# Compact, Portable, Thermal-Noise-Limited Optical Cavity with Low
Acceleration Sensitivity

Megan Lauree Kelleher  1,2,$\dagger$ Charles A.McLemore  1,2  Dahyeon Lee  1,2
Josue Davila-Rodriguez  2,4 Scott A. Diddams  1,2,3  and Franklyn Quinlan 2
1Department of Physics, 390 UCB, University of Colorado, Boulder, CO 80309,
USA
2National Institute of Standards and Technology, 325 Broadway, Boulder, CO
80305, USA
3Electrical, Computer and Energy Engineering, 425 UCB, University of Colorado,
Boulder, CO 80309, USA
4current address: Stable Laser Systems, Boulder, CO 80301, USA
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
††journal: oe††articletype: Research Article

We develop and demonstrate a compact (less than $6$ mL) portable Fabry-Pérot
optical reference cavity. A laser locked to the cavity is thermal noise
limited at $2\times 10^{-14}$ fractional frequency stability. Broadband
feedback control with an electro-optic modulator enables near thermal-noise-
limited phase noise performance from $1$ Hz to $10$ kHz offset frequencies.
The additional low vibration, temperature, and holding force sensitivity of
our design makes it well suited for out-of-the-lab applications such as
optically derived low noise microwave generation, compact and mobile optical
atomic clocks, and environmental sensing through deployed fiber networks.

##  1 Introduction

Lasers frequency-locked to the resonance of a vacuum-gap Fabry-Pérot (FP)
optical cavity have reached fractional frequency instability lower than
$1\times 10^{-16}$ at $1$ s of averaging time [1, 2]. The frequency
instability is inherited from the fluctuations in cavity length as sampled by
an optical beam. The sampled length fluctuations can be reduced to less than
the diameter of a proton. Such extraordinary performance in laser frequency
stability has aided in the advancement of state-of-the-art laboratory optical
atomic clocks, with applications in the redefinition of the SI second [3] and
tests of fundamental physics [4]. The best fractional frequency stabilities in
cavity-stabilized lasers are achieved by exploiting long cavity lengths (up to
$\sim 48$ cm [5, 1]), operation at cryogenic temperatures [2], and extensive
environmental isolation by way of vibration-insensitive mounting and multiple
layers of thermal isolation [6, 7]. However, many out-of-the-lab applications
of stable lasers, such as portable optical atomic clocks[8], earthquake
detection using undersea optical fiber[9], and low phase noise microwave
generation via optical frequency division (OFD) [10, 11], benefit from the
high stability available in the optical domain, but are incompatible with the
size, weight, and infrastructure requirements of large or cryogenic cavity
systems. Furthermore, short cavities have the potential for inherently low
acceleration sensitivity without the need for active vibration
stabilization[12].

The search for laser frequency reference cavities that are rigidly held, have
reduced size and weight, and can operate in harsh and unpredictable
environments has led to the development of both solid-state dielectric
resonators [13, 14, 15, 16, 17], and compact vacuum-gap FPs [18, 19, 20, 21,
22, 6, 23]. Solid-state dielectric resonators are impressively small,
typically millimeter-scale, and, in some cases, can be manufactured at scale.
However, these resonators suffer from higher thermorefractive noise and
temperature sensitivity that has limited the fractional frequency instability
to the $10^{-13}$ level and above [17]. By placing the optical mode in vacuum
and using low expansion materials, compact and rigidly held FPs can reach
fractional frequency stabilities $\sim 1\times 10^{-15}$ with a cavity volume
near $60$ mL [18]. Notably, simulations of the noise of vacuum-gap FPs predict
that their size can be reduced to only a few milliliters while maintaining
fractional frequency stability performance well below the $10^{-13}$ level of
the best solid-state dielectric resonators; indeed, a recent demonstration of
a 10 mm-long, $8$ mL-volume cavity reached $6\times 10^{-15}$ [24], albeit
without testing of holding force or acceleration sensitivity. Thus, there
remains a compelling performance space that can be achieved with a compact FP
optical frequency reference, provided that a laser locked to the FP can
operate with noise at or near the cavity thermal noise limit, and the FP has
low acceleration and holding force sensitivity.

Here we present a simple, rigidly held cylindrical vacuum-gap FP cavity
reaching a fractional frequency stability of $2\times 10^{-14}$, capable of
supporting applications in low phase noise microwave generation via optical
frequency division (OFD)[10, 11], distributed optical fiber sensing [25, 26,
27], and mobile optical clocks[8]. The cavity is composed of fused silica (FS)
mirrors and an ultralow expansion (ULE) glass spacer which is only 6.3 mm
long. The cavity volume is $5.2$ mL. A laser locked to the cavity operates at
the cavity thermal noise limit for noise offset frequencies ranging from $\sim
1$ Hz to $\sim 10$ kHz. To our knowledge, the phase noise level at 10 kHz, at
approximately $-108$ dBc/Hz on the optical carrier, is one of the lowest
reported for any vacuum-gap FP [28, 18], or dielectric resonator [29, 30]. If
paired with an optical frequency comb, the laser system can support state-of-
the-art microwave phase noise that is comparable to the lowest phase noise
achieved to date for offset frequencies above $\sim 100$ Hz. Measurements of
the cavity’s low sensitivity to holding force indicate the cavity may be
reliably held on its end faces. Though other groups have simulated and studied
its impact [31, 32, 19, 20], these are the first direct measurements of
holding force sensitivity of which we are aware. Additionally, the cavity
acceleration sensitivity for three mechanical axes was measured to be $5\times
10^{-11}g^{-1}$, $3\times 10^{-10}g^{-1}$, and $6\times 10^{-10}g^{-1}$. A
variation on our design also allowed us to explore trade-offs in holding force
sensitivity, noise, and long-term stability for a cavity composed only of ULE
and larger diameter mirrors. In the following section, we discuss the cavity
design and laser locking approach. In Section 3, we present cavity holding
force and acceleration sensitivity simulations and measurement results, and
measurements on the phase noise and frequency stability of a laser locked to
the cavity. In Section 4 we summarize and conclude.

##  2 Cavity Design & Laser Locking

Figure 1: a) Photograph of the optical cavity, which is $27$ mm in diameter.
The spacer is 6.3 mm thick and the mirrors are $12.7$ mm in diameter and
$6.35$ mm thick. b) Photograph of the optical cavity in its mounting
structure. The mounting structure sits on top of a Macor spacer and is placed
inside of a heat shield, also pictured. c) Exploded cartoon view of the
holding structure. The cavity is suspended in the Invar mounting structure by
two Viton o-rings. These o-rings are held in place by two Invar holders. Note
the notch on the holder that prevents the o-ring from rotating when the
structure is assembled. Finally, threaded retaining rings are applied to
rigidly hold the cavity in place.

Compact and portable FP reference cavity designs must balance noise
performance, temperature sensitivity, acceleration sensitivity, tolerance to
mirror misalignment [19], sensitivity to changes in the holding force, and
manufacturability. We have chosen a simple cylindrical geometry for ease of
manufacture and its high degree of mechanical symmetry (see Fig. 1).
Mechanical symmetry is important for maintaining low acceleration sensitivity,
and has been exploited in many compact cavity geometries, including cubes
[20], spheres [19], pyramids [21], and other compact cylinders [18]. The ULE
spacer material is chosen for its low thermal expansion and measures $27$ mm
in diameter and $6.3$ mm in length. The spacer has a single vent hole with a
diameter of $3$ mm for evacuating the cavity. The mirror substrates are FS
with a diameter of $12.7$ mm and a standard thickness of $6.35$ mm. Both
mirrors have a radius of curvature (ROC) of $1$ m with a flat outer contact
annulus for optical contact bonding to the spacer. The total cavity volume is
only $5.2$ mL. The mirrors were manufactured and polished in the same batch,
leading to nearly equal optical contact areas. As we show below, the asymmetry
introduced by the single vent hole increases the acceleration sensitivity of
the cavity.

The mirror material was chosen by considering the thermal noise and
temperature sensitivity. Fused silica has a higher mechanical quality factor
than ULE, and results in lower thermal noise [33]. For the lowest temperature
sensitivity, the cavity should be operated at its zero-crossing temperature
($T_{zc}$), where the linear coefficient of thermal expansion (CTE) passes
through zero [34]. The $T_{zc}$ of ULE is typically near room temperature.
However, employing FS mirrors shifts the $T_{zc}$ of the cavity to well below
room temperature due to distortions of the mirror shape caused by the
comparatively large radial expansion of the mirrors [35]. This effect is
particularly important for short FP cavities, where the mirror distortions are
a larger fraction of the cavity length. To counteract this effect, ULE backing
rings [35] can be used to shift the cavity $T_{zc}$ back to a convenient
temperature, but it is not always possible to fully compensate the large
$T_{zc}$ shift of compact cavities in this way. High $T_{zc}$ ULE [24] can
also be used but is not typically specified to the required precision for a
repeatable compact cavity manufacturing process. Here, using a ULE spacer with
unknown $T_{zc}$, we designed our system to rely on temperature stabilization
and shielding of the cavity enclosure. This was driven by our targeted
application of low noise microwave generation and undersea fiber optic cable
measurements, where short-term phase and frequency fluctuations are of greater
concern than long-term cavity length drift.

The $1$ m ROC of the mirrors was chosen to maximize the optical spot size on
the mirrors, $w$, while maintaining reasonable tolerance to mirror
misalignment. For an optical cavity whose thermal noise is dominated by
Brownian noise in the coatings (as is the case here), the phase noise power
scales as $1/w^{2}$ [36], and for cavities where the ROC is much larger than
the cavity length (L), $w^{2}$ is proportional to $\sqrt{\text{ROC}}$.
However, the larger ROC results in a larger optical axis displacement due to a
mirror tilt or displacement, and the shift in the optical axis away from the
cavity mechanical axis increases the effective acceleration sensitivity of the
cavity [37]. Again assuming $\text{ROC}\gg L$, the beam displacement on the
mirror surfaces $d$ is given by $d\approx 0.5\times\text{ROC}\times\theta$,
where $\theta$ is the mirror tilt angle [38]. The residual angle between the
faces of our spacer is $<50~{}\mu$rad, leading to a maximum beam displacement
near $25~{}\mu$m. Furthermore, sag in the mirror surface for large ROC mirrors
is extremely small when the mirror diameter is small. For example, the sag at
the center of a $12.7$ mm diameter mirror with $1$ m ROC is only $\sim
20~{}\mu$m. This makes creating a contact annulus with the required roughness
and surface figure without spoiling the smoothness of the center of the mirror
extremely difficult. Indeed, the largest ROC we could obtain on $12.7$ mm
diameter mirrors with an annulus for optical contact bonding was 1 m.

The cavity’s rigid holding structure was designed to minimize the effect of
holding force variation on cavity length, shown in Fig. 1. We used finite
element analysis (FEA) software to design a holding geometry that can provide
first-order insensitivity to the holding force. Given the short length of our
cavity spacer, the cavity is held on the spacer end faces with Viton o-rings.
The o-rings are held against the cavity with a backing plate, behind which is
a threaded piece that screws into a holding mount. The backing plates and
holding mount are made of low expansion Invar to minimize temperature changes
of the holding structure from coupling to changes to the cavity holding force.
The base of the holding mount is composed of Macor to reduce the thermal
conductivity from the outer vacuum enclosure. Additionally, a polished
aluminum heat shield is placed around the cavity and holding structure to
reduce the radiative heat transfer to the cavity. The volume enclosed by the
heat shield is $\sim$40 mL.

Figure 2: a) Simplified block diagram of the optical cavity system. A $1550$
nm laser is sent through a fiberized AOM then EOM. The light is then split in
fiber. Some of the stabilized light is sent to the optical cavity. The $1550$
nm laser is stabilized to the optical cavity resonance using feedback to the
laser PZT, AOM, and EOM. A frequency diplexer (not shown) is used at the rf
input of the EOM so that the EOM can be used for both the PDH sideband
generation (at $50$ MHz) and feedback control. The transmission detector can
provide a RIN servo error signal, but is only utilized during flip tests to
stabilize the intracavity power because the alignment is affected by the
optical breadboard flexing. b) Stabilized laser light is sent on to be
compared to two different optical references. One is another cavity-stabilized
laser at 1550 nm. The other is a comb that is frequency-referenced to a
cavity-stabilized 1156 nm laser [39]. Software defined radio (SDR) is used to
track the phase of the signals, which are then cross-correlated to remove
uncommon noise in the frequency combs and beat note detection.

##  3 Results

The laser frequency was locked to the cavity with the Pound-Drever-Hall (PDH)
technique [40], as shown in Fig. 2. Cavity ring-down measurements yielded a
finesse of 600,000, providing a steep discriminator slope for laser locking.
We employed a broadband locking scheme utilizing feedback to an electro-optic
modulator (EOM) to achieve nearly cavity thermal noise-limited performance
over a broad offset frequency range [41, 42, 28]. Light from a 1550 nm
commercial fiber laser is routed through a fiber-coupled acousto-optic
modulator (AOM) frequency shifter, followed by a fiber-coupled EOM and a 90/10
fiber coupler, then is launched into free space to interrogate the cavity. A
free-space circulator directs light reflected from the cavity to a
photodetector used for PDH frequency stabilization. A separate photodetector
placed at the back end of the cavity is used to stabilize the intracavity
power, though during normal operation it is not needed as it was determined
that our laser’s intensity noise does not contribute to the measured phase
noise. Laser frequency stabilization to the cavity is implemented through
three feedback paths: a piezoelectric transducer controlling the laser cavity
length is used for low bandwidth/large dynamic range frequency corrections,
the AOM for mid-bandwidth (to few 100 kHz) feedback, and the EOM is used for
high bandwidth (up to $1$ MHz) feedback. The same EOM is also used to impart
50 MHz phase modulation sidebands on the laser light for the PDH error signal
generation. The optical power impinging on the cavity is $\sim 600~{}\mu$W,
leaving greater than $10$ mW of frequency-stabilized output from the 90/10
coupler. Since we use a single EOM for both PDH phase modulation and feedback
control, the stabilized light output will have 50 MHz sidebands. This will be
well outside the bandwidth of any frequency comb lock, and is not anticipated
to impede any applications using a comb. If stabilized light without PDH
sidebands is desired, PDH phase modulation and feedback control can be
separated by using two EOMs.

To measure the laser phase noise, the stabilized laser output was split and
heterodyned against two separate ultrastable optical frequency references. The
heterodyne beat notes were simultaneously digitally sampled by software
defined radio (SDR), and the phase fluctuations were extracted [43]. The phase
noise cross-spectrum was then obtained by averaging the complex product of the
Fourier transforms of the individual phase records [18]. This allowed us to
reject the noise of the optical phase references and proved to be particularly
important to reveal the low phase noise of our system for offset frequencies
$>1$ kHz.

###  3.1 Holding Force and Acceleration Sensitivity

Sensitivity to holding force was measured and compared to simulations, as
shown in Fig. 3. The cavity is held by rings on the end faces of the spacer,
and can be first-order insensitive to changes to the holding force. This can
be understood by considering the behavior of a linear elastic cylinder
compressed on the end faces either near the center or the outer diameter. For
a small ring-radius holding force, the length of cylinder’s central axis will
reduce as the holding force is increased. In contrast, when squeezed at a
large diameter near the cylinder’s rim, distortions in the spacer shape cause
the length along the central axis to increase. A first-order holding force-
insensitive point on cavity length lies in between these two extremes. This
simple description is complicated by the fact that spacer distortions are
coupled to mirror distortions, such that the mirror diameter, thickness and
the contact area between the mirror and spacer impact the holding force
sensitivity. Moreover, depending on the spacer’s aspect ratio, the holding
force insensitive radius may be too close to the outer rim to be practicable.

To simulate this holding force sensitivity for our design, an axisymmetric
cavity model was built in FEA simulation software, and a simulated force was
applied to the ULE spacer end faces along a $\sim$0.2 mm-wide ring that is
equal and opposite on both ends of the cavity. Cavity length changes were
calculated as a function of the holding force ring radius. As expected, the
holding force sensitivity depends on many parameters such as mirror thickness
and diameter, spacer thickness and diameter, and contact area. For an optical
contact annulus width of 6.4 mm, a holding-force zero-crossing is predicted
for a holding diameter just above 20.5 mm, slightly larger than our largest
o-ring. Perhaps more importantly, simulations of this cavity geometry show a
weak dependence of the holding force sensitivity to the o-ring diameter. This
can benefit manufacturability because it can be difficult to obtain an o-ring
with diameter that exactly matches that of the zero-crossing. To further
verify our holding force results, we simulated and measured the holding force
sensitivity of an all-ULE cavity consisting of 25.4 mm diameter mirrors with 1
m ROC, also shown in Fig. 3. In this case, the holding rings are placed on the
backside of the mirrors. For both cavity designs, the dependence of holding
force sensitivity on the contact area is displayed using the shaded sections.
For each simulation, the upper bound of the shaded region represents a larger
contact area, and the lower bound represents a smaller contact area. The
primary cavity design (FS-ULE) has a lesser dependence on contact area and is
more resilient to manufacturing error in this way.

Figure 3: a) Photographs of the primary cavity and the supplemental all-ULE
cavity. The cavities have identical spacers, but different mirrors. b)
Experimental results showing the holding force sensitivity’s dependence on
o-ring size, demonstrating excellent agreement with the finite element
analysis results, which are shown as shaded regions. The y-error bars are a
combination of the standard deviation in the frequency shift measurement and
the uncertainty in the mass of the weight applied. This does not include any
systematic offsets. The x-axis uncertainty is an estimate based on the
thickness of the o-ring (1 mm). Simulations incorporated a range of mirror-
spacer contact areas, where the contact annulus width varied from 3.2 mm to
2.5 mm in the FS-ULE cavity, and 6.4 mm to 5.1 mm in the all-ULE cavity.

To measure the holding force sensitivity, the cavity was mounted in air with
the optical axis aligned vertically (parallel to the force due to gravity). A
small weight of known mass was applied to the top of the cavity for each
available holding diameter. The change in frequency of a laser locked to the
cavity was observed, and the resulting fractional frequency shift per newton
was calculated. The weight was applied and removed a minimum of eight times to
establish statistics on the reproducibility of holding force sensitivity
measurement. Potential systematic errors, such as a small angle between the
optical axis and the force due to gravity, are ignored.

The results are consistent with the simulations, verifying our understanding
of the cavity and its design. The primary cavity design with $12.7$ mm FS
mirrors has a fractional frequency shift per newton of less than $2.4\times
10^{-8}$ with all of the o-ring diameters that were tested. This behavior is
contrasted with the secondary cavity with $25.4$ mm ULE mirrors, where the
holding force sensitivity has a stronger dependence on o-ring diameter. These
simulations also agree with the holding force sensitivity measurements.

We assess the holding force sensitivity by considering a change in holding
force due to a temperature change of the rigid mounting structure. By
accounting for the CTE of the Invar structure and the elastic properties of
the o-rings, we estimate the temperature-dependent holding force change on the
cavity as $\sim 3$ mN/ ∘C. This leads to a fractional cavity length change of
$<10^{-10}/{}^{\circ}$C. This is several orders of magnitude smaller than the
CTE of the FS-ULE cavity of $\sim 10^{-7}/{}^{\circ}$C. This is also smaller
than what one could expect when operating near the cavity $T_{zc}$, where the
CTE for a 1∘C change is $\sim 10^{-9}/{}^{\circ}$C. Thus, the temperature
induced cavity length change will be dominated by the cavity itself as opposed
to the holding force changes.

Figure 4: a) Diagram of the cavity indicating the mechanical axis along which
the cavity is flipped. In the red flip test, the cavity vent hole goes from up
to down. In the blue flip test, the optical axis is flipped, and in the black
flip test, the mechanical axis which is perpendicular to both the vent hole
and the optical beam is flipped. b) Flip test measurements of the primary
cavity acceleration sensitivity. The cavity is flipped along three mechanical
axes inducing a 2g acceleration change while a laser stays locked to the
cavity. The change in frequency is observed. Linear drift, attributed to
cavity temperature change, is removed from the data. The worst axis is
$6\times 10^{-10}\frac{\Delta L}{L}$ per g, and this axis corresponds to
flipping the vent hole up and down. The asymmetry caused by the vent hole is
likely contributing to higher acceleration sensitivity along this axis.

Previous work on cavities ranging from 5 cm to 20 cm in length with cubic,
spherical, and cylindrical geometries have demonstrated passive (without
vibration feedforward correction) acceleration sensitivities in the
$10^{-11}g^{-1}$ to mid $10^{-10}g^{-1}$ range [23, 44, 20, 45, 22, 46, 18].
We simulated and measured the acceleration sensitivity of our FS-ULE cavity
under a 2g static flip test [20, 18, 44]. We placed the cavity in vacuum onto
a rotatable optical breadboard that allowed us to flip the cavity along three
mechanical axes. The results of this measurement are in Fig. 4. The measured
acceleration sensitivities along the three axes are $5\times 10^{-11}g^{-1}$,
$3\times 10^{-10}g^{-1}$, and $6\times 10^{-10}g^{-1}$. The axis with the
highest sensitivity corresponds to a flip that changes the cavity vent hole
from pointing up to pointing down. This is the largest mechanical asymmetry in
the design, and FEA simulations indicate that the acceleration sensitivity
should be at best $\sim 3\times 10^{-10}g^{-1}$ along this mechanical axis.
Furthermore, we note that this low level of acceleration sensitivity was
achieved without active alignment of the cavity supports. The cavity is simply
centered in the mount by visual inspection and the Viton o-rings are centered
to the mount using groves in the holding structure. Simulations indicate that
additional minor asymmetries would result in higher acceleration sensitivity.
For example, offsets in the mirrors along the vent hole direction can cause
$\sim 6\times 10^{-10}g^{-1}$ per millimeter offset, and offsets in the
holding rings along the vent hole direction result in $\sim 8\times
10^{-11}g^{-1}$ per millimeter offset. A similar geometry with smaller vent
holes evenly spaced radially should perform better due to a higher degree of
symmetry. In the high symmetry cavity case, alignment tolerance will dominate,
likely at the $10^{-11}g^{-1}$ level assuming sub-millimeter alignment
tolerance, but this depends on a wide parameter space and could likely be
further optimized by changing variables like mirror thickness.

###  3.2 Phase Noise and Fractional Frequency Stability

Figure 5: Measured phase noise of the free-running commercial laser in blue,
cavity-stabilized laser phase noise in green, the cavity thermal noise limit
in black, an optical frequency divider in gold [11], and state-of-the-art
photodetector phase noise at 10 GHz in brown [47]. Note that the noise of the
cavity-stabilized laser stays near the thermal noise limit to $\sim 10$ kHz
offset. Because the phase noise of the cavity-locked laser above 1 kHz offset
frequency is at or below a state-of-the-art photodetector level, this cavity
can serve as a compact, low-noise reference for OFD microwave generation.

Phase noise of the cavity-stabilized laser, the predicted cavity thermal noise
limit, and the laser’s free-running noise, are shown in Fig. 5. The integrated
timing jitter from 1 MHz to 1.3 Hz is 12 fs. The laser phase noise closely
follows the cavity thermal noise for offset frequencies between 1 Hz and 10
kHz. The predicted thermal noise is calculated using FEA software [48] and
includes Brownian noise of the spacer, mirror coatings, and mirror substrates,
as well as thermoelastic noise of the substrates and spacer. Only the Brownian
noise of our small spacer differed significantly from simple analytic models.
Thermo-optic noise was calculated analytically [49, 50, 51], but did not make
a substantial contribution to the total thermal noise. Broadband noise
reduction of the free-running laser noise was realized using feedback to the
EOM, resulting in a gain bandwidth $>$ 1 MHz. At 10 kHz offset, this large
feedback bandwidth provides $>$ 60 dB suppression of the free-running laser
noise. To our knowledge, the measured phase noise level at 10 kHz is one of
the lowest reported for any vacuum-gap FP (despite the cavity’s small size)
[28, 18] or dielectric resonator [29, 30] .

When coupled with an optical frequency comb, this cavity can support low-noise
microwave generation through OFD [10]. The phase noise contribution of our
cavity on a 10 GHz carrier is shown on the right axis of Fig. 5. For
comparison, the phase noise of a state-of-the-art 10 GHz photodetector [47],
and one of the lowest noise OFD systems demonstrated to date are also shown
[11]. Importantly, our compact cavity can support microwaves comparable to
that of the lowest noise OFD signals produced yet-to-date for offset
frequencies greater than $\sim$100 Hz. Above $\sim$1 kHz, the microwave noise
contribution from the cavity is below that of the projected photodetector
noise, and thus will not adversely impact the signal. Above 10 kHz offset
frequencies, the cavity supports 10 GHz phase noise below -180 dBc/Hz. Of
course, the residual noise of the optical frequency comb will also contribute
to the final microwave phase noise; however, these results show that an
extremely compact cavity can enable microwave signals whose noise is
competitive with those systems that are referenced to much larger-size optical
cavity systems.

Figure 6 shows the measured fractional frequency stability of our primary (FS
mirrors and ULE spacer) cavity, given in terms of the Allan deviation (ADEV).
For comparison, the ADEV of the all-ULE cavity is also shown. The ADEV is
calculated using the phase record of the SDR measuring the beat note of our
cavities against a comb that is frequency-referenced to a cavity-stabilized
1156 nm laser. For the primary cavity, an external AOM driven by a direct
digital synthesizer (DDS) with a linearly chirped frequency correction was
used to compensate for the 136 Hz/s linear drift, which is likely due to the
CTE of this cavity. A residual drift of 14 Hz/s remained after compensation,
limited by the frequency resolution of the DDS. The fractional frequency
stability of the primary cavity (black curve) is near its calculated thermal
noise limit (dashed black curve) at 0.1 s averaging time. The slightly larger
thermal noise limit of the all-ULE cavity is shown in dashed blue. The ADEV of
the all-ULE cavity was measured at both room temperature near 23 ∘C and at 55
∘C, demonstrating the large range of temperatures at which this cavity can
operate (our setup did not allow for cooling the cavity below room
temperature). At room temperature, the frequency is nearly thermal noise
limited from 0.1 s to 1 s with a linear drift of 3 Hz/s. At 55 ∘C, the
frequency stability reaches the calculated thermal noise limit from 0.3 s to
0.8 s, and exhibits a slightly lower drift rate of 2 Hz/s. No $T_{zc}$ was
found with this cavity below 55 ∘C, though we note the ULE used for this
spacer is legacy material from previous experiments [34], and its material
properties are not well known. Still, despite the higher thermal noise and
lack of $T_{zc}$, the all-ULE design can be a valuable compromise where long
term-stability and low drift are desirable.

Figure 6: Measured fractional frequency stability expressed as an Allan
deviation (ADEV) of a laser locked to the primary cavity and the all-ULE
cavity. An external AOM was used to compensate for a linear drift of 136 Hz/s
on the locked laser. There was a residual drift of 14 Hz/s at the time of the
measurement. The all-ULE cavity did not have a drift canceling AOM and the
drift rate was substantially lower, 3 Hz/s at room temperature and 2 Hz/s at
55 ∘C. Note that the all-ULE Cavity is thermal noise limited at 1 s at 55 ∘C.

##  4 Conclusions

We have demonstrated a compact optical frequency reference cavity design, with
a cavity volume of $5.2$ mL, compatible with out-of-the-lab applications which
can support ultralow noise microwave generation through OFD. The design uses a
simple cylindrical geometry for ease of manufacturing and is rigidly held.
Using a frequency-locking technique utilizing EOM feedback with $>$1 MHz
bandwidth, we demonstrated near thermal noise-limited optical phase noise
performance, reaching -110 dBc/Hz at 10 kHz offset. Despite the higher thermal
noise floor resulting from its compact size, this system provides one of the
lowest phase noise results at 10 kHz offset for any optical reference cavity
of which we are aware. For larger offsets, the noise remains below -100
dBc/Hz. Moreover, the cavity demonstrated near thermal noise limited
fractional frequency stability of $2\times 10^{-14}$ at 0.1 s. A lower
frequency drift rate was achieved with an all-ULE cavity variation of the
design.

Additionally, a low holding force sensitivity of the cavity was measured,
demonstrating agreement with simulations. Importantly, the holding force
sensitivity showed minimal dependence on the holding radius. Measurements of
the acceleration sensitivity ranged from $\sim 6\times 10^{-10}g^{-1}$ for the
cavity’s mechanical axis that displays the largest asymmetry to $\sim 5\times
10^{-11}g^{-1}$ for the least sensitive mechanical axis. This low acceleration
sensitivity was achieved with a simple holding geometry and minimal alignment
of the cavity into the mount.

Further improvements to the cavity performance are straightforward. The
acceleration sensitivity can be reduced by implementing a more symmetric vent
hole pattern in the cavity spacer. An all-ULE version of the 12.7 mm cavity
design could combine the low dependence of the holding force diameter with low
frequency drift (at a minimal cost to the low phase noise). With its
demonstrated low noise performance, our cavity design fills an important gap
in the performance-size trade space, enabling compact out-of-the-lab systems
with improved phase and frequency stability.

Funding This work is funded by AFRL and NIST.

Acknowledgments We thank R. Fox for providing the ULE used in these
experiments, and the NIST Yb optical atomic clock group for providing
frequency-stabilized light at 1156 nm. We also thank Andrew Ludlow, Tobias
Bothwell, and Laura Sinclair for their insightful comments on the manuscript.

This work is a contribution of an agency of the U.S. government and is not
subject to copyright in the USA. Commercial equipment is identified for
scientific clarity only and does not represent an endorsement by NIST.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this study are not
publicly available at this time but may be obtained from the authors upon a
reasonable request.

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