A microelectromechanically controlled cavity optomechanical sensing system

Microelectromechanical systems (MEMS) have been applied to many measurement problems in physics, chemistry, biology and medicine. In parallel, cavity optomechanical systems have achieved quantum-limited displacement sensitivity and ground state cooling of nanoscale objects. By integrating a novel cavity optomechanical structure into an actuated MEMS sensing platform, we demonstrate a system with high-quality-factor interferometric readout, electrical tuning of the optomechanical coupling by two orders of magnitude and a mechanical transfer function adjustable via feedback. The platform separates optical and mechanical components, allowing flexible customization for specific scientific and commercial applications. We achieve a displacement sensitivity of 4.6 fm Hz−1/2 and a force sensitivity of 53 aN Hz−1/2 with only 250 nW optical power launched into the sensor. Cold-damping feedback is used to reduce the thermal mechanical vibration of the sensor by three orders of magnitude and to broaden the sensor bandwidth by approximately the same factor, to above twice the fundamental frequency of ≈40 kHz. The readout sensitivity approaching the standard quantum limit is combined with MEMS actuation in a fully integrated, compact, low-power, stable system compatible with Si batch fabrication and electronics integration.

3 readout in a single-chip, fiber-pigtailed, low-power, compact and stable microsystem compatible with batch fabrication at low cost. Conceptually similar to other MEMS sensors [9][10][11][12][13], the device includes a movable surface where samples can be attached or unknown forces or torques can be applied, while the mechanical response is now detected optically with precision and bandwidth improved by orders of magnitude compared to a more conventional electrostatic or piezoresistive readout. Importantly, the ability to actuate the device electrostatically is retained. The combination of extremely low noise readout and MEMS electrostatic tuning results in several new system capabilities. The optical cavity resonance is tunable through 5.54 nm, useful for adjusting the device to operate with a fixed wavelength optical laser source, such as a compact and relatively inexpensive stabilized laser diode. The optomechanical coupling and thus the readout sensitivity is tunable by over two orders of magnitude for optimizing the sensor gain and dynamic range.
In a fully integrated MEMS nanophotonic device, we apply electronic feedback and demonstrate cold damping of the mechanical degree of freedom [22,[36][37][38][39] by more than three orders of magnitude. While not reducing the input-referred force noise, i.e. the Langevin force acting on the sensor, the near-critical damping stabilizes the sensor position and also flattens the frequency-dependent sensor gain, allowing us to use the sensor effectively over a very broad frequency range without severe dynamic range constraints. To demonstrate this, we measure the force noise acting on the sensor for frequencies up to 100 kHz, more than 2.5 times the fundamental resonance frequency of 38.76 kHz. Even with the low mechanical responsivity in this high frequency range the input referred readout noise remains significantly below the thermal mechanical force noise being measured.
The ability to dampen the mechanical noise can strongly reduce the backaction of the sensor onto a system being measured. Moreover, our noise level is ≈2.3 times the standard quantum limit (SQL) [21,40] for our mechanical system, approaching the fundamental readout limits. With future parameter improvements, cooling the sensor to the quantum mechanical ground state while maintaining the high readout bandwidth [41] (i.e. in the 'bad' cavity limit) may be achieved. Figure 1 shows the scanning electron micrographs and the illustrated cross-sections of the sensing system. The detailed fabrication process is described in appendix A. The design details are described in appendix B. The 15 µm diameter, 240 nm thick stationary silicon microdisc optical resonator is used to detect the out of plane mechanical motion of a 200 nm thick, 19 µm outer diameter silicon nitride ring suspended a variable distance Z above. The resonant optical modes of the microdisc are excited and measured by an integrated single-mode on-chip waveguide, which has been fiber-pigtailed for robust and low-loss coupling of light into and out of the device. The microdisc is fixed to the substrate via a silicon nitride anchor, while the ring is attached to a MEMS actuator. The actuator is a silicon cantilever fixed to the substrate via silicon nitride anchors on one side and connected to the ring on the other side. The cantilever consists of two electrically separate parts that are mechanically joined by a dielectric silicon nitride bridge. When a voltage is applied between the first part of the cantilever and the substrate (1 µm below), the attractive electrostatic force bends it towards the substrate. The second part of the cantilever, maintained at the ground potential, serves as a mechanical lever arm to achieve a larger range of is a cross-section taken through the key elements of the mechanical transducer and optical sensor (i.e. SiN x anchors, silicon (Si) MEMS actuator, SiN x ring and Si disc). In these crosssection illustrations, the Z-axis scale has been adjusted for clarity. motion at the ring location. The electrostatic tuning of Z as a function of the applied dc voltage is shown in figure 2(a), measured with a white light interferometer.

Results
Having the movable parts mechanically separated from the optical readout components (disc and waveguide) allows us to independently choose the key parameters of both optical sensing and mechanical transduction. The silicon disc resonator operates with an intrinsic quality factor of nearly 10 6 , approaching the highest values achieved in silicon nanophotonic resonators [42,43], and contributing to the optical sensor's high sensitivity to the ring's motion. Through adjustment of the waveguide-cavity separation, the loaded quality factor of the system can be tuned, determining the optical bandwidth and ensuring that the out-coupled optical signal into the waveguide is sufficiently strong to allow near shot-noise-limited detection. The mechanics is similarly largely unconstrained by the optical readout. The parameter space of mechanical frequency, stiffness and the ring displacement range can be adjusted depending on the specific sensing application in hand. For example, in this work, wide readout tunability afforded by the relatively large ring displacement range was the dominant consideration. For other applications, working at a pre-determined disc-ring gap that is chosen based on the desired readout gain and dynamic range would allow using stiffer cantilevers with higher mechanical frequencies. The sensor platform is also compatible with torsional, parallel plate or elastic membrane mechanical elements and integrated magnetic, thermal or piezoelectric actuators. As illustrated in figure 1(c), in a thin microdisc resonator a significant portion of the optical mode energy is located in the evanescent tails below and above the disc. When the gap Z between the nitride ring and the disc is reduced, the mode shape and frequency are strongly modified. Similar to a change in the position of one of the mirrors in a Fabry-Perot cavity, a small change in the position of the nitride ring shifts the optical resonance frequency linearly with mechanical displacement. The shift can be measured by sensing the modulation of continuous wave (cw) light tuned to near cavity resonance, accomplished either by simple amplitude measurement on the shoulder of the resonance or by a suitable phase-sensitive measurement, such as polarization spectroscopy [44] or the Pound-Drever-Hall technique [45]. Figure 2(b) shows the transmission spectrum of one of the optical modes that is used for subsequent experiments throughout the paper; the loaded optical Q = 3.8 × 10 5 is estimated from the linewidth of the mode as shown in the figure. Identified from the polarization-sensitive coupling loss of the fiber-waveguide couplers and the free spectral range of the mode family of a given radial index, this mode is of transverse magnetic (TM; no magnetic field in the direction of propagation) polarization, so that the electric field is predominantly oriented normal to the plane of the microdisc. Figure 2(c) shows the resonant wavelength as a function of the applied voltage, measured by varying the laser wavelength, while the cavity is locked onto the laser via feedback (see figure 3 and the discussion below). An active tuning of the cavity resonance over the range of 5.54 nm ± 0.02 nm is achieved. The uncertainty is the laser wavelength tuning resolution in this measurement. The system is highly repeatable and stable. The tuning curve is reproducible within the wavelength accuracy of 0.02 nm of the laser after 2 weeks. This corresponds to an upper bound on the average slow drift of the gap of 0.1 nm per day. The sensor readout gain is proportional to the optomechanical coupling g OM = dω c /dZ , where ω c is the optical resonance frequency. It is calculated from ω c and Z measurements in figures 2(a) and (c), and shown in figure 2(d). The optomechanical coupling is increased by a factor of more than 200 (from g OM /2π = 65 MHz nm −1 to g OM /2π = 13.4 GHz nm −1 ) when the applied voltage is increased from 2.5 to 8 V, thus broadly adjusting the gain of the readout. The tuning can also be used to match the cavity to a pre-set working wavelength, e.g. so that an inexpensive fixed wavelength 7 diode laser source can be used for sensing. A detailed study of g OM versus Z is described in appendix C.
We use the cavity optomechanical sensor to study the dynamics of the fundamental mechanical cantilever bending mode (inset of figure 1(a)). The device is placed in a vacuum chamber, where a pressure of ≈0.3 Pa is maintained to reduce the mechanical loss due to air damping. In an optical transmission measurement (figure 3), a cw wave from a tunable laser is launched into the device, with an optical power of ≈250 nW (estimated by measuring the output power and the coupling loss) in the waveguide right before the microdisc, and ≈40 nW out-coupled to a photodetector with a gain of ≈1.9 × 10 6 V W −1 and the 3 dB bandwidth of 200 kHz. The wavelength is tuned to a pre-specified transmitted power set point as shown in figure 2(b). A small motion of the cantilever results in a linear modulation of the transmitted optical power via optomechanical coupling. The resulting photodetector voltage proportional to mechanical displacement is recorded with an electrical signal analyzer. The calibration to obtain the absolute value of the mechanical displacement at the center of the ring is described in appendix D.
We have measured two characteristics of our device under various conditions: the mechanical displacement noise spectral density and the sensor transfer function, shown in figures 4(a) and (b), respectively. The transfer function quantifies the linear response of the MEMS sensor to an external applied force at various frequencies. It is obtained by applying an additional small sinusoidal voltage to the actuator to mimic an external force and recording the resulting displacement as shown in figure 3. It is calibrated in terms of both the applied force and the displacement at the center of the ring, where the actuator mechanical stiffness in the Z-direction is estimated from the finite-element modeling (FEM) to be k 0 = 0.038 N m −1 and is dominated by the fundamental mechanical mode.
The blue curves of figures 4(a) and (b) are taken under a fixed actuator bias voltage of 3 V, with the inset of figure 4(a) showing the high-resolution data near the resonance frequency. The fundamental mechanical mode has a frequency of 38.52 kHz, which agrees well with 38.7 kHz from FEM. The inset of figure 4(b) shows the normalized transfer function under the same bias, measured with a sufficiently low optical power (an additional 26 dB attenuation compared to figure 4(a)) to extract the intrinsic mechanical Q. At this power level, radiation pressureinduced optomechanical excitation or damping is negligible, so that the values measured for both the blue and red detuning of the laser from the center of the cavity resonance agree with each other. The mechanical frequency and the intrinsic Q are estimated to be ≈38.76 kHz and ≈1400, respectively. We attribute the relatively low intrinsic mechanical Q to the oxidization of the silicon surface, although there might be other loss mechanisms.
The black curve in figure 4(a) shows the noise background of the measurement, obtained by detuning the laser from the optical resonance. Very similar values are obtained by completely turning the laser off, indicating that the dominant noise source in our experiment is the detector dark noise. At the moderate optomechanical coupling achieved at 3 V dc bias, the sensor is dominated by the mechanical noise at all frequencies below about 50 kHz and by the readout noise at higher frequencies. As shown below, with higher optomechanical gain the input referred readout noise is further reduced and the sensor becomes limited by the mechanical noise over an even larger bandwidth of 0-100 kHz.
While the high mechanical Q is desirable, corresponding to lower losses and lower thermal mechanical noise, the resulting highly non-uniform transfer function severely limits the sensor dynamic range if broadband application is considered, e.g. when small forces off-resonance have to be measured in the presence of on-resonance forces. The on-resonance forces and thermal noise are mechanically amplified and can exceed the linear dynamic range of the readout set in our measurement by the linear portion of the shoulder of the optical resonance curve. In fact, as the optomechanical coupling is increased with increasing the bias voltage, e.g. g OM /2π = 0.24 GHz nm −1 at 5 V, our estimated thermal root mean square (rms) noise at room temperature of (k b T /k 0 ) 1/2 ≈ 0.47 nm will result in a cavity shift comparable to the cavity linewidth of 4.1 pm.
We overcome this limitation by introducing a cold damping feedback loop to make the system transfer function more uniform across the frequency spectrum and drastically reduce the cantilever displacement noise. Cold damping of the mechanical mode is realized by applying to the integrated actuator an electrical feedback signal derived from the optical readout signal. The feedback loop illustrated in figure 3 consists of two components implemented with two amplifier/filter chains. The top chain responds at frequencies from dc to 100 Hz and provides a negative proportional-integral feedback to the actuator to lock the cavity onto the laser and follow the laser wavelength when it drifts or is tuned intentionally. The set point is ≈43% of the off-resonance power on the red side of the optical mode (as indicated in figure 2(b)), the illustrates the transfer function with a higher damping gain setting, where the mechanical Q is approaching 1. Considering the measured intrinsic mechanical Q of ≈1400, the mechanical mode is damped by more than three orders of magnitude, decreasing the noise-driven rms sensor position variation by the same factor.
Stabilizing the sensor with feedback allows working at increased dc bias, reaching higher optomechanical coupling for higher readout gain and lower readout noise, which in turn also ensures that negligible excess noise is injected back into the system by the feedback. Figure 5 shows the displacement noise spectra and the measurement noise backgrounds at increasing dc bias voltages and readout gains. The displacement sensitivity increases by more than a factor of 15 when the bias voltage is increased from 4.5 to 6.75 V, and is limited by the photodetector noise to (4.6 × 10 -15 ± 0.6 × 10 -15 ) m Hz −1/2 (the average value across the spectrum; errors represent one standard deviation throughout the rest of the paper (appendix D)) at a bias voltage level of 6.75 V. It should be noted that this displacement sensitivity is only 2.3 times larger than the SQL (S SQL x =h Q/k 0 ) for our cantilever, and one can expect a further increase of the measurement sensitivity by using a photodetector with lower noise, and bringing the nitride ring even closer to the microdisc. However, due to the cavity nonlinearity and the heat generated by light, the sensitivity is not improved by simply increasing the laser power. As is evident from the figures, at high voltage, the displacement signal is well above the detector noise over the whole dc to 100 kHz frequency range, which makes the system suitable for dynamic process measurements up to the frequency in excess of 2.5 times the fundamental mechanical frequency of the device.
The degree of damping can be adjusted independently of the bias voltage, as illustrated by the two sets of curves in figures 5(a) and (b). In addition to damping, this simple feedback choice also results in a reduction of the closed-loop mechanical stiffness and a corresponding decrease of the resonance frequency. Employing a more sophisticated feedback scheme with a highbandwidth ( 100 kHz) proportional-integral-derivative (PID) controller, or another general linear controller, may enable one to engineer flat or other desired transfer functions in future, but this is beyond the scope of this work. The increased displacement at low frequencies is caused by 1/ f excess force noise in our devices.
Using the measured transfer functions corresponding to the displacement noise data in figure 5(b), we have obtained the spectra of the force noise at the input of our sensor, shown in figure 6 (see appendix D for detailed calculations). The force noise spectra consist of a 1/ f noise dominant at lower frequencies and flat, white noise appearing at high frequencies. Based on the linear dependence on the actuator dc bias voltage, we attribute the 1/ f force noise to 1/ f electrical noise on the cantilever, probably due to poor electrical contacts between metallic probes and doped silicon electrical pads of the device. The white noise component agrees well with the expected intrinsic thermal Langevin force noise (dashed curve) of 1.3 × 10 -15 N Hz −1/2 calculated using the fluctuation dissipation theorem with the mechanical loss based on a measured intrinsic Q of ≈1400. The dotted line in figure 6 corresponds to the force sensitivity limit imposed by the optical readout noise. The photodetector limited force sensitivity is estimated to be (5.3 × 10 -17 ± 0.2 × 10 -17 ) N Hz −1/2 at ≈25 kHz (resonant frequency with damping feedback) with a 6.75 V dc bias.

Summary and discussion
We have developed a novel integrated MEMS sensing platform enabled by cavity optomechanics. We demonstrate a displacement readout sensitivity of (4.6 × 10 -15 ± 0.6 × 10 -15 ) m Hz −1/2 and a force readout sensitivity of (5.3 × 10 -17 ± 0.2 × 10 -17 ) N Hz −1/2 with the optical power of only 250 nW. The sensitive low-noise optomechanical readout in combination with electrostatic actuation is used to cold-dampen the mechanical mode by more than three orders of magnitude, achieving a corresponding suppression of the displacement noise, flattening of the overall sensor transfer function and expanding the sensor bandwidth to 100 kHz, exceeding the initial mechanical resonant bandwidth of f /Q = 28 Hz by a factor of ≈3500 and the fundamental mechanical resonance frequency by a factor of ≈2.5. Because of the dramatically improved readout sensitivity, the force sensitivity of the system is set by the thermal mechanical noise limit even at these high frequencies. This new regime in MEMS sensing is demonstrated in a compact, fiber-pigtailed, low-power, stable microsystem compatible with silicon batch fabrication at low cost and CMOS integration. The flexibility to independently choose optical and mechanical parameters of the described platform allows the system to be tailored for various sensing tasks.
In future, at least one order of magnitude improvement in the thermal force noise can be expected from optimizing the processing to lower mechanical dissipation, reaching mechanical Q factors of 10 5 [46] reported for similar types of soft, sensitive, low-frequency Si cantilever devices. Our sensing and integration approach will be able to take full advantage of such a thermal noise improvement for fast, broadband force measurements that are even more sensitive.
Our current position readout noise is a factor of 2.3 ± 0.3 above the SQL at the resonance frequency for our mechanical system and can be further improved by increasing the detected optical power, utilizing a lower noise detector and further increasing the optomechanical coupling. With cryogenic cooling and increased mechanical Q for longer decoherence time, this approach may enable SQL-level readout with the bandwidth wider than the inverse decoherence time. In this regime, quantum mechanical behavior of the system is observable. In contrast to optical cooling where a 'good' cavity (cavity linewidth narrower than the mechanical resonance frequency) is required, it has been suggested that cold damping can be used as a mechanism for cooling mechanical oscillators to the quantum ground state [41] in the 'bad cavity limit' where the optical resonance linewidth exceeds the mechanical oscillation frequency. If verified, this can be of great significance for silicon optomechanical systems operating at MHz mechanical frequencies and below. Such mechanical resonance frequencies are important for a number of sensing applications, but operation in the resolved sideband regime is challenging due to the high optical quality factors (Q > 10 8 ) required.

Appendix A
In this appendix, we describe the device fabrication process. We start with a silicon-on-insulator (SOI) wafer with a 240 nm top silicon layer and a 1 µm buried oxide (BOX) layer. The Si layer is patterned via electron beam lithography and reactive ion etching (RIE) with a SF 6 + C 4 F 8 recipe down to the BOX layer to produce high-Q silicon microdiscs, access waveguides for coupling light to/from the microdisc and the actuators. The width of the waveguide is 500 nm and the gap between the waveguide and the disc is 300 nm. The waveguide is linearly tapered down to a width of 125 nm at its end for low loss coupling to/from optical fibers. A sacrificial silicon dioxide layer (≈600 nm) and a low stress silicon nitride layer (≈200 nm) are sequentially deposited in a low-pressure chemical vapor deposition (LPCVD) furnace, patterned via optical lithography, and dry etched to form the nitride ring above the microdisc, nitride anchors and bridge to mechanically attach various structures and an electrical pad to ground the substrate. A photolithography step, buffered oxide etch (BOE) and ion implantation (Boron) process are used to dope the first part of the cantilever and the pads for electrical contacts to a doping level of >10 19 cm −3 for low contact resistivity. Another 1 µm oxide layer was deposited using a lowtemperature oxide (LTO) furnace. The wafer was annealed for 1 h at 1000 • C in an ambient N 2 environment. Another photolithography, a metal (Ni) hard mask deposition and liftoff, an RIE and a TMAH wet etching process were used to define fiber V-grooves in the Si substrate. Silicon dioxide layers are removed by 49% HF wet etching to undercut and release the movable structures. A critical point drying process is used to avoid stiction between parts due to capillary forces. In a self-aligned region at the end of each grove the on-chip waveguide inverse tapers are coupled to an optical fiber placed in the V-groove, actively aligned and cured into place with ultraviolet (UV) light curable epoxy. The fiber-to-fiber loss of the pigtailed device is 16 dB.

Appendix B
In this appendix, we describe the design choices and details. While our integrated MEMS actuator is conceptually similar to a simple cantilever, there are several features that warrant a more detailed explanation. The design goal was to achieve stable, high-bandwidth actuation with the range of 0.5 µm or more under the application of dc voltage. This enables both widerange tuning and fast feedback. Additional goals were that the tilt of the ring be minimized, that the fundamental mechanical mode be coupled well to the ring translation, the fundamental resonance frequency be above 10 kHz and the higher-order modes be well separated from the fundamental.
To simplify the fabrication, it is advantageous to make the mechanical device out of the same 260 nm thick silicon on an insulator layer that is used for the microdisc. The simple geometry of the Si cantilever clamped on one side satisfies our mechanical requirements. The cantilever length is chosen to have high enough resonance frequency without excessive tilt at the ring location. The cantilever width is chosen such that the stiffness in maximized, while the structure remains a cantilever and not a plate. To make the cantilever free to move, the buried oxide under the Si structure needs to be removed completely in the release etch process step (by HF undercut). To minimize this release process time, an array of holes was patterned through the cantilever structure. We chose to attach the fixed end of the cantilever to the substrate by an array of silicon nitride anchor posts. This provides electrical isolation from the substrate, a stable and stiff mechanical attachment and does not rely on precise timing of the release etch step.
Electrostatic actuation was chosen for the combination of bandwidth, stability and lowpower dissipation. The choice of the Si substrate as the stationary electrode gives a simple yet fully integrated solution, with a relatively large capacitance of nearly parallel plates with only a 1 µm gap, resulting in a sufficient force at low voltage. The 1 µm gap is large enough for low optical loss in the microdisc and waveguide, while small enough not to complicate the various dry etching process steps. However, the gap limits the range of stable cantilever actuator motion under a fixed applied voltage. Because capacitance is a nonlinear function of the actuator coordinate, a well-known electrostatic instability arises, whereby when the applied voltage and actuator position exceed certain values, the equilibrium given by the electrostatic and restoring elastic forces is no longer stable. The two conductors move toward each other until mechanical contact occurs. In our case, this prevents us from using the full length of the cantilever as an electrostatically active surface-given our 1 µm gap, we would not be able to achieve the desired range of stable motion at the ring location at the tip of the cantilever. Additionally, making the full cantilever active would result in electric fields at the dielectric ring and the electrically isolated microdisc location, which could lead to slow electrical charging and the corresponding mechanical drift. Therefore, we have chosen to divide the cantilever into two electrically separated but mechanically connected parts. The two parts attach to each other through an insulating bridge of silicon nitride. Several discrete, interleaved bridge attachment points are used to provide additional strain relief, reducing the effects of any residual stress in the nitride on the device shape.
The voltage is applied between the grounded substrate and the electrically active Si part at the base of the cantilever. The second, electrically passive, part of the cantilever is grounded by an electrical connection to the grounded fixed SOI sheet, provided by the two thin flexible Si springs near the tip of the cantilever. These springs also contribute mechanical torque, further reducing the ring tilt as the cantilever moves. The ring and disc charging is avoided by attaching the ring to the grounded part of the cantilever. This grounded part of the cantilever also works as a lever arm to translate the small range of stable actuation achieved at the end of the electrically active part to a required larger range of electrical actuation at the ring location.
The ring is attached to the cantilever at two opposite points for a more symmetric and tiltfree initial position. The center of the disc is rigidly attached to the substrate via a silicon nitride anchor to allow for longer and less precise timing of the HF release process step.
The suspended Si waveguide near the microdisc is rigidly attached by small bridges to the fixed, grounded SOI sheet to eliminate any possibility of charging or mechanical motion. The scattering losses at these bridges appear to be negligible for the purposes of our measurement. between the ring and the disc. As the evanescent fields decay exponentially away from the silicon layer, the optomechanical coupling factor can be quite large. We note that a similar optomechanical coupling mechanism has been demonstrated by other groups in the 'double disc' geometry [30,47], consisting of a pair of silica or silicon nitride discs in which the WGMs are distributed between the two discs and the relative flapping motion between the discs leading to the optomechanical coupling.
We performed numerical simulations of the optomechanical coupling using the finite element method. As a full 3D simulation including the entire device geometry would require a very large computational domain, we focus on the silicon nitride ring/silicon disc portion in which optomechanical interaction takes place, and we make the simplifying assumption that it is perfectly azimuthally symmetric. The eigenmodes of the system will then have an exp(im) dependence (m is the azimuthal mode index and is the azimuthal angle), allowing the problem to be reduced to an effectively 2D one [42]. In addition, we focus on TM polarized modes, whose electric field is predominantly oriented normal to the plane of the microdisc, as these modes should be especially sensitive to changes in the ring-disc gap. Indeed, simulations (not shown) indicate that g OM is as much as a factor of five times larger for TM modes than TE modes. Figure C.1(a) shows the predicted g OM as a function of the ring-disc gap, assuming a silicon nitride ring thickness of 250 nm and a width of 4.5 µm, extending 2 µm past the silicon disc edge. The silicon nitride refractive index was taken as n = 2.1 and the silicon refractive index was taken as n = 3.4. Figures C.1(b) and (c) show a cross-sectional view of the electric field amplitude for first-and second-order radial modes in a device with a gap Z = 380 nm. For gaps of the order of 300 nm, g OM /2 > 10 GHz nm −1 can be achieved, as we have also observed in experiments ( figure 2(d)). However, we have not been able to achieve precise quantitative agreement with simulations, as g OM seems to increase more quickly with decreasing gap in experiments than expected. This may be due to a number of factors (including imprecise knowledge of the exact ring geometry, uniformity of the gap between the ring and the silicon disc, and the refractive index of the silicon nitride layer). Finally, if smaller gaps can be achieved, g OM /2 can be expected to exceed 100 GHz nm −1 .

Appendix D
In this appendix, we describe the displacement and force spectra calculation process. The directly measured voltage signal is converted into the displacement noise by dividing it by the optomechanical gain. The optomechanical gain is a product of the optomechanical coupling g OM and the slope of the cavity expressed as dV det /dω, where ω is the optical frequency. Note that g OM is reported in figure 2(d), while the slope is calibrated in two different ways. In a first method, we directly measure the detector voltage signal corresponding to a fixed known frequency modulation (rms amplitude: 7.8 MHz, at 1 kHz) of the laser wavelength. Note that if the damping feedback is engaged, the cantilever will respond to the varying detector voltage by moving, and so the closed-loop detector modulation will be larger, resulting from both the cantilever motion and the wavelength shift. The factor of modulation enhancement is directly related to the cantilever stiffness change and is equal to ( f / f ) 2 , where f and f are the open-loop and closed-loop resonance frequencies, respectively. The frequencies are obtained by fitting the measured transfer functions. In a second method, the optomechanical gain is directly calculated from the transfer function. A known low-frequency (1 kHz) voltage is applied to the actuator, resulting in a known displacement based on figure 2(a) in the absence of feedback. Under closed-loop control the cantilever becomes effectively softer and the response at low frequency increases. The relative stiffness change is obtained by fitting the system transfer function to extract the closed-loop resonance frequency and then taking the ratio ( f / f ) 2 . The optomechanical gain is calculated by dividing the photodetector signal by the known cantilever displacement. Note that, due to the small difference between dc and 1 kHz and the difficulty of measuring a signal at dc, we use the signal at 1 kHz to estimate that at dc. At all the voltage levels in figure 5, the differences between the two calibration methods are within 20%.
Transfer functions are calculated as a ratio of a given mechanical displacement to an applied force causing it. The displacement is obtained from the photodetector signal as described above. The equivalent force is obtained from the actuator ac voltage. An ac voltage at low frequency will move the cantilever by a known distance Z based on the data in figure 2(a). The cantilever would also move by the same distance if a force F = k 0 Z were applied to the ring at the end of the cantilever. In the absence of feedback, k 0 is taken to be 0.038 N m −1 based on FEM. The force equivalent to a given voltage is assumed to be independent of frequency based on the broadband nature of electrostatic actuation. This allows us to convert the measured ratios V det /V ac ( f ) into the calibrated transfer functions.
The force noise spectra are obtained by dividing the calibrated displacement noise spectra by the calibrated transfer functions.
All the uncertainties quoted in the paper are one standard deviation unless specified otherwise. The quoted uncertainty for the displacement sensitivity is calculated based on the variation of the photodetector limited background over the frequency range and the variation between the two calibration methods. The force sensitivity uncertainty is extracted through statistical analysis of the photodetector limited force background in the frequency range of 24-26 kHz. The uncertainty of the ratio of displacement sensitivity to SQL is dominated by the displacement sensitivity uncertainty.