Optics 049 · Interference, coherence, cavities, and metrology

Sagnac Gyroscope

A connected 3D fiber-coil gyroscope links true radius and area projection, a balanced four-port coupler and two photodiodes to paired phase-bias acquisition, signed principal readout and reciprocal-phase cancellation.

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 $\text{valid model regime}$

Validated scientific apparatus

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 rotation: the signed sample difference reverses, but unbiased brightness does not. Increase coil radius: the actual coil grows and phase grows with area. Tilt its normal away from the gold rotation axis: the projected response shrinks. Set bias or polarization overlap to zero to lose signed readout. The 3D photodiodes show the positive bias plateau; both bias records are below. Outside the highlighted principal branch the displayed estimate is ambiguous. Drag to orbit; play moves direction markers only.

First-order co-rotating circular-turn Sagnac model; ideal lossless coupler/circulator and sampled differential modulation. No closed-loop gyro, dispersion, winding thickness, lead area, thermal drift, noise or precision claim. At most 24 representative turns; enlarged hardware in the instrument frame. Slowed markers are not photons or a physical delay movie.

Physics tutorial

Audit the physics of Sagnac Gyroscope

BackgroundTwo counterpropagating waves share the same co-rotating fiber, but acquire different propagation lags. For a slowly rotating coil the nonreciprocal phase is $\phi_S=\frac{8\pi\boldsymbol\Omega\cdot\mathbf A_{\mathrm{eff}}}{\lambda_0c}$ . The vacuum wavelength and vacuum light speed belong in this relation. The effect measures rotation relative to inertial space, not a mechanical gyroscope torque or a contradiction of special relativity.

Why it mattersHow can two beams in the same fiber detect rotation, and why can raw brightness not tell its direction?

Start with the essentials

Focus question: How can two beams in the same fiber detect rotation, and why can raw brightness not tell its direction?
One-sentence intuition: A raw cosine response is even in rate. To recover its sign, compare two synchronized differential-bias plateaus: $S=p_{-b}-p_{+b}=V\sin\phi_S\sin b$ . This is still an open-loop periodic signal. Without a known branch, an inverse sine cannot uniquely determine a large rate.

Core mathematical model

Area vector, not fiber length alone

\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 world-up axis. The mint coil normal tilts toward the left. The same radius and tilt set both geometry and area. Equal circular turns and negligible lead area are assumed; the single-turn preset is a fiber loop, not a free-space mirror ring. Large angular rates can cross many optical fringes while tangential speeds remain tiny compared with light speed.

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 co-moving guide the first-order segment law reduces by Stokes to projected area. Relativistic velocity addition accounts for the moving medium; simply replacing c by c/n gives the wrong Sagnac coefficient. Ordinary transit time does depend on group index. The transit summary prescribes an illustrative 1.468, excludes 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 Jones amplitudes before taking power. Two equal-amplitude returned linear polarizations have real overlap V. Orthogonal polarization removes interference, but sends half the power to each port: no absorption is introduced. The ideal circulator sends the common-port return to its front-facing diode; the unused coupler port has a second diode. 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 end-located ideal phase modulator is encountered at different times by the two waves. A square wave with period twice the transit time and modulator amplitude half the selected differential bias produces opposite plateau biases. The acquisition excludes switching edges and assumes constant rate across each pair. It is not a static phase plate, a complete electronic timing simulator or closed-loop phase feedback. The 3D diodes show the positive plateau; the two recorded levels are separate below.

Signed readout has a 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 calculated from the two power records, not the injected phase. It equals the rate only if the injected phase is within the highlighted principal interval and the denominator and projected area are nonzero. Outside that interval a folded value is explicitly ambiguous. At zero bias, zero overlap or orthogonal area there is no signed estimate. No noise floor, calibrated precision or fringe-order recovery is claimed.

Static reciprocal phase cancels, not all noise

\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}

The common-phase control rotates both normalized diagnostic phasors by the same angle. Their overlap and every detector sample stay fixed. These phasors exclude the acquisition bias and do not encode absolute field amplitude. Time-dependent temperature changes sampled at different times can violate cancellation; Shupe drift, Kerr effects, polarization nonreciprocity, loss and detector noise are outside this ideal model.

Common difficulties

Brightness is not a signed speedometer

Typical misconceptionA bright detector reveals rotation direction, and more turns always give an unambiguous rate.

Better mental modelUnbiased brightness is the same for opposite rates. Paired bias samples resolve sign near zero, but more area also narrows the unique open-loop branch. Polarization mismatch reduces contrast without changing the physical Sagnac lag.

Run the experiment

01
Scene 1: Signed fiber-coil readout
Reverse the rate with the shortcut. Compare unbiased brightness, paired records and signed signal.
What to observe: Both directions follow one guided route, exactly reversed. The apparatus is in the instrument frame; gold marks rotation axis, mint marks area normal.
02
Scene 2: Single-turn unbiased limit
Grow the coil radius, then tilt its normal by a right angle. Watch the same geometry govern area and phase.
What to observe: At most 24 representative windings are drawn with schematic pitch. Physical turn count drives the solver; hardware and beam widths are enlarged. Sweeps are ideal references, not reconstructed measurements.
03
Scene 3: Reciprocal phase cancellation
Remove bias or make polarization orthogonal; then try many turns and a large rate to reveal ambiguity.
What to observe: Play moves slowed direction markers only. The bias records, diagnostic phase and principal estimate remain identical when paused or reduced motion is enabled.