Sagnac Effect - TIme 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. To act as a ring the trajectory must enclose an area. 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. This arrangement is also called a Sagnac interferometer.
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.

Theory

The shift in interference fringes in a ring interferometer can be viewed intuitively as a consequence of the different distances that light travels due to the rotation of the ring.(Fig. 3)[12] The simplest derivation is for a circular ring of radius R, with a refractive index of one, rotating at an angular velocity of \omega , but the result is general for loop geometries with other shapes. If a light source emits in both directions from one point on the rotating ring, light traveling in the same direction as the rotation direction needs to travel more than one circumference around the ring before it catches up with the light source from behind.

Likewise, the light traveling in the opposite direction of the rotation will travel less than one circumference before hitting the light source on the front side.

The time difference is shown here.

Related formulas

Variables

Δttime difference (s)
πpi
Rradius of circular ring with a refractive index of one (m)
ωangular velocity (1/s)
cSpeed of light