FROM CURRENT LITERATURE

A NEW MEASUREMENT OF THE SPEED OF LIGHT²

The essence of the method for measuring the speed of light used by Anderson is as follows. A beam of light, modulated with the aid of a Kerr cell, is split into two beams that travel along different paths. These beams are directed onto a photocell, whose current is amplified by means of a suitable circuit. The measured electrical voltage is minimal if the intensity oscillations of the two beams lag one behind the other by an odd number of half-periods. By measuring the path difference at the minimum of voltage, one can find the speed of light \(c\) from the equation

\[ c=\frac{2fs}{n}, \]

where \(n\) is the phase difference in half-periods; \(s\) is the optical path difference, and \(f\) is the frequency of modulation of the beam.

The measuring scheme is shown in Fig. 1. The beam came from a 1000 W lamp with air cooling (\(A\)). This lamp gave a luminous flux of 27,000 lm. For many measurements a mercury high-pressure lamp of approximately the same power was also used. Passing through the lens \(L_1\) and the polarizer \(P_1\), the light was directed into the Kerr cell \(K\), cooled by water. Very pure nitrobenzene was used as the liquid dielectric of the cell.

A high-frequency voltage of 90 W power was applied to the cell. The radio-frequency voltage across the cell exceeded 1000 V. The beam of light emerging from the cell and passing also through another polarizer \(P_2\) and lens \(L_2\) was modulated, i.e. its intensity varied with time periodically at each point of the path, and also, of course, was periodically distributed along the path (at a given instant of time), as is shown schematically in Fig. 1. By the lens \(L_2\) the beam of light was projected onto a circular aperture 2 mm in diameter in a metal plate. By thin transverse hairs this aperture was divided into quarters. By the semitransparent mirror \(M_6\) the beam was split into two. The reflected

Fig. 1

the part passed through diaphragm $D$ and lens $L_3$. The lens was set so that its distance to the circular aperture was equal to the focal length; therefore, after passing through lens $L_3$ the beam is practically parallel. Diaphragm $D$ serves to compare the amplitude of the intensity of the beam reflected by mirror $M_6$ with the beam transmitted by this mirror.

After passing through $L_3$, the beam was reflected from mirror $M_3$ and travelled the return path to $M_6$, and then through lens $L_4$ entered receiver $F$.

The second part of the primary beam, transmitted by the semitransparent mirror, was directed either to mirror $M_1$, or to mirror $M_2$.

$M_1$ and $M_2$ are concave mirrors; the first has a focal length of $5$ m, the second $2.5$ m. Mirrors $M_1$ and $M_2$ are mounted together so that each in turn could be set in its focal position.

If mirror $M_1$ is in the reflecting position, then the beam goes along the path $M_1M_7M_5$ and, having been reflected from $M_5$, goes back along the path $M_5M_7M_1M_4M_6$, whence, through $L_4$, it enters the receiver. If mirror $M_2$ is in the reflecting position, then the beam goes along the path $M_2M_6$, i.e. is reflected back, bypassing the long path $M_1M_5$.

The surfaces of the mirrors of which we have spoken are very accurate (to one wavelength).

If we denote by $S$ the path $M_1M_7M_5$; by $x$ the path $M_6M_1$ and by $y$ the path $M_6M_3$, then the condition of minimum intensity when mirror $M_1$ is set in the reflecting position can be written in the form

$$ 2S+2x-2y=(2n+1)\frac{\lambda}{2}, $$

where $\lambda$ is the wavelength corresponding to the modulation frequency.

If mirror $M_2$ stands in the reflecting position (so that the beam goes along the path $M_6M_2M_6$), then the minimum intensity as a result of the addition of the beam reflected and transmitted by $M_6$ can be reached at some new position of the plane movable mirror $M_3$, indicated in Fig. 1 as $M_3''$.

The condition for the minimum can now be written as:

$$ 2x+2\Delta S-2y-2\Delta y=(2m+1)\frac{\lambda}{2}. $$

Here $\Delta S$ is the distance of the mirrors $M_2$ and $M_1$ from one another, and $\Delta y$ is the difference between the positions of the movable mirror $M_3$.

Subtracting the second equality from the first, we obtain

$$ 2S-2\Delta S+2\Delta y=p\lambda, \tag{1} $$

where $p$ is an integer (the difference of two odd numbers gives an even number). The last equality shows in what sense the experimental arrangement described above eliminates inconvenient distance measurements. The segments $S$, $\Delta S$ and $\Delta y$ can be measured with high accuracy.

The entire experiment was carried out automatically. The movable mirror $M_3$ was set into translational motion (back and forth) by means of a motor.

When the whole system is switched on, the following actions are performed automatically. The movable mirror and the photographic film, which records the change in the intensity of the photoelectric current as a function of the position of mirror $M_3$, i.e. actually as a function of the difference in path between the beam intensities, simultaneously begin to move. After the motion in one direction ends (the film and the mirror also stop simultaneously), the direction of motion of the motor is automatically reversed and the frame of the film is changed.

Measurements are made with mirror $M_2$ (path $M_6M_2M_6$) and with mirror $M_1$ (path $M_6M_1M_5M_1M_6$).

The difference of the minima of the recorded current curves is nothing other than $\Delta y$ in equation (1). The coordinates of the minima are determined with respect to the mark formed on the film by the image of a fine wire.

The distance between the mirrors \(M_1\) and \(M_2\) and the path \(M_1M_7M_5\) are measured directly, since, owing to a number of special devices, the accuracy of the measurement of distances is very high.

Anderson describes in considerable detail the receiving device \(F\), which is an 11-stage electron multiplier with a sensitivity of \(2\ \mathrm{A/lm}\) (instead of \(30\ \mu\mathrm{A}\) for an ordinary vacuum photoelectric cell). Special measures were taken to eliminate fluctuations.

Let us now dwell on one special measurement error. If the superposed rays do not coincide exactly (even while being parallel), then, as is seen from Fig. 2, which shows the cross-section of the cathode and the first stages of the electron multiplier, between the electrons emitted from points \(C\) and \(D\) an additional path difference arises, which may distort the real phase shift in the intensities of the superposed rays. With careful operation this possible error can be made very small, but it cannot be eliminated.

Fig. 2

The speed measured directly is the group velocity in air, different from the speed of light in vacuum (in the experiment described, the path of the rays in glass is common to both rays and may be disregarded). Therefore the measured value \(c_a\) must be divided by the group refractive index

\[ \mu_g=\mu-\lambda\frac{d\mu}{d\lambda}. \]

According to the measurements of Meggers and Peters\(^2\) for air,

\[ (\mu-1)10^7=2726.43+2.288(\lambda\cdot10^{-8}+0.355)\lambda^4\cdot10^{-16}. \]

Differentiating this expression, we find \(\mu_g\). The average correction for the group velocity is \(84\ \mathrm{km/sec}\).

As a result of 2,895 measurements carried out during 1939/40, the following value of the speed of light in vacuum was obtained:

\[ c=299\,766\pm14\ \mathrm{km/sec}. \]

Anderson’s measurements refute assumptions\(^3\) that the speed of light decreases or changes periodically with time. The work of the last twenty years leads to one and the same (within the limits of errors) value of the speed of light in vacuum. Nor is any regular change observed in the monthly mean value of the speed in 1939/40.

A. I. Kitaigorodsky

LITERATURE

1. Anderson, JOSA, 31, 187, 1941.
2. Meggers and Peters, Bull. Bureau of Standard, 14, 697, 1918.
3. Birge, Nature, 134, 771, 1934; Miller, Rev. Mod. Phys., 5, 3, 1933.