Fibre-Optic Gyroscope Test Stand (3D)

The Sagnac effect

Two beams run around the same loop of fibre in opposite directions. If the loop rotates at Omega about its axis, the beam travelling with the rotation needs slightly longer, and the beam travelling against it slightly less. The transit-time difference is dt = 4 N A Omega / c^2 for N turns of area A, which as a phase is

dphi = 2 pi c dt / lambda = 8 pi N A Omega / (lambda c) = 4 pi R L Omega / (lambda c)

with coil radius R and fibre length L = 2 pi R N. The refractive index of the glass does not appear: the Fresnel drag in the moving medium cancels it, which the first check verifies with relativistic velocity addition.

From phase to a measurement

At the coupler the two beams add: P = (Pmax / 2) [1 + cos(dphi + phi_b)]. With no bias the cosine is flat near zero, so small rotations are invisible and the sign is lost. A pi/2 bias puts the operating point on the steepest part, where the output is proportional to sin(dphi). A real gyroscope applies a square-wave phase modulation at the eigenfrequency f_p = c / (2 n L) (half the transit time of the coil) and demodulates the difference of the two half-periods; the second check simulates exactly that in the time domain.

Range and noise

The output is unambiguous while |dphi| < pi/2, so the largest rate is pi / (2 K) with the scale factor K = 4 pi R L / (lambda c). The photon shot noise limits the phase to sigma_phi = 1 / sqrt(N_ph) with N_ph = eta P tau / (h nu) detected photons, hence a minimum detectable rate sigma_phi / K. The slider "excess noise" multiplies this to mimic intensity noise, backscatter and electronics.

Earth rate

The Earth turns once per sidereal day (86 164.09 s), 15.041 deg/h. A coil with a vertical axis at latitude phi senses Omega_E sin(phi), one with a horizontal axis pointing north senses Omega_E cos(phi); that is how a gyrocompass finds north.

Simplifications

Ideal reciprocal coil; no Kerr effect, Shupe thermal drift, polarization errors or back-scattering; no closed-loop feedback; the noise model counts shot noise only, scaled by the excess factor; the table speed is displayed boosted only if you ask for it.

What to try

1. Press Add Earth rotation with the table stopped: a coil at rest still reads the planet's rotation. Switch the axis between vertical and horizontal-north and change the latitude.
2. Select No bias with a small rate: the output barely moves and the measured rate is always positive. Change to the pi/2 bias to recover sign and sensitivity.
3. Raise the averaging time or the power and watch the noise band of the measured rate shrink as 1/sqrt(N).
4. Increase turns or radius: the scale factor and the range limit trade against each other.
5. Push the rate beyond the unambiguous limit and watch the measurement fold back.