# Carrier-resolved real-field theory of multi-octave frequency combs

Danila N. Puzyrev  Department of Physics, University of Bath, BA2 7AY, UK
Centre for Photonics and Photonic Materials, University of Bath, BA2 7AY, UK
Dmitry V. Skryabin  Department of Physics, University of Bath, BA2 7AY, UK
Centre for Photonics and Photonic Materials, University of Bath, BA2 7AY, UK
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Optical frequency combs are pillars of precision spectroscopy, and their
microresonator realisation serves applications where miniaturisation and large
tooth separation are important. Microresonator combs cover an enormous range
of time scales varying from the femtosecond periods of optical oscillations to
milliseconds corresponding to the kilohertz linewidth of the comb teeth. Here,
we develop and implement the carrier-resolved real-field model for multi-
octave frequency combs, which allows for near ab-initio capture of all the
time scales involved. As an example, we consider a microresonator that has a
mix of second and third-order nonlinearities and uses periodic poling. By
applying the real-field approach, we demonstrate how to surpass traditional
limitations and model the spectral broadening and soliton mode-locking across
three optical octaves.

††journal: optica

Nonlinear optical frequency conversion was observed at the beginning of the
laser era [1], and since then, it has been one of the most active areas of
applied and fundamental research [2]. The development of high power
femtosecond and attosecond laser systems has opened numerous opportunities to
use nonlinear processes for the generation of electromagnetic spectra between
XUV and THz frequency bands [3, 4, 5]. This has created a demand for an
efficient theoretical and computational framework to model ultra-broad
spectra, which could not be described by the mid-20th century theoretical
tools of nonlinear optics. A variety of generalized envelope equations [6, 7,
8, 9, 10], and a class of carrier-resolved models, also known as
unidirectional pulse-propagation equations (UPPE) [11, 12, 13, 14, 15], have
met this demand.

Optical frequency combs and associated soliton pulses in microresonators have
recently become a fast-growing area and a contender in the generation of
temporally ultra-short and spectrally ultra-broadband waveforms [17, 16]. The
advantages of microresonators include a small footprint, small operational
powers, and dispersion engineering used for spectral tuning of the generated
combs. Microresonator comb spectra spanning across multiple octaves from mid-
infrared to visible have been observed in [18, 19, 20, 21, 22]. Contrary to
filamentation and supercontinuum experiments using high-energy ultra-short
laser pulses, a typical microresonator experiment requires milliwatts of CW
power, while short pulses are generated inside the resonator.

A numerical model well suited to describe comb generation in microresonators
can be formulated using a coupled-mode theory [23, 24, 25]. This approach
leads to a set of ordinary differential equations for amplitudes of the
resonator modes evolving in time. Fourier transforming known coupled-mode
equations to physical space, one finds the driven-damped nonlinear Schrödinger
equation, which is known in the resonator context as the Lugiato-Lefever
equation, or other envelope models [25, 26, 27, 28, 29]. Approximations
underpinning the envelope models rely on the selection of a reference carrier
frequency and wave-number and their subtraction from the rest of the spectrum.

Our aim here is to bypass these approximations and present a carrier-resolved
real-field theory for multi-octave frequency combs. We want this theory to
capture desirable features of the carrier-resolved models known for
propagation in bulk crystals, gases, and waveguides [12, 13, 14, 15]. One such
property is using the real-valued electric field, ${\cal E}$, oscillating with
the required spectrum of electromagnetic frequencies. The real field can be
conveniently used to express the second, ${\cal E}^{2}$, and third, ${\cal
E}^{3}$, order effects without omitting any of nonlinear terms, which is an
unavoidable shortcoming of the envelope based methods [29, 28]. More
generally, the carrier-resolved approach has not been yet formulated for a
broad class of the resonator systems where pump, gain and dissipation are
intrinsic for device operation [30, 31, 32, 33], and where current trend
towards the generation of increasingly extreme waveforms [3, 4] requires
development of novel theoretical methods beyond the envelope Haus-master and
Ginzburg-Landau equations [30].

To develop the carrier-resolved theory of ring microresonators, we proceed
with the initial steps of the formalism described in [29]. This formalism
seeks the electric field components, ${\cal E}_{\alpha}$, obeying Maxwell
equations as a superposition of the resonator modes $F_{\alpha
j}(r,z)e^{ij\vartheta-i\omega_{j}t}$, where $j$ is an azimuthal mode number,
$\alpha$ can be either $x$, $y$ or $z$, $\vartheta=[0,2\pi)$ is the polar
angle and $r$ is the distance in the $(x,y)$-plane, $z$ axis is perpendicular
to the resonator plane, and $\omega_{j}$ are the resonance frequencies. We
account only per one mode for every $j$, and assume that this mode is
dominated by one polarization direction which allows us to omit the subscript
$\alpha$,

${\cal
E}=\sum\limits_{j=j_{\min}}^{j_{\max}}b_{j}B_{j}(t)F_{j}(r,z)e^{ij\vartheta-i\omega_{j}t}+c.c.,$
(1)

and use scalar approximation for nonlinear polarization. Here, $B_{j}(t)$ are
the unknown modal amplitudes. If ${\cal E}$ and $b_{j}B_{j}$ are measured in
V/m, then choosing $b_{j}^{2}=2/\epsilon_{0}cn_{j}S_{j}$, makes $|B_{j}|^{2}$
to have units of Watts, where $n_{j}$ is the refractive index and $S_{j}$ is
the transverse mode area. Ultimately, our approach is superior over the
envelope and prior coupled-mode methods because it explicitly uses a
continuous span of the resonator modes extending over two, three or more
octaves, and true real fields to evaluate the nonlinear terms, which thereby
captures all possible nonlinear wave-mixing processes.

In a resonator, every mode experiences damping with the rate $\kappa_{j}$,
while the comb generation is achieved by pumping one of the modes, e.g.,
$j=M$, with laser light having frequency $\omega_{p}$ close to $\omega_{M}$.
Substituting Eq. (1) to Maxwell equations, we find a set of equations for
modal amplitudes,

$\begin{split}&i\partial_{t}B_{j}=-i\tfrac{1}{2}\kappa_{j}\big{(}B_{j}-\widehat{\delta}_{j,M}{\cal
P}e^{i\delta t}\big{)}\\\
&-\frac{\omega_{j}}{2n_{j}^{2}b_{j}}\frac{e^{i\omega_{j}t}\iiint
F_{j}e^{-ij\vartheta}{\cal N}({\cal E})r{\rm d}r{\rm d}z{\rm
d}\vartheta}{\iiint F_{j}^{2}r{\rm d}r{\rm d}z{\rm d}\vartheta}.\end{split}$
(2)

Here, $\delta=\omega_{M}-\omega_{p}$ is the frequency detuning, and
$\widehat{\delta}_{j,M}=1$ for $j=M$ and is zero otherwise. ${\cal
P}=\sqrt{{\cal F}\eta{\cal W}/\pi}$ is the pump parameter, where ${\cal F}$ is
finesse, ${\cal W}$ is laser power at the coupling point and $\eta$ is
coupling coefficient.

Figure 1: Black (red) lines show power of the pump mode, $|B_{M}|^{2}$, vs
detuning, $\delta$, when $\tilde{\chi}^{(2)}+\chi^{(3)}$ (only $\chi^{(3)}$)
nonlinearities are accounted for. $Q$ is the number of poling periods, and
${\cal W}$ is laser power. Shading shows intervals investigated for frequency
comb generation using the carrier resolved model, Eq. (7). Nonlinear
parameters are $\gamma_{2,2M+Q}/2\pi=8.39$GHz/$\sqrt{\rm W}$ and
$\gamma_{3,M}/2\pi=3.79$MHz/W, linewidth is $\kappa/2\pi=100$MHz, coupling
parameter and finesse are $\eta=0.5$ and ${\cal F}=3940$. Figure 2:  (a,b)
Spectrum (Fourier transform, $t\to\omega$, of ${\cal E}_{0}(t,\vartheta=0)$)
and real field profile, ${\cal E}_{0}(t,\vartheta=0)$, of the Kerr soliton
generated in strongly mismatched resonator, $Q=10$, $M=404$,
$\varepsilon_{2}/2\pi=30$THz. (c,d) Three-octave spectrum corresponding to the
randomly fluctuating electric field generated in quasi-phase-matched
resonator, $Q=89$, $M=404$, $\varepsilon_{2}/2\pi=-445$GHz, $j_{\min}=147$,
$j_{\max}=1478$. Pump is ${\cal W}=200$mW in both cases. Total integration
time is $191$ns in (a) and $143$ns in (b). Spectrum was computed over last
$40$ns in (a) and $95$ns in (b). Round trip time is $2.538$ps.

${\cal N}$ is the nonlinear part of material response, which in our case
includes second-, $\tilde{\chi}^{(2)}$, and third-order, $\chi^{(3)}$,
nonlinear susceptibilities,

${\cal N}({\cal E})=\tilde{\chi}^{(2)}{\cal E}^{2}+\chi^{(3)}{\cal E}^{3}.$
(3)

Second-order susceptibility can vary in $\vartheta$ if periodic poling is
applied for quasi-phase-matching, i.e.,
$\tilde{\chi}^{(2)}(\vartheta)=\tilde{\chi}^{(2)}(\vartheta+2\pi/Q)$, where
$Q$ is the number of periods along the ring [34],

$\tilde{\chi}^{(2)}(\vartheta)=\chi^{(2)}g(\vartheta)=\chi^{(2)}\sum_{m}g_{m}e^{-imQ\vartheta},\leavevmode\nobreak\
m=\pm 1,\pm 2,\dots,$ (4)

and $g(\vartheta)$ varies between $+1$ and $-1$. The no-poling case
corresponds to $g(\vartheta)=1$. Quasi-phase-matching for the modes $M$ and
$2M+Q$ is achieved, when $Q$ is selected to minimize the frequency mismatch
parameter,

$\varepsilon_{2}=2\omega_{M}-\omega_{2M+Q}.$ (5)

Detailed derivation of Eq. (2) can be found in [29], see Eq. (3.2) and Eq.
(6.1). Equation (2) is also structurally the same as Eq. (24) in [24], where
${\cal N}$ was the Kerr part of refractive index. A standard way to simplify
and then solve Eq. (2) is to introduce several groups of modes centred around
the pump and its harmonics and simplify nonlinear terms by limiting nonlinear
wave-mixing processes to the ones that resonantly excite selected modes, which
was also the approach used in [29]. However, this method is compromised when
the individual modal groups approach the octave width and omitted non-resonant
nonlinear terms start playing a role.

Figure 3: $\chi^{(2)}$ solitons generated in quasi-phase-matched resonator,
$Q=89$, $M=405$, $\varepsilon_{2}/2\pi=-445$GHz, for relatively small powers.
(a-c) Spectrum, section of the soliton train and real field profile of a
single soliton for ${\cal W}=20$mW. (d-f) Spectrum, its zoomed interval
showing the resolved comb lines and real field profile of a single soliton for
${\cal W}=7$mW. Total integration time is $190$ns in (a) and $143$ns in (b).
Spectrum was computed over last $80$ns in (a) and $88$ns in (d). Round trip
time is $2.538$ps.

Equation (2) is free from the above limitation. It treats all the modes on
equal footing and accounts for all nonlinear wave-mixing processes by dealing
with ${\cal E}^{2}$ and ${\cal E}^{3}$ without simplifications. However,
substituting Eq. (1) to the right-hand side of Eq. (2) yields tens of
thousands of nonlinear coefficients varying with $j$ within around one order
of magnitude, and the equations appear to be entirely not practical even
before numerical integration in time is approached. To derive a model with the
practical number of nonlinear coefficients and, at the same time, to retain
all terms inside ${\cal E}^{2}$ and ${\cal E}^{3}$, we make a simplification
that for every $j$ there is the same coefficient for all nonlinear mixing
terms entering the respective equation, but the coefficients vary with $j$.
This is achieved by assuming $\iiint{\cal E}^{3}F_{j}e^{-ij\vartheta}r{\rm
d}r{\rm d}z{\rm d}\vartheta\approx$ $b_{j}^{3}\iint F_{j}^{4}r{\rm d}r{\rm
d}z\int_{0}^{2\pi}{\cal E}_{0}^{3}e^{-ij\vartheta}{\rm d}\vartheta$, and
$\iiint g(\vartheta){\cal E}^{2}F_{j}e^{-ij\vartheta}r{\rm d}r{\rm d}z{\rm
d}\vartheta\approx$ $b_{j}^{2}\iint F_{j}^{3}r{\rm d}r{\rm
d}z\int_{0}^{2\pi}g(\vartheta){\cal E}_{0}^{2}e^{-ij\vartheta}{\rm
d}\vartheta$, where

${\cal
E}_{0}=\sum_{j=j_{\min}}^{j_{\max}}B_{j}e^{ij\vartheta-i\omega_{j}t}+c.c.,$
(6)

is the real-valued carrier-resolved field, where the transverse profile of the
modes has been factored out. Equations (2) are then reduced to

$\begin{split}i\partial_{t}B_{j}=&-\tfrac{1}{2}\kappa_{j}\big{(}B_{j}-\widehat{\delta}_{j,M}{\cal
P}e^{i\delta t}\big{)}\\\ &-\gamma_{2,j}\int_{0}^{2\pi}g(\vartheta){\cal
E}_{0}^{2}e^{i\omega_{j}t-ij\vartheta}\frac{{\rm d}\vartheta}{2\pi}\\\
&-\gamma_{3,j}\int_{0}^{2\pi}{\cal
E}_{0}^{3}e^{i\omega_{j}t-ij\vartheta}\frac{{\rm
d}\vartheta}{2\pi},\end{split}$ (7)

which retain all possible nonlinear wave-mixing processes and oscillations at
the carrier frequencies across all the relevant spectrum. Here,

$\begin{split}&\gamma_{3,j}=\frac{\omega_{j}b_{j}^{2}\pi}{n_{j}^{2}}\frac{\iint\chi^{(3)}F_{j}^{4}r{\rm
d}r{\rm d}z}{\iint F_{j}^{2}r{\rm d}r{\rm d}z},\leavevmode\nobreak\
[\gamma_{3,j}/2\pi]=\frac{\rm Hz}{\rm W},\\\
&\gamma_{2,j}=\frac{\omega_{j}b_{j}\pi}{n_{j}^{2}}\frac{\iint\chi^{(2)}F_{j}^{3}r{\rm
d}r{\rm d}z}{\iint F_{j}^{2}r{\rm d}r{\rm d}z},\leavevmode\nobreak\
[\gamma_{2,j}/2\pi]=\frac{\rm Hz}{\rm W^{1/2}},\end{split}$ (8)

are nonlinear coefficients. Numerical data presented in the main text further
assume that $\gamma_{3,j}=\gamma_{3,M}$ and $\gamma_{2,j}=\gamma_{2,2M+Q}$,
while the spectra accounting for dispersion of $\gamma_{2,j}$, $\gamma_{3,j}$
are included in [36] for comparison.

Equation (7) is our main result. Structurally, it can be matched with the
carrier-resolved pulse propagation models, see, e.g., Eq. (85) in [12] and
references therein. Apart from the pump and loss terms, a difference to be
noted is that Eq. (7) describes the evolution of spatial harmonics (resonator
modes) in time. At the same time, aforementioned carrier-resolved [12] and
multi-octave envelope models [9, 10, 28] deal with the evolution of
discretized frequency spectrum in the propagation coordinate. Hence, unlike
the above references, to reconstruct the frequency spectrum, we need to store
the data in time and then Fourier transform them to frequency,
$t\rightarrow\omega$, as post-processing. An advantage of our method for
describing resonator systems generating narrow spectral features is that the
frequency grid is not preset, and frequencies are captured as generated.

As a practical example, we consider a thin-film lithium niobate (LN)
microresonator with a geometry close to implemented in [35]. The ridge width
at the top surface is 1.45$\mu$m, ridge height is $360$nm, wall angle is
$60^{o}$, total LN thickness is $560$nm and the internal radius is 50$\mu$m,
so the repetition rate at $\omega_{M}/2\pi\approx 193$THz is $394$GHz.
Experimental measurements demonstrated that such geometry without periodic
poling and pumped with the $\sim$100mW of power produces 100THz wide Kerr
frequency combs with two strong dispersive waves emitted close to 150 and
250THz [35]. For the data sets with $\omega_{j}$, $\gamma_{2,j}$,
$\gamma_{3,j}$, and $g(\vartheta)$, see [36].

The nonlinear part of Eq. (7) is formulated in a pseudo-spectral form [12],
i.e., it requires going back to physical space to compute ${\cal E}_{0}$.
Fully spectral representation is also possible but computationally less
efficient [29]. In solving Eq. (7) numerically, the integrals and
transformation from the real field, ${\cal E}_{0}$, to the mode amplitudes,
$B_{j}$, and back, $\vartheta\leftrightarrow j$, were implemented using FFT.
The evolution in time was computed using the third-order Runge-Kutta method.
The number of modes was set to embrace the third harmonic. In that case, the
maximal frequency of oscillations produced by the $\gamma_{3,j}$-terms is well
estimated by $\varepsilon_{3}=3\omega_{M}-\omega_{3M}$. $\varepsilon_{3}$ is
zero in the absence of dispersion (equidistant resonator modes), revealing
that the maximal suitable time step is determined not by the optical frequency
but by the resonator dispersion, which is enhanced in integrated photonics due
to tight structural confinement. In our case, $|\varepsilon_{3}|/2\pi\approx
75$THz. In practice, stability and convergence of numerical code were achieved
for steps $\leq\pi/4|\varepsilon_{3}|\approx 1.7$fs. If the third-harmonic
modes are set to zero, then the near-phase-matched resonators can be
integrated using time steps $\sim 100$fs, determined by the minimal
frequencies produced by the $\gamma_{2,j}$-terms, $|\varepsilon_{2}|/2\pi$,
see Eq. (5). Setting second and third harmonics to zero increases permitted
steps to above $1$ps.

In our resonator, modes $M=404,405$ ($\omega_{M}/2\pi\approx 193$THz) are
quasi-phase-matched with $2M+Q$ for $Q=87,88,89$, so that $|\varepsilon_{2}|$
reaches its minimum, where it becomes order of $100$GHz. We start presenting
our simulation results from the strongly mismatched case $Q=10$. Bistable
response of the pumped resonance for power ${\cal W}=200$mW, which includes
(black line) and excludes (red line) $\chi^{(2)}$ nonlinearity is shown in
Fig. 1(a). One can see that this arrangement corresponds to the quasi-Kerr
regime since second-order nonlinearity makes little impact. Therefore, we
expect to find Kerr-soliton combs with two dispersive waves as were observed
in [35]. The shaded area in Fig. 1(a) shows an interval of the pump detuning
where we have found solitons and breathers by solving Eq. (7). A typical
spectrum and carrier-resolved soliton pulse are shown in Figs. 2(a), (b).

Selecting $Q=89$, we bring the resonator to quasi-phase-matching. Figures
1(b)-1(d) show the respective bistable curves for ${\cal W}=200$mW, $20$mW and
$7$mW of pump power. For all three of them, tilts of the resonance happen
dominantly through the $\chi^{(2)}$ nonlinearity and the soliton and comb
generation in the shaded intervals of detunings were found outside the Kerr-
only resonance (red line). Spectra computed in the $200$mW case, continuously
extend across three octaves, see Fig. 2(c). They correspond to the incoherent
randomly varying fields, which carrier-resolved form is shown in Fig. 2(d).

Reducing power to $20$mW and then to $7$mW we have observed formation of
$\chi^{(2)}$ solitons and breathers, see Fig. 3. For higher powers, these
solitons are surrounded by strong dispersive waves emitted around the pump and
second-harmonic. Dispersive wave emission is greatly reduced for powers less
than $10$mW, where the soliton combs become spectrally narrow. Spectral
resolution in our approach is set by the inverse integration time, which was
$88$ns for Fig. 3 providing $11$MHz resolution. A section of the recorded
pulse train, zoomed-on spectra, and real-field profiles of individual solitons
are also shown.

In summary: We developed the real-field carrier-resolved model to describe
optical frequency comb generation in ring microresonators. Our model is
capable of reproducing frequency combs with the multi-octave spectral width,
while simultaneously accounting for every mode physically present in the
resonator and all nonlinear wave-mixing terms entering the nonlinear
polarization specified via the real-valued electric field, see Eq. (3). For
example, if we consider $\chi^{(2)}$ nonlinearity and two envelope functions
$A_{1}$ and $A_{2}$ centred around pump and its second-harmonic, the opening
of the second-order nonlinear polarization
$\chi^{(2)}(A_{1}e^{i\phi_{1}}+A_{2}e^{i\phi_{2}}+c.c.)^{2}$ yields a
combination of ten terms. Here, $\phi_{1,2}$ are the phases oscillating with
optical frequencies. Omission of any term fails to be justifiable if the
spectra of $A_{1,2}$ approach an octave width, while our method is free from
this constraint and solves the problem for nonlinearities of arbitrary orders.
Overall, our results open an avenue for theoretical research into a growing
family of resonator systems requiring carrier-resolved multi-octave modelling
of ultrashort pulses [32, 33, 37].

Acknowledgments Authors acknowledge financial support from the Royal Society
(SIF/R2/222029) and thank A. Villois for discussions. Disclosures The authors
declare no conflicts of interest. Data Availability Statement Data included in
this study are openly available [36]. Author contributions DNP - coding and
data acquisition. DVS - concept and theory.

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MethodReferences

# Versatile Optical Frequency Division with Kerr-induced Synchronization at
Tunable Microcomb Synthetic Dispersive Waves

Grégory Moille<EMAIL_ADDRESS>Joint Quantum Institute,
NIST/University of Maryland, College Park, USA  Microsystems and
Nanotechnology Division, National Institute of Standards and Technology,
Gaithersburg, USA  Pradyoth Shandilya  University of Maryland at Baltimore
County, Baltimore, MD, USA  Alioune Niang  University of Maryland at Baltimore
County, Baltimore, MD, USA  Curtis Menyuk  University of Maryland at Baltimore
County, Baltimore, MD, USA  Gary Carter  University of Maryland at Baltimore
County, Baltimore, MD, USA  Kartik Srinivasan<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of Maryland, College Park, USA
Microsystems and Nanotechnology Division, National Institute of Standards and
Technology, Gaithersburg, USA

(August 31, 2024)

###### Abstract

Kerr-induced synchronization (KIS) provides a new key tool for the control and
stabilization of the repetition rate of a cavity soliton frequency comb. It
enables direct external control of a given comb tooth of a dissipative Kerr
soliton (DKS) thanks to its capture by an injected reference laser. Efficient
KIS requires its coupling energy to be sufficiently large, and hence both the
comb tooth and intracavity reference power must be optimized, which can be
achieved through higher-order dispersion that enables phase-matched dispersive
waves (DWs), where comb teeth are on resonance. However, such a design is
highly restrictive, preventing arbitrary use of reference wavelengths away
from the DW(s). In particular, for large spectral separations from the main
pump the cavity dispersion yields large detuning between comb teeth and their
respective cavity resonances, thereby decreasing the coupling energy and
rendering KIS to be highly inefficient or practically impossible. Here, we
demonstrate an alternative KIS method where efficient synchronization can be
tailored at arbitrary modes as needed. Using a multi-color DKS created from
multi-pumping a microresonator, a synthetic DW at the second-color wavepacket
can be selectively created where otherwise dispersion is far too large for KIS
to be experimentally feasible. Since a unique group velocity for both colors
exists thanks to cross-phase modulation, the repetition rate disciplining of
the secondary color wavepacket through its KIS automatically translates into
the DKS microcomb control. We first investigate this color-KIS phenomenon
theoretically, and then experimentally demonstrate its control and tuning of
the soliton microcomb repetition rate. As a consequence, we demonstrate
optical frequency division that is uncoupled from the main pump that generates
the DKS.

Figure 1:  Concept of color-Kerr-induced synchronization.  a Kerr-induced
synchronization (KIS) concept for optical frequency division (OFD). A
dissipative Kerr soliton is generated by driving a microring resonator with
the main pump. An injected reference laser captures the nearest comb tooth.
Any variation of frequency of the two locked pumps will be divided onto the
repetition rate of the DKS by a factor $\Delta\mu$ being the comb tooth
separation between the two pumps, and hence the DKS group velocity is
controlled through the reference pump. b Outside of synchronization, the
reference pump does not capture a comb tooth. However, cross-phase modulation
(XPM) binds the intracavity color triggered by this auxiliary pump, making it
travel at the same group velocity as the DKS. It results in a two-component
comb sharing the exact same repetition rate, yet offset in carrier envelope
offset from one another by the sum of the integrated dispersion value at this
resonance $D_{\mathrm{int}}(\mu_{\mathrm{aux}})$ and the auxiliary detuning
$\Delta\omega_{\mathrm{sec}}$. In this case, the repetition rate is not
disciplined through OFD. However, a new phase-matching condition occurs, where
a synthetic dispersive wave is created. c In color-KIS, we harness both
effects. The secondary color’s creation of a DW enables both large comb tooth
power and an on-resonance reference pump for efficient KIS – this time at the
secondary color. However, as XPM binds the two colors together, any tuning of
the repetition rate imprinted onto the secondary color is automatically
replicated onto the DKS. Hence, the auxiliary and reference pump can be
independently optically locked for OFD, while the main DKS pump remains free
ruining, since the relative comb color offset $\Omega$ absorbs any frequency
noise from the main pump.

Optical frequency combs (OFCs) are, in their low noise state, an essential
component in the metrology toolbox for frequency and time measurement [1, 2].
They can be used as a phase-coherent frequency multiplier/divider between the
microwave and optical domains for optical frequency synthesis [3], optical
clockworks [4, 5], time transfer [6, 7], and ultra-low noise microwave
generation [8, 9], while also increasingly finding application in more
widespread consumer technologies such as distance ranging [10]. Integrating
such OFCs on chip has been a recurrent quest over the last decade, in
particular with the demonstration of dissipative Kerr soliton (DKS) microcombs
which convert a continuous wave (CW) pump into a periodically-extracted pulse
train [11]. DKSs at their core rely on the dispersion compensation by Kerr
nonlinearity present in several CMOS-compatible materials, such as silicon
nitride (Si3N4), allowing for low-cost and mass-scale fabrication of high
quality factor microring [12] or Fabry-Perot resonators [13] through which
DKSs can be generated with low pump power [14]. Although the combs’ technical
requirements may vary from one application to another, the control and/or
stabilization of their repetition rate is in general a key element [2]. In
particular, optical frequency division (OFD) relies on pinning two comb teeth
to stable lasers, leveraging the fundamental property that their noise will be
divided onto the repetition rate by their comb tooth separation [15]. Although
on-chip operation for the laser stabilization [16] and the comb generation
[17, 18] can be achieved, the overall architecture mostly relies on lab-scale
equipment for the comb-laser locking with only one degree of freedom (i.e. the
main pump). Recently, Kerr-induced synchronization (KIS) has been demonstrated
as an all-optical approach to address the technical challenge of optically
locking the comb [19]. In KIS, the injection of another reference pump laser
into the comb-generating resonator results in the closest comb tooth snapping
onto the reference [Fig. 1(a)], enabling external control of two comb teeth
through both pumps. This method has been successfully shown to enable all-
optical OFD through locking of both pumps [19, 20], with performance on par
with other techniques [17, 21].

Although KIS greatly simplifies an optical clockwork [19, 22], it can be
challenging to implement for other OFD applications. First, it fixes the main
pump, rendering the system stiff if low phase-noise is to be achieved. This is
contrary to a typical external dual-pinning architecture, which uncouples the
DKS generation from the locked reference lasers. Thus, OFD with KIS prevents
the main pump from being directly freely tuned across the resonance to
generate the DKS, or using the promising approach of self-injection locking
for turnkey-DKS generation [23, 24, 25], which complicates further
integration. Secondly, this KIS phase locking efficiency relies on the KIS
coupling energy, which can be expressed as the geometrical mean of the intra-
cavity power of the comb tooth and the reference pump upon synchronization
[19, 20]. The injection of the reference laser at a dispersive wave (DW)
enhances the KIS energy since the comb tooth is resonant, maximizing both the
tooth and reference intracavity power. However, DWs are not always present,
especially in microcombs with directly detectable, low repetition rates that
are often achieved in resonators with pure quadratic dispersion. Without a DW,
KIS can occur, but at a significantly higher reference pump power [20]. In
addition, though the larger OFD factor provided by placing the reference far
from the main pump would be preferred, the smaller KIS coupling energy due to
the resonator dispersion is a hindrance, as the KIS bandwidth could be
narrower than the laser linewidth, making synchronization practically
unachievable. Here we propose a new scheme that uncouples the main pump
soliton – which can remain free-running – with the all-optical OFD enabled
through KIS, by using a multi-color cavity soliton system [26].

Figure 2: Theoretical study of the color-KIS effect. a Two-wave LLE simulation
results for the DKS (gray) and secondary color (blue), using the dispersion
profile similar to experimental system (red, right scale). The auxiliary pump
is held fixed and on-resonance at $\mu=-90$. The reference pump, which is
injected at the (high frequency) second-color DW, is adiabatically swept. The
power is normalized to 1 mW. b Repetition rate variation of the DKS against
time, with a linear ramp of the reference pump detuning. Once close enough to
the secondary-color comb tooth, the reference pump grabs it and modifies the
secondary color repetition rate $\omega_{\mathrm{rep}}$ (blue). Through XPM,
the DKS repetition rate is also modified. We compare it against direct KIS at
the same mode $\mu_{\mathrm{ref}}=-45$ (red), resulting in an opposite
$\omega_{\mathrm{rep}}$ slope, with a KIS bandwidth (BW) about 10$\times$
smaller than the color-KIS BW with the same reference pump power. c and d show
evolution of the DKS and secondary color in the resonator angle against the
slow time $t$ (where $\approx$ 1 ps corresponds to one soliton round trip),
hence the reference detuning. Their trajectories follow the same path as their
group velocities are bound, even when the synchronization occurs at the
secondary color. Outside of the color-KIS, their drift in the slow time $t$ is
the same highlighting the absence of impact of the reference onto the soliton
group velocity. In the second-color synchronization regime, the drift becomes
dependent with the reference detuning, hence, color-KIS provides a means to
change the DKS repetition rate.

Physical concept of the color-KIS - Let’s remind of the multi-pumped DKS
operation outside of synchronzisation. In this case, the second injected laser
does not become a DKS comb tooth [Fig. 1(b)], yet cross-phase modulation (XPM)
is sufficient to lock the group velocity of the auxiliary and DKS components
in the cavity [27]. Thus, these two traveling waves only differ by their phase
velocity mismatch, translating into the output waveguide as two frequency comb
components with the same comb teeth spacing yet with a shift of carrier-
envelope offset (CEO) from one another. Here, we refer to each component (i.e.
the auxiliary and the DKS wave) as colors, defined by their respective phase
velocity (CEO in the spectral domain). Interestingly, using this scheme
enables a new phase matching condition at the second color [28], whose
location is highly flexible since it can be widely tuned by hopping between
cavity modes, or finely tuned through tuning within a cavity mode, without
disrupting the DKS state. This results in the creation of DW(s) that can be
harnessed, for instance, for spectral extension [26, 29], where high power
comb teeth can be generated at frequencies of interest. As these new
dispersive waves are not solely governed by cavity dispersion, but are instead
tunable based on the secondary laser frequency (i.e. tuning the CEO difference
between the colors), we refer to them as synthetic dispersive waves. Here, we
harness both KIS and the synthetic dispersive waves produced in multi-color
solitons [Fig. 1(c)]. By injecting an auxiliary laser on resonance with a
cavity mode, hence shifted from its nearest comb tooth because of the
resonator dispersion, we create a second color at the same group velocity as
the DKS, yet with a spectral shift of tens of gigahertz from the DKS carrier-
envelope offset frequency. We show that the second color, which automatically
presents at least one DW, can undergo a power efficient KIS by a reference
pump (a third laser injected into the resonator). Through XPM, the KIS of the
second color is automatically transferred onto the DKS, enabling control
and/or stabilization of the DKS repetition rate. Importantly, in this scheme
the main pump does not require any locking, since all its noise will be
absorbed by the spectral offset between the two comb components while the
repetition rate remains locked through the OFD between the auxiliary and
reference lasers. Also, as this color-KIS does not impose specific
requirements on the resonator dispersion, it widely expands the range of
devices and scenarios in which KIS can be utilized to synchronize DKS
microcombs to optical references.

Theoretical study of the color-KIS – We aim to study this effect
theoretically. The addition of another driving field could be introduced in
two fashions which modify the Lugiato-Lefever equation (LLE) that is commonly
used to study damped and driven Kerr nonlinear resonators within a mean-field
limit. First, it can be simply introduced in the single LLE by adding another
pump term that accounts for the frequency offset
$D_{\mathrm{int}}(\mu_{\mathrm{aux}})$ due to the cavity dispersion from the
fixed frequency grid spaced by $D_{1}$ (defined by the DKS repetition rate),
as shown in [30]. Although this multi-pump LLE approach captures the Kerr-
induced synchronization phenomenon [19], it does not fully capture the multi-
color aspect. Instead, a set of coupled equations can be used, the first one
at the DKS phase while the second presents a phase offset, with the two
coupled through XPM. To account for the full nature of the system and the
nonlinear mixing in the phase domain, another set of equations shifted by an
integer multiple of this phase offset, hence accounting for parametric
nonlinear phase mixing [31], can be introduced. However, the creation of a so-
called two-dimensional comb [32] is not necessary to understand the concept of
color-KIS that we focus on in this work, and hence it is sufficient to focus
on the DKS color $a_{\mathrm{dks}}$ driven by the main pump with power
$P_{\mathrm{main}}$, and the secondary color $a_{\mathrm{sec}}$ pumped by the
auxiliary laser with power $P_{\mathrm{aux}}$ such that:

$\displaystyle\frac{\partial a_{\mathrm{dks}}}{\partial t}$
$\displaystyle=(-\frac{\kappa}{2}+i\delta\omega_{\mathrm{main}})a_{\mathrm{dks}}+i\Sigma_{\mu}D_{\mathrm{int}}(\mu)A_{\mathrm{dks}}\mathrm{e}^{i\mu\theta}$
$\displaystyle+i\gamma
L(|a_{\mathrm{dks}}|^{2}+2|a_{\mathrm{aux}}|^{2})a_{\mathrm{dks}}$ (1)
$\displaystyle+i\sqrt{\kappa_{\mathrm{ext}}P_{\mathrm{main}}}$
$\displaystyle\frac{\partial a_{\mathrm{sec}}}{\partial t}$
$\displaystyle=\left(-\frac{\kappa}{2}+i\delta\omega_{\mathrm{sec}}-iD_{\mathrm{int}}(\mu_{\mathrm{aux}})\right)a_{\mathrm{sec}}$
$\displaystyle+i\Sigma_{\mu}D_{\mathrm{int}}(\mu)A_{\mathrm{sec}}\mathrm{e}^{i\mu\theta}$
$\displaystyle+i\gamma
L(2|a_{\mathrm{dks}}|^{2}+|a_{\mathrm{sec}}|^{2})a_{\mathrm{sec}}$ (2)
$\displaystyle+i\sqrt{\kappa_{\mathrm{ext}}P_{\mathrm{aux}}}e^{\mathrm{i\mu_{aux}\theta}}$

with $A_{\mathrm{dks}}$ and $A_{\mathrm{sec}}$ the respective Fourier
transforms of $a_{\mathrm{dks}}$ and $a_{\mathrm{sec}}$ in the azimuthal angle
$\theta$/mode $\mu$ of the microring. The resonator properties that we are
using are from the experimental system such that $\gamma=2.3$ W-1$\cdot$m-1 is
the effective nonlinearity calculated at the main pump [19, 26],
$\kappa/2\pi=1.1$ GHz the total loss rate,
$\kappa_{\mathrm{ext}}=\frac{1}{2}\kappa$ the external coupling rate (i.e.,
$Q_{c}=Q_{i}=500\times 10^{3}$ the coupling and intrinsic quality factor
respectively), $L=2\pi\times 23$ µm the resonator circumference, and
$D_{\mathrm{int}}(\mu)$ is the integrated dispersion accounting for the DKS
repetition rate $\omega_{\mathrm{rep}}/2\pi\approx 1$ THz. The external
parameters we experimentally control are the main and auxiliary pump power
$P_{\mathrm{main}}=200$ mW and $P_{\mathrm{aux}}=5$ mW, respectively,
$\mu_{\mathrm{aux}}$ is the auxiliary pumped mode, and
$\delta\omega_{\mathrm{main}}=\omega(\mu=0)-\omega_{\mathrm{main}}$ and
$\delta\omega_{\mathrm{sec}}=\delta\omega_{\mathrm{aux}}-\delta\omega_{\mathrm{main}}=\omega(\mu_{\mathrm{aux}})-\omega_{\mathrm{aux}}$
are the detuning of the main and secondary color respectively, with
$\delta\omega_{\mathrm{aux}}$ the detuning of the auxiliary pump relative to
its resonance mode and $\omega(\mu)$ the resonance frequency at mode $\mu$.

Since the two colors are coupled through XPM, they exhibit the same group
velocity, resulting in two equally-spaced comb components offset from one
another by $\Omega$. As presented in refs [32, 31], this multi-color LLE
formalism allows for straightforward comprehension of the phase matching
resulting in the creation of DWs. Since the secondary color is offset from the
main DKS by
$\Delta\Omega=D_{\mathrm{int}}(\mu_{\mathrm{aux}})+\delta\omega_{\mathrm{sec}}$,
any zero crossing of $D_{\mathrm{int}}(\mu)-\Omega$ will exhibit synthetic DW
features at this color. With an anomalous dispersion regime being defined by a
concave quadratic integrated dispersion, at least one DW will be created at
the secondary color (e.g., at $-\mu_{\mathrm{aux}}$). For the geometry we
study in the experiments presented later, higher-order dispersion is present,
resulting in a roll-off in $D_{\mathrm{int}}$ with an eventual zero-crossing
and generation of a DKS-DW. We auxiliary pump away from the DKS-DW, with the
higher-order dispersion providing a phase-matching point that is on the same
side of the main pump [Fig. 2(a)].

We simulate the multi-color LLE system with a main pump at $\mu=0$, an
auxiliary pump at $\mu_{\mathrm{aux}}=-90$, and dispersion profile presented
in Fig. 2(a). We introduce the reference laser that triggers the
synchronization through the inclusion in Eq. 2 of an additional driving force
$\sqrt{\kappa_{\mathrm{ext}}P_{\mathrm{ref}}}\mathrm{exp}\left(i\Delta\Omega+i\mu_{\mathrm{ref}}\theta\right)$,
with $\Delta\Omega=\delta\omega_{\mathrm{sec}}-\delta\omega_{\mathrm{ref}}$,
$\delta\omega_{\mathrm{ref}}$ the reference pump detuning, and
$\mu_{\mathrm{ref}}=-45$ being the mode (relative to the main pump) at which
we send the reference pump. We adiabatically and linearly sweep the detuning
of the reference pump while seeding the initial solution with the soliton and
secondary color solution for $a_{\mathrm{DKS}}(\theta,t=0)$ and
$a_{\mathrm{sec}}(\theta,t=0)$, respectively, and keeping both main and
auxiliary pump fixed.

When out-of-synchronization, the simulation highlights that the auxiliary and
DKS colors travel at the same group velocity [Fig. 2(c-d)] since the DKS
drifts in $\theta$ at the same rate as the secondary color. Once the reference
laser is within the KIS bandwidth, it captures the secondary-color comb tooth,
stretching or compressing the microcomb and changing the secondary-color
repetition rate [Fig. 2(b)] with a slope given by the optical frequency
division (OFD) factor set by
$\Delta\mu=\mu_{\mathrm{ref}}-\mu_{\mathrm{aux}}=45$, and hence controlling
its group velocity. Because of the XPM binding, the DKS must follow any
variation of group velocity imprinted onto the secondary color, which is
apparent using the two-wave LLE formalism as they drift together. Hence, even
if the the synchronization is performed on the secondary-color, the DKS
repetition rate can be controlled. We can compare this scheme against a direct
KIS at the same mode $\mu_{\mathrm{ref}}$, where the DKS comb does not present
any DW and its shape closely follows a $\mathrm{sech}^{2}$ spectral envelope.
The KIS bandwidth $\Delta\omega_{\mathrm{kis}}=2\omega_{\mathrm{ref}}\Delta\mu
E_{\mathrm{kis}}$, with
$E_{\mathrm{kis}}=\sqrt{\frac{\kappa_{\mathrm{ext}}}{\kappa^{2}}P_{\mathrm{ref}}|A_{X}(\mu_{s})|^{2}}/E_{\mathrm{X}}$,
$E_{X}$ being the total energy with $X$ the soliton or secondary color, and is
directly proportional to the energy of the synchronization mode. Hence, for a
fair comparison, we have chosen this particular configuration as it
conveniently results in the same OFD factor $\Delta\mu$ for the direct and
color-KIS, with only a change of sign changing the direction of the
$\omega_{\mathrm{rep}}$ trend against reference pump detuning. In the direct-
KIS scenario, the available comb tooth energy
$|A_{\mathrm{sec}}(\mu_{s})|^{2}$ is considerably lower, by about 17 dB
compared to the synthetic DW. In addition, the intracavity reference pump does
not experience resonance enhancement (and could be considered single-pass) due
to the close to 40 GHz offset of the comb and its nearest cavity mode – with
assumed losses resulting in an offset of more than 40 resonance linewidths –
preventing any significant resonant power build-up. This translates to a KIS
bandwidth about 10 times smaller than the color-KIS bandwidth, with 130 MHz
and 1.53 GHz synchronization bandwidths, respectively [Fig. 2(b)].

Figure 3:  DKS and new DW generated via auxiliary pumping. a Integrated
dispersion from wavemeter-calibrated resonance frequencies (50 MHz
uncertainty, within marker size) measurement of the fundamental transverse
electric mode (circles) and from the finite-element method dispersion modeling
(solid line), showcasing close agreement and used in Fig. 2. The phase
matching with the auxiliary pump for where creation of a new synthetic DW
appears is shown in the gray dashed line. b Single-pumped DKS microcomb
spectrum (gray) obtained using the counterpropagative cooler method [29, 33].
The secondary-color (blue) is generated by the auxiliary pump, creating a new
synthetic DW where the reference pump can be sent to trigger the color-KIS.
The DKS and secondary color are offset from each other by about +40 GHz (left
inset), in accordance with the integrated dispersion value at the auxiliary
pump. For subsequent measurements in Fig. 4, we measure the repetition rate of
only the DKS using an electro-optic comb apparatus that modulates two adjacent
DKS comb teeth (right inset; green is the electro-optically modulated
spectrum). The power is normalized to 1 mW. c and d are optical spectra for
different auxiliary pumped modes in the 190 THz band and 230 THz band,
respectively. Compared to the single-pumped DKS (gray), the available comb
tooth power for KIS is increased by 25 dB and 32 dB, respectively.  Figure 4:
Repetition rate control and optical frequency division through color-KIS.
Characterization of the repetition rate entrainment (top) and beat note
between the reference and secondary color (bottom) for different mode spacing
$\Delta\mu$ between auxiliary and reference pumps. The repetition rate
$\omega_{\mathrm{rep}}$ is only measured at the DKS. Both
$\Delta\omega_{\mathrm{rep}}$ and $\Delta\Omega$ are extracted from
simultaneous temporal measurement and calibrated according to Appendix S.2.
The color-KIS regime is highlighted in the green area, where a linear change
of $\omega_{\mathrm{rep}}$ following the $\Delta\mu$ OFD factor is observed
(dashed lines). We emphasize that this trend is neither offset nor scaled
thanks to accurate measurement of the zero detuning. Simultaneously,
$\Delta\Omega$ is locked to zero and hence the reference pump is capturing its
nearest secondary-color comb tooth. a Measurements in the case of a fixed
auxiliary pump in the 190 THz band and a swept reference pump in the 230 THz
range. b Measurements in the case of a fixed auxiliary pump in the 230 THz
band and a swept reference pump in the 190 THz range. The change in sign of
the slope of $\Delta\omega_{\mathrm{rep}}$ within the color-KIS window between
a and b is consistent with the change in sign of $\Delta\mu$.

Experimental nonlinear characterization of the microring –  Experimentally, we
use silicon nitride (Si3N4) microring with a nominal thickness of $H=670$ nm
embedded in SiO2 with a designed ring width of $RW=890$ nm, presenting a
similar dispersion profile as the one used in simulations in Fig. 2. The
higher-order dispersion results in $D_{\mathrm{int}}(\mu)$ roll-off, enabling
the creation of new synthetic dispersive wave phase-matching in the 230 THz
range (i.e., 1300 nm range) for auxiliary pumping in the 190 THz range (i.e.,
1550 nm range), and vice-versa. The experimental dispersion is measured using
wavemeter-calibrated resonance frequencies of the first order transverse
electric mode [Fig. 3a]. The experimental setup is discussed in detail in
Appendix S.1. To generate the DKS, we use a 283 THz pump with about 150 mW on-
chip power. The DKS is adiabatically accessed using a 308 THz
counterpropagating cooler laser that pumps a fundamental transverse magnetic
mode [Fig. 3b], which thermally stabilizes the resonator and enables long-term
operation of the DKS state [34, 33]. To generate the secondary color, we use
continuously tunable lasers (CTLs) spanning between 184 THz to 198 THz with
about 2 mW of on-chip power, and 227 THz to 232 THz with about 1 mW of on-chip
power. One of these lasers acts as the auxiliary pump that creates the second
color (i.e., is fixed), while the other is used as the reference. It is
apparent that no direct-KIS could occur between the DKS and the secondary
color since the two comb components are separated by about 40 GHz (resolved
with an optical spectrum analyzer; see left inset to Fig. 3b). We further
verify the flexibility and efficiency of the creation of new synthetic DWs by
auxiliary pumping different modes in the 190 THz band and 230 THz band. Figure
3c-d highlights the creation of a new DW in these other laser bands, and how
the synthetic DW spectral position, which will determine where the KIS
reference laser is applied, can be flexibly controlled by tuning the auxiliary
laser frequency. A significant improvement of 25 dB (190 THz band auxiliary
pump) or 32 dB (230 THz band auxiliary pump) of the comb tooth power available
at the new DW where the color-KIS will happen is observed compared to the
nearest DKS comb tooth. Along with the possibility to send the reference on-
resonance, it confirms the theoretical investigation previously conducted that
color-KIS will be more highly efficient than conducting a direct-KIS at these
modes.

DKS repetition rate entrainment from color-KIS –  To characterize the color-
KIS we measure at the same time the repetition rate $\omega_{\mathrm{rep}}$
and the frequency offset between the reference laser and the secondary color
$\Delta\Omega$ (see experimental setup in Appendix S.1). As our focus is on
DKS repetition rate control, the repetition rate of the system is measured
only at the DKS color. This is done by measurement around 272 THz, where the
secondary color presents negligible power in comparison to the DKS, as evident
through a comparison of the spectra with and without the auxiliary/reference
pumps [Fig. 3b]. We use an electro-optic comb apparatus [19, 35] to modulate
two adjacent DKS comb teeth and frequency translate
$\omega_{\mathrm{rep}}\approx 1$ THz to a detectable microwave beat note; an
example spectrum is shown in the right inset to Fig. 3b. We sweep the
reference laser using an electrical ramp signal sent to the CTL piezo element,
enabling small detuning to be studied, and measure both
$\omega_{\mathrm{rep}}$ and $\Delta\Omega$ by recording simultaneously their
temporal trace with a fast oscilloscope. We then process these data to create
a spectrogram where the temporal dependence of $\omega_{\mathrm{rep}}$ and
$\Delta\Omega$ are obtained. The detuning calibration is obtained through
beating the reference laser against the secondary color comb tooth while
bypassing the resonator (i.e., no nonlinearity, hence no synchronization),
calibrating the amplitude and the zero of the detuning. For more details, the
complete procedure is described in Appendix S.2.

First, we use the 190 THz CTL as the auxiliary pump generating the secondary
color, and keep its detuning fixed. We use the 230 THz CTL to act as the
reference and it is frequency swept close to the secondary color comb tooth
[Fig. 4a]. We use the different modes that have been presented in Fig. 3c-d,
resulting in a variation of $\Delta\mu=25$ to $\Delta\mu=41$. A clear change
in the repetition rate following a linear trend with the reference detuning
occurs when $\Delta\Omega$ is locked to zero, consistent with conventional KIS
behavior, except in these measurements $\Delta\Omega=0$ is at the second color
while the measurement of the repetition rate variation
$\Delta\omega_{\mathrm{rep}}$ is at the DKS color, highlighting the XPM
interplay linking the color-KIS to the DKS. We verify the linear trend of the
repetition rate entrainment by overlaying the reference laser detuning divided
by $\Delta\mu$, which closely matches $\Delta\omega_{\mathrm{rep}}$. We note
here that this linear trend is plotted without offset or scaling, since the
zero detuning is accurately calibrated using the technique presented in
Appendix S.2 and since the OFD factor is a simple comb tooth counting. We can
then switch the system, where the 230 THz CTL becomes the fixed auxiliary pump
creating the secondary color while the 190 THz CTL becomes the swept reference
pump [Fig. 4b]. Similar results as previously described are obtained, yet this
time the OFD coefficient $\Delta\mu=\mu_{\mathrm{ref}}-\mu_{\mathrm{aux}}$ is
negative. The repetition rate entrainement slope becomes negative when
$\Delta\Omega=0$, and is once again accurately captured by a simple linear
trend accounting for $1/\Delta\mu$. We note that this experiment has been
conducted while maintaining the same exact DKS state, highlighting the little
impact the secondary color and its synchronization has on the stability of the
DKS, further providing flexibility to the system for on-the-fly reconfigurable
OFD.

Figure 5:  Color-KIS noise optical frequency division onto the DKS repetition
rate. Optical frequency division characterization of the repetition rate phase
noise for different values of $\Delta\mu$. The auxiliary laser and reference
laser are both locked to a 40 MHz free-spectral range Mach-Zehnder
interferometer, with the 230 THz range CTL carrying most of the optical noise.
Once divided by the appropriate OFD factor $\Delta\mu^{2}$, the laser phase
noise (shown in red) overlaps with the DKS $\omega_{\mathrm{rep}}$ phase noise
(shown in dark blue to light blue for decreasing $\Delta\mu$) according to the
KIS principle. Here, the stability of the DKS repetition rate comes from KIS
of $\omega_{\mathrm{rep}}$ of the secondary-color, which is then transferred
to the DKS $\omega_{\mathrm{rep}}$ through XPM. The power is referenced to
that of the carrier, namely, dBc/Hz

DKS repetition rate phase noise from color-KIS –  We investigate the phase
noise of the DKS repetition rate while the second color becomes synchronized.
For the different OFD factors presented in Fig. 4, we enter the color-KIS
regime and lock both auxiliary and reference lasers to independent fiber Mach-
Zehnder interferometers (MZIs) with a 10 MHz free spectral range, while the
main pump and cooler pump are free running. Using a third similar MZI, we
measure the optical phase noise of both pump lasers. We note that in this
measurement we account for the noise of both lasers; however, since the 230
THz laser carries larger phase noise while locked than the 190 THz laser (see
Appendix S.3), only accounting for the former to compare against
$\omega_{\mathrm{rep}}$ phase noise enables verification of OFD. For each
value of $\Delta\mu$, the repetition rate is well-matched to the laser phase
noise divided by the OFD factor $\Delta\mu^{2}$ [Fig. 5]. Although the
repetition rate entrainement presented in Fig. 4 already demonstrated OFD
based on the color-KIS technique, these additional phase noise measurements
highlight that the XPM-mediated transfer of synchronization of the second
color onto the DKS does not carry additional noise, so that color-KIS can be
used to stabilize a DKS repetition rate indirectly. We further note that for
$\Delta\mu=30$, we have swept the reference pump over a range of about 120 MHz
at a 75 Hz rate following a triangular waveform, and confirmed that no
additional noise is present in the repetition rate phase noise. This is
consistent with $\omega_{\mathrm{rep}}$ being locked for both colors, while
the frequency offset between the two combs is not. Essentially, while in the
color-KIS regime, all of the phase noise of the main pump is transferred onto
the carrier-envelope offset noise of the DKS.

Large tunability of color-KIS frequency –  Finally, we address the large
tunability that the color-KIS enables. In contrast to the standard-KIS scheme,
the color-KIS does not occur on the fixed frequency markers defined by the
repetition rate and the main pump (i.e., the CEO frequency). Instead, the
secondary colors live at a CEO offset from the main pump, which can be widely
tuned through mode hopping between cavity resonances, as previously
demonstrated [Fig. 3(c)-(d)]. Additionally, a finer tuning of the CEO
frequency offset of the second color can be obtained through auxiliary pump
detuning from its resonance, resulting in the tuning of the synthetic DW [28].
In particular, since the quality factor and coupling regime requirements are
much less stringent than for DKS generation, one can harness the natural
overcoupling at lower frequencies than the main pump from straight-waveguide
coupling [36], resulting in large secondary color CEO offset tunability while
still dropping significant auxiliary power from the waveguide to the cavity.
The repetition rate of the DKS remaining mostly unperturbed with the auxiliary
pump detuning enables one to tune the secondary-color comb tooth at which the
color-KIS occurs. We demonstrate this effect experimentally using a OFD factor
$\Delta\mu=34$ between the auxiliary and reference pump in the same DKS
presented previously [Fig. 6]. The reference laser is swept, similarly to Fig.
4, while the auxiliary pump is fixed. The detuning of the auxiliary pump
$\delta\omega_{\mathrm{aux}}$ enables all the comb teeth of the secondary
color to also be shifted by $\delta\omega_{\mathrm{aux}}$, thus shifting the
color-KIS window accordingly. We demonstrate a tuning of over 10 GHz of the
KIS window with $\approx$5 mW of on-chip power for both pumps, respectively,
corresponding to a tunability of about ten times the optimized color-KIS
bandwidth at this power when both pumps are at the center of their respective
resonances. This tunability is only limited by the available power of the
auxiliary pump, and could be further increased by using a more powerful laser
while remaining in the thermal stability range of the DKS. Such tunability is
a key feature of the color-KIS scheme, as it enables synchronization to occur
at a precise target frequency, for instance to match a fixed frequency ultra-
stable laser.

Figure 6:  Tunability of the secondary color comb tooth for color-KIS
frequency versatility – Using the same DKS previously presented, we sweep the
reference laser around 226 THz while the auxiliary pump is parked around the
cavity mode at 192 THz, leading to an OFD factor $\Delta\mu=34$ consistent
with the $\Delta\omega_{\mathrm{rep}}$ entrainment slope against the reference
detuning $\delta\omega_{\mathrm{ref}}$. The auxiliary pump detuning
$\delta\omega_{\mathrm{aux}}$ is tuned in steps over 10 GHz and the color-KIS
frequency window is tuned accordingly, demonstrating the possibility to match
a precise target frequency at which the synchronization shall happen.

Discussion –  In conclusion, we have demonstrated a novel way to efficiently
access the Kerr-induced synchronization regime, enabling cavity soliton
repetition rate control, and optical frequency division with all-optical
synchronization of the comb. Using the fundamental group velocity binding
between a DKS color and a secondary color created by an auxiliary laser
traveling in the same microring resonator, we create new phase-matching
conditions which enable new synthetic DWs that can be used for
synchronization, and whose spectral locations can be flexibly tuned. Compared
to synchronization to a reference laser at the same frequency using a single-
pumped DKS, the comb tooth power is orders of magnitude larger while providing
on-resonance operation for the reference laser to efficiently transfer its
power inside the resonator. We show theoretically that the KIS bandwidth for a
similar situation between direct and color-KIS is increased by at least an
order of magnitude. Experimentally, we have demonstrated this effect using an
integrated microring resonator, where repetition rate entrainment according to
the optical frequency division factor between the auxiliary and the reference
pump is verified. The spectral separation between the auxiliary and reference
pump, and hence the division factor, can be widely controlled with respect to
the underlying cavity dispersion without disrupting the stability of the DKS.
We also show that the locking of the group velocity of the second-color
through auxiliary and reference pump locking in the color-KIS regime is
transferred onto the DKS repetition rate in a noiseless fashion, despite the
indirect control of the DKS through the XPM-coupled secondary color. Finally,
we have demonstrated the versatility of the color-KIS approach in comparison
to the standard KIS one, where the synchronization frequency (and window) can
be largely tuned without impacting the DKS. Our work enables a new efficient
way to control the large repetition rate of integrated microresonator combs,
in particular in situations where the cavity dispersion could otherwise
prevent direct KIS.

We further discuss a few important implications of the color-KIS effect. For
example, KIS-enabled optical frequency division has recently been shown in the
context of low noise microwave generation with a greatly simplified
architecture [21]. Our color-KIS scheme enables further important
simplifications. Indeed, it is interesting to note that the main pump does not
play a role in the repetition rate control and stabilization since the CEO
offset between colors $\Delta\Omega$ is unlocked and therefore uncoupled from
optical frequency division approach. This enables the main pump to be freely
tuned as needed to generate the DKS, without further conditions on its
stability. In addition, although we have mostly focused on demonstrating the
control of the single repetition rate of the different colors, it is important
to point out that the secondary color forms its own frequency comb whose
repetition rate can also be detected and stabilized. As a result, the color-
KIS scheme brings the possibility to coherently phase lock an auxiliary laser
to an ultra-narrow linewidth laser, through the repetition rate feedback onto
the auxiliary laser. As noted above, in color KIS, such a scheme is
independent of the main pump; indeed, the main DKS essentially acts as an
intermediary whose XPM interactions with the second color enable a single
group velocity to exist. Finally, the creation of synthetic DWs can be
extended to other colors, which has been demonstrated to expand the comb
bandwidth well beyond an octave [26], and hence the color-KIS scheme may find
application in the stabilization of such ultra-broadband integrated frequency
combs. Beyond the above, our work could find direct application in distance
ranging while enabling new techniques for fundamental studies of DKS dynamics
in micrometer-scale resonators.

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Acknowledgments
The photonic chips were fabricated in the same fashion as the ones presented
in [19].

The Scientific colour map batlow [37] and subsequent color set is used in this
study to prevent visual distortion of the data and exclusion of readers with
colour-vision deficiencies [38].

We acknowledge partial funding support from the Space Vehicles Directorate of
the Air Force Research Laboratory, the Atomic–Photonic Integration programme
of the Defense Advanced Research Projects Agency, and the NIST-on-a-chip
program of the National Institute of Standards and Technology. P.S and C.M.
acknowledges support from the Air Force Office of Scientific Research (Grant
No. FA9550-20-1-0357) and the National Science Foundation (Grant No.
ECCS-18-07272). We thank Sean Krzyzewski and Marcelo Davanço for insightful
feedback.

Author Contributions
G.M. and K.S. led the project. G.M. designed the resonators and performed the
measurements and simulations. P.S. and C.M. helped with the theoretical and
numerical understanding. A.N. and G.C. helped with the experimental
understanding. K.S. helped with data analysis. G.M. and K.S. wrote the
manuscript, with input from all authors. All the authors contributed and
discussed the content of this manuscript.

Competing Interests
G.M., C.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.

Code availability
The simulation code is available from the authors through the pyLLE package
available online [39], with a modification that is available upon reasonable
request, using the inputs and parameters presented in this work.

##  Appendix S.1 Experimental Setup

Figure S.1:  Experimental setup. A main pump around 283 THz is obtained from
an amplified 1060 nm CTL providing about 150 mW of on-chip power to generate a
DKS, which is then adiabatically accessed thanks to the amplified and
counterpropagating 980 nm CTL cooler pump that thermally stabilizes the
resonator. We introduce an auxiliary pump and a reference pump to create the
secondary color and trigger the color-KIS with 1300 nm and 1550 nm CTLs. We
measure the DKS repetition rate using two cascaded EO phase modulators which
we apply to two adjacent Kerr comb lines around 1100 nm. EO-generated
sidebands of the two DKS teeth span across the microcomb repetition rate, with
high-order sidebands that come close to merging. We filter around this
frequency, allowing retrieval of the beat note with a slow ($<$50 MHz
bandwidth) APD. The beating between the secondary color and the reference pump
is obtained with negligible DKS power thanks to the WDM. Both the repetition
rate and the reference beat note are measured in the time domain and then
processed and calibrated to obtain the data presented in this work. Other
elements such as power meters and photodiodes to verify transmission levels
are not displayed to avoid over-complicating the schematic. CTL: continuously
tunable laser, YDFA: ytterbium-doped fiber amplifier, TA: tapered amplifier,
PC: polarization controller, WDM: wavelength demultiplexer, RF:
radiofrequency, OSA: optical spectrum analyzer, PD: photodiode, APD: avalanche
photodiode, EO: electro-optic

The experimental setup used in this work is depicted in Fig. S.1. We use a
1060 nm continuously tunable laser (CTL) amplified with an ytterbium-doped
fiber amplifier (YDFA) to generate enough power at 283 THz to pump the
microring to harness the resonator’s third-order nonlinearity. An on-chip
power of about 150 mW is sufficient to reach a dissipative Kerr soliton (DKS)
state in the fundamental transverse electric mode (TE) of the 23 µm Si3N4
microring resonator. To help access this state, we use a 980 nm CTL amplified
using a taper amplifier (TA) and set to be counterpropagative and cross-
polarized relative to the main pump, so that this cooler pump only thermally
stabilizes the resonator for adiabatic access to the DKS [34, 33] while
minimizing potential nonlinear mixing with the main pump and/or the DKS. We
use a combinations of wavelength demultiplexers (WDMs) at the input and output
to avoid any injection of the cooler (main pump) into the main pump (cooler).
The auxiliary and reference laser from the 1300 nm and 1550 nm CTLs are
injected in the resonator by combining them with the main pump using a WDM,
and are also set in the TE polarization. We tap 50 % of the output to probe
the microcomb optical spectrum using an optical spectrum analyzer (OSA). We
use another WDM at the other 50 % output to separate wavelengths close to 1100
nm, which are electro-optically (EO) modulated at
$\Omega_{\mathrm{RF}}=17.87892$ GHz. The EO-generated sidebands of two
adjacent DKS comb teeth fill the DKS repetition rate over $N=56$ EOcomb teeth,
providing a measure of the DKS repetition rate
$\omega_{\mathrm{rep}}=N\times\Omega_{\mathrm{RF}}-\delta\omega_{\mathrm{rep}}=1001.21952$
GHz $-\leavevmode\nobreak\ \delta\omega_{\mathrm{beat}}$ (the minus sign is
configuration dependent and has been verified for our experiments), with
$\delta\omega_{\mathrm{beat}}$ the EOcomb beat note that can be detected by a
50 MHz avalanche photodiode (APD), similar to the methods presented in [35,
19]. The other WDM output is used to measure the beat note $\Delta\Omega$
between the secondary color and the reference pump. Both signals are processed
in the time domain using a high speed oscilloscope. We trigger the scope with
the ramp signal used to sweep the piezo element of the CTL, obtaining a
temporal trace which can be processed and converted into a spectrogram,
enabling instantaneous frequency measurement. The calibration of this
technique is further discussed in the next section.

##  Appendix S.2 Repetition rate and detuning calibrations from temporal
traces

Figure S.2:  Repetition rate, detuning beat note and calibration protocols.
a-b Simultaneous measurement of the repetition rate $\omega_{\mathrm{rep}}$
and the beat note between the reference and the second color $\Delta\Omega$.
We record the temporal trace while sweeping the reference laser with a
triangular waveform that actuates the CTL piezo element. The direction of the
sweep is known from the CTL characteristics: high-to-low voltage results in
high-to-low reference laser frequency, and the oscilloscope is triggered on
the raising slope of the triangular waveform. Applying a consecutive fast
Fourier transform (through the scipy python package [40]) allows us to
retrieve the instantaneous frequency of the signal. c Calibration of the
detuning to convert the temporal frequency trace from the spectrogram maxima
in a and b, performed for each $\Delta\mu$ presented in Fig. 4. We bypass the
resonator with the reference laser and beat it directly against the second
color comb tooth. We record the temporal trace and apply a fast Fourier
transform to retrieve the instantaneous frequency. In this calibration
nonlinear effects associated with the reference laser cannot occur, and the
detuning against the comb tooth can be directly obtained, including the zero
reference for the detuning. The obtained frequency beat is flipped if needed
according to knowledge if the detuning is positive or negative based on the
triangular waveform applied.

The EOcomb beat note $\delta\omega_{\mathrm{beat}}$, which directly lets us
precisely characterize the repetition rate $\omega_{\mathrm{rep}}$, and the
reference pump and secondary-color beat $\Delta\Omega$ are measured in the
optical domain using a fast oscilloscope [Fig. S.2a-b], with a temporal
resolution of 320 ps. The piezo element is swept at a 50 Hz rate around the
detuning of interest. We record simultaneously the temporal trace of
$\delta\omega_{\mathrm{beat}}$ and $\Delta\Omega$, which we process using the
scipy.signal.spectrogram python package [40]. We use a binning slice of
$30\times 10^{3}$ points, allowing a frequency resolution of about 104 kHz.
The fast Fourier transform (FFT) windowing is performed using a Hanning
filter, with an overlap of $10\times 10^{3}$ points. We obtain a spectrogram
with a clear peak for each time sample that correspond to either
$\delta\omega_{\mathrm{beat}}$, from which we extract $\omega_{\mathrm{rep}}$
using the RF synthesizer frequency and the number of comb teeth as previously
described, or $\Delta\Omega$.

However, the detuning needs to be calibrated accurately. We harness the fact
that the frequency comb, if unsynchronized, can act as a local oscillator. To
this extent, for each of the different $\Delta\mu$ measurements shown in Fig.
4, we proceed to calibrate the detuning by bypassing the microresonator with
the reference and directly beating it with the closest secondary color comb
tooth (i.e., no nonlinearity present since we bypass the ring, and hence no
KIS) as shown in Fig. S.2c. Once again, we proceed to obtain the spectrogram
from the temporal trace, from which we can calibrate the detuning over time,
with a precise calibration of the zero-detuning (i.e. position of the
secondary color comb tooth). We note that we obtain the absolute value of the
detuning and need to process to change its sign based on the direction of the
triangular waveform applied to the CTL piezo element. From knowledge of the
CTL characteristics, we know that a high-to-low voltage results in a high-to-
low frequency, hence the detuning is negative for times before the calibration
frequency reaches zero, since the oscilloscope is triggered for positive slope
with the piezo ramp signal.

##  Appendix S.3 Laser phase noise measurement

Figure S.3:  Phase noise characterization of the auxiliary and reference pump.
a Schematic of the experimental setup to measure the auxiliary or reference
laser phase noise. We lock the laser to an unbalanced fiber Mach-Zehnder
interferometer (MZI) with a 10 MHz free spectral range. We measure the phase
noise of the laser using another 40 MHz MZI, and a low noise 125 MHz
photodiode. b Schematic explaining the relationship between the noise measured
with the 40 MHz FSR MZI and the laser frequency noise. c Phase noise of the
1300 nm laser (blue) and 1550 nm laser (red). The phase noise of the 1300 nm
laser is larger than the 1550 nm laser, and if accounting for additive noise
between them (gray) shows that it is predominantly noise from the 1300 nm
laser. The power is referenced to that of the carrier, namely, dBc/Hz

We lock the 1300 nm and 1500 nm CTLs using independent 10 MHz free spectral
range (FSR) Mach-Zenhder interferometers (MZIs), and using a Pound-Drever-Hall
technique allows us to actuate the current and piezo element of the laser
(respectively fast and slow feebdack) to lock to the side of a fringe of the
MZI [Fig. S.3]. To measure their respective phase noise, we use another MZI
with a 40 MHz FSR. Setting the laser to be at a quadrature point of this MZI,
we measure the residual MZI noise using a real time electrical spectum
analyzer (RSA). Understanding the noise obtained from the MZI [Fig. S.3b], we
can retrieve the frequency noise of the laser following:

$\displaystyle
S_{\mathrm{\nu}}=\left[\frac{\Delta\nu_{\mathrm{MZI}}}{\pi}\sin^{-1}\left(\frac{U_{\mathrm{MZI}}}{A_{\mathrm{mzi}}}\right)\right]^{2}\quad\mathrm{[Hz^{2}/Hz]}$

with $U_{\mathrm{MZI}}=\sqrt{50\times 10^{-3}\times 10^{S_{\mathrm{MZI}}/10}}$
the voltage obtained from spectral measurement, where $S_{\mathrm{MZI}}$ is
the optical power in decibels normalized to 1 mW of the MZI noise at
quadrature (i.e., in dBm), the spectrum analyzer impedance is assumed to be 50
$\Omega$, $A_{\mathrm{mzi}}=584$ mV is the amplitude of the sinusoidal
modulation with frequency of the MZI, and $\Delta\nu_{\mathrm{MZI}}=40$ MHz is
the free spectral range of the MZI used for noise measurement.

We can obtain the phase noise from the frequency noise, where the power is
referenced to that of the carrier, namely, dBc/Hz:

$\displaystyle
S_{\varphi}=10\log_{10}\left(\frac{S_{\mathrm{\nu}}}{\nu^{2}}\right)\quad\mathrm{[dB/Hz]}$

When comparing the phase noise of the two locked lasers, the 1500 nm laser
exhibits lower noise Fig. S.3. When accounting for additive noise since the
two lasers are locked to independent MZIs, as the 1300 nm laser being more
noisy than the 1500 nm laser, the noise of the 1300 nm laser is predominant.
We therefore consider OFD of the 1300 nm laser noise when assessing the
repetition rate noise under color-KIS.