# Ultra-Broadband Kerr Microcomb Through Soliton Spectral Translation

Gregory Moille<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of
Maryland, College Park, MD, USA  Microsystems and Nanotechnology Division,
National Institute of Standards and Technology, Gaithersburg, MD, USA  Edgar
F. Perez  Joint Quantum Institute, NIST/University of Maryland, College Park,
MD, USA  Microsystems and Nanotechnology Division, National Institute of
Standards and Technology, Gaithersburg, MD, USA  Jordan R. Stone  Joint
Quantum Institute, NIST/University of Maryland, College Park, MD, USA
Microsystems and Nanotechnology Division, National Institute of Standards and
Technology, Gaithersburg, MD, USA  Ashutosh Rao  Microsystems and
Nanotechnology Division, National Institute of Standards and Technology,
Gaithersburg, MD, USA  Institute for Research in Electronics and Applied
Physics, University of Maryland,College Park, MD, USA  Xiyuan Lu  Microsystems
and Nanotechnology Division, National Institute of Standards and Technology,
Gaithersburg, MD, USA  Institute for Research in Electronics and Applied
Physics, University of Maryland,College Park, MD, USA  Tahmid Sami Rahman
Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA
Yanne Chembo  Institute for Research in Electronics and Applied Physics,
University of Maryland,College Park, MD, USA  Kartik Srinivasan
<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of
Maryland, College Park, MD, USA  Microsystems and Nanotechnology Division,
National Institute of Standards and Technology, Gaithersburg, MD, USA

(August 27, 2024)

###### Abstract

Broad bandwidth and stable microresonator frequency combs are critical for
accurate and precise optical frequency measurements in a compact and
deployable format. Typically, broad bandwidths (e.g., octave spans) are
achieved by tailoring the microresonator’s geometric dispersion. However,
geometric dispersion engineering alone may be insufficient for sustaining
bandwidths well beyond an octave. Here, through spectral translation induced
by the nonlinear mixing between the soliton and a secondary pump, we greatly
expand the bandwidth of the Kerr soliton microcomb far beyond the anomalous
geometric dispersion region on both sides of the spectrum. We show that such
nonlinear mixing can be summarized through the concept of synthetic
dispersion, highlighting the frequency matching of the nonlinear process.
Through detailed numerical simulations, we show that the synthetic dispersion
model captures the system’s key physical behavior, in which the second pump
enables the non-degenerate four-wave mixing process of Bragg scattering, which
spectrally translates the soliton and produces new dispersive waves on both
sides of the spectrum, all while preserving low-noise properties across the
full comb bandwidth. We experimentally demonstrate these concepts by pumping a
silicon nitride microring resonator at 1063 nm and 1557 nm to enable the
spectral translation of a single soliton microcomb so that a total comb
bandwidth close to 1.6 octaves is realized (from 137 THz to 407 THz). We
further examine the low-noise characteristics of the resulting comb state,
through beat note measurements across the spectrum, measurements of the comb
tooth spacing in both the primary and spectrally-translated portions, and use
of an electro-optic comb to probe the region in which the primary and
spectrally-translated comb portions overlap. Such ultra-broadband microcombs
provide new opportunities for full microcomb stabilization in optical
frequency synthesis and optical atomic clocks, while the synthetic dispersion
concept can extend microcomb operation to wavelengths that are hard to reach
solely through geometric dispersion engineering.

Microresonator frequency combs are promising for chip-scale metrology
applications including coherent range measurements Riemensberger  _et al._
(2020), spectroscopy Picqué and Hänsch (2019), and optical clocks Yu  _et al._
(2019); Newman  _et al._ (2019). These applications are typically realized in
the dissipative Kerr soliton (DKS) regime of microcomb operation Herr  _et
al._ (2016), and often rely on stabilization of the comb repetition rate and
carrier-envelope offset frequency, the latter usually through a $f$-2$f$
interferometer Udem  _et al._ (2002); Diddams  _et al._ (2020). $f$-2$f$
stabilization requires at least an octave of comb bandwidth, which can be
achieved through geometric dispersion engineering Coen  _et al._ (2012);
Okawachi  _et al._ (2014) to create coherent dispersive waves (DWs) that
broaden the comb spectrum Brasch  _et al._ (2016); Li  _et al._ (2017);
Pfeiffer  _et al._ (2017); Karpov  _et al._ (2018); Yu  _et al._ (2019).
Although DWs significantly increase microcomb bandwidth, the power of these
enhanced comb teeth is still orders of magnitude lower than the pump, so that
the $f$-2$f$ technique remains challenging. For example, an end-to-end comb
bandwidth of one octave is insufficient for self-referencing using the
(centrally located) high-power pump. More complicated resonator cross-sections
Moille  _et al._ (2018) and stacks of different materials Zhang  _et al._
(2012); Dorche  _et al._ (2020) have been proposed to alter the dispersion in
support of ultra-broadband combs, yet remain to be demonstrated
experimentally. Other approaches for super-octave microcomb generation include
combining $\chi^{(2)}$ and $\chi^{(3)}$ effects Chen  _et al._ (2020), but
such broadband combs usually present spectral gaps Szabados  _et al._ (2020),
and the suitability of such combs Chen  _et al._ (2020); Hendry  _et al._
(2020); Bruch  _et al._ (2020) for metrology has not been shown.
Here, we present a low-noise microcomb whose span extends across 1.6 octaves -
without spectral gaps - while bridging the telecom with near-visible
wavelengths. This is made possible through dual pumping, in which the second
pump enables the $\chi^{(3)}$ process of four-wave mixing Bragg scattering
(FWM-BS) Yulin  _et al._ (2004); Xu  _et al._ (2013); Li  _et al._ (2016) to
significantly broaden the typical DKS state, by spectral translation of the
soliton into other spectral bands. Using the dual pump scheme, we demonstrate
that the DKS teeth, acting as the signal in the FWM-BS process, can be
translated to new frequencies and effectively create new DWs on both sides of
the original DKS spectrum, broadening its bandwidth by more than a factor two.
The parametric nature of the FWM-BS process is such that phase coherence is
expected to be maintained, which we probe experimentally through a series of
noise measurements. In particular, heterodyne beat notes across the spectrum,
measurements of the comb tooth spacing in both the original DKS portion and
the spectrally translated portion, and a measurement of the relative noise
between the comb teeth in the overlap between the two microcomb portions, are
all consistent with the picture that FWM-BS spectrally translates the soliton
– thereby preserving its repetition rate – to the spectral region surrounding
the second pump, and the resulting 1.6 octave comb operates in a low-noise
state. The incorporation of the FWM-BS spectral translation mechanism allows
for a tunability and engineering of new DWs well beyond the limits imposed by
geometric and material dispersion on conventional singly-pumped microresonator
DKS states. To better understand the potential of this system, we introduce
the new concept of synthetic dispersion, which captures the underlying physics
and predicts the comb behavior as a function of resonator geometry and pump
frequencies. Simultaneously, we perform a detailed numerical study using a
single multi-pump Lugiato-Lefever Equation Chembo and Menyuk (2013); Taheri
_et al._ (2017) that accounts for the full set of $\chi^{(3)}$ processes
occurring in the resonator and validates the novel concept of synthetic
dispersion. The synthetic dispersion framework is further validated by close
correspondence with our experimental measurements of ultra-broadband
microcombs created by FWM-BS spectral translation of a DKS state.

Figure 1: $|$ Principle of synthetic dispersion through four-wave mixing Bragg
scattering (FWM-BS). a, Integrated dispersion (panel I) of a ring resonator
with two zero-crossings. Under driving by a single pump called the primary
pump (pp), only one DW at high frequency is created (the potential low
frequency DW is not generated due to power considerations), as shown in the
green spectrum in panel II. This part of the spectrum is referred to as the
primary portion. Inclusion of another pump, the synthesis pump (sp), allows
for the FWM-BS condition to be respected, especially considering the negative
(lower frequency) idler process (panel I). It effectively translates comb
teeth surrounding the primary pump to spectral positions surrounding the
synthesis pump, creating a synthesized portion of the frequency comb (panel
II, purple spectrum). b, The same negative idler frequency and momentum
conservation condition is met for larger detuning between the primary pump and
the signal, as the higher order dispersion coefficients allow for a roll-off
of the integrated dispersion, and therefore another mode at the same
integrated dispersion value as at the synthesis pump exists (panel I). Such
efficient FWM-BS effectively creates a new DW at low frequency (panel II). c,
Now considering the higher frequency idler due to FWM-BS, both the the mode
number matching condition and frequency matching conditions are changed (panel
I). The frequency matching condition now imposes that the integrated
dispersion at the idler must be of equal and opposite sign to the value at the
synthesis pump, and therefore a DW in the primary portion of the comb must
already exist (e.g., at 355 THz) to allow a change of sign of
$D_{\mathrm{int}}$ (panel I). In our case, this frequency matching condition
is met twice, but the mode matching condition permits only the high-frequency
mode to undergo FWM-BS. In the same manner as previously, it effectively
creates a new DW at high frequency (at 410 THz), extending the frequency comb
bandwidth on this side of the spectrum (panel II). d, Introduction of a
synthetic dispersion $D_{\mathrm{synth}}$ (purple, right axis) that captures
the nature of the translated portions as new DWs at low and high frequencies
and exhibits a clear difference from the integrated dispersion of the
resonator (solid green, right axis). The resulting LLE simulated comb spectrum
with a single pump (green) and a dual pump (purple) exhibit a clear difference
of bandwidth. The synthetic integrated dispersion emulates the dual pump
system by an effective single pump system (dark purple line) and shows close
agreement with the dual pump simulation. 0 dB is referenced to 1 mW, i.e.,
dBm.

Spectral translation and the synthetic dispersion framework–A microresonator’s
integrated dispersion $D_{\mathrm{int}}$ represents the variation of the cold
cavity resonance frequencies away from an equidistant frequency grid (i.e. the
DKS comb teeth) spaced by $D_{1}/2\pi$, the free spectral range (FSR) around
the primary pump Kippenberg  _et al._ (2018), hereafter annotated with the
label $pp$.
$D_{\mathrm{int}}(\mu)=\omega_{\mathrm{res}}(\mu)-\omega_{\mathrm{\textsc{dks}}}(\mu)=\omega_{\mathrm{res}}(\mu)-\left(\omega_{\mathrm{pp}}+D_{1}\mu\right)$,
with $\mu$ defined as the mode order relative to the pumped mode
$\mu_{\mathrm{pp}}=0$ (i.e. $D_{\mathrm{int}}(\mu_{\mathrm{pp}})=0$),
$\omega_{\mathrm{res}}(\mu)$ being the cavity resonance frequency of the mode
$\mu$, and $\omega_{\mathrm{DKS}}(\mu)$ being the $\mu^{\text{th}}$ DKS comb
tooth frequency. Hence, when the cavity resonances match the DKS comb teeth
frequencies, i.e., $D_{\mathrm{int}}(\mu_{\mathrm{\textsc{dw}}})=0$, a
resonant enhancement happens, leading to the DW creation at the mode position
$\mu_{\mathrm{\textsc{dw}}}$ (Fig. 1a). In this case (and for the
microresonators we study below), $D_{\mathrm{int}}$ is such that only one
primary DW ($DW^{\prime}$) is created, as the second zero crossing (on the low
frequency side) is too far from the primary pump to yield appreciable energy.
The use of an auxiliary pump enables straightforward access to soliton states,
through an effective temperature compensation mechanism that bypasses the
thermal bistability Lu  _et al._ (2019); Zhou  _et al._ (2019); Zhang  _et
al._ (2019). Moreover, simultaneous spectral broadening of the comb,
attributed to cross-phase modulation (XPM) effects, has also recently been
observed Zhang  _et al._ (2020), though the magnitude of the effect was
limited. Here, we consider a dual-pumped system in a regime where much more
significant spectral broadening is realized. We pinpoint the origin of the
strong increase in comb bandwidth as originating from an interband four-wave
mixing Bragg scattering (FWM-BS) process. FWM-BS is mediated by the
combination of a strong, secondary pump (hereafter referred to as the
synthesis pump $sp$) and the primary pump $pp$, and results in phase-coherent
spectral translation of comb teeth across wide spectral gaps determined by the
difference in pump frequencies. While FWM-BS of a single frequency continuous
wave input has been demonstrated in a microcavity Li  _et al._ (2016), and its
role in soliton-DW mixing in the context of optical fibers has previously been
studied in intraband Xu  _et al._ (2013) and interband cases Yulin  _et al._
(2004), here we show how it can play a critical role in the creation of ultra-
broadband microresonator frequency combs.

We consider a FWM-BS framework where the signal can be any comb tooth of the
primary soliton ($pDKS$) resulting from the primary pump $pp$ (i.e., the comb
that would be obtained through a single pump, called the primary portion), and
is converted into an idler that is another spectral component of the comb -
hereafter called the synthesized portion ($sDKS$) - through application of the
synthesis pump $sp$. This process must respect the fundamental criteria of
energy and momentum conservation, which in a ring resonator translate to
frequency matching and azimuthal mode number matching
$\omega_{\textsc{sDKS}}=\omega_{\mathrm{pDKS}}\pm|\omega_{\textsc{pp}}-\omega_{\mathrm{sp}}|$
and
$\mu_{\textsc{sDKS}}=\mu_{\mathrm{pDKS}}\pm|\mu_{\textsc{pp}}-\mu_{\mathrm{sp}}|$,
respectively. Using the integrated dispersion previously defined, these
fundamental conditions can be summarized in a single equation (see Extended
Data Section S-1.)

$\displaystyle\Big{(}\mu_{\mathrm{pDKS}}\pm\mu_{\mathrm{pp}}\Big{)}D_{1}=$
$\displaystyle\Big{(}D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{\pm})\pm
D_{\mathrm{int}}(\mu_{\mathrm{sp}})\Big{)}+$
$\displaystyle\Big{(}\mu_{\mathrm{sp}}\pm\mu_{\mathrm{sDKS}}^{\pm}\Big{)}D_{1}$
(1)

This results in a simple pair of conditions for FWM-BS based on the idler that
is considered:

$\displaystyle\begin{cases}\Delta
D_{\mathrm{int}}^{-}=D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{-})-D_{\mathrm{int}}(\mu_{\mathrm{sp}})=0\\\
\Delta\mu^{-}=\mu_{\mathrm{sDKS}}^{-}-\mu_{\mathrm{sp}}=\mu_{\mathrm{pDKS}}-\mu_{\mathrm{pp}}\end{cases}$
(2) $\displaystyle\begin{cases}\Delta
D_{\mathrm{int}}^{+}=D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{+})+D_{\mathrm{int}}(\mu_{\mathrm{sp}})=0\\\
\Delta\mu^{+}=\mu_{\mathrm{sDKS}}^{+}-\mu_{\mathrm{pp}}=\mu_{\mathrm{pDKS}}-\mu_{\mathrm{sp}}\\\
\end{cases}$ (3)

Here the superscript $\pm$ denotes the two kinds of idler that can result from
a FWM-BS process, one at higher frequency (+) and the other at lower frequency
(-) than the signal. Due to phase matching considerations, only one of each
would match the FWM-BS condition for a given signal.

The first case to consider is for a signal that is close to the main pump
(Fig. 1a). Only the low frequency idler satisfies the condition in Eq. 2.
Therefore, the FWM-BS process translates a DKS comb tooth close to the primary
pump into a comb tooth close to the synthesis pump. As the mismatch in the
integrated dispersion $\Delta D_{\mathrm{int}}^{-}$ (i.e. mismatch in the
fundamental energy conservation of the FWM process) increases with the mode
spacing $\Delta\mu^{-}$, the efficiency of the FWM-BS process decreases,
giving rise to the Lorentzian spectral shape around the synthesis pump.
However, comb teeth close to the primary pump are not the only ones that can
be efficiently spectrally translated. Due to the higher order dispersion
coefficients leading to a roll-off of the integrated dispersion, the energy
conservation condition $\Delta D_{\mathrm{int}}^{-}=0$ is met again for large
enough mode spacing (Fig. 1b)). Therefore, the mode frequency exhibiting the
same integrated dispersion value as at the synthesis pump would exhibit
efficient FWM-BS. This process extends the frequency comb bandwidth on the low
frequency side, with a new DW generated near 160 THz. The system we consider
presents a large asymmetry in the integrated dispersion, and therefore the
condition $\Delta D_{\mathrm{int}}^{-}=0$ is only met once. However, even for
symmetric integrated dispersion in which there is an additional $\Delta
D_{\mathrm{int}}^{-}=0$ on the high frequency side, no corresponding high
frequency DW will be generated through this negative idler process. This is
because the phase matching criterion from Eq. 2 requires the mode spacing
between the synthesis pump and the synthesized portion to match the separation
between the primary pump and the primary portion, and such a separation is too
large for the primary portion to contribute an adequate power signal for the
FWM-BS process.

That being said, Eq. 3 indicates that a higher frequency idler can be
generated for any modes whose integrated dispersion is equal and opposite in
sign to the integrated dispersion value at the synthesis pump. A comb tooth
from the synthesis portion $\mu_{\mathrm{sDKS}}^{+}$ must respect momentum
conservation, which in this case means that the mode spacing between the
signal (comb tooth from the primary portion) and the synthesis pump must match
the mode spacing between the idler (comb tooth from the synthesized portion)
and the primary pump. Once again, due to the higher order dispersion that
allows a zero crossing of the integrated dispersion, negative values of
$D_{\mathrm{int}}$ are possible, and an efficient FWM-BS process can happen at
frequencies beyond the original high-frequency DW of the single pumped DKS.
This results in a new DW near 409 THz, extending the bandwidth of the
frequency comb to higher frequencies, i.e. toward the visible. Interestingly,
another frequency matching condition in our case would occur at low frequency
(near 138 THz), as another zero crossing of the integrated dispersion happens
on this side of the spectrum. However, momentum conservation is not respected,
as the idler must have a higher mode number than the signal, and therefore no
FWM-BS happens here.

Figure 2: $|$ Spectral translation to create ultra-broadband microcombs. A
microring resonator with $RW$=1117 nm is pumped by a primary pump at 282 THz
and synthesis pump at 192 THz. The comb behavior surrounding the primary pump
is spectrally translated to the synthesis pump portion. a, Primary comb
generation with low primary pump power near threshold. The comb spacing is
equal to seven free spectral ranges (FSRs) and this spacing is reproduced
around the synthesis pump, where idlers are created seven FSRs away from the
synthesis pump, highlighting the mixing process between the two pumps and the
primary portion comb teeth. b, Primary comb generation at a higher primary
pump power where, as previously, the spectral spacing in the primary portion
is matched by that in the synthesis portion, as expected by the FWM-BS theory.
c Two-soliton state, where the characteristic 8 FSR modulation in the comb
envelope is replicated near the synthesis pump. The inset shows the LLE-
calculated two-soliton pulse arrangement that results in the simulated comb
envelope shown in red. We highlight the missing comb tooth in the primary
portion ($\Delta\mu$=-4), whose absence is translated onto the synthesized
portion of the comb, respecting the FWM-BS phase matching condition. d, Single
soliton state, where the impact of the synthesis pump is to expand the comb
bandwidth to 1.6 octaves and create new DWs on both ends of the spectrum. The
spectrum agrees with the generalized LLE solution using the dual-pump model
(red line), and greatly exceeds the expected spectrum if just the primary pump
is applied (dashed green line). The phase coherent nature of the generated
comb is further verified through beat note measurements with narrow linewidth
lasers throughout the comb spectrum (four left insets). The noise floor for
each measurement is shown in dashed lines, and is higher in the O-band due to
use of an additional RF amplifier. The rightmost inset shows the LLE
simulation of the expected time-domain behavior under dual pumping (red) and
if only the primary pump is applied (green). The horizontal bars at the bottom
of the graph compare the span achieved here with octave-spanning DKSs from
Refs. Spencer  _et al._ (2018); Yu  _et al._ (2019). We note that the low
frequency portion of the spectrum exhibits OSA artefacts, at 146 THz, 159 THz,
and $<$135 THz; the shortest DW at 141 THz is not impacted by these artefacts.
0 dB is referenced to 1 mW, i.e., dBm.

Using this fundamental property that FWM-BS translates the primary DKS comb
teeth into new spectral regions while maintaining the comb tooth spacing, we
introduce $D_{\mathrm{synt}}$ (Fig. 1d), a synthetic dispersion that captures
the essence of the FWM processes we have presented. $D_{\mathrm{synt}}$ is an
effective integrated dispersion that incorporates the combined influence of
the geometric integrated dispersion and the synthesis-pump-induced FWM-BS
processes. $D_{\mathrm{synt}}$ is essentially a piece-wise shifted version of
$D_{\text{int}}$, with the two being equal in the spectral region surrounding
the primary pump, and differing in the regions where FWM-BS causes a
broadening of the spectrum and the generation of new DWs. In these regions, we
simply use the FWM-BS conditions for creating the new DWs to determine how to
shift $D_{\text{int}}$ (either up or down) so that its zero crossings are
appropriately located. Stitching together the different regions of the
synthetic dispersion is accomplished by taking into account FWM-BS power
considerations, and in particular, where FWM-BS is less efficient. In our
case, this is at the midpoint between the primary pump and the synthetic pump,
and in between the primary DW (i.e., that generated by the original DKS) and
the high frequency FWM-BS DW. This approach provides a linear approximation of
where the DWs will be created ($D_{\mathrm{synt}}$=0) and helps estimate the
spectral extent of the frequency comb. To test its validity, we simulate the
behavior of the system using a generalized version of the Lugiato-Lefever
Equation (LLE), described in detail in the Extended Material Eq. 12. This
version of the equation has not been subjected to typical simplifications, and
in particular, the pumped modes remain as phase terms relative to the center
of the frequency domain, so that the evolution of the intracavity electric
field under multiple driving fields can be studied Taheri  _et al._ (2017).

Ultra-broadband microcombs – To experimentally study the above phenomena, we
perform measurements on Si3N4 microring resonators whose design and basic
characterization are described in the Methods and Extended Data. The
resonators are pumped in two bands, with a primary pump around 1063 nm and a
synthesis pump around 1557 nm, and the coupling enabled by a tailored pulley
waveguide geometry that realizes a relatively flat coupling rate across a wide
spectral range Moille  _et al._ (2019) (see Extended Data Section S-4). We
first show the spectral translation nature of the dual pump system for
different microcomb states. Figure 2a-b shows the spectral behavior for
primary combs generated just above threshold and at a higher power,
respectively, where in both cases we observe that the comb tooth spacing (7
FSR and 10 FSR, respectively) surrounding the primary pump is retained around
the synthesis pump and the higher frequency region between 350 THz to 400 THz.
This behavior persists as we reach the soliton regime, in which the synthesis
pump provides both a new nonlinear mixing mechanism (FWM-BS) as well as
thermal stabilization, with clear signatures of soliton steps observed
(extended data Fig. S3). Figure 2c shows the results for a two-soliton state,
and in Fig. 2d, spectral translation of a single soliton state and the
generation of additional DWs that greatly expand the comb spectrum is
demonstrated. In each of these states, the spectral separation between the
comb lines remains the same between the primary portion and the synthesized
one, illustrating that the comb lines from the synthesized portion are
unlikely to be due to the synthesis pump alone, and instead are a result of
the mixing between both pumps and the primary portion comb lines. In addition,
the clear translation of the two-soliton comb envelope modulation pattern onto
the synthesis component confirms this point. This spectral pattern (due to the
relative phase of the two pulses circulating in the cavity) effectively acts
as a modulation of the signal power in the FWM-BS process, and the replication
of this pattern in the synthesized portion of the comb follows the expectation
for FWM-BS that the generated idler power is linearly proportional to the
input signal power. For example, a comb tooth 4 FSRs below the primary pump
($\Delta\mu=-4$) is absent (likely due to an avoided mode crossing), and this
is replicated by the absence of a comb tooth 4 FSRs below the synthesis pump.
This also emphasizes the phase-matching condition and is consistent with the
FWM-BS framework described previously.

Figure 3: $|$ Ultra-broadband soliton spectral translation: geometric
dependence. Geometric dispersion impacts the location of the generated DWs,
much like the case in singly-pumped DKS devices. Here, microrings are pumped
at 192 THz and 282 THz with a pump power of 200 mW and 250 mW respectively,
for ring widths (RWs) from 1080 nm to 1125 nm. In each case, an ultra-
broadband microcomb is generated in which the soliton comb teeth surrounding
the primary pump are spectrally translated by the synthesis pump. Single pump
DKS states are also showcased for each RW, where thermal stability was
obtained through cross-polarized counterclockwise pumping at 305 THz. The
solid black lines represent the expected single soliton spectra calculated
through the dual-pumped generalized LLE and single-pump LLE for their
corresponding counterpart experiments. The dashed lines represent the
microcomb envelope prediction using the synthetic dispersion and single-pump
LLE. The light grey solid line are visual guides showcasing the shift of the
DWs with RW. 0 dB is referenced to 1 mW, i.e., dBm. Figure 4: $|$ Overlap of
the primary and synthesized portions of the comb spectrum. a, Spectrally-
translated single DKS spectrum with the highlighted overlap region. The
integrated dispersion computed at the primary pump frequency (282 THz) is
shown in green, with an annotation highlighting the expected frequency offset
between the DKS primary portion comb tooth and the cavity resonance at the
synthesis pump frequency. b, Zoom-in of the overlap region, highlighting the
primary portion of the DKS (blue) and the synthesized portion (red). The beat
note of pairs of adjacent comb teeth separated by the overlap-offset frequency
$\delta f$ are displayed in the insets. c, Measured comb tooth spacing for
both the primary and synthesized portion in the overlap region, exhibiting a
uniform value across the spectrum that is ring width dependent. The linear FSR
around the main pump is measured and reported in the gray dashed line, and is
close to the measured repetition rate of the DKS. The error bars represent the
variance of the measured $f_{\mathrm{rep}}$ for different pairs of comb teeth.
d, Measurement of the overlap-offset frequency $\delta f$ and the predicted
value of the integrated dispersion about the primary pump, evaluated at the
synthesis pump frequency ($D_{\mathrm{int}}(\mu_{\mathrm{sp}})$). The overlap-
offset frequency is uniform across the overlap region, and is well-predicted
by $D_{\mathrm{int}}(\mu_{\mathrm{sp}})$, indicative of the intrinsic detuning
between the DKS teeth and the cavity resonances frequencies. The error bars
represent the variance of the measured $\delta f$ for different pairs of comb
teeth. 0 dB is referenced to 1 mW, i.e., dBm.

When reaching the single soliton state, the comb extends from 137 THz to 407
THz, a span allowing f-3f stabilization and a significant increase in
bandwidth relative to state-of-the-art DKS microcombs Spencer  _et al._
(2018); Yu  _et al._ (2019). The comb envelope is in good agreement with the
predictions of the generalized LLE model, which incorporates both the primary
and synthesis pumps, in terms of the overall comb envelope, the spectral
positions of the different DWs, and the $>$80 dB dynamic range in comb tooth
power across the ultra-broadband spectral range. Finally, we note that the LLE
provides insight about the nature of the intracavity field (Fig. 2c-d,
insets). In the time domain, it predicts a two soliton pulse (Fig. 2c) and a
single soliton pulse (Fig. 2d) with an estimated pulse duration (full-width at
half-maximum) of 16 fs. The pulses sit on a modulated background
characteristic of DWs on both sides of the spectrum, and the pulse itself
shows some amount of structure. In contrast, we also plot the expected time-
domain behavior for a singly-pumped soliton state with a single DW (Fig. 3d),
where the background modulation is larger on one side of the pulse, and the
pulse itself shows no additional structure. This highlights the continuous
wave nature of the translated portion of the spectrum, which does not create
new pulses, but rather increases the background modulation in the same manner
as DWs do. We note that the pulsed behavior from the time-domain simulations
is consistent with coherence across the whole 1.6 octave bandwidth of the
comb.
To probe the comb coherence in the single soliton case, we perform beat note
measurements with narrow linewidth tunable lasers (at 970 nm, 1270 nm, 1420
nm, and 1520 nm) positioned at different locations within the comb spectrum,
covering many different parts of the comb with different spectral shapes. In
each case (inset to Fig. 2c), the beat note is a single tone, which is a
signature of the phase-coherent nature of the frequency comb Herr  _et al._
(2012); Raja  _et al._ (2019) and is to be contrasted with the multiple beat
notes that might be expected for modulation instability processes that lead to
sub-comb formation. The beat notes at 1270 nm and 970 nm evaluate the primary
portion of the DKS, and are thus expected to exhibit clear single tones,
reflecting the single pulse nature of the DKS in the resonator. The beat notes
at 1420 nm and at 1520 nm, which lie in spectrally-translated portion near the
synthesis pump, support the hypothesis of the binding of the synthesis pump
with the single soliton and the coherence of the synthesized component of the
comb. Later, we will strengthen these conclusions through heterodyne
measurements between the two portions of the comb.

Soliton spectral translation: geometric dependence — To explicitly demonstrate
the impact of the synthesis laser on spectral translation of soliton
microcombs, in Fig. 3 we compare spectra generated when both primary and
synthesis pump lasers drive nonlinear processes in a series of microresonators
to the case where only the primary pump drives soliton generation. Thermal
stabilization in the latter case is achieved by a counterpropagating cross-
polarized laser at 980 nm; the opposite propagation direction and orthogonal
polarization ensure that its impact on nonlinear dynamics is minimized. For
each microring, we see that effect of the synthesis laser is to mediate
spectral translation and new DW generation, while leaving the portion
surrounding the original soliton state essentially unchanged, as predicted
from Fig. 1. The microrings differ only in their ring widths, which range from
1080 nm to 1125 nm, and since the microring cross-section strongly influences
the integrated geometric dispersion and the resulting synthetic dispersion
under dual pumping, we expect this $RW$ variation to impact the generated comb
spectra and the DW positions. Each device exhibits an ultra-broadband
spectrum, and as expected, all DWs tune with ring width. The generalized LLE
described in the Extended Data Section S-2 provides good agreement (solid
black lines in Fig. 3) with the obtained experimental spectra and reproduces
the observed DW tuning. In addition, the single-pumped synthetic dispersion
simulations (dashed black lines in Fig. 3) provide similar predictions for the
comb envelope and DW positions, highlighting the utility of our heuristic
model and its use as a predictive tool to design ultra-broadband frequency
combs. However, it is important to note that both the single-pump LLE and the
more generalized $N$-pump LLE rely on the basic assumption of a single fixed
frequency grid, indexed by the mode number $\mu$, through Fourier transform of
the fast time temporal profile. Hence these models do not capture any
frequency discrepancy between the DKS teeth and the nearest cavity resonances.

Figure 5: Relative phase noise between the primary and synthesis portions. a,
Schematic of the measurement setup through which the relative phase noise of
two comb teeth in the stitching region, one from the primary comb portion and
one from the synthesis comb portion, is measured with the assistance of a 1300
nm EO comb. b, Spectrum of the microcomb device under investigation. c, Two
microcomb teeth from the stitching region (blue) plotted along with the EO
comb (purple). The EO comb teeth are not fully resolved due to the limited
spectral resolution of the optical spectrum analyzer in comparison to the EO
comb repetition frequency (6.0156 GHz). d, Beat note of the EO comb with the
two microcomb teeth. e, Mixing of the two beat notes produced between the EO
comb and the microcomb teeth to retrieve the relative phase noise of the
microcomb teeth, without the (free-running) 1300 nm laser phase noise
contribution.

Composite frequency comb overlap frequency offset — Closer inspection of Fig.
2d and every spectrum in Fig. 3 reveals an important feature: the primary
portion and the synthesized portion do not overlap perfectly, resulting in a
composite frequency comb. Although we are pumping the same mode family in both
bands, the overlap region in the comb spectrum (Fig. 4a and b) exhibits pairs
of adjacent comb teeth with an overlap-offset frequency $\delta
f=f_{\mathrm{sDKS}}-f_{\mathrm{pDKS}}$ that is smaller than the repetition
rate of the DKS. We now consider whether $\delta f$ remains the same across
this ‘stitching’ region.
We characterize the comb tooth spacing, i.e. the repetition rate
$f_{\mathrm{rep}}$ in the primary and synthesized portions of the comb, for
the state shown in Fig. 4a. This is done by measuring beat notes between a
helper laser and the comb teeth, with the helper laser wavelength determined
by a wavemeter (accuracy $\approx$ 50 MHz). Through measurement across the
helper laser tuning range (228 THz to 238 THz), we find that (Fig. 4b)
$f_{\mathrm{rep}}$ is uniform and equal for both the primary and synthesized
portions, and its value is close to the FSR around the primary pump
(equivalently, $\delta f$ is uniform throughout the overlap region). We then
repeat measurement of $f_{\mathrm{rep}}$ and $\delta f$ for devices with
different $RW$, and find that these conclusions hold, with the specific
measured values dependent on $RW$ (Fig. 4c-d). These conclusions are
understood by the fact that the FWM-BS process must respect frequency
matching, and thus the soliton repetition rate will be transferred to the
synthesized portion of the comb.
In addition, contrary to the strong coupling in a dual pump system that has
been theoretically studied Taheri  _et al._ (2017); Hansson and Wabnitz
(2014), the synthesis pump here does not change the DKS repetition rate or
shift the frequency of resonance enough to enter the regime where Arnold
tongues and sychronization might exist Skryabin  _et al._ (2021). We believe
that such locking cannot happen in our devices because the overlap frequency
shift cannot be compensated by the Kerr shift at the synthesis pump
($\approx-1.3$ GHz). Thus, this overlap-offset frequency between the primary
and synthesized portions of the comb can be understood as the fundamental
discrepancy between the soliton comb tooth frequency and the cavity resonance
frequency Guo  _et al._ (2017). As such, we expect that $\delta f$ should be
given by the value of the primary pump integrated dispersion evaluated at the
synthesis pump frequency $D_{\mathrm{int}}(f_{\mathrm{sp}})$, in the absence
of strong overlap between the primary and the synthesized elements, which is
our case since the synthesis pump is placed outside of the primary soliton
spectral envelope. In Fig. 4d, we compare $\delta f$ and the theoretical value
of $D_{\mathrm{int}}(f_{\mathrm{sp}})$, and find that they are in good
agreement and within the expected Kerr shift (couple of GHz) induced by the
synthesis pump.

Probing the relative noise between the two portions of the comb — Thus far we
have shown that the spectrum produced by our dual-pump system is consistent
with the picture in which FWM-BS mediates soliton spectral translation, with
new DWs generated on both the low and high frequency sides of the spectrum.
Moreover, we have explicitly shown that the comb tooth spacing is translated,
and there is an overall shift between the two portions of the spectrum. In
addition, the narrow heterodyne beat notes across the spectrum suggest that
the primary and synthesis portions of the comb each exhibit low noise, but so
far, their relative noise has not been experimentally considered. Now, if FWM-
BS is indeed the dominant process in the spectral broadening of the primary
DKS state into an ultra-broadband comb, we should expect that no added phase
noise (other than that of the synthesis laser) will be accrued on the
synthesis component teeth, and that there will be low noise between the
synthesis portion and the primary portion of the comb. To investigate this, we
seek to mix two comb teeth from the overlap region, where one comb tooth comes
from the primary portion of the comb and the other tooth comes from the
synthesis portion. Because of the large frequency offset $\delta f\approx$ 200
GHz, as observed earlier, we are unable to directly measure such a large beat
frequency. Instead, we use an electro-optic modulation (EO) comb, generated
through phase modulating an independent 1300 nm laser, to span across $\delta
f$ with a measurable (6.0156 GHz) repetition rate (Fig. 5a-c), and the beat
note of both microcomb teeth with respect to the EO comb is recorded (Fig.
5d). As the 1300 nm laser that seeds the EO comb is free-running and not
locked to the microcomb (which itself is free running), the obtained beat note
is relatively broad. However, by mixing together the two beat notes in a
specific configuration, we can suppress this helper laser phase noise. In
particular, if the EO comb teeth are correctly positioned on the same side of
the DKS comb teeth, for instance both at larger frequency, the two beat notes
can be mixed together in order to suppress the phase noise of the 1300 nm
helper laser (Fig. 5a). The obtained signal after mixing will only correspond
to the beating between the two component teeth of the DKS down-shifted by a
fixed frequency that is related to the microwave signal generator that drives
the EO comb (and which has very low phase noise in comparison to the free-
running laser systems). The expected behavior based on an underlying nonlinear
process dominated by the FWM-BS is corroborated in experiment, where a narrow
single tone, indicating a low-noise state between the two portions of the
comb, is observed (Fig. 5e). Given that a DKS soliton (the primary portion of
the comb) is a low-noise state, this suggests that the spectral translation
process has resulted in a synthesis portion that is also low-noise. Though
stabilization of the DKS and the pump lasers has not been performed, limiting
the ability to draw a final conclusion about the phase coherence between the
two portions of the comb, considering the earlier measurements and theoretical
analysis, which show that the repetition rate in both portions of the comb is
unvarying and identical to within our measurement capability, suggests - as
expected from the physical picture of FWM-BS - that the soliton spectral
translation process results in a single, fully phase-coherent comb. The
frequency of each comb tooth is then known once there is knowledge of the
repetition rate, the carrier-envelope offset frequency, and the overlap-offset
frequency $\delta f$, whose value is determined by the integrated dispersion
value at the synthesis pump.

Hence, based on the experimental demonstrations throughout this work, it seems
likely that the ultra-broadband microcombs we have demonstrated can be used
for metrological purposes, such as $f-2f$ self-referencing, provided that,
along with the comb repetition frequency and carrier-envelope offset
frequency, the overlap-offset frequency $\delta f$ is measured. In such a
scenario, where both pumps can be independently stabilized, the large power
available from the synthesis pump would be of particular appeal, as it could
be efficiently frequency doubled and, through proper dispersion engineering of
the resonator, could be made resonant with the high frequency DW.
Discussion — Zhang and colleagues have recently realized a similar dual
pumping configuration, in which an auxiliary laser at 1330 nm spectrally
broadens a 1550 nm soliton microcomb down to wavelengths of around 1275 nm
Zhang  _et al._ (2020). The spectrum extends from 1275 nm to 1720 nm (taken at
the -50 dB points relative to the maximum), an extent of 60 THz, and the
auxiliary laser is responsible for about 22 THz of spectral extension on the
high frequency side, resulting in a about a factor 1.5$\times$ increase of the
bandwidth of the comb. In contrast, here our 1557 nm auxiliary laser causes a
broadening of the comb spectrum on both low and high frequency sides of the
original soliton spectrum centered at 1063 nm, with a comb extending from 737
nm to 2190 nm, an extent of 270 THz, and the auxiliary laser is responsible
for about 93 THz and 79 THz of spectral extension on the low and high
frequency sides, respectively (173 THz broadening in total), in our case
extending the comb by more than a factor 2.6. Perhaps more important than the
characteristics of the comb broadening is its fundamental physical origin. In
Zhang  _et al._ (2020), spectral broadening is generally attributed to XPM
effects, which leaves open many questions about the relationship between the
original comb and the spectrally broadened region - in particular, whether any
offset between the two regions is indicative of independent frequency combs.
Here, we show that the dominant process behind spectral broadening in our
system is FWM-BS, so that soliton comb teeth generated by the primary pump are
spectrally translated to both the low frequency and high frequency sides of
the spectrum. In addition, we show that the overlap offset frequency between
the primary region of the comb and the spectrally translated one is inherent
to the microring resonator geometry, and is not a signature of two independent
frequency combs. In particular, the natural discrepancy between the DKS comb
teeth and cavity resonance frequency causes the synthesis pump to be offset
from the nearest DKS comb tooth by the value of the integrated dispersion at
this pumped mode, which we confirm through measurement of the overlap offset
frequency as a function of ring geometry. Finally, we have performed several
measurements that indicate that the FWM-BS process in our system directly
translates primary DKS comb teeth to another spectral window, resulting in a
comb state with low-noise in each portion as well as in the overlap region,
pointing to the potential use of such ultra-broadband frequency combs for
metrological purposes. This allows us to introduce a simplifying tool for
designing this new kind of frequency comb, by summarizing the non-linear
interaction and the position of all the DWs generated by the dual-pump system
through the synthetic dispersion. Such coherent ultra-broadband frequency
combs through DKS spectral translation could find many applications, in
particular, by harnessing the high power of the synthesis pump, which would
aid in $f$-2$f$ self referencing, for monolithic integration of $f$-3$f$
within a $\chi^{(3)}$ platform, and by pushing the limit of coherent DW
generation further into the visible.

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Methods
Device design— We use 775 nm thick Si3N4 ring resonators, which were
fabricated at Ligentec SA, with a fixed ring radius of 23 µm, a ring width
($RW$) that is varied between 1088 nm and 1140 nm across the devices, and a
surrounding silica cladding. The access waveguides for coupling to/from the
rings are tapered down to 200 nm at the facets, resulting in about 6 dB and 5
dB insertion losses per facet at 192 THz and 282 THz respectively. We use a
pulley waveguide with a width of $W=550$ nm, a length of $L_{\mathrm{c}}=9$
µm, and a gap $G=370$ nm. The expected frequency-dependent coupling, computed
using the coupled mode theory formalism developed in ref. Moille  _et al._
(2019), exhibits a resonance-free spectrum and $Q_{c}$ that varies within one
order of magnitude over an octave (Extended Data Fig. S2).

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

Acknowledgments
The ring resonators were fabricated at Ligentec Inc. The authors thanks
Alfredo de Rossi and Sylvain Combrié for fruitful discussions.

Funding
The authors acknowledge funding from the DARPA APHI, DARPA ACES, DARPA DODOS,
and NIST-on-a-chip programs. A.R. and X.L. acknowledge support under the
Cooperative Research Agreement between the University of Maryland and NIST-
PML, Award no. 70NANB10H193.

Author contributions
G.M developed the theoretical framework, performed the simulations, designed
the ring resonators and conducted the experiments. E.P., J.R.S., and T.S.R.
helped with the experiments, A.R. helped with ring resonator design, and Y.C.
and X.L contributed in the understanding of the physical phenomenon. G.M. and
K.S. wrote the manuscript, with input from all authors, and K.S supervised the
project. All the authors contributed and discussed the content of this
manuscript.

Competing interests
The authors declare no competing interests.

Additional Information
Correspondence and requests for materials should be addressed to G.M. and K.S.

## Extended Data

###  Four-Wave Mixing Bragg Scattering Matching Conditions

Four-wave mixing (FWM) relies on the basic conditions of energy and momentum
conservation, which in a microring resonator translate to frequency ($\omega$)
and azimuthal mode number ($M$) conservation. In the case of FWM Bragg
Scattering (FWM-BS), the signal photon (which in our case is a comb tooth from
the primary portion of the comb, labeled $pDKS$) can be scattered to two
spectrally-translated idlers, labeled as $sDKS^{\pm}$, such that these
fundamental energy/momentum conservation equations become:

$\begin{cases}&\omega_{\mathrm{pDKS}}\pm\omega_{\mathrm{pp}}=\omega_{\mathrm{sDKS}}^{\pm}\pm\omega_{\mathrm{sp}}\\\
&M_{\mathrm{pDKS}}\pm M_{\mathrm{pp}}=M_{\mathrm{sDKS}}^{\pm}\pm
M_{\textsc{sp}}\end{cases}$ (4)

The frequency shift in this spectral translation process is set by the
difference in the frequencies of the primary pump $pp$ and synthesis pump
$sp$, and both a higher frequency idler ($\omega_{\mathrm{sDKS}}^{+}$) and a
lower frequency idler ($\omega_{\mathrm{sDKS}}^{-}$) relative to the signal
can be generated, if they are phase- and frequency-matched Li  _et al._
(2016).

In a multi-mode resonator, such as the ring resonators studied here, the
integrated dispersion ($D_{\mathrm{int}}$) effectively characterizes the
discrepancy between the modal frequencies and that of a zero-dispersion
resonator, i.e., one in which all resonances are separated by the same free
spectral range (FSR). $D_{\mathrm{int}}$ is usually referenced at the pumped
mode, so that we introduce the relative mode number with respect to the pumped
mode $\mu=M-M_{\mathrm{pp}}$ (i.e $\mu_{\mathrm{pp}}=0$), with:

$D_{\mathrm{int}}(\mu)=\omega_{\mathrm{res}}(\mu)-\left(\omega_{\mathrm{pp}}+D_{1}\mu\right)$
(5)

where $\omega_{\mathrm{res}}(\mu)$ are the cavity resonance frequencies, and
$D_{1}=2\pi FSR$, computed at the primary pumped mode.

From the the above equations, we can compute the frequency of each resonance
of interest from the integrated dispersion, assuming that the signal
$\mu_{\mathrm{pDKS}}$ is part of the primary DKS and that the idler
$\mu_{\mathrm{sDKS}}^{\pm}$ is part of the spectrally-translated (synthesized)
portion of the DKS, and hence on a fixed frequency grid relative to the
synthesis pump. We then have:

$\displaystyle\omega_{\mathrm{pDKS}}=\omega_{\mathrm{pp}}+\mu_{\mathrm{pDKS}}D_{1}$
$\displaystyle\omega_{\mathrm{sp}}=\omega_{\mathrm{pp}}+\mu_{\mathrm{sp}}D_{1}+D_{\mathrm{int}}(\mu_{\mathrm{sp}})$
(6)
$\displaystyle\omega_{\mathrm{sDKS}}^{\pm}=\omega_{\mathrm{pp}}+\mu_{\mathrm{sDKS}}^{\pm}D_{1}+D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{\pm})$

This allows us to rewrite the energy/momentum conservation condition for FWM-
BS of Eq. 4 in a single equation such that:

$\displaystyle\left(\mu_{\mathrm{pDKS}}\pm\mu_{\mathrm{pp}}\right)D_{1}=$
$\displaystyle\Big{(}D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{\pm})\pm
D_{\mathrm{int}}(\mu_{\mathrm{sp}})\Big{)}+$
$\displaystyle\left(\mu_{\mathrm{sp}}\pm\mu_{\mathrm{sDKS}}^{\pm}\right)D_{1}$
(7)

Because of the asymmetric nature of the integrated dispersion (due to odd
order dispersion coefficients), both high and low frequency idlers will be
obtained for different input signals to respect the above equation. First, we
study the case of the lower frequency idlers, as described in Fig 1(a-b).
Equation S-1 in this case becomes:

$\displaystyle\left(\mu_{\mathrm{pDKS}}-\mu_{\mathrm{pp}}\right)D_{1}=$
$\displaystyle\Big{(}D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{-})-D_{\mathrm{int}}(\mu_{\mathrm{sp}})\Big{)}+$
$\displaystyle\left(\mu_{\mathrm{sDKS}}^{-}-\mu_{\mathrm{sp}}\right)D_{1}$ (8)

For efficient idler generation, we start by assuming that momentum is
conserved to allow FWM-BS. In the frequency comb, this is trivial as it
involves the simple condition that the number of resonator modes separating
the primary pump and the signal $\mu_{\mathrm{pDKS}}-\mu_{\mathrm{pp}}$ must
be the same as the number of modes separating the synthesis pump and the idler
$\mu_{\mathrm{sDKS}}^{-}-\mu_{\mathrm{sp}}$. This will be respected due to the
large number of comb teeth that can act as the signal. Assuming these two
conditions, frequency- and phase-matching for FWM-BS becomes:

$\displaystyle\begin{cases}D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{-})=D_{\mathrm{int}}(\mu_{\mathrm{sp}})\\\
\mu_{\mathrm{sDKS}}^{-}-\mu_{\mathrm{sp}}=\mu_{\mathrm{pDKS}}-\mu_{\mathrm{pp}}\end{cases}$
(9)

This implies that any resonator mode, as long as it is at a lower frequency
than the the largest frequency comb tooth, will undergo FWM-BS if it exhibits
the same integrated dispersion value as that at the synthesis pump frequency.
The power of this idler will be limited by the primary comb tooth that
realizes momentum conservation. In our case, we have the synthesis pump at a
lower frequency than the primary pump, so that this condition describes both
the frequency comb shape around the synthesis pump and the newly created DW at
lower frequency.

On the other hand, the higher frequency idler respects the equation:

$\displaystyle
D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{+})+\left(\mu_{\mathrm{sDKS}}^{+}-\mu_{\mathrm{pp}}\right)D_{1}=$
$\displaystyle-
D_{\mathrm{int}}(\mu_{\mathrm{sp}})+\left(\mu_{\mathrm{pDKS}}-\mu_{\mathrm{sp}}\right)D_{1}$
(10)

Momentum conservation will be respected for any spacing between the primary
comb tooth and synthesis pump $\mu_{\mathrm{pDKS}}-\mu_{\mathrm{sp}}$ that
matches the idler and primary pump spacing
$\mu_{\mathrm{sDKS}}^{+}-\mu_{\mathrm{pp}}$, leading to the condition:

$\displaystyle\begin{cases}D_{\mathrm{int}}(\mu_{\mathrm{sDKS}}^{+})=-D_{\mathrm{int}}(\mu_{\mathrm{sp}})\\\
\mu_{\mathrm{sDKS}}^{+}-\mu_{\mathrm{pp}}=\mu_{\mathrm{pDKS}}-\mu_{\mathrm{sp}}\end{cases}$
(11)

Therefore, any resonator mode, as long as it is at higher frequency than the
the lowest frequency comb tooth, and is equal and opposite in sign to the
integrated dispersion value at the synthesis pump frequency will undergo a
FWM-BS process. Interestingly, a primary portion DW must occur between the
primary pump and the idler, as an integrated dispersion zero-crossing must
happen for the above condition to hold. In Fig. 1(d), this DW occurs at 358
THz, for example.

Figure S1: $|$ Tuning of the newly generated DWs by tuning of the synthetic
dispersion. a, Fixed geometric integrated disperson (green) and resulting
synthetic dispersion (red to blue) for different synthesis pump frequencies
and a fixed primary pump frequency. b, Results of generalized LLE simulations
showing how the comb spectrum changes as a function of synthesis pump
frequency, and how the position of the newly generated DWs is predicted by the
zero crossings of the synthetic dispersion curves in a. The solid green curve
shows the spectral envelope when only the primary pump is applied.

###  Lugiato Lefever Model

The Lugiato-Lefever Equation (LLE) can be derived from basic couple mode
theory Hansson  _et al._ (2014); Chembo and Menyuk (2013), which is similar to
the temporal approach using the nonlinear Schrödinger equation (NLSE) with the
assumption of periodic boundary conditions and a slowly varying envelope Coen
_et al._ (2012). In many treatments, the detuning of the pump is simplified
and introduced as a linear term in the equation, making it convenient for
solving a single pump system. However, it has been demonstrated in ref. Taheri
_et al._ (2017) that avoiding such approximation allows for solving a system
under $N$ driving forces. Instead of choosing the frequency grid of the
resonant modes with an origin defined by a single pump, the grid is determined
by the spectral domain under study (with the origin placed at the center of
the grid). The generalized LLE becomes:

$\displaystyle\frac{\partial A(t,\tau)}{\partial t}=$
$\displaystyle\left(-\frac{\alpha}{2}-i\gamma L|A|^{2}\right)A$ (12)
$\displaystyle+$ $\displaystyle
it_{\mathrm{r}}\mathrm{FT}^{-1}\left[D_{\mathrm{int}}^{0}\tilde{A}(t,\omega)\right]$
$\displaystyle+$
$\displaystyle\sum_{\mathrm{p}}\kappa_{\mathrm{ext}}|_{\mathrm{p}}\mathcal{F}_{p}\mathrm{e}^{i\sigma_{\mathrm{p}}}$

where the subscript $p$ refers to either the synthesis pump or the primary
pump ($p=\\{\mathrm{pp};\mathrm{sp}\\}$), the index $0$ represents the center
of the simulation domain and hence $D_{\mathrm{int}}^{0}$ is the integrated
dispersion of the resonator computed at the center of the simulation domain
and $\mu_{0}=0$, $\tilde{A}(t,\omega)=\mathrm{FT}\left[{A}(t,\tau)\right]$ is
the Fourier Transform of the field,
$\mathcal{F}_{p}=-i\sqrt{P_{\mathrm{in_{p}}}}$ represents the $p$th driving
field (either the primary or synthesis pump),
$\sigma_{\mathrm{p}}=\delta\omega_{p}t+D_{\mathrm{int}}^{0}(\mu_{p})t-\mu_{p}\theta$,
represents the phase shift of the $p$ driving field, from which it is obvious
that the part relative to the fast time (or resonator angle $\theta$)
corresponds to the spectral shift away from the center of the simulation
domain, and the slow time $t$ phase corresponds to the dispersion shift away
from the uniform spectral grid and the relative detuning of the $p$ pump from
its pumped mode, $\kappa_{\mathrm{ext}}|_{\mathrm{p}}$ is the waveguide-
resonator coupling rate for the $p$ pumped mode, $\alpha$ is the total loss of
the resonator, $\gamma$ is the non-linear coefficient (here assumed constant),
$L$ is the resonator round-trip length, and $t_{\mathrm{r}}$ is the round trip
time. We note that the synthetic dispersion model replaces
$D_{\mathrm{int}}^{0}$ with $D_{\mathrm{synth}}$ and a single pump field for a
symmetric domain.

Figure S2:  $|$ Pulley scheme for broadband resonator-waveguide coupling. a,
Calculated coupling quality factor $Q_{\mathrm{c}}$ for a waveguide width =
550 nm and a ring resonator with $RR=23$ µm and $RW=1100$ nm, for three
different pulley lengths $L_{\mathrm{c}}=\\{9$, $10$, $11$} µm.
$L_{\mathrm{c}}=9$ $\mu$m provides nearly equal $Q_{\mathrm{c}}$ at the
primary and synthesis pump frequencies and limited variation in
$Q_{\mathrm{c}}$ across the full comb spectrum, In contrast, conventional
straight coupling, shown in black, results in severe overcoupling at the
synthesis pump wavelength and severe undercoupling at high frequencies. b,
Experimental ultra-broadband DKS spectrum for $RW=1103$ nm, illustrating the
different regions of interest for in-coupling and out-coupling to/from the
ring resonator. 0 dB is referenced to 1 mW, i.e., dBm.

###  Dispersive Wave Tuning Independent of Geometric and Material Dispersion

In this section we further justify use of the term ’synthetic dispersion’ to
capture the net effects of the new nonlinear-wave mixing processes that occur
in the dual pump system. Usually, dispersion is driven by two main components.
The first is material dispersion, where the chromatic dependence is such that
the dispersion becomes more normal the closer the wavelength is to the band-
gap, and is a significant challenge to reaching short wavelengths in
integrated frequency combs. The second is geometric dispersion, where the
wavelength-dependent confinement of the light within a guided mode geometry
provides a modification of the phase velocity of light different from the bulk
material trend, therefore providing a counter-balance to material dispersion.
Thus, for a fixed platform (i.e. a chosen set of photonic materials), the
typical view is that the geometry must be changed in order to modify the
dispersion of the resonator, and hence the resulting spectral shape of the
generated DKS. In this work, we have shown that by introducing the synthesis
pump, we provide a new degree of freedom in controlling the comb spectrum.
Apart from the significant spectral broadening mediated by the FWM-BS
processes we have described, control of the synthesis pump frequency results
in tuning of the DWs ( Fig. S1b). This is a consequence of a change in the
synthesis pumped mode resulting in new modes that match the FWM process. The
net result is that DW positions are modified without requiring modification of
the geometry or material. This tuning of the DW positions with synthesis pump
frequency is well-reproduced by the synthetic dispersion ( Fig. S1a), whose
zero-crossings closely matched the DW positions.

###  Coupling Design

In order to in-couple two pumps separated by close to 100 THz and out-couple
close to a 300 THz bandwidth comb, straight waveguide coupling cannot be used
as it exhibits a coupling quality factor $Q_{\mathrm{c}}$ (inversely
proportional to the coupling rate $\kappa_{\mathrm{ext}}$) varying by orders
of magnitude over an octave Moille  _et al._ (2019). However, pulley couplers
exhibiting phase mismatch can achieve much more spectrally flat coupling over
the bandwidth of interest. In our system, we use a pulley waveguide with a
width of $W=550$ nm, a length of $L_{\mathrm{c}}=9$ µm, and a gap $G=370$ nm.
Such coupling, computed using the coupled mode theory (CMT) formalism
developed in ref. Moille  _et al._ (2019), exhibits a resonance-free spectrum
and $Q_{c}$ that varies within one order of magnitude over an octave (Fig.
S2). This allows for efficient extraction of the whole ultra-broadband dual-
pumped frequency comb without any gap in the spectrum, thanks to the
resonance-less coupling. However, this design is quite sensitive to phase-
mismatch, and as a result the geometric parameters that define the coupling,
as longer pulley lengths would exhibit coupling resonances that either limit
coupling of the long wavelength pump (case $L_{c}=10$ µm) or the spectral
region in between the primary pump and the synthesis pump would (case
$L_{c}=11$ µm). However, the optimal flat coupling over a large bandwidth
comes at the cost that the coupling quality factor $Q_{c}$ is lower than the
intrinsic quality factor ($Q_{i}\approx 10^{6}$), resulting in over-coupled
devices.

Figure S3: $|$ Experimental setup. Two amplified lasers at 282 THz (1063 nm)
and 192 THz (1557 nm) pump a microring resonator. The output signal is split
between different paths that allow the observation of the spectrum with an OSA
and the comb power exhibiting the DKS step signature (experimental data
labeled as ‘Soliton detection’). One output path is used to measure the DKS
comb tooth frequency through beat-note measurement with different frequency
lasers and a wavemeter for absolute wavelength measurement. Pwr: Powermeter,
CTL: Continuous Tunable Laser, PC: Polarization Controller, PD: PhotoDiode,
OSA: Optical Spectrum Analyzer, ESA: Electrical Spectrum Analyzer, WDM:
Wavelength Demultiplexer. Figure S4: $|$ Linear measurement of the ring
resonator. a, Transmission spectrum around 1550 nm for $RW=1088$ (I) and
$1103$ nm (II), with a zoom-in of a representative resonance highlighted by
the black dashed line in II displayed in panel III, with the nonlinear least
squares fit shown in the red dashed lined. The 42 MHz MZI that serves as a
calibration is displayed in green. b, Transmission spectrum around 1050 nm for
the same resonators, namely $RW=1088$ (I) and $1103$ nm (II), with the fit of
a representative resonance highlighted by the black dashed line in II. The
nonlinear least squares fit to the zoomed-in data is shown as a red dashed
lined in III. The 83 MHz MZI that serves as a calibration for the 1060nm band
is displayed in green.

###  Characterization setup

The characterization setup used in measurement of the microcomb devices is
shown in Fig. S3. The resonator is pumped in the transverse electric (TE)
polarization in two bands, with a primary pump at 1063 nm amplified by a
ytterbium-doped fiber amplifier and a synthesis pump at 1557 nm amplified with
an erbium-doped fiber amplifier, which are both coupled to the chip using a
wavelength division multiplexer and a lensed fiber. The on-chip primary pump
power, i.e. at 282 THz, is $P_{\mathrm{pp}}=200$ mW (7.5 dB insertion loss per
facet) and the on-chip synthesis pump power $P_{\mathrm{sp}}=250$ mW (6 dB
insertion loss per facet). The generated microcomb is out-coupled and split
between an optical spectrum analyzer and another path allowing for beat-note
measurements and comb-power detection. As expected with the synthesis pump
scheme, appropriate choice of the frequency of each laser thermally stabilizes
the ring resonator and results in a relatively pure Kerr response (i.e.,
thermal effects mitigated) with a clear signature of soliton steps for
different soliton orders (inset to Fig. S3), ultimately reaching the lowest
order single soliton state.

###  Linear Measurements

Linear measurements to determine the mode family and the quality factor of the
ring resonator, in both the 1550 and 1060 nm band, are performed by sweeping a
continuous tunable laser (CTL) and recording the transmission spectrum (Fig.
S4). A Mach-Zehnder Interferometer (MZI), with a free spectral range of 42 MHz
and 83 MHz for the 1550 nm and 1060 nm band respectively, is used to calibrate
the laser sweeps. This allows retrieval of the quality factor of the
resonances in both bands, which are in the over-coupled regime as expected
from the broadband-flat coupling described in the previous section.The modes
in the spectra shown in Fig. S4 exhibit $Q_{\mathrm{i}}\approx 1.1\times
10^{6}$ and $Q_{\mathrm{c}}\approx 3.5\times 10^{5}$ for the average intrinsic
and coupled quality factors, respectively.


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