Chaotic Quivering of Micron-Scaled On-Chip Resonators Excited by Centrifugal Optical Pressure

Opto-mechanical chaotic oscillation of an on-chip resonator is excited by the radiation-pressure nonlinearity. Continuous optical input, with no external feedback or modulation, excites chaotic vibrations in very different geometries of the cavity (both tori and spheres) and shows that opto-mechanical chaotic oscillations are an intrinsic property of optical microcavities. Measured phenomena include period doubling, a spectral continuum, aperiodic oscillations, and complex trajectories. The rate of exponential divergence from a perturbed initial condition (Lyapunov exponent) is calculated. Continuous improvements in cavities mean that such chaotic oscillations can be expected in the future with many other platforms, geometries, and frequency spans.

Figure 1

Experimental setup (a) and results (b)–(d): (a) Continuous optical power is fiber coupled to an on-chip microresonator in which the centrifugal radiation pressure excites mechanical vibration. The calculated vibrational mode for the spherical cavity used here is presented in a rendering where the deformation describes the cavity shape at maximum mechanical amplitude. The deformation is exaggerated in the figure. The experiment is done at room temperature and pressure. $R$ signifies the cavity radius. (b) Oscillation in spherical cavity starts periodic (c), doubles its period (d), and then turns aperiodic (continuous spectra) as input power increases. Inset in (b) is a micrograph of the spherical resonator used in this experiment. Panels b, c, and d were taken at pump power ($|E_p{|}^2$) of 66, 79, and 83 mW. The sphere radius is $12\mu \mathrm{m}$, its optical $Q$ is $7\times {10}^6$, and its mechanical $Q$ is 112.

Figure 2

Experimental results: As the input power increases, the periodic oscillation (a) of a toroidal microcavity doubles its period twice (b) and then turns aperiodic (c). Oscillation of the output power is measured in the frequency domain as well as in the temporal domain. Phase-space plots of the first derivative of the measured output power in time versus the measured output power are also shown. The inset in (a) shows a toroid like the one investigated here. Panels a, b, and c were taken at pump power ($|E_p{|}^2$) of 11, 21, and 28.7 mW. The mechanical $Q$ is 250, and the optical $Q$ is ${10}^7$. Major and minor toroid radii are 14.5 and $3\mu \mathrm{m}$.

Figure 3

Theoretical calculation. (a): $\log [\delta \mathrm{\text{output power}}]$ vs time for low input power exhibits a flat evolution. Parameters are the same as those of the experiment in Fig. 2 and were measured to be $a=1.4\times {10}^6$, $b=1.2\times {10}^{17}$, $c=9779$, $d=1.2\times {10}^8$, $e=1.3\times {10}^8$, $f=1.1\times {10}^{20}$, $g=2.2\times {10}^{10}$ all in MKS. Mechanical- and optical-damping is weak (b), increasing the input power (to be the same as in experiment with details in Fig. 2c) causes exponential divergence of $\delta \mathrm{\text{output power}}$ . Insets are the calculated evolution in phase space and time exhibiting similar behavior to the one measured in Fig. 2.