# Terahertz Voltage-controlled Oscillator from a Kerr-Induced Synchronized
Soliton Microcomb

Usman A. Javid  Joint Quantum Institute, NIST/University of Maryland, College
Park, USA  Microsystems and Nanotechnology Division, National Institute of
Standards and Technology, Gaithersburg, USA  Michal Chojnacky  Joint Quantum
Institute, NIST/University of Maryland, College Park, USA  Sensor Science
Division, National Institute of Standards and Technology, Gaithersburg, USA
Kartik Srinivasan  Joint Quantum Institute, NIST/University of Maryland,
College Park, USA  Microsystems and Nanotechnology Division, National
Institute of Standards and Technology, Gaithersburg, USA  Corresponding
<EMAIL_ADDRESS><EMAIL_ADDRESS>Grégory Moille
Joint Quantum Institute, NIST/University of Maryland, College Park, USA
Microsystems and Nanotechnology Division, National Institute of Standards and
Technology, Gaithersburg, USA  Corresponding
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

The generation of controlled and arbitrarily tunable terahertz radiation,
essential for many applications, has proven challenging due to the complexity
of experimental setups and fabrication techniques. We introduce a new strategy
involving control over a terahertz repetition rate integrated frequency comb,
using Kerr-induced synchronization, that results in a terahertz-voltage-
controlled oscillator. By modulating the reference laser, we can transfer any
microwave waveform onto the microcomb repetition rate via a linear transfer
function based on optical frequency division. The resulting frequency comb
with a terahertz carrier can be created using integrated components, with a
bandwidth constrained only by the synchronization bandwidth and high coherence
resulting from the low-noise soliton microcomb in the Kerr-induced
synchronized state.

††journal: ol

##  1 Introduction

Figure 1: Kerr induced synchronization for dynamic comb repetition rate tuning
for terahertz generation – a Kerr-induced synchronization (KIS) starts with a
dissipative Kerr soliton (DKS) traveling in a microring resonator, which is
generated by a main pump laser. A reference laser is sent to the same
microring resonator, whose circumference is set to produce a free spectral
range near 1 THz. When this reference laser is sufficiently close to one comb
tooth (within the KIS bandwidth), it will capture this comb tooth. The comb,
already pinned by the main pump as a comb tooth, then becomes double-pinned.
Since the KIS bandwidth is non-null, the reference can be tuned to change the
comb repetition rate or can be fixed to lock the comb repetition rate. The
repetition rate of the comb can be detected by sending two adjacent comb lines
to a photomixer, creating a low-noise terahertz signal. b Thanks to the large
KIS bandwidth, sending a frequency-modulated reference laser into the
microring resonator enables the repetition rate to be modulated at the same
rate. As such, the terahertz signal detected by the photomixer from the
beating of the now modulated two adjacent comb lines will present spectral
components around the nominal repetition rate frequency. This process creates
a coherent, low-noise terahertz comb that harnesses the intrinsic optical
frequency division property of the frequency comb.

Technological developments in the terahertz spectral region (0.1 THz to 10
THz) have recently expanded rapidly [1, 2, 3, 4], bridging the so-called
“terahertz gap”. This range is of particular interest for a wide variety of
applications, including molecular spectroscopy [5, 6], remote sensing [7],
medical imaging [8], and climate monitoring [9, 10], while enabling the new
generation of 6G wireless networks [11, 12] to meet the ever-increasing demand
for ultra-fast wireless communication. Terahertz radiation may be generated in
two distinct manners: electronically, by the up-conversion of microwave
frequencies through electro-optic multiplication [13], or optically, by mixing
two separated optical tones detected by an ultra-fast photomixer [14] or
through the use of nonlinear crystals via optical rectification [15]. The
noise associated with the radio frequency and optical waves directly impacts
the quality of the generated terahertz signal, which can present challenges
when low-noise operation is required. In such applications, an alternative
approach involves harnessing the intrinsic low-noise properties of an optical
frequency comb, which creates a phase-coherent link between the optical and
microwave domains [16]. A microwave signal can be extracted from
photodetection of the comb repetition rate (or a harmonic of it), with low-
noise performance realized through optical frequency division (OFD), where the
repetition rate frequency noise is given by the mutual noise of two optically
locked comb teeth divided by the integer comb tooth spacing ($\Delta\mu$)
between them. OFD has been established as an effective method for ultra-low
noise synthesis of microwave signals [17, 18, 19, 20]. To extend OFD to the
terahertz regime, scaling down the comb size is critical, as the comb
repetition rate is inversely proportional to the cavity size. Such downscaling
is feasible with a chip-integrated microring resonator made of $\chi^{(3)}$
nonlinear material, which supports dissipative Kerr soliton (DKS) generation
[21]. The traveling DKS in the microring is periodically extracted at every
round-trip, creating a pulse train output corresponding to a low-noise
microcomb. When the microcomb’s repetition rate is detected, it enables
successful transduction from the optical domain to the millimeter wave [22,
23, 24]. As such soliton microcombs can also be battery operated [25], the
size, weight, and power (SWaP) of this terahertz generation approach can be
compelling.

While terahertz synthesis via microcombs presents a promising pathway, on-chip
control and tuning of the terahertz repetition rate of the DKS remain
challenging. Approaches that rely on changing the properties of the microring
resonator, through thermo-refractive [26, 27], electro-optic [28], or
piezoelectric [29] elements, enable limited repetition rate tuning while
mostly impacting the cavity resonance frequencies, with the latter forcing the
main pump to be re-tuned for maintaining the soliton state. In contrast, a new
approach has recently been introduced, termed Kerr-induced synchronization
(KIS) [30], which enables all-optical tuning of the DKS repetition rate. Here,
a secondary reference laser is injected into the microring and synchronizes
the cavity soliton. Under the correct conditions, such synchronization results
in the capture by the reference laser of the nearest comb tooth, with tuning
of the reference laser within the KIS bandwidth resulting in direct control of
the DKS repetition rate. Thanks to the significant KIS bandwidth (up to a few
gigahertz), this enables DKS repetition rate modulation. This approach solely
modifies the DKS, without affecting the resonator, making high-speed
modulation while maintaining the low-noise DKS state achievable. In this work,
we demonstrate that KIS can be used to create a terahertz voltage-controlled
oscillator through voltage-controlled modulation of the reference laser
frequency. We show that the modulation of the reference laser frequency is
directly transduced onto the DKS repetition rate via OFD, enabling the
generation of low noise arbitrary waveforms centered at around 1 THz. Our
results suggest that terahertz generation via KIS of DKS microcombs can be
used in applications such as spectroscopy, ranging, and imaging.

Figure 2: 1 THz octave spanning frequency with sideband Kerr-induced
synchronization – The microring used (inset left) is made of Si3N4 with a ring
width of 860 nm and an outer ring radius of 23 µm embedded in silica. The
counter-propagating and cross-polarized cooler pump, relative to the main pump
at 283 THz, helps thermally stabilize the resonator for adiabatic access to
the DKS state, while minimizing its nonlinear interaction with the soliton
(teal spectra). The comb exhibits a DW at 191.5 THz (blue spectrum) which we
leverage for energy-efficient Kerr-induced synchronization (green spectra).
Contrary to previous work, we modulate the reference laser at
$\Omega_{\mathrm{mod}}=18$ GHz and offset it from the comb tooth by the same
amount for the lower frequency sideband to achieve synchronization (see left
inset). This approach allows the reference to remain fixed and potentially
locked to a stable cavity for low-noise operation, while enabling sideband
frequency modulation with minimal residual amplitude modulation. Once the
sideband is synchronized, the same repetition rate modulation scheme can be
applied. A similar phenomenon can be observed for the short DW at 389 THz,
which is amplified through self-balancing [31].

##  2 Concept

First, it is important to recall the fundamental properties of KIS and
understand how it can be harnessed to create tunable terahertz oscillators
[fig. 1a]. A continuous wave (CW) main pump creates a DKS inside a ring
resonator cavity. By also injecting a weak reference pump laser, the
$\chi^{(3)}$ nonlinearity of the resonator enables the DKS and the intracavity
reference laser to bind their group velocities through cross-phase modulation
(XPM) [32, 33]. When the reference laser frequency is close enough to a comb
tooth that is $\mu_{s}$ modes away from the main pump, the XPM is sufficient
for the soliton and reference to also synchronize their phase velocities [30].
This synchronization corresponds to the closest comb tooth snapping onto the
reference laser, making them indistinguishable. Since the comb is now doubly
pinned by the main pump and the reference pump – fixing the group velocity of
the traveling DKS in the resonator – the repetition rate noise of the comb is
converted from the pump lasers’ noise via OFD [30, 34]. These two pumps can be
optically locked to cavity references, enabling an ultra-low noise repetition
rate. Since the DKS round trip time can be made to be in the picosecond
regime, a high quality terahertz signal can be generated by detecting the
beating of two adjacent comb lines. One important property of KIS is its
bandwidth $\Delta\Omega$, determined by the KIS coupling energy, which is the
geometric mean of the intracavity energy of the reference and synchronized
comb tooth, respectively [30]. Therefore, the synchronization happens within a
frequency window around the comb tooth, for which the reference will capture
this tooth. Hence, the soliton must adapt its group velocity – thus the comb
repetition rate – for the comb tooth captured by the reference to still be on
a fixed frequency grid. Along with the potential for better phase matching to
improve dispersive wave (DW) generation [31], KIS enables direct frequency
modulation of the reference pump to be coherently transferred onto the comb
repetition rate [fig. 1b]. In this regime, the DKS acts as a transducer where
the up-shifted microwave signal at $\Omega_{\mathrm{RF}}$ is imprinted on the
reference frequency modulation, via a voltage controlled oscillator (VCO)
combined with an electro-optic modulator, which is then frequency down-shifted
to the terahertz domain. The system therefore becomes analogous to a frequency
modulator, where a frequency comb in the terahertz domain can be generated,
where each comb line $n$ around the repetition rate carrier frequency will be
separated by $\Omega_{\mathrm{RF}}$ and will exhibit an amplitude
$A_{n}=J_{n}(\frac{\delta\Omega}{\Omega_{\mathrm{RF}}})$, with $J_{n}$ being
the $n$th Bessel function. The modulation amplitude $\delta\Omega$ of the
repetition rate $\omega_{\mathrm{rep}}$, which is directly linked to the
modulation amplitude of the reference by a factor $\mu_{s}$, leads to the
terahertz comb and can have a maximum value of half of the KIS bandwidth
$\Delta\Omega$ divided by the OFD factor $\Delta\mu$, and will define the
maximum possible excursion of the modulation, while the inverse of the cavity
photon lifetime will determine the maximum speed at which the modulation
$\Omega_{\mathrm{RF}}$ can be applied.

##  3 Results

###  3.1 Optical characterization of the microcomb

To experimentally study this effect, we use a $H=670$ nm thick silicon nitride
(Si3N4) microring resonator with a ring width $RW=860$ nm and an outer ring
radius of $RR=23$ µm, where the DKS round-trip time is around 1 ps, hence
creating a $\omega_{\mathrm{rep}}/2\pi\approx 1$ THz frequency comb repetition
rate. The dispersion of the fundamental transverse electric mode of this ring
is engineered such that for a pump in the 283 THz range (1061 nm range), the
dispersion is anomalous with additional higher order dispersion terms allowing
for phase matching around 191.5 THz (1566 nm) and 389 THz (770 nm), for which
the DKS can emit DWs [fig. 2], creating an octave spanning comb [30, 35]. To
generate the soliton, we use a counterpropagating cooler pump in the
transverse-magnetic polarization to minimize the nonlinear coupling with the
main pump while enabling temperature stabilization of the resonator for
adiabatic DKS access with 140 mW of on-chip main pump power [36, 37]. The
short DW is not properly extracted in the output bus waveguide since the
coupling dispersion using a $L_{c}=17$ µm pulley-like coupler [38] has been
engineered mostly for balancing the critical coupling regime at frequencies
between the main pump down to the lower frequency DW at 191.5 THz. To trigger
KIS, we send a weak reference laser pump close to the maximum of the low
frequency DW. The advantage of this choice is two fold: the comb tooth being
per definition close to a cavity mode resonance, the reference intracavity
field can be maximized, while the resonance also enhances the comb tooth
energy. Consequently, the KIS coupling energy
$E_{\mathrm{kis}}=\sqrt{\frac{\kappa_{\mathrm{ext}}}{\kappa^{2}}P_{\mathrm{ref}}E_{\mu
s}}/E_{\mathrm{dks}}$ is maximized [30, 39], with
$\kappa_{\mathrm{ext}}=2\pi\times 100$ MHz, $\kappa=2\kappa_{\mathrm{ext}}$
the total loss rate ($\kappa_{\mathrm{ext}}$ is the waveguide coupling rate),
$P_{\mathrm{ref}}$ the on-chip reference pump power, $E_{\mu s}\approx
1.2\times 10^{-4}$ J the energy of the comb tooth at the synchronized mode
(whose index $\mu_{s}=-91$ is with respect to the main pump), and
$E_{\mathrm{dks}}\approx 10^{-2}$ J the intracavity energy of the DKS obtained
from the Lugiato-Lefever equation [40]. The KIS bandwidth being directly
related to the KIS energy such that
$\Delta\Omega=4\mu_{s}\omega_{\mathrm{rep}}E_{\mathrm{kis}}$, working at the
DW enables us to obtain a large KIS bandwidth of 1.2 GHz with
$P_{\mathrm{ref}}\approx 200$ µW sufficient to obtain a large enough frequency
modulation contrast. Although KIS using direct modulation of the laser
frequency (through current modulation) is possible, the targeted excursion
range of a few gigahertz will also induce amplitude modulation. Though KIS
should be robust against resulting variations in the reference energy,
residual amplitude modulation will pose a challenge when locking to a
reference cavity. Instead, here we propose to use an electro-optic modulator
to create modulation sidebands at a large frequency separation
$\Omega_{\mathrm{mod}}=\pm 18$ GHz from the reference laser. By positioning
the reference laser at $\approx\Omega_{\mathrm{mod}}$, one of the modulation
sidebands can synchronize the soliton [fig. 2]. Such a large frequency
separation between the reference and its sideband is necessary to prevent
other types of synchronization dynamics that arise if $\Omega_{\mathrm{mod}}$
is comparable to $\Delta\Omega$ [39]; instead, setting
$\Omega_{\mathrm{mod}}\gg\Delta\Omega$ allows us to treat the reference and
its sideband as independent oscillators in the context of DKS synchronization.
This approach allows for direct radiofrequency control of the sideband
frequency that synchronizes DKS, while maintaining a fixed reference laser
that can be locked to a cavity reference to reduce its linewidth and the
resulting DKS repetition rate noise via OFD. For the remainder of the
manuscript, we will refer to this reference sideband simply as the reference,
given that the reference carrier from which it is derived will remain
unchanging and its contribution to the nonlinear dynamics of the system can be
neglected. We adjust the reference laser power for the synchronization
sideband to have an on-chip power of $200\ \mu W$.

Figure 3:  Instantaneous repetition rate modulation characterization – a
Experimental setup schematic. The reference laser is modulated at
$\Omega_{\mathrm{mod}}+\Omega_{\mathrm{RF}}$ using an arbitrary waveform
generator (AWG) that drives a voltage control oscillator (VCO), with the
output fed to an electro-optic phase modulator to create and control the
sideband reference that synchronizes the comb. The repetition rate is detected
using an electro-optic comb (EOcomb) apparatus, with the output beat note
detected by an avalanche photodiode (APD) and measured with a fast
oscilloscope. b The sinusoidal AWG signal (top) at 100 kHz is transferred onto
the reference through electro-optic modulation, and is in turn transduced onto
the $\approx$999.35901 GHz repetition rate carrier through KIS (bottom). There
are $\Delta\mu=-91$ comb teeth between the main pump and the reference, so the
reference frequency excursion is divided onto the repetition rate by -91. The
red lines represent the AWG average (top), which is optical frequency divided
(OFD’d) onto the repetition rate (bottom), highlighting the linear transfer of
the KIS transduction. c Similar to the sinusoidal AWG in (b), except with a
triangular signal. The resolution of the sharp edge, which contains higher
Fourier frequencies, highlights the large modulation bandwidth of the KIS
transduction and its linearity.  Figure 4: Terahertz frequency-comb
generation. a Spectrum of the modulated DKS repetition rate (measured with the
EOcomb apparatus), hence the terahertz comb, through direct actuation via KIS
with a modulated reference, for a modulation amplitude of
$\delta\Omega=\mu_{s}^{-1}\times 341.4$ MHz, with $\mu_{s}=-91$ the mode at
which KIS happens. The $y$ axis corresponds to the modulation frequency
$\Omega_{\mathrm{RF}}$ ranging from 10 kHz to 4 MHz, hence impacting both the
modulation excursion and the spacing between the terahertz comb teeth and
their central carrier. The DKS repetition rate, hence the terahertz comb
carrier, is at 999.40016 GHz ±10 kHz, uncertainty from the
$\omega_{\mathrm{rep}}$ linewidth with free runing pumps [30], while the
different spectra are recorded at a bandwidth of 100 Hz. b Individual
terahertz comb spectra for modulation frequencies $\Omega_{\mathrm{RF}}=100$
kHz (iv), $300$ kHz (iii), $1$ MHz (ii), and $3$ MHz (i) as indicated by the
dashed lines in a. The number and amplitude of terahertz comb teeth increases
with $\Omega_{\mathrm{RF}}$ decreasing, since
$\nicefrac{{\delta\Omega}}{{\Omega_{\mathrm{RF}}}}$ increases from (i)-(iv),
with from $\nicefrac{{\delta\Omega}}{{\Omega_{\mathrm{RF}}}}=114$ (i); $341$
(ii); $1138$ (iii); $3414$ (iv). c Theoretically obtained terahertz comb
spectrum at the same modulation frequencies and amplitudes as the experiment
in a, using the experimentally calibrated response of the VCO (see
supplementary material).

###  3.2 Instantaneous response of the repetition rate

We then study the modulation of the instantaneous repetition rate through the
dynamic KIS. To modulate the reference (sideband), we source a voltage
controlled oscillator (VCO) with an arbitrary waveform generator (AWG) to
produce the microwave signal that drives an electro-optic phase modulator at
$\Omega_{\mathrm{mod}}+\Omega_{\mathrm{RF}}$. This apparatus controls the
sideband frequency and synchronizes the DKS comb tooth [fig. 3a]. We calibrate
the transfer function of the AWG+VCO onto the modulation of the reference to
account for the VCO’s exhibited nonlinear response with frequency (see the
supplementary material for more information). The output repetition rate is
measured using two cascaded electro-optic phase modulators [41, 30, 33, 39,
42], both modulated at 17.84706 GHz, enabling the overmodulation of two
adjacent Kerr comb lines and forming an electro-optic comb (EOcomb), which can
extract the DKS repetition rate. We use a narrow filter to select only the
closest translated Kerr comb teeth, separated by less than 50 MHz, which can
be detected by an avalanche photodiode (APD). From the beat we calculate
$\omega_{\mathrm{rep}}=\pm\Omega_{\mathrm{rep}}+N\times\omega_{EO}\approx
2\pi\times 1$ THz, where $\Omega_{\mathrm{rep}}$ is the detected beat of the
two translated Kerr comb teeth, $\omega_{\mathrm{EO}}$ is the EOcomb driving
frequency, and $N=56$ is the number of of EOcomb teeth, which can be
accurately measured with a $2\times 10^{-8}$ digit precision (i.e., 10 kHz)
when the DKS is synchronized with the main and reference pumps free running,
according to their respective laser linewidths and the OFD factor $\Delta\mu$
(locking the lasers enables $10^{-12}$ precision [30]). Importantly, we
measure the DKS repetition rate far from the reference portion of the comb
spectrum around 279 THz (1075 nm) to avoid cross-talk with the reference
modulation, while the main pump is kept fixed yet unlocked (i.e., free
running).

We temporally record the signal from the APD with a fast real-time
oscilloscope with a 40 ps sampling interval, from which we reconstruct a
spectrogram of the instantaneous repetition rate frequency against time for a
100 kHz sinusoidal microwave modulation [fig. 3b] and for a triangular
modulation [fig. 3c] (see [39] for details on the experimental apparatus). We
observe that the repetition rate follows the driving waveform accurately
around its nominal carrier frequency at $\omega_{\mathrm{rep}}=999.35901$ GHz
±10 kHz. After applying the expected OFD factor $\Delta\mu=-91$ corresponding
to the comb tooth distance between the the main pump and the reference, it is
clear that the repetition rate closely follows the microwave drive,
demonstrating the faithful transduction into the millimeter-wave regime thanks
to KIS of the DKS. We note that the spectrogram resolution is only limited by
the sampling rate/memory buffer size of the fast oscilloscope forcing a trade-
off between temporal and spectral resolution. This can limit resolution of
temporally fast and spectrally narrow responses, like the $C^{1}$
discontinuity corners when driven by the triangular AWG signal [fig. 3c]. Yet,
such a waveform is accurately reproduced even at the discontinuous change of
slope – which contains higher Fourier frequences than the 100 kHz pattern
frequency due to the nonlinearity of the response at that particular time –
highlighting the fast response time of the KIS transduction and its linearity.
To understand the limits at which KIS can modulate the repetition rate, it is
first important to recall that group velocity locking happens in low-
dispersion systems through a simple XPM interaction without comb tooth
capture, resulting in a mutli-color soliton system [32, 43]. On the other
hand, KIS is a phase-locking mechanism in addition to the previously described
group velocity locking (hence, the term synchronization). The phase velocities
of the DKS and the reference lock, making them indistinguishable [30, 39].
Their respective carrier envelope offset frequencies align while exhibiting
the same repetition rate, and hence the reference captures its closest comb
tooth [30, 34]. This occurs since the phase slip time between the soliton and
the intracavity reference $\tau_{\mathrm{slip}}=2\pi
c\left(\frac{1}{\omega_{\mathrm{ref}}}+\frac{1}{\omega_{\mathrm{pmp}}}\right)\frac{1}{\Delta
v_{\varphi}}\approx 140$ ps is longer than the characteristic nonlinear time
$\tau_{\mathrm{NL}}=\nicefrac{{2\pi}}{{\gamma L|A|^{2}}}\approx 30$ ps, with
$\gamma=2$ W-1$\cdot$m-1 the effective nonlinearity in our system, $\Delta
v_{\varphi}$ the mismatch of phase velocity, $\omega_{\mathrm{pmp,ref}}$ the
frequency of the pump and reference, respectively, and $|A|^{2}$ the DKS peak
power obtained from the Lugiato-Lefever equation [40]. Hence, the nonlinearity
is fast enough to phase lock the reference and the DKS and yield KIS. It is
important to note that the change of repetition rate for the KIS-modulated
reference essentially occurs because the main pump is (loosely) pinned, and
hence a change of carrier-envelope offset frequency $\omega_{\mathrm{ceo}}$
also changes $\omega_{\mathrm{rep}}$ according to the OFD factor $\Delta\mu$
[39]. Once an intracavity variation of the reference occurs, the modification
of the repetition rate will be distributed onto the DKS at a timescale
$\tau_{\mathrm{NL}}$. However, the reference being an external field, the
cavity is loaded on a timescale of the photon lifetime
$\tau_{\mathrm{phot}}=\nicefrac{{1}}{{\kappa}}\approx 1$ ns
$\gg\tau_{\mathrm{NL}}$. Hence, $\tau_{\mathrm{phot}}$ ultimately limits the
speed of the modulation of the reference to actuate the comb repetition rate.
Here, this limit is about 1 GHz, which is far faster than the triangular
modulation frequency applied in fig. 3b. Interestingly, to speed up the
maximum modulation frequency, the cavity photon lifetime could be engineered
such that the main pump remains in a critical coupling regime to optimize DKS
generation while the reference mode could be overcoupled. Here the pulley
coupling scheme allows for a quasi-dispersion-less coupling over the octave
for which the comb spans [38], balancing the coupling rate of both pumps.
However, a much simpler straight bus coupling scheme would naturally allow for
stronger coupling at the reference (longer wavelength) than at the pump. This
would enable reduction of the quality factor by order of magnitudes (and
correspondingly the photon lifetime at the reference), while also increasing
the KIS energy coupling $E_{\mathrm{kis}}$ and the resulting synchronization
bandwidth [39].

Figure 5: Terahertz comb tooth noise measurement. Comparison of the DKS
repetition rate phase noise (hence the terahertz comb carrier phase noise) in
the static KIS regime (dark blue) and the noise of one of the terahertz comb
teeth in the dynamic KIS regime of repetition rate modulation (teal). Their
close overlap demonstrates the coherence of the produced terahertz comb
through dynamic KIS repetition rate modulation. The terahertz comb tooth noise
is measured using a homodyne technique, where the frequency down-shifted
repetition rate $\Omega_{\mathrm{rep}}$ from the electro-optic comb apparatus,
through filtering, allows us to measure the first negative terahertz comb
tooth (inset). The AWG and VCO noise are negligible compared to the terahertz
comb noise (about -120 dB/Hz at 1 kHz), since the experimental apparatus uses
free-running lasers to synchronize the DKS. Further improvement of the
terahertz comb noise can be readily available from optically locking the pump
and reference laser to reference cavities. The power is referenced to that of
the carrier, namely, dBc/Hz

###  3.3 Terahertz comb demonstration

The sinusoidal frequency modulation of the repetition rate is interesting in
the spectral domain as it leads to the formation of narrowly-spaced frequency
combs with a carrier frequency at the Kerr optical frequency comb repetition
rate $\omega_{\mathrm{rep}}$, thus leading to a terahertz comb. To investigate
this, we characterize the obtained repetition rate-modulated frequency comb
using a real-time electronic spectrum analyzer (RSA). We then scan the VCO
frequency from $\Omega_{\mathrm{RF}}=$10 kHz to 4 MHz with a modulation
amplitude $\delta\Omega=341.4/\Delta\mu$ MHz [fig. 4a], which is much smaller
than the $\Delta\Omega=1.2/\Delta\mu$ GHz KIS bandwidth once projected onto
the repetition rate, allowing us to work in the same linear regime as
previously described. At low frequencies, narrowly-spaced lines are observed,
which space apart from the center as the modulation frequency increases. These
lines are equally spaced at integer multiples of the modulation frequency from
the carrier as expected from the spectrum of a frequency modulated sinusoid.
At low frequencies, we can clearly see formation of a terahertz frequency comb
with a center frequency of 999.40016 GHz ±10 kHz [fig. 4b]. This terahertz
comb exhibits reduced comb line power as $\Omega_{\mathrm{RF}}$ decreases,
since reducing $\Omega_{\mathrm{RF}}$ not only reduces the repetition rate of
the obtained comb, but also changes the modulation strength, thus impacting
the terahertz comb span and comb line amplitudes. This is expected since, as
previously described, the terahertz comb tooth of order $N$ from the carrier
frequency follows the well-known Bessel function relationship
$A_{n}=J_{n}(\frac{\delta\Omega}{\Omega_{\mathrm{RF}}})$. Hence, for large
modulation frequency, the spacing between terahertz comb teeth is obviously
larger, but these comb teeth also present a lower amplitude [fig. 4b-i]. The
terahertz comb becomes more dense with teeth whose amplitude increases with
reduced $\Omega_{\mathrm{RF}}$ [fig. 4b ii to iv]. Interestingly, since the
first order Bessel function $J_{0}$ presents zeros, one could also suppress
the terahertz comb carrier, essentially removing the frequency component of
the DKS repetition rate corresponding to the unmodulated KIS. For terahertz
comb generation applications, the ability to arbitrarily control the bandwidth
of the comb and its line spacing is a critical technical point compared to
other techniques. Indeed, generation of terahertz frequency combs often
requires complex processes such as mode-locking in quantum cascade lasers
[44], or heterodyning a femtosecond laser at a terahertz photomixer [45]. In
DKS combs (without KIS), terahertz comb generation can be realized by post-
processing the Kerr comb. For instance, we could pass one of the Kerr comb
lines through a modulation circuit before beating it with an adjacent Kerr
comb tooth or we could beat two Kerr combs with slightly different repetition
rates, as done in dual-comb spectroscopy. In contrast, KIS allows modulation
of the repetition rate within the resonator using an all-optical scheme with
off-the-shelf components to create an arbitrary comb profile while leveraging
the low-noise characteristics of DKS-based optical frequency combs.
Furthermore, a terahertz comb obtained by beating two combs or lines of one
comb spaced at multiples of the repetition rate [45] will only yield a
discrete set of accessible terahertz frequencies which could form a comb, but
would not allow the continuous tuning expected from a terahertz VCO for
arbitrary frequency control. To demonstrate the linear and continuous tuning
of the repetition rate, we theoretically model the spectrum expected from a
sinusoidally-modulated carrier signal (Fig. 4(c)). Here, we have plotted the
power in each harmonic of the terahertz comb given by the previously described
Bessel function model. By fitting the experimental data for
$\delta\Omega_{\mathrm{rep}}$, and calibrating the VCO excursion
$\delta\Omega$, we find that it agrees with the OFD factor $\Delta\mu=$-91. In
this calculation we have also accounted for the limited temporal response of
the driving electronics. The VCO has a response limited to a modulation
frequency of 0.5 MHz, after which its excursion significantly drops. We have
calibrated the frequency dependence of this excursion and applied a frequency
dependent correction factor in the model in fig. 4c, with a detailed
description and relevant data provided in the supplementary material. The
results match the experimental data in Fig. 4a well, indicating that the DKS
repetition rate has a purely sinusoidal modulation, demonstrating the
linearity of the reference laser frequency transduction onto the repetition
rate.

###  3.4 Terahertz comb noise characterization

As described previously, the method we propose for generation of a terahertz
comb is highly flexible compared to other methods, while also harnessing the
low-noise repetition rate of the DKS. To support the latter point, we must
also characterize the terahertz comb tooth noise. To do so, we homodyne the
output of the APD with a mixer, creating up- and down-shifted copies of the
spectrum (from $\Omega_{\mathrm{rep}}$ to DC and 2$\Omega_{\mathrm{rep}}$)
[fig. 5]. The down-shifted spectrum is quite weak since the mixer typically
requires a stronger local oscillator and the resulting signal-to-noise ratio
is insufficient for detection by a phase noise analyzer (PNA). We filter out
one comb line adjacent to the spectral center
($2\Omega_{\mathrm{rep}}-\Omega_{\mathrm{RF}}$) in the up-shifted spectrum
using a software-based band-pass filter and amplify it. The phase noise of
this comb line is measured by referencing against the AWG’s internal clock
signal at 10 MHz using a PNA. The generated noise curve is scaled down by 6 dB
due to the frequency doubling that also doubles repetition rate noise present
in all comb teeth (hence quadrupled in the electrical domain) [fig. 5]. We
obtain a phase noise of -25 dB/Hz at 1 kHz and -68 dB/Hz at 10 kHz offset
frequencies (the power is referenced to that of the carrier, namely, dBc/Hz).
We also compare the terahertz comb tooth noise with the noise of the
repetition rate in the KIS regime with the modulation turned off, referenced
to the same clock signal. The plots show that the terahertz comb teeth carry
the repetition rate noise as both noise curves overlap quite well. This
demonstrates that the dynamic regime of KIS did not introduce any additional
noise. Furthermore, since the DKS repetition rate is the dominant noise
source, existing methods to lock and stabilize the DKS can be put to use to
reduce the noise in the terahertz comb, for example see [46].

##  4 Discussion

Since we have demonstrated highly flexible terahertz comb generation with
coherence and potential for ultra-low noise operation once the two lasers that
optically pin the OFC are locked, it is important to discuss and compare our
approach with the current state-of-the-art, particularly those works
leveraging integrated microcombs for tunable terahertz signal generation.
Until now, DKS repetition rate control has mainly been exhibited through three
effects: thermal tuning to modify the cavity refractive index via the thermo-
refractive effect, piezo actuation for modulation of the cavity’s physical
dimensions, and electro-optic modulation using $\chi^{(2)}$ compatible
materials. The first option through thermal tuning, in particular with
integrated heaters, has been shown to tune the repetition rate significantly,
from hundreds of megahertz to a few gigahertz [47, 26]. Two main effects can
be leveraged from thermo-refractive actuation. First, a change in dispersion
occurs since the effective refractive index, being frequency-dependent, arises
from the difference in confinement with frequency, where the ring core and
cladding materials exhibit different thermo-refractive coefficients. Hence,
the FSR may change [26]. Second, it alters the detuning of the main pump from
the cavity, which may result in a change in the repetition rate, particularly
in the presence of asymmetric dispersion or the Raman effect. Regardless, the
maximum speed for thermal tuning is typically limited to the ten kilohertz
range since the thermal lifetime of the integrated system is slow. This speed
is far too limiting in controlling the terahertz comb. On the other hand,
electro-optic [48] and piezo-electric effects [29] can provide fast tuning,
with on-chip EO modulators demonstrated with bandwidths exceeding 100 GHz
[49]. In this case, the dispersion remains mostly intact, negligibly impacting
$\omega_{\mathrm{rep}}$. Instead, the cavity resonance frequencies are
modulated, which in turn influences the detuning of the main pump, thereby
causing a variation in the repetition rate. In these devices, repetition rate
tuning on the order of a few megahertz can be achieved [28, 29], which is
similar to the tens of megahertz achievable through KIS. Unlike the EO effect,
the piezoelectric effect can only efficiently modulate the resonator at fixed
mechanical frequencies determined by its size. This is quite limiting as well
for terahertz voltage-controlled-oscillator functionality. In addition, even
though these three different methods are based on different physical effects,
they all rely on the actuation of the cavity within which the DKS lives, by
tuning its resonance frequencies. Hence, the overall excursion is limited by
the DKS pump frequency detuning existence window, since a large frequency
tuning would result in loss of the cavity soliton.

In sharp contrast, the KIS scheme demonstrated here does not rely on cavity
actuation, but rather direct actuation of the group velocity of the cavity
soliton (without impacting the cavity). This is a key difference since there
is no impact on the DKS existence window (since the pump frequency is
unchanged), while repetition rate actuation relies only on intracavity
nonlinear dynamics instead of increased complexity in the device architecture
for on-chip electrodes or incorporation of electrically/mechanically-active
materials. The overall speed at which the DKS can be modulated is only limited
by the cavity linewidth, which without further optimization has an upper limit
of about 1 GHz, which is on par with EO and piezo-based systems for DKS
modulation. This bandwidth can be improved through coupling engineering to
reduce the photon lifetime, which has been shown to significantly increase the
KIS bandwidth [33].

##  5 Conclusion

In conclusion, we have introduced and experimentally demonstrated a new
concept to create an integrated terahertz voltage-controlled-oscillator that
transduces a microwave signal carried by an electro-optically modulated
reference laser into the the terahertz regime, through dynamic Kerr-induced
synchronization of a cavity soliton. We have shown its ability to arbitrarily
tune frequency around the DKS repetition rate near 1 THz, yielding a terahertz
comb with 13 MHz bandwidth and consisting of about 130 comb teeth that carry
the same noise as the DKS repetition rate, which is obtained from the main and
reference pump noise through optical frequency division. Our work presents a
proof-of-concept approach for a tunable and flexible terahertz oscillator,
where different metrics – while already on par with other techniques for
modulating DKS microcombs – could be readily improved by locking the two pumps
for better noise performance or coupling engineering for faster modulation
speed. Moreover, our demonstration showcases direct DKS actuation rather than
dynamic modification of the cavity in which the DKS lives, bypassing certain
critical limitations in DKS-based optical systems for microwave and terahertz
signal generation.

### Acknowledgments

The photonic chips were fabricated in the same fashion as those presented in
[30]. The Scientific colour map batlow [50] and subsequent color set is used
in this study to prevent visual distortion of the data and exclusion of
readers with colour-vision deficiencies [51]. We acknowledge partial funding
support from the Space Vehicles Directorate of the Air Force Research
Laboratory and the NIST-on-a-chip program of the National Institute of
Standards and Technology. We thank David Long and Yichen Shen for insightful
feedback.

### Author Contributions

U.A.J and M.C. performed the experiment and processed the data. G.M.
conceptualized the project and designed the resonators. G.M. and K.S. helped
with data processing and the understanding of the phenomenon. K.S. and G.M.
led the project and helped with data analysis. All the authors wrote,
contributed and discussed the content of this manuscript

### Competing Interests

G.M. and K.S have submitted a provisional patent application based on aspects
of the work presented in this paper.

### Data availability

The data that supports the plots within this paper and other findings of this
study are available from the corresponding authors upon request.

## References

  * [1] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “What should 6G be?” Nature Electronics 3, 20–29 (2020).
  * [2] H. Elayan, O. Amin, B. Shihada, R. M. Shubair, and M.-S. Alouini, “Terahertz band: The last piece of RF spectrum puzzle for communication systems,” IEEE Open Journal of the Communications Society 1, 1–32 (2020).
  * [3] A. Leitenstorfer, A. S. Moskalenko, T. Kampfrath, J. Kono, E. Castro-Camus, K. Peng, N. Qureshi, D. Turchinovich, K. Tanaka, A. G. Markelz, M. Havenith, C. Hough, H. J. Joyce, W. J. Padilla, B. Zhou, K.-Y. Kim, X.-C. Zhang, P. U. Jepsen, S. Dhillon, M. Vitiello, E. Linfield, A. G. Davies, M. C. Hoffmann, R. Lewis, M. Tonouchi, P. Klarskov, T. S. Seifert, Y. A. Gerasimenko, D. Mihailovic, R. Huber, J. L. Boland, O. Mitrofanov, P. Dean, B. N. Ellison, P. G. Huggard, S. P. Rea, C. Walker, D. T. Leisawitz, J. R. Gao, C. Li, Q. Chen, G. Valušis, V. P. Wallace, E. Pickwell-MacPherson, X. Shang, J. Hesler, N. Ridler, C. C. Renaud, I. Kallfass, T. Nagatsuma, J. A. Zeitler, D. Arnone, M. B. Johnston, and J. Cunningham, “The 2023 terahertz science and technology roadmap,” Journal of Physics D: Applied Physics 56, 223001 (2023).
  * [4] D. M. Mittleman, “Perspective: Terahertz science and technology,” Journal of Applied Physics 122 (2017).
  * [5] M. C. Beard, G. M. Turner, and C. A. Schmuttenmaer, “Terahertz Spectroscopy,” The Journal of Physical Chemistry B 106, 7146–7159 (2002).
  * [6] P. Jepsen, D. Cooke, and M. Koch, “Terahertz spectroscopy and imaging – Modern techniques and applications,” Laser & Photonics Reviews 5, 124–166 (2011).
  * [7] J. Liu, J. Dai, S. L. Chin, and X.-C. Zhang, “Broadband terahertz wave remote sensing using coherent manipulation of fluorescence from asymmetrically ionized gases,” Nature Photonics 4, 627–631 (2010).
  * [8] X. Yang, X. Zhao, K. Yang, Y. Liu, Y. Liu, W. Fu, and Y. Luo, “Biomedical Applications of Terahertz Spectroscopy and Imaging,” Trends in Biotechnology 34, 810–824 (2016).
  * [9] N. Born, D. Behringer, S. Liepelt, S. Beyer, M. Schwerdtfeger, B. Ziegenhagen, and M. Koch, “Monitoring Plant Drought Stress Response Using Terahertz Time-Domain Spectroscopy,” Plant Physiology 164, 1571–1577 (2014).
  * [10] L. T. Wedage, B. Butler, S. Balasubramaniam, Y. Koucheryavy, J. M. Jornet, and M. C. Vuran, “Climate Change Sensing through Terahertz Communication Infrastructure: A Disruptive Application of 6G Networks,” IEEE Network pp. 1–1 (2023).
  * [11] T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nature Photonics 10, 371–379 (2016).
  * [12] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal, A. Alkhateeb, and G. C. Trichopoulos, “Wireless communications and applications above 100 GHz: Opportunities and challenges for 6G and beyond,” IEEE access : practical innovations, open solutions 7, 78729–78757 (2019).
  * [13] K. Sengupta, T. Nagatsuma, and D. M. Mittleman, “Terahertz integrated electronic and hybrid electronic–photonic systems,” Nature Electronics 1, 622–635 (2018).
  * [14] H. Ito, T. Furuta, F. Nakajima, K. Yoshino, and T. Ishibashi, “Photonic Generation of Continuous THz Wave Using Uni-Traveling-Carrier Photodiode,” Journal of Lightwave Technology 23, 4016 (2005).
  * [15] M. C. Hoffmann, K.-L. Yeh, J. Hebling, and K. A. Nelson, “Efficient terahertz generation by optical rectification at 1035 nm,” Optics Express 15, 11706–11713 (2007).
  * [16] S. A. Diddams, K. Vahala, and T. Udem, “Optical frequency combs: Coherently uniting the electromagnetic spectrum,” Science 369, eaay3676 (2020).
  * [17] S. Diallo and Y. K. Chembo, “Optimization of primary Kerr optical frequency combs for tunable microwave generation,” Optics Letters 42, 3522–3525 (2017).
  * [18] TM. Fortier, A. Rolland, F. Quinlan, FN. Baynes, AJ. Metcalf, A. Hati, AD. Ludlow, N. Hinkley, M. Shimizu, T. Ishibashi _et al._ , “Optically referenced broadband electronic synthesizer with 15 digits of resolution,” Laser & Photonics Reviews 10, 780–790 (2016).
  * [19] T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nature Photonics 5, 425–429 (2011).
  * [20] E. Hamidi, D. E. Leaird, and A. M. Weiner, “Tunable Programmable Microwave Photonic Filters Based on an Optical Frequency Comb,” IEEE Transactions on Microwave Theory and Techniques 58, 3269–3278 (2010).
  * [21] T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361, eaan8083 (2018).
  * [22] J. Greenberg, B. M. Heffernan, and A. Rolland, “Terahertz microcomb oscillator stabilized by molecular rotation,” APL Photonics 9 (2024).
  * [23] T. Tetsumoto, T. Nagatsuma, M. E. Fermann, G. Navickaite, M. Geiselmann, and A. Rolland, “Optically referenced 300 GHz millimetre-wave oscillator,” Nature Photonics 15, 516–522 (2021).
  * [24] T. Tetsumoto, F. Ayano, M. Yeo, J. Webber, T. Nagatsuma, and A. Rolland, “300 GHz wave generation based on a Kerr microresonator frequency comb stabilized to a low noise microwave reference,” Optics Letters 45, 4377–4380 (2020).
  * [25] B. Stern, X. Ji, Y. Okawachi, A. L. Gaeta, and M. Lipson, “Battery-operated integrated frequency comb generator | Nature,” Nature 562, 401–405 (2018).
  * [26] G. Moille, D. Westly, E. F. Perez, M. Metzler, G. Simelgor, and K. Srinivasan, “Integrated buried heaters for efficient spectral control of air-clad microresonator frequency combs,” APL Photonics 7 (2022).
  * [27] R. Niu, M. Li, S. Wan, Y. R. Sun, S.-M. Hu, C.-L. Zou, G.-C. Guo, and C.-H. Dong, “kHz-precision wavemeter based on reconfigurable microsoliton,” Nature Communications 14, 169 (2023).
  * [28] Y. He, R. Lopez-Rios, U. A. Javid, J. Ling, M. Li, S. Xue, K. Vahala, and Q. Lin, “High-speed tunable microwave-rate soliton microcomb,” Nature Communications 14, 3467 (2023).
  * [29] J. Liu, H. Tian, E. Lucas, A. S. Raja, G. Lihachev, R. N. Wang, J. He, T. Liu, M. H. Anderson, W. Weng _et al._ , “Monolithic piezoelectric control of soliton microcombs,” Nature 583, 385–390 (2020).
  * [30] G. Moille, J. Stone, M. Chojnacky, R. Shrestha, U. A. Javid, C. Menyuk, and K. Srinivasan, “Kerr-induced synchronization of a cavity soliton to an optical reference,” Nature 624, 267–274 (2023).
  * [31] G. Moille, M. Chojnacky, U. A. Javid, P. H. Shandilya, C. R. Menyuk, and K. Srinivasan, “Octave Spanning Microcomb Dispersive Wave Optimization from Self-Balanced Soliton under Kerr-Induced Synchronization,” in “Frontiers in Optics,” (2023), pp. FTh3E–2.
  * [32] Y. Wang, F. Leo, J. Fatome, M. Erkintalo, S. G. Murdoch, and S. Coen, “Universal mechanism for the binding of temporal cavity solitons,” Optica 4, 855 (2017).
  * [33] G. Moille, P. Shandilya, A. Niang, C. Menyuk, G. Carter, and K. Srinivasan, “Versatile Optical Frequency Division with Kerr-induced Synchronization at Tunable Microcomb Synthetic Dispersive Waves,” arXiv 2403.00109 (2024).
  * [34] T. Wildi, A. Ulanov, N. Englebert, T. Voumard, and T. Herr, “Sideband injection locking in microresonator frequency combs,” APL Photonics 8, 120801 (2023).
  * [35] S.-P. Yu, T. C. Briles, G. T. Moille, X. Lu, S. A. Diddams, K. Srinivasan, and S. B. Papp, “Tuning Kerr-Soliton Frequency Combs to Atomic Resonances,” Physical Review Applied 11, 044017 (2019).
  * [36] S. Zhang, J. M. Silver, L. Del Bino, F. Copie, M. T. M. Woodley, G. N. Ghalanos, A. Ø. Svela, N. Moroney, and P. Del’Haye, “Sub-milliwatt-level microresonator solitons with extended access range using an auxiliary laser,” Optica 6, 206 (2019).
  * [37] H. Zhou, Y. Geng, W. Cui, S.-W. Huang, Q. Zhou, K. Qiu, and C. Wei Wong, “Soliton bursts and deterministic dissipative Kerr soliton generation in auxiliary-assisted microcavities,” Light: Science & Applications 8, 50 (2019).
  * [38] G. Moille, Q. Li, T. C. Briles, S.-P. Yu, T. Drake, X. Lu, A. Rao, D. Westly, S. B. Papp, and K. Srinivasan, “Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs,” Optics Letters 44, 4737–4740 (2019).
  * [39] G. Moille, U. A. Javid, M. Chojnacky, P. Shandilya, C. Menyuk, and K. Srinivasan, “AC-Josephson Effect and Sub-Comb Mode-Locking in a Kerr-Induced Synchronized Cavity Soliton,” arXiv 2402.08154 (2024).
  * [40] G. Moille, Q. Li, L. Xiyuan, and K. Srinivasan, “pyLLE: A Fast and User Friendly Lugiato-Lefever Equation Solver,” Journal of Research of the NIST 124, 124012 (2019).
  * [41] T. E. Drake, T. C. Briles, J. R. Stone, D. T. Spencer, D. R. Carlson, D. D. Hickstein, Q. Li, D. Westly, K. Srinivasan, S. A. Diddams _et al._ , “Terahertz-rate Kerr-microresonator optical clockwork,” Physical Review X 9, 031023 (2019).
  * [42] J. R. Stone and S. B. Papp, “Harnessing Dispersion in Soliton Microcombs to Mitigate Thermal Noise,” Physical Review Letters 125, 153901 (2020).
  * [43] G. Moille, C. Li, J. Stone, M. Chojnacky, P. Shandilya, Y. K. Chembo, A. Dutt, C. Menyuk, and K. Srinivasan, “Two-Dimensional Nonlinear Mixing Between a Dissipative Kerr Soliton and Continuous Waves for a Higher-Dimension Frequency Comb,” arXiv (2023).
  * [44] S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nature Photonics 5, 306–313 (2011).
  * [45] T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, and T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Applied Physics Letters 88 (2006).
  * [46] K. Liu, N. Chauhan, J. Wang, A. Isichenko, G. M. Brodnik, P. A. Morton, R. O. Behunin, S. B. Papp, and D. J. Blumenthal, “36 Hz integral linewidth laser based on a photonic integrated 4.0 m coil resonator,” Optica 9, 770–775 (2022).
  * [47] X. Xue, Y. Xuan, C. Wang, P.-H. Wang, Y. Liu, B. Niu, D. E. Leaird, M. Qi, and A. M. Weiner, “Thermal tuning of Kerr frequency combs in silicon nitride microring resonators,” Optics express 24, 687–698 (2016).
  * [48] Y. He, R. Lopez-Rios, U. A. Javid, J. Ling, M. Li, S. Xue, K. Vahala, and Q. Lin, “High-speed tunable microwave-rate soliton microcomb,” Nature Communications 14, 3467 (2023).
  * [49] X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4, 096101 (2019).
  * [50] F. Crameri, “Scientific colour maps,” Zenodo (2023).
  * [51] F. Crameri, G. E. Shephard, and P. J. Heron, “The misuse of colour in science communication,” Nature Communications 11, 5444 (2020).

##  Appendix S.1 Additional data

Terahertz comb spectra obtained at different modulation amplitudes are plotted
in Fig. S.1, along with theoretical fits. The corresponding center frequencies
are also provided.

Figure S.1: Terahertz frequency comb spectra for three different modulation
amplitude values $\delta\Omega$: (a) 113.8 MHz (b) 227.6 MHz, and (c) 455.2
MHz , controlled with peak-to-peak applied voltage to the VCO, with modulation
frequency varied from 0.1 MHz to 4 MHz. The offset frequency of the central
line is provided with each plot. The corresponding theoretical plots are
provided in (d)-(f). Figure S.2: Experimental setup for terahertz comb
generation and characterization. Detailed explanation is provided in the
accompanying text. CTL: Continuously-tunable laser, YDFA: Ytterbium-doped
fiber amplifier, VOA: variable optical attenuator, PC: Polarization
controller, AWG: Arbitrary waveform generator, VCO: Voltage-controlled
oscillator, PD: Photo-detector, PM: Power meter, WDM: Wavelength-division
multiplexer, LF: Lensed fiber, OSA: Optical spectrum analyzer, LPF: Long-pass
filter, SG: Signal generator, FBG: Fiber Bragg grating filter, APD: Avalanche
photo-diode, RSA: Real-time spectrum analyzer.

##  Appendix S.2 Experimental setup and repetition rate measurement

Figure S.2 shows the experimental setup. The experiment uses three lasers. One
laser pumps the soliton at 1061 nm (CTL) with 140 mW of optical power provided
after an Ytterbium-doped fiber amplifier (YDFA) and the second CTL thermally
stabilizes the resonator to easily access the soliton detuning regime, by
pumping in the opposite direction with orthogonal polarization at 981 nm, with
350 mW of optical power provided after a tapered amplifier (TA). A third laser
is used to synchronize a soliton comb tooth at 1566 nm with
$\approx\leavevmode\nobreak\ $0.2 mW on-chip power in the first sideband of
the EO modulator. These lasers are combined using fiber-optic wavelength
division multiplexers (WDMs) and sent into lensed fibers that couple the light
into the chip. At the output, 10 % of light is tapped out and sent to an
optical spectrum analyzer (OSA) to capture the comb spectrum and the rest is
split using WDMs to reject the lasers. A section of the comb spectrum on the
red side of the pump laser is filtered out using a free-space long-pass filter
at 1070 nm.

Figure S.3: Calibration of the VCO excursion. (a) A frequency shearing Mach-
Zehnder interferometer with the VCO driven EO modulator in one arm and a
second EO modulator driven by a second signal generator (SG). The modulation
signal in the lower arm is a single frequency 18 GHz sine wave while the VCO
applies a frequency modulated signal with the same 18 GHz center frequency and
an excursion $\delta\Omega$ at a frequency $\Omega_{\mathrm{RF}}$ varied from
0.1 MHz to 4 MHz. (b) Measured beat signal spectra at the output of the
interferometer at different values of $\Omega_{\mathrm{RF}}$. (c) A sample
theoretical fit of the generated beat note spectrum at $\Omega_{\mathrm{RF}}=$
0.5 MHz. These fits are used to determine the value of $\delta\Omega$ at each
modulation frequency. (d) The theoretical fits of the data in (b) provide a
correction factor for $\delta\Omega$ at each modulation frequency, shown in
blue.

As mentioned in the main article, the 1 THz repetition rate was indirectly
measured by down-shifting it to the megahertz domain and using a fast optical
detector. These modulators are driven at a frequency of 17.84706 GHz and close
to saturation with a power of 3 W (35 dBm) using RF amplifiers. The relative
phase between the modulators is adjusted to maximize the bandwidth of the
generated EO comb. The modulation frequency is chosen such that the 56th
multiple of this number approximately matches the $\approx 1$ THz repetition
rate. This guarantees that one comb line from each side overlaps in the
middle, with a difference small enough to be picked up with an InGaAs APD with
50 MHz bandwidth. The spectrum in this overlap region is filtered out with a
0.1 nm tunable bandpass fiber bragg grating filter (FBG) and fed to the APD.
The output of the APD is sent to a real-time spectrum analyzer (RSA) which
gives the down-shifted terahertz frequency comb spectra.

##  Appendix S.3 VCO calibration

The VCO used in the experiment has a response time limited to a bandwidth of
0.5 MHz, after which its excursion starts to drop. This makes the effective
modulation amplitude smaller at a given applied voltage at frequencies larger
than 0.5 MHz, and is observed in the measured excursion of the DKS repetition
rate, which reduces drastically at high frequencies. Therefore, a calibration
of the VCO excursion for each frequency is needed. To do this, we set up a
frequency shearing Mach-Zehnder interferometer (MZI), with the VCO in one arm
and another microwave signal generator (SG) in the other arm, as shown in Fig.
S.3(a). The signal generator is driven at a frequency of 18 GHz, the same used
in the experiment as the separation between the reference laser central line
and its first EO sideband. The VCO signal is swept with the same 18 GHz center
frequency and an amplitude corresponding to a sine wave with a peak-to-peak
amplitude of 50 mV, applied using the AWG. The measurement then consists of
determining the VCO modulation amplitude $\delta\Omega$ corresponding to this
voltage. To do so, the frequency of the sine wave $\Omega_{\mathrm{RF}}$ is
varied from 0.1 MHz to 4 MHz, and a telecom band CW laser is sent into one
input of the interferometer. The light is modulated in the two arms and then
recombined at the output. This mixing downshifts the EO modulation imparted by
the VCO to DC, where we can measure the frequency excursion with a 150 MHZ
bandwidth APD. The output of the APD is fed into the RSA. Figure S.3(b) plots
the measured spectra from the beating of the SG and VCO revealing the exact
excursion for varying modulation frequency. The MZI output produces an EO comb
corresponding to the Fourier transform of a frequency modulated sine wave at
18 GHz down-shifted to DC. Its bandwidth can be measured and fitted to obtain
the modulation amplitude that produces an identical spectrum. This will give
us the modulation amplitude $\delta\Omega$ corresponding to the 50 mV signal
we applied to the VCO. Immediately, we can see that the bandwidth of the EO
combs remains the same up till 0.5 MHz after which it slowly starts to
decline. After 1 MHz the decline is drastic. Using the same Bessel function
expansion of a frequency modulated signal as we did in the main article, we
fit each of these spectra. Below 0.5 MHz, the VCO excursion maintains its
value, which we fit to obtain a modulation amplitude $\delta\Omega=$ 59.6 MHz.
Using this, we obtained the $\delta\Omega=$ 341.4 MHz for the terahertz comb
plot in Fig. 4(a) in the main article, which was taken at a 300 mV peak-to-
peak sine wave. For the plots in Fig. S.1, the modulation amplitudes are 113.8
MHz for 100 mV, 227.6 MHz for 200 mV, and 455.2 MHz for 400 mV peak-to-peak
sine waves. For the smallest excursion at 100 mV in Fig. S.1(a), the fitted
modulation amplitude of the repetition rate is evaluated as
$\delta\Omega_{\mathrm{rep}}=$ 1.2504 MHz, giving us a ratio of
1.2504/113.8=1/91.011. This is in excellent agreement with the OFD factor
$|\Delta\mu|$=1/91. For the larger modulation amplitudes (200 mV, 300 mV and
400 mV) in Fig. S.1(b),(c), the fitted modulation amplitudes remain within an
error of 3.3% from the OFD factor.

The fitted modulation amplitudes for each frequency give us a frequency
dependent correction factor to apply in our theoretical model, which we have
used to obtain the theoretical plots in Fig. 4(c) in the main article and in
Fig. S.1 here. This correction factor vs frequency is plotted in Fig. S.3(d)
and labelled as VCO calibration. To verify that the frequency response is
solely responsible for the reduced excursion of the repetition rate, we have
calculated a fitting error for the measured therahertz comb spectra for Fig.
S.1(a) without any correction applied at several frequencies. These are also
shown in Fig. S.3(d) along with the VCO calibration. The two data agree well,
indicating that this is indeed the source of the reduced frequency response.