Fig. 1. Sketches of bidirectional ML gyro configurations (left) and their corresponding beat-note responses (right). The × corresponds to a pulse crossing that does not introduce phase coupling. G is the gain, and Δϕ=Δ/τrt is the differential round-trip phase shift. Note that the time unit in Eq. (1) has been normalized to τrt. (a) Linear (i.e., no dead band) response. (b) When a scattering interface is placed at the crossing point, a square-root (i.e., dead band) response is observed (data from Ref. [21]).

Fig. 2. Polar plots of the imaginary versus real part of the beat field near (left) and far (right) from the EP (dead band). Near the EP the beat note stems from amplitude modulation, while far from the EP it is caused by pure phase modulation.

Fig. 3. Beat-signal spectrum from a numerical solution of Eq. (1) (top), and experimentally measured beat-note signal (bottom) showing the clustering of frequency harmonics near the dead band (left) and their absence for larger Δ (right).

Fig. 4. Beat-signal spectrum from a numerical solution of Eq. (1) showing the clustering of harmonics near the EP (dead band).

Fig. 5. Gyro beat-note response curve changes with κ˜ and s. All large circles are beat frequencies numerically solved from Eq. (1) with κ˜=0.05 and s=0 (blue), s=0.03 (orange), s=0.05 (yellow), and s=0.06 (purple). The green circles are with κ˜=0 and s=0.05. The red-dashed curves correspond to the eigenvalue beat frequency 2Δω determined from Eq. (5). When saturable gain is included, the green circles shift to the positions of the cyan crosses because the COG (rather than the most prevalent peak) of the spectrum must be used. An example of data matching the κ˜=0 case can be found in Ref. [21].

Fig. 6. Numerical solution (blue circles) to Eq. (1) and analytic prediction (red dashes) of Eq. (6) showing an EP at Δ=0 for κ˜=0.05 with a saturable gain in one resonator, α^1=0.051, αL=0, β=1, γ=0, and a constant loss in the other, α2=−|κ˜|=−0.05. Inset: because the gain difference depends on Δ, it only cancels the coupling exactly at Δ=0. The time dependence of the fields in polar coordinates is shown in Visualization 1 for Δ=0.01.