Figure 1

Scanning electron micrographs of the device: a 130 nm thick ${\mathrm{SiO}}_2$ membrane forms the pendulum, which consists of a $10\mu \mathrm{m}$ diameter disk and a $50\mu \mathrm{m}\times 0.43\mu \mathrm{m}$ tether. (a) Overview: The pendulum is suspended from a $60\mathrm{\text{−}}\mu \mathrm{m}$ wide ${\mathrm{SiO}}_2$ annulus. The annulus (wrinkled area) and the pendulum are etched into a ${\mathrm{SiO}}_2$ film for which the Si substrate directly underneath has been removed. The dark background in the center is a clear opening of the substrate. (b) Close-up view of the pendulum, which is deflected 10–15° out of the plane of the substrate due to residual stress of the film.Reuse & Permissions

Figure 2

Optical trapping of a membrane disk. (a) Schematic of the experimental setup. The sample chip is enclosed inside a vacuum chamber. We trap the disk in an optical standing wave formed by a single laser beam at 1.064 μm and its reflection from the disk ($M_1$) and a mirror ($M_2$, $\mathrm{\text{reflectivity}}=0.98$). We monitor the thermal motion of the pendulum by the deflection of an off-axis probe beam (blue) reflected from the disk onto a quadrant photodiode (${\mathrm{PD}}_1$) and transmitted intensity onto ${\mathrm{PD}}_2$. (b) Calculated trap frequency $f_{\mathrm{opt}}$ vs membrane thickness $d_m$ normalized to $f_{\mathrm{opt}}$ at $d_m/\lambda \to 0$ for a fixed optical power. For $d_m\sim \lambda$, we solve for the steady electric-field amplitude on the left and right of the membrane.Reuse & Permissions

Figure 3

Displacement power spectrum of vibrational modes. (a) Spectrogram of vertical angular displacement [inferred from deflection measurement, Fig. 2a] versus trapping power, $P$. Several vibrational mode branches are evident, e.g., “$a$” and “$b$”. As the optical trap power is increased (in discrete steps), the frequencies of the three lowest modes ($a_1$, $b_1$, $c_1$) increase as $\sqrt{f_{0i}^2+{\alpha }_{i}P}$ (see text). Two avoided crossings are visible here: (◊) formed by the c.m. mode and the violin mode; ($\times$) formed by the c.m. mode and an annulus mode. At higher powers, the c.m. mode is a hybrid of pendulum and violin motion ($b_2$). Other visible modes in the spectrum are associated with vibrations of the annulus. Overall, the c.m. frequency shifts from 6.2 to 145 kHz when 4.3 W of optical power is applied. (b) Finite-element-simulated spectrum of an optically trapped pendulum (black) suspended from an annulus (blue) that is in turn anchored to a substrate [as in Fig. 1a] for qualitative comparison to (a). The red lines in both (a) and (b) are drawn for a trapping slope of $\alpha =4500\frac{{\mathrm{kHz}}^2}{\mathrm{W}}$. (c) Simulated mode shapes for different optical trapping forces in (a). ($a_1$) the pendulum mode, also called the c.m. mode, ($b_1$) the violin mode, and ($c_1$) the torsional mode.Reuse & Permissions

Figure 4

Mechanical $Q$ factor of the trapped pendulum. (a) Finite-element calculation of the ratio between the energy stored in the optical potential ($U_o$) and the mechanical potential ($U_m$) vs trap frequency. The two branches correspond to “$a$” and “$b$” vibrational modes, respectively, in Fig. 3b. $Q$ vs trapping frequency for two vibrational modes in the trap, independently recorded on ${\mathrm{PD}}_1$ (circles) and ${\mathrm{PD}}_2$ (triangles). In addition, two data sets with slightly different optical trapping alignments are shown for “$a$” (light and dark green) and “$b$” (light and dark magenta). The $Q$ of the c.m. mode increases 60-fold from its natural value, $Q_i=1.1(2)\times {10}^4$ ($P=0$, data in red square) with a slope that is consistent with $(f/f_0{)}^2$ scaling (solid line) before turning over near the avoided crossing (◊) with a violin mode. Beyond this avoided-crossing, $Q$ increases again followed by a minimum near the annulus avoided-crossing ($\times$) and then a steep increase. We also show the expectation of a $Q$-increase that scales as $f/f_0$ (dashed line) for comparison. We find the measured $Q$ increase to be in qualitative agreement with the calculated $1+\frac{U_o}{U_m}$ in (a).Reuse & Permissions