Visualize the Sagnac effect: rotation-induced phase shift in counter-propagating light beams, fiber optic gyroscope principle
In 1913, Georges Sagnac demonstrated that counter-propagating light beams in a rotating loop acquire a path-length difference. When the interferometer rotates with angular velocity Ω, the beam traveling with the rotation traverses a longer path, while the counter-rotating beam traverses a shorter path. The resulting phase difference is Δφ = 8πAΩ/(λc), where A is the enclosed area, λ the wavelength, and c the speed of light. Crucially, the effect depends only on the enclosed area and rotation rate — not on the refractive index or fiber length.
Phase shift: Δφ = 8πNAΩ/(λc) for N fiber loops enclosing area A. Effective time difference: Δt = 4NAΩ/c². Fringe shift: Δm = Δφ/(2π) = 4NAΩ/(λc). For a circular loop of radius R: A = πR², so Δφ = 8π²R²ΩN/(λc). The sensitivity scales with enclosed area — this is why fiber optic gyroscopes use many turns of fiber coiled into a compact package.
Fiber optic gyroscopes (FOG) detect rotation by measuring the Sagnac phase shift in a coil of optical fiber — used in aircraft, spacecraft, and autonomous vehicles for inertial navigation. Ring laser gyroscopes (RLG) use a triangular or square cavity where counter-propagating laser beams create beat frequencies proportional to rotation rate. The Michelson-Gale experiment (1925) used a massive Sagnac interferometer to detect Earth's rotation. Modern applications include geophysics, seismology, and precision tests of general relativity.
Adjust the angular velocity Ω slider to rotate the interferometer clockwise (positive) or counterclockwise (negative). The main panel shows two counter-propagating beams — note how the co-rotating beam (red) is stretched and the counter-rotating beam (blue) is compressed as the loop turns. The interference fringe pattern shifts in real time. Adjust R, λ, and N to explore sensitivity scaling. Use presets for common configurations.