Fig. 1. Schematics and experimental setup. (a) Schematic showing the four fields (E1,cw, E1,ccw, E2,cw, E2,ccw) circulating within the two coupled resonators with two inputs. (b) Experimental setup used to observe the linear and nonlinear interactions of light within the coupled twin-resonator system. ECDL, external cavity diode laser; EDFA, erbium-doped fiber amplifier; VA, variable attenuator; PD, photodiode. (c) Side view photograph of the two coupled rod-resonators used in our experiments with the coupling region highlighted in the white dashed box. (d) Schematic showing the rod–taper coupling as well as the inter-resonator coupling mechanism used in our experimental setup. The fixed resonator is denoted as Res.1, whereas the invertedly clamped resonator is Res.2. (e) Image of the resonators, tapered fibers, and the translational stages used in our experiment.

Fig. 2. Theoretical calculation of the resonance splitting in a coupled resonator system. Top [(a), (b), (c)] and bottom [(d), (e), (f)] panels correspond to coupled resonator systems with zero and nonzero differences in cold cavity resonance frequencies respectively. (a), (d) Eigenvalue analysis reveals resonance splitting dependence on inter-resonator coupling. The solid, dashed, and dotted lines show the resonance frequency splitting at different loss ratios of the two resonators. (b), (e) Power coupled into one of the resonators with respect to the frequency detuning. In each case, a new resonance appears for increasing values of the coupling J. (c), (f) Numerical analysis of Eq. (1) shows an increasing resonance splitting with increasing inter-resonator coupling. Used parameter: κ1=1  MHz.

Fig. 3. Mode hybridization at different resonator coupling strengths. The top row shows theoretical results while the bottom row displays experimental data. The graphs are transmission spectra of the two resonators, plotted as a function of the laser frequency detuning from the cold resonance (the resonance of an uncoupled resonator at low optical powers) of resonator 1. (a)–(c) Theoretically calculated transmission profiles for different values of J, while keeping other parameters of Eq. (3) constant (κ1=11.3  MHz, κex,1=5.96  MHz, κ2=10.9  MHz, κex,2=4.71  MHz, δω0,1−δω0,2=2.5κ1,h1,cw=h2,cw=3.4κ0,1). For the simulated scans, J=0 (a), J=κ1 (b), J=1.3κ1 (c). (d) Experimentally obtained transmission spectra when the two resonators are uncoupled. (e), (f) Resonance splitting appears with increasing inter-resonator coupling.

Fig. 4. Circulating field intensities at high power in identical twin-resonators while the input pump scans from high to low frequency at fixed power. (a)–(c) High circulating powers lead to symmetry breakings that result in sudden changes in the power distribution within the resonator modes. The three panels show different combinations of dominant modes after the symmetry breaking bifurcations. For better visualization, we added a tiny vertical offset to some of the colored graphs in order to avoid them being plotted on top of each other in the symmetric regions.

Fig. 5. Resonance spectra of two asymmetric microresonators at high input power. (a)–(c) Simulated transmission profiles as a function of the laser frequency offset are plotted with varying J while other parameters of Eq. (1) are constant (κ12=2κex,1,2=314  MHz, δω0,1−δω0,2=κ1,h1,cw=h2,cw=3.4κ1). The different values of inter-resonator coupling strength J are (a) 0, (b) 1.2κ1, (c) 2.2κ1. (d)–(f) Transmission plots of the resonators while increasing the inter-resonator coupling from uncoupled (d) to maximum inter-resonator coupling (f), where the long monotonous region is horizontally compressed in the shaded region. The green region in (b), (c), (e), and (f) shows the overlap of the first split-resonances (higher-frequency ones) of the two resonators, where the yellow shaded region in (c) and (f) highlights the field interactions of the split resonance at lower frequencies.