Universal symmetry-breaking dynamics for the Kerr interaction of counterpropagating light in dielectric ring resonators

Spontaneous symmetry breaking is an important concept in many areas of physics. A fundamentally simple symmetry-breaking mechanism in electrodynamics occurs between counterpropagating electromagnetic waves in ring resonators, mediated by the Kerr nonlinearity. The interaction of counterpropagating light in bidirectionally pumped microresonators finds application in the realization of optical nonreciprocity (for optical diodes), studies of $\mathcal{PT}$-symmetric systems, and the generation of counterpropagating solitons. Here, we present comprehensive analytical and dynamical models for the nonlinear Kerr interaction of counterpropagating light in a dielectric ring resonator. In particular, we study discontinuous behavior in the onset of spontaneous symmetry breaking, indicating divergent sensitivity to small external perturbations. These results can be applied to realize, for example, highly sensitive near-field or rotation sensors. We then generalize to a time-dependent model, which predicts different types of dynamical behavior, including oscillatory regimes that could enable Kerr-nonlinearity-driven all-optical oscillators. The physics of our model can be applied to other systems featuring Kerr-type interaction between two distinct modes, such as for light of opposite circular polarization in nonlinear resonators, which are commonly described by coupled Lugiato-Lefever equations.

Figure 1

A simplified schematic of a ring resonator setup in a symmetry-broken state. Amplified laser light $(\lambda =1550$ nm) is split into two paths of equal powers, which are then sent in opposite directions into a microresonator. (b) Above a critical power threshold, the parity symmetry is broken, characterized by different resonant frequencies in the two directions such that one direction of light is preferentially coupled into the cavity. Panel (a) shows the case for preferential coupling in the clockwise direction. CW = clockwise, CCW = counterclockwise.

Figure 2

Illustration of Eq. (2), each ellipse for a different (constant) value of the detuning parameter $\mathrm{\Delta }$. Each curve comprises all the values of $p_{1,2}$ at which the two Lorentzians, Eq. (1), intersect. When $p_1\ne p_2$, the symmetry is broken, the threshold of which occurs where an ellipse intersects the straight line. These curves are traced out by continuously increasing the pump power, $\tilde{p}$.

Figure 3

Illustration of Eq. (9), in the case of balanced pump powers, ${\tilde{p}}_1={\tilde{p}}_2=\tilde{p}$. Each curve corresponds to a different choice of pump powers. Each pair of circulating powers, $(p_1,p_2)$, corresponds to a particular pair of detunings, $({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$.

Figure 4

Symmetry-broken solutions under balanced pumping. (a) Tilted resonances for various pump powers, showing the symmetry-broken region. Dark solid curves indicate stable solutions, faint curves show unstable solutions, and dashed curves correspond to oscillatory behavior. (b) Amplitude of the difference in the coupled powers. The gray area corresponds to time-oscillating solutions (white denotes symmetric states). (c) Isolated curve for $\tilde{p}=3.8$, from (a). (d) Real part of the stability eigenvalue, $\lambda$. (e) Imaginary part of the stability eigenvalue. Note that there always exists at least one eigenvalue with a nonzero imaginary part, implying strong susceptibility to oscillations.

Figure 5

Detuning scans of the oscillatory regimes of Eq. (10) between two Hopf bifurcations [lower solutions of Fig. 4]. The field intensity is sampled at its maximum during the oscillation. A single value of $p_1$ for a given $\mathrm{\Delta }$ means that only one (maximal) value of $p_1$ intersects the Poincaré section during its trajectory with a single periodicity. In contrast, two values of $p_1$ for a specified $\mathrm{\Delta }$ means that the maximum of $p_1$ is cycling between two distinct values with an overall period that has doubled. Further period doublings are observed for larger values of $\mathrm{\Delta }$, which eventually transition into chaos. The arrows indicate the direction of scanning the detuning. (a) Scan for $\tilde{p}=3.4$ (showing no dependence on direction). Forward (b) and backward (c) scans for $\tilde{p}=3.8$.