# Fizeau–Foucault apparatus

Fizeau–Foucault apparatus is a term sometimes used to refer to two types of instrument historically used to measure the speed of light.

The confusion between the two instruments arises in no small part because Hippolyte Fizeau and Léon Foucault had originally been friends and collaborators, working together on such projects as using the Daguerreotype process to take images of the Sun between 1843–1845. At the instigation of François Arago, they began a new project to measure the speed of light. Sometime in 1849, however, it appears that the two had a falling out, and they parted ways pursuing separate means of performing this measurement.[1]

## Fizeau's determination of the speed of light

Figure 1: Schematic of the Fizeau apparatus. The light passes on one side of a tooth on the way out, and the other side on the way back, assuming the cog rotates one tooth during transit of the light.

In 1848, Hippolyte Fizeau determined the speed of light between an intense light source and a mirror about 8 km distant. The light source was interrupted by a rotating cogwheel with 720 notches that could be rotated at a variable speed of up to hundreds of times a second. (Figure 1) Fizeau adjusted the rotation speed of the cogwheel until light passing through one notch of the cogwheel would be completely eclipsed by the adjacent tooth.[Note 1] Given the rotational speed of the wheel and the distance between the wheel and the mirror, Fizeau was able to calculate a value of 313000 km/s for the speed of light. It was difficult for Fizeau to visually estimate the intensity minimum of the light being blocked by the adjacent teeth,[2] and his value for light's speed was about 5% too high.[3]

The early-to-mid 1800s were a period of intense debate on the particle-versus-wave nature of light. Arago had suggested in 1838 that a differential comparison of the speed of light in air versus water would serve to prove or disprove the wave nature of light. In his experiments, Fizeau was not particularly concerned with obtaining an accurate absolute value for the speed of light, but was primarily concerned in performing a differential comparison of its value in different media. In 1850, Fizeau set up an experiment in which he split a beam of light into two beams, passing one through water while the other traveled through air. He found that the speed of light was greater as it traveled through air, validating the wave theory of light.[1]

In 1872–1876, Marie Alfred Cornu repeated Fizeau's experiment with various improvements. His final experiment was run over a path nearly three times as long as that used by Fizeau, and yielded a figure of 300400 km/s that is within 0.2% of the modern value.[4]

## Foucault's determination of the speed of light

Figure 2: In Foucault's experiment, lens L forms an image of slit S at spherical mirror M. If mirror R is stationary, the reflected image of the slit reforms at the original position of slit S regardless of how R is tilted, as shown in the lower annotated figure. However, if R rotates rapidly, the time delay due to the finite speed of light traveling from R to M and back to R results in the reflected image of the slit at S becoming displaced.
Figure 3: Schematic of the Foucault apparatus. Left panel: Light is reflected by a rotating mirror (left) toward a stationary mirror (top). Right panel: The reflected light from the stationary mirror bounces from the rotating mirror that has advanced an angle θ during the transit of the light. The telescope at an angle 2θ from the source picks up the reflected beam from the rotating mirror.

Between 1850–1862, Léon Foucault made improved determinations of the speed of light substituting a rotating mirror for Fizeau's toothed wheel. (Figure 2) The apparatus involves light from slit S reflecting off a rotating mirror R, towards a distant stationary mirror M, then being reflected back to reform an image of the original slit. If mirror R is stationary, then the slit image will reform at S regardless of the mirror's tilt. The situation is different, however, if R is in rapid rotation.

As the rotating mirror R will have moved slightly in the time it takes for the light to bounce off the stationary mirror (and return to the rotating mirror), it will thus be deflected away from the original source, by a small angle, as illustrated in Figure 3.[5] If the distance between mirrors is h, the time between the first and second reflections on the rotating mirror is 2h/c (c = speed of light). If the mirror rotates at a known constant angular rate $\omega$, the angle θ is swept in the same time as the light roundtrip, so:

$\frac{\theta}{\omega} = \frac {2h}{c}=t \ .$

In other words the speed of light is calculated from the observed angle θ, known angular speed and measured distance h as

$c = \frac {2 \omega h}{\theta } \ .$

As seen in Figure 3, the displaced image of the source (slit) is at an angle 2θ from the source direction. Technical limitations prevented Foucault from separating mirrors R and M by more than about 20 meters. Despite this limited path length, Foucault was able to measure the position of the displacement of the slit (less than 1 mm[2]) with considerable accuracy. In addition, unlike the case with Fizeau's experiment (which required gauging the rotation rate of an adjustable-speed toothed wheel), he could spin the mirror at a constant, chronometrically determined speed. Foucault's 1862 figure for the speed of light (298000 km/s) was within 0.6% of the modern value.[6]

As with the case of his former partner, Foucault in 1850 was more interested in settling the particle-versus-wave debate than in determining an accurate absolute value for the speed of light.[4] Foucault measured the differential speed of light versus water by inserting a tube filled with water between the rotating mirror and the distant mirror. His experimental results, obtained only shortly after Fizeau had published his results on the same topic, were viewed as "driving the last nail in the coffin" of Newton's corpuscle theory of light when it showed that light travels more slowly through water than through air.[7] Newton had explained refraction as a pull of the medium upon the light, implying an increased speed of light in the medium.[8] The corpuscular theory of light went into abeyance, completely overshadowed by the wave theory until 1905, when Einstein presented heuristic arguments that under various circumstances, such as when considering the photoelectric effect, light exhibits behaviors indicative of a particle nature.[9]

In contrast to his 1850 measurement, Foucault's 1862 measurement was aimed at obtaining an accurate absolute value for the speed of light, since his concern was to deduce an improved value for the astronomical unit.[4]

## Michelson's refinement of the Foucault experiment

Figure 4. Michelson's 1879 repetition of Foucault's speed of light determination incorporated several improvements enabling use of a much longer light path.

As seen in Figure 2, Foucault placed the rotating mirror R as close as possible to lens L so as to maximize the distance between R and the slit S. As R rotates, an enlarged image of slit S sweeps across the face of the distant mirror M. The greater the distance RM, the more quickly that the image sweeps across mirror M and the less light is reflected back. To increase the amount of reflected light, Foucault actually used five concave reflectors rather than the single mirror illustrated, but even so, he could not increase the RM distance beyond about 20 meters without the image of the slit becoming too dim to accurately measure.[10]

Between 1877 and 1931, Albert A. Michelson made multiple measurements of the speed of light. His 1877–1879 measurements incorporated several refinements on Foucault's original arrangement. As seen in Figure 4, Michelson placed the rotating mirror R near the principal focus of lens L (i.e. the focal point given incident parallel rays of light). If the rotating mirror R were exactly at the principal focus, the moving image of the slit would remain upon the distant plane mirror M (equal in diameter to lens L) as long as the axis of the pencil of light remained on the lens, this being true regardless of the RM distance. Michelson was thus able to increase the RM distance to nearly 2000 feet. To achieve a reasonable value for the RS distance, Michelson used an extremely long focal length lens (150 feet) and compromised on the design by placing R about 15 feet closer to L than the principal focus. This allowed an RS distance of between 28.5 to 33.3 feet. He used carefully calibrated tuning forks to monitor the rotation rate of the air-turbine-powered mirror R, and he would typically measure displacements of the slit image on the order of 115 mm.[10] His 1879 figure for the speed of light, 299944±51 km/s, was within about 0.05% of the modern value. His 1926 repeat of the experiment incorporated still further refinements and yielded a figure of 299,796±4 km/s, only about 4 km/s higher than the current accepted value.[6] Michelson's final 1931 attempt to measure the speed of light in vacuum was interrupted by his death.

## Notes

1. ^ Spinning the cogwheel at 3, 5 and 7 times this basic rotation rate also resulted in eclipsing of the reflected light.

## References

1. ^ a b Hughes, Stephan (2012). Catchers of the Light: The Forgotten Lives of the Men and Women Who First Photographed the Heavens. ArtDeCiel Publishing. pp. 202–223. Retrieved 3 July 2015.
2. ^ a b Michelson, Albert A. (1879). "Experimental Determination of the Velocity of Light". Proceedings of the American Association for the Advancement of Science: 71–77. Retrieved 3 July 2015.
3. ^ Abdul Al-Azzawi (2006). Photonics: principles and practices. CRC Press. p. 9. ISBN 0-8493-8290-4.
4. ^ a b c Lauginie, P. (2004). "Measuring Speed of Light: Why ? Speed of what?" (PDF). Proceedings of the Fifth International Conference for History of Science in Science Education. Retrieved 3 July 2015.
5. ^ Ralph Baierlein (2001). Newton to Einstein: the trail of light : an excursion to the wave-particle duality and the special theory of relativity. Cambridge University Press. p. 44; Figure 2.6 and discussion. ISBN 0-521-42323-6.
6. ^ a b Gibbs, Philip. "How is the speed of light measured?". The Original Usenet Physics FAQ. Retrieved 1 July 2015.
7. ^ David Cassidy, Gerald Holton, James Rutherford (2002). Understanding Physics. Birkhäuser. ISBN 0-387-98756-8.
8. ^ Bruce H Walker (1998). Optical Engineering Fundamentals. SPIE Press. p. 13. ISBN 0-8194-2764-0.
9. ^ Niaz, Mansoor; Klassen, Stephen; McMillan, Barbara; Metz, Don (2010). "Reconstruction of the history of the photoelectric effect and its implications for general physics textbooks" (PDF). Science Education 94 (5): 903–931. Retrieved 1 July 2015.
10. ^ a b Michelson, Albert A. (1880). Experimental Determination of the Velocity of Light. Nautical Almanac Office, Bureau of Navigation, Navy Department. Retrieved 2 July 2015.