# Synthetic FM triplet for AM-free precision laser stabilization and
spectroscopy

Dhruv Kedar  1,2,* Zhibin Yao  1,2 Ivan Ryger  1 John L. Hall  1 and Jun Ye1
1JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA
2 Equal contributions<EMAIL_ADDRESS>
###### Abstract

The Pound-Drever-Hall (PDH) cavity-locking scheme has found prevalent uses in
precision optical interferometry and laser frequency stabilization. A form of
frequency modulation spectroscopy, PDH enjoys superior signal-to-noise
recovery, large acquisition dynamic range, wide servo bandwidth, and robust
rejection of spurious effects. However, residual amplitude modulation at the
signal frequency, while significantly suppressed, still presents an important
concern for further advancing the state-of-the-art performances. Here we
present a simplified and improved scheme for PDH using an acousto-optic
modulator to generate digital phase reference sidebands instead of the
traditionally used electro-optic modulator approach. We demonstrate four key
advantages: (1) the carrier and two modulation tones are individually
synthesized and easily reconfigured, (2) robust and orthogonal control of the
modulated optical field is applied directly to the amplitude and phase
quadratures, (3) modulation synthesis, demodulation, and feedback are
implemented in a self-contained and easily reproducible electronic unit, and
(4) superior active and passive control of residual amplitude modulation is
achieved, especially when the carrier power is vanishingly low. These distinct
merits stimulate new ideas on how we optimally enact PDH for a wide range of
applications.

††journal: optica††articletype: Research article

##  1 Introduction

Precision optical interferometry builds on transferring the stability of a
length reference to an optical frequency. Generally an optical oscillator is
stabilized to the reference using the Pound-Drever-Hall (PDH) method [1, 2, 3,
4]. In this scheme, modulation sidebands around a carrier act as phase
reference to quantify frequency deviations of the carrier relative to the
center of the reference’s spectral resonance. The versatility of this method
has allowed its powerful application in a wide range of optical
interferometers, from tabletop experiments seeking to produce low noise
oscillators [5, 6, 7, 8, 9] to km-scale gravitational wave interferometers
[10, 11, 12, 13, 14]. While alternative methods have been developed to allow
homodyne detection of the reference’s error signal [15, 16, 17], PDH detection
is by far the most commonly used and achieves the most robust performance as
the carrier and phase reference sidebands propagate along the same optical
path and share the same wavefront.

A key feature of the PDH technique is that the heterodyne beatnote between the
cavity leakage reference and the incident field is formed with two out-of-
phase modulation sidebands. In the presence of the resonance, this frequency
discrimination scheme converts a frequency modulated (FM) spectrum into an
amplitude modulated field, offering detection sensitivity limited only by the
photon shot noise. Away from the cavity resonance, PDH should ideally produce
a null baseline signal. However residual amplitude modulation (RAM), arising
from any optical components in the beam path, can produce a fluctuating
baseline. Reduction of the spurious RAM has therefore been a desirable goal in
FM spectroscopy, including the development of ultrastable optical
interferometers/oscillators [18, 19, 20, 21].

In a typical PDH realization, an electro-optic modulator (EOM) is driven with
a radio frequency (RF) signal $\Omega$ to produce a series of phase modulation
sidebands around the incident optical carrier,

$\displaystyle Ee^{i(\omega t+\beta\text{sin}\Omega t)}$
$\displaystyle=Ee^{i\omega
t}\Big{(}J_{0}(\beta)+\sum_{n=1}^{\infty}J_{n}(\beta)e^{in\Omega t}$ (1)
$\displaystyle+\sum_{n=1}^{\infty}(-1)^{n}J_{n}(\beta)e^{-in\Omega t}\Big{)}.$

An expansion of the phase modulation term expresses the incident field as a
sum of sidebands, each with an amplitude characterized by Bessel functions
$J_{n}(\beta)$ and modulation depth $\beta$. While the EOM has the convenient
feature of placing the first order sidebands at $\pm\Omega$, 180∘ out of phase
with each other, it has the undesirable feature that deep modulation generates
higher order sidebands at frequencies $\pm n\Omega$ with $n>1$ (Fig. 1 blue
inset). The presence of these higher order sidebands can reduce the frequency
discrimination slope, introduce additional RAM, and limit the dynamic range of
the carrier used to probe the desired resonance.

Figure 1: Optical and electronic experimental schema for AOM and EOM
modulation configurations. The black, blue, and red dashed lines denote the
electronic, fiber, and free-space optical connections, respectively.
$\phi_{+}$: voltage controlled phase shifter; $A_{+}$: voltage controlled
attenuator; $\Sigma$: RF power combiner; $V_{DC}$: Bias DC voltage applied on
EOM for active RAM cancellation; PD: photo detector; NPBS: non polarizing beam
splitter; PBS: polarizing beam splitter. Either the AOM or EOM configuration
is used for PDH-based laser frequency stabilization.

Here we report a new PDH approach that generates only the three useful tones
for frequency discrimination, which are translated into the optical domain
with an acousto-optic modulator (AOM). This allows for arbitrary control of
each tone strength without bringing other unrelated noise terms from high-
order sidebands. This feature also provides a higher signal to noise ratio
(SNR), limited by shot noise. Furthermore, EOMs typically generate significant
RAM in the context of state-of-the-art optical oscillators, owing to the
natural crystal birefringence. It is often desirable to use an active
cancellation scheme to prevent RAM from contaminating the amplitude modulation
generated by the cavity or atomic and molecular resonances [22, 23, 24, 25,
26]. Active cancellation schemes that leverage the birefringence of the
crystal address only the phase quadrature but not the amplitude imbalance
between sidebands. A number of recent research efforts have additionally
investigated methods to reduce the EOM’s intrinsic RAM [27, 28, 29, 30, 31,
32, 33, 34, 35]. Synthesizing the PDH triplet electronically and translating
it into the optical domain by an AOM leads to a lower level of RAM, and
provides a natural basis for feedback corrections in the RF domain. These
electrical feedback actions are direct and orthogonal, leading to an ideal
cavity interrogation beam, limited only by the shotnoise in the guide error
signal from RAM PD. We also note that all components of frequency synthesis,
demodulation, and control can be integrated into a compact electronic unit,
leading to significant reduction of both the cost and complexity of performing
FM spectroscopy.

General realizations of PDH locking rarely venture into the regime of
$\beta>1$, but recent work with state-of-the-art optical oscillators has
identified cases where a suppressed carrier and high modulation tones
(generated with $\beta>1$) can lead to lower frequency drift [36] in
dielectric coatings, or reduced light-induced effects in crystalline coatings
[37, 38]. The use of low optical carrier power has been necessary in achieving
the best long term stability of optical oscillators. This is desirable in
demonstrating the role of cryogenic crystalline cavities for all-optical
timescales [39] and searches for novel fundamental physics [40]. In the
traditional EOM-based FM approach, it is usually more challenging to reduce
and stabilize RAM when $\beta\gg 1$. To make further advances in long-term
stability, the independent and arbitrary amplitude control of the carrier and
modulation sidebands is therefore a highly desirable feature. We realize this
goal using the versatile control offered by our synthetic FM to demonstrate
that we can achieve active RAM control at the fractional $10^{-7}$ level
relative to the cavity linewidth even with a vanishingly small optical
carrier.

Figure 2: (a): The cavity transmission of EOM (top) and AOM (bottom)
configurations when the $P_{c}/P_{tot}=0.5\%$ in the case of modulation index
of EOM $\beta\approx 2.3$. Each inset plot shows the measured carrier
transmission as the modulated laser is swept in frequency. (b): Comparison of
the shot noise limited signal to noise ratio (SNR) between the EOM and AOM
modulation schemes. The total incident power $P_{tot}$ is fixed, and the
fraction of carrier power $\frac{Pc}{P_{tot}}$ is varied. A modulation index
of $\beta=1$ roughly corresponds to $\frac{Pc}{P_{tot}}\approx 0.5$. The
measured results: black dots; calculation: red line.

##  2 Methods

Figure 1 shows the entire experimental setup, including the PDH locking
schematic and active RAM servo for both AOM and EOM configurations. In the AOM
digital configuration, a carrier $f_{0}$ and equally spaced modulation
sidebands $f_{0}\pm\Omega$ (with $f_{0}=80$ MHz and $\Omega=5$ MHz throughout
this text) are generated via a Direct Digital Synthesizer (DDS). Crucially, we
adjust the output phase of one of the DDS channels such that one of the
modulation sidebands is 180${}^{\text{o}}$ out of phase with the carrier,
mimicking the electric field output of a phase modulated EOM. All three tones
are referenced to a common 1 GHz clock and a fourth channel on the DDS
synthesizes the RF local oscillator (LO) that is used in the demodulation
process. The three RF tones drive a fiber coupled AOM so that the first
diffracted orders (at different frequencies) are coupled together and
propagate in the same fiber. Upon outcoupling through a collimator, they then
all share an identical spatial path.

We split the light into two paths - one is incident on a photo detector (PD)
used for measurement and active stabilization of RAM, and the other is
incident on a detector for PDH locking placed in cavity reflection. Each
photodetector has resonant gain at our modulation frequency of 5 Mhz, further
suppressing pickup of any high RF signals. In this work the PDH PD is instead
used to characterize the out-of-loop (OoL) RAM of the entire system when the
laser frequency is tuned sufficiently far away from the cavity resonance and
the corresponding baseline fluctuations are measured as the OoL RAM. The
cavity used in the experiment is 6 cm long with finesse 497,000 corresponding
to a cavity linewidth of 5 kHz. We compare this scheme to traditional EOM-
based PDH by replacing the AOM with a LiNbO3 fiber-coupled waveguide EOM
(bottom of Fig. 1), which was used in our previous cavity-locking efforts. All
other optical elements remain the same.

One intrinsic advantage of synthesizing the modulation triplet via three
separate tones is the absence of higher order sidebands generated when phase
modulating an EOM (Fig. 2 (a)). Consequently, there is a higher signal to
noise offered in the former case since there are no additional shot noise
contributions from the higher modulation harmonics. We illustrate this effect
in Fig. 2(b) where a high modulation index $\beta\approx 2.3$ (corresponding
to a small ratio of carrier power $P_{c}$ over total power $P_{tot}$) is used
for the operation for cryogenic silicon cavities with very small long-term
frequency drift [36, 39, 38]. In the regime of $P_{c}/P_{tot}\ll$ 1, the shot
noise limited SNR for PDH detection via an EOM is only 75% of that with the
optimal modulation scheme of AOM.

Figure 3: The measurement of the free running RAM, (a) and (c): The power
spectral density of RAM-induced frequency noise for high (top,
$P_{c}/P_{tot}=0.6$) and low (bottom, $P_{c}/P_{tot}=0.005$) carrier power
ratio. The red and blue dashed lines denote the shot noise limited frequency
noise while the black dotted line denotes Si6 noise as measured in [38]. (b)
and (d): RAM-induced fractional frequency instability for the corresponding
plots above using the AOM and EOM modulation schemes. Free running RAM for the
AOM configuration exhibits greater than an order of magnitude improved
performance over EOM at longer integration times in both situations. The right
axis plots the data as fractional noise relative to the cavity linewidth.

Next we compare the passive stability of RAM between the EOM and AOM methods
(Fig. 3). In EOM-based PDH, a misalignment of the input field polarization to
the principal axes of the modulator leads to a polarization-dependent
modulation due to differing refractive indices of the two crystal axes.
Thermal variations in the crystal birefringence can then be a significant
source of RAM and are often the dominant contribution. The modulation RF field
additionally has a spatially inhomogeneous distribution inside the EOM
crystal, resulting in another source for RAM. Conversely, triplet synthesis
with an AOM is not subject to the same dispersive effects between the carrier
and sidebands. Both methods are sensitive to intra-crystal etalons, which can
be mitigated by an angle cut of the crystal faces, but notably the fiber-AOM
is found to be relatively insensitive to drifts in the input polarization of
the incident light field. The primary source of RAM in the AOM scheme is
generated during the triplet synthesis itself. Differential amplitude and
phase fluctuations of the synthesizer outputs will appear as noise in the I
and Q quadratures at the modulation frequency $\Omega$ but are suppressed
through the operation of the two RAM servo loops acting directly on the
amplitude and phase quadratures of the two sidebands. Replacing optical issues
by a stable electronic system in this manner is an advantageous exchange as
the free running RAM can be reduced by careful attention to the RF synthesis,
which exhibits a much more robust RAM suppression over a long time scale, as
depicted in Fig. 3.

State-of-the-art optical oscillators exhibit $<$1$\times 10^{-6}$ (sub-ppm)
fractional stabilization to the cavity linewidth, which requires active
stabilization of the RAM. In EOM-based PDH, active stabilization can be
achieved by temperature controlling the electro-optic crystal and biasing it
with a DC voltage that affects the light polarization [30, 31, 33]. Two issues
are therefore present with this modulation method: the modulation device
itself generates unacceptable RAM, and the actuators for RAM stabilization are
not aligned to the in-phase (I) and quadrature (Q) components of the electric
field.

Figure 4: (a) and (c): The power spectral density of RAM-induced frequency
noise for high (top, $P_{c}/P_{tot}=0.6$) and low (bottom,
$P_{c}/P_{tot}=0.005$) carrier power ratio. All measurements actively
stabilize the RAM to noise detected at the RAM PD. The measured signals are
out of loop RAM contributions detected by the PDH PD. The black dotted line
presents the current level of noise measured on the 6 cm silicon cavity (see
[38] for details). Dashed lines denote the shot noise limited frequency noise.
(b) and (d): RAM-induced fractional frequency instability for the
corresponding plots above using the AOM and EOM modulation schemes.

Even with the superior passive RAM stability demonstrated by the AOM triplet
scheme, we can easily improve the performance further by phase locking three
low-phase noise crystal oscillators to each other with an integer frequency
synthesis based on a common clock oscillator, frequency division and single
sideband modulation techniques. Here we present a unique RAM stabilization
scheme to demonstrate that this is not necessary. Specifically, we implement
dual quadrature I/Q demodulation of the signal detected on the RAM PD. Rather
than tune crystal parameters to affect the output field, we directly feedback
to the RF outputs of the DDS. An attenuator represented by a voltage-
controlled channel conductance of a junction field effect transistor on the
$f_{0}+\Omega$ channel output receives a control voltage addressing the error
signal of the mixer’s I output. Phase noise at $\Omega$ is actively removed by
dynamic tuning of the resonance of an inductor-capacitor (represented by two
varactor diodes in anti-series configuration) circuit placed at the output of
the $f_{0}+\Omega$ channel. Orthogonality between the mixer’s I and Q outputs
is maintained by means of a wire-and-core 3dB quadrature hybrid coupler. The
amplitude and phase noise feedback loops are both closed with bandwidths of
120 kHz. The in-loop performance of these stabilization loops is well below
the detection shot noise, demonstrating that high suppression of the RAM
signal is easily achievable with this orthogonalized and simple controller
scheme. A comparison of active RAM stabilization between the AOM and EOM
schemes is shown in Fig. 4 and we discuss the implications below.

##  3 Results

We benchmark performance of the AOM modulation scheme against EOM-based PDH by
using the optical setup illustrated in Fig. 1 to compare free running and out-
of-loop (OoL) RAM performances. This figure of merit will ultimately determine
the contribution of RAM to the cavity-stabilized laser frequency instability.
We compare the two methods first in the regime of conventional PDH, where the
modulation index is set to $\beta\approx 1$ and the PDH error signal is
maximized by setting the sideband power to nearly half of the carrier power.
We set the carrier power $P_{c}$ to be equivalent in both setups (in this
case, $P_{c}=3.3$ $\mu$W and $P_{tot}=5.5$ $\mu$W, or $\beta=1.1$) such that
the corresponding frequency discrimination slopes for both approaches are
nearly identical.

Notably, we observe a significant improvement in free running RAM control with
the AOM when compared to the EOM (Fig. 3). Simply adopting the presented
scheme without feedback enables control of the optical oscillator to the high
$10^{-17}$, a significant standard when compared to state-of-the-art results.
High-ppm level stabilization to the cavity linewidth is obtainable using
synthetic FM from the 0.1-100 s timescale relevant for many metrology
experiments. The lower AOM RAM performance with feedback disabled is even more
pronounced when greatly reducing the carrier power relative to the sideband
power ($P_{c}/P_{tot}=0.005$, Fig. 3(b)). Enabling RAM stabilization, the
resulting OoL RAM signals as measured on the PDH PD are compared in Fig. 4,
displayed in (a) power spectral density and (b) modified Allan deviation. The
fractional frequency noise at bandwidth above 0.1 Hz is photon shot noise
limited to the same value in both methods. Even in this conventional PDH
regime the AOM-based triplet synthesis demonstrates a lower instability at
long integration times, indicating a slightly improved performance at lower
Fourier frequencies. Electronic noise contributions to RAM stabilization
(detection noise, finite servo gain) are well below the measured OoL traces,
and the measured instability is attributed to noise on the optical field. The
Allan deviation data from the two schemes converge to a similar level of noise
at averaging times of $\tau>4000$ s as shown in Fig. 4(b), suggesting the
presence of path length fluctuations that are common to both setups between
the in-loop and out-of-loop detectors. Basically, for $\beta\approx 1$, the
two schemes perform at a similar level. The AOM triplet offers only a slight
advantage in performance, but a significant advantage in experimental
construction and implementation.

For cryogenic silicon cavities operated under 4 K and 16 K [36, 38, 39], long-
term cavity instability is observed to improve with significantly reduced
optical carrier power that is coupled to the inside of the cavity. Thus, we
now consider the regime of $P_{c}/P_{tot}\ll 1$. The carrier power is reduced
to $P_{c}=100$ nW, and we again keep the total optical power is identical for
both setups, $P_{tot}=20$ $\mu$W (Fig. 4(c,d)). Note that the signal-to-noise
ratio obtained from the AOM modulation is now 22$\%$ higher than that received
via EOM-based PDH, and the detected RF signal can be arbitrarily increased
with the independent control of sideband amplitude. At high Fourier
frequencies where the fractional frequency noise power spectral density is
consistent with the photon shot noise limit, both methods show similar
instability that is now 10 dB higher owing to the reduced carrier power.
However, performance at the low Fourier frequencies shows a dramatic advantage
offered by the AOM triplet scheme, where the OoL RAM limitation is still well
below the measured cavity noise. This extreme limit of low $P_{c}$ is exactly
the regime where one needs to operate cryogenic silicon cavities in order to
enjoy the unparalleled short and long term stability [41].

We again verify that the OoL RAM is not limited by electronic contributions.
Direct orthogonalized phase and amplitude control to the frequency triplet
synthesized with the AOM method offers order of magnitude improved performance
over the EOM scheme, where the high modulation index has generated large
amplitude modulation tones at frequencies $\pm 2\Omega$ and $\pm 3\Omega$. The
significant improvement of the AOM scheme indicates that there is a
fundamental difference between the ways that the two methods respond to any
out-of-loop noise process. A prominent example is the presence of a parasitic
etalon that could be located in the out-of-loop path probed by the PDH
detector. Such an effect will generate an amplitude and phase imbalance
between the modulation sidebands. Noise originating from the etalon
instability will be accentuated in the presence of higher order modulation
harmonics. Even with the weak carrier power, RAM is still suppressed to the
0.3 ppm level in the AOM configuration, and such performance demonstrates the
potential for long term laser frequency instablity well below $1\times
10^{-16}$, especially when higher cavity finesse is employed.

##  4 Conclusion

In this article, we present a new method of modulation, in which the PDH
frequency triplet is digitally synthesized and frequency-shifted into an
optical field with FM by means of an AOM, working with the emitted field of a
stable singlemode laser. This new configuration eliminates higher order
sidebands generated via phase modulation and consequently offers a higher shot
noise limited SNR. Direct and independent feedback to amplitude and phase of
the modulated electric field are performed in the RF domain rather than the
optical. The versatility and robustness of this scheme present a economic way
to control the carrier power independent of the sideband amplitude, allowing
one to couple a flexible amount of optimal optical power into the optical
cavity.

We note that this modulation scheme can be extended to experiments requiring a
dynamic sweep of the FM tone. Varying the modulation frequency in an EOM
results in an altered modulation depth but no such effect occurs in our
triplet synthesis scheme. Additionally, we want to point out that that
retroreflecting the free space signal before the cavity and allowing it to
repass the fiber AOM will heterodyne beatnote at a frequency of $2f_{0}$,
which can be used for path length stabilization. A similar approach enacted in
a PDH setup employing an EOM would require a separate AOM for this purpose and
we can effectively combine two stabilization loops with a single actuator.
Dedicated research efforts have sought to improve the innate RAM associated
with EOMs but here we demonstrate a significant improvement in the passive
stability of RAM when using the AOM triplet. With active RAM cancellation we
also surpass the stability obtained with an EOM, especially in the regime of
low carrier power. This is a direct path forward to achieving a lower
frequency drift in cryogenic crystalline cavities employing dielectric
coatings and reduction of light-induced noise in those with crystalline
coatings. The compact and robust nature of this approach also makes PDH and
other forms of FM spectroscopy more easily accessible in a wide range of
experiments.

Acknowledgments The authors thank T. Schibli and J. Sherman for careful
reading of the manuscript and C.-Y. Ma for insightful discussions.

Funding This work is supported by NIST, DARPA, AFRL, and the National Science
Foundation Physics Frontier Center (NSF PHY-1734006).

Disclosures The authors declare no conflicts of interest.

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

## References

  * [1] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Applied Physics B 31, 97–105 (1983).
  * [2] R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Review of Scientific Instruments 17, 490–505 (1946).
  * [3] D. Hils and J. L. Hall, “Ultra-stable cavity-stabilized lasers with subhertz linewidth,” in _Frequency Standards and Metrology,_ A. De Marchi, ed. (Springer Berlin Heidelberg, Berlin, Heidelberg, 1989), pp. 162–173.
  * [4] E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” American Journal of Physics 69, 79–87 (2001).
  * [5] C. Salomon, D. Hils, and J. L. Hall, “Laser stabilization at the millihertz level,” J. Opt. Soc. Am. B 5, 1576–1587 (1988).
  * [6] B. Julsgaard, A. Walther, S. Kröll, and L. Rippe, “Understanding laser stabilization using spectral hole burning,” Optics express 15, 11444–11465 (2007).
  * [7] T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, and J. Ye, “A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity,” Nature Photonics 6, 687–692 (2012).
  * [8] D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson, J. Ye, F. Riehle, and U. Sterr, “$1.5\text{ }\text{ }\mu\mathrm{m}$ lasers with sub-10 mhz linewidth,” Phys. Rev. Lett. 118, 263202 (2017).
  * [9] W. Zhang, J. M. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. L. Hall, T. Legero, D. G. Matei, F. Riehle, U. Sterr, and J. Ye, “Ultrastable silicon cavity in a continuously operating closed-cycle cryostat at 4 k,” Phys. Rev. Lett. 119, 243601 (2017).
  * [10] The LIGO Scientific Collaboration, “Exploring the sensitivity of next generation gravitational wave detectors,” Classical and Quantum Gravity 34, 044001 (2017).
  * [11] The LIGO Scientific Collaboration, “Advanced LIGO,” Classical and Quantum Gravity 32, 074001 (2015).
  * [12] The KAGRA Collaboration, “Large-scale cryogenic gravitational-wave telescope in japan: Kagra,” Journal of Physics: Conference Series 610, 012016 (2015).
  * [13] F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, F. Barone, J.-P. Baronick, M. Barsuglia, A. Basti, F. Basti, T. S. Bauer, V. Bavigadda, M. Bejger, M. G. Beker, C. Belczynski, D. Bersanetti, A. Bertolini, M. Bitossi, M. A. Bizouard, S. Bloemen, M. Blom, M. Boer, G. Bogaert, D. Bondi, F. Bondu, L. Bonelli, R. Bonnand, V. Boschi, L. Bosi, T. Bouedo, C. Bradaschia, M. Branchesi, T. Briant, A. Brillet, V. Brisson, T. Bulik, H. J. Bulten, D. Buskulic, C. Buy, G. Cagnoli, E. Calloni, C. Campeggi, B. Canuel, F. Carbognani, F. Cavalier, R. Cavalieri, G. Cella, E. Cesarini, E. Chassande-Mottin, A. Chincarini, A. Chiummo, S. Chua, F. Cleva, E. Coccia, P.-F. Cohadon, A. Colla, M. Colombini, A. Conte, J.-P. Coulon, E. Cuoco, A. Dalmaz, S. D’Antonio, V. Dattilo, M. Davier, R. Day, G. Debreczeni, J. Degallaix, S. Deléglise, W. D. Pozzo, H. Dereli, R. D. Rosa, L. D. Fiore, A. D. Lieto, A. D. Virgilio, M. Doets, V. Dolique, M. Drago, M. Ducrot, G. Endrőczi, V. Fafone, S. Farinon, I. Ferrante, F. Ferrini, F. Fidecaro, I. Fiori, R. Flaminio, J.-D. Fournier, S. Franco, S. Frasca, F. Frasconi, L. Gammaitoni, F. Garufi, M. Gaspard, A. Gatto, G. Gemme, B. Gendre, E. Genin, A. Gennai, S. Ghosh, L. Giacobone, A. Giazotto, R. Gouaty, M. Granata, G. Greco, P. Groot, G. M. Guidi, J. Harms, A. Heidmann, H. Heitmann, P. Hello, G. Hemming, E. Hennes, D. Hofman, P. Jaranowski, R. J. G. Jonker, M. Kasprzack, F. Kéfélian, I. Kowalska, M. Kraan, A. Królak, A. Kutynia, C. Lazzaro, M. Leonardi, N. Leroy, N. Letendre, T. G. F. Li, B. Lieunard, M. Lorenzini, V. Loriette, G. Losurdo, C. Magazzù, E. Majorana, I. Maksimovic, V. Malvezzi, N. Man, V. Mangano, M. Mantovani, F. Marchesoni, F. Marion, J. Marque, F. Martelli, L. Martellini, A. Masserot, D. Meacher, J. Meidam, F. Mezzani, C. Michel, L. Milano, Y. Minenkov, A. Moggi, M. Mohan, M. Montani, N. Morgado, B. Mours, F. Mul, M. F. Nagy, I. Nardecchia, L. Naticchioni, G. Nelemans, I. Neri, M. Neri, F. Nocera, E. Pacaud, C. Palomba, F. Paoletti, A. Paoli, A. Pasqualetti, R. Passaquieti, D. Passuello, M. Perciballi, S. Petit, M. Pichot, F. Piergiovanni, G. Pillant, A. Piluso, L. Pinard, R. Poggiani, M. Prijatelj, G. A. Prodi, M. Punturo, P. Puppo, D. S. Rabeling, I. Rácz, P. Rapagnani, M. Razzano, V. Re, T. Regimbau, F. Ricci, F. Robinet, A. Rocchi, L. Rolland, R. Romano, D. Rosińska, P. Ruggi, E. Saracco, B. Sassolas, F. Schimmel, D. Sentenac, V. Sequino, S. Shah, K. Siellez, N. Straniero, B. Swinkels, M. Tacca, M. Tonelli, F. Travasso, M. Turconi, G. Vajente, N. van Bakel, M. van Beuzekom, J. F. J. van den Brand, C. V. D. Broeck, M. V. van der Sluys, J. van Heijningen, M. Vasúth, G. Vedovato, J. Veitch, D. Verkindt, F. Vetrano, A. Viceré, J.-Y. Vinet, G. Visser, H. Vocca, R. Ward, M. Was, L.-W. Wei, M. Yvert, A. Z. żny, and J.-P. Zendri, “Advanced virgo: a second-generation interferometric gravitational wave detector,” Classical and Quantum Gravity 32, 024001 (2014).
  * [14] The Einstein Telescope Collaboration, “The einstein telescope: a third-generation gravitational wave observatory,” Classical and Quantum Gravity 27, 194002 (2010).
  * [15] T. Hansch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Optics communications 35, 441–444 (1980).
  * [16] D. Shaddock, M. Gray, and D. McClelland, “Frequency locking a laser to an optical cavity by use of spatial mode interference,” Optics letters 24, 1499–1501 (1999).
  * [17] F. Diorico, A. Zhutov, and O. Hosten, “Laser-cavity locking at the $10^{-7}$ instability scale utilizing beam elipticity,” arXiv:2203.04550 (2022).
  * [18] G. C. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (fm) spectroscopy: theory of lineshapes and signal-to-noise analysis,” Applied Physics B 32, 145–152 (1983).
  * [19] E. A. Whittaker, C. M. Shum, H. Grebel, and H. Lotem, “Reduction of residual amplitude modulation in frequency-modulation spectroscopy by using harmonic frequency modulation,” JOSA B 5, 1253–1256 (1988).
  * [20] S. Kasapi, S. Lathi, and Y. Yamamoto, “Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers,” J. Opt. Soc. Am. B 17, 275–279 (2000).
  * [21] K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Arai, and R. X. Adhikari, “Residual amplitude modulation in interferometric gravitational wave detectors,” JOSA A 31, 81–88 (2014).
  * [22] N. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” JOSA B 2, 1527–1533 (1985).
  * [23] J. Ye, L.-S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic and molecular physics: demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6–15 (1998).
  * [24] A. Foltynowicz, I. Silander, and O. Axner, “Reduction of background signals in fiber-based nice-ohms,” JOSA B 28, 2797–2805 (2011).
  * [25] E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that use electro-optic modulators,” Measurement Science and Technology 20, 025302 (2008).
  * [26] L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Review of Scientific Instruments 83 (2012).
  * [27] J. L. Hall, W. Zhang, and J. Ye, “Accurate removal of ram from fm laser beams,” in _2015 Joint Conference of the IEEE International Frequency Control Symposium and the European Frequency and Time Forum,_ (2015), pp. 713–716.
  * [28] K. L. Dooley, M. A. Arain, D. Feldbaum, V. V. Frolov, M. Heintze, D. Hoak, E. A. Khazanov, A. Lucianetti, R. M. Martin, G. Mueller _et al._ , “Thermal effects in the input optics of the enhanced laser interferometer gravitational-wave observatory interferometers,” Review of Scientific Instruments 83 (2012).
  * [29] J. Sathian and E. Jaatinen, “Reducing residual amplitude modulation in electro-optic phase modulators by erasing photorefractive scatter,” Optics express 21, 12309–12317 (2013).
  * [30] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, C. Hagemann, T. Legero, U. Sterr, F. Riehle, G. D. Cole, and M. Aspelmeyer, “Reduction of residual amplitude modulation to 1 $\times 10^{-6}$ for frequency modulation and laser stabilization,” Opt. Lett. 39, 1980–1983 (2014).
  * [31] H. Shen, L. Li, J. Bi, J. Wang, and L. Chen, “Systematic and quantitative analysis of residual amplitude modulation in pound-drever-hall frequency stabilization,” Phys. Rev. A 92, 063809 (2015).
  * [32] J. F. Diehl, C. E. Sunderman, J. M. Singley, V. J. Urick, and K. J. Williams, “Control of residual amplitude modulation in lithium niobate phase modulators,” Optics Express 25, 32985–32994 (2017).
  * [33] J. Gillot, S. F. Tetsing-Talla, S. Denis, G. Goavec-Merou, J. Millo, C. Lacroûte, and Y. Kersalé, “Digital control of residual amplitude modulation at the 10−7 level for ultra-stable lasers,” Opt. Express 30, 35179–35188 (2022).
  * [34] J. Bi, Y. Zhi, L. Li, and L. Chen, “Suppressing residual amplitude modulation to the 10−7 level in optical phase modulation,” Appl. Opt. 58, 690–694 (2019).
  * [35] Z. Li, W. Ma, W. Yang, Y. Wang, and Y. Zheng, “Reduction of zero baseline drift of the pound–drever–hall error signal with a wedged electro-optical crystal for squeezed state generation,” Optics Letters 41, 3331–3334 (2016).
  * [36] J. M. Robinson, E. Oelker, W. R. Milner, W. Zhang, T. Legero, D. G. Matei, F. Riehle, U. Sterr, and J. Ye, “Crystalline optical cavity at 4 k with thermal-noise-limited instability and ultralow drift,” Optica 6, 240–243 (2019).
  * [37] J. Yu, D. Kedar, S. Häfner, T. Legero, F. Riehle, S. Herbers, D. Nicolodi, C. Y. Ma, J. M. Robinson, E. Oelker, J. Ye, and U. Sterr, “Excess noise in highly reflective crystalline mirror coatings,” arXiv:2210.15671 (2022).
  * [38] D. Kedar, J. Yu, E. Oelker, A. Staron, W. R. Milner, J. M. Robinson, T. Legero, F. Riehle, U. Sterr, and J. Ye, “Frequency stability of cryogenic silicon cavities with semiconductor crystalline coatings,” Optica 10, 464–470 (2023).
  * [39] W. R. Milner, J. M. Robinson, C. J. Kennedy, T. Bothwell, D. Kedar, D. G. Matei, T. Legero, U. Sterr, F. Riehle, H. Leopardi, T. M. Fortier, J. A. Sherman, J. Levine, J. Yao, J. Ye, and E. Oelker, “Demonstration of a timescale based on a stable optical carrier,” Phys. Rev. Lett. 123, 173201 (2019).
  * [40] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye, “Precision metrology meets cosmology: Improved constraints on ultralight dark matter from atom-cavity frequency comparisons,” Phys. Rev. Lett. 125, 201302 (2020).
  * [41] E. Oelker, R. Hutson, C. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. Robinson, G. Marti, D. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, “Demonstration of $4.8\times 10^{-17}$ stability at 1 s for two independent optical clocks,” Nature Photonics 13, 714–719 (2019).