handbook of modern sensors

318 8 Velocity and Acceleration
Fig. 8.13. Sagnac effect.
ring having refractive index n and radius R. One beam goes in a clockwise (CW)
direction, and the other goes in a counterclockwise (CCW) direction. The amount of
time it takes light to travel within the ring is t = 2πR/nc, where c is the speed of
light. Now, let us assume that the ring rotates with angular rate in the clockwise
direction. In that case, light will travel different paths at two directions. The CW beam
will travel l cw = 2πR+ Rt, and the CCW beam will travel l ccw = 2πR–Rt.
Hence, the difference between the paths is
l = 4πR2 . (8.16)
nc
Therefore, to accurately measure , a technique must be developed to determine l.
There are three basic methods known for the path detection: (1) optical resonators,
(2) open-loop interferometers, and (3) closed-loop interferometers.
For the ring laser gyro, measurements of l are made by taking advantages of
the lasing characteristics of an optical cavity (i.e., of its ability to produce coherent
light). For lasing to occur in a closed optical cavity, there must be an integral number
of wavelengths about the complete ring. The light beams, which do not satisfy this
condition, interfere with themselves as they subsequently travel the optical path. In
order to compensate for a change in the perimeter due to rotation, the wavelength λ
and frequency ν of the light must change:
− dv
v = dλ
λ = dl
l . (8.17)
Equation (8.17) is a fundamental equation relating frequency, wavelength, and
perimeter change in the ring laser. If the ring laser rotates at a rate , then Eq.
(8.16) indicates that light waves stretch in one direction and compress in the other
direction to meet the criteria for the lasing of an integral number of wavelengths about
the ring. This, in turn, results in a net frequency difference between the light beams.
If the two beams are bit together (mixed), the resulting signal has frequency is
F = 4A
λnl , (8.18)
where A is the area enclosed by the ring.