Self-sustained oscillation and dynamical multistability of optomechanical systems in the extremely-large-amplitude regime

Optomechanics is concerned with the coupling between optical cavities and mechanical resonators. Most early works are concentrated in the physics of optomechanics in the small-displacement regime and consider one single optical cavity mode participating in the optomechanical coupling. In this article, we focus on optomechanics in the extremely-large-amplitude regime in which a mechanical resonator is coupled with multiple optical cavity modes during the oscillation. We explicitly show that the mechanical resonator can present self-sustained oscillations with limit cycles in the shape of sawtooth-edged ellipses and can exhibit dynamical multistability. By analyzing the mechanical oscillation process and the accompanying variation of the optical cavity occupation, we develop an energy-balanced condition to ensure the stability of self-sustained oscillation. The effect of the mechanical nonlinearities on the dynamics of the mechanical resonator is also investigated.

Figure 1

Schematic of a generic optomechanical system. An optical cavity of static equilibrium length $L_0$ with a fixed, partially reflecting mirror on the left side and a movable, perfectly reflecting mirror on the right side is driven by an external laser of frequency ${\omega }_l$ and power $P$. In the ELAR, multiple optical cavity modes of frequencies ${\omega }_n$ and decay rates ${\kappa }_n$ can be excited and coupled with the mechanical resonator of intrinsic frequency ${\omega }_m$, mass $m$, and damping rate $\gamma$.

Figure 2

Limit cycles in the phase space of the mechanical resonator scanned by $x$ and $p=m\dot{x}$ with different values of $P$. The parameters are ${\omega }_m={10}^7$ Hz, $m=5\times {10}^{-15}$ kg, $\gamma ={10}^{-2}{\omega }_m,{\kappa }_n=\kappa ={10}^2{\omega }_m,{\lambda }_l=1000$ nm, $L_0=N{\lambda }_l/2$, and $N=10000$. The values of $P$ are (a) $P=1$ W, (b) $P=4.1$ W, (c) $P=7$ W, (d) $P=10$ W, (e) $P=11$ W, (f) $P=12$ W, (g) $P=15$ W, and (h) $P=17$ W, as specified by the vertical black dashed lines in Fig. 4.

Figure 3

The variation of the minimum (a) and maximum (b) limit cycles as the power of the external driving power changes in the range of 4.1 to 18 W, in which the mechanical resonator presents dynamical multistability.

Figure 4

(a) The dependence of the frequencies ${\omega }_n$ of the optical cavity modes on the displacement $x$ of the mechanical resonator. (b) The variation of the optical cavity occupation with the oscillation of the mechanical resonator. Here, the power of the external driving laser is $P=11$ W. The other parameters are same as those in Fig. 2.

Figure 5

(a) Definition of the average amplitude as $\overline{A}=\sqrt{(A_{\min }^2+A_{\max }^2)/2}$. (b) Attractor diagram on a plane of $\overline{A}$ and $P$. Here, the parameters are same as those in Fig. 2. The red (dense) dots show the numerically exact results captured by solving the original coupled dynamical equations, Eqs. (3) and (4). The black (sparse) dots are the approximately estimated results (see the Appendix) from the energy-balanced condition, Eq. (12). The vertical black dashed lines specify some values of $P$ distributed in different regions. Limit cycles for these values of $P$ are plotted in Fig. 2.

Figure 6

The dependence of the average amplitude $\overline{A}$ on $\alpha$. Here, the power of the external driving laser is set to be $P=7$ W. The other parameters are same as those in Fig. 2.