= Ring laser gyroscope
A \b[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>, $\boldsymbol\Omega$ its <angular velocity> and $\lambda$ the nominal vacuum wavelength. At small rotation speed the round-trip time difference has magnitude $|\Delta T|\simeq4|\mathbf A\cdot\boldsymbol\Omega|/c^2$. 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 gives
$$
\boxed{f_S=\frac{4\mathbf A\cdot\boldsymbol\Omega}{\lambda P}.}
$$
This 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>.<Regarding the Proof for the Existence of a Luminiferous Ether Using a Rotating Inteferometer Experiment by Georges Sagnac>[Original rotating-interferometer experiment; translation, pp.77–80]{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 $\Omega=0.010\,\mathrm{rad/s}$,
$$
f_S=\frac{4(0.010)(0.010)}{(632.8\times10^{-9})(0.40)}\simeq1580\,\mathrm{Hz}.
$$
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 $1.153\,\mathrm{\mu m}$ 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.<Rotation Rate Sensing with Traveling-wave Ring Lasers by W. M. Macek and D. T. M. Davis, Jr.>[Original first-page apparatus paragraph and Figure1, Applied Physics Letters2, pp.67–68, 1963; reproduced in the linked INFN presentation]{ref}
For two coherent output powers $I_1,I_2$, the ideal combined signal includes
$$
I(t)=I_1+I_2+2\sqrt{I_1I_2}\cos(2\pi f_S t+\phi).
$$
One 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.
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