# A low repetition rate optical frequency comb

Francesco Canella  Dipartimento di Fisica, Politecnico di Milano, piazza
Leonardo da Vinci 32, 20133 Milan, Italy  Max-Planck-Institut für
Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany  Istituto di
Fotonica e Nanotecnologie - Consiglio Nazionale delle Ricerche, piazza
Leonardo da Vinci 32, 20133 Milan, Italy<EMAIL_ADDRESS>Johannes Weitenberg  Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-
Straße 1, 85748 Garching, Germany  Fraunhofer-Institut für Lasertechnik ILT,
Steinbachstraße 15, 52074 Aachen, Germany  Muhammad Thariq  Max-Planck-
Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
Karlsruhe School of Optics and Photonics, Karlsruher Institut für Technologie,
Schlossplatz 19, 76131 Karlsruhe, Germany  Fabian Schmid  Max-Planck-Institut
für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany  Institute
for Quantum Electronics, ETH Zürich, Otto-Stern-Weg 1, 8093 Zurich,
Switzerland  Paras Dwivedi  Max-Planck-Institut für Quantenoptik, Hans-
Kopfermann-Straße 1, 85748 Garching, Germany  Fakultät für Physik, Ludwig-
Maximilians-Universität München, Schellingstraße 4, 80799 Munich, Germany
Gianluca Galzerano  Istituto di Fotonica e Nanotecnologie - Consiglio
Nazionale delle Ricerche, piazza Leonardo da Vinci 32, 20133 Milan, Italy
Theodor W. Hänsch  Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-
Straße 1, 85748 Garching, Germany  Fakultät für Physik, Ludwig-Maximilians-
Universität München, Schellingstraße 4, 80799 Munich, Germany  Thomas Udem
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748
Garching, Germany  Fakultät für Physik, Ludwig-Maximilians-Universität
München, Schellingstraße 4, 80799 Munich, Germany  Akira Ozawa  Max-Planck-
Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
<EMAIL_ADDRESS>
###### Abstract

Reducing the pulse repetition rate of an optical frequency comb increases the
pulse energy for a given average power. This enhances the efficiency of
nonlinear frequency conversion and it facilitates extending the accessible
wavelength range, for example into the extreme ultraviolet (XUV). The
resulting spectrally dense frequency comb can still be used for precision
spectroscopy of narrow atomic or molecular transitions. In this article, we
demonstrate a low-noise infrared frequency comb with a repetition rate as low
as $40\text{\,}\mathrm{k}\Hz$ using a Yb:KYW mode-locked laser, pulse picking,
and subsequent amplification. The frequency comb structure is confirmed by
generating a beat note with a continuous wave reference laser. A comb mode is
actively stabilized to the reference laser, and the integrated rms phase noise
from $20\text{\,}\Hz$ to $20\text{\,}\mathrm{k}\Hz$ is measured to be
$195\text{\,}\mathrm{mrad}$.

††journal: Optica

##  1 Introduction

Optical frequency combs have revolutionized the field of optical frequency
metrology [1, 2, 3] and are indispensable tools for high-precision laser
spectroscopy to study fundamental physics [4, 5, 6] and for optical frequency
standards [7, 8, 9, 10, 11]. The first application of frequency combs was to
measure the frequency of a continuous wave laser that is then used for
precision spectroscopy [12, 13, 3]. It is also possible to use the frequency
comb itself to excite target transitions and perform spectroscopy with ultra-
wide spectral coverage, fast detection times, and high sensitivities [5, 14,
15, 16, 17, 18].

An optical frequency comb consists of many spectral modes that are equally
spaced by the repetition rate of the generating mode-locked pulse train which
is typically in the $\text{\,}\mathrm{MHz}$ to the $\text{\,}\mathrm{GHz}$
range for conventional solid-state and fiber laser-based oscillator designs.
While combs with mode spacings of several tens of GHz have found applications
for calibration of astronomical spectrographs [19, 20], a pulse train with a
relatively low repetition rate below $10\text{\,}\mathrm{MHz}$ can have
correspondingly higher pulse energy and is therefore advantageous for several
applications, including for example driving efficient nonlinear processes [21,
22, 23]. Nonlinear frequency conversion of optical frequency combs can enable
precision spectroscopy in wavelength ranges where continuous-wave lasers are
not available, such as the extreme ultraviolet (XUV) [24, 25, 26]. For
example, our planned experiment of precision spectroscopy of the 1S-2S
transition in He+ ions requires an optical frequency comb at 60.8 nm [27, 28].
The Ramsey-type frequency comb, which consists of pairs of intense pulses, can
be an alternative method to address transitions at XUV wavelengths [29].
Precision spectroscopy of the nuclear transition of 229Th at around 149 nm may
find application as a nuclear optical clock [30, 31, 32, 33].

Intra-cavity high-order harmonic generation allows the generation of high-
power XUV frequency combs suitable for direct frequency comb spectroscopy [34,
35, 36, 24, 37, 38]. In this scheme, the high-order harmonic generation
process is performed inside an enhancement cavity using a gaseous or solid
medium placed at the focus of the cavity. Special care must be taken to avoid
detrimental effects of the high average power and intensity, such as thermal
lensing [39], plasma phase shift [40, 41, 42], misalignment due to elevated
temperature [39], and damage to the optics [43]. In addition, enhancement
cavities for ultrashort pulses have to be carefully designed to tailor the
intra-cavity dispersion [44, 45].

Alternatively, by operating a comb at a lower repetition rate, a similarly
high pulse energy could be achieved with modest average power even without an
enhancement cavity. Lowering the repetition rate results in a frequency comb
with a smaller mode spacing. For precision spectroscopy, the mode spacing
should be at least several times larger than the linewidth of the transition
under investigation in order to obtain a comb-mode resolved spectroscopy
signal. For instance, a signal linewidth of about $1\text{\,}\mathrm{kHz}$ is
expected for the 1S-2S transition in He+ at 60.8 nm [27, 28], while the
nuclear transition of 229Th at about 149 nm has a natural linewidth of about
$20\text{\,}\mathrm{\SIUnitSymbolMicro Hz}$ [30, 31]. In principle, a comb
with a few kHz mode spacing would be sufficient for these applications.

High pulse energy ultrafast lasers at such low repetition rates are
conventionally generated using master oscillator power amplifiers (MOPAs),
where the chirped pulse amplification (CPA) scheme is employed and have found
various applications including attosecond physics [46, 47], laser particle
acceleration [48, 49], and ultrafast transient spectroscopy [50]. By actively
controlling the round-trip phase-shift of the oscillator, each pulse can have
an identical carrier-envelope phase (CEP), which is particularly important for
the study of field-sensitive phenomena driven by few-cycle pulses [51]. Using
the balanced optical cross-correlator method [52], the pulse-to-pulse timing
jitter of the CPA laser system can be controlled precisely. For example, in
Ref. [53] a sub-100 fs timing jitter is demonstrated at
$1\text{\,}\mathrm{kHz}$ repetition rate. A pulse timing jitter of 100 fs
still introduces a relative frequency uncertainty of 10-10 at a repetition
rate of $1\text{\,}\mathrm{kHz}$. This corresponds to a broadening of the
optical comb modes that is larger than the mode spacing and could wash out the
comb structure in the frequency domain. Although such CEP-stabilized low-
repetition-rate laser systems have been shown to be suitable for studying
ultrafast phenomena, they do not guarantee a low-noise frequency comb
structure.

In this work, we report a low-noise optical frequency comb that operates at
tunable repetition rates from 40 kHz to 40 MHz using a Yb:KYW mode-locked
oscillator. Mode-locked lasers oscillating directly at sub-MHz repetition
rates would require very long cavities which may not be practical. Instead,
conventional mode-locked lasers and pulse pickers are used to generate optical
frequency combs at low repetition rates [54, 55]. The associated power loss is
compensated by re-amplifying the pulse train such that higher pulse energy is
achieved. A comb mode is actively stabilized to an ultra-stable continuous
wave (cw) reference laser. The phase noise of the stabilized mode is
characterized with respect to the reference laser and is shown to result in a
narrow linewidth suitable for exciting narrow transitions.

##  2 Modelling of the pulse-picking process

In this section, we first model the pulse picking process by treating the
pulse picker as an ideal amplitude modulator.

For mathematical convenience, we model the output of the mode-locked laser as
a train of Gaussian-shaped pulses with a repetition rate $f_{\rm rep}$. The
temporal pulse spacing is $T=f_{\rm rep}^{-1}$. The electric field $E(t)$ of
the laser pulse train can be described as

$E(t)=A\sum_{k=-\infty}^{\infty}\exp{\left[-\frac{\left(t-k\,T\right)^{2}}{\Delta
t^{2}}\right]}e^{i\left(\omega_{0}t+\varphi\left(t\right)\right)},$ (1)

Figure 1: Low-repetition rate frequency comb generation using an AOM-based
pulse picker. Every $m$-th pulse is diffracted by the AOM to the 1st
diffraction order, while the others remain in the 0th order and are dumped.
After the pulse picking, the pulse-to-pulse time interval increases by the
factor $m$, while the mode spacing in the frequency domain becomes $m$ times
smaller. The AOM is driven by an RF carrier which is amplitude modulated with
a rectangular-shaped gate-pulse train.

where $A$ is the field envelope peak amplitude, $\omega_{0}$ is the angular
frequency of the carrier, and $\Delta t$ is the pulse duration defined by the
$1/e$ half-width of the field amplitude. The residual phase fluctuation after
comb stabilization is represented by $\varphi(t)$. For simplicity, the CEO
frequency of the comb is assumed to be zero in eq.(1) by assuming that the
carrier frequency $\omega_{0}$ is an integer multiple of the pulse repetition
rate $2\pi/T$. Including a finite CEO frequency in the calculation is
straightforward, and does not change the results. We introduce pulse picking
by assuming that the laser pulses pass through an ideal amplitude modulator
with rectangular-shaped gating which selects every $m$-th pulse. We call $m$
the pulse picking factor. The pulse picker reduces the average power and the
repetition rate by a factor $m$ and hence the power in each of the modes by a
factor $m^{2}$. Under the approximation that the electric field amplitude at
the rising and falling edge of the rectangular-shaped gating is negligible,
the pulse-picked field $E_{\rm p}(t)$ can be written as

$\begin{split}E_{\rm
p}(t)&=A\sum_{k=-\infty}^{\infty}\exp{\left[-\frac{\left(t-m\,k\,T\right)^{2}}{\Delta
t^{2}}\right]}e^{i\left(\omega_{0}t+\varphi\left(t\right)\right)}\\\ &\equiv
E_{\rm p,0}(t)e^{i\varphi\left(t\right)},\end{split}$ (2)

where $E_{\rm p,0}(t)$ is defined as the noiseless component of the pulse-
picked field. The spectrum of the pulse-picked field $E_{\rm p}(t)$ is given
by

$\tilde{E}_{\rm p}(\omega)=\int_{-\infty}^{\infty}E_{\rm p}(t)e^{-i\omega
t}{\rm d}t\approx\tilde{E}_{\rm
p,0}(\omega)+\frac{i}{2\pi}\,\left(\tilde{E}_{\mathrm{p},0}\ast\tilde{\varphi}\right)(\omega),$
(3)

where the approximation is valid for small rms phase noise with a vanishing
mean, i.e. $e^{i\varphi\left(t\right)}\approx 1+i\,\varphi\left(t\right)$. In
Eq. (3), the convolution is defined as
$\left(\tilde{E}_{\mathrm{p},0}\ast\tilde{\varphi}\right)(\omega)=\int_{-\infty}^{\infty}\tilde{E}_{\rm
p,0}(\omega^{\prime})\tilde{\varphi}(\omega-\omega^{\prime}){\rm
d}\omega^{\prime}$, with $\tilde{E}_{\rm p,0}(\omega)$ and
$\tilde{\varphi}(\omega)$ are Fourier transforms of $E_{\rm p,0}(t)$ and
$\varphi(t)$, respectively. The main Fourier components of $\tilde{E}_{\rm
p,0}(\omega)$ and $\tilde{\varphi}(\omega)$ are in the optical and the radio
frequency domain respectively. The spectrum of the noiseless component is
given by:

$\displaystyle\tilde{E}_{\rm p,0}(\omega)=$
$\displaystyle\,A\,\\!\\!\int_{-\infty}^{\infty}\\!\sum_{k=-\infty}^{\infty}\\!\\!\exp{\left[-\frac{\left(t-m\,k\,T\right)^{2}}{\Delta
t^{2}}\right]}e^{i(\omega_{0}-\omega)t}{\rm d}t$ $\displaystyle=$
$\displaystyle\frac{2\pi^{3/2}\Delta t}{mT}A\exp{\left[-\frac{1}{4}\Delta
t^{2}\left(\omega-\omega_{0}\right)^{2}\right]}$
$\displaystyle\times\sum_{n=-\infty}^{\infty}\delta\left(\omega-\frac{2\pi
n}{mT}\right).$ (4)

As expected, the spectrum after pulse picking consists of comb modes with a
mode spacing of $1/mT=f_{\rm rep}/m$, as expressed by the sum over the delta
functions $\delta(\omega-2\pi n/mT)$. The newly created comb modes can be
considered as the sidebands at subharmonics of the original repetition rate
introduced by the amplitude modulation that selects every $m$-th pulse (see
Supplement for a detailed derivation). These additional sidebands fill in the
gaps between the modes of the original comb. The Gaussian spectral envelope is
maintained (in the limit of $\Delta t\ll\tau$), also for the newly added
modes, since the picked pulses are still Gaussian in the time domain.

The spectral intensity can be calculated as $\tilde{I}_{\rm
p,0}(\omega)\propto|\tilde{E}_{\rm p,0}(\omega)|^{2}$ demonstrating that the
comb-mode power scales with $m^{-2}$ as explained above. Note that
$|\tilde{E}_{\rm p,0}(\omega)|^{2}$ includes the squares of the delta
functions, i.e. infinite power density at the frequencies of the modes
compatible with an infinite number of pulses. Using Eq. (4) in the last term
of Eq. (3), we find that the phase noise is convolved into all comb modes,
including the new modes created by pulse picking. The phase noise spectrum
$\tilde{\varphi}(\omega)$ for radio frequencies $|\omega|>2\pi/mT$ is folded
into $|\omega|<2\pi/mT$. In the time domain, the effect can be considered as
an aliasing where the phase noise at $|\omega|>2\pi/mT$ is undersampled by the
pulse train. If the phase noise spectrum $\tilde{\varphi}(\omega)$ is flat and
extends up to the original repetition frequency, reducing the repetition rate
by a factor of $m$ will result in a $m$-fold increase in the power spectral
density (PSD) of the phase noise in the frequency range of $|\omega|<2\pi/mT$.
On the other hand, the integrated phase noise does not change after pulse
picking because the $m$-fold increase in the PSD is canceled by an inverse
reduction of the maximum frequency, i.e. the Nyquist frequency reduces by a
factor $m$. This is consistent with Eqs. (1) and (2) which contain identical
noise terms $e^{i\varphi\left(t\right)}$, and the rms phase noise is expected
to be identical before and after pulse picking.

The model described here shows that the phase noise already present in the
original pulse train affects the frequency comb structure equally before and
after pulse picking, regardless of the repetition rate. A trivial requirement
is that the linewidth of the comb modes prior to pulse picking should be
narrower than the mode spacing after pulse-picking to avoid washing out the
comb structure. In addition, the actual implementation of the pulse picker and
the subsequent amplification should be low-noise to preserve the comb
structure at lower repetition rates. Our work described in sections 3 and 4
aims to experimentally confirm that this is possible.

Pulse picking can be implemented using an acousto-optical modulator (AOM) or a
combination of an electro-optic modulator (EOM) and a polarizer. In Fig. 1, we
show a conceptual scheme of an AOM-based pulse picking. The AOM is driven at a
carrier frequency $f_{\rm AOM}$ (typically tens or hundreds of
$\text{\,}\mathrm{MHz}$) which is amplitude modulated with gating pulses. Here
we assume a rectangular-shaped gate function with repetition rate $f_{\rm
gate}$ (time spacing $T_{\rm gate}=f_{\rm gate}^{-1}$) and a width of
$\tau$.111In the Supplement, we discuss the effect of gating timing jitter.
The RF drive signal for the AOM can be described as

$V_{\rm RF}(t)=\sum_{n=-\infty}^{\infty}A_{\rm RF}\left(t-nT_{\rm
gate}\right)e^{i\omega_{\rm AOM}t},$ (5)

where $A_{\rm RF}$ is the rectangular-shaped gate function, and $\omega_{\rm
AOM}=2\pi f_{\rm AOM}$.
The modulator diffracts every $m$-th pulse to the 1st diffraction order, and
the rest is sent to a beam dump. The Fourier transform of Eq. (5) shows that
the spectrum of the RF signal consists of narrow lines spaced by $f_{\rm
gate}$, similar to an optical frequency comb. The frequency of the RF modes is
given by

$f_{{\rm RF},n^{\prime}}=n^{\prime}\,f_{\rm gate}+f_{\rm AOM},$ (6)

with an integer $n^{\prime}$. The frequency of the optical comb modes
$f_{n,n^{\prime}}$ after pulse picking is given by the sum of the original
comb mode frequencies and the AOM RF frequencies:

$\displaystyle f_{n,n^{\prime}}$ $\displaystyle=n\,f_{\rm rep}+f_{\rm
CEO}+f_{{\rm RF},n^{\prime}}$ (7) $\displaystyle=n\,f_{\rm
rep}+n^{\prime}\,f_{\rm gate}+f_{\rm CEO}+f_{\rm AOM},$

where $f_{\rm CEO}$ is the CEO frequency of the original comb. To obtain an
equidistant comb structure after pulse-picking, the ratio $f_{\rm rep}/f_{\rm
gate}$ must be an integer. In the time domain, this condition translates to
the requirement that $T_{\rm gate}=m\,T$, with $m$ introduced in Eq. (2). The
CEO of the pulse-picked comb remains unchanged after pulse picking if $f_{\rm
AOM}/f_{\rm gate}=q$ is an integer. With an integer $q$, we can define a new
mode-index $\tilde{n}\equiv n^{\prime}+q+n\,m$ to find a compact expression
for the mode frequencies of the pulse-picked comb

$f_{\tilde{n}}=\tilde{n}\,\frac{f_{\rm rep}}{m}+f_{\rm CEO}.$ (8)

In the special case of zero CEO frequency ($f_{\rm CEO}=0$), a pulse train
with a constant CEP can be obtained which finds interesting applications in
attosecond physics [56, 57, 58].

##  3 Experiment

Figure 2: Schematic of the experimental setup. The 40 MHz optical frequency
comb (dotted box) consists of a mode-locked Yb:KYW oscillator, a solid-state
amplifier using Yb:LuAG as gain medium, and an AOM frequency shifter for fast
control of the stabilized comb mode frequency. Slower control acts on PZT-
actuated mirrors of the laser cavity (PZTs). The output of the Yb:KYW
oscillator is centered at $1030\text{\,}\mathrm{nm}$ with a bandwidth of
$14\text{\,}\mathrm{nm}$. The error signal for phase stabilization of one of
the comb modes is obtained from beat notes between that mode and an ultra-
stable cw reference laser emitting at 1033 nm. The cw reference laser is
amplified by a semiconductor optical amplifier (BOA1050P Thorlabs, not shown)
before sending it to the second and third beat note detection units. An AOM-
based pulse picker reduces the comb’s repetition rate to
$($40\text{\,}\mathrm{MHz}$)/m$, where $m$ is the pulse picking factor. After
the pulse picker, the pulses are re-amplified by a second Yb:LuAG amplifier.
The third feedback loop controls a PZT-actuated mirror in the beamline and
reduces the phase noise due to fluctuations in the beam path length.

Figure 2 shows our setup for generating and testing a low repetition rate
optical frequency comb. A home-built Yb:KYW oscillator is mode-locked by soft-
aperture Kerr-lensing and generates a $40\text{\,}\mathrm{MHz}$ pulse train.
The output spectrum is centered at $1030\text{\,}\mathrm{nm}$ and has a FWHM
bandwidth of $14\text{\,}\mathrm{nm}$. The average output power is
$26\text{\,}\mathrm{mW}$. The FWHM temporal pulse duration is measured to be
$89\text{\,}\mathrm{fs}$ using the intensity autocorrelation method
(autocorrelation length $137\text{\,}\mathrm{fs}$), assuming sech2 pulse
shape.

Two PZT-actuated mirrors are installed in the laser cavity to control the
cavity length and are used to stabilize the frequency comb (PZT stands for
lead zirconate titanate). One has a bandwidth of about
$10\text{\,}\mathrm{kHz}$, while a second is used to compensate for slow
drifts. The laser cavity is placed on a vibration-isolated, temperature-
stabilized aluminum baseplate and installed inside an air-tight aluminum
housing [59].

At the laser output, the repetition rate $f_{\rm rep}$ is detected with a fast
photodiode (Thorlabs DET01CFC, not shown in Fig. 2). The laser has an
auxiliary output that is taken from the reflection of an intracavity optic
element with a compromised spectral phase compared with the main output. This
second output has about $60\text{\,}\mathrm{mW}$ of power and is sent into a
heterodyne beat detection setup with a continuous wave (cw) reference laser.
The cw reference laser operates at $1033\text{\,}\mathrm{nm}$ and is
stabilized to an ultra-stable reference cavity. The rms phase noise of the
reference laser was measured to be 10.2 mrad integrated from
$10\text{\,}\mathrm{kHz}$ to $10\text{\,}\mathrm{MHz}$ with respect to the
reference cavity [60].

In the heterodyne beat detection setup, about 100 comb modes are filtered out
around the frequency of the cw reference laser using an interference filter
(Alluxa A4017) and an etalon (LightMachinery OP-6204-M,
$7.3\text{\,}\mathrm{GHz}$ FWHM bandwidth). The beat signal between the
frequency comb and the reference laser is detected using balanced
photodetectors (Koheron PD100B) which suppresses the contribution of classical
amplitude noise [61]. The RF beat signal is filtered to isolate the beat note
between the closest comb modes and the reference laser. Then the signal is
phase-compared to a $10\text{\,}\mathrm{MHz}$ signal generated by a signal
generator (Marconi Instruments 2022C). The signal controls the PZT actuators
of the Yb:KYW laser via a home-built loop filter. This way, one of the comb
modes is phase-stabilized to the cw reference laser.

The frequency comb is amplified by a solid-state double-pass Yb:LuAG amplifier
pumped by a multimode diode laser operating at around
$935\text{\,}\mathrm{nm}$. With an input seed power of
$26\text{\,}\mathrm{mW}$, we obtain $250\text{\,}\mathrm{mW}$ at the output
when the pump power is set to $7.2\text{\,}\mathrm{W}$. Since the gain
bandwidth of Yb:LuAG is about $5\text{\,}\mathrm{nm}$ [62], a significant gain
narrowing effect reduces the bandwidth of the amplifier output to
$2.7\text{\,}\mathrm{nm}$. The peak gain at $1030\text{\,}\mathrm{nm}$ is
approximately $16.6\text{\,}\mathrm{dB}$. The amplifier output is sent to an
AOM (AA Opto Electronic MT110) that is used to stabilize one of the comb modes
in combination with the PZTs in the laser cavity. The first-order diffraction
of the AOM downshifts the entire frequency comb of the laser by about
$110\text{\,}\mathrm{MHz}$. The diffraction efficiency of the AOM is about
$70\text{\,}\mathrm{\char 37\relax}$. The second beat signal between the
frequency comb and the reference cw laser is obtained after the AOM. The beat
signal is compared with the $10\text{\,}\mathrm{MHz}$ frequency reference and
is sent to a loop filter (Vescent Photonics D2-125). The loop filter’s output
is sent to a voltage-controlled oscillator (VCO, Pasternack Enterprises Inc.
PE1V31008) which generates the RF signal that drives the AOM. The control
bandwidth is estimated to be >$100\text{\,}\mathrm{kHz}$, limited by the time
required for the acoustic wave inside the AOM to reach the laser beam.

Figure 3:  Time domain traces of pulses after pulse picking and the second
amplifier ("Output" in Fig. 2). The negative signal is due to ringing. a) At a
repetition rate of $40\text{\,}\mathrm{MHz}$ without pulse picking ($m=1$). b)
At a repetition rate of $4\text{\,}\mathrm{MHz}$ which corresponds to a pulse-
to-pulse interval of $250\text{\,}\mathrm{ns}$ and a pulse picking factor of
$m=10$. The magnified inset shows a trace averaged over 1000 acquisitions
where residual $40\text{\,}\mathrm{MHz}$ pulses are visible with about
$28\text{\,}\mathrm{dB}$ of supression. c) Pulses at a repetition rate of
$400\text{\,}\mathrm{kHz}$, corresponding to $m=10^{2}$
($2.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$ pulse-to-pulse interval). d)
Pulses at $40\text{\,}\mathrm{kHz}$ repetition rate, corresponding to a pulse-
to-pulse interval of $25\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and
$m=10^{3}$. The oscilloscope sampling rate is 40 Gs/s for all traces.

The following pulse picker AOM (AA Opto Electronic MT200-A0.4-1064) is driven
at a carrier frequency of $f_{\rm AOM}=$200\text{\,}\mathrm{MHz}$$. It selects
every $m$-th pulse by amplitude modulating the carrier with a rectangular-
shaped envelope. A rectangular gate signal with a pulse width of
$\tau=$32\text{\,}\mathrm{ns}$$ at a frequency of $f_{\rm gate}=f_{\rm rep}/m$
is used. The gate pulse width is less than $2T$ as required to select
individual pulses. An RF switch (Minicircuit ZASWA-2-50DR+) is used for the
modulation. The diffraction efficiency of the AOM is measured to be
$>$$60\text{\,}\mathrm{\char 37\relax}$, and the pulses remaining in the 0th
order are sent to a beam dump. The gate signal is generated by a delay
generator (Alphanov Tombak) using the repetition rate signal from the Yb:KYW
oscillator as the timing source. The RF carrier signal to drive the AOM is
derived from the 5th harmonic of the repetition rate which is generated in the
detection with a high-bandwidth photodiode. A band-pass filter with a
$3\text{\,}\mathrm{dB}$-bandwidth of $10\text{\,}\mathrm{MHz}$ is used to
isolate the 5th harmonic. In this way, the gate signal, the repetition rate,
and the AOM RF carrier are phase synchronized. The RF switch produces a
modulation with 10-90% rise/fall times of $5\text{\,}\mathrm{ns}$. Measuring
the energy of the picked pulses for different gating delays with respect to
the pulses reveals the time response of the AOM. The
10-$90\text{\,}\mathrm{\char 37\relax}$ rise and fall time of the AOM was
measured to be $7.5\text{\,}\mathrm{ns}$. This is dominated by the time it
takes for the acoustic wave to cross the focused laser beam at the point of
interaction. From the speed of sound within the modulator material ($\rm
TeO_{2}$) of $4200\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$ and the laser beam
diameter of $2w_{0}=$39\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the rise/fall
time is estimated to $6\text{\,}\mathrm{ns}$.

The picked pulses are sent to a second Yb:LuAG amplifier which is similar in
design to the first amplifier. When the pump power is set to
$14\text{\,}\mathrm{W}$, the amplifier output power is
$2.5\text{\,}\mathrm{W}$, $374\text{\,}\mathrm{mW}$, $43\text{\,}\mathrm{mW}$,
and $8\text{\,}\mathrm{mW}$ at a repetition rate of $40\text{\,}\mathrm{MHz}$,
$4\text{\,}\mathrm{MHz}$, $400\text{\,}\mathrm{kHz}$, and
$40\text{\,}\mathrm{kHz}$, respectively.

The output pulse train of the second amplifier is measured by a fast InGaAs
photodiode (Thorlabs DET01CFC, $1.2\text{\,}\mathrm{GHz}$ bandwidth) and a 2.5
GHz oscilloscope (LeCroy WavePro 7Zi). The results are shown in Figure 3 at
repetition rates of $40\text{\,}\mathrm{MHz}$, $4\text{\,}\mathrm{MHz}$,
$400\text{\,}\mathrm{kHz}$, and $40\text{\,}\mathrm{kHz}$.

We find that the pulse-picked beam still contains a tiny fraction of the pulse
train at the original repetition rate of $40\text{\,}\mathrm{MHz}$. This is
not due to incomplete suppression of the RF carrier, but is caused by
scattering within the AOM material. We could suppress the optical power of
this component by $28\text{\,}\mathrm{dB}$ compared to the picked pulses. This
was achieved by carefully adjusting the size and position of an iris
surrounding the pulse-picked laser beam.

A PZT-actuated mirror is introduced in the beamline after the second amplifier
to compensate for possible low-frequency phase fluctuations due to free-space
parts of the setup, e.g. mirrors or breadboard vibrations. In addition, the
phase noise introduced by the pulse picker and amplifier is partially
compensated. A portion of the beam after the PZT actuated mirror is sent to
the third beat detection setup. The beat signal is then phase compared to a
reference at $11.3\text{\,}\mathrm{MHz}$ from an electronic synthesizer
(Marconi Instruments 2022C) and is used as an error signal to drive the PZT
via a home-built loop filter. The in-loop error signal shows a peak at about
$4\text{\,}\mathrm{kHz}$ when the feedback gain is too high, indicating a
control bandwidth of approximately $4\text{\,}\mathrm{kHz}$. This is fast
enough to significantly suppress low-frequency phase noise caused by
mechanical and acoustic vibrations.

##  4 Results and discussion

Figure 4: Beat note spectra acquired from the third beat note detection unit
at different repetition rates of (a) $40\text{\,}\mathrm{MHz}$, (b)
$4\text{\,}\mathrm{MHz}$, (c) $400\text{\,}\mathrm{kHz}$, and (d)
$40\text{\,}\mathrm{kHz}$. A resolution bandwidth (RB) of
$10\text{\,}\mathrm{kHz}$, $1\text{\,}\mathrm{kHz}$,
$100\text{\,}\mathrm{Hz}$, and $10\text{\,}\mathrm{Hz}$ was used,
respectively. In all plots except for panel (d), two peaks which correspond to
$f_{\rm rep}/m$ and $2f_{\mathrm{rep}}/m$ are visible in addition to two peaks
that correspond to the beat frequencies. The repetition rate signal is
strongly suppressed by balanced photodetection. The trace acquired for
$40\text{\,}\mathrm{kHz}$ shows frequencies between the 17th and the 18th
harmonics of $f_{\rm rep}$ to avoid the elevated noise floor of the
measurement setup at low frequencies. The RF spectrum at
$40\text{\,}\mathrm{MHz}$ repetition rate without pulse picking shown in (a)
contains several peaks between the repetition rate peaks and the beat notes.
Most of these are due to the mixing of the strong repetition rate and beat
note signals on the photodiode.

The evaluated beat note spectra are obtained from the third beat note
detection unit (Fig. 2) and shown in Fig. 4. The acquisitions were made at
repetition rates of $40\text{\,}\mathrm{MHz}$ (no pulse picking),
$4\text{\,}\mathrm{MHz}$, $400\text{\,}\mathrm{kHz}$, and
$40\text{\,}\mathrm{kHz}$ with resolution bandwidths of
$10\text{\,}\mathrm{kHz}$, $1\text{\,}\mathrm{kHz}$,
$100\text{\,}\mathrm{Hz}$, and $10\text{\,}\mathrm{Hz}$, respectively. A clear
comb structure is maintained even at the reduced repetition rate of
$40\text{\,}\mathrm{kHz}$, where a narrow beat note peak is still visible with
a signal-to-noise ratio above $30\text{\,}\mathrm{dB}$. For all repetition
rates, the linewidth of the beat signal is limited by the resolution bandwidth
of the spectrum analyzer (Agilent E4445A).

Figure 5: Beat note power level as a function of the pulse picking factor $m$.
The dashed line shows a trend proportional to $m^{-2}$, as expected from the
pulse picking model. The trend line is drawn assuming the
$40\text{\,}\mathrm{MHz}$ point as reference.

Fig. 5 shows the RF power of the beat notes for different pulse picking
factors $m$. The electrical beat note power is proportional to the optical
power contained in a single comb mode and scales with $1/m^{2}$, as expected
from the spectral intensity $I_{\rm p,0}(\omega)$ of our model in Eq. (4).

Figure 6: Power spectral density (PSD) of phase noise at different repetition
rates obtained by the GATOR technique (solid traces). Straight horizontal
lines indicate the average of the noise floor of the measurement (only cw
reference processed in the same way as the beat signal) for each PSD trace.
The rms integrated phase noise is also shown (dashed lines). The endpoint of
the integration is at the Nyquist frequency, i.e. $f_{\rm rep}/2m$. Higher
frequency noise contributions are aliased such that the endpoints of the
dashed lines give the total phase noise. These endpoints are comparable for
all the picked pulse trains. The strong peak at about 20 MHz on the 40 MHz
repetition rate trace is due to a neighboring beat note.

The power spectral density (PSD) of the phase noise is obtained by Fourier
transforming the recorded time trace, assuming that $\varphi(t)$ is small.
Analyzing the heterodyne beat note allows to bring optical phase noise to the
RF domain, where it can be recorded and analyzed. In a sense, the convolution
of Eq.(3) converts the RF noise of $\tilde{\varphi}(\omega)$ into the optical
domain, while heterodying brings it back into the RF domain. The time traces
contain $2.05\times 10^{8}$ samples, and the Blackman window was applied
before performing the Fourier transform. The sideband spectrum of the Fourier
transform trace around the beat frequency gives the PSD of the phase noise
after normalization to the peak amplitude of the beat-note. Since the beat
signal becomes weaker when lowering the repetition rate of the comb, the phase
noise PSD experiences a relative increase in the noise floor. To overcome this
issue, the gated optical noise reduction (GATOR) technique described by
Deschênes et al. in Ref. [63] was used to evaluate the data. In the time
domain, the beat note of a frequency comb with a cw laser can be understood as
the pulse train sampling the cw wave, i.e. the beat note signal is available
only for the duration of the pulses. The GATOR technique strongly suppresses
the noise from the cw reference laser and the detection setup by evaluating
the beat signal only within narrow time windows around the comb pulses. In the
frequency domain, the GATOR method effectively averages the spectrum of the
beat notes between the cw laser and several different comb modes. This
increases the signal-to-noise ratio compared to beat detection with a single
comb mode. We used a software implementation of GATOR by introducing temporal
gate windows around each of the pulses. The width of the gate windows was set
to $40\text{\,}\mathrm{ns}$ for all repetition rates, limited by the bandwidth
of the detector. The resulting phase noise PSDs are shown with solid lines in
Fig. 6. To evaluate the noise floor, we repeated the same measurement without
comb pulses and reperformed the same analysis. Since for traces in which the
pulse separation time is lower than the gate window width the GATOR does not
affect the data, we used for simplicity the ungated data to compute the
$40\text{\,}\mathrm{MHz}$ of Fig. 6.

The PSD of the original frequency comb at $40\text{\,}\mathrm{MHz}$ repetition
rate contains low-frequency phase noise up to about $1\text{\,}\mathrm{kHz}$
due to uncompensated environmental vibrations. Two broad noise peaks at about
$20\text{\,}\mathrm{kHz}$ and $100\text{\,}\mathrm{kHz}$ are servo bumps from
feedback loops controlling the fast PZT actuator in the laser cavity and the
AOM used for phase stabilization, respectively. The servo bump of the PZT for
path length stabilization is around $400\text{\,}\mathrm{Hz}$. The peak at
$17.3\text{\,}\mathrm{MHz}$ in the $40\text{\,}\mathrm{MHz}$ trace is due to
the neighboring beat note. Other spurious signals that we attribute to radio
frequency pickups are visible at frequencies beyond
$100\text{\,}\mathrm{kHz}$. The noise floor is determined by the amplitude
noise of the cw reference laser, the noise of the photodetectors, and the
electronics used in the beat detection setup. It increases for lower
repetition rates due to reduced carrier power of the beat signal. The low-
frequency noise below $1\text{\,}\mathrm{kHz}$ is the same for repetition
rates of $40\text{\,}\mathrm{MHz}$, $4\text{\,}\mathrm{MHz}$, and
$400\text{\,}\mathrm{kHz}$. At a repetition rate of $40\text{\,}\mathrm{kHz}$,
some of the noise structure below 1 kHz appears 5 to 10 dB higher than other
repetition frequencies, which contributes to the increase of the rms
integrated phase noise of about 50 mrad. This might be due to the different
gain settings of the path length stabilization feedback. No other significant
increase in phase noise was observed above the noise floor for the pulse-
picked frequency combs. The FWHM linewidth of the frequency comb is limited by
the resolution bandwidth of the measurement ($10\text{\,}\mathrm{Hz}$) for all
repetition rates investigated.

The rms phase noise integrated from $20\text{\,}\mathrm{Hz}$ to half the
repetition rate $f_{\rm rep}/2m$ was calculated from the PSD spectra. They are
$100\text{\,}\mathrm{mrad}$, $190\text{\,}\mathrm{mrad}$,
$156\text{\,}\mathrm{mrad}$, and $195\text{\,}\mathrm{mrad}$ for repetition
rates of $40\text{\,}\mathrm{MHz}$, $4\text{\,}\mathrm{MHz}$,
$400\text{\,}\mathrm{kHz}$, and $40\text{\,}\mathrm{kHz}$, respectively. The
contributions of the harmonic peaks of $f_{\rm rep}$ and the beat notes are
not representing the phase noise of the comb mode and are therefore excluded
from the integration. For all repetition frequencies, the measurement noise
floor contributes significantly to the rms integrated phase noise. Therefore,
our measurement gives the upper limit of the phase noise of the pulse-picked
comb.

The measurements presented here are based on in-loop signals. In our case,
this is sufficient to determine the noise of the low repetition rate laser
relative to the cw reference laser. The phase noise is not less than the
measurement noise floor at any frequency. Therefore, our in-loop phase noise
measurement adequately reflects the upper limit of the phase noise of our low
repetition rate laser system relative to the cw reference laser. Note that the
path length stabilization used here should be implemented in any application
of the low repetition rate frequency comb.

It is interesting to consider what limits the lowest possible repetition rate.
In the laser system discussed here, the high repetition rate frequency comb is
stabilized before pulse-picking using a method described in Sect. 3. The
amplifier setup after pulse picking is prone to additional noise and more
difficult to actively stabilize. This is because the error signal for path
length stabilization is taken from low-repetition-rate pulses and the
available feedback bandwidth is limited to half the repetition rate. In the
time domain, the servo system needs to receive at least two pulses before it
can know if the phase shift has changed. As a result, the phase noise at
Fourier frequencies significantly lower than half the repetition rate can be
efficiently suppressed, while noise at higher frequencies remains
unsuppressed. Note that feedback systems based on Proportional-Integral-
Differential (PID) controllers require their control bandwidth to be much
larger than the frequency of the noise in order to achieve a small phase delay
that allows high feedback gain at lower frequencies while suppressing
oscillation at higher frequencies [64]. The amazing accuracy of optical
frequency combs comes from their steady-state operation, i.e. a regular pulse
train that finds the same environment for each pulse. Steady-state operation
is disturbed when the pulse-to-pulse interval is longer than the typical time
scale of the phase noise introduced by the environment, and the disturbance
cannot be actively stabilized due to limited feedback bandwidth. This problem
is significant for repetition rates below 10 kHz where acoustic vibrations of
many standard optical components are significant. The lowest possible
repetition rate depends on the type of amplifier and the design of the setup.
Our amplifiers are pumped with a cw pump source and therefore pulse-to-pulse
phase variations are minimized. It also does not rely on the CPA scheme which
can introduce wavelength-dependent phase fluctuations. Our demonstration shows
that the compact and simple solid-state amplifiers used here introduce
sufficiently low phase noise to maintain the comb structure at a repetition
rate as low as 40 kHz.

To increase the feedback bandwidth significantly beyond the repetition rate
after pulse picking, a probe laser with a higher repetition rate or even a
continuous wave laser could be superimposed on the beamline to monitor the
phase variations. However, we expect limitations of this method because the
probe beam would have a different peak intensity and spectrum. Intensity and
spectrum-dependent phase shifts expected in an optical amplifier, pulse
compressor, and nonlinear frequency conversion cannot be easily accounted for
in this way.

##  5 Conclusions and future prospects

In this paper, we have demonstrated a low repetition rate optical frequency
comb based on a Yb:KYW solid-state mode-locked oscillator using an AOM pulse
picker. The repetition rate is adjustable over three orders of magnitude from
$40\text{\,}\mathrm{MHz}$ to $40\text{\,}\mathrm{kHz}$. One of the modes of
the frequency comb is tightly phase-locked to a cavity-stabilized ultra-low
noise cw laser by measuring the heterodyne beat note between them and
providing feedback to the cavity length of the oscillator, an external AOM,
and a PZT actuated mirror in the beamline. We have characterized the phase
noise of the frequency comb at repetition rates of $40\text{\,}\mathrm{MHz}$,
$4\text{\,}\mathrm{MHz}$, $400\text{\,}\mathrm{kHz}$ and
$40\text{\,}\mathrm{kHz}$ with respect to the reference cw laser. The results
confirm that a narrow linewidth comb structure is preserved even after pulse
picking. Using the power spectral density of the phase noise obtained using
the GATOR technique, the integrated rms phase noise was evaluated to be
$195\text{\,}\mathrm{mrad}$ at $40\text{\,}\mathrm{kHz}$ repetition rate. For
the first time to the best of our knowledge, we demonstrate optically-
stabilized low-noise frequency combs at repetition rates as low as a few tens
of kHz.

The pulse energy and average power were $200\text{\,}\mathrm{nJ}$ and
$8\text{\,}\mathrm{mW}$ at a repetition rate of $40\text{\,}\mathrm{kHz}$. A
solid-state amplifier similar to the one used in this work can be added to
increase the pulse energy to a >10 $\text{\,}\mathrm{\SIUnitSymbolMicro J}$
level. Frequency combs with such high pulse energy and moderate average power
are expected to be useful for driving high harmonic generation processes and
generating XUV frequency combs [65]. The low noise and narrow linewidth comb
modes shown here indicate that low repetition rate frequency combs are a
promising option for high-resolution spectroscopy at exotic wavelengths. Dual
comb spectroscopy at XUV wavelengths can be an interesting application of the
low repetition rate XUV frequency combs.

When direct frequency comb spectroscopy is performed using frequency combs
with a low repetition rate, it may be difficult to determine the comb mode
number. If the pulse picking factor is large and the uncertainty of the line
center determination is small enough, the comb mode number can be
unambiguously determined by repeating the measurement for different pulse
picking factors.

The single-pass pulse picking scheme used in this study is inefficient for
reducing the repetition rate because most of the original pulses are unused.
In the future, we plan to perform pulse picking in a femtosecond buildup
cavity to avoid the loss of average power [66, 67]. Intracavity pulse picking
will also serve as a narrow spectral filter to efficiently suppress phase and
amplitude noise at frequencies greater than half the resonance width.

## Funding

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 742247). T.W. Hänsch acknowledges support from the Max-
Planck Foundation.

## Data availability

Data supporting the results presented in this paper are available from the
authors upon reasonable request.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement for supporting content.

## References

  * [1] T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
  * [2] D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
  * [3] S. A. Diddams, K. Vahala, and T. Udem, “Optical frequency combs: Coherently uniting the electromagnetic spectrum,” Science 369, eaay3676 (2020).
  * [4] A. Beyer, L. Maisenbacher, A. Matveev, R. Pohl, K. Khabarova, A. Grinin, T. Lamour, D. C. Yost, T. W. Hänsch, N. Kolachevsky, and T. Udem, “The Rydberg constant and proton size from atomic hydrogen,” Science 358, 79–85 (2017).
  * [5] A. Grinin, A. Matveev, D. C. Yost, L. Maisenbacher, V. Wirthl, R. Pohl, T. W. Hänsch, and T. Udem, “Two-photon frequency comb spectroscopy of atomic hydrogen,” Science 370, 1061–1066 (2020).
  * [6] S. Patra, M. Germann, J.-P. Karr, M. Haidar, L. Hilico, V. I. Korobov, F. M. J. Cozijn, K. S. E. Eikema, W. Ubachs, and J. C. J. Koelemeij, “Proton-electron mass ratio from laser spectroscopy of HD+ at the part-per-trillion level,” Science 369, 1238–1241 (2020).
  * [7] S. A. Diddams, T. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland, “An optical clock based on a single trapped 199Hg+ ion,” Science 293, 825–828 (2001).
  * [8] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506, 71–75 (2014).
  * [9] I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, and H. Katori, “Cryogenic optical lattice clocks,” Nature Photonics 9, 185–189 (2015).
  * [10] 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 Photonics 11, 48–52 (2017).
  * [11] 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 $10^{-18}$,” Physical Review Letters 123, 033201 (2019).
  * [12] T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Absolute optical frequency measurement of the cesium ${\mathit{d}}_{1}$ line with a mode-locked laser,” Phys. Rev. Lett. 82, 3568–3571 (1999).
  * [13] R. Holzwarth, A. Nevsky, M. Zimmermann, T. Udem, T. W. Hänsch, J. von Zanthier, H. Walther, J. C. Knight, W. J. Wadsworth, P. Russell, M. N. Skvortsov, and S. N. Bagayev, “Absolute frequency measurement of iodine lines with a femtosecond optical synthesizer,” Applied Physics B 73, 269–271 (2001).
  * [14] M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, “Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection,” Science 311, 1595–1599 (2006).
  * [15] S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445, 627–630 (2007).
  * [16] C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hänsch, “Frequency comb vernier spectroscopy for broadband, high-resolution, high-sensitivity absorption and dispersion spectra,” Phys. Rev. Lett. 99, 263902 (2007).
  * [17] I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 013902 (2008).
  * [18] N. Picqué and T. W. Hänsch, “Frequency comb spectroscopy,” Nature Photonics 13, 146–157 (2019).
  * [19] T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321, 1335–1337 (2008).
  * [20] C.-H. Li, A. J. Benedick, P. Fendel, A. G. Glenday, F. X. Kärtner, D. F. Phillips, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “A laser frequency comb that enables radial velocity measurements with a precision of 1 cm s-1,” Nature 452, 610 – 612 (2008).
  * [21] B. R. Masters, P. T. C. So, C. Buehler, N. P. Barry, J. D. B. Sutin, W. W. Mantulin, and E. Gratton, “Mitigating thermal mechanical damage potential during two-photon dermal imaging,” Journal of Biomedical Optics 9, 1265 – 1270 (2004).
  * [22] S. Hädrich, M. Krebs, A. Hoffmann, A. Klenke, J. Rothhardt, J. Limpert, and A. Tünnermann, “Exploring new avenues in high repetition rate table-top coherent extreme ultraviolet sources,” Light: Science & Applications 4, e320–e320 (2015).
  * [23] F. Köttig, D. Schade, J. R. Koehler, P. S. J. Russell, and F. Tani, “Efficient single-cycle pulse compression of an ytterbium fiber laser at 10 MHz repetition rate,” Opt. Express 28, 9099–9110 (2020).
  * [24] A. Cingöz, D. C. Yost, T. K. Allison, A. Ruehl, M. E. Fermann, I. Hartl, and J. Ye, “Direct frequency comb spectroscopy in the extreme ultraviolet,” Nature 482, 68–71 (2012).
  * [25] A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nature Photonics 6, 407–407 (2012).
  * [26] A. Ozawa and Y. Kobayashi, “vuv frequency-comb spectroscopy of atomic xenon,” Phys. Rev. A 87, 022507 (2013).
  * [27] M. Herrmann, M. Haas, U. D. Jentschura, F. Kottmann, D. Leibfried, G. Saathoff, C. Gohle, A. Ozawa, V. Batteiger, S. Knünz, N. Kolachevsky, H. A. Schüssler, T. W. Hänsch, and T. Udem, “Feasibility of coherent xuv spectroscopy on the 1S$-$2S transition in singly ionized helium,” Phys. Rev. A 79, 052505 (2009).
  * [28] J. Moreno, F. Schmid, J. Weitenberg, S. G. Karshenboim, T. W. Hänsch, T. Udem, and A. Ozawa, “Toward XUV frequency comb spectroscopy of the 1 S-2 S transition in $\hbox{He}^{+}$,” The European Physical Journal D 77, 67 (2023).
  * [29] L. S. Dreissen, C. Roth, E. L. Gründeman, J. J. Krauth, M. Favier, and K. S. E. Eikema, “High-precision Ramsey-comb spectroscopy based on high-harmonic generation,” Phys. Rev. Lett. 123, 143001 (2019).
  * [30] L. von der Wense and C. Zhang, “Concepts for direct frequency-comb spectroscopy of ${}^{229\text{m}}$Th and an internal-conversion-based solid-state nuclear clock,” The European Physical Journal D 74, 146 (2020).
  * [31] E. Peik, T. Schumm, M. S. Safronova, A. Pálffy, J. Weitenberg, and P. G. Thirolf, “Nuclear clocks for testing fundamental physics,” Quantum Science and Technology 6, 034002 (2021).
  * [32] C. Zhang, P. Li, J. Jiang, L. von der Wense, J. F. Doyle, M. E. Fermann, and J. Ye, “Tunable VUV frequency comb for ${}^{229\text{m}}$Th nuclear spectroscopy,” Opt. Lett. 47, 5591–5594 (2022).
  * [33] S. Kraemer, J. Moens, M. Athanasakis-Kaklamanakis, S. Bara, K. Beeks, P. Chhetri, K. Chrysalidis, A. Claessens, T. E. Cocolios, J. M. Correia, H. D. Witte, R. Ferrer, S. Geldhof, R. Heinke, N. Hosseini, M. Huyse, U. Köster, Y. Kudryavtsev, M. Laatiaoui, R. Lica, G. Magchiels, V. Manea, C. Merckling, L. M. C. Pereira, S. Raeder, T. Schumm, S. Sels, P. G. Thirolf, S. M. Tunhuma, P. V. D. Bergh, P. V. Duppen, A. Vantomme, M. Verlinde, R. Villarreal, and U. Wahl, “Observation of the radiative decay of the ${}^{229}\mathrm{Th}$ nuclear clock isomer,” (2022).
  * [34] C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hänsen, “A frequency comb in the extreme ultraviolet,” Nature 436, 234–237 (2005).
  * [35] R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, “Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity,” Phys. Rev. Lett. 94, 193201 (2005).
  * [36] A. Ozawa, J. Rauschenberger, C. Gohle, M. Herrmann, D. R. Walker, V. Pervak, A. Fernandez, R. Graf, A. Apolonski, R. Holzwarth, F. Krausz, T. W. Hänsch, and T. Udem, “High harmonic frequency combs for high resolution spectroscopy,” Phys. Rev. Lett. 100, 253901 (2008).
  * [37] I. Pupeza, C. Zhang, M. Högner, and J. Ye, “Extreme-ultraviolet frequency combs for precision metrology and attosecond science,” Nature Photonics 15, 175–186 (2021).
  * [38] J. Seres, E. Seres, C. Serrat, E. C. Young, J. S. Speck, and T. Schumm, “All-solid-state vuv frequency comb at 160 nm using high-harmonic generation in nonlinear femtosecond enhancement cavity,” Opt. Express 27, 6618–6628 (2019).
  * [39] H. Carstens, N. Lilienfein, S. Holzberger, C. Jocher, T. Eidam, J. Limpert, A. Tünnermann, J. Weitenberg, D. C. Yost, A. Alghamdi, Z. Alahmed, A. Azzeer, A. Apolonski, E. Fill, F. Krausz, and I. Pupeza, “Megawatt-scale average-power ultrashort pulses in an enhancement cavity,” Opt. Lett. 39, 2595–2598 (2014).
  * [40] S. Holzberger, N. Lilienfein, H. Carstens, T. Saule, M. Högner, F. Lücking, M. Trubetskov, V. Pervak, T. Eidam, J. Limpert, A. Tünnermann, E. Fill, F. Krausz, and I. Pupeza, “Femtosecond enhancement cavities in the nonlinear regime,” Phys. Rev. Lett. 115, 023902 (2015).
  * [41] D. R. Carlson, J. Lee, J. Mongelli, E. M. Wright, and R. J. Jones, “Intracavity ionization and pulse formation in femtosecond enhancement cavities,” Opt. Lett. 36, 2991–2993 (2011).
  * [42] T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
  * [43] H. Wang, L. Amoudry, K. Cassou, R. Chiche, J. Degallaix, K. Dupraz, W. Huang, A. Martens, C. Michel, H. Monard, D. Nutarelli, L. Pinard, C. Tang, L. Yan, and F. Zomer, “Prior-damage dynamics in a high-finesse optical enhancement cavity,” Appl. Opt. 59, 10995–11002 (2020).
  * [44] A. Schliesser, C. Gohle, T. Udem, and T. W. Hänsch, “Complete characterization of a broadband high-finesse cavity using an optical frequency comb,” Opt. Express 14, 5975–5983 (2006).
  * [45] N. Lilienfein, C. Hofer, S. Holzberger, C. Matzer, P. Zimmermann, M. Trubetskov, V. Pervak, and I. Pupeza, “Enhancement cavities for few-cycle pulses,” Opt. Lett. 42, 271–274 (2017).
  * [46] F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
  * [47] G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. D. Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006).
  * [48] O. Lundh, J. Lim, C. Rechatin, L. Ammoura, A. Ben-Ismaïl, X. Davoine, G. Gallot, J.-P. Goddet, E. Lefebvre, V. Malka, and J. Faure, “Few femtosecond, few kiloampere electron bunch produced by a laser–plasma accelerator,” Nature Physics 7, 219–222 (2011).
  * [49] A. J. Gonsalves, K. Nakamura, J. Daniels, C. Benedetti, C. Pieronek, T. C. H. de Raadt, S. Steinke, J. H. Bin, S. S. Bulanov, J. van Tilborg, C. G. R. Geddes, C. B. Schroeder, C. Tóth, E. Esarey, K. Swanson, L. Fan-Chiang, G. Bagdasarov, N. Bobrova, V. Gasilov, G. Korn, P. Sasorov, and W. P. Leemans, “Petawatt laser guiding and electron beam acceleration to 8 GeV in a laser-heated capillary discharge waveguide,” Phys. Rev. Lett. 122, 084801 (2019).
  * [50] D. Polli, P. Altoè, O. Weingart, K. M. Spillane, C. Manzoni, D. Brida, G. Tomasello, G. Orlandi, P. Kukura, R. A. Mathies, M. Garavelli, and G. Cerullo, “Conical intersection dynamics of the primary photoisomerization event in vision,” Nature 467, 440 – 443 (2010).
  * [51] A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611–615 (2003).
  * [52] T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28, 947–949 (2003).
  * [53] S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nature Photonics 5, 475–479 (2011).
  * [54] T. S. Khwaja and T. H. Yoon, “Phase-stable and selectable repetition-rate division of an optical frequency comb,” Physical Review A 102, 1–8 (2020).
  * [55] T. Saule, S. Holzberger, O. De Vries, M. Plötner, J. Limpert, A. Tünnermann, and I. Pupeza, “Phase-stable, multi-µJ femtosecond pulses from a repetition-rate tunable Ti:Sa-oscillator-seeded Yb-fiber amplifier,” Applied Physics B: Lasers and Optics 123, 1–7 (2017).
  * [56] N. Ishii, K. Kaneshima, K. Kitano, T. Kanai, S. Watanabe, and J. Itatani, “Carrier-envelope phase-dependent high harmonic generation in the water window using few-cycle infrared pulses,” Nature Communications 5, 3331 (2014).
  * [57] O. de Vries, T. Saule, M. Plötner, F. Lücking, T. Eidam, A. Hoffmann, A. Klenke, S. Hädrich, J. Limpert, S. Holzberger, T. Schreiber, R. Eberhardt, I. Pupeza, and A. Tünnermann, “Acousto-optic pulse picking scheme with carrier-frequency-to-pulse-repetition-rate synchronization,” Optics Express 23, 19586 (2015).
  * [58] F. Calegari, G. Sansone, S. Stagira, C. Vozzi, and M. Nisoli, “Advances in attosecond science,” Journal of Physics B: Atomic, Molecular and Optical Physics 49, 062001 (2016).
  * [59] F. Schmid, J. Moreno, J. Weitenberg, P. Russbüldt, T. W. Hänsch, T. Udem, and A. Ozawa, “An ultra-stable high-power optical frequency comb,” Manuscript submitted for publication (2023).
  * [60] F. Schmid, J. Weitenberg, T. W. Hänsch, T. Udem, and A. Ozawa, “Simple phase noise measurement scheme for cavity-stabilized laser systems,” Opt. Lett. 44, 2709–2712 (2019).
  * [61] H. R. Carleton and W. T. Maloney, “A balanced optical heterodyne detector,” Appl. Opt. 7, 1241–1243 (1968).
  * [62] M. Siebold, M. Loeser, F. Roeser, M. Seltmann, G. Harzendorf, I. Tsybin, S. Linke, S. Banerjee, P. D. Mason, P. J. Phillips, K. Ertel, J. C. Collier, and U. Schramm, “High-energy, ceramic-disk Yb:LuAG laser amplifier,” Opt. Express 20, 21992–22000 (2012).
  * [63] J. D. Deschênes and J. Genest, “Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot-noise limit of a single comb mode,” Physical Review A - Atomic, Molecular, and Optical Physics 87, 1–6 (2013).
  * [64] J. Bechhoefer, “Feedback for physicists: A tutorial essay on control,” Rev. Mod. Phys. 77, 783–836 (2005).
  * [65] A. Schönberg, H. S. Salman, A. Tajalli, S. Kumar, I. Hartl, and C. M. Heyl, “Below-threshold harmonic generation in gas-jets for th-229 nuclear spectroscopy,” Opt. Express 31, 12880–12893 (2023).
  * [66] Y. Vidne, M. Rosenbluh, and T. W. Hansch, “Pulse picking by phase-coherent additive pulse generation in an external cavity,” Opt. Lett. 28, 2396–2398 (2003).
  * [67] R. J. Jones and J. Ye, “High-repetition-rate coherent femtosecond pulse amplification with an external passive optical cavity,” Opt. Lett. 29, 2812–2814 (2004).

## Supplement

In Eq.(2) we assumed that the gating pulses are much longer than the laser
pulses. In this supplement, we give a more rigorous derivation of the
frequency comb structure after pulse picking without making this assumption.
We also consider a possible timing jitter of the gating pulses. We show that
this jitter has very little effect on the picked frequency comb, as expected
from the time domain description.

We assume a rectangular shaped gating pulse

$r(t)=\left\\{\begin{array}[]{ll}1&-\tau/2<t<+\tau/2\\\
0&\,\textrm{otherwise}\\\ \end{array}\right.$ (S9)

with a temporal full width $\tau$. A train of gating pulses separated in time
by $mT$ is then given by

$g(t)=\sum_{q=-\infty}^{\infty}r\left(t-qmT-\tau_{q}\right),$ (S10)

where we added a timing jitter of $\tau_{q}$ to the $q$-th gating pulse. The
Fourier transform of the gating pulse train is:

$\tilde{g}(\omega)=\tau\,\mathrm{sinc}\\!\left(\frac{\omega\tau}{2}\right)\,\sum_{q=-\infty}^{\infty}\,e^{-i\omega(qmT+\tau_{q})}.$
(S11)

The sinc function is defined as $\mathrm{sinc}(x)=\sin(x)/x$. The Fourier
transform of the original pulse train Eq.(1) is:

$\tilde{E}(\omega)=\sqrt{\pi}\Delta t\,A\,\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\omega-\omega_{0}\right)^{2}\right)\,\sum_{k=-\infty}^{\infty}e^{i(\omega-\omega_{0})kT}.$
(S12)

We have assumed that the carrier envelope frequency vanishes, i.e.
$\omega_{0}kT$ is an integer multiple of $2\pi$, so the $\omega_{0}$ term in
the sum can be dropped. The phase fluctuation $\varphi(t)$ in Eq.(1) is
assumed to be zero. With the phase fluctuation $\varphi(t)$ included, the
following discussion still applies to the noise less component of the spectrum
$\tilde{E}_{\rm p,0}(\omega)$. The Fourier transform of the picked pulse train
is then given by the following convolution:

$\displaystyle\tilde{E}_{\rm p}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left(\tilde{E}\ast\tilde{g}\right)(\omega)$ (S13)
$\displaystyle\\!=\\!$ $\displaystyle\\!\frac{\sqrt{\pi}\tau\Delta
t}{T}A\sum_{n=-\infty}^{\infty}\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\frac{2\pi
n}{T}-\omega_{0}\right)^{2}\right)\,\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)$
$\displaystyle\times\,\sum_{q=-\infty}^{\infty}e^{-i\omega qmT-i(\omega-2\pi
n/T)\tau_{q}}.$

Here, the Fourier representation of the Dirac comb

$\sum_{n=-\infty}^{\infty}\delta(x-ny)=\frac{1}{y}\sum_{p=-\infty}^{\infty}e^{i2\pi
px/y},$ (S14)

was used to replace the sum over $k$ with a sum of
$\delta(\omega^{\prime}-2\pi n/T)$, which allows to evaluate the convolution
integral.

We will now discuss the two limiting cases that are of interest. In both cases
we assume that the pulse duration is much smaller than the gating width
($\Delta t\ll\tau$). This is a reasonable assumption for picosecond and
femtosecond pulse trains.

  1. 1.

Small timing jitter: $\tau_{q}\ll\tau$

Using the approximation of

$\displaystyle\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)e^{-i(\omega-\frac{2\pi n}{T})\tau_{q}}$
$\displaystyle=$ $\displaystyle\frac{i}{\left(\omega-\frac{2\pi
n}{T}\right)\tau}\left(e^{-i\left(\omega-\frac{2\pi
n}{T}\right)\left(\tau_{q}+\frac{\tau}{2}\right)}-e^{-i\left(\omega-\frac{2\pi
n}{T}\right)\left(\tau_{q}-\frac{\tau}{2}\right)}\right)$ (S15)
$\displaystyle\approx$
$\displaystyle\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right),$

we obtain

$\displaystyle\tilde{E}_{\rm p}(\omega)\\!$ $\displaystyle\\!\approx\\!$
$\displaystyle\\!\frac{\sqrt{\pi}\tau\Delta
t}{T}A\sum_{n=-\infty}^{\infty}\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\frac{2\pi
n}{T}-\omega_{0}\right)^{2}\right)\,\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)$ (S16)
$\displaystyle\times\,\frac{2\pi}{mT}\,\sum_{n^{\prime}=-\infty}^{\infty}\delta\left(\omega-\frac{2\pi
n^{\prime}}{mT}\right),$

where we have used the Fourier representation of the Dirac comb again. The
series of $\delta$-functions represents the dense frequency comb modes, and
the sinc function represents the spectral intensity of the modes added by the
amplitude modulation of the pulse picker. The gating pulse jitter $\tau_{q}$
disappears in this approximation. Exploting the finite width of the sinc-
function we can approximate the argument of the Gaussian envelope as:

$\displaystyle\tilde{E}_{\rm p}(\omega)\\!$ $\displaystyle\\!\approx\\!$
$\displaystyle\\!\frac{\sqrt{\pi}\tau\Delta t}{T}A\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\omega-\omega_{0}\right)^{2}\right)\sum_{n=-\infty}^{\infty}\,\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)$ (S17)
$\displaystyle\times\,\frac{2\pi}{mT}\,\sum_{n^{\prime}=-\infty}^{\infty}\delta\left(\omega-\frac{2\pi
n^{\prime}}{mT}\right)$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2\pi^{3/2}\Delta t}{mT}A\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\omega-\omega_{0}\right)^{2}\right)\sum_{n^{\prime}=-\infty}^{\infty}\delta\left(\omega-\frac{2\pi
n^{\prime}}{mT}\right)$ $\displaystyle=$ $\displaystyle\mathrm{Eq.(4)}$

The first line is a very good approximation as long as $\Delta t\ll\tau$,
which is the regime we are interested in (optical pulse duration much shorter
than the width of the gating pulses). For the sum over the sinc-function we
used

$\sum_{n=-\infty}^{\infty}\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)=\frac{T}{\tau},$ (S19)

which can be proven using the Poisson summation formula. Hence the various
sinc contributions to the spectral envelope add up to a constant. Remarkably,
the spectral envelope of the picked frequency comb retains the Gaussian shape
of the original comb. This is expected from the approximate time domain
treatment in the main text.

Instead of taking the limit of a narrow sinc function, we can consider the
opposite case of a narrow Gaussian spectral envelope. In this case, we would
replace $\omega$ in the sinc function by $\omega_{0}$, i.e., by the central
value of the (narrow) Gaussian spectral envelope, such that the gating pulses
generate the spectral envelope of the comb. We mention this limit only for the
sake of completeness, although it is by no means in an interesting regime for
the topic of this work: The associated long pulse durations (nanoseconds) do
not effectively drive nonlinear processes.

  2. 2.

Non-negligible timing jitter:

To describe the noise added by the timing jitter $\tau_{q}$ we consider the
frequencies that contribute separately by setting

$\tau_{q}=\tau_{0}\sin(\omega_{\mathrm{noise}}qmT).$ (S20)

Using the Jacobi-Anger expansion

$e^{iz\sin(\theta)}=\sum_{k=-\infty}^{\infty}J_{k}(z)\,e^{ik\theta},$ (S21)

we obtain:

$\displaystyle\tilde{E}_{\rm p}(\omega)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{\sqrt{\pi}\tau\Delta
t}{T}A\sum_{n=-\infty}^{\infty}\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\frac{2\pi
n}{T}-\omega_{0}\right)^{2}\right)\,\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)$ (S22)
$\displaystyle\times\,\sum_{q,k=-\infty}^{\infty}J_{k}\left(\left(\omega-\frac{2\pi
n}{T}\right)\tau_{0}\right)\,\,e^{-iq(\omega-k\omega_{\mathrm{noise}})mT}.$
$\displaystyle\\!\approx\\!$ $\displaystyle\\!\frac{\sqrt{\pi}\tau\Delta
t}{T}A\exp\\!\left(-\frac{\Delta
t^{2}}{4}\left(\omega-\omega_{0}\right)^{2}\right)\,$
$\displaystyle\times\,\frac{2\pi}{mT}\sum_{n,n^{\prime},k=-\infty}^{\infty}\\!\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)J_{k}\left(\left(\omega-\frac{2\pi
n}{T}\right)\tau_{0}\right)\,\,\delta\left(\omega-\frac{2\pi
n^{\prime}}{mT}-k\omega_{\mathrm{noise}}\right).$

Again, we have assumed that the Gaussian spectral envelope is much broader
than the sinc-function, i.e. $\Delta t\ll\tau$. In the time domain this means
that the optical pulses have no tail that the gating pulses could cut away. We
also used Eq.(S14) again. Additional sidebands at harmonics of the noise
frequency $k\omega_{\mathrm{noise}}$ appear in the argument of the $\delta$
function. If we consider a particular noise sideband with indices $n^{\prime}$
and $k$, the remaining sum over $n$ can be evaluated using the Poisson
summation formula:

$\sum_{n=-\infty}^{\infty}\\!\mathrm{sinc}\\!\left(\left(\omega-\frac{2\pi
n}{T}\right)\frac{\tau}{2}\right)J_{k}\left(\left(\omega-\frac{2\pi
n}{T}\right)\tau_{0}\right)=0,\\\ \ $ (S23)

which is valid for $k\neq 0$ when $\tau_{0}<\tau/2$ and $\tau_{0}+\tau/2<T$.
These conditions mean that the edges of the gating pulses do not touch the
optical pulses. The timing jitter noise is completely canceled in this case.
This noise immunity is what is readily expected from the time domain picture,
and it is reassuring to see it in the frequency domain as well, albeit in a
much more complicated way.