PHOTOGRAPHIC METHOD FOR OBTAINING LINES OF EQUAL INTENSITY IN AN INTERFERENCE PATTERN*

The accuracy attainable in measurements by interferometric methods is determined mainly by the possibilities of determining the position and shape of the interference fringes. With ordinary visual and photographic methods of observation, the position of an interference fringe is determined with an error of the order of 0.1 of the fringe width. To increase the accuracy of measurements, one may proceed further in two ways. The first way consists in narrowing the fringes relative to the distance between them. This is achieved in multiple-beam interference devices and makes it possible, under favorable conditions, to obtain an increase in accuracy by 2–3 orders of magnitude. However, in a number of measurement problems, for example in interferometric measurement of angles, multiple-beam interferometers are inapplicable. It then becomes necessary to take the other path—seeking methods for more precise location of broad two-beam fringes. A well-known success in this direction can be achieved by using photoelectric recording methods. However, for a number of reasons, which we shall not discuss here, the direct use of photoelectric receivers has not yet led to a significant—

*) E. Lau, J. Rienitz, C. Roose, Feingerätetechnik 2, No. 3, 101 (1953).

to a considerable increase in accuracy. Fundamentally new possibilities along this path were recently opened by G. S. Gorelik and Bernstein*) as a result of combining photoelectric methods of recording with modulation of the interference pattern. Although such a technique makes it possible to obtain record-high accuracies, which do not yield to the accuracies of multi-beam devices, the complexity of the apparatus and the need to modulate the interference pattern substantially limit the range of its application. Therefore the problem of increasing the accuracy of measuring ordinary two-beam interferograms (in particular, photographs of interference patterns) remains very topical.

From this point of view, the method described below for obtaining lines of equal intensity, developed by the authors of the paper under review, is of some interest.

In two-beam interferometry the form of the fringes is determined by the simple relation

\[ I=\cos^2\alpha=\frac{1+\cos 2\alpha}{2}, \]

where \(\alpha\) is the phase shift between the interfering rays. Thus, the intensity varies according to a strictly sinusoidal law with half the period. Suppose now that we have made a positive and a negative of a photographic image of one and the same interference pattern, having the same photographic density. If the characteristic curve of the photographic emulsion were strictly rectilinear, then, with accurate superposition of the positive and negative on one another, we would obtain a field of uniform density, i.e. the complete disappearance of the interference pattern. In reality the characteristic curve is not linear. Therefore, when the positive and negative are superposed, the interference pattern does not disappear completely—comparatively narrow lines of increased transparency remain, corresponding, as is easy to see, to the lines of equal intensity on the original interference pattern.

Fig. 1. Intersecting interference fringes (enlarged).

If, by contact printing, a photographic image of these lines is obtained and the process of making and mutually superposing the positive and negative is again repeated, the sharpness of the equal-intensity lines will increase again—the lines will prove incomparably narrower than the original interference fringes and will reproduce their form with great accuracy (but not their position).

Thus the purely photographic method described makes it possible to increase by 2–3 orders of magnitude the accuracy of determining the form (kinks, bends, etc.) of two-beam interference fringes photographed in the usual way on a photographic plate. If the negative and positive have different photographic densities, then the equal-intensity lines will also be obtained, but they will be located in other places.

The authors used this method to check the sinusoidality of the form of two-beam interference fringes. If one and the same photographic plate is exposed twice to one and the same interference pattern, and on the second exposure the plate is turned through \(90^\circ\), then two systems of intersecting fringes are obtained (Fig. 1).

The lines of equal intensity in this case will correspond to the condition \(\cos 2\alpha+\cos 2\beta=\mathrm{const}\), where \(\alpha\) and \(\beta\) are the relative phase shifts

) See, for example, UFN 47, 631 (1952) and 49*, 631 (1953).

interfering rays for two intersecting systems of interference fringes at the given point (Figs. 2 and 3).

Fig. 2. Theoretically calculated arrangement of lines of equal intensity for the case of Fig. 1.

Fig. 3. Theoretically calculated lines of equal intensity for the case of Fig. 1.

In Figs. 4–7 are shown the lines of equal intensity of the first (Fig. 4), second (Figs. 5 and 6), and third (Fig. 7) order, i.e. those obtained by one-, two-, and threefold repetition of the process described. From the figures

Fig. 4. Lines of equal intensity of the first order, obtained from Fig. 1.

Fig. 5. Lines of equal intensity of the second order, obtained from Fig. 4.

Fig. 6. Lines of equal intensity of the second order, obtained from Fig. 4 with other optical densities of the negative and positive.

it is clearly visible how sharp these lines become and how well they reveal the inhomogeneities of the interference pattern. The original interference pattern (Fig. 1) was obtained with the aid of a Krug–Lau interference microscope in the light of the mercury line \(\lambda = 5461\) Å.

In Figs. 8 and 9 the projection superposition of these lines on the diagram of Fig. 3 is shown. The excellent agreement of the photographs with the precomputed

the diagram indicates that the shape of the interference fringes corresponds, with a high degree of accuracy, to a sinusoid.

Fig. 7. Lines of equal density of the third order, obtained from Fig. 6.

Fig. 8. Superposition of lines of equal intensity (Fig. 6, indicated by an arrow) on the diagram of Fig. 3.

Fig. 9. Superposition of lines of equal intensity (Fig. 5) on the diagram of Fig. 3.
(a—corresponds to arrow 1 in Fig. 5; b—corresponds to arrow 2 in Fig. 5).

The described method is unlikely to find wide application; however, in a number of cases it may prove very useful. In particular, it can be used for a more detailed study of photographs of nonrepeatable or difficult-to-repeat interference patterns.

R. G.