Fizeau–Foucault apparatus

Figure 1: 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.

Figure 2: 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.

The Fizeau–Foucault apparatus (1850) (Figure 1) was designed by the French physicists Hippolyte Fizeau and Léon Foucault for measuring the speed of light. The apparatus involves light reflecting off a rotating mirror, toward a stationary mirror some 20 miles (35 kilometers) away. As the rotating mirror 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. 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 dθ/dt, the angle θ is given by:

$\theta = \frac {d \theta}{dt} \frac {2h}{c} \ .$

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

$c =\frac {d \theta}{dt} \frac {2h}{\theta} \ .$

The detector is at an angle 2θ from the source direction because the normal to the rotating mirror rotates by θ, decreasing by θ both the angle of incidence of the beam and its angle of reflection.

Foucault based his apparatus on an earlier experiment by Fizeau (Figure 2) who, in 1849, used two fixed mirrors, one partially obscured by a rotating cogwheel. Fizeau's value for light's speed was about 5% too high.

The Fizeau experiment to measure the speed of light in water has been 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. Newton predicted refraction as a pull of the medium upon the light, implying an increased speed of light in the medium. However, Fizeau showed the speed of light in water to be less than in air, not more, by inserting a tube of water in the light path.

Figure 3: Fizeau interferometer for measuring the effect of water movement upon the speed of light. Light reflected from the tilted beam splitter is made parallel using a lens and split by slits into two beams, which traverse a tube carrying water moving with velocity v. Each beam travels a different leg of the tube, is reflected at the mirror at left, and returns through the opposite leg of the tube. Thus, both beams travel the same path, but one in the direction of flow of the water, and the other opposing the flow. The two beams are recombined at the detector, forming an interference pattern that depends upon any difference in time traveling the two paths.

Figure 4: Schematic of the Fizeau–Foucault apparatus and the improvement by Michelson. The original setup used a less brilliant light source, so that a clear scale had to be used so that one could stare into the beam.

Fizeau interferometer

Main article: Fizeau interferometer

Fizeau also improved interferometry, and today the Fizeau interferometer still is used in a standard method for determining imperfections in surfaces and lenses.

Speed of light in moving media

Using an interferometer, Fizeau determined how the velocity of light is affected by a moving medium, as shown in an idealized form in Figure 3. Water flows in a pipe at velocity v. According to the non-relativistic view at the time of the experiment, the speed of light should be increased when "dragged" along by the water, and decreased when "overcoming" the resistance of the water. That is, the speed of light w in one arm is:

$w_+=\frac{c}{n}+v \ ,$

and in the other arm is

$w_-=\frac{c}{n} - v \ ,$

with n the index of refraction of water, and c/n the velocity of light in stationary water. As shown in the diagram, one path for the light should take a longer time than the other (clockwise time > counterclockwise time).

The interference pattern between the two beams when the light is recombined at the observer depends upon the transit times over the two paths. Thus, the speed of light in the water can be found as a function of the speed of the water. Fizeau found

$w_+=\frac{c}{n}+(1-\frac{1}{n^2})v \ .$

This surprising result was instrumental in Einstein's thinking about relativity, and can be derived as an approximation to the law of special relativity expressing addition of velocities, namely:

$w =\frac{ \frac{c}{n} +v }{ 1+ \frac {c/n}{c}\frac{v}{c} } = \frac{\frac{c}{n} +v}{1+\frac{v}{nc}}$

 $\approx \left( \frac{c}{n}+v\right) \left(1-\frac{v}{nc}\right)$ $=\frac{c}{n}-\frac{v}{n^2} +v -\frac{v^2}{nc}$ $=\frac{c}{n}\left( 1-\left(\frac{v}{c}\right)^2\right)+v \left(1-\frac{1}{n^2}\right)\ ,$

which is accurate when v/c << 1, and agrees with the formula based upon Fizeau's measurements, which satisfied the condition v/c << 1. In deriving this result, a truncated series expansion for small x was used: 1 ⁄ (1+x) ≈ 1 − x + x² −x³ ⁄ (1+x) ≈ 1 − x.

External links

• Diagram showing the original experimental design found in the second volume of Foucalt's collected works: Volume Two - Recueil des travaux scientifiques de Léon Foucault 1878.
• Speed of Light Formal Report (The Foucault Method)
• Light in moving media