THREE-SLIT INTERFEROMETER

In a number of cases there arises a need to determine a phase shift whose magnitude differs from a given one by only a small fraction of the wavelength of light. As is known, two-beam interferometers make it possible to measure a phase shift with an accuracy not exceeding approximately \( \frac{1}{20} \) of a wavelength. Incomparably smaller quantities—down to \( \frac{1}{1000} \) of a wavelength—can be measured with the aid of multibeam devices, for example of the Fabry–Perot interferometer type. However, such an increase in measurement accuracy is achieved in these devices, essentially, by successive multiple accumulation of the phase shift, so that the phase shift between the extreme beams effectively participating in the formation of the interference pattern reaches a very considerable magnitude. Therefore the use of multibeam techniques is possible only in those cases where the phase shift being measured is amenable to multiplication, which is by no means true in all cases of practical interest. From this point of view the three-slit interferometer described below[^1], which provides measurement of a phase shift with an accuracy of the order of \( \frac{1}{200} \) of a wavelength without the necessity of accumulating it, represents a substantial contribution to interference technique. The idea of the method is as follows.

Fig. 1.

The interference pattern produced by three equally spaced slits (Fig. 1) may be regarded as the result of superposition of the interference pattern from the two outer slits on a coherent light field produced by the middle slit and weakening or strengthening this pattern depending on the phase relationship. In the case where the radiation emerging from all three slits is in phase and the width of the middle slit is equal to the sum of the widths of the outer (identical) slits, all odd maxima will be extinguished, while the even ones will be strengthened (Fig. 2, a), i.e. an interference pattern with twice as large a period as in the case of two slits will be obtained. However, if the beam passing through the middle slit has (relative to the beams passing through the outer slits) a phase shift of

\[ \pm \frac{\pi}{2} \]

(for example, a compensator has been introduced into its path—see Fig. 1), then both the even and the odd maxima will be strengthened to the same degree (see Fig. 2, b). If, however, the phase shift of the middle beam (relative to the outer ones)

Figure labels:
a) Middle target; outer targets; odd maximum; even maximum.
b) Resultant odd maximum; middle target; resultant even maximum; odd maximum from the outer targets; even maximum from the outer targets; \(\pi/2\).
c) Resultant odd maximum; middle target; resultant even maximum; odd maximum from the outer targets; even maximum from the outer targets; \(\delta\).

Fig. 2. Vector diagram illustrating the formation of an interference pattern from three targets:
\(a\) — in-phase radiation from the three targets, \(b\) — the phase of the radiation of the middle target is shifted by \(\dfrac{\pi}{2}\), \(c\) — the phase of the radiation of the middle target is shifted by \(\dfrac{\pi}{2}-\delta\).

is equal to \(\frac{\pi}{2}+\delta\), where \(\delta \ll 1\), then the intensities of the even and odd maxima change in different directions: an increase in the intensity of the even maxima will be accompanied by a decrease in the intensity of the odd ones, or conversely, depending on the sign of \(\delta\) (Fig. 2, c). In this case the intensities of the maxima in the case of small \(\delta\) are determined by the expression

\[ I_{\pm}=2E_0^2(1\pm\delta), \tag{1} \]

where \(E_0\) is the amplitude of the wave arriving from one of the outer slits. Consequently, the relative difference in the intensities of adjacent maxima is

\[ \frac{\Delta I}{I}=\frac{I_+-I_-}{2E_0^2}=2\delta . \tag{2} \]

The eye is readily able to detect a relative change in the intensity of maxima of \(\sim 5\%\), which corresponds to \(\delta \cong 0.025\) radian, i.e. about \(\frac{1}{200}-\frac{1}{300}\) of a wavelength.

The author then proposes the following procedure for measuring the phase shift. The optical paths of rays coming from all three slits are identical only if they converge at the focus of the lens. For rays converging at a point located on the optical axis but at a distance \(r\) from the focus, the optical path of the ray coming from the middle slit will differ from the optical path of the rays coming from the outer slits by an amount equal, to a first approximation, to

\[ \Delta l=\frac{1}{2}d^2\left(\frac{1}{F}-\frac{1}{F+r}\right), \tag{3} \]

where \(d\) is the distance between the outer slits and \(F\) is the focal length of the lens. Therefore, if the radiation from all three slits is in phase, two positions of the eyepiece should be observed (clearly determined experimentally), corresponding to \(\Delta l=\pm \frac{\lambda}{4}\), at which the intensities of adjacent bands are equal. Conversely, in the focal plane the odd maxima will be extinguished. The distance between the two indicated positions is rather large. Thus, for \(\lambda \cong 5\cdot 10^{-5}\ \text{cm}\) (green light), \(F=45\ \text{cm}\), and \(d=0.15\ \text{cm}\), it is about \(5\ \text{cm}\). If the ray coming from the middle slit is shifted in phase, then the positions of the eyepiece corresponding to equality of the intensities of adjacent maxima will be shifted accordingly. By measuring these shifts of the eyepiece with a scale, it is not difficult to determine the phase shift, the accuracy of setting the eyepiece corresponding, as stated above, to approximately \(\frac{1}{200}-\frac{1}{300}\) of a wavelength.

The same principle was subsequently used\(^2\) for a modification of the Michelson interferometer, providing an increase in the accuracy of measurements with the latter to the same order of magnitude, \(\frac{1}{200}-\frac{1}{300}\) of a wavelength. The modification consists in introducing an additional semitransparent plate and an additional mirror in such a way as to create a path of rays analogous to the case considered above of a three-slit interferometer. The authors indicate that in practice this can be achieved by replacing one of the mirrors of the Michelson interferometer (the reference mirror) with a mirror coated with a thin transparent wedge of dielectric, on the surface of which a semitransparent layer of silver is deposited. The latter

in this case simultaneously also performs the role of an additional mirror and the role of an additional beam splitter. There is no doubt that further development of this principle as applied to various types of two-beam interferometers will make it possible to substantially broaden the possibilities of interferometric technique.

V. Yur’ev

REFERENCES CITED

1. F. Zernike, J. Opt. Soc. Am. 40, No. 5, 376 (1950).
2. M. Bottema and F. Zernike, J. Opt. Soc. Am. 41, No. 11, 870 (1951).