,22; moreover, the last-named author observed the orienting action of the surface on particles of precipitated silver. Finally, recent investigations of thin metallic films with the aid of the electron microscope have brought complete clarity to the question of their structure ^5,7,13,23. These investigations have vividly demonstrated that in very thin layers the substance forming the film is not distributed uniformly over the substrate surface, but forms separate particles, isolated from one another, whose sizes and shape are determined, apparently, by the character of the surface forces ^11,22, and also by the conditions under which the film-formation process was carried out ^5,13. As the amount of deposit increases, bridges arise between the individual particles; then the particles begin to unite into agglomerates, and gradually the granulated structure of the coating gives way to an initially loose, but increasingly compact, continuous structure.

What has been said is illustrated by Fig. 1, which shows electron microphotographs of silver films of various thicknesses, obtained by evaporation of silver in vacuum.

It turned out that the structure of a film depends very strongly on the rate at which the process of its preparation is conducted. This is clearly illustrated by Fig. 2, which presents electron microphotographs of two films of almost identical thickness, prepared at different rates.

To an even greater extent, the structure of a film depends on the temperature of the substrate surface ^5,7,17. Thus, if the substrate surface during deposition, or after it, was heated to 200–300° C, then, as the film thickness increases, the granulated structure is not transformed into a continuous one, but acquires a distinctly expressed sponge-like cellular character (Fig. 3). The latter is manifested, in particular, in a noticeable increase in light scattering and in an increase of the critical film thickness at which it becomes electrically conducting (up to 400 Å for silver) ^5,7.

The structure formed in the process of preparing the film does not remain unchanged, but tends toward further transformation (the aging process), connected, apparently, with surface migration of the metal atoms and leading to a substantial change in the optical properties of the film ^17,24,25.

A number of authors ^5,7 note that as the thickness of the film increases, its surface becomes more and more uneven; moreover, the surface irregularities (in particular, individual, rather numerous large crystallites) reach dimensions of the order of the wavelength of visible light. This, evidently, explains the comparatively strong diffuse reflection from opaque films (with thickness of the order of 1000 Å and more). Of course, the character of the film

depends most strongly on the nature of the deposited metal. This is illustrated by Fig. 4 (according to data in\(^{13}\), the structure of chromium films, when viewed in an electron microscope, appears continuous even at a thickness of 20 Å).

Of the metals whose films are used for interferometric purposes, the best studied are the most commonly used—silver and aluminum. Let us note that whereas in silver films the granular structure is very pronounced and the grain sizes are comparatively large, aluminum films are considerably smoother and the sizes of individual crystallites are much smaller than the wavelength of visible light (Fig. 3). Even at very small thicknesses, aluminum films appear almost homogeneous. As a consequence, light scattering in aluminum films obtained at room temperature is appreciably weaker than in silver films. However, when deposited on a heated substrate, aluminum exhibits just as distinct a cellular structure as silver\(^{5,7}\).

Thus, thin films do not constitute a homogeneous layer of metal and, from the optical point of view, must be regarded as a turbid medium*). Consequently, one can speak only of a certain effective film thickness over a sufficiently large area (as compared with the wavelength). We shall return to this circumstance once more below, in connection with the problem of studying the microrelief of surfaces.

Since a film is a “two-dimensional colloid,” its optical properties must therefore be determined by the character of scattering by the granules forming the film. In doing so, one must take into account the existence of peculiar resonance phenomena that occur when light is scattered by small particles with finite dielectric permittivity (see, for example,\(^{36}\)). The possible influence of resonance effects of this kind on the optical properties of thin films has been considered by a number of authors\(^{27,28,29}\). Owing to the smallness of the inhomogeneities themselves and to the fact that the distance between them is less than or of the order of the wavelength of light, scattering by individual granules will be coherent to one degree or another, and direct application of the theory of light scattering by inhomogeneous inclusions is inadmissible here. The theory of coherent scattering in colloidal systems has been developed by a number of authors\(^{26,30,31,32,33,34}\). Consideration shows that in this case as well a turbid medium can be characterized by certain effective indices of refraction \(n_{\mathrm{eff}}\) and absorption \(k_{\mathrm{eff}}\), or (which is the same thing) by an effective complex dielectric permittivity \(\varepsilon=(n+ik)^2\).

*) It should be borne in mind that the dimensions of the inhomogeneities are, generally speaking, considerably smaller than the wavelength of visible light and become comparable with it only in relatively thick layers, the most commonly used for interferometric purposes.

Fig. 1. Electron micrographs of silver films of various thicknesses deposited on glass by evaporation in vacuum. Deposition time: 2 sec.

Fig. 2. Difference in the structure of silver films of approximately the same thickness, produced over 20 minutes (a) and 75 minutes (b).

a

b

Fig. 3. Dependence of the film structure on the temperature of the underlying surface. a — silver films of various thicknesses at a temperature of 300°C; b — aluminum films of thickness 500 Å at various temperatures. The cellular structure formed at high temperature is clearly visible. The scale in Figs. a and b is the same.

Fig. 4. Films of different metals: a—silver, thickness 85 Å; b—gold, thickness 75 Å; c—antimony, thickness 180 Å; d—copper, thickness 120 Å; e—nickel, thickness 75 Å. Film deposition time: 2 seconds.

MULTIPLE-BEAM INTERFEROMETRY

Maxwell-Garnett^30, proceeding from the idea of metallic spheres chaotically arranged at a small distance from one another, and taking into account the dipole moments induced on the spheres by the light wave, as well as the mutual influence of neighboring dipoles, related \(n_{\mathrm{eff}}\) and \(k_{\mathrm{eff}}\) to \(n\) and \(k\) of the massive metal from which the spheres are formed. According to^30, the imaginary part of \(\varepsilon_{\mathrm{eff}}\) is

\[ 2n_{\mathrm{eff}}k_{\mathrm{eff}}= \frac{6qb}{(1-qa)^2+4q^2b^2}, \tag{1,1} \]

and the real part of \(\varepsilon_{\mathrm{eff}}\) is

\[ n_{\mathrm{eff}}^2-k_{\mathrm{eff}}^2= \frac{3(1-qa)^2}{(1-qa)^2+4q^2b^2}-2, \tag{1,2} \]

where \(q\) is the specific volume occupied by the metal \((0\leq q\leq 1)\),

\[ a=\frac{(k^2-n^2+1)(k^2-n^2-2)+4n^2k^2} {(k^2-n^2-2)^2+4n^2k^2} \]

and

\[ b=\frac{3nk}{(k^2-n^2-2)^2+4n^2k^2}. \]

In the case of thin films, \(q\) may be related to their structure and thickness \(t\).

A comparison of Maxwell-Garnett’s theory with experiment was carried out by Goos^35 and Sennett and Scott^13. According to Goos, the imaginary part of the effective dielectric constant of the film is related to its thickness \(t\), transparency \(T\), and absorptivity \(A\) by the relation

\[ n_{\mathrm{eff}}k_{\mathrm{eff}} = n'\frac{\lambda}{4\pi t}\frac{A}{T}, \tag{1,3} \]

where \(\lambda\) is the wavelength of light and \(n'\) is the refractive index of the transparent plate on which the layer is deposited. It follows from (1,1) that \(n_{\mathrm{eff}}k_{\mathrm{eff}}\), and hence also the absorption in the layer \((A)\), will have a maximum under the condition

\[ q^2>\frac{1}{a^2+4b^2}. \tag{1,4} \]

Comparison with experiment for a series of metallic films^13 shows that, in cases where this inequality is satisfied, maxima are indeed observed on the curves of the dependence of the absorption \(A\) in a thin film on its thickness \(t\). In the opposite case no maxima are observed on the curve.

Assuming \(b\ll a\) (as is usually the case), the condition for the existence of a maximum can be rewritten in the form:

\[ k^2-n^2>0.5. \tag{1,5} \]

In this case the maximum of \(k_{\mathrm{eff}}n_{\mathrm{eff}}\) should be observed at \(q\simeq \frac{1}{a}\) and be proportional to \(\frac{1}{qb}\).

A typical dependence^18,55 of \(n_{\mathrm{eff}}\) and \(k_{\mathrm{eff}}\) of a silver film on its thickness \(t\) is shown in Fig. 5.

In Fig. 6 are shown curves of the dependence of \(k_{\mathrm{eff}} n_{\mathrm{eff}}\) for silver films\(^{13}\), calculated according to the Maxwell–Garnett theory (a), and found from the experimentally measured values of \(T\), \(A\), and \(t\) (b) for different wavelengths (in Å). A comparison of Figs. 6a and 6b indicates qualitative agreement between the Maxwell–Garnett theory and experiment. Qualitative agreement of the experimental data with the Maxwell–Garnett theory for silver films was also found by Goos\(^{35}\).

Analogous maxima on the curves of the dependence of absorption in a film on its thickness were observed for films of silver, gold, and other metals\(^{17,36,37,38,39}\), including films of Na and Ca, which had a granular structure visible under the microscope\(^{40}\), and films of mercury, cadmium, and zinc formed at low temperatures\(^{41}\).

Fig. 5. Dependence of \(n_{\mathrm{eff}}\) and \(k_{\mathrm{eff}}\) on the thickness of a silver film for different wavelengths.

Incidentally, we note that, despite the significant role of scattering in the formation of the optical properties of the film, scattering itself is small; the fraction of scattered light for slowly deposited films of thickness 150 Å at \(\lambda = 5500\) Å is only about 1.5%. (For this film \(A\) is close to 40%.) For films with lower absorption the scattering is still smaller\(^{13}\). Thus, in thin films we are dealing not with apparent, but with actual absorption of energy by the substance of the film itself.

Convincing, albeit indirect, arguments in favor of the fact that the optical properties of very thin films are determined mainly by scattering from inhomogeneities of the layer are also provided by studies of the change in the phase shift upon reflection from a film\(^{42,43}\) (see below).

However, it should be borne in mind that the presence of maxima on the curves of the dependence of the absorptivity of a film on its thickness in a number of cases may also be due to any of the other effects mentioned above (resonance phenomena, changes in the free path length of electrons), as well as to purely interference effects, as was shown by a number of the authors cited above. Therefore, the qualitative comparison with the Maxwell–Garnett theory presented in Fig. 6 is not a decisive argument in favor of this theory. In other words, the relative role of the various factors considered above that influence the dependence of the optical properties of a film on its thickness still remains unclear. It can hardly be doubted,

that for different films this role is different and depends substantially on the nature of the film-deposition process. However, there is every reason to believe that the dominant factor is

Fig. 6. Comparison of experimentally obtained curves of the dependence of \(k_{\mathrm{eff}} n_{\mathrm{eff}}\) for silver films (b) with curves calculated according to the Maxwell–Garnett theory (a).

the granularity of the film structure. If the remaining factors cannot be ignored, they should nevertheless be regarded as secondary, if only because their manifestations must likewise be due primarily to the graininess of the film.

6. Optical properties of silver films

From what has been said above it is evident that the notion of a thin film as a homogeneous formation has a very conditional meaning—it is applicable only on the average for comparatively large areas of the film surface, in any case for areas whose linear dimensions exceed the wavelength of light by several times. At the same time, characterizing the optical properties of a film by its effective refractive and absorption indices is, from the standpoint of practical applications, of little expediency, since these quantities themselves vary noticeably with the thickness of the film and depend substantially on the nature of the underlying surface and on the manner in which the process of depositing the film is carried out. Therefore a film is usually characterized by its amplitude or energy coefficients of reflection, transmission, and absorption. These quantities are also individual for different films and depend substantially both on the physicochemical properties of the underlying surface and on the refractive indices of the media that bound the film. Since the media bounding the film on the two sides usually have different refractive indices (for example, the glass on which the film is deposited and air), the coefficients of reflection, transmission, and absorption of the film for rays incident on the film from outside and from the side of the underlying surface are also usually different.

In what follows we shall adhere to the following notation: the amplitude reflection coefficient—$re^{i\alpha}$,
the amplitude transmission coefficient—$\tau e^{i\beta}$,
the energy reflection coefficient—$R=r^2$,
the energy transmission coefficient (transparency)—$T=\tau^2$,
the energy absorption coefficient—$A=1-(R+T)$.

Quantities referring to the light beam incident on the film from outside (from its front side) will be written without primes; a prime will denote quantities referring to a beam incident on the film from the side of the underlying surface.

Below we give the principal data on metallic (predominantly silver) films used in interferometry. Because of the profound individual differences among films, these data should be regarded only as typical, suitable exclusively for orientational considerations. In each particular case the optical properties of the films must be determined separately, experimentally.

In interferometric devices one usually encounters two kinds of application of semitransparent metallic coatings. Either (in two-beam interferometry) the film serves to divide the beam into two components according to the poss—

MULTIPLE-BEAM INTERFEROMETRY

...possibility of equal intensity, or (in multiple-beam interferometry) the requirement of the highest possible reflectivity is imposed on it. In both cases (especially in multiple-beam interferometry) the film must, as far as possible, have low absorption; at the same time the film must possess sufficient physicochemical stability. In the visible region of the spectrum, these requirements are best satisfied by silver, which also accounts for its most frequent use in interferometric devices. If absorption losses are not so substantial, but high corrosion resistance is required, and also when working in the ultraviolet region of the spectrum, where the reflectivity of silver drops sharply \(^{46,47}\) (Fig. 7), aluminum coatings are used. Much more rarely—only when special chemical and mechanical strength is needed—films of rhodium are used, although they have still greater absorbing power. Platinum films, which, as is known, possess high spectral neutrality \(^{44}\), are practically not used at all, precisely because of their very considerable absorption \(^{45}\).

Fig. 7. Reflection coefficient of silver and aluminum in bulk samples.

With the development of the technique of cathode sputtering and evaporation of metals in vacuum, chemical methods of depositing metallic films, which as a rule gave films of considerably poorer quality, have practically fallen out of use. In recent years cathode sputtering has likewise been used less and less frequently. However, the preference given to the method of evaporation in vacuum is apparently due purely to technological considerations. In any case, there is as yet no basis for assuming that the optical properties of films obtained by this method are higher than the properties of films obtained by cathode sputtering, especially since the actual mechanism of cathode sputtering, as is known, is not...

differs so strongly from the mechanism of thermal evaporation as it may seem at first glance.^184

In this connection, let us note that for a long time it was generally considered impossible to obtain silver films with a reflection coefficient \(R>0.9\) and low absorption. This was first achieved in 1934 by M. F. Romanova, A. D. Rubtsov, and G. I. Pokrovskii^24 at the State Optical Institute precisely by cathode sputtering. Having ensured a high

Fig. 8. Typical dependence of \(R\), \(R'\), \(T\), and \(A\) on film thickness (for a silver film deposited on glass).

purity of the conditions and having selected the proper regime for carrying out the process, the authors mentioned obtained silver films with \(R=95\text{–}97\%\) (and even \(R=98.7\%\) for an opaque film), \(T=4.0\text{–}2.5\%\), and \(A=1\text{–}0.4\%\) in the visible region. Films of such quality have not been obtained since then by any investigator, although many of them came rather close to the indicated values of \(R\) and \(T\).

Figure 8 shows the typical dependence of \(R\), \(R'\), \(T\), and \(A\) of a silver film deposited on glass on its thickness.^43 The reflection coefficient for rays incident from the side of the backing surface (\(R'\)) is, over the whole range of thicknesses, somewhat smaller than for rays incident on the film from outside (\(R\)). This difference is especially noticeable at small thicknesses \((<150\text{–}200\,\text{Å})\),

where \(R'\) here has a distinctly pronounced minimum, lying in the thickness interval from 20 to 100 Å. The film thickness corresponding to the minimum of \(R'\) depends on the wavelength, the technology of film deposition, and the nature of the underlying surface. The minimum value of \(R'\) is also different for different films; usually it amounts to fractions of a percent, often approaching

Fig. 9. Dependence of \(R\) and \(T\) on film thickness for various wavelengths (silver film on glass. Duration of the deposition process 2 sec.).

zero \(^{4, 13, 16, 18, 25, 35, 42, 43, 48—52}\). In other words, in films of thickness 20–100 Å the reflection of rays incident on the film from the glass side proves to be practically completely suppressed over the entire visible region of the spectrum.

This phenomenon can, generally speaking, be used for antireflection purposes in optics; however, as noted by a number of authors, such thin films are highly variable (for example, \(^{17, 25, 42}\), etc.). The maximum absorption and the minimum transparency of the film are usually located at the same thicknesses at which the minimum \(R'\) is observed. We note that approximately at

of the same film thickness, its electrical conductivity becomes measurable^17. Since \(R'\) at small film thicknesses differs substantially from \(R\), \(T'\) and \(A'\) must therefore also differ from \(T\) and \(A\). However, a special comparison of \(T'\) and \(A'\) with \(T\) and \(A\) apparently has not been carried out, and as yet there are no data for judging the actual difference between these quantities.

The dependence of \(T\) and \(R\) on the film thickness for various wavelengths, and on the wavelength for various thicknesses, is shown in Figs. 9–10.

Fig. 10. Dependence of \(R\) and \(T\) on wavelength for different film thicknesses (silver film on glass. Duration of the deposition process 2 sec.).

In Fig. 11 are shown the dependences of \(T\) and \(R\) on the film thickness for silver films deposited at different rates^13. The well-known fact is clearly apparent that the faster the film deposition process is carried out, the higher its reflection coefficient, and the decrease in the reflection coefficient occurs mainly not at the expense of an increase in transparency, but at the expense of an increase in absorption. An analogous phenomenon is also observed for aluminum films^53.

Figs. 12 and 13 show the dependence of the absorption coefficient \(A\) (defined as \(1 - R - T\)) on the thickness of the silver film for various wavelengths and various rates of the deposition process. The dependence of the absorptive capacity of an aluminum

Fig. 11. Dependence of \(R\) and \(T\) on the thickness of a silver film for different durations of the spraying process.

Fig. 12. Dependence of the absorbing power of a silver film on its thickness for different wavelengths.

of the film on its thickness for various wavelengths is shown in Fig. 14. The maximum of the absorption curve at a thickness of \(\sim 1000\) Å

Fig. 13. Dependence of the absorptivity of a silver film on its thickness for various durations of the deposition process.

Fig. 14. Dependence of the absorptivity of an aluminum film on its thickness for various wavelengths.

corresponds to a minimum in reflection as a result of interference effects in the film. In the visible and ultraviolet regions of the spectrum, for aluminum films it is difficult to obtain \(A\) less than 10%; usually it is considerably greater than this value.

From the figures it is clear that, if \(R \approx T\), the absorptivity of a silver film is comparatively large, especially for not very good films deposited at a slow rate. (In the latter case, if \(R \approx T\), then \(A \approx T\).) Thus, semitransparent silver films, which are the best among metallic films, nevertheless fall far short of the requirements imposed on beam splitters in two-beam interferometry. Below we shall see that multilayer dielectric coatings meet these requirements much better, and apparently are destined to displace metallic films in two-beam interferometry.

Fig. 15. Typical dependence of the reflection coefficient of a silver film on its transparency.

For thicker silver films (\(R \gg T\)), the absorptivity decreases with increasing film thickness (Figs. 12 and 13; see \(^{13,16,17,18,25,35,43}\) and others). Typical dependences of \(R\) and \(R+T\) on the transparency \(T\) for a good silver film are shown in Figs. 15 and 16.

The optical properties of the remaining metals have been studied considerably less well. The reflection coefficient of a rhodium film is significantly lower than that of silver; at \(\lambda = 0.8\mu\) it is only about 80% \(^{62}\). There are indications \(^{63}\) that films of aluminum–magnesium alloys have a high reflection coefficient (up to 93–94%). But for the time being this report remains unconfirmed.

A number of authors have studied the process of change in the optical properties of a film with time (the aging effect) under various conditions (see, for example, \(^{17,24,25,42,54}\) and others). As a rule, aging of the film leads to a substantial deterioration of its optical qualities,

in particular, to a considerable increase in absorption, which is clearly illustrated by Fig. 16. This process limits, in particular, the period during which silver films can be used in interference devices. Thus, in the work of M. F. Romanova, A. D. Rubtsov, and G. I. Pokrovsky\(^ {24}\) it is indicated that a Fabry–Perot interferometer with the best of the silver coatings they obtained could be used for 1–1.5 months, after which the quality of the films deteriorated so much that they had to be renewed.

Fig. 16. Dependence of \(R+T\) on the transparency of the film for different wavelengths. Solid lines—a freshly prepared silver film. Dashed line—the same film after three weeks during which the film was kept in comparatively clean air.

The influence of the thermal regime on the optical properties of silver layers was studied by P. M. Morozov and M. M. Butelov\(^ {17}\). The influence of subsequent treatment of the film (in particular, polishing) on its optical properties was studied by Dell\(^ {64}\).

For purposes of interferometry, knowledge of the phase shifts undergone by an electromagnetic wave reflected from a film or passing through it is of the highest importance. Phase shifts upon reflection have been investigated by a number of authors\(^ {42,43,55-60}\) (see also\(^ {25}\)). Figure 17 shows the results of measurements of \(\alpha\) and \(\alpha'\) for a silver film deposited on glass, as functions of its thickness \(t\) for two wavelengths\(^ {43}\). As is seen from the figure, for a film thickness \(>400\,\text{\AA}\) the phase shift practically does not differ from the value corresponding to massive silver. Figure 18 shows the phase shift upon reflection from the glass–silver-film boundary according to data\(^ {42}\).

In this figure the dependence of the optical constants of the film on its thickness is clearly evident. In some cases it was observed that the experimental curve (dashed) at a certain layer thickness passed through the origin (disappearance of reflection; \(R'=0\)). Noteworthy is the circumstance that at a certain layer thickness (\(\sim 60\,\text{\AA}\)) \(\alpha'\) changes sign (compare with Fig. 17).

Fig. 17. Dependence of the phase shift upon reflection from a silver film deposited on glass on its thickness. \(a\) — phase shift upon reflection from the air–film boundary; \(a'\) — phase shift upon reflection from the glass–film boundary. (The phase shift is expressed in fractions of \(\pi\).)

Fig. 18. Calculated and measured values of the phase shift upon reflection from the reverse side of a silver layer. The phase shift is represented by the angle \(\alpha'\) between the abscissa axis and the radius vector drawn from the origin to the given point of the curve (corresponding to the given layer thickness). The length of the radius vector is proportional to the intensity of the light reflected by the film. The numbers indicate the film thickness in angstroms. The solid curve was calculated under the assumption that \(n_{\mathrm{eff}}\) and \(k_{\mathrm{eff}}\) have the same values as for bulk silver. The dashed curve represents measurements.

Fig. 19. Dependence of the phase shift \(\beta\) of a light wave as it passes through a silver film on the film thickness (for \(\lambda = 5900\ \text{Å}\)).

Fig. 20. Dependence of the phase shift \(\beta\) of a light wave as it passes through a silver film on wavelength for various film thicknesses.

In transmitted light the phase shift \(\beta\) was measured, as far as we know, only by Ishiguro and Kuwahara \(^{52,61}\).

Figs. 19 and 20 show the results obtained by them. What was measured directly was the phase shift of a wave that had passed through the film relative to a wave that had bypassed the film. Therefore the quantities plotted in Fig. 19 along the ordinate differ from the total change in the phase \(\beta\) of the wave when it traverses the film thickness (expressed in fractions of \(\pi\)) by the amount \(\dfrac{2t}{\lambda}\), where \(t\) is the film thickness and \(\lambda\) is the wavelength, as is indicated in the figure. We note that for thicknesses \(<30\,\text{Å}\) the relative phase shift changes sign—the wave passing through the film lags behind the wave bypassing it.

2. DIELECTRIC FILMS

The structure of thin dielectric films is much more diverse than the structure of metallic films. This is due not only to the greater variety of physicochemical properties of the substances used to form films, but also to the variety of technological methods for depositing them. Along with evaporation in vacuum, a whole series of physicochemical methods are widely used here, such as chemical (or electrochemical) treatment of a film (for example, oxidation of a metallic surface or etching of a glass surface), successive adsorption of monomolecular layers, application of a solution to a rotating surface, and many others (see, for example, \(^{1,2}\)). As a result, films of one and the same substance may have completely different structures (crystalline or amorphous) and may possess a completely different degree of looseness, as a consequence of which the refractive index of the film substance is subject to very considerable variations at one and the same film thickness \(^{1,2}\).

A characteristic feature of dielectric films is the inhomogeneity of their properties through the film thickness \(^{1,2}\). Thus, according to \(^{65,66}\), films of fluoride compounds obtained by evaporation in vacuum have a microcrystalline structure; moreover, in thin films \((t \lesssim 500\,\text{Å})\) the crystallites are arranged without order, whereas in thicker layers the upper rows of crystallites are oriented in a definite way relative to the surface. The layer itself has a distinctly pronounced porosity, which depends strongly on the character of the process by which the film is deposited \(^{1,67,68}\). The dimensions of individual crystallites reach \(100\text{—}150\,\text{Å}\) \(^{69}\). Si and SiO, when evaporated in vacuum, form homogeneous amorphous films \(^{7}\); the refractive index of SiO films varies from 1.5 to 1.6 depending on the deposition rate. When heated to \(700^\circ\text{C}\), the Si film crystallizes. The dependence of the optical properties of a Ge film on its thickness

was studied by N. V. Rappon[^70] and others.[^7],[^71] A sharp change in the refractive index was found at a film thickness of the order of 500 Å, apparently caused by a rearrangement of the film structure.* Investigation of SiO$_2$, ZnS, MgF$_2$, and Al$_2$O$_3$ films by means of an electron microscope and electron diffraction showed[^7],[^72],[^73] that at thicknesses less than approximately 1000 Å they are amorphous. Determination of the density of an MgF$_2$ film gave the value[^74] $2.56 \pm 0.04\ \mathrm{g/cm^3}$ instead of $\sim 3.0\ \mathrm{g/cm^3}$ for the massive substance. A study of the phase shift upon reflection from films of ZnS, MgF$_2$, cryolite, and CaF$_2$ showed,[^42] that if for MgF$_2$ and cryolite the experimental points lie well on the theoretical curve, which indicates the uniformity of the film, then for ZnS and CaF$_2$ distinct deviations are observed for thicknesses smaller than 1000 Å. The dependence observed by the authors[^42],[^77] of the refractive index of a CaF$_2$ film on its thickness ($n = 1.43$ for the massive body, $n = 1.27$ for $t = 3000$ Å and $n = 1.23$ for $t = 40\,000$ Å) indicates, in their opinion, a high degree of porosity of this film (compare[^1]). An analogous phenomenon is observed in cryolite films.[^77] Dell[^64] notes that films with a small refractive index sometimes scatter light weakly, whereas films with a large value of $n$ usually do not scatter light. The dependence of the optical properties of dielectric films on the method of their preparation is emphasized by many authors ([^1],[^2],[^3],[^7],[^75],[^76], etc.). It is noted that not only the refractive index but also the dispersion depends on the film thickness (MgF$_2$ films).[^162]

Some authors have observed the phenomenon of aging (for example, in CaSiO$_3$ and ZnS it is manifested in an increase of $R$—from 26.5% to 34.3% for a ZnS film over the course of $3\frac{1}{2}$ weeks—and in a change in the optical thickness of the film; the author relates this to crystallization of the film[^74],[^78]; compare also[^64]). On the other hand, there are indications of the great stability of certain dielectric films, which preserved their optical properties practically unchanged over the course of 1–1.5 years.[^124],[^155]

A number of dielectric films (for example, SiO$_2$ or Al$_2$O$_3$) possess high physicochemical stability, which makes it possible to use them as anticorrosion coatings (for example, Al$_2$O$_3$ on aluminized mirrors), and their protective action is often combined with effective use of their optical properties (see below).

* Germanium films are of considerable interest in the infrared region of the spectrum, owing to the fact that for $\lambda > 10\,000$ Å their absorption is very small. At the same time the refractive index is very large ($n = 4.3$ for $\lambda = 60\,000$ Å, $5.2$ for $\lambda = 8000$ Å, and $2.3$ for $\lambda = 4000$ Å). With decreasing $\lambda$ the absorption increases, and the imaginary part of the dielectric constant has a clearly expressed maximum. An analogous maximum (at somewhat larger $\lambda$) is observed for Si.[^71]

a. Methods for calculating thin films

Methods for determining the thickness and optical constants of thin films are treated in detail in monographs 1, 2; see also 79, 96, 148. Here we shall note only that the determination of the optical constants of a film presents serious methodological difficulties connected with the peculiarities of the dependence of these constants on the film thickness. It is precisely for this reason that early attempts to develop methods for determining the constants 85, 86 proved untenable, since they assumed that \(n\) and \(k\) were independent of the film thickness. This even gave Brous 87 grounds, in 1939, to express the opinion that the problem of separately determining the thickness and the optical constants of a film by optical means might prove insoluble. The first path toward its solution was indicated by I. V. Obreimov 79 in 1944, and the corresponding methods were developed in the Soviet Union 1, 80–84 (in the foreign literature, studies in this direction appear only beginning in 1947). These methods are based chiefly on the use of the laws of reflection of light from a thin film (intensity, polarization, etc.); in other words, they rely on a sufficiently developed theory of interference phenomena in films. We shall now turn to consideration of the methods used by this theory.

Although the theory of interference phenomena in films is, in a certain sense, the basis for the application of thin films in interferometry and thus has a direct bearing on the subject of the present survey, we have considered it advisable to refrain from discussing it in detail. This is dictated by two circumstances. First, the theory of the propagation of electromagnetic waves in thin films has developed so significantly in recent years that it deserves an independent review. The inclusion of an extensive theoretical chapter in a review devoted to experimental methods and devices used in interferometry would, moreover, excessively broaden its already wide scope.

The second circumstance is of a much more fundamental nature. The strongly expressed dependence of the properties of a film on the technological peculiarities of the process by which it is made leads to the fact that each particular film possesses individuality and must be subjected to an independent optical investigation. Therefore the practical application of thin films is possible only when a high degree of reproducibility is ensured. It is for this reason that only comparatively recently, in connection with the development of technological processes and precision methods of control, dielectric films began to enter the practice of interferometric technique. This same circumstance to a considerable extent deprives theoretical consideration of practical significance—its application to the calculation of concrete

films still remains very unpromising. At present, the role of the theory of light propagation in thin films is reduced almost exclusively to qualitative analysis, which is very important in the development of new devices, but still comparatively little suited for use under concrete conditions. The reader will find a detailed exposition of the theoretical questions in monographs1,2. We shall confine ourselves to a brief outline of the methods of calculation.

The calculation of a light beam that has passed through a thin film (or has been reflected from it) with refractive index \(n_2\), separating media with refractive indices \(n_1\) and \(n_3\), may be based on successively tracing the fate of each of the rays reflected and refracted at each of the film boundaries, with subsequent consideration of the interference among all these derived rays. In doing so, the amplitude coefficients of reflection and transmission must be specified (or computed from the known \(n_1\), \(n_2\), and \(n_3\)): \(r_{ik} e^{i\alpha_{ik}}\) and \(\tau_{ik} e^{i\beta_{ik}}\). In the case of a single-layer nonabsorbing film, such a consideration leads to Airy’s formulas (see, for example,100). Below we shall reproduce this derivation in connection with the theory of the Fabry–Perot interferometer. In the case where the angle of incidence \(\vartheta\) differs from zero, the problem becomes mathematically more complicated, since a separate consideration of the fates of both alternatively polarized components is necessary.

Fig. 21.

Such a consideration was first carried out for single-layer coatings by A. G. Vlasov1 and N. F. Timofeeva99, who investigated in detail the problems of applying thin films to the antireflection of optics (see below), as well as the question of measuring their optical constants. In doing so, A. G. Vlasov showed that both single-layer and multilayer coatings can (just like the boundary between two dielectrics) be characterized by amplitude coefficients of reflection and transmission, with only the difference that the dependence of these coefficients on the angle of incidence turns out to be different from that following from Fresnel’s formulas; the character of this dependence is determined by the thicknesses and refractive indices of the layers. Later this method of calculation was repeatedly applied by other authors, some of whom used the matrix apparatus proposed by Jones97 and Parke98.

Another method for considering interference in thin films consists in solving Maxwell’s equations under given boundary conditions (see, for example,¹⁰⁰), taking into account (at each boundary) two incident and two reflected waves (Fig. 21).

Using this method, one can, in particular, show¹⁰¹ that a multilayer coating is optically equivalent to a single-layer one with some effective refractive index, the magnitude of which depends on the angle of incidence \(\vartheta\) (this assertion is, in essence, equivalent to the conclusion of A. G. Vlasov mentioned above). In comparison with the method of ray multiplicity, this method offers no special advantages for solving practical problems and is used no more often than the latter.

Several years ago a modification of this method was proposed and became widespread, based on a formal

Fig. 22. Diagram of an equivalent transmission line.

analogy between the equations determining the propagation of plane light waves in isotropic layered media and the equations describing the propagation of alternating currents in a network¹⁰²–¹⁰⁸. The essence of the method is as follows.

Let us consider a transmission line with distributed parameters (Fig. 22). The equations relating the voltage between the conductors \(V(z)=V_0(z)e^{i\omega t}\) and the current in them \(I(z)=I_0(z)e^{i\omega t}\) have the form:

\[ \frac{dV}{dz}=-ZI,\qquad \frac{dI}{dz}=-YV, \tag{2,1} \]

where \(z\) is the distance along the line, \(Z=R-i\omega L\) is the impedance, \(Y=G-i\omega C\) is the total shunt conductance, \(R\) is the resistance per unit length of the line, \(L\) is its inductance per unit length, \(G\) is the leakage conductance per unit length, and \(C\) is the shunt capacitance per unit length.

The solution of (2,1) has the form:

\[ I^{+}(z)=I^{+}(0)e^{-ikz},\qquad I^{-}(z)=I^{-}(0)e^{ikz}, \tag{2,2} \]

\[ \begin{aligned} V^{+}&=Z_0 I^{+}, &\qquad Y_0 V^{+}&=I^{+},\\ V^{-}&=-Z_0 I, &\qquad Y_0 V^{-}&=I^{-}, \end{aligned} \tag{2,3} \]

where \(k\), \(Z_0\), and \(Y_0\) are the propagation constant, the characteristic impedance, and the characteristic shunt conductivity, respectively, with

\[ k=-i\sqrt{YZ},\qquad Z_0=\frac{1}{Y_0}=\sqrt{\frac{Z}{Y}}. \tag{2,4} \]

For a plane electromagnetic wave propagating in a homogeneous isotropic medium, Maxwell’s equations can be rewritten in the form:

\[ \frac{\partial \mathbf E}{\partial z} = -i\,\frac{\mu\omega}{c}\,[\mathbf H,\mathbf n], \]

\[ \frac{\partial}{\partial z}[\mathbf H,\mathbf n] = -i\left(\frac{\varepsilon\omega}{c}-i\frac{4\pi\sigma}{c}\right)\mathbf E, \tag{2,5} \]

where \(\mathbf n\) is the unit vector in the direction of propagation of the wave. Putting

\[ Z=i\frac{\mu\omega}{c} \]

and

\[ Y=i\left(\frac{\varepsilon\omega}{c}-i\frac{4\pi\sigma}{c}\right), \tag{2,6} \]

we have:

\[ \frac{\partial \mathbf E}{\partial z} = -Z[\mathbf H,\mathbf n]; \qquad \frac{\partial}{\partial z}[\mathbf H,\mathbf n] = -Y\mathbf E, \tag{2,7} \]

which differs from (2,1) by the replacement of \(V\) by \(\mathbf E\) and \(I\) by \([\mathbf H,\mathbf n]\). The propagation constant (wave number)

\[ k=-i\sqrt{YZ} = \frac{\omega}{c}\sqrt{\varepsilon\mu-i\frac{4\pi\sigma\mu}{\omega}} = \frac{\omega n}{c}, \tag{2,8} \]

where

\[ n=\sqrt{\varepsilon\mu-i\frac{4\pi\sigma\mu}{\omega}} \tag{2,9} \]

is the complex refractive index of the medium. The characteristic impedance

\[ Z_0=\sqrt{\frac{Z}{Y}} = \sqrt{\frac{\mu\omega}{\varepsilon\omega-i4\pi\sigma}} \tag{2,10} \]

and

\[ Y_0=\sqrt{\frac{\varepsilon\omega-i4\pi\sigma}{\mu\omega}} = \frac{n}{\mu}. \tag{2,11} \]

If \(\mu=1\), which in any case holds in the region of optical frequencies, then

\[ Y_0=n. \tag{2,12} \]

Transforming (2,5) and taking into account (2,2) and (2,4), we have:

\[ [\mathbf n,\mathbf E]=Z_0\mathbf H,\qquad \mathbf E=Z_0[\mathbf H,\mathbf n], \tag{2,13} \]

or, according to (1,17):

\[ n\mathbf E=[\mathbf H,\mathbf n],\qquad \mathbf H=n[\mathbf n,\mathbf E]. \tag{2,14} \]

Let us note that the characteristic impedance \(Z_0\) is directly connected with the complex Umov–Poynting vector\({}^{102}\):

\[ Z_0^{*}=\frac{\mathbf{E}^{2}}{[[\mathbf{E},\mathbf{H}^{*}]]^{2}} . \tag{2,15} \]

Thus, the problem of the propagation of light in a layered medium can formally be reduced to the problem of the propagation of alternating current in a loaded transmission line with distributed parameters, and can be solved by methods developed for the calculation of networks. In this case a characteristic shunting conductance of the equivalent line is put in correspondence with the refractive index of the medium \(n\). In a number of cases such a device makes it possible to simplify the treatment somewhat. However, the calculation of any complicated systems nevertheless proves to be very laborious. Therefore the development of various kinds of graphical methods of calculation, based on one or another of the methods indicated above\({}^{3,106,107,109—113}\), acquires great importance.

Finally, let us point to the theoretical investigations of N. M. Sharapov\({}^{114}\), who considered the question of interference in thin isotropic and anisotropic films, proceeding from the idea of the dipole structure of matter in the spirit of the general molecular theory of the propagation of light in homogeneous media (compare, for example,\({}^{100}\)).

6. Single-layer dielectric films on a transparent dielectric. Antireflection of optics

One of the most important practical applications of thin dielectric films is the so-called antireflection of optics, proposed and first developed (1934) by a group of collaborators of the State Optical Institute under the direction of I. V. Grebenshchikov and A. A. Lebedev, i.e. the reduction of the reflecting power of glasses by coating them with thin films of a dielectric.

The importance of antireflection of glasses is clear from the following. When a light beam passes once through a glass–air boundary, from 4 to 9% of the intensity is lost to reflection (depending on the type of glass; for oblique incidence of the rays the losses are considerably higher).

In modern complex optical instruments, containing a very large number of interfaces, sometimes up to 80% and more of the light is lost to reflection, which very adversely affects the aperture ratio of the apparatus. Moreover, the reflected light, undergoing a number of subsequent reflections, creates a diffuse light background, which has a very substantial masking effect.

No less important, especially for interferometric purposes, is the inverse problem of increasing the reflecting power

glass. For normal incidence, the reflection coefficient at the boundary of two media is, as is known, equal to

\[ R=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2, \tag{2,16} \]

where \(n_1\) and \(n_2\) are the refractive indices of the media.

In the visible region for transparent dielectrics \(n \leqslant 2.5\), as a result of which the reflection coefficient cannot exceed 15%. It is precisely this circumstance that has led to the use of thin metallic films in interferometry, despite their comparatively strong absorption. The use of thin-layer coatings makes it possible both substantially to reduce and to increase the reflection coefficient (when absorption is small), and recently dielectric films have not only been widely used for antireflection treatment of optics, but have also begun to displace metallic coatings in interference devices (see below). A detailed treatment of the theory of the antireflection action of thin films and of the technology of their deposition is contained in monographs\(^{1,2}\) (see also \(^{74,99,115—121,149}\)).

Fig. 23. On the theory of the antireflection action of a thin film.

Here we shall restrict ourselves to the simplest case of normal incidence. Suppose that a plane wave is incident from infinity on a layer of dielectric with refractive index \(n_2\), separating two media with refractive indices \(n_1\) and \(n_3\) (Fig. 23). In media 1 and 2 there are both direct and reflected waves; in medium 3 there is only a direct wave. Writing the boundary conditions for \(z=0\) and \(z=t\) (where \(t\) is the thickness of the layer) and using (2,14), we obtain (see, for example,\(^{102}\)) for the reflected wave:

\[ E_1^-= \frac{(n_1-n_2)(n_2+n_3)+(n_1+n_2)(n_2-n_3)e^{2ik_2t}} {(n_1+n_2)(n_2+n_3)+(n_1-n_2)(n_2-n_3)e^{2ik_2t}}, \tag{2,17} \]

where \(k_2\) is the wave number. Denoting

\[ r_{ik}=\frac{n_i-n_k}{n_i+n_k} \tag{2,18} \]

(\(r_{ik}r_{ik}^{*}=R_{ik}\) is the reflection coefficient at the boundary of media with \(n=n_i\) and \(n=n_k\)), according to (2,16) we have:

\[ E_1^-=\frac{r_{12}+r_{23}e^{2ik_2t}} {1+r_{12}r_{23}e^{2ik_2t}}\,E_1^+, \tag{2,19} \]

i.e. the amplitude reflection coefficient from a dielectric with \(n=n_3\),

coated with a layer of dielectric with \(n=n_2\) and thickness \(t\), is equal to

\[ r e^{i\alpha}=\frac{r_{12}+r_{23}e^{2ik_2t}}{1+r_{12}r_{23}e^{2ik_2t}} . \tag{2,20} \]

For a dielectric in the absence of absorption, \(n_2\) and \(k_2\) are real numbers, and the energy coefficient of reflection is

\[ R=r^2=\frac{(r_{12}+r_{23})^2+4r_{12}r_{23}\sin^2 k_2t}{(1+r_{12}r_{23})^2-4r_{12}r_{23}\sin^2 k_2t}. \tag{2,21} \]

If

\[ t=\frac{2m+1}{2}\lambda, \]

then expression (2,21) takes the form

\[ R=\frac{(n_2^2-n_1n_3)^2}{(n_2^2+n_1n_3)^2}. \tag{2,22} \]

The condition for antireflection (i.e. for reducing \(R\) to zero), therefore, is as follows:

\[ t=\frac{2m+1}{4}\lambda,\qquad n_2=\sqrt{n_1n_3}, \tag{2,23} \]

where \(\lambda\) is the wavelength of light in the film and \(m\) is an arbitrary integer. Thus, the film completely suppresses reflection provided that its optical thickness is an odd multiple of \(\lambda/4\), and the refractive index of the film is equal to the geometric mean of the refractive indices of the media bounding it.

The physical meaning of conditions (2,23) is that the amplitudes of the waves reflected from the outer and inner surfaces of the film are the same, while the phase shift between these waves is equal to an odd number of half-waves; as a result, the two reflected waves, interfering with one another, extinguish each other. The role of the surface film in this case is the same as the role of a load at the junction of two transmission lines.

In the case of oblique incidence, the conditions for minimum reflection turn out to be different for the \(p\)- and \(s\)-components\(^1\), namely:

for the \(p\)-component:

\[ c_2^2 n_1^2=c_1c_3 n_1n_3, \]

and for the \(s\)-component:

\[ c_1c_3 n_2^2=c_2^2 n_1n_3, \tag{2,24} \]

where

\[ c_1=\cos\vartheta,\quad c_2=\frac{1}{n_2}\sqrt{n_2^2-n_1^2\sin^2\vartheta},\quad c_3=\frac{1}{n_3}\sqrt{n_3^2-n_1^2\sin^2\vartheta} \tag{2,25} \]

and \(\vartheta\) is the angle of incidence. With increasing \(\vartheta\), the value of \(n_2\) required to satisfy conditions (2,24) increases for the \(p\)-component and decreases for the \(s\)-component. At the same time the length also changes

wavelength for which the minimum condition is satisfied. The latter is illustrated by Fig. 24, borrowed from ¹.

When applying a film, two parameters can be varied—the thickness of the film \(t\) and its refractive index. However, for \(\vartheta \ne 0\) and a given \(\lambda\), the number of conditions that must be satisfied in order to reduce \(R\) to zero is greater than two, and in the general case reducing \(R\) to 0 for \(\vartheta \ne 0\) is impossible. This becomes possible only when using two-layer coatings¹, as discussed below. For \(\vartheta = 0\), the number of conditions necessary for reducing \(R\) to zero is equal to the number of variable quantities, and in principle reduction of \(R\) to zero can always be achieved by means of a single-layer coating. The difficulties that arise here are only technological in nature. With modern technology for depositing films, obtaining a reproducible film of a specified thickness is not an insoluble problem. It is considerably more difficult to fulfill the second of conditions (2,23). The range of substances practically convenient for use as coating films is still comparatively small.

Table I

Refractive indices of films of some dielectrics

Substance	Refractive index
LiF	1.29
Cryolite \((\mathrm{Na_3AlF_6})\)	1.35
\(\mathrm{MgF_2}\)	1.35–1.40
\(\mathrm{CaF_2}\)	1.21–1.43
\(\mathrm{SiO_2}\)	1.44
\(\mathrm{SiO}\)	1.5–1.6
\(\mathrm{CaSiO_3}\)	1.5–1.69
\(\mathrm{Al_2O_3}\)	1.62–1.67
\(\mathrm{ZnS}\)	2.12–2.37
\(\mathrm{TiO_2}\)	1.8–3
\(\mathrm{SbS_3}\)	\(\sim 3\)
Ge	2.3–5.2

The refractive indices of some of the most frequently used substances are given in Table I.

In some cases coatings made from a mixture of different substances have been used. However, their deposition presents serious technological difficulties.

In practice it proves very difficult, and sometimes impossible, to select for light glasses a film whose refractive index would satisfy with sufficient accuracy the relation \(n_2 = \sqrt{n_1 n_3}\). This entails the fact that in reality \(R_{\min} \ne 0\). Figure 25 shows the dependence of \(R_{\min}\) on the refractive index of the film for a given type of glass¹. Usually \(R_{\min}\) has a value of 0.2–0.4% or higher¹.

The conditions for the minimum of \(R\) (2, 23) in the case of a single-layer film can be fulfilled only for selected wavelengths. Moreover, there will always be wavelengths for which the given film will not decrease but increase the reflection coefficient

\[ \left( t = m \frac{\lambda}{2} \right). \]

Thus, antireflection films are substantially chromatic. However, the visible region of the spectrum is so narrow that

Fig. 24. Dependence of the wavelength at which the reflection coefficient has its minimum value \((\lambda_{\min})\) on the angle of incidence \(\vartheta\) for a single-layer coating.

Fig. 25. Dependence of \(R_{\min}\) on the refractive index of the film for a given refractive index of the glass.

it is possible to select films that have a noticeable antireflection effect over the entire range of visible light.

A typical example of the dependence of \(R\) on \(\lambda\) is given in Fig. 26. In some cases the antireflection role of the film is combined with its protective action, either as an anticorrosion coating\({}^{1}\) or as a light filter for protection against actinic radiation. For example, it was proposed\({}^{117}\) to make windows for cells intended for operation in the infrared region of the spectrum from silver chloride, using as protection against the destructive action of visible light a film of stibnite\({}^{123}\) \((\mathrm{SbS}_3)\), which at the same time has an antireflection effect.

Fig. 26. Typical dependence of the reflection coefficient on wavelength in the case \(n_2 < n_3\). Film of \(\mathrm{MgF}_2\) \((t \simeq 1054\ \text{Å})\) on glass.

Fig. 27. Typical dependence of the reflection coefficient on wavelength in the case \(n_2 > n_3\). Film of \(\mathrm{CaSiO}_3\) \((t \simeq 1325\ \text{Å})\) on glass.

If it is desired to increase the reflection coefficient of a surface, it is likewise coated with a film whose thickness is equal to \(\frac{\lambda}{4}\) (or to an odd multiple of \(\frac{\lambda}{4}\)), but the refractive index of the film is chosen so that it is higher than the refractive index of the underlying surface. The additional phase shift by \(\pi\) upon reflection from the film–underlying-surface boundary changes the conditions for interference of the reflected rays, and the rays reflected from the outer and inner surfaces of the film reinforce one another. As is seen from condition (2,22), which is also valid for this case, as \(n_2\) increases the value of \(R\) also increases. Thus, for \(n_1 = 1\) (air) and \(n_3 = 1.5\) (glass), \(R\) reaches 0.38 at \(n_2 = 2.5\). The typical dependence of \(R\) on \(\lambda\) for \(n_2 > n_3\) is shown in Fig. 27. Here, likewise

as in the case \(n_2 < n_3\), the reflection coefficient reaches an extreme value only in a comparatively narrow interval of wavelengths.

As has already been noted, in two-beam interferometry the quality of the interference fringes depends substantially on the relation between the reflection and transmission coefficients of the semitransparent plate separating the beams. The metallic films usually employed are far from satisfying the optimum conditions. A considerably better approximation to the optimum conditions can be obtained by means of dielectric films, for example a film of \(\mathrm{TiO}_2\). The use of a \(\mathrm{TiO}_2\) coating instead of metallization leads to a substantial improvement in the quality of two-beam interference fringes \(^{125}\).

When thin films are illuminated at angles close to the angle of total internal reflection, peculiar interference phenomena are observed which, in particular, can be used to determine the optical constants of the film \(^{77}\).

3. MULTILAYER DIELECTRIC COATINGS

Multilayer antireflection coatings were first used at the State Optical Institute during the years of the Great Patriotic War \(^{1}\). Subsequently the range of applications of multilayer coatings expanded considerably, and they began to win an ever more significant place in the technology of modern interferometry.

The disadvantages of single-layer coatings are: 1) the sharply pronounced chromaticity of the antireflection effect, 2) the difficulty, and sometimes the impossibility, of selecting a material for making a film satisfying the optimum requirements, 3) the impossibility, in the general case, of ensuring antireflection for \(\vartheta \ne 0\), 4) the impossibility of obtaining \(R \gg 0.4\). It was natural to seek ways of overcoming these shortcomings either: a) in the deposition of films whose properties vary regularly through the thickness (for example, \(^{142}\)), or b) in the use of multilayer coatings.

The first approach has not yet been subjected to detailed development, and almost no definite data are available in this direction. The second approach—multilayer coatings—on the contrary, has been widely developed both theoretically and experimentally and has led to very outstanding successes, to consideration of which we now turn.

The theory of the propagation of light through layered substances has been developed by many authors \(^{1, 2, 101, 103, 107, 115, 126—137}\), including the more complicated case of conducting (metallic) layers \(^{18, 35, 36, 86, 102, 138—147}\). In the latter case the calculations are complicated by the fact that the refractive index is a complex quantity, i.e. contains two constants. Some simplification is achieved in the infrared region of the spectrum, where the real and imaginary parts of the dielectric constant may be assumed equal \(^{147}\).

a. Achromatic antireflection coatings

One of the possible methods for obtaining an effective antireflection coating is the application of a series of successive layers of appropriate thickness with a gradually decreasing refractive index132, 150, 151. However, this method has not been widely developed and in practice has been superseded by the method of applying alternating layers with large and small refractive indices. Thus, at the State Optical Institute two-layer coatings TiO₂ + SiO₂ were used1 for antireflection coating of optical elements. Other two-layer film combinations are also used: barium stearate*) + ZnS116, TiO₂ + MgF₂152, 153, sapphire + quartz153, cryolite + ZnS155, cryolite + TiO₂156, etc.

In principle, with a two-layer film it is possible to achieve complete suppression of reflection for a given wavelength at any given angle of incidence1. In practice, however, this cannot be achieved, apparently owing to the technological impossibility of creating a film that satisfies sufficiently strictly the conditions for extinction of the reflected beam (the prescribed values of \(n\) and \(t\) throughout the entire film). Nevertheless, it is possible1 to obtain a reflection coefficient at the minimum (for white light!) of only about 0.5%. Approximately the same value of \(R\) at the minimum was obtained by Strong153, using a two-layer coating of another type: first the glass was coated with a layer of substance \(\left(\text{of thickness } \frac{\lambda}{2}\right)\), whose refractive index varied regularly through the thickness of the layer from \(n\) at the boundary with the glass (\(n\) is the refractive index of the glass) to \(n^2\) at the outer surface of the layer, and then a layer of thickness \(\frac{\lambda}{4}\) with refractive index \(n\) was applied.

A natural development of the antireflection method using a two-layer coating was the use of multilayer coatings. In the case of applying \(m\) films, the experimenter has at his disposal \(2m\) parameters (\(m\) refractive indices and \(m\) thicknesses), as a result of which it is possible not only to suppress reflection over a wide range of angles, but also to reduce the reflection coefficient of the coated surface to zero for \(m\) wavelengths (at normal incidence), distributed over the spectrum in accordance with the requirements imposed by the particular conditions of use of the reflecting surface.

Thus, the problem of calculating and producing a multilayer coating proves in many respects analogous to the problem of chromatic correction of lenses. As in the latter case, for successful achromatization of an antireflection coating it is necessary to have a suffi—

*) The optical properties of a barium stearate film, which is birefringent, were studied54 in the thickness interval from 25 to 40,000 Å.

exactly by means of an extensive set of substances with different refractive indices (and dispersions). Most often, for the achromatization of antireflection films over a not too wide spectral interval, alternating layers with large and small refractive indices and a regularly increasing thickness are used: \(\frac{\lambda}{4}\), \(\frac{\lambda}{2}\), \(\frac{3\lambda}{4}\), \(\lambda\), and so on.\(^{107,161}\) (see Fig. 28).

Fig. 28. Achromatization of an antireflection film in the visible region of the spectrum. Two-layer film:
\[ n_1=1.41,\quad n_1t_1=\frac{\lambda}{4};\qquad n_2=2.00;\qquad n_2t_2=\frac{\lambda}{2}; \]
\[ n_3=1.62. \]

The antireflection action of multilayer films proves in this case to be so effective that projects have arisen for creating, with their aid, an absolutely black body\(^{161}\) (over a not broad interval of wavelengths and angles).

6. Coatings with a Large Reflection Coefficient

From the point of view of interferometric applications, the greatest interest lies in solving the inverse problem—creating coatings possessing as large a reflection coefficient as possible. The desire to pass from metallic films to dielectric ones is determined by the fact that, in the case of the latter, it is possible to achieve substantially smaller losses of light by absorption in the film, which, as will be seen from what follows, is of decisive importance for improving the quality of interferometric apparatus. In addition, the use of dielectric coatings makes it possible significantly to expan...

...to extend the range of wavelengths for which full-fledged interference devices can be realized (up to wavelengths in the centimeter range)^[161].

To increase the reflectivity of the surface of glass (or another transparent dielectric) with refractive index \(n_3\), one resorts to depositing on it alternating layers with small and large refractive indices (Fig. 29). The optical thickness of these layers is chosen equal to \(\frac{\lambda}{4}\). The best results, apparently, are achieved in the case of an odd number of layers^[124]. However, a number of authors obtain good results using an even number of layers (for example,^[161]).

With the aid of multilayer coatings it has proved possible to attain reflection coefficients close to that of silver layers (and even exceeding it). Thus, the use of alternating films of \(\mathrm{TiO}_2\) and cryolite^[156] made it possible to obtain \(R = 85\%\). With alternating films of \(\mathrm{ZnS}\) and cryolite (Fig. 29), values \(R = 60, 80\), and \(90\%\) were obtained for three-, five-, and seven-layer coatings^[157]. Values \(R = 0.3, 0.6\), and \(0.8\) for white light were obtained^[164] by means of one-, three-, and five-layer coatings. For analogous results see^[122,155,158,159]. Figure 30 gives typical dependences of the transparency of glass with a multilayer coating of \(\mathrm{ZnS}\) and cryolite^[165] on wavelength. The dependence of \(R\) and \(A\) for glass coated with five and seven alternating films of \(\mathrm{ZnS}\) and cryolite on wavelength is shown in Fig. 31. It is clearly seen that, in the case, for example, of a nine-layer coating, the reflection coefficient over a broad range of wavelengths exceeds \(90\%\). At the same time, the absorption is considerably lower than the absorption of silver layers with the same reflection coefficient. However, such a small value of \(A\) occurs only in the visible and infrared regions of the spectrum. In the ultraviolet region of the spectrum, the materials used to form the films usually have strong absorption bands, which for the time being makes the use of dielectric coatings ineffective^[160].

In addition to interference devices, multilayer coatings with large \(R\) have proved useful in powerful projection installations. One of the factors limiting the power of such installations is the heating of the apparatus by the abundant infrared radiation of the light sources used. The use of multilayer coatings makes it possible to create “cold mirrors,” i.e., reflectors having a high reflection coefficient in the visible region of the spectrum and a high transmission coefficient in the infrared region^[161]. Such a “cold mirror,” while almost not attenuating the visible beam reflected into the apparatus, allows almost all infrared radiation to pass outward, thereby preventing overheating of the apparatus.

Fig. 29. Diagram of a multilayer coating with large \(R\).

Fig. 30. Dependence of the transparency of glass with a multilayer coating of alternating layers of ZnS and cryolite on wavelength: \(1\)—single-, \(2\)—three-, \(3\)—five-, \(4\)—seven-, and \(5\)—nine-layer coatings.

Fig. 31. Dependence of \(R\) and \(A\) for glass coated with five and seven alternating films of ZnS and cryolite on wavelength. For comparison, the dependences of \(A\) are given for silver layers with the same reflection coefficient.

To obtain a “cold mirror,” the surface of the glass is first coated[^161] with a thin film of germanium. (The latter, as is known, has a comparatively high reflection coefficient—about 45%—in the visible region of the spectrum and is transparent in the infrared region.) A multilayer dielectric coating is then deposited on the germanium film. A typical dependence of \(R\) and \(T\) for a “cold mirror” on the wavelength in the case of a four-layer coating of alternating MgF\(_2\) and ZnS films is shown in Fig. 32.

Fig. 32. Dependence of \(R\) and \(T\) for a “cold mirror,” coated with four alternating layers of MgF\(_2\) and ZnS, on wavelength. The transparency of the germanium substrate at \(\lambda = 4400\) Å was 17%.

c. Dielectric coatings on metal

The problem of studying the influence of thin dielectric films deposited on the surface of a metal on its reflectivity naturally arose in connection with the use of dielectric films to protect metal mirrors from corrosion. In particular, the most widely used aluminized mirrors (especially in astronomical equipment) are usually coated with aluminum oxide (Al\(_2\)O\(_3\)); moreover, the quality of the resulting mirror depends very strongly on the thickness of the coating and on the method of its formation[^75],[^166]. In addition, a thin-layer dielectric coating causes the appearance of typical interference effects in the dependence of the reflection coefficient of the mirror on wavelength[^75], which must be taken into account in a number of cases.

If the optical thickness of the film is \(nt\), then the position of the interference maxima in the reflected light is determined by the well-known relation

\[ 2nt + \delta_1 - \delta_2 = m\frac{\lambda}{2}, \tag{3,1} \]

where \(\delta_1\) and \(\delta_2\) denote, respectively, the phase shifts upon reflection at the air—film and film—metal boundaries, expressed in angstroms. Whereas \(\delta_1\) is always equal to \(\frac{\lambda}{2}\), \(\delta_2\) depends essentially on the optical constants of the film and the metal:

\[ \tg \delta_2 = \frac{r n n_\mu k_\mu}{n^3 - n_\mu^2 - (n_\mu k_\mu)^2}, \tag{3,2} \]

where \(n_\mu\) and \(k_\mu\) are the real and imaginary parts of the refractive index of the metal. (In the case of an \(\mathrm{Al_2O_3}\) film on Al it was found\(^{75}\) that \(\delta_1-\delta_2\) is practically constant and equal to 250 Å.)

Let us note that the dependence of the reflectivity of a metallic surface covered with a film on the properties of the film underlies a number of methods for determining the optical constants of the latter (see, for example,\(^{2,167}\) and others).

The development of techniques for producing reproducible dielectric films made it possible to pose the question of increasing reflection from a metal by means of single-layer and multilayer dielectric coatings\(^{122,168,170}\). In the cited works it is reported that, by means of dielectric coatings, the reflection coefficient of silver for white light can be increased from 0.98 to 0.995, and that of aluminum from 0.91 to 0.96 and even to 0.98. At the same time the absorption in the metal is substantially reduced. The latter is connected with the fact that only a small fraction of the incident radiation reaches the surface of the metal itself—the reflection is mainly effected by the nonabsorbing dielectric coating.

As an example, we give calculated data relating to the change in the optical properties of a surface coated with a semitransparent layer of silver when a quarter-wave ZnS film is deposited on it (Table II)\(^{155}\).

Table II

Influence of a covering dielectric film on the optical properties of a silvered surface

Layers on the glass surface	\(R\)	\(T\)	\(A\)
Silver	0.86	0.08	0.06
\(\mathrm{Ag}\left(\dfrac{\lambda}{8}\right)+\mathrm{ZnS}\left(\dfrac{\lambda}{4}\right)\)	0.939	0.041	0.02

A peculiar influence of surface coatings on magneto-optical phenomena occurring upon reflection of light from a magnetized metal was discovered and studied by M. M. Noskov, Ya. I. Frenkel, and A. V. Sokolov\(^{171–175}\). The difference in the position of the planes of polarization of light reflected from the outer and inner surfaces of the film leads here to an unusual dependence of the character of the polarization of the reflected light on the thickness of the film.

г. Polarization devices

Differences in the conditions of reflection of the two alternatively polarized components of an obliquely incident light beam from both surfaces of a coating film determine the substantial dependence of the antireflection action of the film on the state of polarization of the light and on the angle of incidence; compare (2, 24) and (2, 25). Therefore thin films exert a very effective polarizing action both in transmitted and in reflected light. A detailed theoretical analysis of this phenomenon was first carried out by A. G. Vlasov¹ (see also³) and formed the basis for a number of methods for determining the thickness and optical constants of a film⁷⁸—⁸⁴,².

At the same time, there arose a desire to use thin films for creating polarizing devices. Their applicability in the infrared region of the spectrum is especially important. As is known, the transparency region of Iceland spar is limited on the long-wavelength side to approximately \(2\mu\), as a result of which ordinary polarizing prisms are no longer suitable for radiation of greater wavelength. Polaroids are likewise inapplicable, since their polarizing ability falls off sharply already at the red boundary of the visible region of the spectrum¹⁷⁶. The problem of producing and analyzing polarized radiation in the infrared region of the spectrum has not yet been satisfactorily solved by other, sometimes very ingenious, devices either. They either possess insufficient polarizing ability, or substantially weaken the intensity of the light, or, finally, prove extremely inconvenient in use. Thin films are free of all these shortcomings. Moreover, with the aid of thin films it becomes possible to create polarizers with a comparatively large working area, amounting to many square centimeters.

The use of thin films for polarimetric purposes proceeds, essentially, in two directions:

1) The creation, from thin films, of devices completely analogous to a glass pile, a description of which may be found in any textbook of optics. In this case one strives, as far as possible, to avoid interference phenomena associated with reflections from successively arranged films.

2) The creation of multilayer polarizers using interference between rays reflected from successive layers.

1) A pile of thin films

Theoretical consideration of the possibility of obtaining polarized light by means of a pile of thin films showed¹⁷⁷ that in this way one can obtain a very high-aperture and almost achromatic polarizer both in reflected and in transmitted light. According to the author, a pile of two glass plates coated on both sides with a film of \(\mathrm{TiO}_2\) provides—

provides a degree of polarization up to 99.7%. For the infrared region of the spectrum (from 2 to 14 μ), a stop of this kind was made1 from selenium films each 4 μ thick.

According to the authors’ measurements, the selenium film shows no appreciable absorption in the indicated wavelength interval at least up to a thickness of 52 μ. The refractive index of selenium is \(n \simeq 2.54\), and, correspondingly, the angle of complete polarization is 68.5°. The results obtained by the authors are as follows.

For a stop of five films the degree of polarization of light incident at the angle of complete polarization did not fall below 94% over the entire wavelength interval from 2 to 14 μ, and over most of this interval it was appreciably higher than this value. For a stop of six films the degree of polarization over the entire investigated spectral region exceeded 98%.

Variations of the angle of incidence of the light beam on the stop within \(\pm 5^\circ\) did not have any noticeable effect on the degree of polarization. Interference effects, quite noticeable with a small number of films, rapidly weakened as this number increased and, in the case of a stop of five or six films, could be ignored.

Another important indicator of the quality of a polarizing device is its transparency. It turned out that a stop of five films transmits 47% of the incident unpolarized light, i.e., 94% of the corresponding polarized component. The transparency of a stop of six films is somewhat lower. The authors describe in detail the technology for making the stop (for a brief abstract, see2).

2) Multilayer interference polarizer3,4,5

The device is an ordinary multilayer coating of alternating layers with large and small refractive indices (ZnS + cryolite), arranged with respect to the light beam incident on it at the angle of complete polarization. The coating is calculated in such a way that the reflection coefficient for the corresponding component is close to unity.

In practice such a device can be realized by successively depositing the layers on the base of a three-sided equilateral rectangular glass prism. After deposition, the layers are covered with a second glass prism, identical to the prism that served as the base (Fig. 33).

A polarizer consisting of 10 layers, each with an area of 16 cm², provided6 a degree of polarization in transmitted light of 99.9%, and in reflected light—about 99%, the reflection coefficient being close to 0.95. The total intensity of the two beams reached 84% of the intensity of the unfiltered radiation, and the losses (16%) had to be attributed mainly

…by means of reflections from the surfaces of glass prisms. The authors believe that the incomplete polarization in the reflected beam is due chiefly to the inexact coincidence of the angle of incidence with the angle of total polarization at the boundary of the glass prism with the multilayer coating, and also to the nonparallelism of the light beam. When white light was used, the polarized components were slightly colored, which indicated insufficient achromatization of the coating.

In the case shown in Fig. 33, with six layers3, the degree of polarization exceeded 98% for beams incident within a cone with an aperture of 10° (with its axis normal to the faces of the prism). The spectral dependence of the polarizing action is characterized by Table III.

Fig. 33. Diagram of an interference polarizer.

Table III

Depolarization of light after passing through the device shown in Fig. 33

Region of the spectrum	Red	Green	Blue
In reflected light	0.0035	0.0065	0.0103
In transmitted light	0.0138	0.0031	0.0545

The use of the described device makes it possible to obtain both alternatively polarized components simultaneously, which has a number of advantages (for example, it makes it possible to measure simultaneously their intensities or directly the depolarization of the light incident on the device). In this respect interference polarizers are closer to devices of the Wollaston-prism type than to polarizing prisms. Having a comparatively large area, they can also be used to obtain images of extended objects in polarized light.

If it is necessary to obtain the maximum luminous flux in a definite state of polarization, one of the components may be reflected at an angle of \(90^\circ\) (in the direction of the other component), and the plane of its polarization may then be rotated through \(90^\circ\) by means of a compensator \(^{181}\). This makes it possible to transform natural light into completely polarized light practically without loss in luminous flux (but with a twofold decrease in its intensity).

(Conclusion in the next issue)

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26. P. Cotton, J. phys. et rad. 11, No. 7, 321—326 (1950).
27. C. Dufour, J. phys. et rad. 11, No. 7, 327—331 (1950).
28. D. Malè, J. phys. et rad. 11, No. 1, 332—336 (1950).
29. N. M. Sharapov, Scientific Notes of Sverdlovsk State University, No. 2, 97—103 (1937); No. 3, 37—40 (1941).
30. F. Abelès, Rev. d’optique 28, No. 1, 11—31 (1948).
31. K. B. Blodgett, Phys. Rev. 55, No. 4, 391 (1939); 57, 921 (1940).
32. M. Hyman Jr. and B. N. Billings, J. Opt. Soc. Am. 37, No. 2, 113—118 (1947).
33. P. Jacquinot, Rev. d’optique 21, 15 (1942).
34. F. Matossi, J. Opt. Soc. Am. 39, No. 11, 926 (1949).
35. J. Strong, J. Opt. Soc. Am. 26, No. 1, 73—74 (1936).
36. A. B. Winterbottom, J. Opt. Soc. Am. 38, No. 12, 1074—1082 (1948).
37. A. G. Samartsev, J. Opt.-Mech. Industry, No. 7, 33—40 (1946).
38. B. N. Billings and M. Hyman Jr., J. Opt. Soc. Am. 37, No. 2, 119—121 (1947); correction: J. Opt. Soc. Am. 37, No. 5, 395 (1947).
39. P. Jacquinot et C. Dufour, J. phys. et rad. 11, No. 7, 427—431 (1950).

1. H. Barell and J. S. Preston, Proc. Phys. Soc. 64B, No. 2, 97—104 (1951).

2. R. V. Subrahmanian, Proc. Ind. Ac. Sci. 13A, 467 (1941).

3. G. N. Ramachandran, Proc. Ind. Ac. Sci. 16A, 336 (1942).

4. R. L. Mooney, J. Opt. Soc. Am. 35, 574 (1945); 36, 254 (1946).

5. D. L. Caballero, J. Opt. Soc. Am. 37, 176 (1947).

6. W. Weinstein, J. Opt. Soc. Am. 37, 576—581 (1947).

7. A. W. Crook, J. Opt. Soc. Am. 38, No. 11, 954—964 (1948).

8. L. Hiesinger, Optik 3, 485 (1948).

9. A. Vasiček, J. Opt. Soc. Am. 37, 623 (1947); 39, 409 (1949).

10. P. Leurgans, J. Opt. Soc. Am. 39, 639 (1949).

11. F. Abelès, Ann. de physique 3, 504 (1949).

12. W. L. Wilcock, J. Opt. Soc. Am. 39, No. 10, 889 (1949), correction of an error made in 133.

13. F. Scandone, J. phys. et rad. 11, No. 7, 337—341 (1950).

14. P. Drude, Optics, ONTI, 1935.

15. R. C. MacLaurin, Proc. Roy. Soc. A78, 296 (1906).

16. M. Czerny, Zeits. f. Physik 65, 600 (1930).

17. R. B. Barns and M. Czerny, Phys. Rev. 38, 338 (1931).

18. C. H. Cartwright and M. Czerny, Zeits. f. Physik 85, 269 (1932).

19. G. Bauer, Ann. der Physik 19, No. 5, 434—464 (1934).

20. Bäz, Phys. Zeits. 40, 394 (1939).

21. Pfister und Roth, Hochfreq. und Elektroakust. 51, 156 (1938).

22. L. N. Hadley and D. M. Dennison, Phys. Rev. 69, 258 (1946).

23. L. N. Hadley and D. M. Dennison, J. Opt. Soc. Am. 37, No. 6, 451—465 (1947); 38, No. 6, 483—496 (1948).

24. A. Vasiček, J. phys. et rad. 11, No. 7, 345—349 (1950).

25. A. Vasiček, J. phys. et rad. 11, No. 7, 342—345 (1950).

26. P. King and L. B. Lockhart, J. Opt. Soc. Am. 36, 513 (1946).

27. L. B. Lockhart and P. King, J. Opt. Soc. Am. 37, 689 (1947).

28. C. H. Cartwright, Phys. Rev. 55, 675 (1939).

29. M. Banning, J. Opt. Soc. Am. 37, 688 (1947).

30. A. Rothen and M. Hanson, Rev. Scient. Instr. 19, 839 (1948); 20 No. 1, 66—72 (1949).

31. C. Dufour, Ann. de physique 6, No. 1—2, 5—107 (1951).

32. C. H. Cartwright and A. F. Turner, Phys. Rev. 55, 595, 1128 (1939).

33. C. Dufour, Rev. d’optique 24, 11 (1945).

34. K. M. Greenland, J. Scient. Instr. 23, 48 (1946).

35. C. Dufour, J. phys. et rad. 9, No. 3, 14 (1948).

36. H. Kuhn, Reports on Progress in Physics 14, 64—94 (1951).

37. A. F. Turner, J. phys. et rad. 11, No. 7, 444—460 (1950).

38. L. G. Schulz, J. Opt. Soc. Am. 41, No. 11, 871 (1951).

39. J. Strong, J. phys. et rad. 11, No. 7, 441—443 (1950).

40. R. Messner, Zeits. Nachr. 4, 253 (1943).

41. H. Banning, J. Opt. Soc. Am. 37, No. 10, 792—797 (1947).

1. Flamant, J. Opt. Soc. Am. 11, No. 7, 380–384 (1950).

2. J. B. Bateman and M. W. Harris, Ann. N. Y. Acad. Sci. 53, No. 5, 1064–1081 (1951).

3. A. F. Turner, J. Opt. Soc. Am. 36, 711 (1946).

4. R. Messner, Optik 2, 228 (1947).

5. K. Hammer and O. Miller, Optik 5, No. 6, 365–370 (1949).

6. M. M. Noskov, DAN 31, No. 2 (1941).

7. Ya. I. Frenkel, ZhETF 12, No. 10, 467 (1942).

8. M. M. Noskov, DAN 53, No. 5, 417 (1946).

9. M. M. Noskov, ZhETF 17, No. 11, 964–966 (1947).

10. M. M. Noskov and A. V. Sokolov, ZhETF 17, No. 11, 967–975 (1947).

11. Handbook of Military Optics, Gostekhizdat, Moscow–Leningrad, 1945.

12. F. Abelès, J. phys. et rad. 11, No. 7, 403–406 (1950).

13. A. Elliott, E. J. Ambrose and R. Temple, J. Opt. Soc. Am. 38, 212 (1948).

14. UFN 35, No. 3, 415 (1948).

15. M. Banning, Phys. Rev. 59, 914 (1941).

16. H. Schröder und R. Schläfer, Zeits. f. Naturforschung, 4a, 576 (1948).

17. H. Schröder, Optik 3, 499 (1948).

18. P. Cotton et P. Rouard, J. phys. et rad. 11, No. 7, 461 (1950).

19. See, for example, N. D. Morgulis, UFN 28, 202 (1946).

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