Foucault's & Michelson's Rotating Mirrors

A mirror spinning hundreds of times a second turns the tiny time light spends in flight into an angle you can measure with a microscope.

Foucault’s rotating mirror#

Light from a source S passes through a glass plate and a converging lens, then strikes a plane mirror M₁ that can rotate fast. M₁ reflects it to a concave mirror M₂ whose radius of curvature R equals its distance from M₁. The central ray strikes M₂ normally and comes straight back to M₁, then back through the lens, and the glass plate reflects it to form an image I of the source.

If M₁ is still, the image doesn’t move. If it spins, it turns through a small angle Δθ in the time 2R/c the light spends going to M₂ and back. The returning beam is then reflected at 2Δθ to its old direction, and the image shifts to I′, a distance s away.

The formula

The light reflected by M₁ seems to come from the image of M₂’s centre in M₁, which moves by R·2Δθ. The lens images this onto the source’s plane with magnification a/(R + b) (a = source to lens, b = lens to M₁). With Δθ = 2Rω/c:

Michelson’s rotating polygon#

Michelson replaced the plane mirror with a polygon of N mirror faces. Light reflects off one face, travels a long path (to a distant concave mirror and back, many kilometres), and returns to strike the polygon at another face before entering a telescope.

If the polygon is still, the image is seen. Spinning it, the returning light generally finds a face at the wrong angle and the image flickers and vanishes. But at one speed the polygon turns exactly one face (2π/N) while the light is away, and the next face sits exactly where the last one was: the image is steady again. With D the path between the two reflections:

Michelson first used an octagonal mirror in 1879; the last of his series, finished by Pease and Pearson in 1935, used a 32-sided mirror.