FROM CURRENT LITERATURE

RECENT MEASUREMENTS OF THE SPEED OF LIGHT AND MICROWAVES IN VACUUM

The speed of propagation of electromagnetic waves in vacuum is one of the most important universal constants, and a possibly exact knowledge of its value is absolutely necessary for many branches of physics.

Until recently, the most reliable value was considered to be that given by Birge[^1] as a result of a careful analysis of all measurements made up to 1941:

\[ c = 299\,776 \pm 4 \ \text{km/sec}. \]

This same value was also adopted in later reviews by Dorsey[^2] and DuMond and Cohen[^3]. However, in recent years this value has been called into question, and at present there are convincing data requiring its substantial change, far beyond the probable error indicated by Birge.

It should be emphasized at once that the probable error indicated by Birge, as well as by all the investigators whose data he used, is the product of a statistical treatment of a very large number of individual measurements. The possible error of an individual measurement in all cases without exception was incomparably larger. Thus, in Michelson’s experiments, completed after his death by Pease and Pearson, the scatter of the values obtained by averaging over groups that included about 500 individual measurements was \(\pm 11\) km/sec, while averaging over groups of six measurements gave a scatter of \(\pm 93\) km/sec. In particular, three series of measurements carried out in December 1932 and in January–February 1933 gave the following values: \(299\,785 \pm 10\), \(299\,765 \pm 10\), and \(299\,785 \pm 10\) km/sec. Similarly, in Anderson’s measurements the scatter of means taken over groups comprising up to 200 individual measurements reached \(\pm 67\) km/sec.

Furthermore, in a number of experiments the possibility of systematic errors was not excluded. For example, in Anderson’s method[^4], which consisted in determining the relative phase shift of the intensity oscillations of two modulated beams that had traversed different distances, both light beams fell on the photocathode of a single photocell, and the change in the length of the path of one of the beams, sufficient to make the photocurrent amplified at the modulation frequency go to zero, was determined. Here the essential assumption is that the flight time of the photoelectrons from the photocathode to the anode is the same for both beams. However, uncontrolled differences in the structure of the light beams and in the positions of the regions of the photocathode illuminated by them may violate this

...condition and introduce a systematic error into the measured value of the speed of light. (In the Bergstrand experiments described below, the possibility of such an error is excluded.)

Finally, it should be pointed out that in almost all cases the directly measured quantity was the propagation velocity of electromagnetic waves in the visible range. The exceptions are the experiments of Rosa and Dorsey (the ratio of electrostatic units to electromagnetic units) and Mercier (standing Hertz waves), and the accuracy of the latter is small.

Measurements of the propagation velocity of electromagnetic waves in the radio range are of no interest from this point of view, since they include the unaccounted-for (and very substantial) influence of the earth’s surface. In the microwave range, on the contrary, this influence is negligibly small, and the propagation velocity, as for light, depends exclusively on the refractive index of the medium. At the same time, for the purposes of radiolocation (including radiogeodesy), knowledge of the propagation velocity of microwaves with the highest possible accuracy is necessary. It was along this path that the first doubts arose about the validity of the value given by Birge.

A comparison of the data of radiogeodetic measurements, carried out according to the so-called “Goby” system [see, for example, \(^{5}\)], with the data of triangulation measurements (whose accuracy is sufficiently high) led to the conclusion \(^{6,7}\) that there exists a systematic error which is largely eliminated if it is assumed that the propagation velocity of microwaves in vacuum is equal to

\[ c = 299\,738\ \text{km/sec}. \]

However, such an assumption did not eliminate the discrepancies completely, and the authors interpreted their results as a determination of the propagation velocity of microwaves under various conditions.

An analogous comparison of triangulation measurements with radiogeodetic measurements (with the “Shoran” system) was carried out by Aslakson \(^{8}\). The method of radiolocation measurements consisted in analytically determining the minimum of the sum of the distances from two ground stations to an aircraft flying between them. In all, 47 straight lines were measured, with lengths approximately from 100 to 550 km. To compute the distances, the value of the propagation velocity of electromagnetic waves in vacuum given by Birge was used. In this case a systematic discrepancy in the distances was found; a careful analysis of it led to the conclusion that the error could be rooted only in an inaccurate value either of the propagation velocity of microwaves in vacuum, or of the refractive index of air. Assuming the former, the author sought a correction to Birge’s value according to the formula:

\[ \Delta c = -\frac{\Delta l}{l}\,c_1, \]

where \(l\) is the distance determined by the radiolocation method, \(\Delta l\) is the difference between the geodetic and radiogeodetic distances, and \(c_1\) is the speed of light in vacuum according to Birge. The value of the speed of microwaves in vacuum corrected in this way turned out to be

\[ c = 299\,792.3 \pm 2.4\ \text{km/sec}. \]

The deviations of the individual values from the mean ranged from 0.2 to 3.6 km/sec, with the exception of the shortest distance, where it reached 8.4 km/sec. If this least reliable value is discarded, the probable error is only \(\pm 1\) km/sec. The same result was obtained by another method of processing the measurement results, on which we shall not dwell. The mean deviation from

most probable value for different measurements of the length of the same and of the same distance, carried out on the same or on different days, was about 2 m.

The development of microwave technique also created another possibility for determining the speed of light in vacuum, realized by Essen and Gordon-Smith ^9, ^10.

It is known that the attenuation of microwaves propagating along a closed metal tube of appropriate dimensions is very small and that a closed long waveguide is a resonator with extremely low attenuation. If the resonator is made in the form of a hollow straight circular cylinder, then the resonant frequency is determined by the expression

\[ f_{\mathrm{res}} = v \sqrt{\left[\left(\frac{r}{\pi D}\right)^2+\left(\frac{n}{2L}\right)^2\right]\left(1-\frac{1}{2Q}\right)}, \]

where \(v\) is the velocity of propagation of microwaves in the medium filling the resonator, \(D\) and \(L\) are the diameter and length of the cavity, \(r\) is a constant characteristic of waves of the given type (its value is found by calculation), \(n\) is the number of half-waves fitting into the length of the resonator, and \(Q\) is the quality factor of the resonator. Thus, measurements of \(f_{\mathrm{res}}\), \(D\), \(L\), and \(Q\), which can be carried out with very high accuracy, make it possible to find the value of the velocity of propagation of microwaves in the medium filling the resonator. In the experiments described, the air was removed from the cavity of the resonator and, in this way, the speed of light in vacuum was determined directly. This freed the result from the additional errors usually introduced by the uncertainty in the value of the refractive index of air.

The resonator was a copper cylinder with an internal diameter of about 7.4 cm and a length of 8.5 cm. Special measurements showed that the effect of the leads and connecting devices was negligibly small. The magnitudes of the waves of type \(E_{010}\) and \(E_{011}\) were measured, for which the experimental conditions were most favorable.

The initial value obtained for the propagation speed was ^9:

\[ c = 299\,793 \pm 9\ \text{km/sec}. \]

Later measurements, the details of which have not yet been published ^10, improved this value:

\[ c = 299\,792.5 \pm 3\ \text{km/sec}. \]

Essen points out that small systematic discrepancies were discovered between measurements carried out under different conditions, and he believes that they were caused by mechanical or electrical defects in the walls of the resonator.

Essen emphasizes in particular the fact that the method of the measurements carried out by him differs fundamentally from the others and that possible errors in this case are due to other factors, which are subject to both experimental and theoretical accounting. At the same time, he stresses that, in contrast to the other measurements, here it is possible to estimate not only the probable but also the limiting absolute error of the measurements; the value of the latter is indicated by him.

In conclusion, let us dwell on the measurements of the speed of light carried out by Bergstrand ^11, ^12, ^13. These measurements are noteworthy, first—

However, by using an ordinary optical method, the author increased the accuracy of the measurements by approximately two orders of magnitude, narrowing the error limits to several hundred meters per second.

Fig. 1. Schematic diagram of Bergstrand’s experiment.

The method used by Bergstrand is essentially identical to the method used (exactly a century earlier) by Fizeau. But in its technical implementation it is closer to Anderson’s method[^4], while being free, however, of the shortcomings of the latter. The general scheme of the experiment is clear from Fig. 1.

The intensity of the light flux emitted by the source \(И\) (a 30-watt incandescent lamp) is modulated at a frequency of \(8.3\) MHz by means of a Kerr cell fed by a generator \(M.Г.\) on a stabilized quartz. With the same frequency and in the same phase (from the same generator), the sensitivity of the photocell \(\Phi\) is varied; the photocell receives light reflected from a plane mirror \(З\), located at a large distance \(l\).

Obviously, between the oscillations of the intensity of the light beam incident on the photocell and the oscillations of the sensitivity of the photocell there will be a phase shift depending on the time \(T\) spent by the light in traversing the path \(2l\) to the mirror and back.

Consequently, if the photocurrent is measured with a sufficiently inertial instrument, averaging the value of the photocurrent over many periods of modulation, then the dependence of the measured current \(i\) on \(T\), and hence on the distance \(l\) to the mirror, will have the form shown in Fig. 2a.

Fig. 2. Dependence of the intensity of the measured photocurrent on the time \(T\) of propagation of light to the mirror and back: a) the phases of modulation of the light beam and of the photocell sensitivity coincide; b) the phase shift is \(180^\circ\); c) the phase of modulation is periodically inverted.

Thus, from the magnitude of the measured current one can judge the time \(T\). However, the accuracy of determining \(T\) proves to be low. To increase the accuracy, Bergstrand applied a null method, using the following ingenious device.

If the intensity of the light beam and the sensitivity of the photocell are modulated not in phase but in antiphase, then the curve of the dependence of the measured current \(i\) on the distance to the mirror will take the form,

shown in Fig. 2, б. Let us now suppose that the phase shift is periodically changed by \(180^\circ\) and, at the same time, the direction of the photocurrent in the receiving instrument is changed. Then, if the phase reversal occurs at a sufficiently high frequency and the instrument is sufficiently inertial, the dependence of the instrument readings on \(l\) will have the form shown in Fig. 2, в. It is not difficult to see that the curve is strictly symmetric with respect to

Fig. 3. Schematic diagram of the receiving and modulating devices in Bergstrand’s experiments:

\(И\) — light source, \(Я. К.\) — Kerr cell, \(П_1\) and \(П_2\) — polarizing prisms (crossed), \(М. Г.\) — modulating generator, \(В. Г.\) — auxiliary generator reversing the phase, \(Г\) — galvanometer, \(Ф.-К.\) — photocathode, \(A\) — photomultiplier anode, \(K\) — capacitors, \(Л\) — lenses, \(З_1\) and \(З_2\) — mirrors (the diameter of mirror \(З_2\) is 45 cm).

the zero of the instrument, and the zero position can be fixed with great accuracy. In practice, reversal of the phase of modulation of the intensity of the light beam was carried out (one hundred times per second) by an auxiliary generator \(В. Г.\) (Fig. 3). At the same time, by means of the auxiliary generator \(В. Г.\), the direction of the photocurrent in the galvanometer \(Г\) was switched with the aid of a lamp circuit.

Figure 3 shows a schematic diagram of the receiving and modulating devices.

The positions of the mirror corresponding to the measured current \(i\) becoming zero are determined by the relation

\[ l_N = k + \frac{2N - 1}{8}\lambda, \]

where \(k\) is an apparatus constant determined experimentally, \(N\) is an integer whose value can be found by two rough determinations of the mirror position, and \(\lambda\) is the wavelength corresponding to the modulation frequency. Under the experimental conditions \(\lambda \simeq 36\) m, and the distance between two mirror positions corresponding to zero galvanometer readings was about 9 m. The values of \(l_N\) are constant to the same extent as the modulation frequency (i.e., \(\sim 1 \cdot 10^{-7}\)), but depend on atmospheric conditions. For \(l = 10\) km this dependence is expressed approximately by the following quantities: 0.9 cm per \(1^\circ\)C and 0.4 cm per 1 mm of pressure.

Under field conditions, setting the mirror in the position exactly corresponding to \(i = 0\) is very difficult, and this condition was achieved by means of a small, strictly controlled change in frequency.

At a distance to the mirror of \(l = 9\) km, the mean error of six measurements performed over the course of \(1/4\) hour was 0.4 cm, and the deviation from the mean on different days did not exceed 3 cm.

In the initial measurements\({}^{11,12}\) for the speed of light in vacuum it was obtained that:

\[ c = 299\,796 \pm 2 \text{ km/sec}. \]

However, later\({}^{13}\) the author discovered an apparatus error, and this value was corrected to

\[ c = 299\,793 \pm 2 \text{ km/sec}. \]

The measurements of 1949 were carried out with improved apparatus and gave the value

\[ c = 299\,792.7 \pm 0.25 \text{ km/sec}. \]

The deviations of individual measurement results from the mean do not exceed 1.1 km/sec. The limits of error in the determination of \(l\), arising from consequences that cannot be fully accounted for in the assessment of various factors, are characterized by the following figures: atmospheric conditions—0.20 cm, modulation frequency—0.25 cm, color of the radiation—0.20 cm, length of the baseline—0.25 cm (for a baseline length of 7 km; for short baselines the error may increase to 1 cm). The author proposes to continue measurements at increased distances.

Thus, the measurements of recent years, carried out by three different methods, consistently lead to the value found by Bergstrand. This value exceeds Berg’s value by 16 km/sec and is far outside the limits of the probable error indicated by him. At the same time, in contrast to the data used by Berg, the measurements of recent years are characterized by an extremely small scatter of the results of individual measurements and by an insignificant probable error with comparatively poor statistics. All this convincingly testifies in favor of the new value of the speed of light in vacuum. The comparison of values obtained after 1905 is given in the table and in Fig. 4.

Results of measurements of the speed of light in vacuum, carried out after 1905.

For earlier measurements, which are now only of historical interest, substantially larger values of the speed of light and incomparably larger errors are characteristic.

Fig. 4. Comparison of the results of measurements of the speed of light in vacuum carried out after 1905.
(The probable error is shown by a dashed line; the greatest possible error by a solid line.)

G. Rosenberg

CITED LITERATURE

1. Birg, Physics 8, 90 (1941).
2. Dorsey, Trans. Amer. Phys. Soc. 5, 34, part I, 109 (1944).
3. Du Mond and Cohen, Rev. Mod. Phys. 20, 82 (1948).
4. Anderson, J. Opt. Soc. Amer. 31, 187 (1941).
5. N. N. Malov, UFN 29, 1 (1936).
6. Jones and Cornford, J. Inst. Electr. Eng. 95, part II (1948).
7. Jones and Cornford, J. Inst. Electr. Eng. 96, part III, 447 (1949).
8. Aslakson, Nature 164, 711 (1949).
9. Essen and Gordon-Smith, Proc. Roy. Soc. Lond. A194, 348 (1948).
10. Essen, Nature 165, 582 (1950).
11. Bergstrand, Nature 163, 339 (1949).
12. Bergstrand, Arkiv Matem. Astron. och Fysik 36A, 20 (1949).
13. Bergstrand, Nature 165, 405 (1950).