# Optical and microwave metrology at the 10-18 level with an Er/Yb:glass
frequency comb

N. V. Nardelli<EMAIL_ADDRESS>and<EMAIL_ADDRESS>National
Institute of Standards & Technology, 325 Broadway, Boulder, CO 80305, USA
University of Colorado, Boulder, Colorado 80309, USA  H. Leopardi  National
Institute of Standards & Technology, 325 Broadway, Boulder, CO 80305, USA
University of Colorado, Boulder, Colorado 80309, USA  T. R. Schibli
University of Colorado, Boulder, Colorado 80309, USA  T. M. Fortier
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>National Institute of
Standards & Technology, 325 Broadway, Boulder, CO 80305, USA

###### Abstract

Optical frequency combs are an essential tool for precision metrology
experiments ranging in application from remote spectroscopic sensing of trace
gases to the characterization and comparison of optical atomic clocks for
precision time-keeping and searches for physics beyond the standard model.
Here we describe the architecture and fully characterize a telecom-band, self-
modelocking frequency comb based on a free-space laser with an Er/Yb co-doped
glass gain medium. The laser provides a robust and cost-effective alternative
to Er:fiber laser based frequency combs, while offering stability and noise
performance similar to Ti:sapphire laser systems. Finally, we demonstrate the
Er/Yb:glass frequency comb’s utility in high-stability frequency synthesis
using two ultra-stable optical references at 1157 nm and 1070 nm and in low-
noise photonic microwave generation by dividing these references to the
microwave domain.

##  I Introduction

Just a few years after the first demonstration of the laser in 1960 Maiman
(1960), 1963 brought the very first modelocked laser Hargrove, Fork, and
Pollack (1964), which used an intra-cavity acousto-optic modulator to
dynamically affect the cavity loss. The result was a highly regular train of
pulses, which was the seed for numerous technologies, such as laser
communication enabling trans-oceanic fiber optics and the modern internet.
Pulse formation via shared phase coherence of laser spectral modes, or
modelocking, yielded shorter pulses and higher pulse energies, permitting
researchers new tools for both applied and fundamental research. High peak
powers benefited applications such as soft-tissue ablation used in laser eye
surgery Stern  _et al._ (1989), inertial confinement nuclear fusion Andiel
_et al._ (2002), which offers an alternative to magnetic confinement fusion,
and two-photon absorption spectroscopy Denk, Strickler, and Webb (1990), which
relies on highly efficient nonlinear interactions facilitated by the high
optical intensity seen in pulses, just to name a few. Ultra-short pulses, with
durations of femtoseconds or attoseconds, made possible ultra-precise
measurements of time and distance, and are increasingly utilized in time-
resolved measurements such as pump-probe experiments Lytle, Parrish, and
Barnes (1985) that yield information about previously inaccessible ultra-fast
processes. In fact, the modelocked laser is responsible for the shortest
human-made event at 43 attoseconds Gaumnitz  _et al._ (2017).

Optical frequency combs (OFCs) based on modelocked lasers take advantage of
the extreme precision with which the optical field can be controlled, and the
nonlinear wavelength conversion made possible with highly energetic optical
pulses, to enable high precision synthesis and control of the modelocked laser
spectrum. Due to strong nonlinear optical interactions within a laser cavity
Spence, Kean, and Sibbett (1991); Hofer  _et al._ (1992); Keller  _et al._
(1996), frequency modes are forced to oscillate with a common phase and at
rigid frequency intervals dictated by the length of the cavity. This phase-
coherence creates a fixed frequency and phase relationship between all
resonant optical frequency modes that extends to microwave signals generated
via photodetection of the laser pulse train. This concept of optical-to-
microwave synthesis makes possible the derivation of microwave timing signals
from optical atomic clocks and the characterization of optical clocks against
the current definition of the SI second, based on a microwave transition in
the 133Cs atom Essen and Parry (1955). Additionally, ultra-low phase noise
microwave signals have been demonstrated that take advantage of the coherent
division of OFCs to divide the frequency and noise of narrow-linewidth cavity-
stabilized optical references used to probe optical clock transitions Zhang
_et al._ (2010); Haboucha  _et al._ (2011); Fortier  _et al._ (2013); Xie  _et
al._ (2017); Kalubovilage, Endo, and Schibli (2020); Nakamura  _et al._
(2020). These room-temperature systems surpass the best cryogenic microwave
systems and may enhance some applications such as low-noise Doppler radar by
improving sensitivity to low-velocity and low-cross section objects Ye, Hall,
and Diddams (2000); Fortier  _et al._ (2011); Xie  _et al._ (2017).

Because the optical spectrum of OFCs can span up to and beyond an optical
octave of bandwidth, the optical spectrum of the modelocked laser is a
coherent source of 100,000s to millions of equally spaced optical modes. The
latter characteristic can enable the comparison of optical atomic clocks whose
transition frequencies span 100s of terahertz, and with resolutions beyond the
current limit imposed by the definition of the Hertz.

The optical-to-optical synthesis of high-accuracy and high-stability optical
frequencies derived from atomic clocks via OFCs has helped to support a number
of scientific and technological applications, including searches for Bosonic
dark matter that are conducted by comparing optical clocks of different
species Rosenband  _et al._ (2008); Safronova  _et al._ (2018). Additionally,
optically derived microwave signals could be used for the synchronization of
radio telescopes in very long-baseline interferometry (VLBI) Clivati  _et al._
(2017), which is used to image cosmic radio sources. A high level of
synchronization also benefits next-generation communication and navigation
networks, such as GPS, enabling faster acquisition of location data with
higher accuracy.

Figure 1: Er/Yb:glass frequency comb experimental setup. a) The modelocked
laser cavity architecture and geometry. b) External optical pulse
amplification and supercontinuum generation using a PM-EDFA and PM-HNLF. c)
Optical interferometers for referencing of the modelocked optical spectrum to
optical references at 1157 nm and 1070 nm. A single optical reference is used
for stabilization of the OFC repetition rate. The offset frequency $f_{0}$ is
detected using a nonlinear $f-2f$ interferometer. The detected optical beats
are counted using zero-deadtime SDR counters. d) Low-noise optical-to-
microwave generation. An interleaver to multiply $f_{\text{rep}}$ from 500 MHz
to 2 GHz and an MUTC photodiode to convert the optical pulse train to the
microwave domain for detection of the pulse repetition rate.

Optical frequency combs based on the Ti:Sapphire laser were used to realize
the first OFCs Ye, Hall, and Diddams (2000) and have been used in some of the
highest performance metrology experiments with atomic clocks Fortier and
Baumann (2019); Collaboration (2021). However, the laser is prone to loss of
modelocking as a result of high sensitivity to the ambient temperature drifts
and cavity misalignment. It also requires a high-power, expensive visible pump
laser (usually 532 nm) to excite emission in the gain bandwidth of the laser
crystal. To address the latter drawback in robustness, the Er:fiber OFC
Tauser, Leitenstorfer, and Zinth (2003) was developed as a more accessible
system operating in the telecom band. OFCs based on Er:fiber lasers require
almost no alignment after constituent fibers are spliced together, and employ
off-the-shelf 980 nm pump diodes that have benefited from decades of
development by the telecommunication industry. High loss in Er:fiber lasers,
coupled with pump lasers with relatively high amplitude noise, results in
resonant optical linewidths of 100 kHz to MHz. Also, due to the lengths of
fiber that are required for intra-cavity gain and modelocking, fiber lasers
generally do not support repetition frequencies higher than a few hundred MHz.
For optical clock comparisons, this decreases the signal-to-noise ratio (SNR)
of the optical beat between CW laser and OFC and makes phase-locking and
detection more difficult. For low-noise microwave generation, lower repetition
rates are associated with higher nonlinearity in photodetection.

In this work, we describe a more recently developed OFC based on a 500 MHz
free-space laser with an Er/Yb co-doped phosphate glass gain medium Nardelli
_et al._ (2022) that is designed to address the issues mentioned above. Due to
the choice of gain medium, with an emission bandwidth near 1550 nm, the laser
benefits from the same 980 nm pump diodes used in the Er:fiber laser.
Modelocking is achieved via a semiconductor saturable absorber mirror (SESAM)
Keller, Knox, and Roskos (1990); Keller  _et al._ (1996), which permits robust
self-starting and pulsed operation without precise cavity alignment. The laser
also exhibits low intrinsic noise due, in part, to the low intra-cavity loss
and the co-doping of Er and Yb that suppresses high-frequency pump intensity
noise. We demonstrate the utility of this OFC in high-precision optical and
microwave synthesis, which benefits next-generation optical atomic clock
comparisons.

##  II Er/Yb:glass OFC description and characterization

In this section, we describe and characterize an Er/Yb:glass frequency comb,
including the details of the modelocked laser oscillator, the amplification
and supercontinuum generation, and the stabilization of the carrier-envelope
offset frequency and the repetition frequency to an optical reference.
Modelocked lasers based on Er/Yb:glass have been previously demonstrated with
a repetition frequency of 1 GHz or less Lesko  _et al._ (2020); Spühler  _et
al._ (2005), however, this laser has also been demonstrated with 10 GHz Resan
_et al._ (2016) and 100 GHz Oehler  _et al._ (2008) repetition frequencies.
Additionally, the SESAM and Er/Yb:glass platform was demonstrated as a 1 GHz
monolithic laser in bulk CaF2 rather than free-space Shoji  _et al._ (2016).

###  II.1 Laser oscillator

####  II.1.1 Er/Yb-doped phosphate glass

An Erbium-doped gain medium constitutes a 3-level laser structure whereby 980
nm radiation drives electrons from the ground state (4I15/2) to the excited
state (4I11/2). Energy is dissipated non-radiatively via a long-lived laser
state (4I13/2) from where electrons are stimulated back to the ground state
via 1.55 $\mu m$ photons. Direct pumping of Er3+ exhibits low efficiency
because of its weak absorption bands. Co-doping with Yb3+ ions facilitates
electron transfer and increased pumping efficiency of the gain medium. This is
because Yb exhibits a very intense absorption band near 980 nm Gapontsev  _et
al._ (1982) and the Yb3+ excited state (4F5/2) is resonant with the 4I11/2
excited state in Er3+.

Despite the low thermal damage threshold of intense optical pumping, phosphate
glass serves as a better gain material for 1.55 $\mu m$ lasers compared to
other glasses and crystals. This is due to the presence of higher phonon
energies that shorten the 4I11/2Er3+ lifetime to 2-3 $\mu s$, thus reducing
the backwards transfer of energy from Er3+ to Yb3+ and increasing pumping
efficiency Gapontsev  _et al._ (1982).

####  II.1.2 Cavity geometry

The Er/Yb:glass optical frequency comb employs a linear free-space laser
cavity with a free spectral range of $f_{\text{FSR}}=f_{\text{rep}}\approx
500$ MHz (600 cm total optical path length), pictured in Figure 1 a). An
advantage of the linear cavity design is that it permits tuning of the
repetition frequency, via translation of the laser output coupler, by $>\pm$ 4
% (from 480 MHz to 520 MHz) with minimal cavity misalignment. The laser glass
medium is phosphate glass co-doped with Er3+ and Yb3+ (QX/Er), approximately 2
mm in length, with doping concentrations 0.8 % wt. Er3+ and 21 % wt. Yb3+, and
an upper state lifetime of 7.9 ms. The relatively long upper state lifetime
permits continuous pumping due to the large energy capacity of the gain
medium, with negligible laser dynamics as no single pulse has enough energy to
deplete the stored energy. We achieve a modelocked output power around 65 mW
using a 0.65 % output coupler centered at 1550 nm and a single 980 nm laser
diode providing 450 mW of pump power. This supports transform limited optical
pulses of 180 femtoseconds with pulse optical bandwidths of approximately 14
nm. Unfortunately, due to the low heat-handling capacity of phosphate glass,
pumping the laser at powers higher than 600 mW introduces the risk of thermal
fractures and cavity misalignment induced via thermal lensing and stress-
induced birefringence.

We choose the cavity geometry to be asymmetric so that it yields two focuses
(beam waist $\approx 35$ $\mu m$): one inside of the glass gain medium to
support sufficient population inversion and nonlinearity, and a second at the
surface of a SESAM, for passive modelocking. Figure 2 depicts how the beam
waist varies over the cavity length, for both tangential and saggital
dimensions, generated with an ABCD matrix calculation Kogelnik and Li (1966).
As seen in Figure 2, the 1550 nm output is a collimated beam with 1/e2
diameter around 0.6 mm. As a consequence of the cavity asymmetry, the
Er/Yb:glass is off-center from the two curved mirrors. The single-mode 980 nm
light is focused into the laser cavity via curved mirror M2 using a 7.5 cm
lens. Each curved mirror has a 1550 nm reflectivity $>99.9\%$, a 980 nm
transmission $>98\%$ and a radius of curvature, $r=50$ mm. A 6 degree angle
between the curved mirrors and the optical axis helps counteract astigmatism
introduced by the Er/Yb:glass, which is held at Brewster’s angle ($\approx 57$
degrees with respect to the optical axis) to minimize light loss due to
Fresnel reflections off the glass surfaces. Astigmatism expands the mode size
in the tangential plane (parallel to the page) while preserving the mode size
in the sagittal plane (perpendicular to the page) Hanna (1969). The latter is
important for preserving cavity stability along both beam axes. While not
employed here, coma aberration may also be compensated by arranging the cavity
so that it has a Z geometry instead of an X geometry Dunn and Ferguson (1977).

Figure 2: Beam waist vs position in the cavity. The sagittal beam axis is
shown in blue and tangential beam axis in orange. Laser mirror, Er/Yb:glass
gain medium and SESAM positions are indicated by dashed black lines.

####  II.1.3 Modelocking characterization

For self-starting modelocking we employ a SESAM designed at the University of
Colorado Boulder and grown at NIST that is composed of a single Er-doped, low-
temperature-grown InGaAs quantum well deposited on top of a Bragg stack with
alternating GaAs and AlAs layers, similar to the device reported by Lee et al.
Lee  _et al._ (2015). Defects are intentionally introduced by growing the
InGaAs at low temperature (475 C), allowing for ultrafast electron-hole
recombination, which can facilitate the formation of intracavity femtosecond
pulses M. Haiml and Weber (1999); Mangold  _et al._ (2013). Slower dynamics
between the conduction and valance bands (several picoseconds) allow intensity
perturbations to grow for optical pulses as they make multiple passes through
the laser cavity, leading to self-starting pulsed operation.

The SESAM has greater than 99 % reflectivity for fluences greater than 100
$\mu$J/cm2, which corresponds to about 4 W of optical power circulating in the
laser cavity. Typically, we operate the laser at about 10 W of circulating
power and a fluence of about 300 $\mu$J/cm2 on the SESAM. Nonsaturable loss
accounts for the lost power, which is $<$ 1 %. We found that SESAMs with
higher nonsaturable loss ($>$ 3 %) did not support stable pulse formation with
this cavity geometry due to the limited gain possible with the 2 mm
Er/Yb:glass.

We achieved stable pulse formation using 0.65 % output coupling with as little
as 250 mW of 980 nm pump power and an optical efficiency near 15 %, decreasing
at higher pump power (see Figure 3). The efficiency can be increased to 35 %
using an output coupler with 1.5 % transmission, but at the sacrifice of
modelocked optical bandwidth. This latter effect is due to the increased Er3+
population inversion due to higher output coupling losses, which leads to a
smaller gain bandwidth. Er/Yb:glass gain bandwidth is maximized for population
inversions around 70 % to 80 % Spühler  _et al._ (2005). From Figure 3 we also
observe that the laser can maintain stable modelocked operation for pump
powers ranging from 250 mW to 550 mW, necessary for long-range control and
stabilization of the carrier-envelope offset frequency (see Section on "OFC
Phase Stabilization").

Figure 3: Modelocked laser output power (thin trace, left axis) and pulse
bandwidth (thick trace, right axis) versus 980 nm pump power.

###  II.2 Amplification and supercontinuum generation

The 500 MHz pulse train generated by the modelocked laser is coupled into
polarization-maintaining (PM) fiber and amplified in a home-built PM Erbium-
doped fiber amplifier (EDFA). The higher repetition rate of the Er/Yb:glass
laser requires amplification to 400 mW to achieve the pulse energies required
($\approx$ 0.8 nJ) for supercontinuum generation in highly nonlinear fiber
(HNLF). As a result, amplification requires three 980 nm pump diodes with a
total power of about 2.5 W to pump a 110 cm length of Erbium-doped fiber,
shown in Figure 1 b).

The amplified pulses then traverse $\approx$1 m of hybrid HNLF Korel  _et al._
(2014), which spreads the $\approx$14 nm input pulse across an optical octave
of bandwidth from 1000 nm to 2200 nm. The HNLF consists of an 11 cm length of
HNLF [4.7 ps/(km$\cdot$nm) dispersion] followed by a 50 cm length of HNLF [2
ps/(km$\cdot$nm) dispersion]. The high-dispersion short length of fiber serves
to generate a broad dispersive wave that contains frequencies from about 1000
nm to 1200 nm, which covers the wavelengths of NIST’s two optical atomic clock
lasers at 1070 nm and 1157 nm Schioppo  _et al._ (2016); Young  _et al._
(1999) and a region around 1035 nm. The low-dispersion longer length of fiber
serves to spread optical power to longer wavelengths to access the region
around 2070 nm. Light from the HNLF at 1035 nm and 2070 nm is used for self-
referenced detection of the carrier-envelope offset frequency Telle  _et al._
(1999); Reichert, amd T. Udem, and Hänsch (1999); Jones  _et al._ (2000).

Figure 4 a) depicts the optical spectrum, with dotted lines indicating both
the clock frequencies and the f-2f frequencies. A line at 1542 nm is included,
which indicates the frequency of the 87Sr lattice clock’s cryogenic Silcon
cavity stabilized laser at JILA Matei  _et al._ (2017). This supercontinuum
spectrum can be shaped and tuned to a small degree by controlling the power
and dispersion of the pulses launched into the HNLF. Higher pulse energies
push optical power out to the $f$ and $2f$ frequencies at the expense of power
at the 1157 nm clock laser frequency. We are able to further shape the
spectrum by adding short lengths of fiber between the EDFA and the HNLF to
pre-chirp and temporally compress pulses as they traverse the HNLF. The latter
enables a longer interaction length and more efficient spectral broadening.

The optical heterodyne beats between comb and optical clock light and the
carrier-envelope offset frequency are shown in Figure 4 b) and c),
respectively. Typically, an SNR of greater than 30 dB in a 300 kHz bandwidth
ensures that a beat signal is adequately counted and stabilized with minimal
cycle/phase slips. Our single-branch optical amplifier and HNLF architecture
is sufficient to meet the latter criteria. However, one can potentially
improve SNR by shaping the optical spectrum using a nonlinear pulse
propagation solver to optimize the HNLF length, or by directly engineering the
HNLF dispersion with patterned Bragg gratings Westbrook  _et al._ (2004), or
engineered nonlinear waveguides to enhance specific regions of the spectrum
Carlson  _et al._ (2017).

Figure 4: a) Optical pulse spectrum as measured at different points in the OFC
experimental setup: at the output of the modelocked laser (blue), and after
amplification with the PM EDFA (green), after broadening with HNLF (black).
Dashed lines indicate the frequencies of the optical references at 1157 nm
(orange), 1070 nm (red) and 1542 nm (purple) and the $f$ and $2f$ frequencies
used for carrier-envelope offset detection (black). In the bottom, RF spectra
of the heterodyne beat signals between the comb and optical references, b),
and of the $f-2f$ interferometer, c). Spectra are measured with a resolution
bandwidth of 300 kHz.

Additionally, amplitude noise is important to consider given the potential
impact of amplitude-to-phase conversion in nonlinear optical and electronic
components. For instance, when using a frequency comb to synthesize low-noise
microwave signals, nonlinearity in the photodetector converts amplitude noise
on the optical pulse train to phase noise on the detected microwave carriers
Taylor  _et al._ (2011); Fortier  _et al._ (2013).

Figure 5 a) depicts the suppression of relative intensity noise (RIN) at
different points in the optical frequency comb, from the 980 nm pump, to the
modelocked laser and to the output of the HNLF. As seen in Figure 5 a), RIN on
the 1550 nm pulse train output from the laser cavity appears to be suppressed
compared to the RIN of the 980 nm pump laser. The laser cavity RIN is further
suppressed by operating the EDFA in saturation, which clamps the output power.
From Figure 5 b) it is clear that RIN is not equal among the different
frequency bands of interest, with the 1157 nm band displaying much more
intensity noise than the other bands. This is likely due to the fast roll-off
in wavelength-dependent intensity around 1157 nm as seen in Figure 4 a). Also
seen in these plots is a large feature around 10 MHz, which is due to the
transimpedance amplifier of the photodetector circuit.

The Er/Yb co-doping renders this laser relatively insensitive to pump laser
intensity noise since the Yb ions primarily absorb the pump photons and the Er
ions primarily emit photons Taccheo  _et al._ (1998). Three-level lasers, such
as the Er:fiber laser, are directly pumped and so are more sensitive to pump
intensity noise. The roll-off around 1 kHz is a result of the Yb ion upper-
state lifetime and energy transfer rate to Er ions, which acts as a low-pass
filter for intensity noise Lee and Schibli (2014). The higher-frequency RIN
originates instead from cavity losses due to acoustic and mechanical
environmental noise.

Figure 5: a) Relative intensity noise as measured at different points in the
OFC experimental setup: the 980 nm pump laser (blue), the modelocked laser
output (orange), the pulse train at the output of the HNLF, filtered around
1550 nm (green). b) Relative intensity noise measured at the output of the
HNLF, filtered around 1157 nm, 1550 nm and 1070 nm. The measurement noise
floor in a) and b) is shown in black.

###  II.3 OFC Phase Stabilization

Pulse formation in the modelocked laser forces all resonant longitudinal
cavity modes to share the same phase. This shared phase coherence allows the
optical modes to be connected to one another via a simple and fixed frequency
relationship, known as the comb equation. As a result, the $N^{\text{th}}$
mode of the comb is defined by the mode number and the pulse repetition
frequency and the laser offset frequency, such that

$\nu_{N}=Nf_{\text{rep}}+f_{0}.$ (1)

Note that we use the symbols $f$ and $\nu$ to denote RF and optical
frequencies, respectively. The latter equation has a multitude of important
consequences regarding OFC operation and measurement capabilities Fortier and
Baumann (2019). The definition of OFC optical modes with RF frequencies yields
a direct optical-to-microwave frequency link, which permits SI traceability of
all OFC optical modes, multiplication of microwave frequencies to the optical
domain, and division of optical frequencies to the microwave domain.
Additionally, shared phase coherence across the OFC optical modes also permits
the comparison of optical references separated by more than 100 THz in
frequency. The latter is possible since the regularly spaced optical modes of
the comb can be used to measure the relative frequency differences, as well as
the relative phase deviations between multiple optical references.
Stabilization of the modelocked laser is achieved by detecting and then phase-
locking $f_{0}$ and $\nu_{N}$ to high stability frequency references.

Repetition rate detection and stabilization The $M^{\text{th}}$ mode of the
laser repetition rate is stabilized to an optical reference by detecting and
controlling the difference frequency (i.e., beat frequency),
$f_{b,1157}=\nu_{1157}-(Mf_{\text{rep}}+f_{0})<500$ MHz, between the nearest
neighbor comb mode, $M$, and a cavity-stabilized laser at 1157 nm,
$\nu_{1157}$. The 1157 nm laser serves as the local oscillator to the 171Yb
optical lattice clock via doubling to the clock transition frequency near 578
nm.

As shown in Figure 1 c), light output from the HNLF is split across various
wavelength ranges and sent to optical interferometers. To minimize noise due
to uncommon fiber-optic paths, light from the HNLF is coupled to free space.
To a lower extent than fiber paths, uncompensated free-space paths contribute
added instability to the detected heterodyne beat signals between the comb and
optical references as a result of refractive index changes due to air
currents, as well as from length changes due to expansion of the optical
breadboard. To mitigate the effects of the additive instability, comb light
that is transmitted through the dichroic mirror is interfered on a single beam
splitter with the combined light from two ultra-stable optical references at
1157 nm and 1070 nm. In this configuration, the total uncompensated free-space
path length is 20 cm.

Optical filters are used in the detection of all optical beat signals to
reduce the light noise in photodetection from non-contributing optical comb
modes. The 1157 nm beat signal is filtered and amplified before being mixed
with a 100 MHz synthesized frequency derived from a H-Maser. The filtered
error signal is fed back to a piezo-electric actuated mirror to compensate
frequency deviations and phase noise due to instabilities in laser cavity
length. To maximize the actuator bandwidth, a miniature piezo stack is affixed
to a lead-filled copper mount, whose asymmetric geometry is chosen to damp
acoustic resonances Briles  _et al._ (2010). Due to the light-weight SESAM
($\approx$ 3 mm $\times$ 3 mm $\times$ 1 mm), and sturdy PZT mounting
structure, we observe a closed loop $f_{b,1157}$ servo bandwidth of >150 kHz.
Referencing of the optical beat signal to a H-maser reference and filtering of
the subsequent error signals is achieved using a PID algorithm in a field
programmable gate array (FPGA) Pomponio, Hati, and Nelson (2020). Because the
algorithm is implemented digitally, we are able to control and optimize the
proportional (P), integral (I) and differential (D) gains and corner
frequencies via a computer interface. The design of high bandwidth feedback
actuators helps to suppress high-frequency comb noise and support higher
performance frequency synthesis.

Offset frequency detection and stabilization We detect the offset frequency
using a nonlinear self-referencing technique that compares the frequency-
doubled low-frequency portion of the octave-spanning spectrum generated from
the HNLF to the fundamental high-frequency portion of the spectrum. The latter
technique is necessary since $f_{0}$ is only manifest on the optical carrier
phase, which is inaccessible via direct photodetection.

Using the self-referencing technique, the offset frequency is detected as
$f_{0}=2\nu_{N}-\nu_{2N}=2(Nf_{\text{rep}}+f_{0})-(2Nf_{\text{rep}}+f_{0})$.
The optical spectrum around 1035 nm and 2070 nm is split off with a dichroic
mirror and focused into a periodically poled lithium niobate (PPLN) waveguide
where light at 2070 nm is frequency-doubled to 1035 nm. The interference
between the fundamental 1035 nm light and doubled 2070 nm light is detected as
$f_{0}$. Control and stabilization of the detected $f_{0}$ beat signal is
achieved via comparison to a H-maser synthesized RF frequency. The resulting
error signal is sent through a second FPGA PID loop filter and fed back to the
Er/Yb:glass laser 980 nm pump current. Changes in the 980 nm pump power modify
the gain of the Er/Yb:glass laser, which permits tuning of $f_{0}$, whereby
$\approx$ 10 MHz of tuning is achieved by changing the 980 nm pump power by 50
mW.

###  II.4 OFC stability and noise characterization

Once the modelocked laser offset frequency and repetition rate are stabilized,
the resultant frequency comb represents a versatile tool for high precision
optical and microwave frequency metrology. Assessing its stability and
accuracy at different timescales is necessary to ensure it can support the
most stringent of measurements. Also, because the relevant timescales differ
by application, below we evaluate the stabilized Er/Yb:glass laser performance
for timescales < 1 s by measuring the single-sideband phase noise (SSPN), and
for time scales > 1 s by assessing the frequency stability of the time record
of frequency counted data. The latter metrics can be used to evaluate both the
"in-loop" and "out-of-loop" performance. The in-loop noise indicates how well
the laser phase-locked loop is able to suppress the noise of the modelocked
laser when it is stabilized to a low-noise reference. The out-of-loop
measurement assesses the performance of the full system, including the OFC
synthesis, uncompensated noise by the feedback loop, and the performance of
the frequency reference.

####  II.4.1 OFC performance at short timescales: phase noise

Figure 6 a) and b) show the RF spectrum (1 kHz bandwidth) of the in-loop
stabilized $f_{0}$ and $f_{b,1157}$ beat signals. The resolution-limited
coherent carriers at 102 MHz and 179 MHz indicate that the $f_{b,1157}$ and
$f_{0}$ are phase locked with an additive noise linewidth less than 1 kHz.

Higher resolution frequency and phase noise measurements can be achieved by
demodulating the microwave carriers. Figure 6 c) shows the single-sideband
phase noise spectrum as measured using an RF network analyzer. The SSPN
represents the power spectral density of the noise sidebands to one side of
the RF spectrum. As mentioned previously, the phase noise power spectral
density of the in-loop signal measures the additive noise of the Er/Yb:glass
laser when it is stabilized to a low-noise frequency reference. As seen in
Figure 6 b), we observe a servo bandwidth for the $f_{0}$ feedback loop near
60 kHz. The bandwidth of the $f_{0}$ feedback loop was increased beyond the 10
kHz upper state lifetime limit of the Er/Yb:glass gain medium by adjusting the
differential gain and corner frequency of the FPGA-based PID filter. We also
observe (Figure 6 a)) a 160 kHz $f_{\text{rep}}$ feedback bandwidth of the
piezo-actuated cavity mirror on the $f_{b,1157}$ SSPN. While control and
stabilization of $f_{\text{rep}}$ via the cavity length results in minimal
frequency deviations on $f_{0}$, we observe that stabilization of $f_{0}$, via
modulation of the 980 nm pump power pump, writes noise onto the laser
repetition as observed by the shared 60 kHz sidebands on the $f_{0}$ and
$f_{b,1157}$ beat signals and SSPN. Additionally, we observe a large, broad
noise feature on the $f_{b,1157}$ SSPN between 100 Hz and 10 kHz, which is a
result of unsuppressed cavity length noise due to acoustic and mechanical
vibration of the modelocked laser.

Using the SSPN we calculate the additive integrated phase noise, pulse-to-
pulse timing jitter and noise effective linewidth. We estimate that the root-
mean-squared (RMS) integrated phase noise from 1 Hz to 1 MHz is 80 mrads and
188 mrads for $f_{0}$ and $f_{b,1157}$, respectively. This accounts for
respective RMS timing jitters of 44 attoseconds (on 290 THz) and 115
attoseconds (on 259 THz). From the integrated phase noise, we can also
determine that the phase-locked linewidths are less than 1 Hz, limited by
resolution of the phase noise analyzer. The linewidth of $f_{b,1157}$ is
ultimately limited by the high-stability 1157 nm reference laser (around 10
mHz) and the linewidth of $f_{0}$ is ultimately limited by the H-maser
reference (around 1 $\mu$Hz).

Figure 6: RF spectra of the a) stabilized optical beat between the comb and
the optical reference (left), $f_{b,1157}$ and b) carrier-envelope offset
frequency, $f_{0}$. The RF spectra are measured with a resolution bandwidth of
1 kHz. c) Single-sideband phase noise spectra of the in-loop $f_{b,1157}$ beat
signal (orange) and in-loop $f_{0}$ beat signal (blue). The latter yielded a
RMS integrated phase noise (from 1 Hz to 1 MHz) of the 188 mrads and 80 mrads,
respectively.

####  II.4.2 OFC performance at longer timescales: frequency counting

Optical frequency comparisons of like and unlike species of atomic clock are
necessary in the characterization and development of optical atomic clock
technology. While like clocks can be compared directly to one another, inter-
species clock comparisons require an OFC to bridge the vast differences that
separate transition frequencies. Aside from the application to the development
of time and frequency references, the relative comparison of clock transition
frequencies permits clocks to be high-sensitivity sensors of space-time
variation for relativistic geodesy McGrew  _et al._ (2018); Grotti  _et al._
(2018); Delva, Denker, and Lion (2019) and for searches for physics beyond the
standard model Rosenband  _et al._ (2008); Safronova  _et al._ (2018).

For clock comparisons of unlike species, the additive noise of the optical
frequency comb is required to be lower than the clocks being compared.
Currently, the best optical clocks have demonstrated fractional frequency
accuracies around $10^{-18}$ and fractional instabilities around $10^{-16}$
after 1 second of averaging Huntemann  _et al._ (2016); Huang  _et al._
(2022); Brewer  _et al._ (2019); McGrew  _et al._ (2018); Bothwell  _et al._
(2019). Below, we demonstrate that the Er/Yb:glass comb is able to achieve an
additive measurement instability at least one order of magnitude lower than
the current best optical atomic clocks comparison.

As mentioned previously, frequency counting can be used to assess the
stability of the OFC at timescales greater than 1 s. For the results presented
here, signals from the OFC were measured using home-built, zero-dead time,
frequency counters employing software-defined radio (SDR), which offers a
reconfigurable platform for digital signal processing. Our SDR counter is
configured using two channels, which allows for cancellation of noise
introduced by the SDR’s numerically controlled oscillator, thus permitting
zero-dead time frequency counting with 12-digits of frequency resolution and
additive frequency offsets of $10^{-13}$ on a counted 10 MHz signal. Details
regarding our SDR frequency counters can be found in Ref. Sherman and Jördens
(2016). Zero-dead time counting permits phase correlations to be maintained
between frequency measurements, which helps to supports accurate frequency
statistics for signals limited by white phase noise. The latter is important
for characterizing the signals from optical frequency combs, which are
expected to exhibit phase coherence in optical-to-optical, and in optical-to-
microwave synthesis.

To assess the long-term performance of the OFC, the sampled input signal to an
SDR counter was averaged over 1 s to deduce its average frequency. Consecutive
measurements produce a time record of frequencies, averaged over 1 s and
spaced by 1 s. Using the latter time record of frequency measurements, we
calculate the two-sample overlapping Allan deviation, which yields the
fractional frequency stability, versus measurement time. For signals dominated
by white frequency noise and white phase noise, the fractional frequency
stability improves with longer averaging time, $\tau$, with respective slopes
of $\tau^{-1/2}$ and $\tau^{-1}$.

As seen in Figure 7 the Allan deviation of the in-loop $f_{0}$ and
$f_{b,1157}$ beat signals both follow slopes at, or close to, $\tau^{-1}$ for
all averaging times, indicating shot or thermal noise as the limiting noise
source. While in-loop signals mainly assess the quality of the laser feedback
actuators, for OFCs with single-branch architectures Leopardi  _et al._ (2017)
they can provide relatively good estimates for the total residual, or additive
noise, that the modelocked laser contributes in optical synthesis. This is
possible if all optical fibers after the modelocked laser are within the
stabilization loops and their noise is compensated via feedback to the laser.

Er/Yb:glass laser OFC: sources of additive instability In this OFC,
uncompensated free-space paths, and optical and electronic components are the
main source of differential noise when detecting a common wavelength. Shown in
Figure 7 are the additive instabilities from the SDR, the RF synthesizer and
microwave electronics (RF amplifiers, mixers and filters). We can evaluate the
noise due to uncommon free-space optical paths in the Er/Yb:glass OFC system
by detecting the $f_{b,1157}$ beat on two photodetectors. The in-loop
$f_{b,1157}$ beat signal is used to stabilize the modelocked laser cavity
length, and the second beat signal $f^{\prime}_{b,1157}$ measures the 20-cm
differential free-space paths, and noise added in photodetection between
optical interferometers, seen in Figure 1 c). Figure 7 shows the stability of
the frequency detected on the in-loop detector (orange) and on the second
detector (blue). From the latter measurement we observe that 20-cm of enclosed
and uncommon free-space optical paths contribute a frequency instability near
$7\times 10^{-19}$ at 1 s. Noise resulting from changes in the index of
refraction of air reduces the phase coherence, which alters the slope in the
Allan deviation from $\tau^{-1}$ to a slope closer to $\tau^{-1/2}$.

Figure 7: Fractional frequency stability of the in-loop $f_{b,1157}$ (orange),
$f^{\prime}_{b,1157}$ (blue) and $f_{0}$ (green) beat signals as calculated
via Allan deviation. The $f^{\prime}_{b,1157}$ beat measures the noise
contribution from 20 cm of out-of-loop optical free-space paths. Frequency
noise from the H-maser referenced synthesizer and associated electronics to
the optical beat signal (grey). Resolution limit contributed by the SDR
frequency counter (black dashed line).

##  III Stability and accuracy in optical frequency synthesis

At NIST, optical frequency combs are used to characterize and compare the
171Yb and 87Sr neutral lattice clocks McGrew  _et al._ (2018); Bothwell  _et
al._ (2019) and the 27Al+ quantum logic clock Brewer  _et al._ (2019). While
the clocks operate on transition frequencies of 518 THz, 489 THz and 1.12 PHz,
respectively, OFCs measure the clock local oscillator frequencies of 259 THz
(1157 nm), 195 THz (1542 nm), and 280 THz (1070 nm).

While optical synthesis via four wave mixing is expected to maintain the
equidistance of the modelocked laser optical modes and preserve their phase
coherence, tests to evaluate the noise in optical synthesis are necessary to
assess the full performance of the OFC for optical atomic clock comparisons.
Nonlinear techniques have been used to evaluate synthesis between OFC modes
separated by one octave using a single comb Stalnaker  _et al._ (2007);
Leopardi  _et al._ (2017); Martin  _et al._ (2009); J. Stenger and Telle
(2002). However, assessing synthesis errors over smaller bandwidths generally
requires comparing the relative output of two OFCs. Below we evaluate the
additive instability in optical synthesis of the Er/Yb:glass comb between 1070
nm and 1157 nm via comparison to a second frequency comb based on a
Ti:sapphire modelocked laser Fortier, Bartels, and Diddams (2006), see Figure
8.

Figure 8: Setup to characterize the Er/Yb:glass frequency comb in optical-to-
optical and optical-to-microwave synthesis. Light is distributed from optical
references at 1070 nm and 1157 nm over phase-stabilized fiber links to the
Er/Yb:glass comb and a Ti:sapphire comb. Both combs are stabilized to the 1157
nm optical reference and light synthesized at 1070 nm from the two OFCs is
compared using a common 1070 nm optical reference. Optically derived microwave
at 10 GHz signals, generated via detection of the OFC optical pulse trains
detecting the comb pulse trains on MUTC photodiodes, are also compared.

The reference cavities used to stabilize the 1157 nm and 1070 nm clock lasers
are shown in Figure 8 and their light was distributed over independent phase-
stabilized optical fiber links Ma  _et al._ (1994). In the measurement, both
OFCs were phase-locked to the 1157 nm reference. Although the OFCs have
different offset frequencies and repetition rates, stabilization of both comb
spectra to the 1157 nm reference should faithfully, and identically, transfer
the optical reference phase and frequency information to all OFC synthesized
optical modes, including those near 1070 nm. Errors in synthesis can be
measured by comparing the detected beat signals between the 1070 nm reference
and the nearest mode for each comb. The latter comparison was achieved by
filtering and amplifying each 1070 nm beat signal and deriving an RF signal at
the difference frequency of the two beats using a double balanced mixer. The
difference frequency was counted on an SDR frequency counter and the Allan
deviation was calculated (black points in Figure 9). Because the optical
references were free-running, we set the repetition frequency of the
Ti:sapphire comb, $f_{\text{rep,TiS}}$, to be exactly twice that of the
Er/Yb:glass comb, $f_{\text{rep,Er/Yb}}$, so that the scaling factor between
1157 nm and 1070 nm frequency modes for the two combs was identical. This
allowed the frequency drift of the optical references to cancel in the
stability measurement.

Figure 9: Comparison of fractional frequency instabilities for various
optical-to-optical comparisons. Comparison of the noise in optical-to-optical
synthesis from 1157 nm to 1070 nm between the Ti:sapphire and Er/Yb:glass OFCs
(black points). Optical beat note between two 50 m 1157 nm fiber links (orange
trace). Optical beat note between two 100 m 1070 nm fiber links (blue trace).
The current best inter-species optical clock comparison Kim  _et al._ (2021)
between the 27Al+ ion clock and 171Yb optical lattice clock using differential
spectroscopy (red points) fit to a white frequency noise trend line
($1.6\times 10^{-16}~{}\tau^{-1/2}$), shown with red dashed line. The purple
squares represent a high-stability, cryogenic Si optical reference cavity
Matei  _et al._ (2017).

From the figure, the data represented by the black points exhibits a 1-s
instability near $3\times 10^{-17}$, averaging down to the $10^{-20}$ level,
limited by the total measurement time. It is important to note that the
observed stability represents the combined noise of the four optical fiber
links, linking the optical references to the OFCs, as well as the noise in
synthesis from two OFCs. As a result, the observed instability represents an
upper limit to the residual noise in optical synthesis. More specifically, we
found that the latter optical synthesis measurement was partly limited by the
frequency stability of the Doppler-stabilized fiber links. The fiber link
stabilities were assessed by measuring a beat signal between light from the
two 1157 nm links and a separate beat signal between light from the two 1070
nm links. The Allan deviations for both sets of links are plotted in orange
and blue, respectively, in Figure 9. These demonstrate that the Doppler-
cancelled fiber links contributed instabilities near $10^{-17}$ at 1 second
averaging times, averaging down to levels near $10^{-20}$ at longer times.

Also shown for comparison in Figure 9 is the current best stability achieved
for an inter-species optical clock comparison Kim  _et al._ (2021), which
yielded a 1-s stability near $2\times 10^{-16}$. As seen from the Figure, the
current level of the Doppler stabilized fiber links and optical synthesis with
the OFC is sufficient to support the current best atomic clock comparisons.
Currently, the 1-s instability of most inter-species clock comparisons is
limited by the clock local oscillator. The development of cryogenic optical
reference cavities Wiens  _et al._ (2014); Matei  _et al._ (2017) and
differential spectroscopy of atomic clocks Kim  _et al._ (2021) offer a path
toward lower clock and measurement instability, respectively. As a result,
optical synthesis with the OFC, as well as the dissemination of clock signals
via optical fiber, will also need to improve to achieve better short term
instability.

##  IV Stability and accuracy in optical-to-microwave synthesis

Above we characterized the additive instability of the Er/Yb:glass laser OFC
in optical synthesis and assessed its performance in the context of optical
atomic clock comparisons. Here, we evaluate another important capability of
OFCs, namely, their ability to coherently link the optical domain to the
microwave domain. Briefly, when the OFC is phase locked to a high-stability
cavity-stabilized optical reference, either free-running or locked to an
atomic clock, the frequency stability of the optical reference is transferred
to timing stability in the comb optical pulse train. Photodetection of the
optical pulse train yields a pulse train in the electronic domain. Because the
photodetector is only sensitive to the pulse envelope, the optical pulse
carrier information is lost. As a result, the Fourier transform of the
electronic pulse train only yields harmonics of the pulse repetition rate.

From the comb equation, to first order, the frequency of the comb repetition
rate is proportional to $\nu_{\text{ref}}/N$, where $N$ is the mode number of
the optical mode closest in frequency to the optical reference,
$\nu_{\text{ref}}$. For this reason, the technique of optical-to-microwave
conversion is called optical frequency division (OFD) Zhang  _et al._ (2010);
Haboucha  _et al._ (2011); Fortier  _et al._ (2013); Xie  _et al._ (2017);
Kalubovilage, Endo, and Schibli (2020); Nakamura  _et al._ (2020). Not only is
the optical frequency divided by $N$, but the frequency and phase noise power
spectral density of the optical reference is also reduced by $N^{2}$. A 259
THz optical signal divided to 10 GHz, ideally, will yield a reduction in the
optical phase noise by 88 dB ($20\times\log[259\text{ THz}/10\text{ GHz}$])
and will preserve the optical fractional frequency stability in its conversion
to the microwave domain.

Optical frequency division permits the extraction of microwave timing signals
from optical atomic references. The latter signals can be used for
dissemination of optical atomic clock stability using traditional microwave
networks, for their comparison and characterization against the 133Cs primary
frequency standard, or for use as the local oscillator for advanced radar and
communications systems. Below, we assess the additive instability, phase noise
and frequency errors in optical-to-microwave synthesis.

###  IV.1 Optical-to-microwave synthesis: 10 GHz phase noise

The ability of an optical frequency comb to phase-coherently divide an optical
signal to the microwave domain is crucial because it yields a straight-forward
way to count optical clock oscillations with standard electronics.
Additionally, optical-to-microwave synthesis provides a means to convert the
low phase noise of atomic clock probe lasers to the microwave domain. The
latter conversion helps realize high-spectral purity microwave signals, which
have natural applications to next generation communications, sensing and
ranging.

In this section we assess the spectral purity of 10 GHz microwave signals
derived via the 20${}^{\text{th}}$ harmonic of the Er/Yb:glass laser
repetition rate. As shown in Figure 8, the 10 GHz generated by the Ti:sapphire
comb is used as a reference signal for comparison Fortier  _et al._ (2013). To
isolate the additive noise in optical-to-microwave synthesis, both OFCs were
locked to the same 1157 nm optical reference. Additionally, both combs
employed pulse interleavers that effectively multiply the comb repetition
frequencies to 2 GHz prior to photodetection Haboucha  _et al._ (2011).
Repetition rate multiplication improves the strength of the 10 GHz microwave
harmonic by reducing the number of detected harmonics and helps to mitigate
photodetector nonlinearities Taylor  _et al._ (2011) that occur at high
optical pulse energies. Detection nonlinearities and the microwave carrier
strength were further improved by employing high-power, high-linearity
modified uni-traveling carrier (MUTC) photodetectors Li  _et al._ (2011). The
MUTC PD yields $\sim 12.5$ GHz bandwidth, 50 $\mu m$ active area diameter, and
can be operated with a bias voltage as high as -21V Li  _et al._ (2011, 2010).
The photodetected 10 GHz signals from both combs were sent over 1 m of
uncompensated microwave cable and compared using a double balanced mixer. The
intermediate signal, $\Delta f_{10\text{GHz}}$ near 1 MHz, was measured using
a Symmetricom 53132A phase noise analyzer that uses a 10 MHz H-maser signal as
a reference. The results of this comparison are shown in Figure 10 a), which
characterize the additive noise of all components in Figure 8 except the 1157
nm optical cavity, which is a common source of noise for both frequency combs.

Figure 10: Noise characterization in optical-to-microwave synthesis from 1157
nm to 10 GHz. a) Single sideband phase noise of the difference frequency,
$\Delta f_{10\text{GHz}}$, between the 10 GHz generated via OFD by the
Er/Yb:glass and Ti:sapphire combs. Also shown is the combined phase noise
contributed by the 10 MHz H-maser reference (yellow) and the phase noise test
set to the measurement at 10 GHz. b) Allan deviation of the difference
frequency, $\Delta f_{10\text{GHz}}$, between the 10 GHz generated via OFD by
the Er/Yb:glass and Ti:sapphire combs

At 1 Hz offset from the 10 GHz carrier, we demonstrate a residual phase noise
near -115 dBc/Hz. Frequency drift in the unstabilized microwave cables limits
the phase noise for offset frequencies near and below 1 Hz. Phase noise at
offset frequencies between 10 kHz to 200 kHz is the result of uncompensated
noise in the $f_{0}$ and $f_{\text{rep}}$ locks of both optical frequency
combs. Also shown is the additive noise of the 10 MHz H-maser reference to the
10 GHz phase noise measurement. We observe that the H-maser limits the 10 GHz
comparison for offset frequencies below 10 Hz and contributes a high-frequency
noise floor near -160 dBc/Hz. A lower measurement noise floor can be achieved
by using a two channel phase noise measurement system with cross-correlation
Walls (1992). From past studies Fortier  _et al._ (2013) that use cross-
correlation, we expect a noise floor below -170 dBc/Hz. Assuming a thermal
noise level of -170 dBc/Hz, we estimate a total integrated phase noise (from 1
Hz to 5 GHz) around 318 $\mu$rads, which indicates an RMS timing jitter on 10
GHz of around 7.2 femtoseconds.

###  IV.2 Optical-to-microwave synthesis: additive instability

To assess the additive instability for timescales longer than 1 s, we
calculate the Allan deviation of the compared 10 GHz signals, $\Delta
f_{10\text{GHz}}$. The results are depicted in Figure 10 b). The instability
represents an upper bound to the additive optical-to-microwave synthesis with
the Er/Yb:glass laser OFC. This is because $\Delta f_{10\text{GHz}}$ carries
phase information regarding the noise of the optical fiber links to the OFCs,
the OFC optical-to-microwave synthesis, noise in photodetection and in the
comparison and measurement of the 10 GHz signals. As seen in the Figure, the
Allan deviation of $\Delta f_{10\text{GHz}}$ yields a 1-s additive instability
$<1\times 10^{-15}$, averaging down with a slope near $\tau^{-1}$.

###  IV.3 Optical-to-microwave synthesis: additive synthesis errors

To quantify the frequency errors added by optical-to-microwave synthesis, we
generate and compare 10 GHz signals based on an absolute frequency reference.
For the latter measurement, the Er/Yb:glass laser and Ti:Sapphire laser OFC
were both phase stabilized to the same 171Yb optical atomic clock. Using the
atomic reference, both combs generated electronic signals near 10 GHz. The
Ti:Sapphire OFC generated a second signal near 1 GHz, $f_{\text{rep,TiS}}$,
that was measured against the international ensemble of primary and secondary
standards (PSFS), via the NIST maser ensemble NIS (2021). The difference
between the two 10 GHz signals from the Ti:Sapphire and Er/Yb:glass laser
OFCs, $\Delta f_{10\text{GHz}}$, was simultaneously counted on an SDR-based
frequency counter, using a single H-maser in the NIST maser ensemble. As such,
the Er/Yb:glass laser repetition rate can be evaluated as,
$f_{\text{rep,Er/Yb}}=\left(10\times f_{\text{rep,TiS}}+\Delta
f_{10\text{GHz}}\right)/20$. By simultaneously counting the offset frequencies
and beat signals between the OFCs and their respective 171Yb local oscillator
signals, the comb equation can be used to calculate the 171Yb optical
frequency as measured by both OFCs.

Using this procedure we periodically measured the 171Yb optical atomic clock
frequency vs PSFS using both frequency combs, Figure 11 a). In Figure 11 a),
the error bars are determined by the daily statistical uncertainty, which is
dominated by the noise of reference H-maser ($\sim 10^{-13}$ at 1-s averaging
time), common to both OFCs. Figure 11 b) shows the difference between the
Er/Yb:glass laser and Ti:Sapphire laser OFC measurements of the 171Yb optical
atomic clock frequency, per day. The error bars in b) represent the
statistical error in the measurement of $\Delta f_{10\text{GHz}}$, which is
dominated by the additive noise of the SDR-based frequency counter. From
Figure 11 b), we observe an agreement in the mean value of the daily
measurement of the 171Yb optical clock, between both OFCs, at $-0.87\pm
2.05\times 10^{-18}$, limited by the measurement uncertainty.

Figure 11: a) Comparison in the evaluation of the absolute frequency of the
171Yb lattice clock as measured by the Er/Yb:glass laser OFC (yellow
triangles) and Ti:sapphire OFC (black circles). Data in a) and b) were derived
from unioned data only. Data in a) is offset from the BIPM recommended value
for 171Yb of 518 295 836 590 863.63 Hz. The grey error bars were limited by
the instability of the H-maser that was used for referencing the counted 1 GHz
Ti:sapphire repetition rate signal. b) Shows the fractional frequency
difference between the black and yellow markers in a), demonstrating an
agreement between the OFCs at the $10^{-18}$ level. The blue error bars in b)
were limited by the instability of the H-maser that was used for referencing
the counted $\Delta f_{10\text{GHz}}$ signal.

##  V Conclusion

In summary, the Er/Yb:glass laser represents a versatile and robust system for
laboratory-based, high performance metrology applications. The laser
represents a competitive alternative to OFCs based on Er:fiber and Ti:sapphire
modelocked lasers because it is simple to build and demonstrates self-starting
modelocking due to the SESAM. Additionally, the low net laser cavity loss
permits narrow intrinsic optical linewidths, despite the use of a 980 nm
telecom diode laser for pumping. The latter permits high performance optical
and microwave synthesis with moderate engineering of the laser feedback
actuators and feedback loops.

To convert the Er/Yb:glass laser to a frequency comb, we interfaced the laser
with a home-built PM-Erbium-doped amplifier and a hybrid HNLF to enable
generation of an octave-spanning optical spectrum. Once phase stabilized, we
demonstrated the utility of the Er/Yb:glass laser OFC, and quantified its
performance in terms of precision metrology applications. We have evaluated
the additive noise in optical synthesis, which contributed an upper bound to
the instability at 1-s near 10-17. A lower bound to optical synthesis, below
10-18, was found to be limited by uncompensated free-space optical paths in
the interferometers used to reference the comb to high-stability optical
frequency references. In 10 GHz optical-to-microwave synthesis, we observed an
additive instability below 10-15 for 1-s averaging times, with a residual
single-sideband phase noise level below -115 dBc/Hz at a 1 Hz offset from the
carrier. When phase-locked to a 171Yb optical atomic clock, we estimated that
the Er/Yb:glass laser OFC contributed errors in optical-to-microwave frequency
synthesis near the 10-18 level, limited by the measurement statistics.

Both high-precision optical-to-optical and optical-to-microwave synthesis are
required for the implementation of an all-optical timescale Milner  _et al._
(2019); Yao  _et al._ (2019), an important part of the infrastructure for
redefinition of the SI second to optical atomic references. Additionally,
optically derived timing signals will help realize timing signals that are at
least two orders-of-magnitude more stable and accurate than current timescales
based on microwave oscillators.

##  VI Acknowledgements

We thank J. A. Sherman for help with the SDR frequency counter and R. P. Mirin
for the SESAM.

##  VII Disclaimers

Certain commercial equipment, instruments, or materials (or suppliers, or
software, etc.) are identified in this paper to foster understanding. Such
identification does not imply recommendation or endorsement by the National
Institute of Standards and Technology, nor does it imply that the materials or
equipment identified are necessarily the best available for the purpose.

##  VIII Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

## References

  * Maiman (1960) T. H. Maiman, “Stimulated optical radiation in ruby,” Nature 187, 493–494 (1960).
  * Hargrove, Fork, and Pollack (1964) L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of he–ne laser modes induced by synchronous intracavity modulation,” Applied Physics Letters 5, 4–5 (1964).
  * Stern  _et al._ (1989) D. Stern, R. W. Schoenlein, C. A. Puliafito, E. T. Dobi, R. Birngruber, and J. G. Fujimoto, “Corneal ablation by nanosecond, picosecond, and femtosecond lasers at 532 and 625 nm,” Archives of ophthalmology 107, 587–592 (1989).
  * Andiel  _et al._ (2002) U. Andiel, K. Eidmann, K. Witte, R. Mancini, and P. Hakel, “Astrophysical and inertial-confinement-fusion plasmas generated with millijoule femtosecond laser pulses,” Journal of Modern Optics 49, 2615–2628 (2002).
  * Denk, Strickler, and Webb (1990) W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
  * Lytle, Parrish, and Barnes (1985) F. E. Lytle, R. M. Parrish, and W. T. Barnes, “An introduction to time-resolved pump/probe spectroscopy,” Applied spectroscopy 39, 444–451 (1985).
  * Gaumnitz  _et al._ (2017) T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-x-ray pulses generated by a passively cep-stable mid-infrared driver,” Optics express 25, 27506–27518 (2017).
  * Spence, Kean, and Sibbett (1991) D. Spence, P. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked ti: sapphire laser,” Optics letters 16, 42–44 (1991).
  * Hofer  _et al._ (1992) M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE Journal of Quantum Electronics 28, 720–728 (1992).
  * Keller  _et al._ (1996) U. Keller, K. J. Weingarten, F. X. Kartner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Honninger, N. Matuschek, and J. A. D. Au, “Semiconductor saturable absorber mirrors (sesam’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE Journal of selected topics in QUANTUM ELECTRONICS 2, 435–453 (1996).
  * Essen and Parry (1955) L. Essen and J. V. L. Parry, “An atomic standard of frequency and time interval: a caesium resonator,” Nature 176, 280–282 (1955).
  * Zhang  _et al._ (2010) W. Zhang, M. L. Z. Xu, R. Boudot, Y. Kersalé, G. Santarelli, and Y. L. Coq, “Sub-100 attoseconds stability optics-to-microwave synchronization,” Appl. Phys. Lett. 96, 211105 (2010).
  * Haboucha  _et al._ (2011) A. Haboucha, W. Zhang, T. Li, M. Lours, A. N. Luiten, Y. L. Coq, and G. Santarelli, “Optical-fiber pulse rate multiplier for ultralow phase-noise signal generation,” Opt. Lett. 36, 3654–3656 (2011).
  * Fortier  _et al._ (2013) T. M. Fortier, F. Quinlan, A. Hati, C. Nelson, J. A. Taylor, Y. Fu, J. Campbell, and S. A. Diddams, “Photonic microwave generation with high-power photodiodes,” Opt. Lett. 38, 1712–1714 (2013).
  * Xie  _et al._ (2017) X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. L. Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photon. 11, 44–47 (2017).
  * Kalubovilage, Endo, and Schibli (2020) M. Kalubovilage, M. Endo, and T. R. Schibli, “Ultra-low phase noise microwave generation with a free-running monolithic femtosecond laser,” Opt. Express 28, 25400–25409 (2020).
  * Nakamura  _et al._ (2020) T. Nakamura, J. Davila-Rodriguez, H. Leopardi, J. A. Sherman, T. M. Fortier, X. Xie, J. C. Campbell, W. F. McGrew, X. Zhang, Y. S. Hassan, D. Nicolodi, K. Beloy, A. D. Ludlow, S. A. Diddams, and F. Quinlan, “Coherent optical clock down-conversion for microwave frequencies with $10^{-18}$ instability,” Science 368, 889–892 (2020).
  * Ye, Hall, and Diddams (2000) J. Ye, J. L. Hall, and S. A. Diddams, “Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum,” Opt. Lett. 25, 1675–1677 (2000).
  * Fortier  _et al._ (2011) T. M. Fortier, M. S. Kirchner, F. Quinlan, J. C. B. J. Taylor, N. L. T. Rosenband, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photon. 5, 425–429 (2011).
  * Rosenband  _et al._ (2008) T. Rosenband, D. Hume, P. Schmidt, C. Chou, A. Brusch, L. Lorini, W. Oskay, R. Drullinger, T. Fortier, J. Stalnaker, S. Diddams, W. Swann, N. Newbury, W. Itano, D. Wineland, and J. Bergquist, “Frequency ratio of al+ and hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
  * Safronova  _et al._ (2018) M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, “Search for new physics with atoms and molecules,” Reviews of Modern Physics 90, 025008 (2018).
  * Clivati  _et al._ (2017) C. Clivati, R. Ambrosini, T. Artz, A. Bertarini, C. Bortolotti, M. Frittelli, F. Levi, A. Mura, G. Maccaferri, M. Nanni, M. Negusini, F. Perini, M. Roma, M. Stagni, M. Zucco, and D. Calonico, “A vlbi experiment using a remote atomic clock via a coherent fibre link,” Scientific reports 7, 1–8 (2017).
  * Fortier and Baumann (2019) T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Communications Physics 2, 1–16 (2019).
  * Collaboration (2021) B. A. C. O. N. B. Collaboration, “Frequency ratio measurements at 18-digit accuracy using an optical clock network,” Nature 591, 564–569 (2021).
  * Tauser, Leitenstorfer, and Zinth (2003) F. Tauser, A. Leitenstorfer, and W. Zinth, “Amplified femtosecond pulses from an er:fiber system: Nonlinear pulse shortening and self-referencing detection of the carrier-envelope phase evolution,” Opt. Express 11, 594–600 (2003).
  * Nardelli  _et al._ (2022) N. V. Nardelli, T. M. Fortier, M. Pomponio, E. Baumann, C. W. Nelson, T. R. Schibli, and A. Hati, “10 ghz generation with ultra-low phase noise via the transfer oscillator technique,” APL Photonics 7, 026105 (2022).
  * Keller, Knox, and Roskos (1990) U. Keller, W. H. Knox, and H. Roskos, “Coupled-cavity resonant passive mode-locked ti: sapphire laser,” Optics letters 15, 1377–1379 (1990).
  * Lesko  _et al._ (2020) D. M. B. Lesko, A. J. Lind, N. Hoghooghi, A. Kowligy, H. Timmers, P. Sekhar, B. Rudin, F. Emaury, G. B. Rieker, and S. A. Diddams, “Fully phase-stabilized 1 ghz turnkey frequency comb at 1.56 µm,” OSA Continuum 3, 2070–2077 (2020).
  * Spühler  _et al._ (2005) G. J. Spühler, L. Krainer, E. Innerhofer, R. Paschotta, K. J. Weingarten, and U. Keller, “Soliton mode-locked er: Yb: glass laser,” Optics Letters 30, 263–265 (2005).
  * Resan  _et al._ (2016) B. Resan, S. Kurmulis, Z. Y. Zhang, A. E. H. Oehler, V. Markovic, M. Mangold, T. Südmeyer, U. Keller, R. A. Hogg, and K. J. Weingarten, “10 ghz pulse repetition rate er: Yb: glass laser modelocked with quantum dot semiconductor saturable absorber mirror,” Applied Optics 55, 3776–3780 (2016).
  * Oehler  _et al._ (2008) A. E. H. Oehler, T. Südmeyer, K. J. Weingarten, and U. Keller, “100 ghz passively mode-locked er: Yb: glass laser at 1.5 µm with 1.6-ps pulses,” Optics Express 16, 21930–21935 (2008).
  * Shoji  _et al._ (2016) T. D. Shoji, W. Xie, K. L. Silverman, A. Feldman, T. Harvey, R. P. Mirin, and T. R. Schibli, “Ultra-low-noise monolithic mode-locked solid-state laser,” Optica 3, 995–998 (2016).
  * Gapontsev  _et al._ (1982) V. P. Gapontsev, S. M. Matitsin, A. A. Isineev, and V. B. Kravchenko, “Erbium glass lasers and their applications,” Optics and Laser Technology 14, 189–196 (1982).
  * Kogelnik and Li (1966) H. Kogelnik and T. Li, “Laser beams and resonators,” Applied Optics 5, 1550–1567 (1966).
  * Hanna (1969) D. Hanna, “Astigmatic gaussian beams produced by axially asymmetric laser cavities,” IEEE Journal of quantum electronics 5, 483–488 (1969).
  * Dunn and Ferguson (1977) M. H. Dunn and A. I. Ferguson, “Coma compensation in off-axis laser resonators,” Optics Communications 20, 214–219 (1977).
  * Lee  _et al._ (2015) C. Lee, Y. Hayashi, K. Silverman, A. Feldman, T. Harvey, R. Mirin, and T. Schibli, “Monolithic device for modelocking and stabilization of frequency combs,” Optics Express 23, 33038–33043 (2015).
  * M. Haiml and Weber (1999) F. M.-G. U. K. M. L. P. S. M. Haiml, U. Siegner and E. R. Weber, “Femtosecond response times and high optical nonlinearity in beryllium-doped low-temperature grown gaas,” Applied Physics Letters 74, 1269 (1999).
  * Mangold  _et al._ (2013) M. Mangold, V. J. Wittwer, C. A. Zaugg, S. M. Link, M. Golling, B. W. Tilma, and U. Keller, “Femtosecond pulses from a modelocked integrated external-cavity surface emitting laser (mixsel),” Optics Express 21, 24904–24911 (2013).
  * Korel  _et al._ (2014) I. I. Korel, B. N. Nyushkov, V. I. Denisov, V. S. Pivtsov, N. A. Koliada, A. A. Sysoliatin, S. M. Ignatovich, N. L. Kvashnin, M. N. Skvortsov, and S. N. Bagayev, “Hybrid highly nonlinear fiber for spectral supercontinuum generation in mobile femtosecond clockwork,” Laser Physics 24, 074012 (2014).
  * Schioppo  _et al._ (2016) M. Schioppo, R. C. Brown, W. F. McGrew, N. Hinkley, R. J. Fasano, K. Beloy, T. H. Yoon, G. Milani, D. Nicolodi, J. A. Sherman, N. B. Phillips, C. W. Oates, and A. D. Ludlow, “Ultrastable optical clock with two cold-atom ensembles,” Nature Photon. 11, 48–52 (2016).
  * Young  _et al._ (1999) B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
  * Telle  _et al._ (1999) H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U. Keller, “Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation,” Appl. Phys. B 69, 327–332 (1999).
  * Reichert, amd T. Udem, and Hänsch (1999) J. Reichert, R. amd T. Udem, and T. W. Hänsch, “Measuring the frequency of light with mode-locked lasers,” Optics Communications 172, 59–68 (1999).
  * Jones  _et al._ (2000) D. Jones, S. Diddams, J. Ranka, A. Stentz, R. Windeler, J. Hall, and S. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
  * Matei  _et al._ (2017) 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 $\mu$m lasers with sub-10 mhz linewidth,” Physical Review Letters 118, 263202 (2017).
  * Westbrook  _et al._ (2004) P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, “Supercontinuum generation in a fiber grating,” Applied Physics Letters 85, 4600–4602 (2004).
  * Carlson  _et al._ (2017) D. R. Carlson, D. D. Hickstein, A. Lind, J. B. Olson, R. W. Fox, R. C. Brown, A. D. Ludlow, Q. Li, D. Westly, H. Leopardi, T. M. Fortier, K. Srinivasan, S. A. Diddams, and S. B. Papp, “Photonic-chip supercontinuum with tailored spectra for counting optical frequencies,” Phys. Rev. Appl. 8, 014027 (2017).
  * Taylor  _et al._ (2011) J. Taylor, S. Datta, A. Hati, C. Nelson, F. Quinlan, A. Joshi, and S. Diddams, “Characterization of power-to-phase conversion in high-speed p-i-n photodiodes,” IEEE Photonics J. 3, 140–151 (2011).
  * Taccheo  _et al._ (1998) S. Taccheo, P. Laporta, O. Svelto, and G. D. Geronimo, “Theoretical and experimental analysis of intensity noise in a codoped erbium-ytterbium glass laser,” Applied Physics B: Lasers and Optics 66, 19–26 (1998).
  * Lee and Schibli (2014) C.-C. Lee and T. R. Schibli, “Intrinsic power oscillations generated by the backaction of continuum on solitons and its implications on the transfer functions of a mode-locked laser,” Physical Review Letters 112, 223903 (2014).
  * Briles  _et al._ (2010) T. C. Briles, D. C. Yost, A. Cingöz, J. Ye, and T. R. Schibli, “Simple piezoelectric-actuated mirror with 180 khz servo bandwidth,” Opt. Express 18, 9739–9746 (2010).
  * Pomponio, Hati, and Nelson (2020) M. Pomponio, A. Hati, and C. Nelson, “Fpga-based low-latency digital servo for optical physics experiments,” Joint Conference of the IEEE International Frequency Control Symposium and International Symposium on Applications of Ferroelectrics (IFCS-ISAF) , 1–2 (2020).
  * McGrew  _et al._ (2018) W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, “Atomic clock performance enabling geodesy below the centimetre level,” Nature 564, 87–90 (2018).
  * Grotti  _et al._ (2018) J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland, F. N. Baynes, H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G. A. Costanzo, C. Clivati, F. Levi, and D. Calonico, “Geodesy and metrology with a transportable optical clock,” Nature Physics 14, 437–441 (2018).
  * Delva, Denker, and Lion (2019) P. Delva, H. Denker, and G. Lion, “Chronometric geodesy: methods and applications,” in  _Relativistic Geodesy: Foundations and Applications_ , edited by D. Puetzfeld and C. Lämmerzahl (Springer International Publishing, 2019) pp. 25–85.
  * Huntemann  _et al._ (2016) N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik, “Single-ion atomic clock with 3x10-18 systematic uncertainty,” Phys. Rev. Lett. 116, 063001 (2016).
  * Huang  _et al._ (2022) Y. Huang, B. Zhang, M. Zeng, Y. Hao, Z. Ma, H. Zhang, H. Guan, Z. Chen, M. Wang, and K. Gao, “A liquid nitrogen-cooled ca+ optical clock with systematic uncertainty of 3x10-18.” Phys. Rev. Applied (2022).
  * Brewer  _et al._ (2019) S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, “27Al+ quantum-logic clock with a systematic uncertainty below $10^{-18}$,” Phys. Rev. Lett. 123, 033201 (2019).
  * Bothwell  _et al._ (2019) T. Bothwell, D. Kedar, E. Oelker, J. M. Robinson, S. L. Bromley, W. L. Tew, J. Ye, and C. J. Kennedy, “Jila sri optical lattice clock with uncertainty of 2.0x10-18,” Metrologia 56, 065004 (2019).
  * Sherman and Jördens (2016) J. A. Sherman and R. Jördens, “Oscillator metrology with software defined radio,” Review of Scientific Instruments 87, 054711 (2016).
  * Leopardi  _et al._ (2017) H. Leopardi, J. Davila-Rodriguez, F. Quinlan, J. Olson, J. A. Sherman, S. A. Diddams, and T. M. Fortier, “Single-branch er: fiber frequency comb for precision optical metrology with 10-18 fractional instability,” Optica 4, 879–885 (2017).
  * Stalnaker  _et al._ (2007) J. Stalnaker, S. A. Diddams, T. M. Fortier, K. Kim, L. W. Hollberg, J. C. Bergquist, W. M. Itano, M. Delaney, L. Lorini, W. Oskay, T. P. Heavner, S. R. Jefferts, F. Levi, T. E. Parker, and J. H. Shirley, “Optical-to-microwave frequency comparison with fractional uncertainty of 10-15,” Applied Physics B 89, 167–176 (2007).
  * Martin  _et al._ (2009) M. J. Martin, S. M. Foreman, T. Schibli, and J. Ye, “Testing ultrafast mode-locking at microhertz relative optical linewidth,” Optics Express 17, 558–568 (2009).
  * J. Stenger and Telle (2002) C. T. J. Stenger, H. Schnatz and H. R. Telle, “Ultraprecise measurement of optical frequency ratios,” Physical Review Letters 88, 073601 (2002).
  * Fortier, Bartels, and Diddams (2006) T. M. Fortier, A. Bartels, and S. A. Diddams, “Octave-spanning ti:sapphire laser with a repetition rate >1 ghz for optical frequency measurements and comparisons,” Opt. Lett. 31, 1011–1013 (2006).
  * Ma  _et al._ (1994) L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Opt. Lett. 19, 1777–1779 (1994).
  * Kim  _et al._ (2021) M. E. Kim, W. F. McGrew, N. V. Nardelli, E. R. Clements, Y. S. Hassan, X. Zhang, J. Valencia, H. Leopardi, D. B. Hume, T. M. Fortier, A. D. Ludlow, and D. R. Leibrandt, “Optical coherence between atomic species at the second scale: improved clock comparisons via differential spectroscopy,” arXiv preprint arXiv:2109.09540 (2021).
  * Wiens  _et al._ (2014) E. Wiens, Q.-F. Chen, I. Ernsting, H. Luckmann, U. Rosowski, A. Nevsky, and S. Schiller, “Silicon single-crystal cryogenic optical resonator,” Optics Letters 39, 3242–3245 (2014).
  * Li  _et al._ (2011) Z. Li, Y. Fu, M. Piels, H. P. Pan, A. Beling, J. E. Bowers, and J. C. Campbell, “High-power high-linearity flip-chip bonded modified uni-traveling carrier photodiode,” Opt. Express 19, B385–B390 (2011).
  * Li  _et al._ (2010) Z. Li, H. Pan, H. Chen, A. Beling, and J. C. Campbell, “High-saturation-current modified uni-traveling-carrier photodiode with cliff layer,” IEEE Journal of Quantum Electronics 46, 626–632 (2010).
  * Walls (1992) W. F. Walls, “Cross-correlation phase noise measurements,” in  _Proceedings of the 1992 IEEE Frequency Control Symposium_ (IEEE, 1992) pp. 257–261.
  * NIS (2021) “Frequency evaluation of utc(nist) by nist-yb1 for the period mjd 59249 to 59269,” Published online at https://webtai.bipm.org/ftp/pub/tai/data/PSFS_reports/nist-yb1_59249-59269.pdf (2021).
  * Milner  _et al._ (2019) 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).
  * Yao  _et al._ (2019) J. Yao, J. Sherman, T. Fortier, A. Ludlow, H. Leopardi, T. Parker, W. McGrew, S. Diddams, and J. Levine, “Optical-clock-based time scale,” Phys. Rev. Applied 12, 044069 (2019).

##  IX Appendix / Supplementary

####  IX.0.1 cavity alignment procedure

In this section we detail some of the steps taken to construct the modelocked
Er/Yb:glass laser and specify characteristics that are useful in its
optimization.

Place optics: We first secured the 980 nm pump fiber-to-free-space coupler.
While the laser diode was running at low power we then secured components in
the following order: (1) the $f=75$ mm lens, (2) the curved mirror (M2), (3)
the crystal (at Brewster’s angle, $\approx 57$ degrees) as seen in Figure 1,
and (4) the curved mirror (M2). Translation stages are used on all mirrors,
except the curved mirror (M2) through which the 980 nm pump light is focused.
In addition to a linear translation stage, the Er/Yb:glass gain medium is
placed on a rotation stage to fine tune Brewster’s angle. While actuation
stages can reduce the alignment stability, they also permit higher efficiency
in laser alignment. To reduce their impact to the optical alignment we employ
a low beam height (4 cm), highly rigid optical mounts, and high-performance
stainless actuator stages. A half-waveplate, after the 980 nm pump but before
the focusing lens, may be necessary to tune the 980 nm polarization to
minimize the Fresnel reflection off the Er/Yb:glass gain medium.

For initial alignment, the green fluorescence from the gain medium is used to
estimate the position of the lasing 1550 nm beam and its focal points. The
latter technique is used to position the output coupler (OC) and SESAM, by
centering the green fluorescence at their centers. Note that the SESAM is
placed at the focus of the green fluorescence. To align the laser for CW
operation, tip tilt on the OC and HR was used to overlap the fluorescent
return beams from the two arms of the linear cavity; one between the
Er/Yb:glass and the OC and the other from the Er/Yb:glass to the SESAM or HR.

Once the laser cavity is adequately aligned, the brightness of the green
fluorescence from the crystal will dim noticeably as energy is transferred to
the 1550 nm lasing light. When varying the pump current from threshold to
higher power, the 1550 nm output power should also increase steadily. If the
cavity alignment is sub-optimal, the effects of thermal lensing will be
observed via thermally induced refractive index changes in the crystal. The
latter manifest as sudden drops in output power as the pump power increases.

After loosely positioning the output coupler, we then use a high-reflector in
place of the SESAM to optimize the laser for continuous wave (CW) lasing.
Using an HR in place of the SESAM for initial alignment also helps to avoid
damage due to Q-switching. Once CW power is maximized, the high-reflector is
replaced by the SESAM, using the fluorescence focus for positioning.
Translation of the SESAM, via a linear translation stage, is used to induce
and tune modelocked operation.

Modelock the laser: Once the laser cavity was optimally aligned with the HR,
we replaced the HR with the SESAM. With the 980 nm pump power near 400 mW, the
correct SESAM position and tip-tilt angles were easily found by watching for
the dimming of the green fluorescence. Once lasing, the threshold pump power
was re-optimized by translating the SESAM through the 1550 nm focus. Once
optimized, by increasing the pump power to > 250 mW the laser should generally
begin modelocking. Finally, an optical spectrum analyzer was used to optimize
the modelocked state by characterizing the CW breakthrough and spectral
bandwidth as the SESAM is translated across the 1550 nm focus.