Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity

The theoretical work of Braginsky predicted that radiation pressure can couple the mechanical, mirror eigenmodes of a Fabry-Pérot resonator to its optical modes, leading to a parametric oscillation instability. This regime is characterized by regenerative mechanical oscillation of the mechanical mirror eigenmodes. We have recently observed the excitation of mechanical modes in an ultrahigh $Q$ optical microcavity. Here, we present a detailed experimental analysis of this effect and demonstrate that radiation pressure is the excitation mechanism of the observed mechanical oscillations.

Figure 1

(a) Finite element modeling of the micromechanical resonances of a silica toroid microcavity (diameter of the toroid is $4\mu \mathrm{m}$). The radial and azimuthal mode order are denoted with $n$ and $m$ (where $m=0$ corresponds to rotationally symmetric modes). Shown are the first three rotationally symmetric radial modes ($n=1$, 2, 3, $m=0$) in cross section with the amplitude of motion greatly exaggerated for clarity. In addition, the stress field is indicated using color. Note that the mechanical motion modulates the cavity path length due to a change in the cavity radius, which causes the excitation of optical sidebands. (b) Mechanical oscillation frequency versus the cavity length $L$ for the first three fundamental mechanical modes (dots are experimentally measured frequencies from Ref. 5). (c) Typical frequency spectrum of the optical cavity transmission, revealing the presence of several sidebands that correspond to the first three mechanical eigenmodes (${\omega }_m)$, as well as harmonics of ${\omega }_m$. Here, only the $n=2$ mode is above threshold, whereas the remaining mechanical modes are subthreshold (and observable due to their room temperature thermal displacement noise).

Figure 2

Main panel: The oscillation threshold ($\mu \mathrm{Watts}$) versus the mechanical quality factor of the $n=1$ mode. The solid line is the theoretical prediction based on an inverse $Q$ relation ($P\propto 1/Q_m$). Right inset: Side-view, optical micrograph of the experimental setup, consisting of a silica microprobe in contact with a taper-fiber-coupled microtoroid of $72\mu \mathrm{m}$-principal diameter. Left inset: Spectrum of the optical cavity transmission exhibiting the thermal displacement noise of the $n=1$ mechanical mode. Solid line: The Lorentzian fit to infer the value of $Q_m$.

Figure 3

Main panel: The measured mechanical oscillation threshold ($\mu \mathrm{Watts}$) plotted versus the optical $Q$ factor for the fundamental flexural mode ($n=1$, ${\nu }_m=4.4\mathrm{MHz}$, $Q_m=3500$). The solid line is a one-parameter theoretical fit obtained from the minimum threshold equation by first performing a minimization with respect to coupling and pump wavelength detuning, and then fitting by adjustment of the effective mass ($m_{\mathrm{eff}}^{(1)}=3.3\times {10}^{-8}\mathrm{kg}$). Inset: The measured threshold for the 3rd order mode ($n=3$, ${\nu }_m=49\mathrm{MHz}$, $Q=2500$) plotted versus optical $Q$. The solid line gives again the theoretical prediction with $m_{\mathrm{eff}}^{(3)}=0.5\times {10}^{-10}\mathrm{kg}$. The $n=1$ mode data from the main panel is superimposed for comparison.

Figure 4

(a) Scanning-electron micrograph of the microcavity cross section achieved using focused-ion beam (FIB) preparation. The image reveals the presence of an offset in the toroid with respect to the $2\mu \mathrm{m}$ thick silica support disk (offset of ca. $1.3\mu \mathrm{m}$). (b) Finite element modeling of the fundamental optical mode. The presence of an offset provides a moment arm that effectively enhances coupling of radiation pressure to the $n=1$ transverse motion.