Fiber-optic gyroscope

A fiber-optic gyroscope (FOG) measures rotation rate with no moving parts. Light from a broadband source is split into two beams that travel in opposite directions around a coil of fiber, typically hundreds of meters to a few kilometers long, and are recombined at a detector. When the coil rotates about its axis, the beam traveling with the rotation takes slightly longer to return than the other: this is the Sagnac effect. The resulting phase difference is proportional to the rotation rate. FOGs are used in aircraft and marine inertial navigation, spacecraft attitude control, platform stabilization, surveying and drilling, with performance grades from a few degrees per hour down to about 0.01°/h and below.

Sagnac phase

For a coil of total fiber length $L$ wound on a diameter $D$, rotating at rate $\Omega$ about the coil axis, the phase difference between the counter-propagating beams is

$$\Delta\varphi_S = \frac{2\pi L D}{\lambda c}\,\Omega$$

The refractive index of the fiber does not appear. The factor multiplying $\Omega$ is the scale factor. For $L$ = 1 km, $D$ = 0.1 m and $\lambda$ = 1550 nm it is 1.35 s, so the Earth's rotation rate, 15.04°/h or $7.29 \times 10^{-5}$ rad/s, produces a phase of $9.9 \times 10^{-5}$ rad. A rate of 0.01°/h corresponds to $6.6 \times 10^{-8}$ rad. Measuring phases this small is the central engineering problem; winding more fiber on a larger coil raises the signal, at the expense of size, price and sensitivity to temperature gradients.

The phase is a Sagnac interferometer signal, so the detected power varies as $1 + \cos\Delta\varphi_S$. The open-loop response is unambiguous only while $|\Delta\varphi_S| < \pi$, which for the coil above is up to 2.32 rad/s, or 133°/s.

Minimum reciprocal configuration

The two beams share the same fiber, so almost all disturbances affect them equally and cancel. To keep that cancellation exact, practical gyroscopes use the minimum reciprocal configuration: light enters and leaves the loop through the same single-mode port and passes through a polarizer, so both beams follow identical paths apart from the direction of travel. The main components are:

• a broadband source, usually a superluminescent diode or an erbium-doped fiber ASE source, whose short coherence length suppresses errors from backscatter and the Kerr effect;
• a source coupler that sends the returning light to the photodiode;
• an integrated-optic chip, usually on lithium niobate, combining the polarizer, the Y-junction splitter and a pair of phase modulators;
• the sensing coil, often wound from polarization-maintaining fiber with a small cladding diameter to reduce coil volume.

Bias modulation and closed loop

At zero rotation the cosine response has zero slope, so a small rate would give no first-order signal. A phase modulator at one end of the coil applies a square-wave or sinusoidal bias modulation. Because the two beams pass the modulator at times separated by the loop transit time $\tau = n_g L / c$, they receive different phases, and the output becomes proportional to $\sin\Delta\varphi_S$. The modulation is most effective at the proper frequency $f_p = 1/(2\tau)$: for 1 km of fiber with a group index of 1.468, $\tau$ = 4.90 µs and $f_p$ = 102 kHz.

Closed-loop gyroscopes add a feedback phase ramp (a serrodyne or digital staircase) that cancels the Sagnac phase. The ramp's frequency is then the rate output, which makes the scale factor linear and stable over a wide range and independent of the source power.

Error sources

• Shupe effect. A time-varying temperature gradient reaches the two beams at different times and produces a non-reciprocal phase. Symmetric winding patterns, such as quadrupolar and octupolar windings, place fiber segments equidistant from the coil midpoint next to each other.
• Backscatter. Rayleigh scattering in the coil sends light back toward the detector, where it can interfere coherently with the main signal; the broadband source reduces it.
• Polarization non-reciprocity. Imperfect polarizers and cross-coupling in the coil allow the two directions to take different polarization paths; a polarizer with high extinction and PM fiber control it.
• Magnetic fields. The Faraday effect in twisted fiber produces a non-reciprocal phase, so high-grade coils are magnetically shielded.

The residual errors are specified as bias instability (°/h), angle random walk (°/√h) and scale-factor error (ppm).

Common questions

How does a fiber-optic gyroscope differ from a ring laser gyroscope?

A ring laser gyroscope uses an active laser cavity and measures the beat frequency between two counter-propagating laser modes, which is proportional to rotation. A FOG is passive: an external source and a long fiber coil produce a phase shift. Both use the Sagnac effect and reach navigation grade; the FOG avoids the lock-in effect of ring lasers at low rates and has no high-voltage gas discharge.

Why does a FOG use a broadband source instead of a laser?

Coherent light makes backscattered and polarization-coupled light interfere with the main signal, producing bias drift and noise, and it enhances the non-reciprocal Kerr phase. A source with a coherence length of tens of micrometers averages these effects out.

How accurate is a fiber-optic gyroscope?

Tactical-grade units have bias instabilities of roughly 0.1–10°/h; navigation-grade units reach about 0.01°/h; the best strategic and space instruments are better than 0.001°/h. Coil length, coil diameter, source stability and thermal design set the grade.

References: H. C. Lefèvre, The Fiber-Optic Gyroscope, 3rd ed. (Artech House, 2022); G. Sagnac, C. R. Acad. Sci. 157, 708 (1913); V. Vali, R. W. Shorthill, Appl. Opt. 15, 1099 (1976); D. M. Shupe, Appl. Opt. 19, 654 (1980).