# Optomechanical accelerometers for geodesy

Adam Hines  Andrea Nelson  Yanqi Zhang  Guillermo Valdes  Jose Sanjuan
Jeremiah Stoddart  Felipe Guzman<EMAIL_ADDRESS>Texas A&M University,
Aerospace Engineering & Physics, College Station, TX 77843

###### Abstract

We present a novel optomechanical inertial sensor for low-frequency
applications and corresponding acceleration measurements. This sensor has a
resonant frequency of $4.715(1)\text{\,}\mathrm{Hz}$, a mechanical quality
factor of $4.76(3)\text{\times}{10}^{5}$, a test mass of
$2.6\text{\,}\mathrm{g}$, and a projected noise floor of approximately
$5\text{\times}{10}^{-11}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{\mathrm{Hz}}$
at $1\text{\,}\mathrm{Hz}$. Such performance, together with its small size,
low weight, reduced power consumption, and low susceptibility to environmental
variables such as magnetic field or drag conditions makes it an attractive
technology for future space geodesy missions. In this paper, we present an
experimental demonstration of low-frequency ground seismic noise detection by
direct comparison with a commercial seismometer, and data analysis algorithms
for the identification, characterization, and correction of several noise
sources.

##  I Introduction

Satellite geodesy missions such as the Gravity Recovery and Climate Experiment
(GRACE) [1] and GRACE Follow-on (GRACE-FO) [2] are equipped with low-frequency
three-axis accelerometers used to detect nongravitational accelerations. These
accelerations include forces such as air drag, thruster activation, and
radiation pressure in the twin-satellite constellation, which are essential
measurements to be removed from the missions’ gravity observations. The
accelerometers deployed in GRACE and GRACE-FO are electrostatic in nature,
reaching acceleration sensitivities of
$1\text{\times}{10}^{-10}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{\mathrm{Hz}}$
[3, 4].

One major challenge posed by these accelerometers, however, is the difficulty
to be tested on-ground. These sensors feature quasi-free-falling test masses
with no rigid connections to their electrostatic housing. When not in orbit,
the test masses cannot be electrostatically suspended and therefore cannot
easily be tested. This problem can hence be avoided by suspending a test mass
mechanically, such as with flexures. Optomechanical sensors, for example, can
track the motion of a mechanically suspended test mass using optical readout
methods. While a monolithic suspension can help improve significantly the
accelerometer robustness, it imposes limitations to its performance as a trend
of $1/f^{1/2}$ toward very low frequencies.

In spite of such limitations, we expect the sensitivities of these
optomechanical devices to be at the levels of valuable scientific
observations, while featuring a very small and lightweight footprint. In
addition, their performance is independent of drag conditions (and thus drag-
free or drag compensation controls are not required), insensitive to magnetic
fields, and easy to be thermally controlled due to their small size and
weight. To this end, we propose the use of an optomechanical accelerometer for
use in future geodesy missions, either by replacing current accelerometers or
complementing future baseline instruments as risk-reduction devices, such as
gravitational reference sensors [5, 6] or quantum gravimeters and gradiometers
[7, 8, 9]. In addition, they could be applicable for interplanetary missions
studying the gravity fields around other bodies in our solar system [10].
Finally, they are suitable for ground-based and planetary seismology and
geodesy applications [11, 12], due to the fact that they can be readily
operated under normal gravitational potentials such as Earth’s gravity.

Over the past $10\text{\,}\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}$,
optomechanical sensors have been used for many different high-precision
sensing applications. Force sensitivities on the order of
$10\text{\,}\mathrm{fN}$ at room temperature have been demonstrated using a
$10\text{\,}\mathrm{kHz}$ mechanical resonator with a cavity readout [13].
Optomechanical magnetometers have realized sensitivities better than
$1\text{\,}\mathrm{n}\mathrm{T}\mathrm{/}\sqrt{\mathrm{Hz}}$ over a
$\text{\,}\mathrm{GHz}$ bandwidth using the mechanical modes of barium
titanium silicate microspheres [14]. Furthermore, nano-optomechanical
displacement sensors with operational bandwidths of several
$\text{\,}\mathrm{MHz}$ have been developed with noise floors on the order of
$10\text{\,}\mathrm{f}\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$ [15]. Yet other
applications include atomic force microscopy and acoustic sensing. Finally,
optomechanical sensors have been used for accelerometry, demonstrating an
acceleration noise of $100\text{\,}\mathrm{ng}$ over a
$10\text{\,}\mathrm{kHz}$ bandwidth [16].

However, it is worth mentioning that all the above examples use mechanical
resonators with eigenfrequencies in the $\text{\,}\mathrm{kHz}$ or higher.
While low-frequency accelerometers have been developed and tested for various
applications, most of them are read out electrostatically [10]. Hence,
optomechanical devices have not been investigated as closely, due to the
larger physical dimensions, higher cost, and poorer sensitivities to high-
frequency signals. Yet many fields such as seismology, geophysics, geodesy,
hydrology, and more require the observation of acceleration signals below
$1\text{\,}\mathrm{Hz}$. Previous work on a low-frequency optomechanical
accelerometer demonstrated high mechanical quality factors on the order of
$1\text{\times}{10}^{5}$ with a $2\text{\,}\mathrm{g}$,
$3.7\text{\,}\mathrm{Hz}$ test mass, corresponding to thermal acceleration
sensitivity limits on the order of
$1\text{\times}{10}^{-10}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{\mathrm{Hz}}$
[17]. The drawback of this device was the inability to integrate the optical
readout onto the wafer the resonator was etched from. In this paper, we
present our low-frequency optomechanical accelerometer, consisting of a
$5\text{\,}\mathrm{Hz}$ mechanical resonator whose displacement from
equilibrium is tracked with laser interferometry. This resonator is designed
to be easily integrated with the optical readout, which is advantageous for
maintaining a compact form. We describe the characterization process that
suggests this resonator is a promising candidate for a novel accelerometer for
space geodesy, as well as ground-based planetary and navigation applications.
Using a ringdown measurement, we find the mechanical quality factor of this
device to be $4.76(3)\text{\times}{10}^{5}$, corresponding to a thermal
acceleration noise on the order of
$$5\text{\times}{10}^{-11}\text{\,}\mathrm{m}\mathrm{s}^{-2}$/\sqrt{f}$ below
resonance. Furthermore, we compare acceleration measurements from our
resonator to those from a commercial seismometer to demonstrate the
resonator’s ability to detect seismic motion down to $1\text{\,}\mathrm{mHz}$.

##  II Methods

###  II.1 Optomechanical Accelerometers

Our optomechanical accelerometer is composed of two main components: the
mechanical resonator and the laser interferometric readout. In this section,
we discuss the design of both parts and present characterization measurements
which can be used to project the acceleration noise floor of our device.

#### Resonator—Design and Characterization

We designed a resonator intended to be used for 1D acceleration measurements,
though a triaxial device would be a straightforward extension of the topology
presented here. This resonator was laser-assisted dry-etched from a single
monolithic fused silica wafer and was
$$90\text{\,}\mathrm{mm}$\times$80\text{\,}\mathrm{mm}$\times$6.6\text{\,}\mathrm{mm}$$
in volume with a total mass of $58.2\text{\,}\mathrm{g}$. Our accelerometer
had a smaller form and lighter weight than the GRACE-FO accelerometer,
representing a major advantage for satellite missions. An image of our
resonator is shown in Figure 1. Our design consisted of a
$2.6\text{\,}\mathrm{g}$ parallelogram test mass supported by two
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick leaf-spring flexures.

Figure 1: A diagram of our $5\text{\,}\mathrm{Hz}$ resonator. A penny is added
for reference.

A finite element analysis (FEA) simulation performed in COMSOL predicted a
$5.5\text{\,}\mathrm{Hz}$ resonant frequency, which made our resonator
suitable for measuring noninertial disturbances below $1\text{\,}\mathrm{Hz}$.
At frequencies below resonance, the test mass response $x$ to an external
acceleration $a$ was approximately given by:

$x\approx\frac{a}{\omega_{0}^{2}}$ (1)

where $\omega_{0}$ is the angular resonant frequency. There is a trade-off
between the required displacement sensitivity, how low the resonance can be in
a practical resonator, and its final dimensions. Therefore, a resonance on the
order of $1\text{\,}\mathrm{Hz}$ allows this low-frequency noise to couple
into the test mass motion better than a high resonance device, which in turn
relaxes the required test mass readout precision. Conversely, a lower
resonance device requires a larger resonator, which quickly becomes more
difficult to work due to the low stiffness.

Furthermore, our simulations predicted that all higher-order modes had
frequencies above $130\text{\,}\mathrm{Hz}$, significantly higher than the
lowest resonance by over an order of magnitude. This was desired for
mitigating the cross talk between modes that appeared in our measurements,
which in turn reduced the noise in our data. Figure 2 shows the first two
modes of this resonator: the first being the translational mode of the test
mass and the second being a violin mode of the flexures with a significantly
higher frequency.

Figure 2: The first two eigenmodes of our $5\text{\,}\mathrm{Hz}$ resonator
modeled in COMSOL. Note that the second mode is larger than the first by a
factor of more than 10. Having higher-order modes that are substantially
higher than the first mode is desired for minimizing the cross talk between
modes observed in measurements.

The acceleration experienced by the resonator was recovered by applying a
transfer function, which was defined by the resonance and quality factor, to
the test mass’s displacement data. This transfer function was given by [17]:

$\frac{\tilde{x}(\omega)}{\tilde{a}(\omega)}=\frac{-1}{\omega_{0}^{2}-\omega^{2}+i\omega_{0}\omega/Q}$
(2)

where $\tilde{x}(\omega)$ is the test mass motion, $\tilde{a}(\omega)$ is the
acceleration coupling into the resonator, $\omega_{0}$ is the resonant
frequency, and $Q$ is the mechanical quality factor. Note that when
$\omega<<\omega_{0}$, Equation (2) simplifies to Equation (1). To characterize
the acceleration sensing capabilities of our resonator, we experimentally
determined these parameters using a ringdown test where we deliberately
excited the test mass motion and tracked its oscillations. In the absence of
other perturbations, the oscillation amplitude exponentially decayed, allowing
for an easy calculation of $Q$. Figure 3 depicts the decay envelope of our
ringdown measurement, which was performed over 14 h at a vacuum pressure of
$10\text{\,}\mathrm{\SIUnitSymbolMicro Torr}$. From this measurement, we found
a $Q$-factor of $4.76(3)\text{\times}{10}^{5}$ and an $mQ$-product larger than
$1250\text{\,}\mathrm{kg}$. Furthermore, taking the fast Fourier transform
(FFT) of the raw data yielded a resonant frequency of
$f_{0}=$4.715(1)\text{\,}\mathrm{Hz}$$, in good agreement with our simulated
value of $5.5\text{\,}\mathrm{Hz}$. The transfer function in Equation (2)
could then be evaluated using our experimentally determined parameters. We
applied this transfer function to the displacement measurements in the
frequency domain to convert the signal to acceleration noise.

Figure 3: Decay envelope of our ringdown measurement fitted to an exponential.
The resonant frequency was removed with a low-pass filter, allowing us to
calculate a $Q$-factor of $4.76(3)\text{\times}{10}^{5}$.

#### Laser Interferometer—Design and Characterization

In order to measure the local acceleration noise, we must measure the
displacement of the test mass from its equilibrium. For this, we constructed a
heterodyne displacement interferometer with a differential phase readout.
Cutouts on the test mass and the u-shaped frame of our resonator, shown in
Figure 1, allowed for an easy implementation of mirrors that completed this
laser interferometer. The optical methods and noise rejection schemes used for
this readout are discussed in greater detail in [18], though an overview of
this optical readout is provided in this section. Figure 4 contains a diagram
of our laser interferometer. A fiber-coupled $1064.181\text{\,}\mathrm{nm}$
beam was split and frequency-shifted by two acoustic-optical modulators (AOMs)
to create a $5\text{\,}\mathrm{MHz}$ heterodyne frequency. The frequency
shifted beams were then injected into a series of prisms that created two
distinct Mach–Zehnder type interferometers: one that tracked the test mass
motion and one that acted as a reference.

The measurement interferometer sent one beam to a mirror on the test mass,
$M_{M}$, where the displacement information was imprinted on the phase of the
reflected beam. Similarly, the reference interferometer reflected one beam off
a stationary mirror, $M_{R}$. The second beams for both interferometers
reflected off the same common mirror, $M$. Common mode noise sources, such as
temperature fluctuations in the prisms, coherently affect both
interferometers. Therefore, a differential phase measurement rejected some
environmental noise and allowed for high-sensitivity displacement sensing.

Furthermore, the laser added noise to our data in the form of laser frequency
noise. To combat this, we also introduced a delay-line interferometer to the
setup. This interferometer allowed us to make independent measurements of the
frequency noise by interfering one beam from the laser with a delayed copy of
itself. By creating a path-length difference of $2\text{\,}\mathrm{m}$ in the
delay-line interferometer arms, laser frequency fluctuations became the
dominant noise source in this fiber interferometer, although its signal-to-
noise ratio (SNR) decreased toward lower frequencies due to fiber noise. With
this measurement, we were able to remove the laser frequency noise from the
resonator data in postprocessing. However, for example, in future space
geodesy missions, a frequency-stabilized laser source would be available,
offering a much better stability than what can be achieved through
postcorrection and therefore eliminating the need for a delay-line
interferometer. The readout displacement noise, taken with a stationary mirror
in place of the test mass mirror, is shown in Figure 5. We find that this
interferometer can measure displacement data with a sensitivity of
$3\text{\times}{10}^{-12}\text{\,}\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$ at
$1\text{\,}\mathrm{Hz}$ and
$7\text{\times}{10}^{-10}\text{\,}\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$ at
$1\text{\,}\mathrm{mHz}$. The peaks near $0.4\text{\,}\mathrm{Hz}$ and
$1\text{\,}\mathrm{Hz}$ are caused by the mechanical resonances of a vibration
isolation platform the interferometer was placed on while testing.

Figure 4: A diagram of the heterodyne readout used for our measurements. In
this schematic, the mirrors $M$ and $M_{R}$ are stationary while $M_{M}$ is
the mirror on the test mass. Figure 5: A plot of the displacement noise in our
heterodyne interferometer. As this measurement was taken without our
resonator, seismic noise was not present in the data. The shaded area
represents the estimated error bars of the spectrum.

###  II.2 Accelerometer Noise Floor

The acceleration sensitivity of our optomechanical accelerometer was
anticipated to be limited by both the thermal noise of the resonator and the
displacement noise from the readout interferometer. In this section, we
estimated both contributions in order to calculate the acceleration noise
floor.

Thermal noise, which represents the ultimate acceleration sensitivity that can
be achieved with our resonator, is caused by gas damping and internal loss
mechanisms within the fused silica. These loss mechanisms include bulk losses,
surface losses, and thermoelastic losses. The thermal acceleration noise can
be derived from theory starting from the equation of motion for a resonator,
given by [17, 19]:

$ma=m\ddot{x}+m\Gamma\dot{x}+m\omega_{0}^{2}(1+i\phi(\omega))x$ (3)

where $m$ is the test mass, $\Gamma$ is the gas damping rate, and
$\phi(\omega)$ is the internal loss coefficient of our fused silica
oscillator. Converting to the frequency-domain and applying the
fluctuation–dissipation theorem [19], we find that the thermal acceleration
noise is given by:

$\tilde{a}_{\mathrm{th}}^{2}(\omega)=\frac{4k_{\mathrm{B}}T}{m\omega}(\omega\Gamma+\omega_{0}^{2}\phi(\omega))$
(4)

where $T$ is the temperature of the test mass and $k_{B}$ is Boltzmann’s
constant. In general, both gas damping and internal losses contribute to the
thermal noise. However, because our testing environment could reach
$\text{\,}\mathrm{\SIUnitSymbolMicro Torr}$ pressures, we operated under the
assumption that gas damping was negligible in comparison to internal losses.
Previous work on a similar resonator estimated that a vacuum pressure of
$10\text{\,}\mathrm{\SIUnitSymbolMicro Torr}$ would be sufficient for making
gas damping negligible compared to other mechanical losses, which is achieved
both in laboratory environments, portable vacuum enclosures, and certainly
space geodesy missions. Therefore, for the remainder of the paper, we let
$\Gamma=0$.

We can calculate $\phi(\omega)$ using known equations for the different loss
mechanisms [20, 21, 22, 23], but this is not ideal for our purposes. The
mechanical losses of fused silica have been studied intensively by the
gravitational wave community for use in low-thermal noise test mass
suspensions. The models developed by these investigations are useful for
optimizing a fused silica sensor’s topology, but the only information we can
experimentally obtain for $\phi(\omega)$ is the $Q$-factor. Therefore, we want
an equation for the thermal noise in terms of $Q$ rather than $\phi(\omega)$.
For this, we assume that the internal losses are constant in the bandwidth of
interest and are given by:

$\phi(\omega)=\frac{1}{Q}$ (5)

Using these assumptions, we simplify Equation (4) to:

$\tilde{a}_{\mathrm{th}}^{2}(\omega)=\frac{4k_{\mathrm{B}}T\omega_{0}^{2}}{mQ\omega}$
(6)

This is the final equation we used to calculate the thermal acceleration
noise. Note that larger $Q$-factors lead to lower thermal noise, highlighting
the importance of fused silica as a material of choice. Fused silica is known
to have very low internal losses at room temperature, with $Q$-factors well
above $1\text{\times}{10}^{6}$ for high-frequency resonators [24, 25, 26, 27,
28] and above $1\text{\times}{10}^{5}$ for low-frequency devices [17].
Research into the material properties of fused silica at cryogenic
temperatures has shown that the mechanical losses $\phi$ of this material
increases substantially as temperature decreases [29]. At
$30\text{\,}\mathrm{K}$, these losses can increase by as much as four orders
of magnitude, which would in turn increase the thermal acceleration noise of
our optomechanical accelerometer by a factor of 100. Moreover, this work
suggested a fused silica resonator could be operated at temperatures as low as
$225\text{\,}\mathrm{K}$ with minimal quality factor degradation of at most a
few percent. Hence, we adopted $225\text{\,}\mathrm{K}$ as the minimum
operating temperature of our accelerometer, which was not anticipated to
impede our device’s performance in the context of space geodesy, as such
systems typically operate near room temperature.

In addition to thermal motion, displacement readout noise from the laser
interferometer contributed significantly to the acceleration noise floor of
our optomechanical accelerometer. For this section, we note that the Laser
Interferometer Space Antenna (LISA) Pathfinder mission has demonstrated an
optical readout with a sensitivity reaching
$35\text{\,}\mathrm{f}\mathrm{m}\mathrm{/}\sqrt{Hz}$ [30], representing an
excellent sensitivity level that can be used to model our own projected noise
floor. To convert this displacement noise to acceleration, we simply applied
the transfer function in Equation (2). This readout noise was assumed to be
incoherent with the thermal motion of the resonator, so the two noises added
in quadrature:

$\tilde{a}_{\mathrm{floor}}^{2}(\omega)=\tilde{a}_{\mathrm{th}}^{2}(\omega)+\big{|}\frac{\tilde{a}(\omega)}{\tilde{x}(\omega)}\big{|}^{2}\tilde{x}_{\mathrm{int}}^{2}(\omega)$
(7)

Equation (7) is plotted in Figure 6a, where we observe the acceleration
sensitivity is anticipated to be
$5\text{\times}{10}^{-11}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{Hz}$
at $1\text{\,}\mathrm{Hz}$ and increases towards low frequencies as
$f^{-1/2}$. Above resonance, the noise floor increases rapidly, dominated by
readout noise, while below resonance the thermal motion is sufficiently low
that the noise floor is
$1\text{\times}{10}^{-9}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{Hz}$
at $1\text{\,}\mathrm{mHz}$.

Figure 6: (a) The acceleration noise floor of our optomechanical accelerometer
including thermal motion from the resonator and readout noise from the laser
interferometer. (b) The corresponding displacement readout noise floor
required to observe this acceleration noise level. Both plots assume a
mechanical $Q$-factor of $4.76\text{\times}{10}^{5}$ and a readout noise
consistent with that achieved by LISA Pathfinder [30].

##  III Results

To demonstrate the acceleration sensing capabilities of our device, we took
simultaneous measurements with the optomechanical accelerometer and a
Nanometrics Trillium Horizon 120-2 (T120H) seismometer [31]. The T120H
seismometer was placed on top of the vacuum chamber that housed the resonator
to ensure seismic noise coupled into both accelerometers coherently. Moreover,
the axis of motion for the resonator was set up to coincide with the x-axis of
the seismometer, allowing for an easy comparison of the two sensors. Photos of
this setup are shown in Figure 7. In this section, we present preliminary
measurements taken with our optomechanical accelerometer and describe the
correction methods we used to remove various noise sources.

Figure 7: (a) An image of our experimental setup, including our fused silica
resonator, heterodyne laser interferometer, and delay-line interferometer. (b)
An image of the commercial seismometer’s setup relative to our vacuum chamber,
which houses the items shown in (a).

###  III.1 Measuring Seismic Noise

Data were taken for 60 h allowing us to observe frequencies as low as
$4.6\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$. The acceleration noise observed
by both devices is shown in Figure 8, where we note the agreement in the
microseismic band between $100\text{\,}\mathrm{mHz}$ and
$500\text{\,}\mathrm{mHz}$. The seismometer detected less noise at frequencies
below this bandwidth, indicating that our optomechanical accelerometer was
limited by environmental noise below $100\text{\,}\mathrm{mHz}$. Even though
our optomechanical accelerometer’s observed noise is significantly higher than
the measured optical readout noise in Figure 5, the laser path length
difference between the measurement and reference interferometers when the
resonator is incorporated in the setup is greater than
$1\text{\,}\mathrm{cm}$. This in turn enlarges the laser frequency noise in
comparison to what was observed in the interferometer stability test.

Figure 8: The acceleration noise measured by our optomechanical accelerometer
and Trillium Horizon 120 seismometer. We observe good agreement between
$100\text{\,}\mathrm{mHz}$ and $500\text{\,}\mathrm{mHz}$. The shaded areas
represent the estimated error bars of the spectra.

###  III.2 Data Post-Correction

The data taken with our optomechanical accelerometer presented in Section
III.1 were initially unable to observe the same acceleration noise as our
T120H seismometer below $100\text{\,}\mathrm{mHz}$ due to a combination of
signals originating from the environment and the optical readout. However,
with careful environmental monitoring, some of this noise could be removed in
postcorrection. In this section we discuss the different noise removal methods
we utilized in our data to achieve a better acceleration sensitivity, and
correspondingly a better agreement with the commercial seismometer.

#### Time-Domain Linear Regression

A portion of the environmental noise in our data could be removed by
performing linear fittings to different sets of environmental data taken at
the same time as the acceleration measurement. In addition to the previously
discussed delay-line interferometer, this environmental data included the room
temperature, vacuum chamber temperature, vacuum pressure, and the heterodyne
signal amplitudes of the measurement, reference, and delay-line
interferometers. Furthermore, we retrieved data on the ambient weather
conditions from Texas A&M University’s Research Farm [32], as the changing
barometric pressure can induce tilting in our optical bench which causes a
projection of the local gravity into the resonator’s axis of motion.

For frequencies below $100\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$, we found
that the barometric pressure dominated the resonator’s acceleration noise,
with a coupling coefficient of
$-29.24\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{Bar}}^{-1}$
found by linear regression in the time domain. To isolate the frequency band
where the pressure noise dominated, this linear regression was performed on
bandpass-filtered data. While effective, the efficacy of this postcorrection
was limited by the data acquisition of the weather station at Texas A&M’s
Research Farm, which reported hourly averages. The ultralow Nyquist frequency
of $139\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ restricted our ability to
correct our data for barometric pressure fluctuations above that frequency. As
part of our ongoing efforts to develop this technology, we will include in-
house measurements of the ambient laboratory air pressure taken with a
commercial barometer sampled at a much higher rate to avoid this problem.

The acceleration noise caused by laser frequency noise was found in a similar
way. Laser frequency fluctuations appeared in our resonator data as a bump
around $12\text{\,}\mathrm{mHz}$, which can be observed in Figure 8. By
applying a bandpass filter with corner frequencies of $5\text{\,}\mathrm{mHz}$
and $40\text{\,}\mathrm{mHz}$ to the resonator and delay-line interferometer
data, we found a coupling coefficient of
$1.1421\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{rad}}^{-1}$. This coupling
coefficient informed us on how stable a laser would need to be in order for
our resonator to be thermally limited. The phase of our delay-line
interferometer, $\phi_{\mathrm{delay}}$, was proportional to our laser
source’s frequency noise, $\nu_{\mathrm{laser}}$, by [18]:

$\phi_{\mathrm{delay}}=\frac{2\pi\Delta L}{c}\nu_{\mathrm{laser}}$ (8)

where $c$ is the speed of light and $\Delta L$ is the path length difference
in the delay-line interferometer, which was $2\text{\,}\mathrm{m}$ in our
case. This suggested that the coupling factor between our resonator’s
acceleration, $a$, and the laser frequency noise $\nu_{\mathrm{laser}}$ was
$4.709\text{\times}{10}^{-14}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{Hz}}^{-1}$.
Using our projected thermal acceleration noise shown in Figure 6, we
anticipated a laser frequency stability of
$$1.06\text{\,}\mathrm{kHz}$/\sqrt{f}$ would be required for observing the
thermal motion of our resonator. This stability is very achievable, and the
frequency stability of the laser onboard GRACE-FO is much better than this
requirement [33].

#### Transfer Function

The linear regression method of noise subtraction that we used for the
barometric pressure and laser frequency corrections assumed that the phase and
amplitude of their respective coupling coefficients were uniform over the
bandwidths we were correcting. However, noise generally does not couple into a
given system uniformly in frequency; both the amplitude and phase of the
coupling factor can be frequency-dependent. The temperature inside our vacuum
chamber for example was a significant source of noise in our experiment, but
it did not couple uniformly in frequency. Because of this, we had to use a
different approach to remove the temperature effects from our data. We
corrected for the vacuum chamber temperature by calculating a transfer
function, $H_{Ta}(\omega)$, between the temperature and acceleration data
[34]. This transfer function is typically estimated by taking the ratio of the
cross-power spectral density (CPSD) of the acceleration and temperature data,
$S_{Ta}(\omega)$, and the power spectral density (PSD) of the acceleration,
$S_{aa}(\omega)$ [35]:

$H_{Ta}(\omega)=\frac{S_{Ta}(\omega)}{S_{aa}(\omega)}$ (9)

For this spectral analysis, we used a Nuttall window function and
$1.98\text{\times}{10}^{5}$ samples. The number of averages increased with
frequency. The lowest frequency bin, $5\text{\,}\mathrm{\SIUnitSymbolMicro
Hz}$, had only one average, which increased to over 1100 averages at
$0.5\text{\,}\mathrm{Hz}$. At $230\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$,
there were 10 averages, and so we adopted this frequency as the approximate
cutoff below which the spectral analysis had too few averages to be reliable.
This analysis was performed with LTPDA, an open-source MATLAB toolbox for data
analysis and signal processing developed and distributed by the LISA
Pathfinder community [36].

The amplitude of the transfer function between the vacuum chamber temperature
and the resonator’s acceleration after removing laser frequency and barometric
pressure noise is shown in Figure 9. We observe a frequency-dependent transfer
function amplitude ranging from
$2\text{\times}{10}^{-3}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{K}^{-1}$ to
$1\text{\times}{10}^{-2}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{K}^{-1}$.
Similar to laser frequency noise, the transfer function presented in Figure 9
can be used to estimate the temperature stability required for the resonator
to be limited by thermal motion. At $10\text{\,}\mathrm{mHz}$, the amplitude
of this transfer function is on the order of
$10\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{K}}^{-1}$,
while the thermal acceleration noise at this frequency is approximately
$4\text{\times}{10}^{-10}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{{}^{2}}\mathrm{/}\sqrt{\mathrm{Hz}}$.
This suggests that a thermally limited acceleration measurement would require
an environmental temperature stability of
$4\text{\times}{10}^{-8}\text{\,}\mathrm{K}\mathrm{/}\sqrt{\mathrm{Hz}}$. This
requirement being very improbable to achieve, we instead investigated the
precise mechanism through which temperature coupled into our setup and reduced
it to relax the temperature stability we needed to reach.

Figure 9: The transfer function between vacuum chamber temperature and
acceleration. Note that between $300\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$
and $50\text{\,}\mathrm{mHz}$, the amplitude varies by a factor of nearly 5,
suggesting that a linear regression would not be suitable for this data set.
The transfer function is not plotted below
$300\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ because the low number of
averages causes large uncertainties. The shaded area represents the estimated
error bars of the spectrum.

The typical transfer function one might expect for temperature coupling is one
that resembles a low-pass filter [34], which was not fully observed in our
setup. This behavior could be due to the fact that temperature was a
ubiquitous effect, present everywhere in the setup. The complicated transfer
function could be composed of several thermal effects occurring simultaneously
in different parts of the experiment. The exact mechanism causing temperature
fluctuations to couple into our setup in this manner is currently under
investigation.

We also calculated the uncertainty in the transfer function, which depends on
the transfer function amplitude, the coherence between vacuum temperature and
acceleration $C_{Ta}(\omega)$, and the number of averages in each frequency
bin [35]. In bins where either the coherence between the two time series was
poor or the number of averages was low, the variance in the transfer function
was large. The coherence, shown in Figure 10, was estimated by taking the
ratio of the magnitude of the cross-spectral density, $|S_{Ta}(\omega)|^{2}$,
and the product of the power spectral densities $S_{aa}(\omega)$ and
$S_{TT}(\omega)$ [35]:

$C_{Ta}(\omega)=\frac{|S_{Ta}(\omega)|^{2}}{S_{aa}(\omega)S_{TT}(\omega)}$
(10)

The coherence between the vacuum chamber temperature and the acceleration data
was larger than 0.8 between $300\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ and
$60\text{\,}\mathrm{mHz}$, indicating that temperature fluctuations were a
dominant noise source in this frequency band. Above $60\text{\,}\mathrm{mHz}$
there was a sharp decrease in the coherence, explaining the corresponding
decrease in transfer function amplitude and the increase in the transfer
function uncertainty.

Figure 10: The magnitude of the coherence between our vacuum chamber
temperature and our resonator’s acceleration data. The coherence is greater
than 0.8 between $0.3\text{\,}\mathrm{mHz}$ and $60\text{\,}\mathrm{mHz}$,
indicating that temperature is a significant noise source. The coherence
function is not plotted below $300\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$
because the low number of averages causes large uncertainties. The shaded area
represents the estimated error bars of the spectrum.

The temperature-induced acceleration noise was calculated by taking the FFT of
the temperature data, applying the transfer function, and then taking the
inverse FFT to return to the time domain. The inverse transform step of this
process, while not strictly necessary if one wants frequency-domain results
[35], is useful for visualizing the data and identifying the next limiting
noise source.

The noise breakdown for this measurement is shown in Figure 11a as well as the
residual after removing the laser frequency, pressure, and temperature noise.
This postcorrection is compared to the measured interferometric readout noise,
from which we see that the resonator’s acceleration data are above the
acceleration noise of the interferometer. This suggests that the noise we are
observing is a real signal with seismic or environmental origins. Figure 11b
also shows the difference between our optomechanical accelerometer data and
the seismometer data. We observe that this difference is larger than the
readout-induced acceleration noise, suggesting that the residual is limited by
noise not originating from the optical readout. Our data do not show any more
significant coherences between our acceleration and environmental monitoring.
As such, we consider the possibility that the residual noise in our
accelerometer data was, at least partially, introduced through postcorrection.
For example, if the self-noise of our temperature sensors was on the order of
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}\mathrm{/}\sqrt{\mathrm{Hz}}$
at $10\text{\,}\mathrm{mHz}$, that self-noise would be added to our
acceleration measurements during postcorrection at a level of approximately
$2\text{\times}{10}^{-7}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{\mathrm{Hz}}$,
ultimately restricting our final sensitivity. This highlights the importance
of investigating and characterizing these noise sources, as reducing their
physical coupling into the experiment offers better results than decohering
our data in postcorrections.

Furthermore, we find that after removing the temperature and barometric
pressure fluctuations, the good agreement between our resonator and
seismometer data extends down to $1\text{\,}\mathrm{mHz}$, which can be
visualized in frequency-space in Figure 11a and in the coherence between the
two data sets in Figure 12. The bump in the postcorrected resonator data
between $100\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ and
$400\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ is likely residual barometric
pressure noise that could not be removed due to the low Nyquist frequency of
the pressure measurement. Moreover, we can visually compare the data from both
accelerometers in the time domain to demonstrate the agreement between the
sensors. In Figure 13, we plotted a 1000 s segment of both data streams and in
doing so, we observed excellent agreement in both the magnitude and phase of
the two devices. These comparisons validated our resonator’s ability to detect
seismic noise above $1\text{\,}\mathrm{mHz}$.

Figure 11: (a) The amplitude spectral density of a seismic measurement taken
with our optomechanical accelerometer alongside the noise contributions from
laser frequency fluctuations, barometric pressure, and vacuum chamber
temperature. Shown also is the residual after removing all three noise
sources, demonstrating a significant reduction in noise from
$0.2\text{\,}\mathrm{mHz}$ to $100\text{\,}\mathrm{mHz}$ and below
$100\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$. (b) A comparison of the
postcorrected resonator data to the seismometer data. The good agreement
between the two devices now extends down to $1\text{\,}\mathrm{mHz}$. The
shaded areas represent the estimated error bars of the spectra. Figure 12: The
magnitude of the coherence between our postcorrected resonator data and
seismometer data both before and after removing environmental noises. It can
be seen that by removing temperature, pressure, and laser frequency
fluctuations from the resonator data, the coherence above
$1\text{\,}\mathrm{mHz}$ improves significantly. The shaded areas represent
the estimated error bars of the spectra. Figure 13: A 1000-s-long subset of
the acceleration time series obtained by our optomechanical accelerometer and
seismometer, in which we see excellent agreement between the two time series.
Both traces are high-pass filtered above $0.8\text{\,}\mathrm{mHz}$ for better
comparison of the frequency band with high coherence.

##  IV Discussion

###  IV.1 Comparative Technology Assessment

In Section II.2, we calculated the noise floor of our optomechanical
accelerometer using estimates of the optical readout noise and the thermal
motion associated with our experimentally measured $Q$-factor. This noise
floor can be compared to existing technologies to assess the viability of our
accelerometer for different applications. In Figure 14, we plotted our
estimated noise floor against the GRACE [3] and GRACE-FO [4] accelerometers,
as well as several ground-based seismometers including the Geotech S-13, the
Nanometrics Trillium 360, and the Nanometrics Trillium 120, which is the one
we utilized in our laboratory for the comparison measurements presented in
Section III.1 [37]. We observe that our accelerometer is anticipated to have a
competitive, if not lower, acceleration noise than the ground-seismometers,
suggesting that our device will be useful for seismometry and ground-based
geodesy studies. Moreover, our accelerometer has a mass of
$58.2\text{\,}\mathrm{g}$, and we anticipate the realization of a highly
compact and lightweight system compared to existing systems. Because of this,
our optomechanical accelerometer is smaller, more portable, and better-suited
for field work than these commercial technologies.

When compared to the GRACE and GRACE-FO accelerometers, our optomechanical
accelerometer is competitive at higher-frequencies around
$1\text{\,}\mathrm{Hz}$ but is expected to be noisier at lower-frequencies.
However, current mass change measurements taken by satellite gravity recovery
missions are limited primarily by temporal aliasing errors, not accelerometer
noise [38]. Hence, despite this slightly higher noise floor at low
frequencies, we expect our optomechanical accelerometers to be able to provide
meaningful data for science observations. Rather, with the advantages
exhibited by our optomechanical accelerometer, including a lower weight, more
compact form, and the ability to be tested on-ground, our accelerometer is
still valuable for satellite geodesy missions.

Figure 14: The projected noise floor of our optomechanical accelerometer
compared to those of other technologies, including the GRACE [3] and GRACE-FO
[4] accelerometers and the Geotech S-13, Trillium 120, and Trillium 360
seismometers [37]. Two traces are plotted for the noise floor of our
accelerometer: our current best estimate using our experimental value of the
$Q$-factor, and our resonator design using a value of $Q$ calculated using
models for the loss mechanisms of fused silica.

###  IV.2 Planned Developments

In the measurements presented in this article, we encountered several noise
sources, such as instabilities in the laser frequency and the optical readout,
limiting our ability to detect low-frequency signals. Future measurements will
be conducted with a frequency-stabilized laser and a compact quasi-monolithic
interferometer to improve our sensor’s low-frequency sensitivity. This
interferometer operates on the same principles as our current readout. It
consists of prisms and beamsplitters bonded together and small enough that
they can be integrated onto the wafer of our resonator, miniaturizing our
current experimental setup while offering better mechanical stability. A photo
of this interferometer is shown in Figure 15a.

Other improvements that can be made to reduce the measurement noise include
thermally isolating the system to dampen temperature fluctuations as well as
investigating noise due to tilt-to-length coupling and nonlinear optical path
length differences.

Furthermore, we will experimentally measure the thermal noise floor of our
accelerometer by performing a Huddle test. The test consists of placing two
identical resonators close to each other. The seismic noise would couple into
both devices coherently. After removing that correlated seismic noise, the
remaining noise will be the uncorrelated noise originated from the resonators
themselves.

These developments and a better understanding of low-frequency environmental
noise sources will allow for other tests to characterize our acceleration
sensing capabilities, e.g., to monitor the lunar and solar tidal acceleration
changes.

Finally, there are several extensions of the work presented in this manuscript
that would make this optomechanical accelerometer technology ready for use
onboard a space geodesy satellite. Specifically, we are currently developing
appropriate mounts and launch-lock mechanisms for the accelerometer that
facilitate its deployment and protect the dynamic test mass during
installation and launch. This cage should include a method of securing the
test mass and flexures to avoid damage while not in use, such as during launch
when significantly higher accelerations and shock are expected in contrast to
science operation. To this end, the higher-order violin modes of the flexures
in our accelerometer should be investigated in greater depth to ensure
compatibility with the vibrational loads present during launch.

Moreover, a triaxial optomechanical accelerometer could be constructed from a
series of resonators operating along different axes, which is required for
observing all noninertial perturbations while in orbit. A concept design of
such a sensor is depicted in Figure 15b, which has a volume of
$$110\text{\,}\mathrm{mm}$\times$110\text{\,}\mathrm{mm}$\times$22\text{\,}\mathrm{mm}$$
and a mass of $0.282\text{\,}\mathrm{kg}$, not including the masses of the
optical readout, which should be low overall as each quasi-monolithic
interferometer assembly weighs only $4.5\text{\,}\mathrm{g}$. This design is a
quasi-monolithic assembly of two layers of fused silica resonators that are
separated by spacers. The top layer holds two orthogonal resonators identical
to the one presented in Section II.1 operating along the $x$ and $y$ axes.
Because these resonators share the same topology as the one presented in this
paper, they will have the same noise floors shown in Figure 6. Furthermore, on
the top layer is a cutout that houses the optical readouts for all three axes.
An interferometer reaches the test mass of the bottom layer through a hole in
the top layer covered by a pentaprism.

Figure 15: (a) A visualization of our quasi-monolithic interferometer
integrated onto the wafer of our resonator. A US nickel is included for scale.
(b) A rendering of a triaxial accelerometer concept. This design consists of
two layers: the top containing resonators oscillating in the $x$ and $y$-axes
and the bottom with a resonator operating along the $z$-axis.

The bottom layer contains a resonator that measures out-of-plane accelerations
along the $z$-axis. However, due to limitations in fabrication, a different
topology is required for measuring the out-of-plane motion. To this end, we
have developed a preliminary design for the bottom layer resonator, which
consists of a $12\text{\,}\mathrm{g}$ rectangular prism test mass supported by
flexures on four sides. These flexures have cross sections of
$$8\text{\,}\mathrm{mm}$\times$0.1\text{\,}\mathrm{mm}$$ and are oriented such
that the test mass oscillates along the $z$-axis. Tuning the lengths of the
flexures as well as the test mass allows for a natural frequency less than
$10\text{\,}\mathrm{Hz}$, which improves the resonator’s low-frequency sensing
capabilities. Like the resonator presented in Section II.1, we require all
higher order modes to have resonances larger than the natural mode by an order
of magnitude. In the topology we have chosen for the bottom layer resonator,
however, tip-and-tilt modes can have a low enough frequency to allow cross
talk between the modes of the resonator.

Figure 16a visualizes this tip-and-tilt mode. To combat this, we offset the
flexure heights relative to the bottom of the wafer. Two flexures are
$1\text{\,}\mathrm{mm}$ above the bottom of the wafer and the other two are
$1\text{\,}\mathrm{mm}$ below the top of the wafer. Having the test mass and
flexures connected at multiple points along the $z$-axis increases the
frequency of this tip-tilt mode to above the 10$\times$ threshold. Figure 16b
depicts a close-up of the test mass and flexures to show this offset.

With the aim of developing a triaxial accelerometer with equal noise floors
along all three axes, an optimization of the bottom layer resonator must be
performed to ensure it has comparable noise to the top layer resonators. This
optimization includes adjusting the test mass and cross section to lower its
resonance and reduce its thermal motion while keeping the stress in the
flexures within a safe operating range.

Figure 16: (a) A close-up of the $z$-axis test mass and its flexures. Note
that the flexures are not at the midpoint of the wafer but have different
offsets to increase the stiffnesses and frequencies of tip-tilt modes. (b) The
tip-tilt mode of the $z$-axis resonator as seen from the underside of the
three-axis resonator. The colors indicate the total displacement, with light
green being the stationary and purple being the largest displacement.

##  V Conclusions

In this paper, we demonstrated that a compact and lightweight optomechanical
accelerometer etched from a monolithic wafer of fused silica with dimensions
of
$$90\text{\,}\mathrm{mm}$\times$80\text{\,}\mathrm{mm}$\times$6.6\text{\,}\mathrm{mm}$$
could detect seismic noise above $1\text{\,}\mathrm{mHz}$ in good agreement
with commercial seismometer technologies. This device had a test mass that was
suspended mechanically by thin flexures, allowing it to be tested on the
ground where some electrostatic devices cannot. As such, we proposed the use
of this technology onboard future space geodesy missions, as well as in
ground-based planetary and geodesy applications, for measuring noninertial
disturbances.

Although a direct measurement of the accelerometer’s noise floor was not
possible due to our testing environment being flooded with signals, there is
presently no evidence of noise sources that would prevent us from observing
the self-noise of our accelerometer with a sufficiently quiet test bed. The
anticipated thermal acceleration noise was approximately
$5\text{\times}{10}^{-11}\text{\,}\mathrm{m}\mathrm{s}^{-2}\mathrm{/}\sqrt{\mathrm{Hz}}$
at $1\text{\,}\mathrm{Hz}$, making this technology competitive with the
accelerometers that have flown on space geodesy missions such as GRACE and
GRACE-FO. As such, our optomechanical accelerometer is expected to be suitable
for satellite geodesy missions, among other applications.

The resonator’s displacement was measured optically with a heterodyne
interferometer and a $1064\text{\,}\mathrm{nm}$ laser that was not frequency-
stabilized. Below $100\text{\,}\mathrm{mHz}$, the acceleration data obtained
by our resonator were dominated by laser frequency noise as well as
temperature and pressure fluctuations. These noise sources can be partially
removed by careful environmental monitoring, significantly increasing the
coherence between our resonator and a commercial seismometer. Future works
will incorporate improvements to our optical readout, laser source, and
environmental monitoring to further enhance our acceleration measurements.

### Author contributions

Conceptualization, Adam Hines and Felipe Guzmán; Formal analysis, Adam Hines
and Jose Sanjuan; Funding acquisition, Felipe Guzmán; Investigation, Adam
Hines, Andrea Nelson, Yanqi Zhang and Guillermo Valdes; Methodology, Adam
Hines, Andrea Nelson, Yanqi Zhang, Guillermo Valdes, Jeremiah Stoddart and
Felipe Guzmán; Software, Jeremiah Stoddart; Supervision, Guillermo Valdes and
Felipe Guzmán; Validation, Adam Hines; Visualization, Adam Hines; Writing –
original draft, Adam Hines; Writing – review & editing, Andrea Nelson, Yanqi
Zhang, Guillermo Valdes, Jose Sanjuan and Felipe Guzmán. All authors have read
and agreed to the published version of the manuscript.

### Funding

This research was funded by: National Geospatial-Intelligence Agency (NGA)
(grant HMA04762010016), National Science Foundation (NSF) (grant PHY-2045579),
National Aeronautics and Space Administration (NASA) (grants 80NSSC20K1723 and
80NSSC22K0281), and Jet Propulsion Laboratory (JPL) (contract 1677619).

### Conflict of Interest

The authors declare no conflict of interest. The funders had no role in the
design of the study; in the collection, analyses, or interpretation of data;
in the writing of the manuscript; or in the decision to publish the results.

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