MODULATION INTERFEROMETRY

For a long time physicists were dominated by the conviction that the wave nature of light “fundamentally” limits the possibility of increasing the resolving power of optical instruments and measurement methods. Therefore, the search for a way out of the resulting “impasse” was directed chiefly toward finding means of replacing visible

light with radiation of shorter wavelength (ultraviolet rays, X-ray radiation, electron beams, neutrons, ions, etc.). However, such a path, generally speaking, though very effective, has in a number of cases proved unsatisfactory. The clearly matured need in recent years for a radical increase in the resolving power precisely of optical methods of measurement, together with the unusual and diverse development of experimental technique, forced a reconsideration of the question of the limitations imposed by the wave nature of light; in doing so, a number of possibilities were revealed which had previously escaped attention and which make it possible substantially to push back this supposedly fundamental limit. We have in mind the methods of so-called multi-beam interferometry (see, for example,\(^{1}\)) with variable phase contrast\(^{2}\), three-beam interferometry\(^{3}\), and others. These methods have made it possible to increase the resolving power of optical methods by 1–3 orders of magnitude and have made accessible to optical observation

Fig. 1.

objects which only recently seemed to lie far beyond the limits of what was attainable (for example, measuring interatomic distances in crystals or detecting surface defects as small as about \(0.5\,\text{\AA}\)).

However, all these new methods are connected with the improvement of only one aspect of optical measurements—the method of obtaining an image (or an interference pattern). An entirely different, fundamentally unrestricted technique for increasing the resolving power was developed in the referential works of Soviet physicists\(^{4,5,6}\). It is based on changing the method of observing the image (or interference pattern).

The essence of the method is as follows. Suppose we are interested in a small displacement (Fig. 1, a) or change of shape (Fig. 1, b) of an interference fringe (spectral line, diffraction figure, etc.) as a result of some changes in the conditions of its formation. Let us isolate, by means of a suitable diaphragm, a small portion of the fringe of interest to us (Fig. 1) and measure, with a photocell, the intensity of the light passing through the diaphragm. Obviously, small displacements (or changes in the shape) of the fringe will entail equally small changes in the amount of light falling on the photocell, and the resolving power (sensitivity) of the device will be limited by the instability of the measurement conditions (brightness of the source, gain coefficient, etc.). Suppose now that the observed pattern is modulated, i.e. that the changes of interest to us occur periodically with some frequency \(F\). Then, by inserting into the measuring scheme a resonant filter with a sufficiently narrow passband, tuned to the same frequency \(F\), we always have the possibility, to the required degree, of freeing ourselves from the influence of random disturbances and of observing arbitrarily small changes in the position or shape of an interference fringe (spectral line, diffraction pattern, etc.).

In fact\(^4\), suppose, for example, that we have a two-beam interferometer, and the phase shift between the interfering beams is modulated according to the law

\[ \Phi=\Phi_0+f(t), \tag{1} \]

where \(f(t)\) is a periodic function of time. The voltage taken from the photocell (proportional to the illumination of the slit) will then be a function of time:

\[ u=A+B\cos[\Phi_0+f(t)], \tag{2} \]

where \(A\) and \(B\) depend on the intensities of the interfering beams and on the sensitivity of the photocell.

Assuming that \(0 \leq f(t) \leq \varepsilon\) (where \(\varepsilon \ll 1\)), relation (2) can be rewritten in the form

\[ u=(A+B\cos\Phi_0)-B\sin\Phi_0 f(t)=P+Q\cdot f(t). \tag{3} \]

The quantities \(A\), \(B\), and \(\Phi_0\) (and consequently \(P\) and \(Q\) as well) do not remain, strictly speaking, unchanged with the passage of time: they are subject to all sorts of random changes caused by a whole complex of diverse circumstances capable of disturbing the stability of the measurement conditions. A resonant filter tuned to the modulation frequency \(F\) (if this frequency is sufficiently high) will cut off both the slow component of the variations of \(P\) and the fast component of the variations of \(Q\). As a result of this, the voltage at the output of the resonant filter, supplied to the measuring instrument, will be sinusoidal with a slowly varying amplitude equal to

\[ U=K(t)\{p_F(t)+[\overline{Q}+q(t)]g\}, \tag{4} \]

where \(K(t)\) is the gain factor, \(p_F(t)\) is the component of the fast fluctuations of the quantity \(P\) passing through the filter, \(\overline{Q}\) is the mean value of \(Q\), \(q(t)\) is the component of the slow variations of \(Q\) passing through the filter, and \(g\) is the coefficient of the first term in the expansion of the function \(f(t)\) in a Fourier series. Since

\[ p_F(t)=w\Delta P, \tag{5} \]

where \(\Delta F\) is the passband width of the filter, and \(w\) is a constant not depending on \(\Delta F\), therefore by a proper choice of \(\Delta F\) the condition \(p_F(t) \ll [\overline{Q}+q(t)]g\) can be achieved, i.e.

\[ U=Sg, \tag{6} \]

where

\[ S=K(t)[\overline{Q}+q(t)]. \]

In other words, with a sufficiently small passband width \(\Delta F\) of the filter it is in principle possible to detect arbitrarily small \(g\), and the accuracy of measuring \(g\) is limited exclusively by the stability of the quantity \(S\). If the latter is not sufficiently stable, then the null method may be applied, which makes it possible to measure \(g\) with fundamentally unlimited accuracy.

The author considers, in particular, the case in which the phase \(\Phi\) during modulation takes alternately the values \(\Phi_0\) and \(\Phi_0+\varepsilon\). Then \(g=\dfrac{2\varepsilon}{\pi}\), and the method makes it possible to measure an arbitrarily small phase shift \(\varepsilon\) with any desired accuracy.

If the question is not one of phase shift but of the splitting of spectral lines in a strong field, then, obviously, the field strength causing the splitting is to be measured.

A vivid example of the practical application of the method described is the determination of the amplitude of oscillations of a membrane by means of a Michelson interferometer1. If one of the mirrors of the interferometer (mounted on the membrane) undergoes harmonic oscillations with amplitude \(z\) and sound frequency \(\omega\), then the amplitude of the first harmonic of the current at the output of the photomultiplier is

\[ I_1 = B \sin (2kz_0) J_1(2kz), \tag{7} \]

where \(B\) is a constant depending on the intensity of the interfering rays,

\[ k = \frac{2\pi}{\lambda} \]

is the wave number, \(z_0\) is the path difference of the interfering rays at \(z = 0\), and \(J_1\) is the Bessel function of the 1st order. With a sufficiently narrow-band filter, the reading of the measuring instrument \(a\) is proportional to \(|I_1|\):

\[ a = b |J_1(2kz)|, \tag{8} \]

where \(b\) is an instrumental constant. Finding (by gradually increasing \(z\)) the value \(a = a_m\) at the first maximum, and taking into account that the first maximum occurs at \(J_1(2kz) = 0.58\), we obtain \(b = \dfrac{a_m}{0.58}\). Substituting this value in (8), it is not difficult to find \(z\) for any \(a\).

Fig. 2.

The authors present (Fig. 2) the results of their determination, by this method, of the dependence of \(a\) on the amplitude \(V\) of the alternating voltage applied to the electromagnet exciting the oscillations of the membrane. Along the ordinate is plotted the quantity

\[ \left|J_1(x)\right| = \frac{0.58\, a_m}{a}, \]

and along the abscissa axis \(x = aV\), where \(a\) is a constant coefficient chosen so that \(a\) becomes zero at \(x = 3.83\), the first root of the function \(J_1(x)\) distinct from zero. (From experiment the value \(a = 4.79\) was found.) The solid line is the graph of the function \(|J_1(x)|\).

For \(x < 3.83\), the experimental points lie well on the theoretical curve. At larger values of \(x\), a systematic discrepancy is observed, evidently due to the nonlinearity of the dependence of \(z\) on \(V\).

Starting from the fact that \(x = 2kz\) and \(\lambda = 4358\) Å, on the basis of the data of Fig. 2 the authors obtained:

\[ \frac{z}{V} = \frac{a\lambda}{4\pi} = 1663 \frac{\text{Å}}{\text{volt}}. \]

The stability of the needle of the output instrument in the experiments described was about \(\pm 2.5\cdot 10^{-3}\alpha_{\mathrm{m}}\). In other words, the apparatus made it possible to detect such \(z\) for which

\[ \frac{\alpha}{\alpha_{\mathrm{m}}} = \frac{J_1(2kz)}{0.58} > 2.5\cdot 10^{-3}. \]

Since, for small \(z\), \(J_1(2kz)\cong kz\), the condition for detection of \(z\) was

\[ z \geqslant 2.3\cdot 10^{-4}\lambda \cong 1\ \text{\AA}. \]

The authors note that this sensitivity—approximately 10 times greater than that attained in measurements of the vibration amplitude of piezo-quartz crystals by methods of multiple-beam interferometry\({}^{7}\)—was obtained by them with an apparatus that was very imperfect both optically and electrically\({}^{1}\), and this is the limiting factor for the method described. Thus, we have before us a technique that opens up entirely new possibilities in optical measurement technology. At the same time it is obvious that this technique is not specifically optical. Somewhat earlier, an analogous modulation method was used to solve certain radio-physics problems, in particular to repeat the Sagnac experiment with radio waves\({}^{6}\). Since this latter experiment is of independent interest, we shall dwell on it separately.

Fig. 3.

As is known, in 1912 Harress, and in 1913 Sagnac, carried out the so-called “vortex experiment,” the aim of which was to measure the speed of light in a reference system moving with acceleration. A rotating system of mirrors was used as such a system, ensuring closure of the contour traversed by a light beam. In the experiments described by I. L. Bernstein, the system of mirrors was replaced by a concentric cable wound on a rotating drum. Radiation with a wavelength of 10 m was directed from the generator to the receiver input along two paths: through the cable (\(U_{\mathrm{cab}}\)) and through a concentrated impedance (\(U_{\mathrm{imp}}\)). The parameters of the circuit were selected so that the vector diagram of the voltages arriving at the receiver input had the form shown in Fig. 3. As a result of interference at the input, a small voltage \(U_0\) was obtained, which, after amplification, was fed to a high-frequency detector. With the aid of a mechanical commutator, the ends of the cable were periodically switched (at a frequency of 12.5 cps), i.e., the direction of the wave traveling along it was changed. If in this process the phase difference \(\varphi\) changed by a small amount \(\delta\varphi\), then \(U_0\) had to vary periodically by \(U_{\mathrm{imp}}\delta\varphi\), which could be detected at the detector output. The detector output voltage was fed to a low-frequency amplifier (12.5 cps) and then to a second detector and a microammeter. The difference in the readings of the microammeter was measured when the direction of rotation of the drum was reversed.

Under the conditions of the experiment described (frequency \(30.3\cdot 10^{-6}\) cps, drum radius 1 m, its rotation speed \(1\div 1.3\ \mathrm{sec}^{-1}\), cable length 244 m), theoretical considerations led one to expect \(\Delta\varphi \cong 3''\). The author carried out several series of measurements (25–40 individual measurements in a se-

figure). A typical pattern obtained in one of the series is presented in Fig. 4. Along the abscissa axis is plotted the magnitude of the phase shift. Along the ordinate axis is the number of observations that gave this value of the shift. The deviation of the mean over the series from the theoretical prediction (dotted line) did not exceed 15% (for this series \(\Delta \varphi_{\text{obs}} = 1.61 \cdot 10^{-5}\), \(\Delta \varphi_{\text{theor}} = 1.57 \cdot 10^{-5}\)). The author explains the large scatter of the data of individual measurements by shortcomings of the apparatus.

Fig. 4.

The author sees the interest of the experiment he performed in the fact that it is the first radiophysical experiment on the electrodynamics of moving bodies, and also in the fact that the propagation of radio waves took place in a medium strongly differing from vacuum (the dielectric constant of the dielectric filling the cable is \(\varepsilon = 2.24\)), which brings this experiment close to Fizeau’s well-known experiment on determining the speed of light in a jet of water.

From the point of view of interferometric technique, of no less interest is the possibility, clearly illustrated by these experiments, of measuring a relative change in the velocity of wave propagation of the order of \(10^{-8}\) and a change in the path difference of the interfering waves of the order of \(10^{-6}\lambda\); moreover, this, again, was not the limit.

In conclusion, let us emphasize once more what G. S. Gorelik showed: “with a given optical or interference device, the resolving power of the measuring installation can be made arbitrarily large by an appropriate choice of the modulation frequency and a sufficient narrowing of the passband of the filter.” The rational use of this method in combination with methods for increasing the resolving power of interference apparatus will undoubtedly make it possible to solve many problems that have hitherto remained inaccessible.

G. Rozenberg

CITED LITERATURE

1. G. V. Rozenberg, Multi-beam interferometry and interference light filters, UFN 47, issue 2, 173 (1952).
2. H. Osterberg and G. E. Pride, J. Opt. Soc. Am. 40, 64–74 (1950).
3. F. Zernike, J. Opt. Soc. Am. 40, 326–328 (1950); see also abstract, UFN 47, issue 2, 146 (1952).
4. G. S. Gorelik, DAN 83, No. 4, 549–552 (1952).
5. I. Ya. Brustein, G. S. Gorelik, and S. A. Pikovsky, DAN 83, No. 4, 553–556 (1952).
6. I. L. Bernstein, DAN 75, No. 5, 635–638 (1950).
7. S. Tolansky and W. Bardsley, Proc. Phys. Soc. 64B, 224 (1951).

1. Reference number as printed on the source page. ↩