Sagnac effect (phase difference)

Description

The Sagnac effect, also called Sagnac interference, named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called a ring interferometer. A beam of light is split and the two beams are made to follow the same path but in opposite directions. On return to the point of entry the two light beams are allowed to exit the ring and undergo interference. The relative phases of the two exiting beams, and thus the position of the interference fringes, are shifted according to the angular velocity of the apparatus. In other words, when the interferometer is at rest with respect to the earth, the light travels at a constant speed. However, when the interferometer system is spun, one beam of light will slow with respect to the other beam of light. This arrangement is also called a Sagnac interferometer. Georges Sagnac set up this experiment to prove the existence of the aether that Einstein’s theory of special relativity had discarded.

A gimbal mounted mechanical gyroscope remains pointing in the same direction after spinning up, and thus can be used as a rotational reference for an inertial navigation system. With the development of so-called laser gyroscopes and fiber optic gyroscopes based on the Sagnac effect, the bulky mechanical gyroscope is replaced by one having no moving parts in many modern inertial navigation systems. The principles behind the two devices are different, however. A conventional gyroscope relies on the principle of conservation of angular momentum whereas the sensitivity of the ring interferometer to rotation arises from the invariance of the speed of light for all inertial frames of reference.

Typically three or more mirrors are used, so that counter-propagating light beams follow a closed path such as a triangle or square. Alternatively fiber optics can be employed to guide the light through a closed path. If the platform on which the ring interferometer is mounted is rotating, the interference fringes are displaced compared to their position when the platform is not rotating. The amount of displacement is proportional to the angular velocity of the rotating platform. The axis of rotation does not have to be inside the enclosed area. The phase shift of the interference fringes is proportional to the platform’s angular velocity ω and is given by a formula originally derived by Sagnac.

The effect is a consequence of the different times it takes a right and left moving light beams to complete a full round trip in the interferometer ring.

Related formulas

Variables

Δ_ϕphase difference (rad)
πpi
λwavelength (m)
cspeed of light (m/s)
ωangular velocity (rad/s)
Aoriented area of loop (m2)