Critical dynamics of an asymmetrically bidirectionally pumped optical microresonator

An optical ring resonator with third-order, or Kerr, nonlinearity will exhibit symmetry breaking between the two counterpropagating circulating powers when pumped with sufficient power in both the clockwise and counterclockwise directions. This is due to the effects of self- and cross-phase modulation on the resonance frequencies in the two directions. The critical point of this symmetry breaking exhibits universal behaviors including divergent responsivity to external perturbations, critical slowing down, and scaling invariance. Here we derive a model for the critical dynamics of this system, first for a symmetrically pumped resonator and then for the general case of asymmetric pumping conditions and self- and cross-phase modulation coefficients. This theory not only provides a detailed understanding of the dynamical response of critical-point-enhanced optical gyroscopes and near-field sensors, but is also applicable to nonlinear critical points in a wide range of systems.

Figure 1

Solutions to Eq. (1) under symmetric pumping conditions ${\tilde{p}}_{1,2}=\tilde{p}$ and ${\mathrm{\Delta }}_{1,2}=\mathrm{\Delta }$, illustrated for $\tilde{p}=1.75$. Solid black and grey lines represent stable and unstable solutions respectively. Between the critical points ${\text{A}}_1$ and ${\text{A}}_2$, the symmetric solution $p_1=p_2$ is unstable and two new symmetry-broken stable solutions appear in which $p_1$ and $p_2$ take the opposite branches shown. Another unstable region lies between ${\text{B}}_1$ and ${\text{B}}_2$, which occurs even for a unidirectionally pumped resonator and here corresponds to a symmetric bistability.

Figure 2

Response of a resonator to a sinusoidally modulated splitting between clockwise and counterclockwise modes (e.g., from a sinusoidal back-and-forth rotation of the resonator). The response is scale invariant close to the critical point. Starting from the steady state at the symmetric critical point $p=1$, $\mathrm{\Delta }=2$, a sinusoidal, purely differential-mode detuning perturbation of the form ${\mathrm{\Delta }}_1\left(t\right)=\mathrm{\Delta }+{\delta }_{\text{d}}\left(t\right)$, ${\mathrm{\Delta }}_2\left(t\right)=\mathrm{\Delta }-{\delta }_{\text{d}}\left(t\right)$, with ${\delta }_{\text{d}}\left(t\right)={\delta }_{\text{d}}^{\text{AC}}cos\omega t$, is applied from time $t=0$. This is done using ${\delta }_{\text{d}}^{\text{AC}}=0.5{\kappa }^3$ and $\omega =0.2{\kappa }^2$ for three values of the scaling parameter $\kappa$, namely 1.0, 0.5, and 0.2. The solid blue line shows ${\delta }_{\text{d}}\left(t\right)/{\kappa }^3$ vs ${\kappa }^{2}t/\pi$, which is the same curve in each case. The dotted lines show the resulting “true” differential-mode circulating powers $p_{\text{d}}^{\text{t}}=(p_1-p_2)/2$ calculated directly using Eq. (2), divided by $\kappa$, for each value of $\kappa$. As $\kappa$ decreases, these converge on a single curve, suggesting that close to the critical point the system's dynamics can be described by effective equations that are scale-invariant with respect to $\kappa$. The solid green line correspondingly shows $p_{\text{d}}/\kappa$ calculated using the scale-invariant leading-order approximation (“LOA”) developed in this paper, which is the limit of the true curves. The dashed lines show the differences between the true and LOA solutions, $D_p=p_{\text{d}}^{\text{t}}-p_{\text{d}}^{\text{LOA}}$, plotted as $2D_p/{\kappa }^3$. The fact that these also converge to a single curve demonstrates that the fractional error in the leading-order approximation scales as ${\kappa }^2$.

Figure 3

Solutions to Eq. (1) under symmetric pump powers ${\tilde{p}}_{1,2}=\tilde{p}=1.75$ as in Fig. 1 but for five values of the differential-mode detuning ${\mathrm{\Delta }}_{\text{d}}=({\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2)/2$, plotted as circulating power $p_1$ vs common-mode detuning ${\mathrm{\Delta }}_{\text{c}}=({\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2)/2$. The case ${\mathrm{\Delta }}_{\text{d}}=0$ and points ${\text{A}}_1$, ${\text{A}}_2$, ${\text{B}}_1$, and ${\text{B}}_2$ are as in Fig. 1, and dotted lines represent unstable solutions. The locus of the edge of the symmetry-breaking-related unstable region for varying ${\mathrm{\Delta }}_{\text{d}}$ is shown as a thick black line.