It follows that the time difference for completing a cycle for the two beams is Δ Δ T = ∫ ∫ ( d t + − − d t − − ) ≈ ≈ 2 c 2 ∮ ∮ v ⋅ ⋅ d x Remarkably, the time difference is independent of the refraction index n and the velocity of light in the fiber.