Abstract

Acceleration measurement is widely used in commercial, scientific, and defense applications, but the resolution and accuracy achievable for demanding applications is limited by the current technology used to build and calibrate accelerometers. We report an optomechanical accelerometer based on a Fabry–Perot microcavity in a silicon chip that is extremely precise, field deployable, and can self-calibrate. The measured acceleration resolution is the highest reported to date for a microfabricated optomechanical accelerometer and is achieved over a wide frequency range (${314}\;{{\rm nm}\cdot {\rm s}^{- 2}}/\surd {\rm Hz}$ over 6.8 kHz). The combination of a single vibrational mode in the mechanical spectrum and the broadband thermally limited resolution enables accurate conversion from sensor displacement to acceleration. This also allows measurement of acceleration directly in terms of the laser wavelength, making it possible for sensors to calibrate internally and serve as intrinsic standards. This sensing platform is applicable to a range of measurements from industrial accelerometry and inertial navigation to gravimetry and fundamental physics.

1. INTRODUCTION

High-precision, high-bandwidth acceleration measurement is central to many important applications, including inertial navigation [1,2], seismometry [3,4], and structural health monitoring of buildings and bridges [5]. Traditional electromechanical accelerometers have largely relied on piezoelectric, capacitive, or piezoresistive transduction to convert the displacement of the accelerometer’s proof mass to an output voltage when an excitation is applied. However, these transduction methods have reached sensitivity and bandwidth limits that are prohibitive for many applications. As a result, optical accelerometers have long been of interest due to the high precision provided by interferometry. These have included accelerometers assembled from macroscale optics [6] as well as those based on fiber optic interferometers [7] and fiber Bragg grating cavities [8]. More recently, cavity optomechanics has opened new avenues of research in both fundamental physics and precision measurement [9] by significantly advancing the sensitivity achievable in detecting attonewton forces [10], magnetic fields [11], and gravitational waves [12]. The development of integrated micro- and nanoscale optomechanical devices has produced accelerometers with significantly better displacement resolution than previously reported. Examples include a zipper photonic crystal optomechanical cavity in silicon nitride [13], a fiber-based microcavity integrated into a fused silica mechanical resonator [14,15], a whispering-gallery-mode accelerometer [16], and a slot-type photonic crystal cavity [17]. These integrated micro- and nanoscale cavities provide displacement resolution in the range of ${1}\;{\rm fm/}\surd {\rm Hz}$ and below due to their low optical loss, which can result in an acceleration resolution on the order of ${1}\;{{\unicode{x00B5}{\rm m}}\cdot {\rm s}^{- 2}}/\surd {\rm Hz}$ and below for acceleration frequencies up to 10 kHz or more [1317].

In addition to high resolution, optomechanical accelerometers promise greater accuracy without the need for calibration because the displacement of the proof mass can be measured directly in terms of the laser wavelength, an accepted practical realization of the meter [18], rather than electrical quantities. To determine the acceleration acting on the sensor from the displacement of its proof mass, the device physics must be accurately known. Therefore, the accelerometer must have a simple, deterministic mechanical response so that the dynamic model can be accurately inverted to convert displacement to acceleration. Ideally, the thermomechanical noise of the accelerometer should exceed the other fundamental noise source, optical shot noise in the displacement measurement, so that the mechanical response can be identified with high fidelity and the acceleration noise floor will be flat over a wide frequency range [1921].

 figure: Fig. 1.

Fig. 1. Optomechanical accelerometer design. (a) Cross section of the accelerometer, including microfabricated cavity optomechanical components, polarization maintaining (PM) fiber optics, and a stainless-steel package. (b) Cross section of the two microfabricated chips. (c) Stitched optical micrograph of the mechanical resonator showing the high-reflectivity mirror coating restricted to the proof mass in order to avoid fouling the microbeams. Inset: Scanning electron micrograph of the silicon nitride microbeams. (d) Scanning electron micrograph of a cleaved concave silicon micromirror. Inset: Close-up of the high-reflectivity mirror coating with quarter-wave periodicity. (e) Image of a packaged and fiber-coupled accelerometer.

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In previous work, the mechanical mode structure has been too complex and difficult to identify to allow reliable, broadband conversion between displacement and acceleration, or shot noise has dominated over most of the bandwidth of the accelerometer, or both, thereby preventing broadband measurement. Here we demonstrate a microfabricated optomechanical accelerometer that reaches the thermodynamic resolution limit over a broad frequency range (${314}\;{{\rm nm}\cdot {\rm s}^{- 2}}/\surd{\rm Hz}$ over 6.8 kHz), greatly exceeding the resolution and bandwidth found in conventional accelerometers. Broadband measurement is necessary for detection of general time-varying signals at the thermodynamic limit, as well as rigorous understanding of the device physics required for advanced applications. In addition, the devices reported here are fully packaged, field-deployable, scalable, operable in air and vacuum—and achieve the highest acceleration resolution reported to date for a microfabricated optomechanical accelerometer. While the focus here is acceleration for vibration measurement, the platform is equally well suited for inertial sensing, seismometry, and gravimetry. In addition, the platform is applicable to many other applied and fundamental physical measurements. For example, optomechanical detection has recently been applied to dark matter detection [22], and approaches using mechanical detection have been proposed that include geometries similar to that presented here [2325].

2. ACCELEROMETER DESIGN

The optomechanical accelerometer and its components are described in Fig. 1. Two silicon microfabricated chips comprise the main sensing elements of the accelerometer. One chip contains a millimeter-scale silicon proof mass suspended on both sides by silicon nitride (${{\rm Si}_3}{{\rm N}_4}$) microbeams, and the other chip has a concave silicon micromirror. Both optical elements have patterned dielectric mirror and antireflective coatings. A hemispherical Fabry–Perot cavity is formed by assembling the chips such that the displacement of the mechanical resonator relative to the concave micromirror can be measured with high precision by interrogating one of the cavity’s optical resonances [Fig. 1(b)]. When an acceleration is applied to the accelerometer package, the mechanical resonator displaces relative to the concave micromirror, which is measured as an intensity change in the light reflected from the cavity and converted to a measured acceleration.

The concave micromirror is fabricated in single crystal silicon using a wet etching process [26,27], resulting in high-quality mirrors with radii of curvature of approximately 410 µm, a depth of 257 µm, and a surface roughness of 1 nm RMS. The mechanical resonator is composed of a single-crystal silicon proof mass that is constrained on both sides by 1.5 µm thick silicon nitride beams [Fig. 1(c)]. This design ensures nearly ideal piston-like displacement in response to an acceleration perpendicular to the chip’s surface and provides large frequency separation between the piston mode and higher-order modes (see Supplement 1). In addition, this design provides low cross-axis sensitivity because the in-plane stiffness of the resonator is 1700 times larger than that along the optical axis based on finite element analysis.

Two accelerometers were used in the presented experiments, which are only principally different in the dimensions of the proof mass and silicon nitride beams as well as the packaging. Device A has a ${3}\;{\rm mm} \times {3}\;{\rm mm} \times {0.525}\;{\rm mm}$ proof mass; beams that are 20 µm wide, 92 µm long, and spaced by 20 µm; a resonant frequency of 9.86 kHz; a mass of approximately 11 mg; and it is packaged as shown in Fig. 1(e). Device B is a bare device mounted without a cover for vacuum compatibility and has a ${4}\;{\rm mm} \times {4}\;{\rm mm} \times {0.525}\;{\rm mm}$ proof mass; beams that are 20 µm wide, 84 µm long, and spaced by 20 µm; a resonant frequency of 8.74 kHz; and a mass of approximately 20 mg. This sensor design can be extended to a range of measurements such as force, pressure, seismology, and gravimetry by simply modifying the mechanical resonator to have the appropriate mass, stiffness, and damping properties for the given application.

3. ACCELEROMETER FABRICATION AND ASSEMBLY

The concave silicon micromirror was fabricated using a slow isotropic wet etching process on a double-side polished, 525 µm thick silicon wafer. First, a 35 µm deep recess was etched using deep reactive ion etching (DRIE), providing space between the moving proof mass and micromirror when assembled. Then the wafer was coated with stoichiometric silicon nitride (300 nm thick) using low-pressure chemical vapor deposition (LPCVD), which serves as a hard mask during wet etching. Circular apertures 300 µm in diameter were patterned in the silicon nitride layer using reactive ion etching (RIE). The wafer was then etched in a mixture of hydrofluoric, nitric, and acetic acids (HNA, 9:75:30 ratio) at room temperature for a predetermined time to achieve the desired depth and radius of curvature, which are approximately 257 µm and 410 µm, respectively, in the presented accelerometers. See [27] for more details.

 figure: Fig. 2.

Fig. 2. Spectra for the optical cavity. (a) Reflected and transmitted spectra for the optical cavity over a single free spectral range (FSR) near 1550 nm. Higher-order transverse modes in addition to the fundamental (${{\rm TEM}_{00}}$) modes are imaged in transmission using an InGaAs camera. (b) A single fundamental mode that is used to transduce the motion of the proof mass is shown, where the optical finesse $F$ is 5430. The red region on the resonance indicates the location for side-locking to the cavity.

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The mechanical resonator was fabricated on a double-side polished, 525 µm thick silicon wafer by patterning both sides of the wafer identically. A 1.5 µm thick, low-stress silicon nitride layer was deposited on the wafer using LPCVD. The proof mass and beam geometry were patterned with optical lithography, and the silicon nitride was etched with RIE. DRIE was then used to etch the beam pattern through the silicon wafer from both sides in subsequent etch steps. After dicing into 1 cm chips, the beams and proof mass were released by undercutting the silicon nitride beams using KOH with a concentration of 30% at 60°C. The anisotropic etch results in a uniform, faceted sidewall on the proof mass that is self-limiting due to the etch resistance of the ${\langle}{111} \rangle $ crystal planes, providing repeatable dimensions for the proof mass.

Dielectric mirror and antireflection coatings with alternating tantalum pentoxide and silicon dioxide layers were applied to the concave micromirrors and mechanical resonators using ion beam sputtering [Fig. 1(d)]. A shadow mask made from an etched silicon wafer was used to selectively deposit the coatings on the proof mass and concave mirror. A pair of the completed chips were aligned and bonded with UV curable adhesive. This is a self-aligned process that requires no adjustment of angle or translation beyond ensuring overlap of the concave micromirror and proof mass. Finally, the chip assembly was aligned to a polarization maintaining fiber collimator within the accelerometer package and bonded using UV curable adhesive [Fig. 1(a)]. Antireflection coatings on the focusing lens and the back of the proof mass are used to reduce parasitic reflections.

4. OPTICAL READOUT

The optical spectrum of the hemispherical cavity was measured in both transmission and reflection as shown for wavelengths near 1550 nm in Fig. 2(a), where the free spectral range (FSR) is 400 GHz (3.21 nm), and higher-order transverse modes can be seen between the dominant fundamental modes. These modes were imaged in transmission on an InGaAs camera, showing intensity profiles characteristic of highly symmetric spatial modes. Modes grouped in columns have similar resonance wavelengths but are not degenerate. Displacement measurements of the mechanical resonator were performed in reflection using a fundamental cavity mode (${{\rm TEM}_{00}}$) near a wavelength of 1551 nm with a linewidth of $\Gamma = {73.7}\;{\rm MHz}$ (FWHM), a finesse of $F = {5430}$, and a mirror reflectivity of $R = {99.89}\%$ as shown in Fig. 2(b). The selection of $F$ was based on the trade-off between sensitivity and dynamic range for measurement with a side-locked laser.

The readout method used for small-amplitude displacement measurement of the optical cavity is shown in Fig. 3(a). A stable fiber laser (FL) with a short-term linewidth near 100 Hz is phase modulated using an electro-optic modulator (EOM), which is driven near 3 GHz to generate sidebands. One sideband is locked to the cavity at the maximum slope point on the side of the optical resonance. Side-locking is achieved with a low bandwidth proportional-integral-derivative (PID) controller ($\approx \;{300}\;{\rm Hz}$). Slow changes in cavity length, largely due to thermal- or humidity-induced drift of the cavity length, are tracked by the laser wavelength, while faster motion of the mechanical resonator generates intensity fluctuations that are used to detect acceleration. The incident optical power is 350 µW, which is expected to displace the proof mass by roughly 100 fm on resonance due to radiation pressure. Though a measurable displacement, this does not affect the results reported here. A static displacement does not change the response function of the accelerometer, which depends only on the resonant frequency and damping.

 figure: Fig. 3.

Fig. 3. Displacement spectral densities and the noise equivalent acceleration. (a) Diagram of the optical cavity readout method used to measure the noise performance of the accelerometer. EOM, electro-optic phase modulator; VOA, variable optical attenuator; OSA, optical spectrum analyzer; VCO, voltage-controlled oscillator; CIR, circulator; BPD, balanced photodetector; ESA, electronic spectrum analyzer; IGA, InGaAs camera; PD, photodetector; LPF, low-pass filter; and PID, proportional-integral-derivative controller. (b) Displacement spectral density for the accelerometer in air. Dashed line: Fit to the thermomechanical noise model. Gray line: Shot noise when the laser sideband is completely detuned from the optical resonance. Black line: Photodetector dark noise. Inset: Log–log plot of displacement spectral density. (c) Comparison between operation in air and in vacuum. Dashed lines: Respective fits to the thermomechanical noise model. (d) Noise equivalent acceleration (NEA). Indicated frequency bands represent the range over which the NEA is within 3 dB of the acceleration thermomechanical noise limit (dashed lines).

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To suppress laser intensity noise, a balanced detection scheme with a bandwidth near 1 MHz was used. The resulting signal from the balanced detector was digitized using a 12-bit spectrum analyzer with a bandwidth of 28 kHz. This approach was used for the sensing resolution measurements presented in Section 5 due to the superior broadband noise performance of the FL. In addition, a widely tunable external cavity diode laser (ECDL) was used in place of the FL for the measurements in this section and in Section 6 due to its wider wavelength tuning range and resulting ability to easily tune to a desired cavity mode under rapidly varying measurement conditions (see Supplement 1). For both lasers, the reflected intensity fluctuations for the side-locked cavity result in a detector voltage $\Delta V$ that is converted to displacement $\Delta L$ using the relation $\Delta L = L\;\Delta V/(\lambda \;S)$, where $L$ is the nominal cavity length, $\lambda$ is the nominal cavity resonance wavelength, and $S = dV/d\lambda$ is the slope of the optical resonance at the lock point (see details in Supplement 1).

5. SENSING RESOLUTION

The displacement noise floor was measured in air and in a vacuum chamber ($P = {133}\;{\rm mPa}$) at room temperature, while the accelerometer was acoustically and vibrationally isolated. The resulting displacement spectral density in air for Device A is shown in Fig. 3(b), where a single vibrational mode is present between 100 Hz and 28 kHz and is driven purely by thermomechanical noise. This is the first demonstration reported of an optomechanical accelerometer operating with a single vibrational mode over such a wide bandwidth. A pure single-mode response is important for the accurate determination of the acceleration acting on the sensor from the displacement of its proof mass using first principles (see Supplement 1). The presence of additional modes and antiresonances between modes would increase the complexity of the model fit from the thermomechanical noise response. In addition, antiresonances are generally not visible in the thermomechanical noise response. Both of these issues can result in significant inaccuracy in the conversion from displacement to acceleration with a multimode model.

A fit of the displacement spectral density to the expected thermomechanical noise response for a simple harmonic oscillator with viscous damping shows close agreement in Fig. 3(b) (see Supplement 1), allowing precise estimates of the resonance frequency ${\omega _0} = {2}\pi \times {9.852}({16})\;{\rm kHz}$, quality factor $Q = {99}({2})$, and mass $m = {10.8}({9})\;{\rm mg}$. This mass estimate derived from the thermomechanical fit is well within the uncertainty of the value of 11.07(53) mg calculated from the dimensions of the silicon resonator and optical coatings (see Supplement 1). The noise floor at the lowest frequencies is set by readout noise that is likely due to laser frequency noise, phase modulation noise from the EOM, or thermal effects. Well above resonance, approaching 28 kHz, the noise floor closely approaches the optical shot noise limit. Importantly, the displacement resolution is limited by thermomechanical noise over most of the measured frequency range. This was achieved by optimizing the optical ($L, F$) and mechanical ($m, Q, {\omega _0}$) parameters so that the thermomechanical noise is above or equal to the shot noise within the bandwidth of interest. One benefit of being broadband limited by thermomechanical noise is that the harmonic oscillator model fit can be very accurate due to a high signal-to-noise ratio, which provides greater precision when converting from proof mass displacement to acceleration.

Comparing the displacement spectral density in air and vacuum for Device B in Fig. 3(c), the increased $Q$ in vacuum, due to a reduction in gas damping, results in larger thermomechanical noise on resonance and less away from resonance, as expected. However, due to the balance between the thermomechanical noise and shot noise, the frequency range over which the spectral density is thermomechanically limited is clearly reduced. The displacement spectral densities in Fig. 3(c) are converted to a noise equivalent acceleration (NEA) by dividing the response by the harmonic oscillator transfer function (see Supplement 1) as shown in Fig. 3(d). As expected, the NEA reaches the acceleration thermomechanical limit, which is independent of frequency (${a_{\rm{th}}} = \sqrt{{4{k_B}T{\omega _0}/mQ}}$, see Supplement 1), wherever the displacement spectral density is limited by thermomechanical noise. Fluctuations are reduced when the damping is lower, providing a lower thermodynamic limit but making it more difficult to reach since the shot noise must be lower than the thermomechanical noise. Due to increased damping in air, the minimum NEA is higher, ${912}\;{{\rm nm}\cdot {\rm s}^{- 2}}/\surd{\rm Hz}$ (${93}\;{{\rm ng}_n}/\surd {\rm Hz}$, ${1}\;{{\rm g}_n} = {9.81}\;{{\rm m}\cdot {\rm s}^{- 2}}$), than in vacuum, ${314}\;{{\rm nm}\cdot {\rm s}^{- 2}}/\surd {\rm Hz}$ (${32}\;{{\rm ng}_n}/\surd {\rm Hz}$). The resolution in vacuum represents the lowest value reported—by 2 orders of magnitude—for a microfabricated optomechanical accelerometer with equivalent bandwidth [13,17]. The achieved resolution is significant in this class of device because microfabrication enables scalable fabrication and embedded devices. The bandwidth over which the NEA is within 3 dB of the acceleration thermomechanical limit is 13.6 kHz and 6.8 kHz for air and vacuum, respectively. This wide range is made possible by the exceptionally low displacement readout noise. Furthermore, the NEA only varies by 1 order of magnitude over the frequency range, which is an improvement of 2 to 4 orders of magnitude compared to previously reported optomechanical accelerometers [13,14]. This reasonably flat NEA is important for making high-precision broadband acceleration measurements since it provides a consistent signal-to-noise ratio over the measurement bandwidth.

6. SENSING PERFORMANCE UNDER EXTERNAL ACCELERATION

As a test of sensing performance for a range of external acceleration frequencies, the optomechanical accelerometer was placed on a piezoelectric shaker table, and the accelerometer output was compared with the motion measured with a homodyne Michelson interferometer [see Fig. 4(a) and Supplement 1]. The frequency of the sinusoidal acceleration generated by the shaker was swept from 1 to 20 kHz. The interferometer was used to measure the displacement of the accelerometer package, which has a 5 mm square gold-on-silicon mirror bonded to it. The resulting displacement amplitude as a function of drive frequency for Device A is shown in Fig. 4(b), where the displacement of the proof mass and package are different because the accelerometer response includes the resonance of the proof mass (9.86 kHz) and the first resonance of the shaker (12.68 kHz), whereas the external interferometer can only detect the shaker resonance. The inset shows that the shaker linearity is better than 1.3% (see Supplement 1). In addition to the large resonances, much smaller structures in the accelerometer displacement data can be seen at 3.9 kHz and 11.6 kHz. They have been linked to the accelerometer packaging and the shaker itself and are dependent on the torque used in mounting the accelerometer onto the shaker.

 figure: Fig. 4.

Fig. 4. Shaker table testing of the accelerometer. (a) Experimental configuration for the shaker table tests. M, mirror; PD, photodetector; BS, nonpolarizing beamsplitter; ISO, optical isolator; and PID, proportional-integral-derivative servo loop. The microcavity readout is shown in Fig. 3(a). (b) Comparison of the normalized displacement measured with the accelerometer and interferometer. (c) Comparison of the normalized acceleration measured by the accelerometer and interferometer. The displacement resolution of the accelerometer is more than 100 times greater than that of the interferometer (${0.1}\;{\rm fm/}\surd {\rm Hz}$ and ${60}\;{\rm fm/}\surd {\rm Hz}$, respectively). As a result, different drive voltages were used, 0.1 mV (blue) and 25 mV (red) for the accelerometer and 5 mV (navy) and 30 mV (green) for the interferometer, respectively. The shaker was found to be highly linear for this drive voltage range [see the inset in (b) at a shaker frequency of 5 kHz and Supplement 1], making this comparison possible.

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The displacement data from the accelerometer was converted to acceleration, and the interferometer displacement data was transformed to acceleration by multiplying by ${({2\pi {f_d}})^2}$, where ${f_d}$ is the drive frequency. Each data set is normalized by the shaker table drive voltage. As shown in Fig. 4(c), there is close agreement between the accelerometer and interferometer throughout the entire 20 kHz bandwidth. The maximum amplitude of acceleration measured in this case is slightly less than ${0.1}\;{{\rm m/s}^2}$. Using a side lock at higher amplitudes can lead to loss of lock or nonlinear response, so an optical comb readout for microcavities has been developed to measure larger amplitudes [28].

The accelerometer’s fundamental resonance does not appear in the acceleration data due to the model inversion, demonstrating that measurement on and even above resonance can be effective for these single-mode devices. The percent deviation of the accelerometer from the interferometer was calculated at each measurement frequency. The standard deviation of this value over the entire frequency range is 15.9% and between 4.5 and 11 kHz it is 9.7% after applying a moving average filter to the interferometer data to reduce noise (see Supplement 1). This comparison confirms that the accelerometer is behaving like a harmonic oscillator (i.e., exhibiting a single, one-dimensional, viscously damped piston mode of the proof mass) and is effective for broadband acceleration measurements. This represents the widest bandwidth demonstrated to date at this error level using a first-principles description based on a single-degree-of-freedom oscillator model. However, this comparison does not accurately indicate the accelerometer performance, as the deviation is dominated by the mechanics of the external reference interferometer and its interaction with the shaker table.

7. CONCLUSION

In conclusion, we have demonstrated a compact, microfabricated optomechanical accelerometer that achieves the thermodynamic limit of resolution over a frequency range greater than 13 kHz, including on, above, and below resonance. Microfabrication enables scalable fabrication and embedded applications, while the highly ideal single-mode structure enables accurate inversion of the mechanical response for accurate measurement. Additionally, broadband measurement at the thermodynamic limit yields a detection resolution nearly independent of frequency, so resonant enhancement is not necessary for detection of weak signals and detection even above resonance is possible with the same noise-equivalent resolution despite a rapidly falling response. The compact size of the sensor enables high-precision measurements outside of laboratory settings, and the optomechanical sensing platform is widely applicable to measurements beyond acceleration, such as force, pressure, and gravity sensing, through straightforward modification of the mechanical resonator.

Funding

National Institute of Standards and Technology (NIST on a Chip Program, 70NANB17H247, 70NANB20H174).

Acknowledgment

This work was partially supported by the NIST on a Chip Program. Y. B. acknowledges support from the National Institute of Standards and Technology (NIST), Department of Commerce, USA (70NANB17H247). F. Z. acknowledges support from the National Institute of Standards and Technology (NIST), Department of Commerce, USA (70NANB20H174). The authors thank Ben Reschovsky for helpful discussions and sharing analysis of the uncertainties derived from fits to thermomechanical spectra. This research was performed in part in the NIST Center for Nanoscale Science and Technology NanoFab.

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

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References

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  1. N. Yazdi, F. Ayazi, and K. Najafi, “Micromachined inertial sensors,” Proc. IEEE 86, 1640–1659 (1998).
    [Crossref]
  2. D. K. Shaeffer, “MEMS inertial sensors: a tutorial overview,” IEEE Commun. Mag.51(4), 100–109 (2013).
    [Crossref]
  3. H. Nakstad and J. Kringlebotn, “Probing oil fields,” Nat. Photonics 2, 147–149 (2008).
    [Crossref]
  4. R. P. Middlemiss, A. Samarelli, D. J. Paul, J. Hough, S. Rowan, and G. D. Hammond, “Measurement of the Earth tides with a MEMS gravimeter,” Nature 531, 614–617 (2016).
    [Crossref]
  5. H. Jo, J. A. Rice, B. F. Spencer, and T. Nagayama, “Development of high-sensitivity accelerometer board for structural health monitoring,” Proc. SPIE 7647, 764706 (2010).
    [Crossref]
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  25. D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” New J. Phys. (2021) in press.
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2021 (2)

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

D. A. Long, B. J. Reschovsky, F. Zhou, Y. Bao, T. W. LeBrun, and J. J. Gorman, “Electro-optic frequency combs for rapid interrogation in cavity optomechanics,” Opt. Lett. 46, 645–648 (2021).
[Crossref]

2020 (2)

F. Monteiro, G. Afek, D. Carney, G. Krnjaic, J. Wang, and D. C. Moore, “Search for composite dark matter with optically levitated sensors,” Phys. Rev. Lett. 125, 181102 (2020).
[Crossref]

Y. Huang, J. G. Flor Flores, Y. Li, W. Wang, D. Wang, N. Goldberg, J. Zheng, M. Yu, M. Lu, M. Kutzer, D. Rogers, D.-L. Kwong, L. Churchill, and C. W. Wong, “A chip-scale oscillation-mode optomechanical inertial sensor near the thermodynamical limits,” Laser Photon. Rev. 14, 1800329 (2020).
[Crossref]

2019 (2)

J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568, 364–367 (2019).
[Crossref]

A. A. Geraci, C. Bradley, D. Gao, J. Weinstein, and A. Derevianko, “Searching for ultralight dark matter with optical cavities,” Phys. Rev. Lett. 123, 031304 (2019).
[Crossref]

2018 (1)

2017 (2)

M. Wu, N. L. Y. Wu, T. Firdous, F. F. Sani, J. E. Losby, M. R. Freeman, and P. E. Barclay, “Nanocavity optomechanical torque magnetometry and radiofrequency susceptometry,” Nat. Nanotechnol. 12, 127–131 (2017).
[Crossref]

Y. Bao, F. Zhou, T. W. LeBrun, and J. J. Gorman, “Concave silicon micromirrors for stable hemispherical optical microcavities,” Opt. Express 25, 15493–15503 (2017).
[Crossref]

2016 (2)

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

R. P. Middlemiss, A. Samarelli, D. J. Paul, J. Hough, S. Rowan, and G. D. Hammond, “Measurement of the Earth tides with a MEMS gravimeter,” Nature 531, 614–617 (2016).
[Crossref]

2015 (1)

O. Gerberding, F. G. Cervantes, J. Melcher, J. R. Pratt, and J. M. Taylor, “Optomechanical reference accelerometer,” Metrologia 52, 654–665 (2015).
[Crossref]

2014 (2)

F. G. Cervantes, L. Kumanchik, J. R. Pratt, and J. M. Taylor, “High sensitivity optomechanical reference accelerometer over 10 kHz,” Appl. Phys. Lett. 104, 221111 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

2012 (2)

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
[Crossref]

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and Q. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

2010 (1)

H. Jo, J. A. Rice, B. F. Spencer, and T. Nagayama, “Development of high-sensitivity accelerometer board for structural health monitoring,” Proc. SPIE 7647, 764706 (2010).
[Crossref]

2008 (1)

H. Nakstad and J. Kringlebotn, “Probing oil fields,” Nat. Photonics 2, 147–149 (2008).
[Crossref]

2005 (1)

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

2003 (1)

T. J. Quinn, “Practical realization of the definition of the metre, including recommended radiations of other optical frequency standards (2001),” Metrologia 40, 103–133 (2003).
[Crossref]

1998 (1)

N. Yazdi, F. Ayazi, and K. Najafi, “Micromachined inertial sensors,” Proc. IEEE 86, 1640–1659 (1998).
[Crossref]

1996 (1)

T. A. Berkoff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 8, 1677–1679 (1996).
[Crossref]

1995 (1)

T. B. Gabrielson, “Fundamental noise limits for miniature acoustic and vibration sensors,” J. Vib. Acoust. 117, 405–410 (1995).
[Crossref]

1993 (1)

T. B. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. Electron Devices 40, 903–909 (1993).
[Crossref]

1982 (1)

A. D. Kersey, D. A. Jackson, and M. Corke, “High-sensitivity fibre-optic accelerometer,” Electron. Lett. 18, 559–561 (1982).
[Crossref]

Afek, G.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

F. Monteiro, G. Afek, D. Carney, G. Krnjaic, J. Wang, and D. C. Moore, “Search for composite dark matter with optically levitated sensors,” Phys. Rev. Lett. 125, 181102 (2020).
[Crossref]

Aggarwal, N.

J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568, 364–367 (2019).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Ayazi, F.

N. Yazdi, F. Ayazi, and K. Najafi, “Micromachined inertial sensors,” Proc. IEEE 86, 1640–1659 (1998).
[Crossref]

Bao, Y.

Barclay, P. E.

M. Wu, N. L. Y. Wu, T. Firdous, F. F. Sani, J. E. Losby, M. R. Freeman, and P. E. Barclay, “Nanocavity optomechanical torque magnetometry and radiofrequency susceptometry,” Nat. Nanotechnol. 12, 127–131 (2017).
[Crossref]

Barker, P. F.

Berkoff, T. A.

T. A. Berkoff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 8, 1677–1679 (1996).
[Crossref]

Bhave, S.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

Blasius, T. D.

A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and Q. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012).
[Crossref]

Bradley, C.

A. A. Geraci, C. Bradley, D. Gao, J. Weinstein, and A. Derevianko, “Searching for ultralight dark matter with optical cavities,” Phys. Rev. Lett. 123, 031304 (2019).
[Crossref]

Brubaker, B.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

Carney, D.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

F. Monteiro, G. Afek, D. Carney, G. Krnjaic, J. Wang, and D. C. Moore, “Search for composite dark matter with optically levitated sensors,” Phys. Rev. Lett. 125, 181102 (2020).
[Crossref]

D. Carney, A. Hook, Z. Liu, J. M. Taylor, and Y. Zhao, “Ultralight dark matter detection with mechanical quantum sensors,” New J. Phys. (2021) in press.

Cervantes, F. G.

O. Gerberding, F. G. Cervantes, J. Melcher, J. R. Pratt, and J. M. Taylor, “Optomechanical reference accelerometer,” Metrologia 52, 654–665 (2015).
[Crossref]

F. G. Cervantes, L. Kumanchik, J. R. Pratt, and J. M. Taylor, “High sensitivity optomechanical reference accelerometer over 10 kHz,” Appl. Phys. Lett. 104, 221111 (2014).
[Crossref]

Churchill, L.

Y. Huang, J. G. Flor Flores, Y. Li, W. Wang, D. Wang, N. Goldberg, J. Zheng, M. Yu, M. Lu, M. Kutzer, D. Rogers, D.-L. Kwong, L. Churchill, and C. W. Wong, “A chip-scale oscillation-mode optomechanical inertial sensor near the thermodynamical limits,” Laser Photon. Rev. 14, 1800329 (2020).
[Crossref]

Cole, G. D.

J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568, 364–367 (2019).
[Crossref]

Corbitt, T.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568, 364–367 (2019).
[Crossref]

Corke, M.

A. D. Kersey, D. A. Jackson, and M. Corke, “High-sensitivity fibre-optic accelerometer,” Electron. Lett. 18, 559–561 (1982).
[Crossref]

Cripe, J.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568, 364–367 (2019).
[Crossref]

Crisosto, N.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

Curtis, E. A.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Derevianko, A.

A. A. Geraci, C. Bradley, D. Gao, J. Weinstein, and A. Derevianko, “Searching for ultralight dark matter with optical cavities,” Phys. Rev. Lett. 123, 031304 (2019).
[Crossref]

Elion, G. R.

A. R. Nelson and G. R. Elion, “Optical accelerometer,” U.S. patent4,567,771 (4February1986).

Eriksson, S.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Firdous, T.

M. Wu, N. L. Y. Wu, T. Firdous, F. F. Sani, J. E. Losby, M. R. Freeman, and P. E. Barclay, “Nanocavity optomechanical torque magnetometry and radiofrequency susceptometry,” Nat. Nanotechnol. 12, 127–131 (2017).
[Crossref]

Flor Flores, J. G.

Y. Huang, J. G. Flor Flores, Y. Li, W. Wang, D. Wang, N. Goldberg, J. Zheng, M. Yu, M. Lu, M. Kutzer, D. Rogers, D.-L. Kwong, L. Churchill, and C. W. Wong, “A chip-scale oscillation-mode optomechanical inertial sensor near the thermodynamical limits,” Laser Photon. Rev. 14, 1800329 (2020).
[Crossref]

Follman, D.

J. Cripe, N. Aggarwal, R. Lanza, A. Libson, R. Singh, P. Heu, D. Follman, G. D. Cole, N. Mavalvala, and T. Corbitt, “Measurement of quantum back action in the audio band at room temperature,” Nature 568, 364–367 (2019).
[Crossref]

Freeman, M. R.

M. Wu, N. L. Y. Wu, T. Firdous, F. F. Sani, J. E. Losby, M. R. Freeman, and P. E. Barclay, “Nanocavity optomechanical torque magnetometry and radiofrequency susceptometry,” Nat. Nanotechnol. 12, 127–131 (2017).
[Crossref]

Gabrielson, T. B.

T. B. Gabrielson, “Fundamental noise limits for miniature acoustic and vibration sensors,” J. Vib. Acoust. 117, 405–410 (1995).
[Crossref]

T. B. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. Electron Devices 40, 903–909 (1993).
[Crossref]

Gao, D.

A. A. Geraci, C. Bradley, D. Gao, J. Weinstein, and A. Derevianko, “Searching for ultralight dark matter with optical cavities,” Phys. Rev. Lett. 123, 031304 (2019).
[Crossref]

Gavartin, E.

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
[Crossref]

Geraci, A.

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and A. Geraci, “Mechanical quantum sensing in the search for dark matter,” Quantum Sci. Technol. 6, 024002 (2021).
[Crossref]

Geraci, A. A.

A. A. Geraci, C. Bradley, D. Gao, J. Weinstein, and A. Derevianko, “Searching for ultralight dark matter with optical cavities,” Phys. Rev. Lett. 123, 031304 (2019).
[Crossref]

Gerberding, O.

O. Gerberding, F. G. Cervantes, J. Melcher, J. R. Pratt, and J. M. Taylor, “Optomechanical reference accelerometer,” Metrologia 52, 654–665 (2015).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Optomechanical accelerometer design. (a) Cross section of the accelerometer, including microfabricated cavity optomechanical components, polarization maintaining (PM) fiber optics, and a stainless-steel package. (b) Cross section of the two microfabricated chips. (c) Stitched optical micrograph of the mechanical resonator showing the high-reflectivity mirror coating restricted to the proof mass in order to avoid fouling the microbeams. Inset: Scanning electron micrograph of the silicon nitride microbeams. (d) Scanning electron micrograph of a cleaved concave silicon micromirror. Inset: Close-up of the high-reflectivity mirror coating with quarter-wave periodicity. (e) Image of a packaged and fiber-coupled accelerometer.
Fig. 2.
Fig. 2. Spectra for the optical cavity. (a) Reflected and transmitted spectra for the optical cavity over a single free spectral range (FSR) near 1550 nm. Higher-order transverse modes in addition to the fundamental (${{\rm TEM}_{00}}$) modes are imaged in transmission using an InGaAs camera. (b) A single fundamental mode that is used to transduce the motion of the proof mass is shown, where the optical finesse $F$ is 5430. The red region on the resonance indicates the location for side-locking to the cavity.
Fig. 3.
Fig. 3. Displacement spectral densities and the noise equivalent acceleration. (a) Diagram of the optical cavity readout method used to measure the noise performance of the accelerometer. EOM, electro-optic phase modulator; VOA, variable optical attenuator; OSA, optical spectrum analyzer; VCO, voltage-controlled oscillator; CIR, circulator; BPD, balanced photodetector; ESA, electronic spectrum analyzer; IGA, InGaAs camera; PD, photodetector; LPF, low-pass filter; and PID, proportional-integral-derivative controller. (b) Displacement spectral density for the accelerometer in air. Dashed line: Fit to the thermomechanical noise model. Gray line: Shot noise when the laser sideband is completely detuned from the optical resonance. Black line: Photodetector dark noise. Inset: Log–log plot of displacement spectral density. (c) Comparison between operation in air and in vacuum. Dashed lines: Respective fits to the thermomechanical noise model. (d) Noise equivalent acceleration (NEA). Indicated frequency bands represent the range over which the NEA is within 3 dB of the acceleration thermomechanical noise limit (dashed lines).
Fig. 4.
Fig. 4. Shaker table testing of the accelerometer. (a) Experimental configuration for the shaker table tests. M, mirror; PD, photodetector; BS, nonpolarizing beamsplitter; ISO, optical isolator; and PID, proportional-integral-derivative servo loop. The microcavity readout is shown in Fig. 3(a). (b) Comparison of the normalized displacement measured with the accelerometer and interferometer. (c) Comparison of the normalized acceleration measured by the accelerometer and interferometer. The displacement resolution of the accelerometer is more than 100 times greater than that of the interferometer (${0.1}\;{\rm fm/}\surd {\rm Hz}$ and ${60}\;{\rm fm/}\surd {\rm Hz}$, respectively). As a result, different drive voltages were used, 0.1 mV (blue) and 25 mV (red) for the accelerometer and 5 mV (navy) and 30 mV (green) for the interferometer, respectively. The shaker was found to be highly linear for this drive voltage range [see the inset in (b) at a shaker frequency of 5 kHz and Supplement 1], making this comparison possible.

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