MULTIPLE-BEAM INTERFEROMETRY

AND INTERFERENCE FILTERS. II

G. V. Rozenberg

Part II. INTERFERENCE DEVICES AND METHODS. FILTERS*

CONTENTS

1. The Fabry–Perot interferometer . . . . . . . . . . . . . . . . . . . . 173
       A. The Fabry–Perot interferometer in transmitted light . . . . . . . . . 174
           a. The interference pattern in transmitted light . . . . . . . . . . . 174
           b. Types of interference fringes . . . . . . . . . . . . . . . . . . . . 177
           c. Transparency of the interferometer . . . . . . . . . . . . . . . . . 184
           d. Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
           e. Half-width of the fringes . . . . . . . . . . . . . . . . . . . . . . 189
           f. Requirements on the mutual arrangement of the plates. Collimation error and figure error . . . . . . . . . . . . . . . . . . . . 195
       B. The Fabry–Perot interferometer in reflected light . . . . . . . . . . . 201
       C. The doubled interferometer . . . . . . . . . . . . . . . . . . . . . . 207
       D. N. R. Baitarchukova’s monochromator and self-registering interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

2. Study of the microrelief of surfaces and determination of optical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

3. Interference filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
       a. Filter in transmitted light . . . . . . . . . . . . . . . . . . . . . . 227
       b. Reflecting interference filters . . . . . . . . . . . . . . . . . . . . 232
       c. Filter using total internal reflection . . . . . . . . . . . . . . . . 234

4. Interference-polarization filters . . . . . . . . . . . . . . . . . . . 239

4. THE FABRY–PEROT INTERFEROMETER

The development of interference devices and methods in recent years, at least insofar as it is connected with the improvement of the technology of making thin films, has proceeded chiefly along the line of devices related to the Fabry–Perot interferometer. In this interferometer, which is essentially a pair of semitransparent layers separated by a dielectric—

* For the beginning, see UFN, vol. XLVII, no. 1, p. 3 (1952).

layer, we shall focus our attention. At the present time the theory of the Fabry–Perot interferometer has been developed rather well, and an exposition of its foundations, as well as the corresponding bibliographical information, can be found in a number of monographs and reviews (see, for example, ^185–191), and, in elementary form, in any textbook of optics. However, a number of essential circumstances have been clarified only quite recently, and it seems necessary to us to dwell in detail on certain questions that bear directly on the paths of development of modern interferometric technique, all the more so since, in the domestic literature, as far as we know, there is no systematic exposition of this theory.

A. THE FABRY–PEROT INTERFEROMETER IN TRANSMITTED LIGHT

a. The interference pattern in transmitted light

The Fabry–Perot interferometer is usually made in the form of two relatively thick plane-parallel plates of glass or quartz, coated on one side with a semitransparent layer of silver or aluminum and facing one another with their metallized surfaces.

The gap between the plates is filled with air, usually (but not always) at atmospheric pressure. Often, instead of an interferometer, the so-called Fabry–Perot etalon is used, which is a strictly plane-parallel plate of glass or quartz, metallized on both sides. In what follows, except in specially specified cases, we shall not distinguish between the interferometer and the etalon.

Fig. 34. On the theory of the Fabry–Perot interferometer.

Let us suppose that a beam of light is incident on the interferometer schematically shown in Fig. 34. On striking each of the semitransparent coatings, the beam is split into a transmitted and a reflected—

coherent, as a result of which, both in transmitted and in reflected light, a set of coherent light beams is obtained with regularly varying intensities (in geometric progression) and phases (in arithmetic progression).

Interfering with one another, these beams form a certain interference pattern, which we must investigate. Let us first turn to the pattern obtained in transmitted light.

Let, as before, \(\rho_j e^{i\alpha_j}\) be the amplitude reflection coefficient of the \(j\)-th semitransparent coating; \(\tau_j e^{i\beta_j}\) its amplitude transmission coefficient, \(R_j=\rho_j^2\), \(T_j=\tau_j^2\), and \(A_j=1-(R_j+T_j)\) the energy coefficients of reflection, transmission, and absorption of the same coating, respectively.* Quantities pertaining to the case in which the ray passes from inside the interferometer outward will be denoted by a prime; quantities pertaining to the reverse direction of the rays are written without a prime. Let us further denote the angle of incidence by \(\vartheta\), the angle of refraction by \(\vartheta'\), the thickness of the dielectric interlayer (the thickness of the interferometer) by \(t\), and the refractive index of the dielectric interlayer relative to air by \(n\) \(\left(n=\dfrac{n_2}{n_0}\right)\).

The relative phase retardation of two rays that have undergone \(s\) and \(s+1\) reflections from the second semitransparent surface (derived rays of the \(s\)-th and \((s+1)\)-st order) is determined by the well-known expression:

\[ \delta=\frac{4\pi}{\lambda}nt\cos\vartheta', \tag{4,1} \]

where \(\lambda\) is the wavelength of light in vacuum. If \(n_2=n_0\), as is usually the case in an interferometer filled with air, then \(\vartheta'=\vartheta\). In what follows, since we shall be interested only in the qualitative picture, we shall not distinguish between these angles.

Calculating the electric-field strength \(E_s\) for the derived ray of the \(s\)-th order and evaluating \(\sum_{s=0}^{s=\infty} E_s\), we find that the intensity of the light beam transmitted through the interferometer is equal to

\[ I=I_0 \frac{T_1T_2'} {(1-\rho_1'\rho_2)^2+4\rho_1'\rho_2\sin^2\left(\dfrac{\alpha_1'+\alpha_2+\delta}{2}\right)}, \tag{4,2} \]

where \(I_0\) is the intensity of the light beam incident on the interferometer.

* From the definition of \(A_j\) it follows that this quantity takes into account not only the true absorption in the semitransparent coating, but also the attenuation of light as a result of scattering. The latter, however, as a rule, is small. See Part I.

Denoting

\[ \tau'_1 \tau'_2 \equiv \sqrt{T_1 T_2}=T,\qquad \rho'_1 \rho'_2 \equiv \sqrt{R_1 R_2}=R \quad \text{and} \quad \alpha'_1+\alpha'_2=\alpha, \tag{4,3} \]

we obtain the well-known expression

\[ I=I_0\frac{T^2}{(1-R)^2+4R\sin^2\left(\frac{\alpha+\delta}{2}\right)}. \tag{4,4} \]

Thus, the character of the interference pattern formed in transmitted light depends not on the individual characteristics of each of the semitransparent coatings, but on their aggregate characteristics: the geometric means of their energy transmission and reflection coefficients and the total phase shift upon two successive internal reflections. This creates the well-known possibilities for maneuvering when applying semitransparent coatings.

For a given interferometer, the maximum intensity in transmitted light is observed when the condition

\[ \sin^2\left(\frac{\alpha+\delta}{2}\right)=0, \tag{4,5} \]

is satisfied, whence the intensity distribution in the interference pattern is determined by the expression

\[ I=\frac{I_{\max}}{1+F\sin^2\left(\frac{\alpha+\delta}{2}\right)}, \tag{4,6} \]

where \(I_{\max}\) is the intensity at the maximum and

\[ F=\frac{4R}{(1-R)^2} \tag{4,7} \]

is the so-called sharpness factor (finesse).

The quantity \(m=\dfrac{\alpha+\delta}{2\pi}\), expressing in units of wavelength the path difference between arbitrary rays of neighboring orders, is called the order of interference. Denoting \(\dfrac{\alpha}{2\pi}=\alpha_0\) and expressing the frequency of the light wave in inverse centimeters \(\left(\nu=\dfrac{1}{\lambda}\right)\), we have:

\[ m=2\nu nt\cos\vartheta+\alpha_0. \tag{4,8} \]

The condition for a maximum of intensity in transmitted light (4,5) is evidently equivalent to the requirement that the order of interference be an integer. Thus, the Fabry–Perot interferometer may be regarded as a light filter that selects only those light beams which satisfy the condition of integrality of \(m\), determined by

according to (4.8). Depending on the specific conditions of illumination of the interferometer, interference fringes of various types are obtained. Some special cases of practical interest are considered below.

6. Types of interference fringes

1. Fringes of equal inclination in monochromatic light \((nt=\operatorname{const}, \nu=\operatorname{const})\) are obtained by placing an extended light source in the focal plane of the condenser \(L_1\) (Fig. 35). The interference fringes are localized at infinity,

Fig. 35. Arrangement for obtaining fringes of equal inclination. \(И\) — light source, \(Ф\text{-}П\) — interferometer, \(Э\) — screen, \(Л_1\) and \(Л_2\) — lenses.

and their observation is carried out with the aid of a second lens \(L_2\), which projects them onto a screen (or photographic plate), or else visually (without the lens \(L_2\)), with the eye accommodated to infinity. In this case an essential condition is the finiteness of the dimensions of the light source, since, as is clear from Fig. 35, the interference pattern is obtained, in essence, as the result of the superposition upon one another of coherent images of the light source produced by the derivative rays of different orders. Equally essential is the strict parallelism of the interferometer plates, since, owing to the incoherence of different parts of the image, the mutual displacement of the interfering images must not exceed the dimensions of the diffraction circle corresponding to the aperture of the light beam.

Differentiating (4.8) and putting \(\Delta m=1\), we obtain for the angular distance between neighboring interference fringes

\[ \Delta\vartheta=-\frac{1}{(m-a_0)\operatorname{tg}\vartheta}. \tag{4.9} \]

Thus (taking into account the dependence of \(m-a_0\) on \(\vartheta\) in (4.8)), the distance

between the fringes decreases with increasing angle of incidence \(\vartheta\), and, consequently, with decreasing order of interference. (We note that the order of interference at \(\vartheta=0\) has a maximum value and decreases as \(\vartheta\) increases.)

The displacement of the fringes with a change in the wavelength of light is given by the relation

\[ \frac{d\vartheta}{\Delta\vartheta}=-(m-a_0)\,d\ln\nu, \tag{4,10} \]

where \(d\vartheta\) is the change in the angular position of a fringe when the frequency changes by \(d\nu\), and \(\Delta\vartheta\) is the angular distance between fringes of neighboring orders for given \(\nu\) and \(m\) (the dispersion of the phase shift \(a_0\) is neglected here). This relation underlies the spectral applications of the Fabry–Perot interferometer.

Fig. 36. Newton’s rings obtained with silvered surfaces. The bends of the fringes reveal defects of the surfaces.

2. Fringes of equal optical thickness in monochromatic light \((\nu=\mathrm{const},\ \vartheta=\mathrm{const})\) are observed when the interferometer is illuminated by a beam of parallel rays sent by a point source placed at the principal focus of a condenser lens. The fringes are localized on the surface of the interferometer and are observed either directly on the surface or by projecting the surface onto a screen or photographic plate (Fig. 36).

In this case the order of interference is related to the optical thickness of the interferometer \(nt\) by the simple relation:

\[ d\ln(m-a_0)=d\ln(nt). \tag{4,11} \]

This relation underlies measurements of the thicknesses and refractive indices of films, as well as the study of the microrelief of various surfaces.

3. Fringes of equal chromatic order \(^{189,192}\) \((m=\mathrm{const},\ \vartheta=\mathrm{const})\), observed in white light and corresponding to the constancy of the product \(n t\), arise under the same conditions as fringes of equal optical thickness and, since

MULTIPLE-BEAM INTERFEROMETRY

the latter being localized on the surface of the interferometer. If the optical thickness of the interferometer is the same over its entire area, then the interferometer obviously performs the function of a light filter transmitting only radiation with definite wavelengths satisfying the integral condition \(m\). Interference light filters of this kind will be discussed in detail below.

Fig. 37. Diagram of an arrangement for obtaining fringes of equal chromatic order. \(I\)—light source, \(D\)—diaphragm (diameter about \(2\) mm), \(F\!-\!P\)—interferometer, \(Sh\)—spectrograph slit, \(L_1\), \(L_2\), and \(L_3\)—lenses.

If, however, the optical thickness of the interferometer varies from point to point, then radiation of different wavelengths will pass through different portions of the interferometer; moreover, the relative change in wavelength corresponding to the maximum transparency of the interferometer will be connected with the relative change in its optical thickness by the simple relation:

\[ d \ln \nu = - d \ln \lambda = = - d \ln (nt). \tag{4,12} \]

Thus, on the surface of the interferometer there will appear the well-known colored patterns, which are a superposition of fringes of equal optical thickness corresponding to different values of \(\nu\). If now the image of the interferometer surface is projected onto the slit of a spectrograph (Fig. 37), then through each point of the slit there will pass only those wavelengths which satisfy the condition for the maximum transparency of the portion of the interferometer projected onto that point of the slit. As a result, in place of a continuous spectrum there will remain only separate interference fringes, which have received the name of fringes of equal chromatic order (Fig. 38). Their position on the wavelength scale is determined

Fig. 38. Fringes of equal chromatic order for the visible region of the spectrum, obtained with the aid of the arrangement for observing Newton’s rings (cf. Fig. 36).

exclusively by the optical thickness of the interferometer. Therefore changes in the thickness of the interferometer along the region projected onto the slit will lead to a curvature (and, correspondingly, to a change in color) of the fringes—the fringes of equal chromatic order reproduce, in their shape, but on a greatly enlarged scale, changes of the optical thickness of the interferometer along the region projected onto the slit (cf. Figs. 36 and 38).

It is not difficult to estimate the magnification thus obtained, as well as the distance between the fringes of neighboring orders.

With the aid of (4.8), equation (4.12) can be transformed to the form

\[ d\lambda = 2\,\frac{\cos \vartheta}{m-a_0}\,d(nt). \tag{4.13} \]

Putting \(\vartheta = 0\) and denoting the dispersion of the spectral apparatus by

\[ \frac{dl}{d\lambda}=D, \]

we find that the displacement of an interference fringe is

\[ dl=\frac{2D}{m}\,d(nt). \tag{4.14} \]

Thus the resulting linear magnification is equal to \(\frac{2D}{m}\). Since the dispersion is usually a very considerable quantity, it is not difficult to obtain a linear magnification of tens and hundreds of thousands of times. Moreover, the magnification is the greater, the smaller the interference order, i.e., the thinner the interferometer.

As for the distance between the fringes of neighboring orders, differentiating (4.8) for constant \(n\), \(t\), and \(\vartheta\), and putting \(\Delta m=1\), we obtain:

\[ \Delta \nu=\frac{1}{2nt\cos\vartheta}, \tag{4.15} \]

i.e., the distance between neighboring fringes on the frequency scale does not depend on the interference order and increases as the thickness of the interferometer decreases. In reality, a very weak but regular increase of \(\Delta \nu\) with decreasing \(\lambda\) is observed, possibly due to the dispersion of \(a_0\) \(^{189}\).

It should be noted that fringes of equal chromatic order possess a feature unique for interference phenomena: the relative displacement of fringes caused by a change in the refractive index does not depend on the thickness of the interferometer and, conversely, the relative displacements of fringes caused by variations in the thickness of the interferometer do not depend on the refractive index of its dielectric interlayer.

1. Nonlocalized fringes of equal monochromatic order \(^{189}\) (\(\nu=\mathrm{const}\), \(m=\mathrm{const}\)) are obtained in a beam of rays diverging from a point light source, without

applications of lenses (Fig. 39). The nature of the occurrence of these fringes is quite obvious if one recalls that the Fabry–Perot interferometer is essentially a light filter that transmits only rays satisfying the requirement of the integral value of \(m\), i.e., in the present case only those rays for which

\[ nt \cos \vartheta = \frac{m-a_0}{2\nu} = \mathrm{const} \qquad (m\text{ is an integer}). \tag{4,16} \]

If a screen is placed behind the interferometer, then, whatever the distance to the screen, a system of fringes will arise on it (circular ones, if \(nt=\mathrm{const}\)). The angular radius of the \(q\)-th fringe in the case of

Fig. 39. Diagram of the formation of nonlocalized fringes of equal monochromatic order. \(И\) is a point light source; \(И_1, И_2\), etc. are virtual sources.

integral value of the interference order at \(\vartheta=0\) is equal to

\[ \vartheta_q = \sqrt{\frac{q}{nt\nu}}, \tag{4,17} \]

while the linear dimensions coincide with the dimensions of the corresponding fringes of equal inclination when using a lens with focal length equal to the distance \(L\) from the source to the screen. By placing the screen at sufficiently large distances \(L\), it is not difficult to obtain fringes whose diameters are measured in tens and even hundreds of centimeters (Fig. 40). The displacement of the fringes upon a change in the thickness of the interferometer is determined by the relation

\[ \frac{d\vartheta}{\Delta \vartheta} = -(m-a_0)\ln(nt), \tag{4,18} \]

where \(\Delta \vartheta\) is, as before, the angular distance between fringes of neighboring orders.

A more rigorous consideration\(^{189}\) shows that the described fringes differ somewhat from fringes of equal inclination. Indeed, the derived rays may be regarded as being produced by virtual equidistant sources separated from one another

at a distance \(2nt\) (Fig. 39), the brightness of these sources decreasing in a geometric progression*). The change in the intensity of the derived rays, due to the inverse-square law, is in this case equivalent to a certain decrease in the reflection coefficient of the mirror surfaces of the interferometer. If the distances to the virtual sources increased in an arithmetic progression, the intensity distribution would correspond exactly to the case of fringes of equal inclination. In reality, however, the increase of the phase shift follows a more complicated law, which leads to

Fig. 40. Nonlocalized fringes of equal monochromatic order (reduced).

a certain displacement, broadening, and asymmetry of the fringes. These distortions will be the more noticeable the smaller \(L\) is. A simple calculation\(^{189}\) leads, in the first approximation, to the following expression for the phase lag of the \(s\)-th derived ray relative to the ray whose phase would correspond to a change in arithmetic progression:

\[ \Delta = (s^2 - 1)\frac{4\pi t}{L}\,q, \tag{4,19} \]

where \(q = m - m_0\) is the number of the interference maximum and \(m_0\) is the order of interference at \(\vartheta = 0\).

Thus, for sufficiently small \(\dfrac{t}{L}\), the result of the interference will be almost identical with that described by formula (4,6).

\[ \text{*) Unlike the case of a diffraction grating or a chain of equidistant coherent emitters (cf., for example, }^{100,247}\text{).} \]

Moreover, it should be borne in mind that, for derivative rays of different order, the optical path length in the interferometer is somewhat different. To a first approximation this can be taken into account\({}^{189}\) if, for the \(q\)-th interference maximum, \(t\) is replaced by

\[ t\left(1+\frac{qs\lambda}{L}\right), \]

where \(s\) is, as before, the order of the derivative ray. (The optical center of gravity of the fringe lies\({}^{189}\) near \(s=5\).) Consequently, the order of interference for the \(q\)-th fringe increases by

\[ \frac{2qst}{L}, \]

in comparison with the order of the corresponding fringe of equal inclination, which leads to a slight decrease in the diameter of the fringes.

1. Fringes of “equal tangential inclination” are observed in strictly parallel monochromatic rays when the plates of the interferometer are bent\({}^{193}\). The scheme of their formation is clear from Fig. 41. The surface of localization of these fringes differs little from the plane passing through the center of curvature of the surfaces of the interferometer. What is significant is the increase in contrast (in comparison with ordinary fringes of equal thickness) owing to the relative increase in the intensity of derivative rays of high orders, which occurs because of the increase of the reflection coefficient as \(\vartheta\) grows. The diameter of the rings is determined by the approximate relation

\[ d=2R\sqrt{\frac{n\lambda\left(q+m_{0}-1\right)}{t}}, \tag{4,20} \]

where \(R\) is the radius of curvature of the interferometer and \(m_{0}\) is the order of the fringe

Fig. 41. Scheme of formation of fringes of “equal tangential inclination.”

Fig. 42. Fringes of “equal tangential inclination,” obtained with a bent mica plate. Splitting of the fringes is due to the birefringence of mica (cf. Fig. 73).

at the center. An example of fringes of equal tangential inclination is shown in Fig. 42.

b. Transparency of the interferometer

The quality of an interferometer is determined mainly by the following characteristics:

1. Transparency, i.e., the ratio of the intensity of the transmitted light at the maximum of transmission to the intensity of the radiation incident on the interferometer \(\left(\dfrac{I_{\max}}{I_0}\right)\).

2. Contrast of the interference pattern, i.e., the ratio of the intensity at the maximum of transmission to the background intensity at the minimum of transmission \(\left(C=\dfrac{I_{\max}}{I_{\min}}\right)\).

3. Half-width of the interference fringes \(w\), i.e., twice the distance between the maximum peak and the point where the intensity is equal to half the maximum.

Fig. 43. Dependence of the transparency of a Fabry–Perot interferometer on the reflectivity of the semitransparent layers forming it. The data for silver, corresponding to different wavelengths, were obtained for films prepared in different ways. Therefore comparison of the curves can have only a conditional significance.

where the intensity is equal to half the maximum.

Let us consider the influence of various factors on the indicated characteristics.

According to (4.4), the transparency of the interferometer is equal to

\[ \frac{I_{\max}}{I_0}=\frac{T^2}{(1-R)^2}. \tag{4.21} \]

If the semitransparent layers do not absorb light \((A=0)\), then \(1-R=T\) and \(I_{\max}=I_0\) (the transparency is equal to unity) independently of the values of \(T\) and \(R\). In other words, in the absence of absorption, the light incident on the interferometer and satisfying condition (4.5) passes through it completely, without reflections or other losses. In the presence of absorption the transparency proves to be the smaller, the larger \(A\), and the decrease of transparency occurs not only at the expense of absorption in the semitransparent layers, but also at the expense of an increase in the fraction of energy reflected by the interferometer. Thus, the intensity of the interference fringes in transmitted light is determined entirely by the absorbing power of the semitransparent coatings forming the interferometer.

The absorption of real silver layers, as was shown in Part I, generally has a maximum at not too large values of \(R\). As the reflectivity increases, the absorption decreases, as a result of which the transmittance of the interferometer should increase. However, at sufficiently large values of \(R\), with increasing layer thickness \(T\) rapidly decreases (for an infinitely thick layer \(T=0\) and \(R=R_\infty<1\)), as a result of which the transmittance of the interferometer must also tend to zero. This is illustrated in Fig. 43,

Fig. 44. Dependence of the transmittance of a Fabry—Perot interferometer on the reflectivity of the semitransparent layers forming it. The asterisks indicate data relating to interferometers whose semitransparent coatings contained no metallic layers, but were formed from 3, 5, and 7 alternating layers of ZnS and cryolite.

showing the dependence of the transmittance of a Fabry—Perot interferometer on the reflectivity of the semitransparent layers forming it. As metallic layers age, their absorptive capacity increases (cf. Fig. 16), as a result of which the transmittance of the interferometer drops sharply\(^{160}\). Since the absorptive capacity of metallic films depends substantially on the technology of their deposition, the transmittance of the interferometer is determined primarily by the technological features of its manufacture.

It was shown above (Part I) that it is possible to substantially reduce absorption by coating the metallic film with several layers.

with layers of dielectric, or even the complete replacement of the metallic film by a multilayer dielectric coating. This leads to a 2–3-fold increase in the transparency of the interferometer155, as illustrated

Fig. 45. a—dependence of the reflection coefficient of a five-layer coating of ZnS and cryolite on wavelength; b—transparency of interferometers with multilayer dielectric coatings. Curves 1 and 2 refer to coatings calculated for different wavelengths. The circles show values corresponding to silver layers having the same reflection coefficients.

by Figs. 44 and 45. The scheme of an interferometer formed from three-layer dielectric coatings is shown in Fig. 46*).

*) On the effective optical thickness of such an interferometer, see, for example,194.

d. Contrast

The intensity of the light passing through the Fabry–Perot interferometer has a minimum value under the condition

\[ \sin^3\left(\frac{\alpha+\delta}{2}\right)=1. \tag{4.22} \]

Consequently, the minimum intensity of the background on which the interference fringes appear is equal to

Fig. 46. Diagram of a Fabry–Perot interferometer in which the metallic semitransparent films are replaced by three-layer coatings of ZnS and cryolite.

Fig. 47. Dependence of \(\dfrac{1}{C}\) on \(R\).

\[ I_{\min}=\frac{I_0T^2}{(1+R)^2} =I_{\max}\left(\frac{1-R}{1+R}\right)^2, \tag{4.23} \]

whence for the contrast of the interference fringes we obtain:

\[ C=\left(\frac{1+R}{1-R}\right)^2. \tag{4.24} \]

Thus, the contrast depends exclusively on the reflecting power of the semitransparent coatings, increasing with increasing \(R\), but does not depend on their absorbing power. The indicated dependence is illustrated in Fig. 47.

Table IV

$R$	0.70	0.75	0.80	0.85	0.90	0.95
$F$	31.1	48	80	151	360	1520
$\dfrac{I_{\min}}{I_0}$ in %	3.22	2.04	1.23	0.66	0.28	0.06

Table IV gives the values of $\dfrac{I_{\min}}{I_0}$ and $F$ for various $R$.

In reality, at large $R$ the contrast turns out to be considerably lower than the calculated one. Thus, Kuhn reports^160 that for monochromatic light, instead of the value $C=200$ expected from calculation, one obtains $C=150$, and instead of $C=500$—approximately one half of this value. Such a decrease in the contrast of real interferometers in comparison with theoretical expectations should apparently be explained by a number of causes whose influence is negligibly small at not too high values of $R$ (and, consequently, of $C$), but begins to have a substantial effect when $R$ increases above approximately 0.93–0.94. These include^155,160,161,188 defects in the polishing of the plates, defects in the semitransparent layers themselves, distortions of the shape of the plates (for example, as a result of stresses acting from the mounts), and also reflections from non-metallized surfaces. As a rule, the larger the area of the plates of the interferometer, the smaller the value of the contrast that can be obtained with it^160. A substantial role in reducing the actually attainable contrast is also played by the nonmonochromaticity of the light used and by diffraction phenomena caused by the finite extent of the beams^187,155. Kuhn believes^160 that, using an interferometer with a silver coating, at the present level of technology it is impossible to obtain a contrast exceeding 300.

The possibilities for a radical increase in contrast are apparently connected with replacing metallic films by multilayer dielectric coatings. Thus, Duffour’s calculations^155 led him to the conclusion that coating both silver layers in the Fabry–Perot interferometer with even a single-layer quarter-wave ZnS film should increase the contrast from 176 to 1040, i.e., by more than a factor of five. An even greater effect may be expected when multilayer coatings are used.

However, it should be borne in mind that the circumstances noted above, in particular defects and inhomogeneities of the coatings themselves, substantially reduce the calculated values^155,161.

d. Half-width of the fringes

The form of the interference fringes is determined by expression (4.6). The phase shift upon reflection \((\alpha)\), if one neglects its dependence on the angle of incidence \(\vartheta\), which is very weak in the small range of angles with which one practically has to deal, affects only the overall displacement of the interference fringes (a change of order, identical for all fringes), but does not affect the character of the intensity distribution*). Therefore the form of the interference fringes is completely determined by the value of the sharpness factor \(F\), i.e., according to (4.7), by the reflectivity of the semitransparent coatings. The intensity distribution in the interference pattern for different values of \(R\) is shown in Fig. 48.

Fig. 48. Dependence of \(\dfrac{I}{I_{\max}}\) on the phase difference of two derived rays of adjacent orders for different values of \(R\).

If \(R\) is sufficiently large, then the derived rays of high orders still have sufficient intensity to take an effective part in the formation of the interference pattern. As a consequence of this (analogously to a diffraction grating), the interference fringes turn out to be the narrower the larger \(R\) is. Thus, in the case of large \(R\), for analyzing the shape of a fringe one may put \(m=m_0+\mu\), where \(m_0\) is an integer corresponding to the order of the maximum of the given fringe, and \(\mu \ll 1\). Then

\[ \sin^2\left(\frac{\alpha+\delta}{2}\right) \equiv \sin^2 m\pi = \sin^2 \mu\pi \simeq \mu^2\pi^2, \tag{4,25} \]

* Corrections for the dispersion of the phase shift \(\alpha\) must be introduced \({}^{55,201}\) in spectroscopic measurements carried out with an accuracy of the order of \(10^{-4}\) Å.

and expression (4.6) is simplified, taking the form

\[ I=\frac{I_{\max}}{1+F\pi^2\mu^2}. \tag{4.26} \]

The intensity of the transmitted light will be equal to one half of the maximum if

\[ F\pi^2\mu_0^2=1 \tag{4.27} \]

or

\[ \mu_0=\frac{1}{\pi\sqrt{F}}=\frac{1-R}{2\pi\sqrt{R}}, \tag{4.28} \]

whence the half-width of the maximum, expressed in fractions of the distance between the maxima of neighboring orders, is

\[ w=2\mu_0=\frac{1-R}{\pi\sqrt{R}}. \tag{4.29} \]

Consequently, the relative half-width of the band, as well as the contrast, depends only on \(R\). For \(1-R\ll 1\) (\(R>0.8\)), which is practically always the case, the dependence between \(w\) and \(C\) assumes a particularly simple form (for more details see, for example, \(^{188}\)):

\[ w\simeq \frac{2}{\pi\sqrt{C}}. \tag{4.30} \]

As noted above, one may introduce the concept of the number of secondary rays \(N_{\mathrm{eff}}\) effectively participating in the formation of the interference pattern. This number, by analogy with the theory of the diffraction grating, is defined as

\[ N_{\mathrm{eff}}=\frac{1}{w}=\frac{\pi\sqrt{R}}{1-R}. \tag{4.31} \]

It is not difficult to estimate the contribution made by the secondary ray with number \(N_{\mathrm{eff}}\) to the total intensity of the interference maximum. The electric-field strength of the light wave \(E_s\), corresponding to the secondary ray of the \(s\)-th order, is proportional to \(R^s\). If \(R^s\ll 1\), then a term of the form \(E_s^2\) is negligibly small in comparison with terms of the form \(E_sE_{s'}\) (where \(s'\ll s\)), proportional to the first power of \(E_s\).

Consequently, the contribution made by the \(s\)-th secondary ray in the case \(s\gg 1\) amounts to an \(R^s\)-th fraction of the contribution made by the secondary ray of zeroth order. The corresponding estimates for \(s=50\) and \(s=100\), made using the approximate formula \(R^s=e^{-(1-R)s}\), are given in Table V.

Table V

Fraction contributed to the intensity of the interference fringe by the 50th and 100th derivative rays (as a percentage of the fraction contributed by the zero ray) for various values of \(R\)

\(s \backslash R\)	0.80	0.84	0.88	0.90	0.92	0.94	0.96	0.98	0.99
50	0.005	0.03	0.25	0.7	1.8	5.0	13	37	60
100	0.00001	0.005	0.03	0.25	0.7	1.8	5.0	13	37

For the derivative ray with number \(N_{\mathrm{eff}}\), in the case of sufficiently large values of \(R\) \((R \gtrsim 0.8)\),

\[ R^{N_{\mathrm{eff}}} = R^{\frac{\pi \sqrt{R}}{1-R}} \simeq e^{-\pi} \simeq 4\%, \tag{4.32} \]

i.e., in fact, derivative rays with a number exceeding \(N_{\mathrm{eff}}\) do not make any appreciable contribution to the formation of the interference pattern.

Table VI

Values of \(w\), \(N_{\mathrm{eff}}\), and \(C\) as functions of \(R\)

\(R\)	\(w \cdot 10^3\)	\(N_{\mathrm{eff}}\)	\(C\)	\(R\)	\(w \cdot 10^3\)	\(N_{\mathrm{eff}}\)	\(C\)
0.98	6.4	155	9800	0.88	42	24	240
0.97	9.6	104	4300	0.86	50	20	170
0.96	13	77	2400	0.84	56	18	130
0.95	16	62	1500	0.82	62	16	100
0.94	19	52	1000	0.80	70	14	80
0.93	23	44	750	0.75	90	11	50
0.92	26	38	550	0.70	110	9	33
0.91	30	33	450	0.65	140	7	22
0.90	34	29	360	0.60	170	6	16

Table VI gives the calculated values of \(w\), \(N_{\mathrm{eff}}\), and \(C\) for various values of \(R\) according to the data of \({}^{196}\). The corresponding dependences of \(w\) and \(N_{\mathrm{eff}}\) on \(R\) are shown in Fig. 49.

As was already noted in the discussion of contrast, increasing \(R\) above a certain limit proves in practice to be ineffective, because defects of the reflecting layers and other factors not taken into account by the approximate theory presented above begin to have an effect (for example, diffraction phenomena \(^{187,155}\)). According to measurements carried out with silvered interferometers \(^{195}\), for not too large \(R\) the values of \(w\) found experimentally and calculated from (4.29) agree well. However, for \(N_{\mathrm{eff}}>20\) (i.e., \(R>0.86\)) the half-width of the bands observed experimentally turns out to be greater than its theoretical value, and by an amount that increases with increasing \(N_{\mathrm{eff}}\). (We note that in this case relation (4.31), which connects \(N_{\mathrm{eff}}\) with \(R\), remains valid, but the equality of \(N_{\mathrm{eff}}\) and \(1/w\) is already violated, since the broadening of the bands is connected not with a decrease of \(N_{\mathrm{eff}}\), but with the action of secondary factors.) The smallest value of \(w\) obtained by the authors \(^{195}\) was 0.02. On the basis of these data, Kuhn \(^{160}\) believes that, with present-day surface-processing technology (polishing of plates, deposition of layers), a value \(R>0.94\) is practically unusable. Approximately the same limiting value is also cited by other authors (for example, \(^{188,189}\)); however, the use of multilayer dielectric coatings makes it possible \(^{155,161}\) not only to obtain, but apparently also to use effectively, somewhat larger values of \(R\).

Fig. 49. Dependences of \(w\) and \(N_{\mathrm{eff}}\) on \(R\).

As for aluminized interferometers, special measurements of \(w\) apparently have not been made. It seems probable \(^{153,160}\) that in this case \(N_{\mathrm{eff}}\) may be close to 15 for the near ultraviolet and even considerably smaller for shorter wavelengths.

Since increasing \(R\) in the case of metallic films leads to a substantial decrease in the transparency of the interferometer, the choice of the thickness of the metallic film is determined by considerations of compro-

between losses in intensity and an increase in the contrast and sharpness of the interference fringes. In doing so it should be borne in mind that, in accordance with what was said above, excessive thickening of the film leads only to excessive losses of light, without any gain in the contrast and sharpness of the fringes.

Since \(R\) and \(A\) depend substantially on the wavelength, different wavelengths will correspond to different optimum thicknesses of the metallic film. One should also take into account the change in \(R\)

Fig. 50. Dependence of \(N_{\mathrm{eff}}\) and \(\dfrac{I_{\max}}{I_0}\) for a Fabry–Perot interferometer formed by silver films three weeks old on the transparency of these films at different wavelengths.

as a result of aging of the film, especially significant during the first weeks (\(\gtrsim 1\%\)) (Fig. 16)\(^{155}\).

In practice it is desirable to replace measurements of the reflection coefficients of metallic films by much simpler, from the technical point of view, measurements of their transparency \(T\). This is possible with a sufficiently carefully developed film-fabrication technology, ensuring a high degree of reproducibility.

Figure 50 shows a typical dependence of \(N_{\mathrm{eff}}\) and \(\dfrac{I_{\max}}{I_0}\) on \(T\), obtained\(^{25}\) for an interferometer with silver layers three weeks after its fabrication. Such curves prove very convenient for selecting the optimum parameters of the interferometer\(^{155}\). Thus, for example, a transparency for which \(\dfrac{I_{\max}}{I_0}=0.4\) will correspond (see Fig. 50)

\(N_{\mathrm{eff}}=67\) in red light and \(N_{\mathrm{eff}}=23\) in green light. For this, silver layers with transparency \(T \cong 0.03\) in red light and \(T \cong 0.06\) in green light are needed.

According to the measurement data \({}^{195}\) (cf. Fig. 12), for sufficiently large values of \(R\) used in multiple-beam interferometry, \(A\) depends comparatively weakly on the film thickness, i.e., \(R+T\), to a first approximation, may be regarded as practically independent of \(T\) (cf. Fig. 16). This makes it possible, in constructing curves of the type shown in Fig. 50, not to measure \(R\) and \(T\) each time for films of different thicknesses, but to confine oneself to measuring one or two standard films \({}^{195}\). In doing so, the dependence of \(R+T\) on wavelength should be taken into account. An example of such a dependence is given in Table VII (cf. Fig. 16).

Table VII

Dependence of \(R+T\) for a silver film on wavelength \({}^{195}\)

\(\lambda\) (in Å)	6800	6000	5500	5000	4500	4000	3800	3600
\(R+T\)	0.97	0.96	0.95	0.94	0.92	0.86	0.82	0.78

Starting from (4.26), it is likewise not difficult to determine the resolving power of the interferometer. According to Rayleigh’s criterion, the limit of resolution is determined by the condition that the intensity at the middle of the band formed by the superposition of two close bands of equal intensity is

\[ \frac{8}{\pi^2} I_{\max}. \]

The corresponding value of the order of interference \(\mu_0\) is then found from the condition

\[ \frac{1}{1+F\pi^2\mu_0^2}=\frac{4}{\pi^2}=0.405, \tag{4.33} \]

whence

\[ \mu_0=\frac{1.21}{\pi\sqrt{F}}. \tag{4.34} \]

The limit of resolution, therefore, is

\[ w_0=2\mu_0=\frac{2\cdot 1.21}{\pi\sqrt{F}}, \tag{4.35} \]

and, according to the definition, the resolving power is

\[ \frac{m_0}{w_0}=\frac{m_0\pi\sqrt{F}}{2\cdot 1.21}, \tag{4.36} \]

i.e.,

\[ \frac{m_0}{w_0}=\frac{m_0}{1.21}N_{\mathrm{eff}}. \tag{4.37} \]

Thus, the resolving power increases with increasing \(R\) and is proportional to the order of interference (and, consequently, to the thickness of the interferometer).

For a number of practical applications considered below, the case is of considerable interest in which, over a substantial extent of the field of view, the order of interference \(\mu\) remains almost unchanged. Let us determine how in this case the intensity of the light passing through the interferometer will depend on the value of \(\mu\). Differentiating (4.6) and using (4.25), we obtain:

\[ \frac{d\ln I}{d\mu} = -\frac{2\pi^{2}\mu F}{1+F\pi^{2}\mu^{2}} . \tag{4.38} \]

Putting \(I=\frac{1}{2}I_{\max}\), i.e., according to (4.28), \(\mu=\frac{1}{\pi\sqrt{F}}\), we have:

\[ d\ln I \simeq -\pi\sqrt{F}\,d\mu . \tag{4.39} \]

A change of \(\ln I\) by \(10\%\) is readily distinguishable by eye. Accordingly, from a change in the intensity of the interference pattern, it is possible without difficulty to detect a change in the order of interference by an amount equal to \(\frac{0.1}{\pi\sqrt{F}}\). Since it is not difficult to make coatings with \(R=0.94\), which corresponds to \(F=1044\), changes in the order of interference of \(1\cdot10^{-3}\) prove to be readily detectable. A further increase of \(R\) (for example, by using multilayer dielectric coatings), entailing a rapid increase of \(F\), should make it possible to lower substantially this limit of detectability of changes in the order of interference in comparison with the value just mentioned (see below. \(^{196}\)).

e. Requirements on the mutual arrangement of the plates. Collimation error and figure error \(^{186,188,189}\)

In the case of interference of two beams (the case usually considered in optics courses), fringes of equal inclination are always localized on the surface of the interferometer (Fig. 51). In reality, for a Fabry–Perot interferometer \(N_{\mathrm{eff}}\gg 1\), and therefore, in order to localize the fringes on the surface of the interferometer and to obtain a sharp interference pattern, certain critical conditions must be fulfilled; we now proceed to consider them. Suppose that a parallel beam of rays falls on a wedge with aperture angle \(\varepsilon\ll 1\) (Fig. 52). Then, at each subsequent reflection of the derived ray, its angle of inclination to the initial direction will increase in an arithmetic progression—the ray of the \(s\)-th order on emergence will be inclined by an angle \(2s\varepsilon\). In accordance with this, the optical path lengths (and, consequently, the phase-

phase shifts) for rays converging at the point \(A\) and forming an interference pattern on the surface of the interferometer will increase, as the order \(s\) of the ray increases, not in an arithmetic progression (as was assumed in deriving relation (4.6)), but according to a more complicated law.

Fig. 51. Localization of fringes of equal thickness in the case \(N_{\mathrm{eff}} = 2\).

As calculations show\(^{189,197}\), in the case \(\vartheta = 0\) the retardation of the derived ray of order \(s\) relative to the ray of zero order, for large values of \(s\), is approximately equal to

\[ 2nst\left(1 - \frac{2s+1}{3}\varepsilon^{2}\right). \]

Thus the \(s\)-th ray experiences an additional retardation (relative to that assumed in deriving relation (4.6)) which, for \(s \gg 1\), may be represented in the form

\[ \frac{4}{3}\,ns^{3}\varepsilon^{2}t. \]

Fig. 52. On the calculation of multibeam interference in a wedge.

The result of the interference will not differ too sharply from that described by formula (4.6) if the retardation of the ray with number \(N_{\mathrm{eff}}\) does not exceed that assumed in deriving this formula by more than half a wavelength. Since in a Fabry–Perot interferometer \(N_{\mathrm{eff}}\), as a rule, is very large, this ...

the requirement can be written in the form of the inequality

\[ \frac{4}{3}\, n N_{\mathrm{eff}}^{3}\,\varepsilon^{3} t \ll \frac{\lambda}{2}. \tag{4.40} \]

Let us denote by \(K\) the number of interference fringes accommodated along \(1\ \mathrm{cm}\) of the interferometer surface, i.e., set

\[ \varepsilon=\frac{K\lambda}{2n}. \tag{4.41} \]

(It is assumed that \(\cos\vartheta=1\).) Substituting (4.41) into (4.40), we find the greatest permissible thickness of the interferometer

\[ t_{\max}=\frac{3n}{2\lambda N_{\mathrm{eff}}^{3} K^{2}}. \tag{4.42} \]

In particular, for \(\lambda=5.5\cdot 10^{-5}\ \mathrm{cm}\), \(n=1\), and \(N_{\mathrm{eff}}=60\) (\(R=0.95\)), we obtain:

\[ t_{\max}\simeq \frac{1}{7.92K^{2}}. \]

Thus, in order to obtain sufficiently sharp fringes, the thickness of the interferometer must not exceed a certain value inversely proportional to the square of the number of fringes located along \(1\ \mathrm{cm}\) of the interferometer surface.

In addition to the supplementary phase lag of higher-order rays considered above, one must also take into account the geometrical displacement of rays of different orders relative to the zero-order ray. If for \(s=1\) this displacement is equal to \(d_1=4t\varepsilon\) (Fig. 52), then for \(s\gg 1\)

\[ d_s \simeq 2s^2 t\varepsilon=\frac{s^2 tK\lambda}{n}. \tag{4.43} \]

Then the greatest displacement of the ray with number \(N_{\mathrm{eff}}\), corresponding to the value \(t\) determined according to (4.42), is

\[ d_{\max}=\frac{3}{2N_{\mathrm{eff}}K}. \tag{4.44} \]

In particular, for \(N_{\mathrm{eff}}=60\),

\[ d_{\max}=\frac{1}{4K}\ \mathrm{mm}. \]

The values of \(t_{\max}\) and \(d_{\max}\) for various \(K\) at \(\vartheta=0\) and \(N_{\mathrm{eff}}=60\) are given in Table VIII.

Table VIII.

Values of \(t_{\max}\) and \(d_{\max}\) corresponding to various \(K\)

\(K\)	1	10	100
\(t_{\max}\) (in mm)	1.26	0.012	0.0001
\(d_{\max}\) (in mm)	0.25	0.025	0.0025

If fringes of equal thickness are observed not by contact reproduction, but with the aid of an optical system (microscope, camera, eye), as is usually the case, then additional requirements arise\({}^{160,186,189}\).

1. All rays with \(s \leq N_{\mathrm{eff}}\) must enter the objective. Consequently, a necessary requirement is

\[ 2\varepsilon N_{\mathrm{eff}} \ll u, \tag{4.45} \]

where \(u\) is the aperture of the objective.

1. The resolving limit of the optical system \(D=\dfrac{\lambda}{u}\) must exceed the dimensions of that region within which semitransparent films still cannot be regarded as homogeneous. In Part I we saw that thin films may be regarded as homogeneous only on the average for regions whose diameters are in any case not less than \(1\,\mu\). Thus the condition \(D>1\,\mu\), or \(u \lesssim 0.5\), must be satisfied.

2. If the purpose of the observations is to determine \(t\) and changes of \(t\) from point to point, then it should be required that \(d_{N_{\mathrm{eff}}} \ll D\); otherwise the contours of the fringes formed by rays collected from an area whose dimensions exceed \(D\) will transmit the surface relief in distorted form. Putting in (4.43) \(s=N_{\mathrm{eff}}\) and taking (4.41) into account, we obtain an upper limit for the aperture:

\[ u \ll \frac{\lambda}{2\varepsilon N^{2}t}. \tag{4.46} \]

1. Further, for distinct observation of the fringes it is necessary that the resolving limit be smaller than the half-width of the interference fringes, i.e.

\[ D \cdot K \ll w=\frac{1}{N_{\mathrm{eff}}}. \tag{4.47} \]

1. Finally, there must be a sufficient number of interference fringes in the field of view, i.e. \(K>K_{\min}\), determined by the magnitude of the field of view.

This imposes additional restrictions on the thickness of the interferometer \(t\) and the angle \(\varepsilon\) (and consequently also on \(K\)). Thus, from condition (1) it follows that the inequality

\[ \varepsilon \ll \frac{u}{2N_{\mathrm{eff}}} \tag{4.48} \]

must be satisfied, or

\[ K \ll \frac{u}{\lambda N_{\mathrm{eff}}}, \tag{4.49} \]

which also corresponds to requirement (4). Substituting the limiting

admissible, according to condition (2), value of \(u\), we obtain:

\[ \varepsilon_{\max}=\frac{1}{4N_{\mathrm{eff}}}, \]

\[ K_{\max}=\frac{1}{2\lambda N_{\mathrm{eff}}}. \]

On the other hand, from condition (3),

\[ t \leq \frac{n}{uK^{2}N_{\mathrm{eff}}^{2}}, \tag{4,50} \]

whence, taking into account (4,41) and (4,45), we find

\[ t_{\max}=\frac{n^{2}}{\lambda N_{\mathrm{eff}}^{3}K^{2}}, \tag{4,51} \]

which also satisfies requirement (4,42).

Putting

\[ N_{\mathrm{eff}}=50\ (R=0.94),\quad n=1\ \text{and}\ \lambda=5\cdot 10^{-5}\ \text{cm}, \]

we have:

\[ \varepsilon_{\max}\simeq 0.005\ \text{radian}, \]

\[ K_{\max}\simeq 200\ \text{cm}^{-1}, \]

\[ t_{\max}=\frac{1}{12.5K^{2}}. \]

And for \(K=K_{\max}\), \(t_{\max}=0.05\,\mu\simeq 0.1\lambda\). For smaller values of \(K\), \(t_{\max}\) is substantially larger; however, requirement (5) restricts the value of \(K\) from below by the quantity \(K_{\min}\), determined by the convenience of observing the interference pattern.

Let us note that the method of calculation presented, based on the approximation of geometrical optics, in the case \(t \lesssim \lambda\), strictly speaking, is not applicable and can have only qualitative significance.

Let us now consider the requirements imposed on the parallelism of the light beam, and thereby also on the dimensions of the light source and the accuracy of adjustment of the optical system.

According to (4,8), the change in the order of interference when the angle of incidence changes by \(d\vartheta\) is equal to

\[ dm=-2\nu nt\sin\vartheta\,d\vartheta. \tag{4,52} \]

Putting \(\vartheta\ll 1\), we have

\[ dm=-\nu nt\,\vartheta^{2}=-\frac{1}{2}(m_{0}-a_{0})\vartheta^{2}, \tag{4,53} \]

where \(m_{0}\) is the order of interference at \(\vartheta=0\). Let us require that

\[ dm=-p\mu_{0}; \]

then

\[ \vartheta=\sqrt{\frac{p\mu_{0}}{\nu nt}}=\sqrt{\frac{p w}{m_{0}-a_{0}}}, \tag{4,54} \]

where \(w=2\mu_0\). Taking the admissible broadening of the band due to collimation error to be of order \(\frac{1}{5}\mu_0\) (i.e., \(p=\frac{1}{5}\)) and taking into account that \(w\simeq 2\cdot 10^{-2}\) (Table VI), we find the maximum admissible aperture angle of the light beam:

\[ \vartheta_{\max}=\sqrt{\frac{4\cdot 10^{-3}}{m_0-a_0}}. \tag{4,55} \]

In particular, for \(\lambda=5\cdot 10^{-5}\) cm and \(n=1\),

\[ \vartheta_{\max}\simeq \frac{10^{-4}}{\sqrt{t}} \]

radians, where \(t\) is expressed in millimeters.

Table IX gives the values of \(\vartheta_{\max}\) for various \(t\) and the corresponding admissible dimensions of the light source (or diaphragm)

Table IX

Values of \(\vartheta_{\max}\) and admissible dimensions of the light source \(d_0\) for various \(t\)

\(t\) (in mm)	1	0.1	0.01	0.001
\(\vartheta_{\max}\) (in degrees)	1/10	1/3	1	3
\(d_0\) (in mm)	0.2	0.6	2	6

\(d_0\), calculated by Tolansky\(^{188,189}\) on the assumption that the focal length of the condenser is equal to 10 cm.

Thus, in the case \(K=10\) and \(t=0.01\) mm, it is possible to use light sources of diameter about 2 mm.

In conclusion, let us consider the requirements on the quality of the surfaces. The angular displacement of the bands caused by a change in the optical thickness of the interferometer, in the case of bands of equal inclination, is given by the relation:

\[ \frac{d\vartheta}{\Delta\vartheta}=-(m-a_0)\,d\ln t, \tag{4,56} \]

whence, putting \(\vartheta=0\) and using (4,8), we find

\[ dt=-\frac{\lambda}{2n}\,\frac{d\vartheta}{\Delta\vartheta}. \tag{4,57} \]

Requiring

\[ \frac{d\vartheta}{\Delta\vartheta}\leq p w\simeq 2\cdot 10^{-2}p \]

(for \(R=0.94\)—see Table VI) and putting \(n=1\), we obtain for the admissible error in the effective thickness of the interferometer

\[ (dt)_{\max}=10^{-2}p\lambda. \tag{4,58} \]

Since the accuracy of manufacture of optical surfaces does not exceed \(10^{-2}\lambda\), the relative broadening of the bands due to errors in the shape of the interferometer plates must amount to

give a value of the order of unity. In other words, as was already noted above (see, for example, \(^{160}\)), for the indicated accuracy of treatment of optical surfaces it is inadvisable to use interferometers with \(R\) exceeding 0.94.

B. FABRY–PEROT INTERFEROMETER IN REFLECTED LIGHT

The interference pattern observed in reflected light differs substantially from that considered above. This difference is mainly due to the fact that the ray of zero order \((s=0)\) stands out sharply among the other derived rays both in its intensity and in its phase shift. At the same time, the theoretical treatment of the phenomenon is also complicated.

A calculation entirely analogous to that carried out for the analysis of the interference pattern in transmitted light leads here to a rather opaque formula for the intensity of the reflected light:

\[ I=I_0\frac{ R_0+F\sin\eta\left[ R_1\sin\eta-\frac{\rho_1\tau_1\tau'_1}{\rho'_1}\sin(\eta+2\eta_0) \right] }{ 1+F\sin^2\eta }, \tag{4.59} \]

where

\[ R_0=\left|\rho_1 e^{i\eta_0}+\frac{\rho'_2\tau'_1\tau_1}{1-\rho'_1\rho'_2}e^{-i\eta_0}\right|^2, \tag{4.60} \]

\[ \eta=\frac{1}{2}(\alpha'_1+\alpha'_2+\delta)=\pi m \tag{4.61} \]

and

\[ \eta_0=\frac{1}{2}(\beta_1+\beta'_1-\alpha_1-\alpha'_1); \tag{4.62} \]

the remaining notation is as before. Thus the character of the interference fringes depends essentially on \(\eta_0\), i.e., on the phase shifts that occur upon reflection and transmission of light by the upper semitransparent layer, and also on the ratio between the reflectivities of both layers for the direct and reverse rays.

In the case when \(\eta_0=\dfrac{\pi}{2}\), \(\rho_1=\rho'_1=\rho_2\), and \(\tau'_1=\tau_1\), formula (4.59) is considerably simplified:

\[ I=I_0F\frac{4A^2+(1-A)\sin^2\eta}{1+F\sin^2\eta}, \tag{4.63} \]

where \(A\) is the absorptivity of the first semitransparent layer. It is not difficult to see that \(I\) has extremal values at the same values of \(\eta\) as in transmitted light; however, maxima in transmitted light will here correspond to minima, and conversely. Thus the pattern in reflected light will be inverted relative to the pattern in transmitted light—at the positions of the narrow

instead of bright bands on a dark background, narrow dark bands on a bright background appear here (Fig. 53). The extreme values of \(I\) are equal to:

\[ I_{\min}=4A^2FI_0=\frac{16A^2R}{(1-R)^2}\,I_0, \tag{4.64} \]

\[ I_{\max}=\frac{(1-A)FI_0}{1+F} =\frac{4(1-A)R}{(1+R)^2}\,I_0. \tag{4.65} \]

Assuming \(A\ll 1\), we obtain the expression for the contrast in reflected light

\[ C_{\mathrm{refl}}\simeq \frac{1}{4(1+F)A^2} =\frac{1}{4A^2}\left(\frac{1-R}{1+R}\right)^2 =\frac{1}{4A^2C_{\mathrm{trans}}}; \tag{4.66} \]

where \(C_{\mathrm{trans}}\) is the contrast in transmitted light. Consequently, the contrast \(C_{\mathrm{refl}}\) rapidly decreases as the absorptive capacity of the outer semitransparent layer increases, and, for a given \(A\ne 0\), as \(R\) increases. Since \(A\) depends on \(R\) (see Figs. 12—14), there exists an optimal thickness of semitransparent coatings at which \(C_{\mathrm{refl}}\) reaches a maximum (cf. Fig. 54). For \(A=0\) (the case corresponding to a dielectric film or plate and usually considered in courses on optics) \(I_{\min}=0\), and the pattern in reflected light turns out to be completely complementary to the pattern in transmitted light:

\[ I=I_0\,\frac{F\sin^2\eta}{1+F\sin^2\eta}. \tag{4.67} \]

Abandoning the assumption that \(\rho_1'=\rho_2\) leads to replacing the term \(4A^2\) in formula (4.63) by a more complicated expression; that is, the difference between \(\rho_1'\) and \(\rho_2\), as well as absorption, affects the contrast of the interference pattern without changing its character. This creates favorable conditions for maneuvering the thicknesses of semitransparent layers\({}^{189}\).

In the case of real semitransparent coatings, the assumption that \(\eta_0=\frac{\pi}{2}\), as well as the assumption of equality of \(\rho_1'\) and \(\rho_1\), are, generally speaking, incorrect.

Nevertheless, for comparatively thick silver layers, as experiment shows\({}^{43, 60, 146, 147, 160, 189, 197, 200}\), the pattern in reflected light proves to be approximately complementary to the pattern in transmitted light. An explanation of this by no means trivial circumstance can be obtained from consideration of Figs. 8, 17, and 19. In fact, at sufficiently large thicknesses of the silver film (\(t \gtrsim 200\,\text{\AA}\)) \(\rho'\) differs practically little from \(\rho\) (Fig. 8). Further, it follows from Fig. 17 that for \(\lambda=0.55\,\mu\) at \(t\gtrsim 200\,\text{\AA}\) \(\alpha\simeq 0.8\pi\) and \(\alpha'\simeq 0.7\pi\), and from Fig. 19, approximating the experimentally found dependence (for \(\lambda=0.59\,\mu\)) at \(t\gtrsim 200\,\text{\AA}\) by a straight line,

\[ \beta \simeq \left(0.13+1.8\cdot 10^{-4}\,t-\frac{t}{\lambda}\right)2\pi \simeq \left(0.26+\frac{0.1}{5900}\,t\right)\pi. \]

a

b

Fig. 53. Fringes of equal thickness obtained in an interferometer formed by a silvered plane-parallel plate and the surface of a quartz crystal. a — in transmitted light, b — in reflected light.

Neglecting the second term containing \(t\), and setting \(\beta_1 = \zeta_1\), we find approximately that \(2\eta_0 = (0.52 - 1.5)\pi = -0.93\pi\), i.e. \(\eta_0 \simeq -\dfrac{\pi}{2}\), and it is practically independent of the film thickness \(t\). However, the inexact coincidence of \(\eta_0\) with \(\dfrac{\pi}{2}\) entails a certain displacement of the interference fringes in reflected light relative to the fringes in transmitted light,\(^{43,199}\) and also an asymmetry in the structure of the fringes themselves.\(^{43, 60, 112, 197–200}\)

For other metals, the question of the value of \(\eta_0\) and of its dependence on the film thickness remains open. Some data, including on the character of the interference fringes in reflected light, are apparently available only for aluminum.\(^{60, 75, 201}\)

Fig. 54. Change in the character of fringes in reflected light upon changing the thickness of the covering semitransparent layer. The thickness of the silver layer changes from zero (left) to 300 Å (right).

As the thickness of the covering semitransparent film decreases below \(t \simeq 200\) Å, the character of the interference fringes in reflected light changes sharply (Fig. 54).\(^{60, 200, 202}\) At first, for not very small thicknesses, a sharp asymmetry arises both of the fringes themselves and of their position relative to the fringes in transmitted light, accompanied by an equally sharp blurring of the fringes; and then the maxima and minima in reflected light change places. One of the causes of such a sharp change in the character of the fringes is undoubtedly the change in \(\eta_0\), which, in the region of silver-film thicknesses smaller than approximately 200 Å, has a very rapid character (cf. Fig. 17). However, the principal cause of the reversal of the fringes should apparently be seen, as K. D. Sinel’nikov and N. V. Rapp\(^{202}\) indicated, in the substantial difference arising at small thicknesses of the silver film between \(\rho\) and \(\rho'\) (cf. Fig. 8), where \(\rho\), even if it does not become zero, in any case becomes considerably smaller than \(\rho'\). (Recall that in the case of an interferometer filled with air, the ray incident from outside undergoes reflection at the glass–film boundary, while the ray incident from inside undergoes reflection at the air–film boundary, i.e. in Fig. 8

\(R'\) corresponds to \(\rho^2\) and \(R=-\rho'^2\). Turning to formula (4.59), it is not difficult to see that, for \(\rho_1 \ll \rho'_1 < \rho'_2\),

\[ I \simeq I_0 \frac{F T_1 T'_1}{1+F\sin^2\eta}\,\frac{\rho_2'}{4\rho_1}, \tag{4.68} \]

independently of the value of \(\eta_0\). Using (4.2), we obtain:

\[ I_{\text{refl}}=\frac{1}{4}\frac{T'_1}{T_2}\frac{\rho'_2}{\rho'_1}\left(1-\rho'_1\rho'_2\right)^2 I_{\text{trans}}, \tag{4.69} \]

i.e., the fringes in reflected light coincide both in position and in form with the fringes in transmitted light, differing from them only in intensity by a constant numerical factor. The intensity of the inverted fringes obtained in this case is very large—of the order of 100%; the half-width, however, reaches approximately 0.2 (and more) orders, which is quite understandable if one takes into account the sharp decrease of \(N_{\text{eff}}\) owing to the decrease of \(\rho_1\). In favor of such an explanation, in particular, is the fact that the phenomenon has a completely analogous character both for silver and for aluminum layers\(^{60,75,161}\). Let us note that the sharp broadening of the fringes observed in the transition region is evidently connected with an increase in the absorption of the film\(^{60}\) (cf. Fig. 12).

The physical meaning of the phenomenon described is quite evident\(^{202}\). As was noted above, the difference of the fringes in reflected light is due to the presence of a dominant ray of zero order with a phase shift sharply different from that of the other derived rays. Reducing \(\rho_1\) to zero means suppressing this ray and, thereby, eliminating the cause that creates the difference between the interference patterns in transmitted and reflected light. This idea forms the basis of Bruce’s\(^{199}\) method of inverting the fringes in reflected light and transforming them into fringes analogous to those observed in transmitted light (very narrow bright fringes on a dark background). The essence of the method consists in the artificial removal of the zero-order ray by cutting it off with the corresponding diaphragm. The geometrical separation of the derived rays necessary for implementing such a procedure is achieved by forming a wedge as a result of a mutual inclination of the interferometer plates (cf. Fig. 52). Since \(N_{\text{eff}}\) is large in this case, the fringes obtained have high contrast (Fig. 55).

Despite a number of shortcomings inherent in this method (including the loss of an important advantage of reflection interferometry—elevated, in comparison with interferometry in transmitted light, “transparency”), it may in some cases prove useful.

The principal difficulty of working in reflected rays consists in the fact that the interference pattern obtained is analogous to the spectrum

absorption, whereas in transmitted light it is analogous to the emission spectrum. Therefore, in reflected light, observation with nonmonochromatic light is made very much more difficult—the dark absorption lines for one of the wavelengths are located on the light background created by radiation of another wavelength, which leads to an extremely strong weakening of the contrast (Fig. 56). It is true that by artificial means one can, when photographing, increase the contrast of the picture (for example, underexposure at the minima), but this is effective only in the case when maxima of approximately equal intensity are superimposed.

Fig. 55. Stripes of equal thickness in reflected light. a — ordinary; b — with removal of the zero-order beam.

Fig. 56. Superposition of interference fringes for different wavelengths in reflected light.

Otherwise, weak maxima will in practice be completely drowned out by the background of the stronger one.

radiation^189. At the same time, in a number of cases (for example, the study of oscillations of opaque objects) fringes in reflected light are the only possible ones, and here using them becomes highly desirable (cf. Fig. 72).

B. THE DOUBLED INTERFEROMETER

The possibility of detecting weak satellites against the background of bright spectral lines is determined by the contrast of the interferometer. It was shown above that the Fabry–Perot interferometer is capable of providing a contrast not exceeding 300, which in a number of cases is far from sufficient.

The most radical way to increase the contrast is apparently to double the interferometers. The idea of doubling is not new^203,204; however, it has begun to find practical application only in recent times^155,188,200—209,212. The essence of the method is that two interferometers are placed successively in the path of the light beam, their optical thicknesses being in an integral ratio as a whole. When fringes of equal inclination are obtained, each of the interferometers may be regarded as an angular filter transmitting radiation of the given wavelength only in those directions \(\vartheta\) which satisfy the condition that the order of interference be an integer. If the transparencies of the interferometer as functions of the angle \(\vartheta\) are respectively \(I_1(\vartheta)\) and \(I_2(\vartheta)\), then the total transparency of two interferometers arranged in series is obviously determined by the product \(I_1(\vartheta)\cdot I_2(\vartheta)\). It is essential here to arrange the interferometers in such a way as to exclude as fully as possible cross interference in the interval between them, since interference effects of this kind lead to an undesirable distortion of the interference pattern (the appearance of ghosts, etc.)). The relative half-width of the fringes \(w\), in a first approximation, is equal to*)

\[ w = w_0 \frac{n_2 t_2}{n_1 t_1}, \tag{4.70} \]

where \(w_0\) is the half-width for a single interferometer, determined by (4.29), and \(n_2 t_2\) and \(n_1 t_1\) are the optical thicknesses of the interferometers \((t_2 > t_1)\). The contrast obtained with a doubled interferometer is equal to the product of the contrasts for the single interferometers. Theoretically, contrasts up to 10,000^155 may be expected. In practice, however, owing to defects in the manufacture of the plates and other disturbances considered above, it is possible to obtain contrast—

*) It is possible to proceed in another way, placing the interferometers in immediate proximity to one another, or else forming a complex interferometer from three semitransparent coatings. In this case cross interference acquires substantial importance. See ^155,208.

**) For a more detailed analysis, see ^188.

…the order of 2000 (for green light \(^{160}\)), which is an order of magnitude greater than that attainable with a single interferometer. As an example, it may be noted that with the aid of a doubled interferometer \(^{160,206}\) the satellite helium lines belonging to \({}^3\mathrm{He}\), present in the sample at a concentration of \(1/700\), were isolated. The decrease in transparency when interferometers are doubled can be compensated to a certain extent (which is important in observing weak radiations) by increasing the transparency (i.e., decreasing the thickness) of the semitransparent coatings used.

It should be borne in mind that, when interferometers are doubled, ghosts appear, due to the fact that the transmission maxima in the thicker interferometer are superposed on a nonzero background produced by the thinner interferometer. The relative intensities of the ghosts for the case of an interferometer-thickness ratio of \(4:1\) are given \(^{188}\) in Table X.

Table X

Relative intensity of ghosts for \(t_2/t_1=4\)

\(m\) for the thin interferometer	\(m\) for the thick interferometer	Relative intensity of the maximum
\(m_1\)	\(m_2=4m_1\)	\(1\)
\(m_1 \pm \dfrac{1}{4}\)	\(m_2 \pm 1\)	\(1/41\)
\(m_1 \pm \dfrac{1}{2}\)	\(m_2 \pm 2\)	\(1/81\)
\(m_1 \pm \dfrac{3}{4}\)	\(m_2 \pm 3\)	\(1/41\)
\(m_1 \pm 1\)	\(m_2 \pm 4\)	\(1\)

A very important question is that of the requirements on the accuracy of matching the interferometers \(^{155,160}\). In order that the position of the maximum of the thicker interferometer coincide with the position of the maximum of the thinner interferometer to within the \(p\)-th part of the half-width of the band of the latter, it is necessary that the deviation of \(n_2t_2\) from the value corresponding to exact coincidence of the maxima not exceed

\[ d(n_2t_2)=p\,\frac{n_2t_2}{n_1t_1}\,w\,\frac{\lambda}{2}. \tag{4.71} \]

Such correction is carried out either by a relative tilting of the interferometers (a difference of \(\vartheta_1\) and \(\vartheta_2\)), or by re-

adjustment of the pressure on the plates from the mounting side, or, finally, by changing the pressure of the air filling the interferometer\(^ {20}\).

In addition, it is necessary that a series of maxima coincide, which, strictly speaking, is fulfilled only when \(n_2t_2\) is an exact multiple of \(n_1t_1\). Let us require that, when the \(m_1\)-th maximum for the thin interferometer fully coincides with the \(m_2\)-th maximum for the thick interferometer, the \(m_1+1\)-st maximum coincide with the \(m_2+k\)-th to an accuracy of a \(p\)-th fraction of the half-width of the fringe for the thick interferometer (\(k\) an integer). Then

\[ \frac{n_2t_2}{n_1t_1}=k\pm q, \tag{4,72} \]

where \(q\) must be less than \(p w\). Thus, the thickness of the second interferometer must be an integral multiple of the thickness of the first, to an accuracy of a quantity of the order \(n_1t_1pw\). If \(w \simeq 2\cdot 10^{-2}\) and \(p=0.1\), then the permissible mismatch of the interferometer thicknesses is about \(0.1\%\).

G. N. R. Batarchukova’s Monochromator and Self-Registering Interferometer

A Fabry–Perot interferometer, like any other spectral instrument\(^ {211}\), can be used for monochromatizing light. A monochromator of this kind was constructed by N. R. Batarchukova\(^ {211,212}\). (See also the article by M. F. Romanova in this same issue.)

The interference pattern (fringes of equal inclination) is projected onto a metal mask with narrow circular slits, whose positions correspond to the positions of interference fringes of different orders (usually only one fringe with the greatest possible order) for a given wavelength. The slits thereby serve as the exit aperture of the monochromator. The wavelength selected by the monochromator can be changed either by changing the mask or (much more simply) by shifting the positions of the images of the fringes, either by changing the magnification of the projecting optical system or by changing the optical thickness of the interferometer. The latter method proves most convenient, since a smooth and readily reproducible adjustment of the optical thickness of the interferometer can be achieved by changing the pressure of the air filling the gap between its plates.

It is very important to maintain the temperature of the interferometer constant (to an accuracy of \(0.1—0.2^\circ\)C); a change in temperature can likewise be used to vary the wavelength transmitted by the monochromator\(^ {212}\).

The large resolving power of the Fabry–Perot interferometer ensures a high degree of monochromatization. Thus, N. R. Ba-

By means of the monochromator described, Tarukova succeeded in resolving the hyperfine structure of the cadmium and mercury lines \(^{211}\). To obtain greater resolving power, the use of a doubled interferometer is possible \(^{213}\). It should be noted that preliminary coarse monochromatization of the light is necessary.

Subsequently the same method was used to construct a self-recording photoelectric interferometer \(^{155,213}\). When the pressure of the air filling the interferometer is varied, the fringes shift relative to the mask. The light passing through the mask is directed onto a photocell, and the readings of the galvanometer measuring the photocurrent are recorded synchronously with the recording of the pressure, which uniquely determines the wavelength of the radiation passing through the mask.

5. STUDY OF THE MICRORELIEF OF SURFACES AND DETERMINATION OF OPTICAL CONSTANTS

One of the most remarkable applications of multiple-beam interferometry is its use in the study of the microrelief of surfaces.

Not being able to consider here the results obtained with its aid, we shall dwell only on the possibilities of the method and the various techniques for carrying it out.

The basic idea is that the surface under investigation is assigned the role of one of the plates of the interferometer, while the role of the other plate is performed by a carefully polished glass or quartz plate, silvered on one side.

Since the optical thickness of the interferometer obtained in this way depends on the relief of the plates, the interference pattern reflects the character of this relief.

To realize the possibilities of multiple-beam interferometry, it is necessary that the number of derivative rays effectively participating in the formation of the interference pattern be sufficiently large. This requires a high reflectivity of the surface under investigation. In the overwhelming majority of cases, however, this requirement is not fulfilled, and it is necessary to resort to an artificial increase of \(R\).

In practice this is achieved by silvering the surface of the body under investigation. In this connection there arises the important question of the extent to which the relief of the surface is reproduced by the relief of the silver film covering it. There apparently are as yet no direct data on the accuracy with which the relief is reproduced. However, comparison of the results of studies of various surfaces shows \(^{189}\) that, in height, the outlines of the silver film exactly repeat the outlines of the underlying surface, and the possible errors of reproduction are considerably smaller than the interatomic distances both of silver and of the material onto which the film is deposited.

Such a result may seem paradoxical, especially in view of the information on the structure of the film that was considered in Part I. It is enough to recall that the dimensions of individual inhomogeneities of the film may reach \(1 \mu\), i.e., exceed the errors in reproducing the relief by 3–4 orders of magnitude. It should be remembered, however, that for large \(R\), when it is possible to obtain information about the relief only with an accuracy down to fractions of the lattice constant, microscopes with high magnification cannot be used: an increase in the accuracy of determining the relief in height is inevitably accompanied by a decrease in the accuracy of determining it in the plane of the surface (see Sec. 4e). Therefore, determinations of height always refer to comparatively extensive regions of the surface, and the more extensive the more accurately the height is determined. In other words, the object of measurement is not the actual height of a given point of the surface, but the mean height of some region, substantially exceeding in its dimensions the dimensions of individual inhomogeneities of the film, which is what ensures the indicated accuracy of reproduction of the relief. At the same time there are grounds to suppose\(^{189}\) that, in the plane of the film, its contours reproduce the contours of the underlying surface with rather considerable distortions, the nature of which is clear from Fig. 57. The magnitude of these distortions is not difficult to estimate if one recalls that, to obtain \(R > 0.9\), a silver film about \(400 \text{ Å}\) thick and more is required. However, owing to the inevitably small magnification of the microscope, these distortions also turn out to lie beyond the resolving power of the method.

Fig. 57. On the distortion of horizontal contours of a surface covered with a film.

In a number of cases, especially in the study of opaque objects, it proves expedient to use, instead of the surface itself, transparent replicas taken from it. Control measurements have shown\(^{214}\) that the error in reproducing the relief in depth then does not exceed \(5 \text{ Å}\), while the errors in the plane of the film are negligibly small.

To ensure high contrast of the fringes and, consequently, high resolving power of the method, it is necessary to observe comparatively strict conditions discussed in Sec. 4e. The principal ones here are the requirements on the thickness of the interferometer and on the collimation of the rays. The insufficient attention formerly paid to these requirements was one of the reasons why the enormous possibilities of the essentially very old method of interference

studies of relief have been revealed only in recent years. Depending on the particular conditions of the problem, one or another type of fringe may prove convenient. Therefore we shall briefly consider some questions specific to fringes of different types.

When working with monochromatic light, the requirements imposed on the degree of its monochromaticity are determined by relation (4.12), according to which the permissible relative width of the spectral interval used must be less than the relative error in measuring the optical thickness of the interferometer. Since the absolute error in measuring \(nt\) is equal to \(p w\), where \(p\) is the smallest distinguishable displacement of a fringe, expressed in fractions of the half-width, then even for \(nt \simeq \lambda\) the greatest permissible width of the spectral interval must not exceed several ångströms; that is, for measurements it is necessary to use light sources with a line spectrum.

Fig. 58. Relief map of the surface of a mica plate (fringes of equal thickness).

In the case of using fringes of equal thickness, a relief map of the investigated surface area is obtained directly (Fig. 58). Such a map gives a sufficiently clear representation of the relief and is convenient for measurements only under the condition that the fringes are located sufficiently densely, i.e. the irregularities of the relief considerably exceed \(\frac{\lambda}{2}\). Otherwise the details of the relief will remain unrevealed. To reveal them one may resort to an artificial crowding of the fringes. For this it is sufficient to create a mutual inclination of the plates, choosing in a suitable manner the angle \(\varepsilon\) between them \(^{189}\) (Figs. 59 and 60). Another method of artificially crowding the fringes consists in successively changing, step by step, the air pressure in the interferometer and photographically superposing the interference patterns obtained in this way upon one another \(^{215}\). For better revelation of the relief it is sometimes expedient to obtain (also by photographic superposition) a crossed grid of two mutually perpendicular systems of fringes, changing the direction of the tilt of the plates \(^{199}\) (cf. Fig. 64). This simultaneously facilitates the determination of the direction of change of the relief. The problem of determining the direction of the slope is substantially

Fig. 59. Fringes of equal thickness in reflected light, artificially condensed with an inclined glass plate. Well-polished steel surface, magnification \(80\times\).

Fig. 60. Artificially condensed fringes of equal thickness. Surface of rolled steel. (The beam is poorly collimated in order better to reveal the relief—cf. Fig. 61.)

is simplified if the beam is poorly collimated. In this case an asymmetry of the fringes arises, manifested in the fact that the fringes have a sharp boundary on the side of smaller thicknesses (rises) and a blurred boundary (shadow) on the side of greater thicknesses (descents) (Fig. 61) ^{160,189}.

It is not difficult to estimate the accuracy of measuring \(nt\) from the displacement of the fringes. The displacement of a fringe can be measured with an accuracy of the order of \(1/10\) of its half-width. Taking \(w=2\cdot 10^{-2}\) and allowing for the fact that the distance between neighboring fringes corresponds to a change in the optical thickness of the interferometer by \(\frac{\lambda}{2}\), we find that measurements can be carried out with an accuracy up to \(10^{-3}\), i.e. for measurements in visible light the error is approximately \(5\,\text{Å}\). In practice such accuracy is by no means always attained; usually the error of an individual measurement is \(10\)—\(20\,\text{Å}\) (see, for example, ^{189,192}). We note that, with such accuracy, it proves possible to measure surface inclinations on areas measuring fractions of a millimeter with an error up to \(0.01\) minute of arc ^{189}. The use, instead of a silver film, of multilayer dielectric coatings, as well as the additional coating of the silver film with a dielectric layer, apparently makes it possible to increase the accuracy of measurements somewhat ^{155}.

Fig. 61. Asymmetry of fringes arising with an increase in the angular dimensions of the light source and revealing the direction of thickness variation: \(a\)—point source, \(b\)—extended source.

If the surface under investigation has irregularities amounting to only an insignificant fraction of the wavelength of light, then, using the considerations developed in Section 4d, it is possible to achieve a substantial increase in the accuracy of measurements ^{189,216}. A beam of parallel monochromatic rays is directed onto an interferometer with nearly parallel plates; their wavelength and angle of incidence are chosen so that, at the mean thickness of the interferometer, the intensity of the light transmitted through it is equal to \(\frac{1}{2}I_{\max}\).

According to (4.39), small changes in the thickness of the interferometer will in this case cause sharp changes in the intensity of the light transmitted through it:

\[ d\ln I=-2\pi \sqrt{F}\,\nu \cos\vartheta\, d(nt). \tag{5.1} \]

For visible light, with \(R = 0.94\), this gives:

\[ d \ln I \simeq 400\,dt, \]

where \(t\) is expressed in microns. Since \(d \ln I\) of the order of 0.1 is readily detectable, we find that a change in thickness of \(2.5\) Å is also detectable. An example of the use of this method is shown in Fig. 62.

Fig. 63 shows\({}^{216}\) a photograph, obtained in this way, of the surface of a mica plate, on part of which (the region \(MM\)) a monomolecular layer of stearic acid (about 24 Å thick) has been deposited. The part of the plate \(P\), free from adsorbate, stands out clearly in intensity. The constancy of the intensity along the band \(MM\) indicates a high degree of uniformity of the layer. (The scratches on the layer \(MM\) are the result of carelessness allowed during preparation of the layer.) If white light is used instead of monochromatic light, a brightly colored relief image of the surface is obtained. Viewing this image through various light filters makes it possible to clarify a number of details. It is sometimes useful to superpose an additional grating of crossed fringes of the usual type (by changing the mutual inclination of the plates) upon a similar high-dispersion pattern (Fig. 64). Further increase of the resolving power of this method was achieved in the following way\({}^{217}\). A mica flake was silvered on both sides. The interferometer thus obtained was cut into two parts, which were then mounted successively one behind the other, resulting in an analogue of a double interferometer. When it was illuminated by a parallel beam of white light, on the resulting colored interference

Fig. 62. Microrelief of the surface of a diamond, revealed from the change in intensity of light passing through an interferometer.

Fig. 63. Change in the intensity of transmitted light as a result of applying a monomolecular layer of stearic acid.

Fig. 64. Superposition of a grid of crossed bands of equal thickness, obtained by changing the inclination of the interferometer plates, with a picture obtained with nearly parallel plates, making it possible to carry out a complete quantitative analysis of the relief.

the image clearly showed the finest details of the structure of the mica plate (Fig. 65).

Since the error in determining the thickness is smaller than the lattice constant, it becomes possible to carry out measurements not only of the lattice constant or of molecular dimensions216, 217, but also to study lattice defects (for example, dislocations) and the mechanism of crystal growth218, 219, 220. The theory of dislocations predicts that screw dislocations emerging at the surface can serve as growth centers, and that spiral terraces should form on the surface of the crystal. Such terraces were in fact observed on the surface of corundum crystals by means of multiple-beam interference techniques

Fig. 65. Interference pattern obtained in a double interferometer from mica flakes.

(Fig. 66). Measurements showed that, in agreement with the theory, the height of the terraces is \(15.1 \pm 2\) Å, i.e. the lattice constant of corundum (according to X-ray structural analysis, the lattice constant of corundum is \(c = 15.1\) Å).

To increase the accuracy of measuring fringe displacements, it is desirable to have a scale finer than the distance between fringes of adjacent orders. Such a scale can be provided, for example, by the doublet splitting of the sodium \(D\)-line189. A special variant of the use of such a scale was developed for determining the thickness of thin films with the aid of equal-inclination fringes221. The measured film covers half the field of view. By changing the distance between the plates of the interferometer, two such neighboring positions of the plates are found for which: (1) the fringe corresponding to the component \(\lambda_1\) in the portion of the field of view not covered by the film is the direct continuation of the fringe corresponding in the film-covered portion of the field of view to the component \(\lambda_2\) (Fig. 67a), and (2), conversely, coincidence is obtained of the fringes corresponding

Fig. 66. Spiral-like growth of a corundum crystal near a screw dislocation.

Fig. 67. On the use of the doublet splitting of the sodium \(D\)-line as a wavelength standard when measuring film thickness.

component \(\lambda_2\) in the uncovered part and component \(\lambda_1\) in the part of the field of view covered by the film (Fig. 67b). If the relative displacement of the interferometer plates in going from position (1) to position (2) is denoted by \(M\), then, as is not difficult to show, the thickness of the film is

\[ t=\frac{M}{4K}\lambda, \tag{5.2} \]

where \(\lambda=\frac{\lambda_1+\lambda_2}{2}\), and \(K\) is the displacement of the interferometer plates in passing between two adjacent positions corresponding to the coincidence of the components \(\lambda_1\) and \(\lambda_2\) (in the absence of the film):

\[ K=\frac{\lambda_1\lambda_2}{2(\lambda_2-\lambda_1)}. \tag{5.3} \]

The use of fringes in reflected light, which is unavoidable when opaque objects are investigated without the aid of replicas, creates additional difficulties, reducible (with the exception of those considered above) to the need to use the microscope objective for combining two functions: obtaining an image and illuminating the object, as is usually the case in metallographic microscopes. However, a full combination of these functions, taking into account the requirements set forth in section 4e (in particular, the parallelism of the illuminating beam of rays), proves impossible, and one has to adopt a compromise. A satisfactory pattern is obtained when magnifications up to \(750\times\) are used\({}^{223}\).

To obtain nonlocalized fringes of equal monochromatic order, it is convenient to use the image of a point light source near the surface of the interferometer with the aid of an ordinary microscope objective. If, in a sufficiently small region encompassed by the light cone, the thickness of the interferometer remains practically the same, then the resulting fringes correspond to fringes of equal inclination, and from their position one can judge the thickness of the interferometer. By moving the illuminating objective, it is thus possible to examine successively the entire region of the interferometer surface of interest.

Of particular interest is the application to the study of the relief of fringes of equal chromatic order. Examples of revealing, with their aid, the surface relief along a line projected onto the slit of a spectrograph are given in Figs. 38 and 68–70. The fact that the fringes in the upper part of Fig. 68 are spaced more densely than in the lower part means, according to (4.15), that the thickness of the interferometer is greater here, and thereby the direction of the cleavage detected by the line discontinuity is determined. To determine the height of the cleavage, one may use either the difference in distances between the fringes

of adjacent orders, or by a shift of the band for the given order. In the case of a large number of bands in the field of view, as occurs in Fig. 68, the bands form, along the cleavage line, a kind of vernier. If \(\Delta\nu\) is the distance on the frequency scale between two pairs of coincident bands, then, as is easy to see, the step height (assuming \(\vartheta=0\)) is

\[ \Delta t=\frac{1}{2n\Delta\nu}. \tag{5,4} \]

Usually exact coincidence of the bands does not occur. Then, choosing an arbitrary spectral interval \(\Delta\nu\), we have

\[ \Delta t=\frac{q_1-q_2}{2n\Delta\nu}, \tag{5,5} \]

where \(q_1\) and \(q_2\) are the numbers (generally speaking, fractional) of the bands located in the given spectral interval above and below the cleavage line.

Fig. 68. Bands of equal chromatic order for a portion of a mica plate. The cleavage line intersecting this region is clearly visible.

Another method for determining a change in thickness, suitable both for the case of a smooth change and for the case of cleavage, is based on relation (4,13). However, to use this relation one must know the order of the band. It is not difficult to find it by using (4,15). If the wavelengths corresponding to bands of adjacent orders are \(\lambda\) and \(\lambda'\), then

\[ m=\frac{\lambda'}{\lambda-\lambda'}. \tag{5,6} \]

It should be noted that the quality of the bands depends essentially on the magnification of the lens projecting the surface under investigation onto the slit of the spectrograph: the greater the magnification, i.e. the smaller the change of relief over the length of the image, the sharper the bands. In exactly the same way, the character of the bands depends on the orientation of the inclination of the surface relative to the slit. If the surface has an inclination along the slit, then the inclination of the lines increases; if, however, the inclination of the surface is perpendicular to the slit, this leads to a decrease in magnification and to blurring of the lines. It is easy to see that if \(x\) and \(y\) are the horizontal—

Fig. 69. Bands of equal chromatic order for a portion of the surface of polished steel shown in Fig. 59. Note the sharp increase in scale.

...the horizontal and vertical distances between adjacent fringes, i.e. the tangent of the angle of inclination of the fringes is equal to \(\dfrac{y}{x}\), then the angle of inclination of the surface along the direction of the target is equal to \(\dfrac{\lambda}{2y}\). Thus the angular magnification is determined by the ratio \(\dfrac{2x}{\lambda}\). According to (4.14), \(x=\dfrac{D}{m}\lambda\), and for the angular magnification we have \(\dfrac{2D}{m}\). For example, with a dispersion of \(200\ \text{Å}/\text{mm}\) and \(m=5\), we obtain an angular magnification equal to \(20\,000\) (\(x=5\ \text{mm}\)).

An example of the application of multi-beam techniques is the measurement\(^{223}\) of the thickness of thin-layer coatings (thickness range from \(10\ \text{Å}\) to several microns) with an error not exceeding 5%. It should be noted that in the case of very thin films, in order to reproduce their thickness correctly when forming a comparatively thick mirror coating (necessary for converting the film into one of the plates of the interferometer), one should use the same material from which the thin film is formed, or first coat it with a thick layer of such material and only then deposit the silver mirror layer\(^{223}\). The errors introduced by the dispersion of the phase jump upon reflection from silver are negligibly small (0.5%).

Fig. 70. Measurement of the thickness of a monomolecular layer of stearic acid by means of fringes of equal chromatic order.

With the aid of the same technique it proved possible, for example, to measure without difficulty the thickness of a monomolecular layer formed during the adsorption of certain fatty acids \((\mathrm{C}_{27}\mathrm{H}_{55}\mathrm{COOH}\) and \(\mathrm{C}_{17}\mathrm{H}_{35}\mathrm{COOH})\)\(^{216}\).

In the first case \(t=30\pm1.5\ \text{Å}\) was obtained, and in the second \(19\pm3\ \text{Å}\). The spectrogram corresponding to the second case is shown in Fig. 70; the shift of the fringe in the section \(CD\), crossing the monomolecular layer of stearic acid, is clearly visible. This displacement in the 27th order is \(\Delta\lambda=2.0\pm0.3\ \text{Å}\). It should be noted that X-ray structural data give sharply different values of \(t\): 37.02 and 24.42 Å. The author explains this discrepancy by the failure to take into account the phase shift upon reflection from silver. In confirmation of this point of view he indicates that the differences in thickness obtained by both

methods coincide: 11 Å in the case of optical measurements and 12.6 Å in the case of X-ray structural measurements.

One of the promising applications of multiple-beam interferometry is also the study of oscillations of the surfaces of various objects, in particular piezoelectric crystals. The application of interference methods to the study of oscillations is not

a

b

Fig. 71. Blurring of equal-thickness fringes during oscillations of the surface of piezoquartz (standing waves):
a — short standing waves, and b — long standing waves.

new227, 228. Nevertheless, multiple-beam interferometry makes it possible here as well to take a noticeable step forward. The essence of the method, as in determining relief, consists in constructing a Fabry–Perot interferometer, one of whose plates is the oscillating surface199, 222, 229, 230. In general, it is expedient to use two types of experiments. If standing waves are being studied, observation is carried out under conditions of continuous illumination. The location of the antinodes is then revealed by the blurring of the interference ...

...bands of equal thickness (Figs. 71 and 72), and the degree of blurring makes it possible to determine directly the amplitude of the oscillations. By shifting and rotating the interference bands (by changing the mutual arrangement of the plates), one can study in detail the structure of standing waves over the entire oscillating surface.

For studying the dynamics of oscillations the stroboscopic effect is used; for this purpose the light source is modulated at the frequency of a wave traveling over the surface. It should be noted that even at oscillation frequencies of the order of \(10^6\) cps and amplitudes of the order of fractions of a micron, the velocities of the oscillating surface elements are so small that the Doppler effect may be neglected. This ensures correct reproduction of the relief of the oscillating surface. If one dispenses with the auxiliary silvered plate and forms the interferometer by silvering both surfaces of the oscillating plate, then it becomes possible to study bulk waves and, in particular, the mechanism of destruction of a crystal by a wave propagating through it*).

A similar method of silvering the thin plate under investigation on both sides is used to study the relative relief of the two surfaces, and also to determine the double refraction in the plate \({}^{189}\). Indeed, the presence of double refraction leads to splitting of the bands (Figs. 42 and 73). In the case

Fig. 72. Blurring of the bands during oscillations:
\(a\) — in transmitted light,
\(b\) — in reflected light,
\(c\) — in reflected light with the zeroth-order beam cut off.

*) A fundamentally different method of interferometry, based on modulation of the interference pattern and the use of a photoelement with a subsequent narrow-band amplifier, was developed by G. S. Gorelik \({}^{269}\). Applied to the study of oscillations \({}^{270}\), this method under the conditions of two-beam (!) interferometry made it possible without difficulty to determine oscillation amplitudes with an accuracy up to \(1\) Å, whereas the accuracy of amplitude determination in the multibeam method described above did not exceed \(10\) Å.

bands of equal chromatic order, taking into account the dispersion of the refractive index and putting \(\vartheta=0\), we have:

\[ nt=\frac{\nu}{2\Delta\nu\left(1-\frac{\lambda}{n}\frac{dn}{d\lambda}\right)}, \tag{5,7} \]

where \(\Delta\nu\) is the distance between the bands of adjacent orders on the frequency scale. According to (4,12), the shift of a band \(d\lambda\), caused by a change in the refractive index \(dn\), is equal to

\[ d\lambda=\frac{\lambda}{n}\,dn. \tag{5,8} \]

Fig. 73. Splitting of bands of equal chromatic order, revealing double refraction in a mica plate.

Neglecting in (5,7) the dispersion term (which usually introduces a correction of the order of \(2\%\))\(^{189}\), we find the value of \(nt\) and, substituting this value into (5,8), obtain:

\[ dn=\frac{d\lambda}{2t\Delta\lambda}. \tag{5,9} \]

Thus, from the splitting of bands of equal chromatic order one can measure \(dn\) with an accuracy of about \(1\cdot10^{-4}\). (In particular, for mica at \(\lambda=5000\,\text{\AA}\), the value \(dn=0.0046\pm0.0001\) was measured)\(^{189}\). According to (5,8), the splitting caused by double refraction does not depend on the order of interference, as has already been noted above and as is clearly seen in Fig. 73.

The method described also makes it possible to measure the relative phase shift upon reflection from a cover film. Measurements of this kind have been carried out repeatedly\(^{42,43}\). Since the phase shift depends on the character of the polarization of the light, at oblique incidence (\(\vartheta\ne0\)) a splitting of the interference bands must occur\(^{189,231}\), the magnitude of which will be a function of the angle of incidence \(\vartheta\). An example of such splitting is shown in Fig. 74.

According to (4,8), the shift of a band caused by a change in the phase shift upon reflection is equal to

\[ dm=d\alpha_0, \tag{5,10} \]

which makes it possible\(^{189}\) to measure \(d\alpha_0\) with an accuracy of the order of \(10^{-3}\).

It remains for us to compare the possibilities of the interference technique for the study of surfaces with those of microscopy.

The linear magnification attainable interferometrically is of the order of 400,000, i.e., considerably higher than even that of an electron microscope. The resolving power of interference methods is also higher than that of the electron microscope (3–5 Å instead of 50 Å). Nevertheless, interference technique cannot replace the electron microscope. Whereas the microscope (including the electron microscope) provides magnification mainly in the plane of the object, the technique of multiple-beam interferometry gives magnification exclusively in depth; moreover, as

Fig. 74. Splitting of Newton’s rings under oblique incidence, revealing the difference in the phase shift of the \(s\)- and \(p\)-components of a light wave upon reflection from a silver film: \(a\)—without a polaroid; \(b\) and \(v\)—through polaroids oriented mutually perpendicular to one another.

we saw above, a large magnification in depth is inevitably accompanied by a small magnification in the plane of the object. Thus it is obvious that the two methods of investigation complement one another.

6. INTERFERENCE LIGHT FILTERS

In recent years, various nonabsorbing light filters have been successfully developed and have increasingly entered the practice of physical experiment, i.e., optical devices that isolate comparatively narrow spectral regions and are based not on the absorption of light, but on other optical phenomena (interference, double refraction, total internal

reflections, scattering, etc.). These devices, in contrast to absorption light filters, have the advantage that the selection of a narrow spectral transmission band occurs with considerably smaller losses of light within the transmission band. In addition, the position of the transmission band itself can, within certain limits, be shifted along the spectrum, which makes it possible to adapt the filter much better to specific experimental requirements. On the other hand, unlike monochromators, the cross section of the filtered light beam and its angular aperture may be very large, which brings them closer to absorption light filters and makes them suitable both for the study of weak radiations and for obtaining optical images in monochromatic light.

Moreover, non-absorption light filters can be made for such regions of the spectrum (for example, the infrared) where the selection of absorption filters proves very laborious or even altogether impossible. In terms of the degree of monochromatization of light, light filters of this type occupy an intermediate position between monochromators and absorption light filters.

Among the rather numerous designs proposed up to the present time, the so-called interference light filters, which are a modification of the Fabry–Perot interferometer, have proved especially simple and convenient in use.

As follows from relation (4.8), decreasing the thickness of a Fabry–Perot interferometer leads to a mutual separation of the interference bands, accompanied by their corresponding broadening (since the relative half-width of the bands and their contrast depend only on \(R\), but not on \(t\)). If the plates of the interferometer are strictly parallel to one another and the dielectric layer separating them has an optical thickness of the order of the wavelength, then the number of interference bands remaining in the visible region of the spectrum becomes insignificant. By a suitable choice of the thickness of the layer (see below), only one band may be left in the visible region of the spectrum. The interferometer is thereby transformed into a light filter possessing a comparatively narrow transmission band. The theory of such an interference light filter (see, for example, \(^{101,147,232}\), etc.) is essentially identical with the theory of the Fabry–Perot interferometer considered above*). Here we shall dwell only on certain questions specific to the use of the interferometer as a light filter.

Usually interference light filters are made in the form of a glass plate with an area of several square centi-

*) Strictly speaking, in the case \(nt \simeq \lambda\) the approximation of geometrical optics is no longer applicable. However, electrodynamic treatment leads in this case to practically identical results \(^{101,147}\).

meters, one side of which is successively coated with a semitransparent silver film, a dielectric film, and a second semitransparent layer, and then, for protection against mechanical damage, is covered with a second glass plate (Fig. 75)\(^{233—236}\).

Obviously, the decisive factor in their manufacture is the technology for depositing thin metallic and dielectric films with specified parameters, unchanged over a comparatively large area of the light filter. Precisely for this reason, only the successful development, in recent years, of the technology for depositing dielectric films has made it possible to realize such filters and to ensure their mass production.

Figure labels: dielectric \((\lambda/2)\); Ag; glass.

Fig. 75. Diagram of an interference light filter.

To ensure the calculated width of the transmission band and the correspondence of the position of this band to the prescribed wavelength, it is necessary that the thicknesses of the layers over their entire extent remain constant and have the specified value with an accuracy up to 0.5% (i.e., 10—30 Å). Achieving such accuracy requires continuous monitoring of the course of the layer-deposition process. At present a number of methods for such monitoring have been developed, based on the dependence of the properties of thin films on their thickness (see Part I), namely, the changes with film thickness in its transparency, reflection coefficient, the character of the polarization upon reflection, the phase shift of the reflected light, etc.\(^{236—240}\).

Let us turn to consideration of the optical characteristics of interference light filters.

a. Filter in transmitted light

Since the semitransparent layers in this case are separated by a dielectric with refractive index \(n_g \ne n_0\) and, generally speaking, possessing absorption, in considering the theory of interference light filters one can no longer neglect the difference between \(\vartheta\) and \(\vartheta'\) in expression (4,1). In addition, the reflection coefficient \(R\) must be multiplied by the coefficient \(q\), equal to the transparency of the dielectric interlayer\(^{232}\). In all other respects, all the relations derived above remain unchanged.

According to (4,8) and (4,29), the half-width of the transmission band is

\[ \delta\nu = \frac{1 - qR}{2\sqrt{qR}\,\pi n t \cos \vartheta'} \tag{6,1} \]

and the distance between neighboring bands

\[ \Delta \nu=\frac{1}{2nt\cos\vartheta'} . \tag{6,2} \]

The magnitude of the phase shift upon reflection from the boundary dielectric layer—silver film apparently has not yet been measured. For a qualitative estimate one may assume that it is equal to the phase shift at the glass—silver-film boundary, i.e. (see Fig. 17) \(a_0=\)

\[ =\frac{a_1+a_2}{2}\simeq 0.6. \]

Then, taking \(\vartheta'=0\) and \(nt=4900\ \text{\AA}=20\,000\ \text{cm}^{-1}\), we find the following positions of the transmission bands: first order \(\lambda=24\,500\ \text{\AA}\), second order \(\lambda=7\,000\ \text{\AA}\), third order \(\lambda=4\,083\ \text{\AA}\).

Fig. 76. Position of the transmission bands of an interference filter as a function of its optical thickness \((\chi_0=0)\).

The positions of the bands of different orders, as functions of the optical thickness of the dielectric interlayer under the assumption \(a_0=0\), are shown in Fig. 76\(^ {189}\). Thus the requirement of uniqueness of the transmission band in the visible region of the spectrum limits the interference order to 2, which in turn sharply limits from below the width of the transmission band. According to (6,1), for the present case (taking \(w=2\cdot10^{-2}\)) \(\delta\nu=204\ \text{cm}^{-1}\), i.e.

\[ \delta\lambda_{7000}\simeq 100\ \text{\AA}\quad \text{and}\quad \delta\lambda_{4083}\simeq 32\ \text{\AA}. \]

Half-widths of bands of this order are in fact observed. We give data for several interference light filters\(^ {147}\) (Table XI).

Table XI

Typical characteristics of interference light filters

\(T_1\) in %	\(T_2\) in %	\(T\) in %	\(\dfrac{I_{\max}}{I_0}\)	\(\delta\lambda\) in Å (in first order)
0.7	0.6	0.65	0.063	210
0.92	0.96	0.94	0.11	70
3.7	3.0	3.3	0.309	81
17.5	18.5	18	0.541	440

MULTIBEAM INTERFEROMETRY

Figure 77 shows the shape of the transmission bands of an interference filter in the first order for various \(R\) in the absence of absorption \((a)\) and for real silver layers \((b)\). The results of photographing a continuous spectrum through interference light filters\(^{234, 189}\) are given in Figs. 78 and 80. From the point of view of the light-filtering action, it is far more important to know not the half-width of the band, but the width of the spectral interval at whose boundaries the band intensity is equal to \(0.1 I_{\max}\). Using (4.26), we find that it is equal to \(3w\) (i.e. \(3\delta\nu\)).

Thus the effective width of the filter transmission band is \(100\text{--}300\ \text{Å}\).

To narrow the transmission band, one may use a doubled filter\(^{189, 241}\), entirely analogous to the doubled interferometer considered in Section 4B. Thus, for example,

Fig. 77. Shape of the transmission band of an interference filter. \(a\)—under the assumption \(A=0\), \(b\)—real filters with silver layers,

by combining a filter that has 1 transmission band in the visible region of the spectrum with a filter that has 21 bands in the same region, one obtains a light filter whose effective transmission band is about \(30\ \text{cm}^{-1}\) and which has very high contrast. However, if the transparency of a single filter is about \(10\text{--}30\%\), then the transparency of the doubled filter will be of the order of \(1\text{--}10\%\), i.e. the gain in monochromaticity of the radiation transmitted by the filter is obtained at the cost of a sharp decrease in the transparency of the light filter.

The loss of transparency is considerably smaller if the doubled filters are in direct contact (for example, if a compound filter is made from alternating metallic and dielectric layers deposited one on another). Secondary interference here\(^{189}\) leads to an increase of transparency by the factor

\[ \frac{1}{1-R^4}, \]

which is close to 5 for \(R=0.9\) and to 8.5 for

\(R = 0.94\), as a result of which the doubled filter absorbs not much more than a single one. However, taking into account the requirements for matching the filters (4.71), the realization of such a doubled filter requires a thoroughly developed technology for applying the layers, whereas the matching of separate filters presents no difficulty and can be carried out by a simple rotation of the filters.

Fig. 78. Transmission of interference filters in transmitted light:
a — helium spectrum; b — photograph of a continuous spectrum obtained through a second-order interference light filter \((\vartheta = 0)\); c — the same, \(\vartheta = 10^\circ\); d — the same, through an interference filter of the fourth and fifth orders \((\vartheta = 0)\); e — the same as d, but \(\vartheta = 60^\circ\); f — the same as d, but through a Polaroid oriented parallel to the plane of incidence of the rays; g — the same as d, but through a Polaroid oriented perpendicular to the plane of incidence.

Indeed, according to (4.8), the position of the transmission band depends substantially on the angle of inclination of the filter \(\vartheta\). When the filter is inclined relative to the incident beam, the transmission band shifts—

is shifted into the short-wavelength region of the spectrum,\(^{243}\) as is illustrated\(^{243}\) by Figs. 78 and 79. Within an angle of inclination of about \(10^\circ\) this shift is small; it increases sharply at larger angles and is accompanied by splitting of the transmission band owing to the difference in the phase shifts upon reflection from silver for the \(s\)- and \(p\)-components of the light beam. Simultaneously with the shift of the bands, the distance between them \((6,2)\) and their half-width \((6,1)\) also increase (cf. Fig. 78).

This same dependence of the position of the transmission band on the angle of incidence limits the aperture of the light beams being filtered.\(^{160,232}\) According to (4,53), deviation of the incident beam of light rays from the normal to the surface of the light filter by a small angle \(\vartheta\) entails a shift of the band \(d\lambda\), determined by the relation

\[ \vartheta^2 = n_2 \sqrt{\frac{2\,d\lambda}{\lambda}} = n_2 \sqrt{\frac{2\,d\nu}{\nu}}, \tag{6,3} \]

where \(n_2\) is the refractive index of the dielectric interlayer of the filter, taking into account here the difference between \(\vartheta\) and \(\vartheta'\). Accordingly (cf. (4,54)), putting \(n_2 = 1.5\), \(m_0 = 2\), \(\alpha_0 = 0\), \(\varpi = 2 \cdot 10^{-2}\), and requiring that \(d\nu \leq p\delta\nu\), we find for the maximum permissible aperture of the light beam

\[ \vartheta_{\max} = 0.15\sqrt{p} \]

and for \(p = 1\), \(\vartheta_{\max} = 8—9^\circ\).

Fig. 79. Dependence of the position of the transmission band for the \(s\)- and \(p\)-components of a light beam on the angle of its incidence on the light filter.

A distinctive variant of an interference filter was developed by A. N. Kartashev and N. M. Syromyatnikova.\(^{244}\) The filter consists of a thin sheet of mica, silvered on both sides. The authors produced filters with an area of \(2—3\ \mathrm{cm}^2\), with transparency of \(15—60\%\), and with a transmission-band half-width of \(40—140\) Å. Splitting of the bands owing to the double refraction of mica does not lead to any noticeable broadening of the transmission band.

As in the case of the interferometer, replacement of silver films by dielectric multilayer coatings\(^{128,153}\), and also coating silver films with dielectric layers that increase \(R\) and decrease \(A\), appears very promising.

Such a filter with a two-layer film of ZnS + cryolite, deposited on silver layers, had, according to Turner^155, at \(\lambda = 5380 \text{ Å}\) a transmittance

\[ \frac{I_{\max}}{I_0} = 0.26 \]

and \(\delta\lambda = 80 \text{ Å}\). Turner^161 indicates that each pair of quarter-wave layers increases the reflection by 8% and decreases the transmittance by 5%. The filter he made with four-layer coatings on silver (on each side) had, in the second order, a transmission band whose half-width was 20 Å at a transmittance of 44%. A half-width of 11 Å was obtained in the same way at a transmittance of 28%.

Polster^268 constructed a filter whose semitransparent coatings consisted of seven alternating layers of ZnS and cryolite each (without a metallic film). For the first-order band, located near 5000 Å, the transmittance of the filter was somewhat above 80% with a band half-width of about 65 Å. However, the filter had rather appreciable secondary transmission bands, shifted relative to the principal one by approximately 1000 Å in each direction.

Fig. 80. \(a\)—transmission band of an interference light filter in transmitted light; \(b\)—absorption band of the same light filter in reflected light. The relative displacement of the bands is due to the fact that in reflected light \(\vartheta \ne 0\).

b. Reflecting interference filters

The spectrum of light reflected from an interference light filter is almost complementary (see Section 4b) to the spectrum of transmitted light (Fig. 80)—in reflected light the filter has comparatively narrow absorption bands. Therefore a reflecting filter is expedient to use in cases where a selective absorbent is required. In particular, there are indications^245 of the use of reflecting filters in the infrared region of the spectrum for bolometric measurements, for imitating an absolutely black body in a narrow frequency range, and for calibrating spectrometers^246. However, in most cases the practical interest lies not in selective absorption, but in selective reflection, and it is desirable to preserve the principal advantage of reflec-

... filters—their high reflectance, several times exceeding the transparency attainable for transmitted light. Therefore, in designing reflecting filters, one widely resorts to the complete or partial reversal of bands described in Section 4Б. This is achieved by making the required semitransparent layer so thin^75,146,147,161 (several tens of angstroms) that its reflection coefficient does not exceed 10–50%. As a result, \(N_{\mathrm{eff}}\) becomes small, and the absorption bands broaden considerably, compressing the reflection bands. If the filter is intended to operate exclusively in reflected light, then the second mirror film is made opaque (in order to reduce light losses). An example of the dependence curve \(I/I_0\) for such a filter is given in Fig. 81. Let us note that the reflectance of an interference filter with one opaque mirror surface at the reflection maximum is close to the reflectance of the metal used to form this surface, but does not exceed it^161. The reflecting filters described are used chiefly in the infrared region of the spectrum and, in part, in the microwave region^146,147,161, silver often being replaced by aluminum^75 or rhodium^245. As an example, we shall mention a reflecting aluminum filter of the first order, cutting off the entire visible region of the spectrum (\(R < 2\%\)) and transmitting the infrared (for \(\lambda > 3\mu\), \(R > 0.95\)) and, in part, the ultraviolet (near \(\lambda = 0.25\mu\), \(R = 0.85\)) regions of the spectrum^75.

Fig. 81. Typical reflectance curve of an aluminum second-order light filter with partially reversed bands.

Further improvement of reflecting filters (as well as of filters for transmitted light) proceeded in two directions. On the one hand, this was the replacement of metallic films by multilayer dielectric coatings or the improvement of the optical properties of metallic films by means of such coatings. The results achieved along this path are, in the main, identical with the results considered above, obtained for filters operating in transmission. We shall note only that removal of the metallic...

films makes it possible to substantially extend the boundaries of the spectral range of applicability of interference filters. On the other hand, this is the use of laminated or layered reflecting filters[^155][^161][^241][^248]. A multiple reflecting filter is, evidently, an analogue of the Lippmann color photographic image. At the same time, reflection from a multilayer structure can be treated in the style of the theory of reflection of X-rays from a crystal lattice, leading to the Wulff–Bragg equation, with semitransparent metallic layers here playing the role of atomic planes. By choosing the thicknesses of the components forming a complex filter to be nonidentical, one can extinguish a number of reflection maxima and obtain a reflecting filter of increased selectivity[^161]. An example of the spectral reflection curve of a two-layer filter is given in Fig. 82.

Fig. 82. Dependence of $\dfrac{I}{I_0}$ for a double reflecting filter.

c. Filter using total internal reflection

The transparency of an interference filter, as follows from (4.21), is determined exclusively by the magnitude of light absorption in the semitransparent coatings. In principle, if these coatings had no absorption, one could expect a transparency of the order of 100%. In reality, the absorption of the silver layers usually used reduces the transparency to 20–40%, and the narrower the transmission band, the stronger this effect. This effect becomes especially significant in the infrared region of the spectrum, which substantially limits the possibilities of using such light filters. One of the ways of effectively overcoming this drawback is the replacement, considered above, of silver layers by multilayer dielectric coatings. Another path, to the consideration of which we now turn, is the use of the phenomenon of total internal reflection in a single-layer dielectric film.

As is known, under total internal reflection occurring at an interface with a medium having a smaller refractive index

refractive index, the light wave penetrates into the latter, and the intensity of this wave decreases with distance from the boundary according to an exponential law249. If a medium with a small refractive index forms a thin interlayer between two media with large refractive indices, then light incident on the interface at the angle of total internal reflection partially penetrates through this interlayer, and to a greater extent the smaller its thickness. Thus, an interlayer of a substance with a small refractive index, when light is incident on it at the angle of total internal reflection, will play the role of a nonabsorbing semitransparent layer, and its degree of transparency is entirely determined by the thickness of the interlayer. This makes it possible to replace the silver layers in an interference filter by interlayers of a substance with a small refractive index350. As a result, the interference filter takes the form shown in Fig. 83.

Fig. 83. Interference filter using total internal reflection.

On the base of a three-sided rectangular prism made of a material with a high refractive index (NaCl251, glass with \(n = 1.72\)161), a thin layer of a transparent dielectric (NaF351, MgF\(_2\)161) is deposited, having a considerably smaller refractive index \((n_1 < n_0)\), then a layer of a substance with a large

with refractive index \(n_2\) (AgCl\(^{251}\), ZnS\(^{161}\)), close to the refractive index of the material from which the prism is made, and, finally, a second layer of substance with refractive index \(n_1\), covered by a second prism (identical with the first), cemented with a cement whose refractive index is also equal to \(n_0\). The angle of the triangular prism is chosen depending on the magnitude of the angle of total internal reflection. In the case considered by Turner\(^{161}\), the angle

Fig. 84. Splitting of the transmission band into \(s\)- and \(p\)-components for the filter shown in Fig. 83: \(a\)—without analyzer, \(b\) and \(c\)—with an analyzer in two mutually perpendicular positions. (A two-layer AgCl film served as the analyzer.) Wavelength scale in microns.

of total internal reflection was \(53.2^\circ\), and the prism chosen was a 60-degree one. In the case shown in Fig. 83, the prism is a 30-degree prism\(^{251}\). It is not difficult to see that the position of the transmission band depends on the thickness \(t\) of the middle layer with the larger refractive index, while the half-width of the band depends on the transparency of the bounding weakly refracting layers, i.e., on their thickness. Substituting the value \(\vartheta'\) in (4.8), we obtain for the order of interference the expression:

\[ m = 2\nu t \sqrt{n_2^2 - \sin^2 \vartheta} + a_0, \]

which determines the position of the filter transmission bands; it should be borne in mind that \(a_0\) depends strongly on the angle \(\vartheta\) (see, for example,\(^{100}\)). For a rigorous theory see \(^{252}\).

As a result of the considerable difference in the phase shifts under total internal reflection for the $s$- and $p$-components, the positions of the transmission bands for these components also turn out to be noticeably displaced relative to one another; consequently the bands acquire a doublet structure, with each component corresponding to a quite definite polarization and capable of being extinguished by an appropriately oriented analyzer (Fig. 84). Fig. 85 illustrates the dependence of the positions of both components on the angle of inclination of the filter.

Let us cite some data. The filter shown in Fig. 83 and intended for the infrared region of the spectrum had dimensions $30 \times 30 \times 50\ \text{mm}^{251}$. The thickness of the sodium fluoride layers was $5.5\mu$. The thickness of the middle layer (AgCl) was $3.8\mu$. Accordingly, the transmission bands lay in the region $4.0$—$5.5\mu$, depending on the angle of incidence of the light beam. The band width was about $0.1\mu$ (with a divergence of the light beam of the order of $3^\circ$), and the distance between the bands corresponding to different polarizations was about $0.5\mu$.

Fig. 85. Dependence of the position of the $s$- and $p$-transmission bands of an interference filter using total internal reflection (Fig. 83) on the angle of incidence of the light beam.

Table XII gives filter data for the visible region of the spectrum$^{161}$.

There are indications$^{253}$ of the possibility of using a filter of this type, made of polystyrene (as the prism and as the interlayer with a high refractive index) and air (as the semitransparent interlayers), for the microwave region ($\lambda = 0.25\ \text{mm}$; transmission band about $0.25\mu$). Although theoretically the transmission-band width of the filter can be made arbitrarily small while maintaining 100% transparency, nevertheless, as follows from Table XII, in practice this proves to be far from the case. With a transparency exceeding $0.93$, it is practically impossible to obtain a transmission band of less than $50\ \text{\AA}$. The explanation for this apparently should be sought$^{161,242}$ in defects of the films. In Part I we saw that thin films are far from homogeneous in their structure, and moreover, as the thickness of the film increases, its surface becomes noticeably rougher. At the same time, the shift

Table XII

Comparison of measured and calculated data for light filters using total internal reflection

No.	Thickness of MgF₂ layers	Thickness of ZnS layer	$\lambda_{\max}$ in mμ, measured	$\lambda_{\max}$ in mμ, calculated	Transparency in %, measured	Transparency in %, calculated	$w$ in Å, measured	$w$ in Å, calculated
1	$3.0\,\dfrac{\lambda}{4}$	$3.48\,\dfrac{\lambda}{4}$	530 460	482 394	93 93	100 100	120 66	110 70
2	$3.0\,\dfrac{\lambda}{4}$	$6.45\,\dfrac{\lambda}{4}$	508 463	487 433	90 90	100 100	50 49	57 42
3	$4.5\,\dfrac{\lambda}{4}$	$4.86\,\dfrac{\lambda}{4}$	633 540	673 556	93 93	100 100	75 60	— —
4	$5.0\,\dfrac{\lambda}{4}$	$5.51\,\dfrac{\lambda}{4}$	654 455	623 411	93 1	100 100	50 —	— —
5	$5.2\,\dfrac{\lambda}{4}$	$3.55\,\dfrac{\lambda}{4}$	511	487	12	100	30	12
6	$5.1\,\dfrac{\lambda}{4}$	$6.43\,\dfrac{\lambda}{4}$	502	478	4	100	30	6

of the transmission band $d\lambda$ is connected with the change in the optical thickness of the interferometer by relation (4,12)^{252}. Consequently, for a first-order band a deviation in optical thickness of 10 Å will cause a shift of the transmission band by 20 Å. The experimentally observed limit of decrease of the half-width of the bands (30 Å)^{161} evidently then indicates surface inhomogeneities of the film of the order of 5–10 Å. This phenomenon may, in particular, serve for the study of inhomogeneities in thin films^{161, 252, 254*}.

For bands of equal inclination we have^{254}:

\[ dt=\frac{t}{\lambda}\,\frac{d\lambda}{d\vartheta}\,d\vartheta, \]

whence, by measuring the band shift $d\vartheta$ and knowing $\dfrac{d\lambda}{d\vartheta}$, one can find the vertical dimensions of the surface inhomogeneities. Measurements^{254} gave a value of $\dfrac{d\lambda}{d\vartheta}$ for the second-order band close to $1\,\dfrac{\text{Å}}{\text{angular minute}}$. Since a displacement of the band by $1'$ proves to be measurable, it becomes possible to detect and measure surface defects with a depth down to $1/2$ Å.

* One should also bear in mind the shift of rays under total internal reflection^{249}.

The greatest drawback of the filter described is undoubtedly the doublet structure of the transmission bands. This drawback, however, can to a certain extent be eliminated[^255] by using, as an intermediate layer with a high refractive index, a birefringent substance. Such a possibility arises in connection with the development of techniques for depositing anisotropic layers of organic substances by evaporation in vacuum. By selecting the material appropriately, one can make it so that, at a certain angle of incidence, the difference in the phase shift upon reflection of the \(s\)- and \(p\)-components is compensated by double refraction, and both transmission bands (corresponding to the two directions of polarization) reunite into a single band, thereby doubling the transparency of the filter. A drawback of such a device is the breakdown of compensation when the angle of incidence changes, as a result of which the bands separate again; this appears as a sharp blurring of the transmission band when wide-angle beams are used. The author points out that it is technically possible to make a filter with a transmission-band half-width of the order of \(10\,\text{\AA}\) and a transparency of about \(70\)—\(80\%\). However, the filter made by the author had a transmission band of \(150\,\text{\AA}\) and a transparency of \(20\%\).

7. INTERFERENCE-POLARIZATION LIGHT FILTERS

In terms of their properties, interference-polarization light filters occupy a position intermediate between monochromators and the interference light filters described above. Like the latter, they make it possible to obtain monochromatized light beams of comparatively large cross section (more than \(30\) mm in diameter)[^256],[^257] and make it possible to photograph extended objects in monochromatic light[^256],[^258]—[^260]. At the same time, the degree of monochromatization attainable with their aid is significantly higher[^256],[^258],[^261], which brings them closer to monochromators[^260].

However, while combining the optical advantages of both, interference-polarization light filters prove to be rather complicated and expensive devices, requiring careful maintenance and adjustment[^256],[^258],[^261]. In essence, they are a set of successive birefringent polarization interferometers with properly chosen values of the phase shift of the interfering rays. Therefore each of the known, still not numerous, light filters of this type has been an object of special design work and possesses a number of individual features.

Usually, as a polarization interferometer forming a separate element (or cascade) of a light filter, one uses a plane-parallel birefringent plate bounded on both sides by polarizers. Let us denote by \(\alpha\), \(\beta\), and \(\gamma\) the directions of the principal axes of the ellipsoid of refractive indices, and by \(n_\alpha\), \(n_\beta\), and \(n_\gamma\) the values of the refractive indices corresponding to these axes \((n_\alpha < n_\beta < n_\gamma)\).* Suppose now that the plate is cut perpendicular to the \(\beta\) axis and that the light wave incident on it through the polarizer \(P_0\) (Fig. 86) along the \(\beta\) axis has amplitude \(A_0\) and is linearly polarized in a plane making angles of \(45^\circ\) with the directions \(\alpha\) and \(\gamma\). On entering the plate, it splits into two mutually perpendicular (along the \(\alpha\) and \(\gamma\) axes) polarized components with amplitudes \(A_0/\sqrt{2}\), which then propagate with phase velocities \(c/n_\alpha\) and \(c/n_\gamma\). At the exit from the plate, where both components are again reunited and interfere, they acquire a relative phase shift

\[ \varphi = \frac{2\pi n l}{\lambda}, \tag{7.1} \]

where

\[ n = n_\gamma - n_\alpha \tag{7.2} \]

and \(l\) is the thickness of the plate. If behind the plate there is a polarizer \(P_1\), oriented in the same way as \(P_0\), then the amplitude of the resultant light wave passing through it will be equal to

\[ A = A_0 \cos \frac{\varphi}{2} = A_0 \cos \frac{\pi n l}{\lambda}. \tag{7.3} \]

Fig. 86. Diagram of an interference-polarization light filter.

Consequently, the energy transmittance of an element consisting of a crystalline plate bounded by two corresponding—

\[ \text{* In the case of a uniaxial crystal, in all subsequent formulas one should make the substitution:} \]

\[ n_\alpha = n_\omega,\qquad n_\beta = n_\omega \quad \text{and} \quad n_\gamma = n_\varepsilon \quad \text{for } n_\varepsilon > n_\omega \]

or

\[ n_\alpha = n_\varepsilon,\qquad n_\beta = n_\omega \quad \text{and} \quad n_\gamma = n_\varepsilon \quad \text{for } n_\varepsilon < n_\omega, \]

where \(n_\omega\) and \(n_\varepsilon\) are the refractive indices for the ordinary and extraordinary rays.

by identically oriented polarizers,

\[ T=\cos^2 \pi m, \tag{7,4} \]

where the order of interference is

\[ m=\frac{nl}{\lambda}. \tag{7,5} \]

Thus the spectrum of light passing through the element will consist of alternating bright and dark bands (Fig. 88), the position of the bright bands being determined by the requirement that \(m\) be an integer, and the distance between them by

\[ \Delta \lambda=\frac{\lambda^2}{nl}=\frac{\lambda}{m}. \tag{7,6} \]

By arranging in succession a series of uniformly oriented elements\(^*\), selected so that the thicknesses of the crystalline plates \(l\) increase in a geometric progression with common ratio equal to two:

\[ l_k=2l_{k-1}, \tag{7,7} \]

we obtain a light filter whose transparency is determined by the expression

\[ T=\prod_{k=1}^{k=N} T_k=\prod_{k=1}^{k=N}\cos^2 2^{k-1}\pi m_1, \tag{7,8} \]

where \(N\) is the number of cascades and \(m_1\) is the order of interference in the thinnest plate. In this case the relative order of arrangement of the cascades is immaterial. Fig. 87 schematically shows a light filter composed of six cascades, and gives the transmission curves both for each of the elements separately and for the entire light filter as a whole \(^{256}\). Fig. 88 gives photographs obtained by photographing a continuous spectrum through separate successive cascades of an interference-polarization light filter and through light filters composed of different numbers of elements \(^{234}\). Since in the visible region there are several transmission bands, complete monochromatization requires the introduction of an additional broadband (for example, absorption) light filter.

By a simple transformation, expression (7,8) can be brought to the form \(^{256,262}\)

\[ T=\frac{\sin^2 2^N \pi m_1}{2^{2N}\sin^2 \pi m_1}, \tag{7,9} \]

which corresponds to the interference of \(2^N\) light beams with equal (in contrast to an interference filter) amplitudes \(\dfrac{A_0}{2^N}\)

\(^*\) In this case the output polarizer of the preceding element simultaneously performs the role of the input polarizer of the following element.

and phases increasing in an arithmetic progression: \(0,\ \varphi_1,\ 2\varphi_1,\ldots,(2^N-1)\varphi_1\)^[356]. In other words, the interference-polarization filter is analogous to a diffraction grating (disregarding the diffraction pattern from a single slit) with a number of rulings equal to \(2^N\).

As follows from (7.9), the transparency of the light filter for correspondingly polarized light is equal to 100%. The distance

Fig. 87. \(a\)—an interference-polarization filter consisting of six elements; \(b\)—the transparency of the individual elements and of the whole filter as a whole.

between the transparency maxima is determined by expression (7.6), in which \(m\) is to be understood as the order of interference in the thinnest plate \((m=m_1)\). If dispersion is taken into account, then in (7.6) an additional factor appears:

\[ \Delta \lambda=\frac{\lambda^2}{n l_1}\, \frac{1}{\left(1-\frac{\lambda}{n}\frac{\partial n}{\partial \lambda}\right)}, \tag{7.10} \]

and this dispersion factor is, generally speaking, close to \(0.9\) and changes comparatively little with variation of \(\lambda\)^[356].

To determine the width of a maximum we shall use the same method as in Section 4e. If \(2^N\) is sufficiently large, then

Fig. 88. Transmission bands of six successive elements of an interferential-polarization light filter (1–6) and of filters composed of a series of successive cascades. At the bottom is shown the dependence of the band positions of a six-cascade filter on temperature.

the maxima are narrow and one may put

\[ m_1=(m_1)_0+\mu, \tag{7,11} \]

where \((m_1)_0\) is an integer and \(\mu \ll 1\). Then, in the immediate vicinity of the maximum,

\[ T=\frac{\sin^2 2^N \pi\mu}{2^{2N}\pi^2\mu^2}. \tag{7,12} \]

Expression (7,12) becomes zero under the condition \(2^N\mu_0=1\), whence the relative width of the maximum is

\[ w=2\mu_0=\frac{1}{2^{N-1}}, \tag{7,13} \]

or, on the wavelength scale, according to (7,10),

\[ \delta\lambda=\frac{\lambda^2}{2^{N-1}n l_1} \left( \frac{1}{1-\frac{\lambda}{n}\frac{dn}{d\lambda}} \right). \tag{7,14} \]

Between the principal maxima, as follows from (7,12), there is a series of secondary maxima, which occur under the condition
\(2^N\pi\mu=(2q+1)\dfrac{\pi}{2}\) \((q=1,2,\ldots)\). Substituting this value into (7,12), we find the transmittance of the light filter at the secondary maximum with number \(q\):

\[ T_q=\left(\frac{2}{(2q+1)\pi}\right)^2. \tag{7,15} \]

Consequently, the intensity of the secondary transmission maxima rapidly decreases as their number increases:

\[ T_q=\frac{1}{22},\quad \frac{1}{61},\quad \frac{1}{121},\quad \frac{1}{200},\quad \frac{1}{238},\quad \text{etc.} \]

(see Fig. 87).

All the calculations presented above referred to the case in which there are no light losses either by absorption or by reflection at the numerous boundaries of the crystalline plates and polarizers. It is not difficult to see that the presence of absorption does not affect the shape and position of the maxima, introducing into expression (7,8) only a constant factor. In practice this factor proves to be considerably less than unity, and the development of means for its possible increase is one of the most important problems in the design of a light filter. Reflection losses can be weakened to a considerable extent by using immersion or by cementing the individual elements of the filter\(^{256,261}\). Likewise, absorption in the crystalline plates can be made negligible by an appropriate choice of the material for their manufacture (quartz, Icelandic

... spar)*) ^{256, 258, 261, 263}. The most substantial are losses of light in the polarizers. For purely design-related reasons (see below), the use of polarizing prisms proves undesirable. Therefore, as polarizers one usually resorts to Polaroids, which, as is known, possess strong absorption, which becomes especially significant if one takes into account the need for a large number of polarizers installed in succession. Thus an increase in the number of cascades, necessary for narrowing the pass band, inevitably leads to a decrease in the transparency of the light filter. Thus, filters with a pass band of 20–40 Å have a transparency of the order of 20–30% ^{256, 257, 264}; a filter with a pass band of 2–3 Å (with polarizing prisms) ^{256} has a transparency of 10–25%, while the transparency of a filter with a pass band of 0.6 Å is only 2% ^{260}.

The spectral region of applicability of interference-polarization light filters is evidently limited by two factors: the transparency of the elements forming it and the possibility of using efficient polarizers. Since Polaroids are suitable only in the visible region of the spectrum ^{176}, in the ultraviolet and near infrared regions the use of polarizing prisms or layered polarizers is unavoidable.

Let us estimate the requirements on the accuracy of manufacture of the crystalline plates. A shift of the wavelength corresponding to the transmission maximum is connected with a change in the thickness of the plate by the relation \(d \ln \lambda = d \ln l\). Requiring that \(d\lambda \ll p\,\delta\lambda\), we obtain, according to (7.14), without taking account of the dispersion term,

\[ dl \ll \frac{2p\lambda}{n}. \]

Putting \(p=0.1\), \(\lambda=5\cdot 10^{-5}\), and \(n=0.02\), we have \(dl \ll 5\mu\), which does not present a serious difficulty for manufacture.

Temperature has an appreciable influence on the position of the filter pass band; changes in temperature affect both the dimensions of the crystalline plates and their refractive indices.

According to (7.1),

\[ d\ln\varphi = \left( \frac{\partial \ln n}{\partial t} + \frac{\partial \ln l}{\partial t} \right)dt + \frac{\partial \ln n}{\partial \lambda}\,d\lambda - d\ln\lambda, \tag{7.16} \]

whence, putting \(d\ln\varphi=0\), we find:

\[ d\ln\lambda = \left( \frac{\partial \ln n}{\partial t} + \frac{\partial \ln l}{\partial t} \right) \left( \frac{1}{1-\frac{\lambda}{n}\frac{\partial n}{\partial \lambda}} \right)dt. \tag{7.17} \]

*) The importance of such a choice of material is discussed in detail in work ^{263}: the light filter constructed by the authors, which had muscovite plates as its crystalline plates, proved to be ineffective precisely because of strong absorption of light in the latter. It should be noted that obtaining a narrow-band light filter requires a considerable thickness of birefringent substance.

Measurements have shown\(^{256}\) that for quartz \(\dfrac{d\ln\lambda}{dt}=-1.04\cdot10^{-4}\), and for Iceland spar \(\dfrac{d\ln\lambda}{dt}=0.56\cdot10^{-4}\). Thus a change in temperature by \(1^\circ\mathrm{C}\) causes a displacement of the transmission band of the filter by \(0.4—0.7\,\text{\AA}\), depending on its design\(^{256,258,260,261,267}\) (Fig. 88). On the one hand, this makes it necessary to regulate carefully the temperature of the light filter; on the other hand, it makes it possible to shift arbitrarily the position of its transmission band within one or two tens of angstroms.

Fig. 89.

Our discussion has concerned a beam of parallel rays incident on the filter along the \(\beta\) axis. In the case of obliquely incident light beams, the picture becomes considerably more complicated. If a light ray, polarized in a plane making an angle of \(45^\circ\) with the \(\alpha\) axis, is incident on a crystalline plate in a direction determined by the polar angles \(\vartheta,\ \psi\) (Fig. 89), then the relative phase shift acquired by the alternatively polarized components upon emergence from the plate, in the first approximation\(^{261}\), is equal to:

\[ \varphi=\frac{2\pi nl}{\lambda}\left[1+\vartheta^{2}K\left(\frac{\cos^{2}\psi}{n_{\gamma}}-\frac{\sin^{2}\psi}{n_{\alpha}}\right)\right], \tag{7,18} \]

where

\[ K=\frac{n_{\alpha}n_{\gamma}-n_{\beta}^{2}}{2\left(n_{\gamma}-n_{\alpha}\right)n_{\beta}^{2}}. \tag{7,19} \]

Thus the isochromats are hyperbolas whose axes are directed along the \(a\) and \(\gamma\) axes, while the directions of the asymptotes are determined by the relation \(\tg^{2}\psi=n_{\alpha}/n_{\gamma}\) (Fig. 90, a and b). It is obvious that the largest permissible spread of the light beam is determined by the largest admissible value of \(\varphi-\varphi_{0}\) for the thickest element of the filter \(\left(\varphi_{0}=\dfrac{2\pi n l_{N}}{\lambda}\right)\). Taking into account that \(d\lambda=\lambda\dfrac{d\varphi}{\varphi}\), and requiring that \(d\lambda\leq p\,\delta\lambda\), we find, according to (7,18) and (7,14):

\[ \vartheta\leq \sqrt{ \frac{p\lambda}{ 2^{N-1} n l_{1}K\left(\dfrac{\cos^{2}\psi}{n_{\gamma}}-\dfrac{\sin^{2}\psi}{n_{\alpha}}\right) } }. \tag{7,20} \]

(neglecting the dispersion term in (7,14) and the term with \(\vartheta^3\) in the expression for \(\varphi\)). Since for materials with a sufficiently small \(K\), apparently unknown^{261}, this angle turns out to be very small \((\sim 1^\circ)^{261,266}\). However, it can be greatly increased by making the filter more complicated. This is achieved in a number of ways.

1. Each of the crystalline plates is cut in half, and its parts are rotated (together with the polarizers) relative to one another about the \(\beta\) axis by \(90^\circ\). Between the parts a quarter-wave plate is introduced, whose axis makes an angle of \(45^\circ\) with the \(\alpha\) axes of both parts of the block. Then the relative phase shift of the two components is equal to

Fig. 90. Isochromats. \(a\)—quartz plate, \(l=65.8\) mm, \(\lambda=5460\) Å; \(b\)—Iceland spar plate, \(l=3.63\) mm, \(\lambda=8460\) Å; \(v\) and \(g\)—both plates superposed one on another (\(v\)—\(\lambda=5460\) Å, \(g\)—\(\lambda=4358\) Å).

\[ \varphi=\frac{2\pi nl}{\lambda}\left[1+\vartheta^2\frac{K}{2}\left(\frac{1}{n_\gamma}-\frac{1}{n_\alpha}\right)\right], \tag{7,21} \]

and, consequently, the admissible aperture of the light beam increases sharply. However, such a device is suitable only for a narrow range of wavelengths, owing to dispersion in the quarter-wave plate.

1. The element is composed of two parts made of materials with different signs of \(K\). If the \(\alpha\) axes in both parts of the element are parallel and the thicknesses are selected so that the relation

\[ \frac{n'l'}{n''l''} = - \frac{ K''\left(\dfrac{1}{n_\gamma''}-\dfrac{1}{n_\alpha''}\right) }{ K'\left(\dfrac{1}{n_\gamma'}-\dfrac{1}{n_\alpha'}\right) }, \tag{7,22} \]

where the quantities pertaining to the first and second parts of the element are denoted by single and double primes, respectively, then

\[ \varphi = 2\pi\left[ \frac{n'l' + n''l''}{\lambda} + \vartheta^2 \left( \frac{n'l'K'}{n'_{\gamma}} + \frac{n''l''K''}{n''_{\alpha}} \right) \right]. \tag{7.23} \]

Since \(K'\) and \(K''\) have different signs, the term with \(\vartheta^2\) can be made small for a wide range of wavelengths. Isochromats for an element composed of plates of quartz and Iceland spar of the corresponding thicknesses are shown in Fig. 90, \(b\) and \(c\).

1. Each element is composed of three crystalline plates, two of which, made of the same material (for example, spar), are rotated with respect to one another about the axis \(\beta\) by \(90^\circ\), while the third—made of a material with the opposite sign of \(K\) (for example, quartz)—is oriented so that its axis \(\alpha\) is parallel to the axis \(\alpha\) of one of the first plates. Then, by choosing the thicknesses, one can always make \(\varphi\) independent of \(\vartheta\) to the accuracy with which equation (7.18) is satisfied.

Indeed, in this case the relative phase shift is equal to

\[ \varphi = 2\pi \frac{n'l'}{\lambda} \left[ 1+\vartheta^2 K' \left( \frac{\cos^2\psi}{n'_{\gamma}} - \frac{\sin^2\psi}{n'_{\alpha}} \right) \right] + \]

\[ + 2\pi \frac{n''l''}{\lambda} \left[ 1+\vartheta^2 K'' \left( \frac{\cos^2\psi}{n''_{\gamma}} - \frac{\sin^2\psi}{n''_{\alpha}} \right) \right] - \]

\[ - 2\pi \frac{n'''l'''}{\lambda} \left[ 1+\vartheta^2 K''' \left( \frac{\sin^2\psi}{n'''_{\gamma}} - \frac{\cos^2\psi}{n'''_{\alpha}} \right) \right]. \tag{7.24} \]

Requiring that the coefficient of \(\vartheta^2\) vanish independently of \(\psi\) and that \(n'l' + n''l'' - n'''l''' = nl\), we find:

\[ \left. \begin{aligned} n'l' &= \frac{nl}{D}\, K'''^{\,2} \left( \frac{1}{n_{\alpha}^{\prime\prime\,2}} - \frac{1}{n_{\gamma}^{\prime\prime\,2}} \right), \\[6pt] n'''l''' &= \frac{nl}{D}\, K'K''' \left( \frac{1}{n'_{\gamma}n''_{\gamma}} - \frac{1}{n'_{\alpha}n''_{\alpha}} \right), \\[6pt] n''l'' &= \frac{nl}{D}\, K'K'' \left( \frac{1}{n'_{\alpha}n''_{\gamma}} - \frac{1}{n'_{\gamma}n''_{\gamma}} \right), \end{aligned} \right\} \tag{7.25} \]

where

\[ D= \begin{vmatrix} \dfrac{K'}{n'_{\gamma}} & \dfrac{K''}{n''_{\alpha}} & \dfrac{K'''}{n'''_{\gamma}} \\[8pt] \dfrac{K'}{n'_{\alpha}} & \dfrac{K''}{n''_{\gamma}} & \dfrac{K'''}{n'''_{\alpha}} \\[8pt] 1 & -1 & 1 \end{vmatrix}. \tag{7.26} \]

(An additional condition \(n_a'n_\gamma'' > n_\gamma'n_a''\) can always be satisfied.)

Strictly speaking, compensation takes place only for a definite wavelength. In practice, however, it is carried out for a comparatively broad spectral range. Let us note that such a complication is necessary only in the thickest elements, since the transmission bands of thin cascades are so broad that their broadening due to the divergence of the light beam is of no significance.

Further, it proves possible to reduce by half the number of polarizers (and, consequently, the absorption) by artificially combining two consecutive cascades into one. For this purpose an element is made up of three plates oriented as shown in Fig. 91. As is not difficult to show\(^{261}\), the transparency of such a composite element for light polarized in the plane \(r\), if the analyzer is parallel to the polarizer, is

\[ T_r = \]

\[ = 1-\sin^2 2\pi \frac{n'l'}{\lambda}\sin^2 \pi \frac{n'''l'''}{\lambda}. \tag{7,27} \]

and if the analyzer is crossed with the polarizer:

\[ T_s = \sin^2 2\pi \frac{n'l'}{\lambda}\sin^2 \pi \frac{n'''l'''}{\lambda}. \tag{7,28} \]

Fig. 91. Diagram of a doubled split element.

Thus, in the case of crossed polarizers the composite element is equivalent to two simple elements and has an aperture characteristic coinciding with that considered above in item 1. In this case one of the elements turns out to be split into two parts, between which the second element is wedged.

Since an interference-polarization filter is a complex and expensive instrument, the possibility of varying the position of the transmission band over wide limits becomes important. Temperature control proves convenient only for a narrow wavelength interval\(^{256,258,261}\). When working in parallel beams, small shifts of the transmission band can also be achieved by changing the angle of incidence.

rays[^258], if, of course, the filter is not designed as a wide-angle one. From the theoretical point of view, the most perfect method of varying the position of the pass band is the use of elements of variable thickness[^256], i.e., the formation of an element from two wedges, like a Babinet compensator. Fig. 92 shows photographs of three successive elements of a filter constructed by Lyot[^256], consisting of two plates made of spar and quartz; moreover, the latter, in turn, consists of two wedge-shaped parts, which makes it possible to vary its thickness.

Fig. 92. Elements of variable thickness, consisting of two quartz wedges and a spar plate (to make the filter wide-angle).

Fig. 93 shows the scheme of a filter consisting of nine such elements[^256], divided into three independent groups. In practice, however, the use of movable wedges proves inconvenient because of the high requirements imposed on the mechanical system (preservation of strict agreement of the thicknesses of all elements).

The second possibility consists in providing each element with a compensator introducing a small, strictly controlled phase difference \(2\pi \xi\)1. In this case the transparency of the light filter is described by the expression:

\[ T=\prod_{k=1}^{k=N}\cos^{2}\pi\,(m_k+\xi_k). \tag{7.29} \]

If \(\xi\) can be varied within the limits from \(-1/2\) to \(+1/2\), then the filter can be tuned to any wavelength. Such com-

compensators can be implemented either in the form of Babinet compensators (with wedges)\(^{261}\), or by using photoelastic phenomena or electro- and magneto-optical phenomena\(^{262}\). If, in this case, \(\xi\) depends on the wavelength, then the agreement of the various elements is disturbed, which is manifested, in particular, in the appearance of stray light. In practice, however, this effect proves negligibly small over a sufficiently wide range of wavelengths\(^{261}\).

Another method is also possible\(^{261}\). Suppose that between two quarter-wave plates, whose \(\gamma\)-axis directions coincide and make an angle of \(45^\circ\) with the plane of polarization of the light wave, a half-wave plate is placed. Then, if the \(\gamma\)-axis of the middle plate is turned through an angle \(\psi\), such a system introduces a phase shift

\[ 2\pi \xi = 4\psi . \tag{7,30} \]

Let us note that the independence of \(\xi\) from \(\lambda\) following from (7,30) is illusory, since dispersion in the compensators is not taken into account.

An analogous result can be achieved if, at the input of the cascade, a quarter-wave plate is installed and rigidly fastened both to the working crystalline plate and to the output polarizer. If the angle of rotation of such a composite element relative to the preceding one is \(\psi\), then its transmittance is

\[ T_k = \cos^3(\pi m - \psi). \tag{7,31} \]

Forming a filter from such composite elements, we obtain for its transmittance the expression

\[ T = \prod_{k=1}^{k=N} \cos^3 2^{k-1}(\pi m_1 - \psi_1), \tag{7,32} \]

where \(\psi_1\) is the angle of rotation of the first element relative to the input polarizer, and the rotation of each subsequent element relative to the preceding one is

\[ \psi_k = 2\psi_{k-1}. \tag{7,33} \]

Such a relation between the rotation angles imposes minimal requirements on the mechanical system.

Finally, there is the possibility, especially for broadband light filters, of using as their elements, instead of crystalline plates, layers of substances possessing induced anisotropy (for example, as a result of the application of electric or magnetic fields). This, like the electro-optical compensators mentioned above, makes it possible to carry out practically inertia-free adjustment of the position of the transmission band and finds application in color-television technology\(^{265}\).

Above we considered the possibilities for increasing the permissible aperture of the light beam and showed that it can be made sufficiently large. Therefore, in practice the aperture of the light beam is limited not by the factors discussed, but by the geometrical parameters of the instrument. Since \(n\) is usually small (\(0.009\) for quartz and gypsum, \(0.017\) for Iceland spar), the elements for any sufficiently narrow-band filter turn out to be very long, and the total length of the filter reaches more than

Fig. 93. Filter with nine elements of variable thickness. \(P_i\)—polarizing prisms, \(L_i\)—lenses, \(O_i\)—objectives.

half a meter\(^{256}\) (Fig. 93). At the same time, the area of the filter cannot be very large, owing to the limited possibilities in selecting material for its manufacture. In this case, a considerable share of the filter length is accounted for by the polarizing prisms\(^{256}\). Thus, in order to increase the working aperture of the filter one has to take the path of replacing polarizing prisms by polarizing films, despite the sharp decrease in the transparency of the light filter. From this point of view, the use\(^{261}\) of thin-film polarizers of the type described in Part I appears very promising. A distinctive solution of this problem was given by Lyot\(^{256}\), who replaced the polarizing prisms with birefringent

...prisms of the Rochon-prism type, as a result of which several separated images are obtained simultaneously (Fig. 94), the number of which depends on the number of cascades. Two of them (the outer ones) will be monochromatized, and the transparency of the light filter for them will be described by formula (7.8). The intermediate images, formed as a result of the superposition of coherent light beams that have undergone the same number of deviations in different prisms, have a complex spectral composition. By geometrically separating, at the output of the light filter, the undeviated image from the others, we obtain a light filter possessing three principal advantages. First, the birefringent prisms are considerably shorter than polarizing ones, which leads to a corresponding reduction

Fig. 94. Filter without polarizers, giving a set of separated images.

in the length of the filter and makes it little different in size from a filter with polaroids. Second, the filtered light is completely emitted from the filter, the consequence of which is an almost complete absence of stray light, always present in light filters of the usual type. Third, this filter does not require an input polarizer and permits measurements to be made in natural light. In the described construction of such a light filter, the angular separation of neighboring images was \(3^\circ07'\). Its transparency was equal to 40% at a passband width of \(3\,\text{\AA}\).

Further improvement of interference-polarization filters is also possible in the direction of replacing crystalline plates by polarization interferometers analogous to the Michelson interferometer, but with a polarizing beam-splitting plate. Their advantage is the large value (and easy adjustability) of \(\varphi\) at comparatively small geometrical dimensions. The calculated data for one of the filters of this type are as follows\(^{281}\): the filter consists of four interferometers creating path differences of 245,760, 122,880, 61,440, and 30,720 wavelengths, ten crystalline elements, and four polarizing prisms. At \(\lambda = 5000\,\text{\AA}\) the passband width is \(0.01\,\text{\AA}\); the angular aperture of the light beam is not more than 0.0012 radian; the transparency is approximately

254  G. V. ROSENBERG

1000 times greater than the transparency of monochromators with the same resolving power.

In conclusion, let us note that up to now the principal field of application of interference-polarization light filters has remained astrophysical research and, in particular, studies of solar activity ^256,260,266,267. This is due primarily to the fact that they make it possible to photograph extended objects whose monochromatic brightness amounts to only 0.03% of the brightness of the background of white radiation.

Figure 95 shows frames from the first Soviet motion picture showing the motion of solar prominences, filmed in monochromatic light by A. B. Severnyi and A. B. Gil’varg with the aid of a filter constructed by them ^266. At present, similar

Fig. 95. Photographs of solar prominences obtained outside an eclipse with the aid of an interference-polarization light filter.

light filters are being used to equip chromospheric telescopes for the Solar Service of the USSR ^260. Recently G. A. Shain and V. F. Gaze ^259 have substantially extended their field of application, showing that with the aid of interference-polarization light filters, when photographing nebulae, one can separate their dust and gas components.

Under laboratory conditions, filters of this type, as yet, apparently have not found wide application.

CITED LITERATURE (continued)*)

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1. In a filter with split elements, it is necessary in this case to preserve equality of the phase shifts in both halves of the split cascade and, consequently, each half must be provided with its own compensator. ↩