Foucault's measurements of the speed of light

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In 1850, Léon Foucault used a rotating mirror to perform a differential measurement of the speed of light in water versus its speed in air. In 1862, he used a similar apparatus to measure the speed of light in the air.

Background

In 1834, Charles Wheatstone developed a method of using a rapidly rotating mirror to study transient phenomena, and applied this method to measure the velocity of electricity in a wire and the duration of an electric spark.<ref name=Wheatstone_1834>Template:Cite journal</ref> He communicated to François Arago the idea that his method could be adapted to a study of the speed of light.

The early-to-mid 1800s were a period of intense debate on the particle-versus-wave nature of light. Although the observation of the Arago spot in 1819 may have seemed to settle the matter definitively in favor of Fresnel's wave theory of light, various concerns continued to appear to be addressed more satisfactorily by Newton's corpuscular theory.<ref name=Lauginie2004>Template:Cite journal</ref> Arago expanded upon Wheatstone's concept in an 1838 publication, suggesting that a differential comparison of the speed of light in the air versus water would serve to distinguish between the particle and wave theories of light.

Foucault had worked with Hippolyte Fizeau on projects such as using the Daguerreotype process to take images of the Sun between 1843 and 1845<ref name=Hughes2012/> and characterizing absorption bands in the infrared spectrum of sunlight in 1847.<ref name=Hearnshaw_1987>Template:Cite book</ref> In 1845, Arago suggested to Fizeau and Foucault that they attempt to measure the speed of light. Sometime in 1849, however, it appears that the two had a falling out, and they parted ways.<ref name="Tobin2003"/>Template:Rp<ref name=Hughes2012>Template:Cite book</ref> In 1848−49, Fizeau used, not a rotating mirror, but a toothed wheel apparatus to perform an absolute measurement of the speed of light in air.

In 1850, Fizeau and Foucault both used rotating mirror devices to perform relative measures of the speed of light in the air versus water.

Foucault employed Paul-Gustave Froment to build a rotary-mirror apparatus<ref name="Foucault"/> in which he split a beam of light into two beams, passing one through the water while the other traveled through air. On 27 April 1850,<ref name="Tobin2003"/>Template:Rp he confirmed that the speed of light was greater as it traveled through the air, seemingly validating the wave theory of light.<ref name=Hughes2012/>Template:Refn

With Arago's blessing, Fizeau employed L.F.C. Breguet to construct his apparatus. They achieved their result on 17 June 1850, seven weeks after Foucault.<ref name="Tobin2003">Template:Cite book</ref>Template:Rp

To achieve the high rotational speeds necessary, Foucault abandoned clockwork and used a carefully balanced steam-powered apparatus designed by Charles Cagniard de la Tour. Foucault originally used tin-mercury mirrors, however at speeds exceeding 200 rps, the reflecting layer would break off, so he switched to using new silver mirrors.<ref name="Tobin2003"/>Template:Rp

Foucault's determination of the speed of light

File:Michelson's illustration of the Foucault experiment - annotated.png

Figure 1: 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.<ref name=Michelson1880/>

1850 experiment

File:Foucault's determination of relative speed of light air vs water.png

Figure 2: Foucault's determination of the relative speed of light in air vs water. Light from a passing through a slit (not shown) is reflected by mirror m (rotating clockwise around c) towards the concave spherical mirrors M and M'. Lens L forms images of the slit on the surfaces of the two concave mirrors. The light path from m to M is entirely through air, while the light path from m to M' is mostly through a water-filled tube T. Lens L' compensates for the effects of the water on the focus. The light reflected back from the spherical mirrors is diverted by beam splitter g towards an eyepiece O. If mirror m is stationary, both images of the slit reflected by M and M' reform at position α. If mirror m is rapidly rotating, light reflected from M forms an image of the slit at α' while light reflected from M' forms an image of the slit at α".

In 1850, Léon Foucault measured the relative speeds of light in air and water. The experiment was proposed by Arago, who wrote,

Template:Quote<ref name="Foucault">Template:Cite book</ref>

The apparatus (Figure 1) involves light passing through slit S, reflecting off a mirror R, and forming an image of the slit on the distant stationary mirror M. The light then passes back to mirror R and is reflected back to the original slit. If mirror R is stationary, then the slit image will reform at S.

However, if the mirror R is rotating, it will have moved slightly in the time it takes for the light to bounce from R to M and back, and the light will be deflected away from the original source by a small angle, forming an image to the side of the slit.<ref name=Baierlein/>

Foucault measured the differential speed of light through air versus water by using two distant mirrors (Figure 2). He placed a 3-meter tube of water before one of them.<ref name="Tobin2003"/>Template:Rp The light passing through the slower medium has its image more displaced. By partially masking the air-path mirror, Foucault was able to distinguish the two images super-imposed on top of one another.<ref name="Tobin2003"/>Template:Rp He found the speed of light was slower in water than in air.

This experiment did not determine the absolute speeds of light in water or air, only their relative speeds. The rotational speed of the mirror could not be sufficiently accurately measured to determine the absolute speeds of light in water or air. With a rotational speed of 600-800 revolutions per second, the displacement was 0.2 to 0.3 mm.<ref name="Tobin2003"/>Template:Rp

Guided by similar motivations as 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.<ref name=Lauginie2004/>Template:Refn His experimental results, announced shortly before Fizeau announced 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.<ref>Template:Cite book</ref> Newton had explained refraction as a pull of the medium upon the light, implying an increased speed of light in the medium.<ref name=Walker>Template:Cite book</ref> The corpuscular theory of light went into abeyance, completely overshadowed by the wave theory.Template:Refn This state of affairs lasted 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.<ref name=Niaz2010>Template:Cite journal</ref>

For his efforts, Foucault was made chevalier of the Légion d'honneur, and in 1853 was awarded a doctorate from the Sorbonne.<ref name="Tobin2003"/>Template:Rp

1862 experiment

File:Speed of light calculation using Foucault's rotating mirror.png

Figure 3: Schematic of the Foucault apparatus. Left panel: Mirror R is stationary. Lens L (not shown) forms an image of slit S on spherical mirror M. The reflected image of the slit reforms at the original position of slit S regardless of how R is tilted. Right panel: Mirror R is rotating rapidly. The reflected light from mirror M bounces from mirror R that has advanced an angle θ during the transit of the light. The telescope detects the reflected image of the slit at angle 2θ relative to the position of slit S.<ref name=Baierlein/>

In Foucault's 1862 experiment, he desired to obtain an accurate absolute value for the speed of light, since his concern was to deduce an improved value for the astronomical unit.<ref name=Lauginie2004/>Template:Refn At the time, Foucault was working at the Paris Observatory under Urbain le Verrier. It was le Verrier's belief, based on extensive celestial mechanics calculations, that the consensus value for the speed of light was perhaps 4% too high. 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 displacement of the slit image (less than 1 mm<ref name=Michelson1879>Template:Cite journal</ref>) 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 measurement confirmed le Verrier's estimate.<ref name="Tobin2003"/>Template:Rp His 1862 figure for the speed of light (298000 km/s) was within 0.6% of the modern value.<ref name=PhysicsFAQ>Template:Cite web</ref>

As seen in Figure 3, the displaced image of the source (slit) is at an angle 2θ from the source direction.<ref name=Baierlein> Template:Cite book</ref>

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 ω, it changes angle during the light roundtrip by an amount θ given by: <math>\theta = \frac {2h \omega}{c}=\omega t \ . </math> The speed of light is calculated from the observed angle θ, known angular speed ω and measured distance h as <math>c = \frac {2 \omega h}{\theta } \ . </math>

Michelson's refinement of the Foucault experiment

File:Michelson's 1879 Refinement of Foucault.png

Figure 4: Michelson's 1879 repetition of Foucault's speed of light determination incorporated several improvements enabling use of a much longer light path.<ref name=Michelson1880/>

It was seen in Figure 1 that 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. Foucault could not increase the RM distance in his folded optical arrangement beyond about 20 meters without the image of the slit becoming too dim to accurately measure.<ref name=Michelson1880>Template:Cite book</ref>

Between 1877 and 1931, Albert A. Michelson made multiple measurements of the speed of light. His 1877–79 measurements were performed under the auspices of Simon Newcomb, who was also working on measuring the speed of light. Michelson's setup 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 and 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.<ref name=Michelson1880/> 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 such as the use of polygonal prism-shaped rotating mirrors (enabling a brighter image) having from eight through sixteen facets and a 22 mile baseline surveyed to fractional parts-per-million accuracy. His figure of 299,796±4 km/s<ref name=Michelson1927>Template:Cite journal</ref> was only about 4 km/s higher than the current accepted value.<ref name=PhysicsFAQ/> Michelson's final 1931 attempt to measure the speed of light in vacuum was interrupted by his death. Although his experiment was completed posthumously by F. G. Pease and F. Pearson, various factors militated against a measurement of highest accuracy, including an earthquake which disturbed the baseline measurement.<ref name=Michelson1935>Template:Cite journal</ref>

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