Engineering & Science
Sagnac Interferometer Phase Shift Formula (Fiber-Optic Gyroscope FOG)
Calculate counter-propagating laser Sagnac phase shift Δφ = (2 * π * L * D * Ω_rad_s) / (λ * c) (radians) in aerospace and subsea inertial navigation fiber-optic gyroscopes.
Measuring spacecraft and submarine rotation with zero moving parts via Georges Sagnac’s 1913 relativistic loop!
When a laser beam splits into clockwise and counterclockwise waves around a rotating fiber coil of total length L and diameter D, the co-rotating wave travels slightly farther before meeting the beam-splitter again! Georges Sagnac (1913) showed that the interference phase difference depends strictly on enclosed coil area sum N * A = L * D / 4 and vacuum wavelength λ (independent of glass refractive index due to Fresnel-Fizeau relativistic drag cancellation!): Δφ = (8 * π * N * A * Ω) / (λ * c) = (2 * π * L * D * Ω_rad_s) / (λ * c)!
Design a telecom-band C-band inertial navigation FOG (λ = 1,550 nm, fiber length L = 500 m wound on diameter D = 0.12 m): a 10 deg/s vehicle yaw rate (0.17453 rad/s) induces Δφ = 0.1416 rad, scaling linearly to 0.0142 rad at 1 deg/s!
| Angular Rotation Rate Ω (L = 500 m, D = 0.12 m, λ = 1550 nm) | Rate in rad/s | Sagnac Phase Shift Δφ (rad) | Navigation Regime |
|---|---|---|---|
| 10.0 deg/s (UAV / Autonomous Turn) | 0.1745 rad/s | 0.1416 rad | High-Dynamic Slew |
| 1.0 deg/s (Precision Attitude Hold) | 0.0175 rad/s | 0.0142 rad | Sub-Milliradian Phase Lock |
| Invalid L <= 0, D <= 0, or λ <= 0 | Undefined | "Invalid FOG Coil Parameters" | Optical Geometry Guard |
Sagnac Interferometric Phase Shift Δφ (Radians)
Evaluates (2 * PI() * L_m * D_m * (Ω_deg_s * PI() / 180)) / ((λ_nm * 1E-9) * 299792458) across fiber length B2 (m), coil diameter B3 (m), angular velocity B4 (deg/s), and laser wavelength B5 (nm).
=IF(OR(B2 <= 0, B3 <= 0, B5 <= 0), "Invalid FOG Coil Parameters", ROUND((2 * PI() * B2 * B3 * (B4 * PI() / 180)) / ((B5 * 1E-9) * 299792458), 4))FOG Scale Factor S_FOG (Radians per rad/s)
Computes the intrinsic optical scale factor 2 * PI() * L * D / (λ * c) in seconds (rad / (rad/s)).
=IF(OR(B2 <= 0, B3 <= 0, B5 <= 0), "Invalid FOG Coil Parameters", ROUND((2 * PI() * B2 * B3) / ((B5 * 1E-9) * 299792458), 4))Try the example
Use the original values or change them to test your understanding.