 Ring laser gyroscope

A ring laser gyroscope measures a component of angular velocity from the frequency difference between two counter-circulating waves of a ring laser. Rotation changes their round-trip propagation times through the Sagnac effect. The active optical cavity converts this propagation difference into different resonant frequencies; combining the output waves gives an electrical beat signal through optical heterodyne detection.

For a planar evacuated ideal ring, let $P$ be its full perimeter, $\mathbf{A}$ its oriented vector area, $\mathbf{\Omega }$ its angular velocity and $\lambda$ the nominal vacuum wavelength. At small rotation speed the round-trip time difference has magnitude $|\Delta T|\simeq 4|\mathbf{A}\cdot \mathbf{\Omega }|/c^2$. With $T_0=P/c$ and nominal optical frequency ${\nu }_0=c/\lambda$, the resonant-frequency difference has magnitude ${\nu }_0|\Delta T|/T_0$. Choosing the output sign consistently with the area orientation givesThis is an ideal first-order scale law, with reciprocal paths and resolved counter-circulating modes. Optical dispersion and actual apparatus behaviour require a fuller response model. The underlying circulation-time difference is the one measured by a Sagnac interferometer.^[ref]

For an original square ring with $0.10\mathrm{m}$ sides, $A=0.010{\mathrm{m}}^2$ and $P=0.40\mathrm{m}$. At $\lambda =632.8\mathrm{nm}$ and normal-axis rotation $\Omega =0.010rad/s$,Tilting the ring changes the measured projection, not the full rotation magnitude. Reconstructing a general rotation vector requires independent directional information. The kilohertz-scale difference is also distinct from the optical carrier frequency and the cavity's free spectral range.

A historical experiment made the frequency comparison concrete. W. M. Macek and D. T. M. Davis, Jr. reported rotation sensing in 1963 using a continuous-wave helium-neon ring laser on the $1.153\mu \mathrm{m}$ line. Their metre-square apparatus used four gas tubes with Brewster windows and four external corner mirrors. The first-page diagram directs the two beams toward combining optics and a photocathode, linking the geometric circulation difference to a detectable beat.^[ref]

For two coherent output powers $I_1,I_2$, the ideal combined signal includesOne such intensity record supplies a beat-frequency magnitude, not automatically the rotation sign, since changing the sign can be absorbed into the unknown phase. A directional readout needs additional phase information or a suitable reference. Mode coupling, frequency bias and detector response must also be tested rather than treating a closed optical path as a complete calibrated sensor.