Sagnac Effect in Resonant Microcavities

We report the Sagnac effect in resonant microcavities. The Sagnac effect is the phase and frequency difference between two counter propagating laser beams in a rotating ring resonator, and has been studied for a long time. It forms the basis for the optical gyroscopes, such as ring laser gyroscope or fiber optic gyroscopes. Conventional theoretical approach for the Sagnac effect has been derived from the assumption that the light propagates one-dimensionally and the wavelength of the light is typically much smaller than the cavity length. However, nowadays, micro-fabrication techniques are developed enough that the cavity size can approach the size of the wavelength of the light. The conventional formalism for the Sagnac effect, therefore, breaks down and needs to be replaced. We derive the Sagnac effect in resonant microcavities theoretically and numerically without the conventional assumption and show that the frequency shift due to the Sagnac effect occurs as a threshold phenomenon for rotation velocity in a rotating microcavity by employing perturbation theory typically used in quantum mechanics. The threshold exists even in the absence of the backscattering, which causes the lock-in phenomenon, and depends on the geometric shape and the symmetry of resonant cavities. It is also shown that the eigenfunctions of a rotating microcavity become rotating waves above the threshold while they are standing waves below the threshold. Our theoretical approach can be applied to the resonant cavities of arbitrary shapes and can make it possible to design compact optical gyroscopes that have a low threshold