Optics 049 · Interference, coherence, cavities, and metrology

Sagnac Gyroscope

How can two beams in the same fiber detect rotation, and why can’t brightness alone tell which way it turns? Send light both ways around a fiber coil. When the coil turns, the beam going with the rotation takes slightly longer to get around than the other; the phase difference is proportional to rotation rate and enclosed area. Brightness follows the cosine of that phase, which is the same for either direction of turn, so a fixed phase bias is added to tell the direction.

In a 3D fiber-coil gyroscope, the coil radius and its projected area, a balanced four-port coupler and two photodiodes work together. Take paired phase-biased samples, read the signed rotation on the principal branch and see reciprocal phases cancel.

Injected Sagnac phase / propagation lag

\phi_S,\Delta t

$0.50$

Paired return-port samples / signed difference

p_{+b},p_{-b},S

$50\%$

Principal rate only / visibility / coil length

\widehat\Omega_{\mathrm{pr}},V,L

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Sagnac Gyroscope

Signed rotation rate (positive: right-handed about gold axis)

\Omega

10\,\frac{\mathrm{deg}}{\mathrm s}

Actual mean coil radius (geometry follows)

r

80\,\mathrm{mm}

Physical turns (at most 24 representative turns drawn)

N

500

Vacuum centre wavelength

\lambda_0

1550\,\mathrm{nm}

Angle between coil normal and rotation axis

\alpha

0^{\circ}

Returning-beam polarization mismatch

\theta

0^{\circ}

Paired nonreciprocal phase-bias amplitude

b

90^{\circ}

Equal static reciprocal phase on both beams

\chi

0^{\circ}

Reverse the rotation and the signed difference between samples reverses, but the brightness without bias does not change. Increase the coil radius: the coil grows, and the phase grows with its area. Tilt the coil normal away from the gold rotation axis and the projected response shrinks. Set the bias or the polarization overlap to zero and the readout loses its sign. The 3D photodiodes show the positive-bias plateau; both bias records appear below. Outside the highlighted principal branch, the displayed estimate is ambiguous. Drag to orbit; Play only moves direction markers.

The model is first-order Sagnac for circular turns that rotate with the instrument. The coupler and circulator are ideal and lossless, and the modulation is sampled differentially. Closed-loop operation, dispersion, winding thickness, lead area, thermal drift and noise are not modeled, and no precision is claimed. At most 24 turns are drawn to represent the coil, and the hardware is enlarged and shown in the instrument's frame. The slowed markers are not photons and do not show the real delay.

Physics tutorial

Audit the physics of Sagnac Gyroscope

BackgroundTwo waves travel in opposite directions through the same fiber, which rotates with the instrument, and they pick up different delays. For a slowly rotating coil the nonreciprocal phase is $\phi_S=\frac{8\pi\boldsymbol\Omega\cdot\mathbf A_{\mathrm{eff}}}{\lambda_0c}$ . This relation uses the vacuum wavelength and the vacuum speed of light. The effect measures rotation relative to inertial space. It has nothing to do with the torque of a mechanical gyroscope, and it does not contradict special relativity.

Why it mattersHow can two beams in the same fiber detect rotation, and why can’t brightness alone tell which way it turns?

Start with the essentials

Focus question: How can two beams in the same fiber detect rotation, and why can’t brightness alone tell which way it turns?
One-sentence intuition: The raw cosine response is the same for opposite rotation rates. To recover the sign, compare two synchronized plateaus of opposite differential bias: $S=p_{-b}-p_{+b}=V\sin\phi_S\sin b$ . The signal is still open-loop and periodic. Unless you know which branch you are on, the inverse sine cannot pin down a large rate uniquely.

Core mathematical model

The phase depends on the coil's area vector

\begin{aligned}A_{\parallel}&=N\pi r^2\cos\alpha\\\Delta t&=4A_{\parallel}\Omega/c^2\\\phi_S&=2\pi c\Delta t/\lambda_0\end{aligned}

Positive rotation is right-handed about the gold vertical axis. The mint coil normal tilts toward the left. The same radius and tilt set both the drawn geometry and the area. The turns are assumed to be equal circles, and the area enclosed by the leads is neglected. The single-turn preset is a fiber loop, not a free-space ring of mirrors. At large angular rates the phase can sweep through many optical fringes while the coil's rim still moves far slower than light.

Why co-rotating glass does not multiply the phase

\begin{aligned}\phi_S&=\frac{4\pi}{\lambda_0c}\oint\mathbf v\cdot d\mathbf l\\\mathbf v&=\boldsymbol\Omega\times\mathbf r\\\phi_S&=\frac{8\pi\boldsymbol\Omega\cdot\mathbf A_{\mathrm{eff}}}{\lambda_0c}\end{aligned}

For a guide that moves with the coil, the first-order law for each segment adds up, by Stokes' theorem, to the projected area. Relativistic velocity addition handles the moving medium; simply swapping in the speed of light in glass gives the wrong Sagnac coefficient. The ordinary transit time does depend on the group index. The transit summary uses an illustrative group index of 1.468, leaves out the leads and does not fit material dispersion.

Two physical ports and polarization overlap

\begin{aligned}V&=\cos\theta\\p(\varphi)&=(1+V\cos\varphi)/2\\p_{\mathrm{other}}&=1-p(\varphi)\end{aligned}

An ideal balanced four-port fiber coupler adds the Jones amplitudes before power is taken. When the two returning waves are linearly polarized with equal amplitude, their overlap is a real number, and it sets the fringe visibility. Orthogonal polarizations do not interfere, and each port then gets half the power; nothing is absorbed. The ideal circulator sends the light returning to the common port onto its front-facing photodiode, and the unused coupler port has a second photodiode. The input is fixed at 1 mW.

Nonreciprocal bias needs a timing difference

\begin{aligned}\tau&=n_gL/c,\quad L=2\pi rN\\b(t)&=m(t)-m(t-\tau)\\p_{\pm b}&=\frac{1+V\cos(\phi_S\pm b)}2\end{aligned}

An ideal phase modulator sits at one end of the coil, so the two waves pass through it at different times. Driving it with a square wave whose period is twice the transit time, at an amplitude of half the chosen differential bias, gives plateaus of opposite bias. The acquisition skips the switching edges and assumes the rate is constant over each pair. This is not a static phase plate, a full simulation of the electronic timing or closed-loop phase feedback. The photodiodes in the 3D view show the positive plateau; the two recorded levels are shown separately below.

The signed readout is valid on the principal branch

\begin{aligned}S&=p_{-b}-p_{+b}\\\widehat\phi_{\mathrm{pr}}&=\arcsin\frac{S}{V\sin b}\\\widehat\Omega_{\mathrm{pr}}&=\widehat\phi_{\mathrm{pr}}/K\\K&=8\pi A_{\parallel}/(\lambda_0c)\end{aligned}

The estimate is computed from the two recorded powers, not from the injected phase. It equals the rate only when the injected phase lies inside the highlighted principal interval and both the denominator and the projected area are nonzero. Outside that interval the folded value is explicitly marked as ambiguous. With zero bias, zero overlap or an area perpendicular to the rotation axis, there is no signed estimate. The model claims no noise floor, no calibrated precision and no recovery of the fringe order.

A static reciprocal phase cancels; other noise may not

\begin{aligned}E_1'&=e^{i\chi}E_1,\quad E_2'=e^{i\chi}E_2\\E_1'^*E_2'&=E_1^*E_2\end{aligned}

A phase common to both waves rotates the two normalized diagnostic phasors by the same angle, so their overlap and every detector sample stay the same. These phasors leave out the acquisition bias and do not carry the absolute field amplitude. A temperature change that the two waves sample at different times can break the cancellation. Shupe drift, Kerr effects, polarization nonreciprocity, loss and detector noise are outside this ideal model.

Common difficulties

Brightness alone cannot tell the direction of rotation

Typical misconceptionA brighter detector shows which way the coil is turning, and more turns always give a rate with no ambiguity.

Better mental modelWithout bias, the brightness is the same for opposite rates. Pairs of biased samples reveal the sign near zero rate, but a larger area also narrows the range of rates that the open-loop readout can identify uniquely. A polarization mismatch lowers the contrast without changing the physical Sagnac delay.

Run the experiment

01
Scene 1: Signed fiber-coil readout
Reverse the rate with the shortcut. Compare the unbiased brightness, the paired records and the signed signal.
What to observe: Both waves follow the same guided route in exactly opposite directions. The apparatus is drawn in the instrument frame: gold marks the rotation axis and mint marks the area normal.
02
Scene 2: Single-turn unbiased limit
Increase the coil radius, then tilt its normal through a right angle. The same geometry controls both the area and the phase.
What to observe: At most 24 representative turns are drawn, with a schematic spacing. The solver uses the physical number of turns, and the hardware and beam widths are drawn enlarged. The sweeps are ideal references, not reconstructed measurements.
03
Scene 3: Reciprocal phase cancellation
Remove the bias or make the polarizations orthogonal. Then try many turns and a large rate to see the readout become ambiguous.
What to observe: Play only moves slowed-down direction markers. The bias records, the diagnostic phase and the principal-branch estimate stay the same when the animation is paused or reduced motion is on.