1. Rotating Paths and Light Races

When two light beams depart from the same point and circle a rotating loop in opposite directions, they return at slightly different times. This subtle disparity—first demonstrated by Georges Sagnac in 1913—reveals how rotation alters the geometry of spacetime. The calculator above preserves the traditional purpose of finding the Sagnac time delay but enriches it with an animation. A ring spins while gold and blue pulses chase each other around the rim. One beam gains on the detector as the platform turns toward it; the other must catch up to a retreating target. Watching the race develop conveys the effect far more vividly than a bare formula, especially when you pause, reset, and adjust parameters to see the time gap widen or shrink.

2. Variables and Units

The simulation assumes a perfectly rigid circular loop enclosing area $A$ and rotating at angular velocity $\Omega$. The radius is $r=\sqrt{\frac{A}{\pi }}$, giving circumference $L=2\pi r$. Light of wavelength $\lambda$ travels at speed $c$=299 792 458 m/s in vacuum. All inputs use SI units: square meters for area, radians per second for rotation, meters for wavelength, seconds for timestep. Negative or non‑finite numbers are rejected, and the timestep $\mathrm{\Delta t}$ is clamped between 10^‑7 and 0.01 s to ensure numerical stability. The beams are treated as point pulses; effects like dispersion, relativity beyond first order, and structural deformation of the loop are neglected.

3. Equations at the Core

The classic Sagnac time difference between co‑rotating and counter‑rotating beams is

$$\mathrm{\Delta t}=\frac{4A\Omega }{c^2}$$

The equivalent path length difference is $\mathrm{\Delta L}=c\mathrm{\Delta t}$, and the fringe shift for wavelength $\lambda$ equals $N=\frac{\mathrm{\Delta L}}{\lambda }$. The animation implements the same physics through explicit Euler integration. Let the angular position of the ring be $\theta$, and the beam angles relative to the ring be $\phi$_cw and $\phi$_ccw. The updates each timestep are

$$\theta \leftarrow \theta +\Omega \mathrm{\Delta t}$$

$$\phi$$_cw←φ_cw+crΔt-ΩΔt

$$\phi$$_ccw←φ_ccw-crΔt-ΩΔt

The beams complete the loop when their relative angles reach $\pm \mathrm{2\pi }$. The times taken yield the simulated $\mathrm{\Delta t}$, which is compared with the analytical expression as an error check.

4. Numerical Scheme and Stability

An explicit Euler method suffices because the beams move uniformly except for the rotating frame. The timestep must be small enough that neither beam advances more than a few degrees per step, especially at large rotation rates. The default $\mathrm{\Delta t}=10^\{-5\}$ s keeps the error below 0.1% for meter‑scale rings at 1 rad/s. Doubling the timestep roughly quadruples the error, illustrating the $O(\Delta t)$ scaling of Euler’s method. The code halts if either beam requires more than ten seconds to finish, preventing runaway loops. All computations remain client‑side for performance and privacy.

5. Worked Example

With $A=1$ m² and $\Omega =1$ rad/s, the analytical delay is $4\times 1\times 1/{\mathrm{3\times 10^8}}^2\approx \mathrm{4.45\times 10^\{-17\}}$ s. Press Play and the gold beam (co‑rotating) returns after roughly 2.094395 s, while the blue beam (counter‑rotating) arrives at 2.094395+Δt, with the measured difference matching the formula to within numerical error. Providing a wavelength of 632.8 nm yields a fringe shift of about 2.1×10^-7. The CSV export records time, beam angles, and ring orientation for further analysis.

6. Comparison Table

The table contrasts the baseline configuration with two variants. Each row corresponds to an animation you can reproduce.

Scenario	A (m²)	Ω (rad/s)	Δt (s)	Fringe Shift
Baseline	1	1	4.45e‑17	2.1e‑7
Larger area	4	1	1.78e‑16	8.4e‑7
Faster rotation	1	5	2.22e‑16	1.1e‑6

7. Reading the Animation

The rotating ring is drawn in gray with a small black notch marking the emitter–detector. The gold pulse moves in the same direction as the rotation, while the blue pulse travels against it. Progress bars below the canvas show fractional completion of each beam—gold for co‑rotating, blue for counter—providing a color‑independent cue. The caption narrates arrival times and computed delays so screen‑reader users receive a descriptive summary. Pausing mid‑race freezes the beams, letting you inspect their positions and the current ring orientation.

8. Limitations and Extensions

The model assumes non‑relativistic rotation ($\mathrm{\Omega r\backslash ll\; c}$) and ignores gravitational or inertial frame corrections. Real ring laser gyroscopes amplify the effect by allowing light to circulate thousands of times or by using fiber coils with enormous effective area. Extending this simulator to include multiple turns, matter‑wave beams, or relativistic metrics would deepen its fidelity. For analytic calculations without animation, see the Sagnac phase shift calculator or review rotational kinematics via the rotational kinematics calculator.

9. References

Primary sources include G. Sagnac, “The demonstration of the luminiferous aether by an interferometer in uniform rotation,” C. R. Acad. Sci. (1913), and M. G. de Bizemont, Ring Laser Gyroscopes: Theory and Engineering. Discussions of numerical integration are drawn from W. Press et al., Numerical Recipes, while modern applications are summarized in D. Titterton & J. Weston, Strapdown Inertial Navigation Technology.