#  Bright-soliton frequency combs and dressed states in $\chi^{(2)}$
microresonators

D.N. Puzyrev  V.V. Pankratov  A. Villois  D.V. Skryabin
<EMAIL_ADDRESS>Department of Physics, University of Bath, Bath, BA2
7AY, UK

(today)

###### Abstract

We present a theory of the frequency comb generation in the high-Q ring
microresonators with quadratic nonlinearity and normal dispersion and
demonstrate that the naturally large difference of the repetition rates at the
fundamental and 2nd harmonic frequencies supports a family of the bright
soliton frequency combs providing the parametric gain is moderated by tuning
the index-matching parameter to exceed the repetition rate difference by a
significant factor. This factor equals the sideband number associated with the
high-order phase-matched sum-frequency process. The theoretical framework,
i.e., the dressed-resonator method, to study the frequency conversion and comb
generation is formulated by including the sum-frequency nonlinearity into the
definition of the resonator spectrum. The Rabi splitting of the dressed
frequencies leads to the four distinct parametric down-conversion conditions
(signal-idler-pump photon energy conservation laws). The parametric
instability tongues associated with the generation of the sparse, i.e.,
Turing-pattern-like, frequency combs with varying repetition rates are
analysed in details. The sum-frequency matched sideband exhibits the optical
Pockels nonlinearity and strongly modified dispersion, which limit the soliton
bandwidth and also play a distinct role in the Turing comb generation. Our
methodology and data highlight the analogy between the driven multimode
resonators and the photon-atom interaction.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II Content and results overview
  3. III Linearised sideband equations and cw-state
  4. IV Rabi frequency, parametric gain and linewidth
  5. V Sum-frequency matching
  6. VI Dressed states
  7. VII Dressed spectrum and energy conservation in parametric-down conversion (PDC)
  8. VIII PDC instability tongues
  9. IX Parametric thresholds
  10. X Envelope and coupled-mode equations for modelocked combs
  11. XI Turing-pattern combs
  12. XII Bright soliton prerequisites
    1. XII.1 Dispersion of dressed states
    2. XII.2 Optical Pockels and cascaded-Kerr nonlinearities
    3. XII.3 CW-state stability and instability
  13. XIII Bright soliton frequency combs
  14. XIV Discussion
  15. XV Summary
  16. XVI Acknowledgement
  17. A Envelope equations
  18. B CW-state: $\chi^{(2)}$ vs $\chi^{(3)}$
  19. C CW-state: $\chi^{(2)}$ only
  20. D Linearization around the cw-state
  21. E Approximate PDC conditions
  22. F Laser power at the PDC thresholds

##  I Introduction

Ring microresonators break through the traditional barriers of frequency
conversion in terms of the power efficiency, generated bandwidth, and
compactness [2, 3]. Together with the rise of the microresonator frequency
conversion, the Kerr-soliton frequency combs are reaching unprecedented for
optical solitons levels of practical relevance [2, 3].

Second-order, $\chi^{(2)}$, nonlinearity is a viable alternative to the Kerr
one. It provides a much stronger nonlinear response but comes with the caveats
of the need to care about the phase and group velocity matching to take full
advantage of it. Refs. [4, 5, 6, 7] have been among the first ones to
demonstrate frequency conversion in the high-quality factor whispering gallery
microresonators with quadratic nonlinearity. Since then, this area has made a
significant progress, see Ref. [8] for the few-year old overview, and Refs.
[9, 10, 11, 12, 13, 14, 15, 16] for some of the recent experimental
contributions. It is also important to mention the work on the $\chi^{(3)}$
dominant frequency conversion in the significantly mismatched $\chi^{(2)}$
resonators, see, e.g., [17, 18, 19, 20], and Appendix B for the relative
weighting of the $\chi^{(2)}$ and $\chi^{(3)}$ effects. Material wise,
lithium-niobate (LN) remains the best-established platform choice for the
small-footprint $\chi^{(2)}$ photonics [9], with the silicon [21] and
aluminium [16] nitrides gaining grounds with accelerating pace.

The whispering gallery system considered below achieves finesses ${\cal F}\sim
10^{4}$. The high finesse, is an important prerequisite for achieving
conditions when the rate of the sideband generating photon energy exchange
driven by the sum-frequency nonlinear terms exceeds the loss rate [22, 23] and
the parametric gain rate [24], which corresponds to the strong coupling regime
between the $\omega$ and $2\omega$ photons. As we have reported recently [24]
and investigate in depth below, this makes the frequency conversion and
soliton generation mechanisms to depart significantly from what has been known
about these effects in the relatively low finesse resonators, ${\cal F}\sim
10^{2}$, which often have no resonator-feedback at one of the generated
harmonics, see, e.g., Refs. [25, 26] for an overview. A couple of most obvious
and important features of the high-${\cal F}$ resonators is the channelling of
the parametric gain into the tongue-like instability domains [24] and the
strong-coupling associated with the dressed states [22] and polaritons [24].

Experimental results on the $\chi^{(2)}$-driven microresonator solitons are
limited for now by the outstanding recent report by Bruch and colleagues [16]
on the solitons due to parametric down-conversion (PDC) in the aluminium-
nitride microring with the finesse $\simeq 10^{3}$. The numerical data
reported in Ref. [16] show the exponentially localised pulse in the half-
harmonic field and the de-localised wave-form in the pump. Ref. [16] poses
several problems, in particular, what are the physical mechanisms facilitating
the compensation of the large group velocity, i.e., the repetition rate,
difference between the pump and half-harmonic fields allowing for the soliton
to form, and if having a better shaped pulse in the pump field is possible.
Identifying conditions for the multi-colour multi-pulse modelocked solitons to
compensate the large group velocity differences is one of the classic problems
[27, 28], that needs to be addressed in the context of the high-Q
microresonators (see Ref. [29] for the no-resonator, i.e., bulk propagation,
analysis).

Results on the bright PDC solitons in the resonators with the group velocity
offset published two decades ago [28] provided a conceptual answer, that the
compensation of the group velocity difference is achieved via the balancing
interplay between the dissipative and nonlinear effects. However, Ref. [28]
was published well before the frequency combs and solitons in the micro-
resonators have become possible. Therefore, it could not anticipate a
combination of the small sizes and high Q-factors of the modern day devices,
which leads to the strong quantisation of the spectra of the operators
underpinning the frequency conversion and soliton formation processes. The
model that fully reflects on these aspects has been recently presented in Ref.
[30], which introduction also includes sufficiently comprehensive coverage of
the recent and historic work on the dissipative $\chi^{(2)}$ solitons in
resonators.

Following our recent work on the theory of $\chi^{(2)}$ microresonators [30,
31, 32, 13, 24], we present here the latest findings obtained in the second
harmonic generation (SHG) setting. One of the prime results included below is
the demonstration of the bright soliton pulses in a microresonator which has
the large repetition rate difference between the fundamental, $\omega_{p}$,
and 2nd harmonic frequencies, $2\omega_{p}$, see Figs. 15-17.

The resonator considered here has the $\simeq 20$GHz repetition rate and the
$1$GHz rate difference, which implies that the linear $\omega_{p}$ and
$2\omega_{p}$ pulses would be on the opposite sides of the resonator only
after about seventy round-trips. The dispersion of the resonator is normal and
its deviation from the bulk dispersion is insignificant. The dispersion value
is $\simeq-100$kHz so that the linear pulse becomes twice as broad after
$10^{5}$ round-trips, and hence the repetition-rate difference is by far the
most dangerous for the bright soliton modelocking, which nevertheless will be
shown to exist across the broad and practical range of the system parameters.

The soliton combs reported below represent a pair of the $\omega_{p}$ and
$2\omega_{p}$ modelocked, and hence the repetition rate locked, bright pulses
which existence and properties are derived and interpreted by examining the
details of how the comb teeth, i.e., the frequency sidebands, are generated
and interact via the sum-frequency and PDC nonlinear mixing processes. The
detailed understanding of this problem has become possible thanks to the
dressed resonator theory [24]. The content of what follows is much wider than
just reporting the soliton modelocking, and it is now useful to give it a
brief overview.

##  II Content and results overview

The coupling between the sidebands, i.e., the resonator mode pairs or the comb
teeth, underpins the formation of any frequency combs and modelocking in
optical resonators. Below we consider the microresonator 2nd harmonic
generation, where the sideband coupling mechanisms embrace the complex
interplay of the parametric process that couples the $\mu$ and $-\mu$
sidebands with the two sum-frequency processes. One is responsible for the
coupling between the $\mu$ sidebands of the fundamental and 2nd harmonic,
while the other does the same but for the $-\mu$ sidebands. The sum-frequency
processes can be phase-matched for the select sidebands with the positive and
negative numbers, $\mu=\mu_{*}$ or $\mu=-\mu_{*}$, where $\mu_{*}$ is the
ratio between the phase-mismatch parameter and the repetition rate difference.
The key resonator parameters and characteristics, including $\mu_{*}$, are
illustrated in Fig. 1, and defined in Tables 1 and 2.

If the microresonator is tuned to operate away from the $\mu=0$ phase-
matching, then the sum-frequency sideband coupling rate exceeds the parametric
gain. It creates an opportunity to redefine the resonator spectrum by
including the sum-frequency associated nonlinear terms. The new spectrum,
i.e., the dressed spectrum, can be calculated analytically, bringing a close
analogy with the dressing of the atomic transitions and the Rabi theory [22,
24, 33].

The condition of the maximal parametric gain [34, 35, 36],

$2\hbar\omega_{p}=\hbar\omega_{\text{signal}}+\hbar\omega_{\text{idler}},$ (1)

should then be redefined using the dressed frequencies [24]. The Rabi
splitting leads to the four distinct PDC conditions, see Eq. (52). Knowing
them allows understanding the complex structures of the parametric instability
tongues and the Turing-pattern like frequency combs across the parameter space
spanned by the pump laser frequency, $\omega_{p}$, and the intra-resonator
power.

The dressed modes around either $\mu_{*}$ or $-\mu_{*}$ exhibit dispersion
with the inverted signs and the values significantly exceeding the bare-
resonator ones. Therefore, only by making $\mu_{*}$ to be sufficiently large,
i.e., by taking it away from the soliton spectral core, creates enough of the
modal bandwidth around $\mu=0$ with the low dispersion allowing the formation
of the two-color bright soliton pulses. The large values of $\mu_{*}$ are
achieved by making the frequency mismatch parameter between the fundamental
and 2nd harmonic modes to exceed the repetition rate difference by a
significant, $\gg 1$, factor, which works out to be the $\mu_{*}$ itself.

##  III Linearised sideband equations and cw-state

We assume the intra-resonator electric fields of the fundamental and 2nd
harmonic to be

$\psi_{\text{f}}e^{iM\vartheta-i\omega_{p}t}+c.c.,~{}\psi_{\text{s}}e^{i2M\vartheta-i2\omega_{p}t}+c.c.,$
(2)

where $|\psi_{\text{f},\text{s}}|^{2}$ have units of power [30], see Appendix
A. $M$ and $2M$ are the absolute (physical) mode numbers with frequencies
$\omega_{0\text{f}}$ and $\omega_{0\text{s}}$. $\vartheta\in(-\pi,\pi]$ is the
angular coordinate and $t$ is time. $\omega_{p}$ is the pump laser frequency
tunable around $\omega_{0\text{f}}$, so that,

$\delta=\omega_{0\text{f}}-\omega_{p}$ (3)

is the respective pump detuning.

The envelopes of the fundamental, $\psi_{\text{f}}$, and second harmonic,
$\psi_{\text{s}}$, are expressed via their mode expansions as

$\psi_{\zeta}=\psi_{0\zeta}(t)+\sum_{\mu>0}\left(\psi_{\mu\zeta}(t)e^{i\mu\vartheta}+\psi_{-\mu\zeta}(t)e^{-i\mu\vartheta}\right),$
(4)

with $\zeta=\text{f},\text{s}$, and $\mu$ is an integer characterising the
relative mode number. The resonator frequencies associated with
$\psi_{\pm\mu\zeta}$ are

$\omega_{\pm\mu\zeta}=\omega_{0\zeta}\pm\mu
D_{1\zeta}+\tfrac{1}{2}\mu^{2}D_{2\zeta},$ (5)

where $D_{1\zeta}/2\pi$ are the repetition rates (free spectral ranges) and
$D_{2\zeta}$ are the dispersions, see Fig. 1 for a schematic illustration. The
photon angular momenta corresponding to $\omega_{\pm\mu\text{f}}$ and
$\omega_{\pm\mu\text{s}}$ are

$\hbar(M\pm\mu),~{}\text{and}~{}\hbar(2M\pm\mu).$ (6)

Figure 1: A schematic illustration of the spectrum of the linear, i.e., bare
resonator, $\Omega=0$, around the pump laser frequency, $\omega_{p}$, and its
2nd harmonic, $2\omega_{p}$. The frequency mismatch parameter
$\varepsilon_{0}$ is tunable and could be comparable with $D_{1\zeta}$.
Linewidths: $\kappa_{\text{f}}/2\pi=1$MHz, $\kappa_{\text{s}}/2\pi=4$MHz
---
Repetition rates: $D_{1\text{f}}/2\pi=21$GHz, $D_{1\text{s}}/2\pi=20$GHz
Dispersions: $D_{2\text{f}}/2\pi=-100$kHz, $D_{2\text{s}}/2\pi=-200$kHz
Nonlinear coefficients:
$\gamma_{\text{f},\text{s}}/2\pi=300\text{MHz}/\sqrt{\text{W}}$
Table 1: The resonator parameters characteristic for a bulk-cut LiNbO3
resonator pumped at $1065$nm [13, 30, 24]. $D_{2\text{f},2\text{s}}<0$
correspond to the normal dispersion.

Physical values of the parameters implemented throughout this work are shown
in Table 1, and are typical for a bulk-cut LiNbO3 resonator. Detunings of the
resonator frequencies $\omega_{\pm\mu\zeta}$ from the pump laser frequency,
$\omega_{p}$, and its 2nd harmonic, $2\omega_{p}$, are
$\omega_{\pm\mu\text{f}}-\omega_{p}$ and
$\omega_{\pm\mu\text{s}}-2\omega_{p}$. A Galilean transformation to the
reference frame rotating with the rate $D_{1}/2\pi$,

$\theta=\vartheta-D_{1}t,$ (7)

converts these detunings to

$\begin{split}\varDelta_{\pm\mu\text{f}}&=(\omega_{\pm\mu\text{f}}-\omega_{p})\mp\mu
D_{1}\\\
&=\delta\pm\mu(D_{1\text{f}}-D_{1})+\tfrac{1}{2}D_{2\text{f}}\mu^{2},\\\
\varDelta_{\pm\mu\text{s}}&=(\omega_{\pm\mu\text{s}}-2\omega_{p})\mp\mu
D_{1}\\\
&=2\delta-\varepsilon_{0}\pm\mu(D_{1\text{s}}-D_{1})+\tfrac{1}{2}D_{2\text{s}}\mu^{2},\end{split}$
(8)

where $\varepsilon_{0}=2\omega_{0\text{f}}-\omega_{0\text{s}}$ is the $\mu=0$
frequency mismatch parameter. Setting

$D_{1}=D_{1\text{f}},$ (9)

we get an immediate access to the difference of the repetition rates,
$D_{1\text{s}}-D_{1\text{f}}$, inside $\varDelta_{\pm\mu\text{s}}$.

Figure 2: CW state of the fundamental (a) and 2nd harmonic (b), see Appendix
C. Pump laser power is ${\cal W}=0.1$mW, see Eq. (58). Mismatch parameter
$\varepsilon_{0}/2\pi=-5$GHz.

If $c$ is the vacuum speed of light, $n_{M+\mu}$ is the effective refractive
index of the mode $M+\mu$, and $R$ is the resonator radius, then

$\omega_{\mu\text{f}}=\frac{c}{R}\frac{(M+\mu)}{n_{M+\mu}},~{}\omega_{\mu\text{s}}=\frac{c}{R}\frac{(2M+\mu)}{n_{2M+\mu}}.$
(10)

Hence,

$\varepsilon_{0}=2\omega_{0\text{f}}-\omega_{0\text{s}}=2M\frac{c}{R}\left[\frac{1}{n_{M}}-\frac{1}{n_{2M}}\right],$
(11)

and requiring $\varepsilon_{0}=0$, yields the anticipated index matching
condition, $n_{M}=n_{2M}$. Refractive index, and hence $\varepsilon_{0}$, can
be fine tuned by, e.g., temperature or electro-optic controls.

CW state of the resonator operation implies
$\psi_{\zeta}=\psi_{0\zeta}=\text{const}_{\zeta}$, i.e.,
$\partial_{t}\psi_{0\zeta}=0$, see Fig. 2 and Appendix C. During the initial
stage of the comb development, the sidebands evolve and grow on top of the
undepleted cw-state, so that the field envelopes can be sought as the cw plus
small perturbations, with the latter taken as a superposition of the resonator
modes,

$\psi_{\zeta}(t,\theta)=\psi_{0\zeta}+\sum_{\mu\geq
0}\left[\widetilde{\psi}_{\mu\zeta}(t)e^{i\mu\theta}+\widetilde{\psi}_{-\mu\zeta}^{*}(t)e^{-i\mu\theta}\right].$
(12)

Here $\widetilde{\psi}_{\pm\mu\zeta}$ are the amplitudes of the growing
sidebands. Complex conjugation of $\widetilde{\psi}_{-\mu\zeta}$ was
introduced to make the equations to follow less cluttered. If Eq. (4) could be
referred to as the mode expansion in the bare resonator representation, then
Eq. (12) is a step towards the dressed resonator theory.

Substituting Eq. (12) to the envelope equations (75), and linearising for
small sidebands, we find that $\widetilde{\psi}_{\mu\zeta}(t)$ are driven by
the sideband combinations with the matched net momenta, see Appendix D. The
resulting equations are

$\begin{split}&i\partial_{t}\widetilde{\psi}_{\mu\text{f}}=(\varDelta_{\mu\text{f}}-i\tfrac{1}{2}\kappa_{\text{f}})\widetilde{\psi}_{\mu\text{f}}-\gamma_{\text{f}}(\underline{\widetilde{\psi}_{\mu\text{s}}\psi_{0\text{f}}^{*}}+\underline{\underline{\widetilde{\psi}_{-\mu\text{f}}\psi_{0\text{s}}}}),\\\
&i\partial_{t}\widetilde{\psi}_{\mu\text{s}}=(\varDelta_{\mu\text{s}}-i\tfrac{1}{2}\kappa_{\text{s}})\widetilde{\psi}_{\mu\text{s}}-2\gamma_{\text{s}}\underline{\widetilde{\psi}_{\mu\text{f}}\psi_{0\text{f}}},\\\
&i\partial_{t}\widetilde{\psi}_{-\mu\text{f}}=(-\varDelta_{-\mu\text{f}}-i\tfrac{1}{2}\kappa_{\text{f}})\widetilde{\psi}_{-\mu\text{f}}+\gamma_{\text{f}}(\underline{\widetilde{\psi}_{-\mu\text{s}}\psi_{0\text{f}}}+\underline{\underline{\widetilde{\psi}_{\mu\text{f}}\psi_{0\text{s}}^{*}}}),\\\
&i\partial_{t}\widetilde{\psi}_{-\mu\text{s}}=(-\varDelta_{-\mu\text{s}}-i\tfrac{1}{2}\kappa_{\text{s}})\widetilde{\psi}_{-\mu\text{s}}+2\gamma_{\text{s}}\underline{\widetilde{\psi}_{-\mu\text{f}}\psi_{0\text{f}}^{*}}.\end{split}$
(13)

Here, $\kappa_{\zeta}$ are the loaded linewidth parameters, and
$\gamma_{\zeta}$ are the nonlinear coefficients measured in Hz/W1/2 [30], see
Table 1.

Eq. (13) are linear in the approximation of the undepleted cw-state, and they
represent one of the fundamental models in nonlinear optics expressing the
interplay of the PDC and sum-frequency processes. The sum-frequency terms are
underlined once and the parametric ones, describing conversion to the $\pm\mu$
sidebands (photon-pair generation), are underlined twice. The momentum
conservation laws corresponding to the parametric conversion and the two sum-
frequency processes are

$\hbar(M+\mu)+\hbar(M-\mu)=\hbar 2M,$ (14)

and

$\hbar(M\pm\mu)+\hbar M=\hbar(2M\pm\mu),$ (15)

respectively.

The eigenvalues and eigenvectors of the matrix acting on the vector
$(\widetilde{\psi}_{\mu\text{f}},\widetilde{\psi}_{\mu\text{s}},\widetilde{\psi}_{-\mu\text{f}},\widetilde{\psi}_{-\mu\text{s}})^{T}$
in the right-hand side of Eq. (13) are known in the explicit form in two
cases: (i) for $\varDelta_{\mu\text{f}}=\varDelta_{-\mu\text{f}}$,
$\varDelta_{\mu\text{s}}=\varDelta_{-\mu\text{s}}$ and
$\kappa_{\text{f}}=\kappa_{\text{s}}$, see, e.g., Refs. [37, 38, 39], and (ii)
for $\kappa_{\text{f}}\neq\kappa_{\text{s}}$ and $\varDelta_{\pm\mu\zeta}=0$,
see Ref. [40]. While $\varDelta_{\mu\text{f}}=\varDelta_{-\mu\text{f}}$ is
satisfied exactly, and $\kappa_{\text{f}}=\kappa_{\text{s}}$ could be assumed,
the $\varDelta_{\mu\text{s}}=\varDelta_{-\mu\text{s}}$ condition is typically
far from being true. In most of the practical cases, the repetition rate
difference,
$\varDelta_{\mu\text{s}}-\varDelta_{-\mu\text{s}}=2\mu(D_{1\text{s}}-D_{1\text{f}})$,
creates one of the dominant frequency scales in Eqs. (8), which can be
comparable only to $\varepsilon_{0}$.

Figure 3:  (a) The sum-frequency matched Rabi oscillations near to the $\mu=5$
avoided crossing (red - fundamental, left axis; green - second harmonic, right
axis), $\Omega_{5}\approx|\Omega|\approx 2\pi\times 68$MHz. (b) is as (a) but
it is plotted over the time interval as is used in (c). (c) The mismatched
Rabi oscillations for the $\mu=4$ sidebands,
$\Omega_{4}\approx|\varDelta_{4}|\approx 2\pi\times 1$GHz. Parameters: (a,b)
$\delta=15.15\kappa_{\text{f}}$, $\varepsilon_{0}/2\pi=-5$GHz, ${\cal
W}=1.7$mW (pump power), $|\psi_{5\text{s}}|^{2}=0.36$mW (sideband power) at
$t=0$. (c) $\delta=6.45\kappa_{\text{f}}$, $\varepsilon_{0}/2\pi=-5$GHz,
${\cal W}=2.11$mW, $|\psi_{4\text{s}}|^{2}=0.33$mW at $t=0$.

##  IV Rabi frequency, parametric gain and linewidth

To address the problem at hands, it is important to understand the balances
between the characteristic frequency scales implicated in Eq. (13).
$\psi_{0\text{s}}$, i.e., the cw 2nd harmonic amplitude, can be controlled by
tuning the frequency mismatch, $\varepsilon_{0}$, so that the parametric gain
coefficient, $\gamma_{\text{f}}\psi_{0\text{s}}$, can be made to be much less
than the sum-frequency associated rate of the energy exchange,
$\gamma_{\zeta}\psi_{0\text{f}}$, see Fig. 2 and compare the frequency scales
along the vertical axes.

If the aim is to create the modal bandwidth sufficient for the soliton
generation at both harmonics, then the moderation of the parametric gain is
useful since it would then keep the energy of the 2nd harmonic pulse under
control, and, hence, could be expected to ease, for the stronger fundamental
pulse, the task of synchronising its repetition rate with the 2nd harmonic.

The sum-frequency driven energy exchange is illustrated in Fig. 3, where one
can see the fast anti-phase oscillations of the $\omega_{\mu\text{f}}$ and
$\omega_{\mu\text{s}}$ sidebands. The frequency of the oscillations is much
larger than the decay rates, so that, in the leading order, the first two
equations in Eqs. (13) can be approximated with [24, 41]

$i\partial_{t}\begin{bmatrix}\widetilde{\psi}_{\mu\text{f}}\\\
\widetilde{\psi}_{\mu\text{s}}\end{bmatrix}\approx\begin{bmatrix}\varDelta_{\mu\text{f}}&-\gamma_{\text{f}}\psi_{0\text{f}}^{*}\\\
-2\gamma_{\text{s}}\psi_{0\text{f}}&\varDelta_{\mu\text{s}}\end{bmatrix}\begin{bmatrix}\widetilde{\psi}_{\mu\text{f}}\\\
\widetilde{\psi}_{\mu\text{s}}\end{bmatrix}+\dots.$ (16)

The second pair of equations resemble the above but with $\mu\to-\mu$.
Parametric gain and losses should then come in the next to leading order,
suggesting the development of a perturbation theory. The Rabi theory, well
known in the semi-classical atom-photon interaction [33], is an obvious and
best suited methodology to describe solutions of Eq. (16). The Rabi formalism
was also previously applied for the resonator-free sum-frequency generation
model [42].

Rabi frequency | $\Omega=\psi_{0\text{f}}\sqrt{8\gamma_{\text{f}}\gamma_{\text{s}}}$
---|---
|  $\approx\psi_{0\text{f}}\times 0.8~{}$GHz/$\sqrt{\text{W}}$
Rabi detuning | $\varDelta_{\mu}=\omega_{p}+\omega_{\mu\text{f}}-\omega_{\mu\text{s}}$
Sum-frequency mismatch | $\varepsilon_{\mu}=\omega_{0\text{f}}+\omega_{\mu\text{f}}-\omega_{\mu\text{s}}$
Effective Rabi frequency | $\Omega_{\mu}=\sqrt{\varDelta_{\mu}^{2}+|\Omega|^{2}}$
PDC frequency mismatch | $\varepsilon_{\mu}^{(j_{1}j_{2})}=\widetilde{\omega}_{\mu\text{f}}^{(j_{1})}+\widetilde{\omega}_{-\mu\text{f}}^{(j_{2})}-2\omega_{p}$
Strong-coupling condition | $\kappa_{\zeta}\ll|\Omega|\ll 8|\varepsilon_{0}|$
Sum-frequency matching |  $\varepsilon_{\pm\mu_{*}}=0,$ $\mu_{*}\approx|\varepsilon_{0}|/|D_{1\text{f}}-D_{1\text{s}}|$
Table 2: Definitions of the key parameters, which complement the ones
illustrated in Fig. 1 and listed in Table 1. $\omega_{p}$ is the laser photon
frequency. $\omega_{\mu\zeta}$ and $\widetilde{\omega}_{\mu\zeta}^{(j)}$ are
the bare and dressed resonator frequencies, respectively ($j_{1,2}=1,2,3,4$;
$\zeta=\text{f},\text{s}$).

The frequency of the oscillations in Fig. 3 would then be the effective Rabi
frequency,

$\Omega_{\mu}=\sqrt{\varDelta_{\mu}^{2}+|\Omega|^{2}}.$ (17)

It is controlled by the difference of the sideband detunings, i.e., the Rabi
detuning,

$\varDelta_{\mu}=\varDelta_{\mu\text{f}}-\varDelta_{\mu\text{s}},$ (18)

and by the coupling coefficient, i.e., by the off-diagonal terms in Eq. (16),
characterised by the complex Rabi frequency, $\Omega$,

$\Omega=\psi_{0\text{f}}\sqrt{8\gamma_{\text{f}}\gamma_{\text{s}}}.$ (19)

The complex 2nd harmonic amplitude can also be expressed via $\Omega$ and the
auxiliary complex frequency $\Omega_{\text{s}}$,

$\gamma_{\text{f}}\psi_{0\text{s}}=\frac{\Omega^{2}}{\Omega_{\text{s}}},~{}\Omega_{\text{s}}=8(2\delta-i\tfrac{1}{2}\kappa_{\text{s}})-8\varepsilon_{0},$
(20)

see Appendix C.

$|\Omega|/2\pi\sim 10^{2}$MHz gives $|\psi_{0\text{f}}|^{2}\lesssim 1$W, which
would be typical inside the resonator. Then, for $\kappa_{\zeta}/2\pi\sim
1$MHz we have $|\Omega|\gg\kappa_{\zeta}$, i.e., the Rabi flops are indeed
much faster than the decay rate. The frequency scale associated with the
parametric gain is set by

$\gamma_{\text{f}}|\psi_{0\text{s}}|=\frac{|\Omega|^{2}}{|\Omega_{\text{s}}|}.$
(21)

Arranging $|\varepsilon_{0}|$ to be close or larger than the repetition rate
difference,

$\frac{|\varepsilon_{0}|}{2\pi}\gtrsim\frac{|D_{1\text{f}}-D_{1\text{s}}|}{2\pi}\simeq
1\text{GHz},$ (22)

makes

$\frac{1}{|\Omega|}\frac{|\Omega|^{2}}{|\Omega_{\text{s}}|}\approx\frac{|\Omega|}{8|\varepsilon_{0}|}\ll
1.$ (23)

Hence, the Rabi frequency is also much larger than the parametric gain rate.
Thus, both the linewidth and the parametric terms are small relative to the
right-hand side of Eq. (16), and the strong-coupling (SC) condition [24],

$\kappa_{\zeta}\ll|\Omega|\ll|\Omega_{\text{s}}|,$ (24)

is satisfied. Ref. [22] reported measurements of the $\chi^{(2)}$ Rabi
splitting ($\sim 1$GHz for the laser power ${\cal W}=80$mW) in the AlN
resonators with $|\Omega|/\kappa_{\zeta}\sim 1$, which should be well out-
performed by the bulk resonators considered here. The $\mu=0$ Rabi
oscillations in the $\chi^{(2)}$ resonators were looked at in Ref. [45], well
before the multimode high-Q $\chi^{(2)}$ microresonators have been
demonstrated.

The notations related to the Rabi theory and also the key quantities used
below to characterise the matching conditions for the sum-frequency and
parametric processes are summarised in Table 2.

Figure 4: (a) Frequency matching for the sum-frequency generation: The Rabi
detuning $\varDelta_{-\mu}=\omega_{p}+\omega_{-\mu\text{f}}-\omega_{-\mu s}$
vs the sideband order number $\mu$, $\delta=5\kappa_{\text{f}}$.
$\varepsilon_{0}/2\pi=10.5$GHz and $\varepsilon_{0}/2\pi=10$GHz correspond to
the mismatched and near-matched ($\mu=\widehat{\mu}=10$) cases, respectively.
(b) Avoided-crossing diagram: The dressed frequencies vs $\delta$ for
$|\Omega|=20\kappa_{\text{f}}$. The straight lines correspond to $\Omega=0$.

##  V Sum-frequency matching

While the cascade of the sum- and difference-frequency events engaging a
sequence of $\mu$’s is critical for the generation of the fully blown combs,
see Section X, the frequency matching for the one-step sum-frequency process
in Eq. (14) is also very important and should be analysed in more details. As
it is well known for the coupled oscillators, the full periodic power transfer
between $\widetilde{\psi}_{\mu\text{f}}$ and $\widetilde{\psi}_{\mu\text{s}}$
is ensured by minimizing the Rabi detuning, i.e., $\varDelta_{\mu}\to 0$,
$\Omega_{\mu}\to|\Omega|$, see Eq. (17).

Here, we are dealing with a system possessing two different effective Rabi
frequencies $\Omega_{\mu}$ and $\Omega_{-\mu}$, and the respective Rabi
detunings, $\varDelta_{\pm\mu}$, can be re-expressed as

$\varDelta_{\pm\mu}=\omega_{p}+\omega_{\pm\mu\text{f}}-\omega_{\pm\mu\text{s}}.$
(25)

If one of $\varDelta_{\pm\mu}=0$ is resolved by an integer, i.e.,

$\varDelta_{\widehat{\mu}}=0,~{}\text{or}~{}\varDelta_{-\widehat{\mu}}=0,~{}\widehat{\mu}\in\mathbb{Z},~{}\widehat{\mu}>0,$
(26)

it implies the exact matching for one of the two co-existing sum-frequency
processes. The examples of the exact matching between $\omega_{-\mu\text{f}}$
and $\omega_{-\mu\text{s}}$ for $\mu=10$ and of the mismatched case are shown
in Fig. 4(a). In the mismatched case, the SC condition, Eq. (24), is perfectly
satisfied, but the power transfer during the Rabi flops is significantly
reduced, cf., Figs. 3(b) and (c).

Figure 5: $\Omega_{\mu}$ and $\Omega_{-\mu}$ Rabi frequencies vs $\mu$. (a)
$\varepsilon_{0}/2\pi=10.5$GHz, and (b) $\varepsilon_{0}/2\pi=-10.5$GHz,
$\delta=14\kappa_{\text{f}}$, $|\Omega|=300\kappa_{\text{f}}$.
$\varDelta_{-\mu}=\omega_{p}+\omega_{-\mu\text{f}}-\omega_{-\mu\text{s}}$ is
nearly matched for $\mu=10$ in (a), and
$\varDelta_{\mu}=\omega_{p}+\omega_{\mu\text{f}}-\omega_{\mu\text{s}}$ is in
(b).

If the real positive $\mu_{*}$ solves one of the $\varDelta_{\pm\mu}=0$
equations, i.e.,

$\varDelta_{\mu_{*}}=0,~{}\text{or}~{}\varDelta_{-\mu_{*}}=0,~{}\mu_{*}\in\mathbb{R},~{}\mu_{*}>0,$
(27)

then an integer, or two, nearest to $\mu_{*}$ provide the sidebands order best
complying with the frequency matching. To find $\mu_{*}$, we introduce a new
parameter

$\varepsilon_{\pm\mu}=\omega_{0\text{f}}+\omega_{\pm\mu\text{f}}-\omega_{\pm\mu\text{s}},$
(28)

cf., Eq. (25), so that,

$\varDelta_{\pm\mu}=\varepsilon_{\pm\mu}-\delta.$ (29)

We note that $\varepsilon_{\pm\mu}$ depend only on the resonator geometry and
refractive index, cf., Eq. (11). Using Eq. (5), we express
$\varepsilon_{\pm\mu}$ as

$\varepsilon_{\pm\mu}=\varepsilon_{0}\pm\mu(D_{1\text{f}}-D_{1\text{s}})+\tfrac{1}{2}\mu^{2}(D_{2\text{f}}-D_{2\text{s}}),$
(30)

Relative smallness of the dispersion, i.e., of the $\mu^{2}D_{2\zeta}$, and
$\delta$ terms, provides an excellent approximation for $\mu_{*}$,

$\mu_{*}\approx\frac{|\varepsilon_{0}|}{|D_{1\text{f}}-D_{1\text{s}}|}.$ (31)

If $\varepsilon_{0}>0$, then $\mu_{*}$ solves $\varDelta_{-\mu}=0$, see Fig.
4(a), and if $\varepsilon_{0}<0$, then $\mu_{*}$ solves $\varDelta_{\mu}=0$.
This point is further facilitated in Fig. 5, where we plot $\Omega_{\mu}$ and
$\Omega_{-\mu}$ vs $\mu$ for different signs of $\varepsilon_{0}$. For
$\varepsilon_{0}>0$, $\Omega_{-\mu}$ has minimum at
$\mu=\mu_{*}\approx\widehat{\mu}$, $\Omega_{-\mu_{*}}=|\Omega|$, and for
$\varepsilon_{0}<0$, $\Omega_{\mu}$ is the one with the minimum. The simple
approximations for $\Omega_{\pm\mu}$ are also inferred from Fig. 5. For
$\varepsilon_{0}>0$, we have

$\begin{split}\Omega_{\mu}\approx\varDelta_{\mu}~{}\text{for~{}all}~{}\mu,&~{}\Omega_{-\mu}\approx\varDelta_{-\mu}~{}\text{for}~{}\mu<\widehat{\mu},\\\
\Omega_{-\widehat{\mu}}\approx|\Omega|~{}\text{for}~{}\mu=\widehat{\mu},&~{}\Omega_{-\mu}\approx-\varDelta_{-\mu}~{}\text{for}~{}\mu>\widehat{\mu}.\end{split}$
(32)

The same approximations for $\varepsilon_{0}<0$ require the
$\Omega_{\pm\mu}\to\Omega_{\mp\mu}$ swap in every part of Eq. (32), see Fig.
5(b).

##  VI Dressed states

Equation (16) has an obvious and important class of solutions with the time-
independent sideband powers - dressed (eigen) states. The Rabi oscillations
stem from a superposition of the dressed states. Dressing the states, i.e.,
working with the superpositions between $\widetilde{\psi}_{\mu\text{f}}$ and
$\widetilde{\psi}_{\mu\text{s}}$, rather than with the modes of the linear
resonator, allows to develop a theory embracing the cases with the arbitrary
(small, large or near one) ratios of the
$|\widetilde{\psi}_{\mu\text{s}}|^{2}$ and
$|\widetilde{\psi}_{\mu\text{f}}|^{2}$ powers [24].

After the re-scaling, $\widetilde{\psi}_{\pm\mu\text{f}}=e^{\pm
i\phi_{\text{f}}}\tilde{a}_{\pm\mu\text{f}}/\sqrt{2}$,
$\widetilde{\psi}_{\pm\mu\text{s}}=e^{\pm
i\phi_{\text{s}}}\tilde{a}_{\pm\mu\text{s}}/\sqrt{\gamma_{2\text{f}}/\gamma_{2\text{s}}}$,
$\phi_{\zeta}=\arg\psi_{0\zeta}$, Eq. (13) become,

$i\partial_{t}|{\tilde{a}_{\mu}}\rangle=\big{(}\widehat{H}_{\mu}+\widehat{V}\big{)}|{\tilde{a}_{\mu}}\rangle.$
(33)

Here
$|{\tilde{a}_{\mu}}\rangle=(\tilde{a}_{\mu\text{f}},\tilde{a}_{\mu\text{s}},\tilde{a}_{-\mu\text{f}},\tilde{a}_{-\mu\text{s}})^{T}$
is the state vector,

$\displaystyle\widehat{H}_{\mu}=\left[\begin{array}[]{cccc}\varDelta_{\mu\text{f}}&-\tfrac{1}{2}|\Omega|e^{-i\phi}&0&0\\\
-\tfrac{1}{2}|\Omega|e^{i\phi}&\varDelta_{\mu\text{s}}&0&0\\\
0&0&-\varDelta_{-\mu\text{f}}&\tfrac{1}{2}|\Omega|e^{i\phi}\\\
0&0&\tfrac{1}{2}|\Omega|e^{-i\phi}&-\varDelta_{-\mu\text{s}}\end{array}\right],$
(38)

$\phi=2\phi_{\text{f}}-\phi_{\text{s}}$, and

$\displaystyle\widehat{V}=\left[\begin{array}[]{cccc}-i\tfrac{1}{2}\kappa_{\text{f}}&0&-\dfrac{|\Omega|^{2}e^{-i\phi}}{|\Omega_{\text{s}}|}&0\\\
0&-i\tfrac{1}{2}\kappa_{\text{s}}&0&0\\\
\dfrac{|\Omega|^{2}e^{i\phi}}{|\Omega_{\text{s}}|}&0&-i\tfrac{1}{2}\kappa_{\text{f}}&0\\\
0&0&0&-i\tfrac{1}{2}\kappa_{\text{s}}\end{array}\right].$ (43)

Figure 6: Diagram illustrating the dressed spectra. (a) shows the pump
frequency tuned to satisfy the intra-branch PDC condition, $\hbar
2\omega_{p}=\hbar\widetilde{\omega}_{\mu\text{f}}^{(1)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(1)}$.
(b) shows how the cross-branch PDC condition is satisfied, $\hbar
2\omega_{p}=\hbar\widetilde{\omega}_{\mu\text{f}}^{(2)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(1)}$.
The red and blue arrows show how the PDC gain and Rabi flops redistribute
power within the spectrum. The black doted lines show the Rabi frequencies,
$\Omega_{\mu}$, $\Omega_{-\mu}$, and highlight their inequality.

Setting

$\displaystyle|{\tilde{a}(t)}\rangle=|{a_{\mu}}\rangle\exp\\{t\lambda_{\mu}-it\beta_{\mu}\\},$
(44) $\displaystyle\lambda_{\mu}\in\mathbb{R},~{}\beta_{\mu}\in\mathbb{R},$

we find

$\left(\beta_{\mu}+i\lambda_{\mu}\right)|{a_{\mu}}\rangle=\left(\widehat{H}_{\mu}+\widehat{V}\right)|{a_{\mu}}\rangle,$
(45)

where $\beta_{\mu}$ is the frequency shift, and $\lambda_{\mu}$ is the
sideband growth rate, such that $\lambda_{\mu}$ transiting from negative to
positive signals instability of the cw-state relative to the excitation of the
$\pm\mu$ pair.

In the SC regime, see Eq. (24), $\widehat{V}$ is a perturbation to
$\widehat{H}_{\mu}$, and therefore, before incorporating $\widehat{V}$, we
look into the details of the dressed states,

$\widehat{H}_{\mu}|{b_{\mu}^{(j)}}\rangle=\beta_{\mu}^{(j)}|{b_{\mu}^{(j)}}\rangle,~{}j=1,2,3,4.$
(46)

The eigenfrequencies, $\beta_{\mu}^{(j)}$, and state vectors,
$|{b_{\mu}^{(j)}}\rangle$, of the four branches of the dressed spectrum are
[24]

$\begin{split}\beta_{\mu}^{(1)}&=\dfrac{1}{2}\big{(}\varDelta_{\mu\text{f}}+\varDelta_{\mu\text{s}}\big{)}+\dfrac{1}{2}\Omega_{\mu},\\\
&|{b_{\mu}^{(1)}}\rangle=|\Omega|e^{-i\phi}|{1}\rangle+\big{(}\varDelta_{\mu}-\Omega_{\mu}\big{)}|{2}\rangle;\\\
\beta_{\mu}^{(2)}&=\dfrac{1}{2}\big{(}\varDelta_{\mu\text{f}}+\varDelta_{\mu\text{s}}\big{)}-\dfrac{1}{2}\Omega_{\mu},\\\
&|{b_{\mu}^{(2)}}\rangle=|\Omega|e^{-i\phi}|{1}\rangle+\big{(}\varDelta_{\mu}+\Omega_{\mu}\big{)}|{2}\rangle;\\\
\beta_{\mu}^{(3)}&=-\dfrac{1}{2}\big{(}\varDelta_{-\mu\text{f}}+\varDelta_{-\mu\text{s}}\big{)}-\dfrac{1}{2}\Omega_{-\mu},\\\
&|{b_{\mu}^{(3)}}\rangle=|\Omega|e^{i\phi}|{3}\rangle+\big{(}\varDelta_{-\mu}-\Omega_{-\mu}\big{)}|{4}\rangle;\\\
\beta_{\mu}^{(4)}&=-\dfrac{1}{2}\big{(}\varDelta_{-\mu\text{f}}+\varDelta_{-\mu\text{s}}\big{)}+\dfrac{1}{2}\Omega_{-\mu},\\\
&|{b_{\mu}^{(4)}}\rangle=|\Omega|e^{i\phi}|{3}\rangle+\big{(}\varDelta_{-\mu}+\Omega_{-\mu}\big{)}|{4}\rangle.\end{split}$
(47)

The branches $(1)$ and $(2)$ describe the Rabi induced coupling between the
$+\mu$ sidebands in the fundamental and 2nd harmonic, and $(3)$, $(4)$ do the
same for the $-\mu$ sidebands. The corresponding dressed frequencies are

$\begin{split}&\widetilde{\omega}_{\mu\text{f}}^{(1),(2)}=\omega_{p}+\mu
D_{1}+\beta_{\mu}^{(1),(2)},\\\
&\widetilde{\omega}_{\mu\text{f}}^{(3),(4)}=\omega_{p}-\mu
D_{1}-\beta_{\mu}^{(3),(4)},\\\
&\widetilde{\omega}_{\mu\text{s}}^{(1),(2)}=2\omega_{p}+\mu
D_{1}+\beta_{\mu}^{(1),(2)},\\\
&\widetilde{\omega}_{\mu\text{s}}^{(3),(4)}=2\omega_{p}-\mu
D_{1}-\beta_{\mu}^{(3),(4)}.\end{split}$ (48)

Taking the explicit expressions for $\varDelta_{\mu}$ and
$(\varDelta_{\mu\text{f}}+\varDelta_{\mu\text{s}})/2$ inside
$\beta_{\mu}^{(j)}$ one would find that $\widetilde{\omega}_{\mu\zeta}^{(j)}$
are the reference frame, i.e., $D_{1}$, independent.

Power distribution between the fundamental and 2nd harmonic sidebands within a
given branch and for a given $\mu$, is determined by the dressing parameters,
$\varDelta_{\pm\mu}^{2}/|\Omega|^{2}$. If the Rabi detuning is relatively
large, i.e., a particular mode is far from being the sum-frequency matched,
see the points away from the zero line in Fig. 4(a), then the corresponding
dressed state, $|{b_{\mu}^{(j)}}\rangle$, tends towards an eigenstate of the
bare, i.e., $\Omega=0$, resonator. The bare states are

$\begin{split}&|{1}\rangle=(1,0,0,0)^{T},~{}e^{i(M+\mu)\vartheta-
it\omega_{\mu\text{f}}},\\\
&|{2}\rangle=(0,1,0,0)^{T},~{}e^{i(2M+\mu)\vartheta-
it\omega_{\mu\text{s}}},\\\
&|{3}\rangle=(0,0,1,0)^{T},~{}e^{i(M-\mu)\vartheta-it\omega_{-\mu\text{f}}}\\\
&|{4}\rangle=(0,0,0,1)^{T},~{}e^{i(2M-\mu)\vartheta-
it\omega_{-\mu\text{s}}},\end{split}$ (49)

where, for the sake of clarity, we explicitly associated each of the state
vectors to the corresponding resonator mode.

The maximal dressing condition, $\varDelta_{\pm\mu}=0$, involves frequencies
of the bare resonator, while the matching points are replaced by the avoided
crossings in the dressed resonator, see Fig. 4(b). The avoided crossing
between $\widetilde{\omega}_{\pm\mu\zeta}^{(1)}$ and
$\widetilde{\omega}_{\pm\mu\zeta}^{(2)}$ exist for every $\mu$, however, most
of them do not come to the practical, tens of MHz, proximity of the
$\delta=0$, apart from the ones nearest to $\pm\mu_{*}$.

Noting the symmetries

$\begin{split}&\beta_{\mu}^{(3)}=-\beta_{-\mu}^{(1)},~{}\text{i.e.,}~{}\widetilde{\omega}_{-\mu\zeta}^{(1)}=\widetilde{\omega}_{\mu\zeta}^{(3)},\\\
&\beta_{\mu}^{(4)}=-\beta_{-\mu}^{(2)},~{}\text{i.e.,}~{}\widetilde{\omega}_{-\mu\zeta}^{(2)}=\widetilde{\omega}_{\mu\zeta}^{(4)},\end{split}$
(50)

we conclude that there are two ways to proceed from this point. First, the
problem could be formulated using the four dressed frequencies, e.g.,
$\widetilde{\omega}_{\mu\text{f}}^{(1)}$,
$\widetilde{\omega}_{\mu\text{f}}^{(2)}$,
$\widetilde{\omega}_{\mu\text{f}}^{(3)}$,
$\widetilde{\omega}_{\mu\text{f}}^{(4)}$ [24]. Second, one could switch to
using the two-branch formulation and deal with
$\widetilde{\omega}_{\mu\text{f}}^{(1)}$,
$\widetilde{\omega}_{-\mu\text{f}}^{(1)}$, and
$\widetilde{\omega}_{\mu\text{f}}^{(2)}$,
$\widetilde{\omega}_{-\mu\text{f}}^{(2)}$. The latter approach is slightly
more intuitive and we choose to follow it here. In either case, the four
frequencies and eigenstates have to be traced. The
$\widetilde{\omega}_{\pm\mu\text{f}}^{(1)}$,
$\widetilde{\omega}_{\pm\mu\text{f}}^{(2)}$ frequencies in the dressed
spectrum are illustrated in Fig. 6. The
$\widetilde{\omega}_{\pm\mu\text{s}}^{(1)}$,
$\widetilde{\omega}_{\pm\mu\text{s}}^{(2)}$ spectra characterize the same
dressed states as $\widetilde{\omega}_{\pm\mu\text{f}}^{(1)}$,
$\widetilde{\omega}_{\pm\mu\text{f}}^{(2)}$. If the former are plotted then
they would make the same spectrum as in Fig. 6 apart from being centred at
$2\omega_{p}$, see Eq. (48).

Figure 7:  Frequency matching parameters
$\varepsilon_{\mu}^{(j_{1}j_{2})}=\widetilde{\omega}_{\mu_{1}\text{f}}^{(j_{1})}+\widetilde{\omega}_{\mu_{2}\text{f}}^{(j_{2})}-2\omega_{p}$
for the four types of the parametric-down conversion (PDC) conditions. The
sideband numbers grouped around the dashed horizontal lines,
$\varepsilon_{\mu}^{(j_{1}j_{2})}=0$, correspond to the MHz mismatches that
can be compensated by the nonlinear effects and lead to the exact PDC
frequency matching, see insets. (a), (b) $\delta=2.55\kappa_{\text{f}}$,
$|\Omega|/2\pi=76$MHz, (c), (d) $\delta=8.1\kappa_{\text{f}}$,
$|\Omega|/2\pi=105$MHz, and $\varepsilon_{0}/2\pi=10$GHz. $\mu\approx 10$, see
the corner points, corresponds to the sum-frequency matching,
$\varDelta_{-\mu}\approx 0$.  Figure 8: Lines corresponding to the exact PDC
frequency matching conditions in the parameter space of the pump detuning,
$\delta$, and of the Rabi frequency, $|\Omega|$, characterising the intra-
resonator cw-power, see Fig. 2. The red lines (numbered 0 to 5) correspond to
the intra-branch PDCs, which are
$\hbar\widetilde{\omega}_{\mu\text{f}}^{(1)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(1)}=\hbar
2\omega_{p}$ in (a) where $\varepsilon_{0}/2\pi=5$GHz, and
$\hbar\widetilde{\omega}_{\mu\text{f}}^{(2)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(2)}=\hbar
2\omega_{p}$ in (b) where $\varepsilon_{0}/2\pi=-5$GHz, $0\leq\mu\leq 5$. The
blue lines (numbered 5 to 11) correspond to the cross-branch PDCs, which are
$\hbar\widetilde{\omega}_{\mu\text{f}}^{(1)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(2)}=\hbar
2\omega_{p}$ in (a), and
$\hbar\widetilde{\omega}_{\mu\text{f}}^{(2)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(1)}=\hbar
2\omega_{p}$ in (b), $\mu\geq 5$. The lines in (a) and (b) use Eq. (54), while
the dots in (a) are derived from the approximate Eq. (55) for $\mu\neq 5$ and
Eq. (92) for $\mu=5$.  Figure 9: Parametric down-conversion (PDC) instability
tongues mapped onto the parameter space span by the laser detuning, $\delta$,
and the Rabi frequency, $|\Omega|$ (see the left axis), or, and equivalently,
by the intra-resonator cw power, $|\psi_{0\text{f}}|^{2}$ (see the right
axis). (a) is for $\varepsilon_{0}/2\pi=5$GHz, and (b) is for
$\varepsilon_{0}/2\pi=-5$GHz. The grey shaded tongues correspond to the intra-
branch PDC conditions marked by the red lines in Fig. 8, and the blue tongues
- to the cross-branch PDCs. Some of the tongues are marked with the respective
$\mu$’s, see Fig. 8 for the complete illustration of the $\mu$ ordering. The
magenta lines show
$|\Omega|=|\psi_{0\text{f}}|\sqrt{8\gamma_{\text{f}}\gamma_{\text{s}}}$ vs
$\delta$ achieved for the laser powers ${\cal W}=66\mu$W, $10$mW, and $100$mW.

##  VII Dressed spectrum and energy conservation in parametric-down conversion
(PDC)

In any parametric system, the parametric resonance is achieved for, usually, a
sequence of the resonance values of the drive frequency [43]. In optical
resonators in general, and in the dressed $\chi^{(2)}$ system, in particular,
this is done by tuning $\omega_{p}$ to the mid-point between the desired
sidebands, e.g., $\widetilde{\omega}_{\mu\text{f}}^{(1)}$ and
$\widetilde{\omega}_{-\mu\text{f}}^{(1)}$. One peculiar feature of our case is
that the dressed resonances depend on the pump power and frequency. Another is
that for the two pairs of frequencies,
$\widetilde{\omega}_{\pm\mu\text{f}}^{(1)}$,
$\widetilde{\omega}_{\pm\mu\text{f}}^{(2)}$, there could be four different
mid-points for the same $\mu$, and hence four conditions providing the maximum
of the parametric gain [24, 44],

$\displaystyle
2\hbar\omega_{p}=\hbar\widetilde{\omega}_{\mu\text{f}}^{(1)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(1)},~{}(\beta_{\mu}^{(1)}=-\beta_{-\mu}^{(1)}),$
(51a) $\displaystyle
2\hbar\omega_{p}=\hbar\widetilde{\omega}_{\mu\text{f}}^{(2)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(2)},~{}(\beta_{\mu}^{(2)}=-\beta_{-\mu}^{(2)}),$
(51b) $\displaystyle
2\hbar\omega_{p}=\hbar\widetilde{\omega}_{\mu\text{f}}^{(1)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(2)},~{}(\beta_{\mu}^{(1)}=-\beta_{-\mu}^{(2)}),$
(51c) $\displaystyle
2\hbar\omega_{p}=\hbar\widetilde{\omega}_{\mu\text{f}}^{(2)}+\hbar\widetilde{\omega}_{-\mu\text{f}}^{(1)},~{}(\beta_{\mu}^{(2)}=-\beta_{-\mu}^{(1)}).$
(51d)

The auxiliary frequency $\widetilde{\omega}_{\mu}=\omega_{p}+\mu
D_{1\text{f}}+\tfrac{1}{2}(\varDelta_{\mu\text{f}}+\varDelta_{\mu\text{s}})$
makes the role of the Rabi splitting in the above more transparent,

$\displaystyle
2\hbar\omega_{p}=\hbar(\widetilde{\omega}_{\mu}+\tfrac{1}{2}\Omega_{\mu})+\hbar(\widetilde{\omega}_{-\mu}+\tfrac{1}{2}\Omega_{-\mu}),$
(52a) $\displaystyle
2\hbar\omega_{p}=\hbar(\widetilde{\omega}_{\mu}-\tfrac{1}{2}\Omega_{\mu})+\hbar(\widetilde{\omega}_{-\mu}-\tfrac{1}{2}\Omega_{-\mu}),$
(52b) $\displaystyle
2\hbar\omega_{p}=\hbar(\widetilde{\omega}_{\mu}+\tfrac{1}{2}\Omega_{\mu})+\hbar(\widetilde{\omega}_{-\mu}-\tfrac{1}{2}\Omega_{-\mu}),$
(52c) $\displaystyle
2\hbar\omega_{p}=\hbar(\widetilde{\omega}_{\mu}-\tfrac{1}{2}\Omega_{\mu})+\hbar(\widetilde{\omega}_{-\mu}+\tfrac{1}{2}\Omega_{-\mu}).$
(52d)

The first pair of conditions, Eqs. (51a), (51b), corresponds to the intra-
branch PDC, which is satisfied by tuning the pump frequency to the mid-point
between the $\widetilde{\omega}_{\mu\text{f}}^{(j)}$ and
$\widetilde{\omega}_{-\mu\text{f}}^{(j)}$. The second pair, Eqs. (51c), (51d),
are the cross-branch PDC conditions. They are satisfied by $\omega_{p}$ being
tuned to the mid-point between the $\mu$ and $-\mu$ sidebands from the two
different branches of the dressed spectrum.

Figure 6 illustrates achieving frequency matching for the intra- and cross-
branch cases, and also shows how the effective Rabi frequencies,
$\Omega_{\pm\mu}$, come into play. $\Omega_{\mu}$ and $\Omega_{-\mu}$ are
generally very different, and coincide only for $\mu=0$, see Fig. 5.

To gain further important insights into the PDC conditions in Eq. (51), we
rearrange them as

$\varepsilon_{\mu}^{(j_{1},j_{2})}=\widetilde{\omega}_{\mu\text{f}}^{(j_{1})}+\widetilde{\omega}_{-\mu\text{f}}^{(j_{2})}-2\omega_{p}$
(53)

and plot $\varepsilon_{\mu}^{(j_{1},j_{2})}$ vs $\mu$, see Fig. 7. The
sideband numbers grouped around the zero lines correspond to the MHz level
mismatches that are more easily compensated by the nonlinear effects providing
$\varepsilon_{\mu}^{(j_{1},j_{2})}=0$. The sidebands with
$\varepsilon_{\mu}^{(j_{1},j_{2})}$ detuned away from the zero by the GHz
offsets (see the black dots in Figs. 7(a), (b)) are cutting-off from the
groups of the PDC capable mode numbers. The cut-off is happening for $\mu$’s
around $\mu_{*}$, corresponding to the sum-frequency matching, see Eq. (31).

Figure 7(a) shows that the intra-branch condition in Eq. (51a) can be
satisfied for $0\leq\mu\lesssim\mu_{*}$, while $\varepsilon_{0}$ is set to be
positive. The second intra-branch condition, Eq. (51b), see Fig. 7(c), is
shifted away from zero by $\approx 2\varepsilon_{0}$, and will be swapped with
the first one for $\varepsilon_{0}\to-\varepsilon_{0}$. Figures 7(b) and 7(d)
show the cross-branch PDC conditions, with one of them being satisfied for
$\mu\gtrsim\mu_{*}$.

The challenge with resolving Eq. (51) analytically for either $\Omega$ or
$\delta$ is in the occurrences of them under the square root sign in the
equation for $\Omega_{\mu}$, see Table 2. However, the algebra is proceedable
[24, 44], and leads to finding that all four PDC conditions are resolved by
$|\Omega|=|\Omega_{\text{pdc}}|$, where

$\displaystyle|\Omega_{\text{pdc}}|^{2}$
$\displaystyle=4(\varDelta_{\mu\text{f}}+\varDelta_{-\mu\text{f}})(\varDelta_{\mu\text{s}}+\varDelta_{-\mu\text{s}})$
$\displaystyle\times\frac{(\varDelta_{\mu\text{f}}+\varDelta_{-\mu\text{s}})(\varDelta_{\mu\text{s}}+\varDelta_{-\mu\text{f}})}{(\varDelta_{\mu\text{f}}+\varDelta_{-\mu\text{f}}+\varDelta_{\mu\text{s}}+\varDelta_{-\mu\text{s}})^{2}}.$
(54)

Plots of $|\Omega_{\text{pdc}}|$ vs $\delta$ for positive and negative
$\varepsilon_{0}$, and their associations with the PDC conditions are shown in
Fig. 8. These plots in Fig. 8 could be compared with the temperature tuning
diagrams of the parametric oscillators, see, e.g., [34, 35, 36], but in our
case, the temperature is assumed fixed, while the tuning parameters are the
pump power expressed via $|\Omega|$, and the pump frequency. The range of
$|\Omega|$s considered by us provides the relatively small intraresonator
powers, order of mW to $<1$W, cf., the left and right axes in Fig. 9.

If $|\varepsilon_{0}|$ dominates over all $\delta$’s, and $\mu_{*}$ falls
between the two nearest integers, i.e., the exact sum-frequency matching point
has been missed, then Eq. (54) simplifies to

$|\Omega_{\text{pdc}}|^{2}\approx-4\varepsilon_{0}\left(\delta-\delta_{\mu\text{f}}\right)\left[1-\frac{\mu^{2}}{\mu_{*}^{2}}\right],~{}\delta_{\mu\text{f}}=-\tfrac{1}{2}D_{2\text{f}}\mu^{2},$
(55)

see Appendix E for details.

Eq. (55) reveals what has been described above based on the numerical plots.
First, one can see that the resonances converge to points
$\delta=\delta_{\mu\text{f}}$ for $\Omega\to 0$, which corresponds to the zero
of the first bracket in the numerator of Eq. (54). Second, the direction of
the nonlinearity induced tilts of the resonances depends on the sign of
$\varepsilon_{0}$ and the value of $\mu$, see Fig. 8. If $\varepsilon_{0}<0$,
then the tilt is towards $\delta>\delta_{\mu\text{f}}$ for $0\leq\mu<\mu_{*}$,
and towards $\delta<\delta_{\mu\text{f}}$ for $\mu>\mu_{*}$.
$\varepsilon_{0}>0$ changes the tilt direction for the two groups of modes.

Figure 8 also shows a good agreement between the exact and approximate
$|\Omega_{\text{pdc}}|^{2}$ vs $\delta$ dependencies. Analytical approximation
for $|\Omega_{\text{pdc}}|$ in the case when the sum-frequency process is
either nearly or exactly matched, i.e., $\mu=\widehat{\mu}\approx\mu_{*}$, is
considered in Appendix E.

To summarise - the intra-branch PDC conditions are satisfied for a compact
group of the sideband numbers,

$0\leq\mu\leq\mu_{*},~{}\text{i.e.,
for}~{}\left[1-\frac{\mu^{2}}{\mu_{*}^{2}}\right]\geq 0,$ (56)

while the cross-branch ones are engaged for the unbound set of sidebands,

$\mu\geq\mu_{*},~{}\text{i.e.,
for}~{}\left[1-\frac{\mu^{2}}{\mu_{*}^{2}}\right]\leq 0.$ (57)

Thus, the sum-frequency matched sideband, $\mu_{*}$ or $-\mu_{*}$, defines the
transition between the two different PDC scenarios.

##  VIII PDC instability tongues

While the PDC frequency matching provides conditions for the maximal
parametric gain, the latter still needs to overcome the dissipation in order
to trigger the exponential growth of sidebands, i.e., to induce the cw
instabilities. Regions of the PDC instabilities for every $\pm\mu$ pair of
sidebands can be computed numerically by solving the eigenvalue problem in Eq.
(45) and plotting the lines $\lambda_{\mu}=0$ [24]. Every $\mu$-specific
instability area is represented by a tongue-like domain shaped around the
respective $|\Omega_{\text{pdc}}|$ vs $\delta$ line, cf., Figs. 8 and 9.

The intra-branch instabilities, Eqs. (51a) and (51b), are coloured in grey in
Fig. 9. While, the cross-branch ones, Eqs. (51c), (51d), are shown in blue.
The magenta lines show $|\Omega|$’s vs $\delta$ corresponding to the cw-state
achieved for three representative values of the laser power, ${\cal
W}=66\mu$W, $10$mW, and $100$mW. The cw-state is expressed via $\Omega$ as per
Eq. (19), while $\Omega$ itself is a solution of

$\Omega=\sqrt{\frac{\kappa_{\text{f}}\kappa_{\text{s}}}{2}}\sqrt{\frac{{\cal
W}~{}}{{\cal
W}_{*}}}\frac{\kappa_{\text{f}}}{\Omega_{\text{f}}}\left[1-\frac{|\Omega|^{2}}{\Omega_{\text{f}}\Omega_{\text{s}}}\right]^{-1},$
(58)

where $\Omega_{\text{f}}=\delta-i\tfrac{1}{2}\kappa_{\text{f}}$,
$\Omega_{\text{s}}=8(2\delta-i\tfrac{1}{2}\kappa_{\text{s}})-8\varepsilon_{0}$,
${\cal W}$ is the laser power in Watts, and ${\cal W}_{*}$ is its scaling, see
Eq. (83). Taking modulus squared of Eq. (58) we find a real cubic equation for
$|\Omega|^{2}$, that can have either one or three positive roots, with the
latter case signalling the cw-bistability, see Fig. 2, and further details in
Appendix C.

Thus, a scan of the laser frequency, $\omega_{p}$, would go along an
individual power-defined path crossing the different PDC domains, see the
magenta lines in Fig. 9. The yellow shading marks the $\mu=0$ tongue embracing
the middle branch of the bistability loop, see Fig. 2. For the Kerr
resonators, the similar tongue diagrams were recently reported in Refs. [47,
50].

Figure 10: Examples of the Turing-pattern frequency combs for
$\varepsilon_{0}/2\pi=5$GHz. First and second columns show the combs emerging
from the intra-branch PDC instability tongues (grey areas in (a,b)), and the
third column shows the cross-branch PDC tongue (blue area in (c)) and the
associated comb. Magenta lines in (a-c) show the cw-states,
$|\Omega|=|\psi_{0\text{f}}|\sqrt{8\gamma_{\text{f}}\gamma_{\text{s}}}$ vs
$\delta$. The respective laser powers ${\cal W}=0.664$mW (a), $103$mW(b) and
$10$mW(c). $|\Omega|/2\pi=0.2$GHz corresponds to the intraresonator power
$|\psi_{0\text{f}}|^{2}\simeq 55$mW. The panels (d,e,f) in the second line
show the ratio of the 2nd harmonic and fundamental powers in the indicated
sidebands. $\mu=-5$ case corresponds to the sum-frequency matching leading to
the best conversion efficiency. The thick full lines are stable solutions, and
the thin lines (full and dashed) are unstable. (g,h,i) are the self-
explanatory spectra of the Turing patterns (red-fundamental, green- 2nd
harmonic). (j,k,l) are the space-time profiles of the corresponding Turing-
patterns. The tilt relative to the vertical axis characterizes deviation of
the pattern repetition rate from $D_{1\text{f}}$.  Figure 11: (a) PDC
instability tongues for $\mu=12-18$, $\varepsilon_{0}/2\pi=5$GHz as in Fig.
9(a). The magenta line shows the cw-state for the laser power ${\cal W}=10$mW.
The left axes is the Rabi frequency, $|\Omega|$, and the right axis is the
intra-resonator cw power, $|\psi_{0\text{f}}|^{2}$. (b) The spectrum computed
from the scan of the cw-frequency along the magenta line in (a). (c) The
integrated intra-resonator powers corresponding to (b). The red line (left
axis) is the fundamental, and the green line (right axis) is the second
harmonic. Panels in the two bottom lines show the comb spectra computed at the
detunings marked with the respective letters in (a), and with the dashed white
lines in (b). In the two bottom rows, the shorter bars are plotted in green
and the longer ones in red.  Figure 12: The top row shows the breather state
corresponding to the point ’f’ in Fig. 11(a). The bottom row is the turbulent
state at the point ’e’ in Fig. 11(a). The 1st and 2nd columns compare the
spatio-temporal dynamics (fundamental) and snapshots of the spatial profiles.
The 3rd and 4th columns show the RF spectra of net powers (3rd),
$|FFT\sum_{\mu}|\psi_{\mu\text{f}}(t)|^{2}|$, and the per-mode RF spectra
(4th), $|FFT\\{\psi_{\mu\text{f}}(t)\\}|^{2}$. One can see that the breather
state consists from the five coherent sub-combs with the different offsets and
the same repetition rates, while the turbulent state shows no modelocking
signs.

##  IX Parametric thresholds

PDC thresholds, i.e., the minimal intra-resonator powers triggering the
exponential growth of the $\pm\mu$ sideband pairs, happen at the tips of the
instability tongues, see Fig. 9. To find the threshold when the system is
confined to the $|\mu|$-specific PDC lines, see Figs. 8, 9, we apply the
degenerate state perturbation theory to Eq. (45) by treating $\widehat{V}$ as
a perturbation to $\widehat{H}_{\mu}$, which is valid in the SC regime. The
generic condition for the parametric gain to overcome losses has been
introduced in Ref. [24],

$V_{\mu}^{(j_{1}j_{2})}\cdot
V_{\mu}^{(j_{2}j_{1})}=V_{\mu}^{(j_{1}j_{1})}\cdot V_{\mu}^{(j_{2}j_{2})},$
(59)

where
$V_{\mu}^{(j_{1}j_{2})}=\langle{b_{\mu}^{(j_{1})}}|\widehat{V}|{b_{\mu}^{(j_{2})}}\rangle$
are the matrix elements of $\widehat{V}$.

Opening up Eq. (59) for $(j_{1},j_{2})=(1,3)$, $(j_{1},j_{2})=(2,4)$,
$(j_{1},j_{2})=(1,4)$, and $(j_{1},j_{2})=(2,3)$ yields four threshold
conditions,

$\displaystyle\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{\mu}-\Omega_{\mu})^{2}}{|\Omega|^{2}}\right]\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{-\mu}-\Omega_{-\mu})^{2}}{|\Omega|^{2}}\right]=\frac{4|\Omega|^{4}}{|\Omega_{\text{s}}|^{2}},$
(60a)
$\displaystyle\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{\mu}+\Omega_{\mu})^{2}}{|\Omega|^{2}}\right]\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{-\mu}+\Omega_{-\mu})^{2}}{|\Omega|^{2}}\right]=\frac{4|\Omega|^{4}}{|\Omega_{\text{s}}|^{2}},$
(60b)
$\displaystyle\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{\mu}-\Omega_{\mu})^{2}}{|\Omega|^{2}}\right]\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{-\mu}+\Omega_{-\mu})^{2}}{|\Omega|^{2}}\right]=\frac{4|\Omega|^{4}}{|\Omega_{\text{s}}|^{2}},$
(60c)
$\displaystyle\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{\mu}+\Omega_{\mu})^{2}}{|\Omega|^{2}}\right]\left[\kappa_{\text{f}}+\kappa_{\text{s}}\frac{(\varDelta_{-\mu}-\Omega_{-\mu})^{2}}{|\Omega|^{2}}\right]=\frac{4|\Omega|^{4}}{|\Omega_{\text{s}}|^{2}}.$
(60d)

Eqs. (60) express the balance between the PDC gain (right) and the net loss
(left). The 2nd harmonic losses, $\kappa_{\text{s}}$ are weighted by the
coefficients characterising the power distribution between the components in
the state vectors, see Eq. (47). The explicit threshold condition presented in
Ref. [24] transforms to Eq. (60c) after making use of the identity
$\frac{|\Omega|}{\Omega_{\mu}-\varDelta_{\mu}}=\frac{\Omega_{\mu}+\varDelta_{\mu}}{|\Omega|}$.

We now note that in the SC regime $|\Omega_{\text{s}}|\approx
8|\varepsilon_{0}|$, see Eq. (23), and hence the right-hand sides in Eq. (60)
are approximated with $|\Omega|^{4}/16|\varepsilon_{0}|^{2}$. To simplify the
threshold conditions we make use of the approximations in Eq. (32) related to
the case of $\varepsilon_{0}>0$ and $\Omega_{-\widehat{\mu}}\approx|\Omega|$,
see Fig. 5(a). Then, for $\mu<\widehat{\mu}$ both coefficients after
$\kappa_{\text{s}}$ in Eq. (60a) are small and can be omitted in the leading
order, so that the threshold is determined by $\kappa_{\text{f}}$ only and is
well approximated by,

$|\Omega_{\text{th}}^{(\mu)}|^{2}\approx
4|\varepsilon_{0}|\kappa_{\text{f}},~{}\mu\neq\widehat{\mu},$ (61)

see the grey tongs in Fig. 9(a). For $\mu>\widehat{\mu}$ the left-hand side of
Eq. (60a) becomes $\approx
4\kappa_{\text{f}}\kappa_{s}|\varDelta_{-\mu}|^{2}/|\Omega|^{2}$, which
creates prohibitively large power thresholds, see the cut-off transitions in
Fig. 7.

For $\mu=\widehat{\mu}$, the threshold is approximated by

$|\Omega_{\text{th}}^{(\widehat{\mu})}|^{2}\approx
4|\varepsilon_{0}|\sqrt{\kappa_{\text{f}}(\kappa_{\text{f}}+\kappa_{\text{s}})}.$
(62)

$\kappa_{\text{s}}$ is now also impacting the threshold, but still in a way
that is not equally important with $\kappa_{\text{f}}$. This is because the
powers of the fundamental and second harmonic are balanced only for
$\widetilde{\omega}_{-\widehat{\mu}\text{f}}^{(1)}$ and
$\widetilde{\omega}_{-\widehat{\mu}\text{s}}^{(1)}$, but not for
$\widetilde{\omega}_{\widehat{\mu}\text{f}}^{(1)}$ and
$\widetilde{\omega}_{\widehat{\mu}\text{s}}^{(1)}$ sidebands. In the latter
pair, the sum-frequency condition is mismatched, and hence the 2nd harmonic
sideband is still very weak and can be disregarded, i.e.,
$\varDelta_{\widehat{\mu}}\approx\Omega_{\widehat{\mu}}$ in Eq. (60a). While,
the approximations that work in the minus bracket are
$\varDelta_{-\widehat{\mu}}\approx 0$, and
$\Omega_{-\widehat{\mu}}\approx|\Omega|$.

The first cross-branch condition, Eq. (60c), has practical threshold at
$\mu=\widehat{\mu}$ as in Eq. (62), and for $\mu>\widehat{\mu}$ as in Eq.
(61), see the blue tongues in Fig. 9(a). The second intra-branch, Eq. (60b),
and second cross-branch, Eq. (60d), conditions do not create practical
thresholds for $\varepsilon_{0}>0$, and play their roles for
$\varepsilon_{0}<0$, see Figs. 8(b), 9(b).

The analytical estimates for the detuning values where the $\mu$-specific
instabilities first happen, i.e., locations of the tips of the instability
tongues and the respective laser powers are derived in Appendix F.

##  X Envelope and coupled-mode equations for modelocked combs

We anticipate that the frequency comb solutions bifurcate from the
$\mu$-specific boundaries of the instability tongues in Fig. 9, cf., Ref.
[47]. Since we are going to continue to number the sidebands within the combs
with $\mu$, we use below the letter $\nu=1,2,3,\dots$ to mark the comb states
with the sideband spacing given by $\nu$. We seek the modelocked combs as the
solutions of the equation that couple all the modes through all the allowed
nonlinear coupling terms, see Eq. (75) in Appendix A.

The modelocked combs are assumed to have the period $2\pi/\nu$, i.e.,
$\psi_{\zeta}(t,\vartheta)=\psi_{\zeta}(t,\vartheta+2\pi/\nu)$, and,
therefore, we use the substitution

$\psi_{\zeta}(t,\vartheta)=\Psi_{\nu\zeta}(\theta_{\nu}),~{}\theta_{\nu}=\nu(\vartheta-
D_{1\nu}t).$ (63)

Here $\theta_{\nu}$ is a new auxiliary coordinate, such that the period $2\pi$
in $\theta_{\nu}$ corresponds to the period $2\pi/\nu$ in $\theta$,
$\Psi_{\nu\zeta}(\theta_{\nu})=\Psi_{\nu\zeta}\left(\theta_{\nu}+2\pi\right)$.
$D_{1\nu}$ is an unknown comb repetition rate generally different from either
$D_{1\text{f}}$ or $D_{1\text{s}}$. If the reference frame is chosen to rotate
with $D_{1\text{f}}$, then $D_{1\nu}\neq D_{1\text{f}}$ would imply the
relative rotation with the $D_{1\nu}-D_{1\text{f}}$ rate, leading to the
tilted spatio-temporal profiles like in the bottom row in Fig. 10

The selection mechanisms of the velocity of the dissipative $\chi^{(2)}$
solitons, equivalent to the selection of $D_{1\nu}$, have been discussed
before the microresonator combs came into the existence [28]. This selection
is a generic aspect also encountered in, e.g., the equations with the higher-
order dispersion terms [46], and in the cases showing the spontaneous symmetry
breaking effects [47].

Substituting Eq. (63) into Eq. (75) we find

$\begin{split}&\delta\Psi_{\nu\text{f}}-i\nu(D_{1\text{f}}-D_{1\nu})\frac{d\Psi_{\nu\text{f}}}{d\theta_{\nu}}-\frac{\nu^{2}D_{2\text{f}}}{2}\frac{d^{2}\Psi_{\nu\text{f}}}{d\theta_{\nu}^{2}}\\\
-&\gamma_{\text{f}}\Psi_{\nu\text{s}}\Psi_{\nu\text{f}}^{*}-\frac{i\kappa_{\text{f}}}{2}\big{(}\Psi_{\nu\text{f}}-{\cal
H}\big{)}=0,\\\
&(2\delta-\varepsilon_{0})\Psi_{\nu\text{s}}-i\nu(D_{1\text{s}}-D_{1\nu})\frac{d\Psi_{\nu\text{s}}}{d\theta_{\nu}}-\frac{\nu^{2}D_{2\text{s}}}{2}\frac{d^{2}\Psi_{\nu\text{s}}}{d\theta_{\nu}^{2}}\\\
-&\gamma_{\text{s}}\Psi_{\nu\text{f}}^{2}-\frac{i\kappa_{\text{s}}}{2}\Psi_{\nu\text{s}}=0.\end{split}$
(64)

As one can see, the use of $\theta_{\nu}$ as an argument has allowed to
conveniently sort the modelocked combs by the spacing, $\nu$, their sidebands
make in the momentum space, since $\nu$ enters Eq. (64) explicitly. In fact,
$\nu$ plays a role of the Bloch momentum which is now quantised, unlike the
one that varies continuously in the theory of the unbound crystal lattices
[48] and resonators [49].

For the frequency combs with the spatial period $2\pi/\nu$, the modes making
non-zero contributions to the $\Psi_{\nu\zeta}$ have numbers $\mu=\nu m$,
where $m=0,\pm 1,\pm 2,\pm 3,\dots$ is another integer,

$\Psi_{\nu\zeta}=\sum_{m=-\infty}^{\infty}\Psi_{m\nu\zeta}e^{im\theta_{\nu}}=\sum_{m=-\infty}^{\infty}\Psi_{m\nu\zeta}e^{im\nu(\vartheta-
D_{1\nu}t)}.$ (65)

Here $\Psi_{m\nu\zeta}$ are constants satisfying an algebraic system of
equations, see Eq. (66). The repetition rate with which the $2\pi/\nu$ state
is reproducing itself while rotating in the resonator is $\nu D_{1\nu}/2\pi$.

The coupling between the different $m\nu$ sidebands is provided by the
sequence, i.e., cascade, of the sum-frequency and difference-frequency events,
which become evident on substituting Eq. (65) into Eq. (64),

$\displaystyle\varDelta_{m\nu\text{f}}\Psi_{m\nu\text{f}}-\frac{i\kappa_{\text{f}}}{2}\big{(}\Psi_{m\nu\text{f}}-\widehat{\delta}_{m,0}{\cal
H}\big{)}$
$\displaystyle-\gamma_{\text{f}}\sum_{m_{1}m_{2}}\widehat{\delta}_{m,m_{1}-m_{2}}\Psi_{m_{1}\nu\text{s}}\Psi^{*}_{m_{2}\nu\text{f}}=0,$
(66a)
$\displaystyle\varDelta_{m\nu\text{s}}\Psi_{m\nu\text{s}}-\frac{i\kappa_{\text{s}}}{2}\Psi_{m\nu\text{s}}$
$\displaystyle-\gamma_{\text{s}}\sum_{m_{1}m_{2}}\widehat{\delta}_{m,m_{1}+m_{2}}\Psi_{m_{1}\nu\text{f}}\Psi_{m_{2}\nu\text{f}}=0.$
(66b)

The sideband detunings, $\varDelta_{m\nu\zeta}$, are defined in Eq. (8), where
$\mu=m\nu$. $\widehat{\delta}_{m,m_{1}\pm m_{2}}=1$ for $m=m_{1}\pm m_{2}$ and
is zero otherwise. Hence, every term inside the nonlinear sums in Eqs. (66a)
and (66b) corresponds to the momentum conservation laws,

$\displaystyle\hbar(M+m\nu)=\hbar(2M+m_{1}\nu)-\hbar(M+m_{2}\nu),$ (67a)
$\displaystyle\hbar(2M+m\nu)=\hbar(M+m_{1}\nu)+\hbar(M+m_{2}\nu),$ (67b)

describing the sum- and difference-frequency cascades. The left- and right-
hand sides of Eq. (67) correspond to the linear and nonlinear terms in Eq.
(66), respectively.

##  XI Turing-pattern combs

The comb equations, Eq. (64) or Eq. (66), have been solved by us with a Newton
method allowing to self-consistently find the sideband amplitudes,
$\Psi_{m\nu\zeta}$, and the comb repetition rate, $D_{1\nu}$. Figure 10 shows
how the comb branches with different periods, $2\pi/\nu$, emanate from the
boundaries of the respective instability tongues.

The sparse combs, $\nu\gg 1$, described by a combination of few noticeable
$\mu=\nu m$ sidebands have been observed and modelled in connection to several
recent $\chi^{(2)}$-resonator experiments, see, e.g., [12, 13, 14, 25]. In the
context of the Kerr microresonators, the combs with the sparse spectra, as in
Fig. 10, are often called - the Turing-pattern frequency combs [51, 52, 47].
The Kerr microresonator instability tongues and their connection to the
Turing-patterns have been reported in Refs. [47, 50]. The prior to the ring
microresonator era results on the spatial pattern formation in the planar
$\chi^{(2)}$ resonators can be found in, e.g., Refs. [53, 54, 55, 56, 57, 58,
59].

A typical bifurcation scenario that we found is that the two Turing-comb
branches split from the two edges of the tongues and extend well beyond the
tongues. The branch crossing into the tongue area can be stable (full lines in
the second row in Fig. 10), while the one deviating outside the tongue (dashed
lines) is always unstable. These two branches coalesce for some detuning
value, see Fig. 10(e). The Turing combs emerging from the intra-branch PDC
conditions for $\nu=4,5$ are shown in the first two columns in Fig. 10, and
the $\nu=16$ cross-branch case is shown in the third column.

The $\nu=5$ tongue in Fig. 10(b) corresponds to the
$\omega_{p}+\omega_{-5\text{f}}-\omega_{-5\text{s}}=0$ sum-frequency matching,
see Eq. (25), and therefore, the power of the $-5$th sideband in the 2nd
harmonic is either comparable or even exceeds the one in the fundamental, see
Figs. 10(e), (h), and compare the y-axis scales in (d),(e), and (f). The
bottom line in Fig. 10 shows how the Turing patterns loop around the resonator
in the reference frame rotating with $D_{1\text{f}}$, so that the most
efficient generation of the 2nd harmonic leads to the largest differences
between the linear and nonlinear repetition rates, $D_{1\text{f}}-D_{1\nu}$,
cf., the pattern angles in Figs. 10(j), (k), and (l).

Figure 11 shows a more global outlook on the frequency conversion processes
happening across the PDC tongues when the laser frequency is scanned and its
power is fixed, and $\varepsilon_{0}>0$. The stable Turing-combs are typically
generated for the relatively small intra-resonator powers achieved for large
positive detunings. When detunings are reduced and the intra-resonator powers
are increasing the instabilities bring the breather states producing the
denser combs as in Figs. 11(f),(g),(i), and then more developed chaotic states
as in Figs. 11(d), (e).

The difference between the breathers and chaos is further elucidated in Fig.
12. The first and 2nd columns in Fig. 12 compare the spatio-temporal and
spatial profiles of the two. The coherence of the breather and the incoherence
of the turbulent state are confirmed by comparing the RF spectra of the net
powers (3rd column in Fig. 12), and the per-mode RF spectra (4th column). The
latter shows $|FFT\\{\psi_{\mu\text{f}}(t)\\}|^{2}$ plotted vs the mode
number, $\mu$, and the RF frequency. The top, i.e., the breather, panel has
five visible tilted lines that are vertically separated by the breather
period. These are the five sub-combs having different offset frequencies.

In the example shown in the top line of Fig. 12, the maximal parametric gain
comes to the sidebands with $\mu=\pm 15$ and to the associated sub-combs, see
the top panel in the last column, so that the breather could be interpreted as
a quasi-soliton created via the non-degenerate PDC and dominated by the three
groups of modes centred around $\mu=-15,0,15$. The power and per-mode RF
spectra in the 2nd line of Fig. 12 are unambiguous about the absence of the
inter-mode coherence in the turbulent state. For studies of the transitions
between the breather and chaotic states associated with the rogue-wave
turbulence in the Kerr resonators, see, e.g., Refs. [61, 62, 60].

Figure 13: Dispersion of the dressed states, see Eq. (68), normalised to
$2\pi$: (a) $\widetilde{\omega}^{(1)}_{\pm\mu\zeta}$, and (b)
$\widetilde{\omega}^{(2)}_{\pm\mu\zeta}$. The red lines correspond to the
positive sidebands, $\widetilde{\omega}^{(1),(2)}_{\mu\zeta}$, and the blue
ones to the negative sidebands, $\widetilde{\omega}^{(1),(2)}_{-\mu\zeta}$.
$-100$kHz, and $-200$kHz are the values of $D_{2\text{f}}$ and $D_{2\text{s}}$
in the bare resonator. The grey shading and stars show the dressed states with
the anomalous dispersion induced by the dressing. In (a), the maximal
anomalous dispersion of $\sim 1$GHz (not shown) is achieved at $\mu=25$ for
the $|{b_{25}^{(3)}}\rangle\approx|{3}\rangle e^{i\phi}-|{4}\rangle$ state.
Parameters are $|\Omega|/2\pi=116$MHz, $\varepsilon_{0}/2\pi=25$GHz,
$\delta=-3.8\kappa_{\text{f}}$.  Figure 14: (a) Parametric instability tongues
in the ($\delta$,$|\Omega|$) plane, for $\varepsilon_{0}/2\pi=25$GHz, cf.,
$\varepsilon_{0}/2\pi=5$GHz in Fig. 9(a). The colorbar shows the number of the
simultaneously unstable sidebands, $N$. The black doted line embraces the
$\mu=0$ instability range, i.e., the middle branch of the bistability loop.
$N$ is clamped by
$\mu_{*}=|\varepsilon_{0}|/|D_{1\text{f}}-D_{1\text{s}}|=25$, $N\leq\mu_{*}$.
(b) $N$ vs $\delta$ for three values of ${\cal W}$ (laser power) computed
along the upper branch of the cw-state. The magenta line (${\cal W}=333\mu$W)
terminates at the tip of the cw resonance, i.e., at the end of the bistability
range. The other two lines are for the higher powers and terminate before the
tip, thereby marking stabilisation of the cw-state.  Figure 15: Families of
the bright solitons computed for the laser power ${\cal W}=333\mu$W and
plotted vs $\delta$. The first line data show the fundamental field, and the
second line is for the second harmonic. Panels (a,b) show the soliton branches
(red and green lines) and the cw-state (magenta lines) vs $\delta$. The
solitons are shown for a range of frequency mismatch parameters,
$\varepsilon_{0}/2\pi=$ 21, 23, 25, 27, and 29GHz. The full lines correspond
to the stable solutions, and the dashed lines to the unstable ones. The
$\varepsilon_{0}/2\pi=25$GHz case in (a) and (b) is the one that shows how the
unstable soliton splits from the cw-state and then connects to the stable
soliton. (c,e) shows how the spatial profile of the soliton changes with
$\delta$ for $\varepsilon_{0}/2\pi=25$GHz. (d,f) are like (c,e) but show the
envelopes of the discrete soliton spectra, cf., Fig. 17.  Figure 16:
Instability of the cw-state and spontaneous birth of the bright soliton. Laser
power ${\cal W}=333\mu$W, $\varepsilon_{0}/2\pi=25$GHz, and
$\delta=-3.8\kappa_{\text{f}}$. (a) is the fundamental field and (b) is the
second harmonic.  Figure 17: Families of the bright solitons computed for the
laser power ${\cal W}=333\mu$W and plotted vs $\varepsilon_{0}$ for
$\delta=-3.9\kappa_{\text{f}}$. The first and second columns show the
fundamental and second harmonic data, respectively. Panels (a,b) show the
multiple soliton branches centred around $\varepsilon_{0}/2\pi\approx\mu_{*}$,
see Eq. (31). (c,d,e,f) show the spectra at the indicated values of
$\varepsilon_{0}$. (g,i) show the respective spatial soliton profiles. The
shaded areas in (c,d,e,f) highlight the interval between the two zero
dispersion points, cf., Fig. 13.

##  XII Bright soliton prerequisites

The strong-coupling regime and the dressed state that naturally emerge from
it, imply that the parametric gain is relatively small, and therefore, to the
leading order, the soliton existence problem could be approached through the
prism of the nonlinearity vs dispersion balance. In this section, we work
through the dispersive and nonlinear properties of the dressed states, and the
stability of the cw state that underpins the existence of the bright soliton
frequency combs. For normal dispersion, $D_{2\zeta}<0$, as we have here, these
properties suggest the soliton existence for $\varepsilon_{0}>0$, which is the
case we describe in details below. Solitons themselves are introduced in
Section XIII.

###  XII.1 Dispersion of dressed states

The microresonator dressed states are parametrised by momentum $\mu$, and
represent families of the quasi-particles, photon-photon polaritons, with the
effective mass inversely proportional to the polariton dispersion [24]. Our
focus here is on the frequency conversion, and therefore the dispersion
terminology is more natural. Dispersions of the first and second branches in
the dressed spectrum are calculated as

$\displaystyle\partial^{2}_{\mu}\widetilde{\omega}_{\pm\mu\zeta}^{(1)}=\partial^{2}_{\mu}\beta_{\pm\mu}^{(1)}=\tfrac{1}{2}(D_{2\text{f}}+D_{2\text{s}})+\tfrac{1}{2}\Omega^{\prime\prime}_{\pm\mu},$
(68a)
$\displaystyle\partial^{2}_{\mu}\widetilde{\omega}_{\pm\mu\zeta}^{(2)}=\partial^{2}_{\mu}\beta_{\pm\mu}^{(2)}=\tfrac{1}{2}(D_{2\text{f}}+D_{2\text{s}})-\tfrac{1}{2}\Omega^{\prime\prime}_{\pm\mu},$
(68b)

where,
$\Omega^{\prime\prime}_{\mu}=\Omega_{\mu+1}+\Omega_{\mu-1}-2\Omega_{\mu}$.
$\Omega^{\prime\prime}_{\mu}$ depends on $|\Omega|$ (i.e., on the sum-
frequency nonlinearity), $D_{2\zeta}$, and on the repetition rate difference,
$D_{1\text{f}}-D_{1\text{s}}$, which means that the dressed state dispersion
is determined by the interplay of all three factors. This is unlike the bare-
state dispersion, trivially given by
$\partial^{2}_{\mu}\omega_{\pm\mu\zeta}=D_{2\zeta}$.

For $\varepsilon_{0}>0$, none of the $\widetilde{\omega}_{\mu\zeta}^{(1)}$
frequencies is sum-frequency matched, while
$\widetilde{\omega}_{-\mu\zeta}^{(1)}$ are quasi-matched for $\mu$ around
$\mu_{*}$, see Section V and Fig. 5. Therefore, the dressed states
corresponding to $\widetilde{\omega}_{\mu\zeta}^{(1)}$ are the quasi-bare
states, $|{b_{\mu}^{(1)}}\rangle\approx|{1}\rangle$, with the dispersion
$\approx D_{2\text{f}}$, see the red line in Fig. 13(a). The dressed states
corresponding to $\widetilde{\omega}_{-\mu\zeta}^{(1)}$ are
$|{b_{-\mu}^{(1)}}\rangle\approx|{3}\rangle$ for $\mu<\mu_{*}$, then, in the
proximity of $\mu_{*}$, they hybridise to the maximally dressed state,
$|{3}\rangle e^{i\phi}-|{4}\rangle$, and then transform to the quasi-bare
$|{4}\rangle$ states.

It is instructive to evaluate the dressed dispersion for $\mu$ around
$\mu_{*}$, where $\varDelta_{-\mu}\approx 0$, $\Omega_{-\mu}\approx|\Omega|$,
see Eqs. (17), (18),

$\begin{split}\Omega^{\prime\prime}_{-\mu}&\approx\frac{\varDelta_{-\mu}^{3}\varDelta_{-\mu}^{\prime\prime}+|\Omega|^{2}\big{(}\varDelta^{\prime}_{-\mu}\varDelta_{-\mu}\big{)}^{\prime}}{\Omega_{-\mu}^{3}}\\\
&\approx\frac{(\varDelta^{\prime}_{-\mu})^{2}}{|\Omega|}\approx\frac{(D_{1\text{f}}-D_{1\text{s}})^{2}}{|\Omega|}\sim
1\text{~{}to~{}}10\text{GHz}.\end{split}$ (69)

The above strongly dominates over $D_{2\zeta}$, so that, the dispersion of
$\widetilde{\omega}_{-\mu\zeta}^{(1)}$ becomes large anomalous. Thus, dressing
the $\widetilde{\omega}_{-\mu\zeta}^{(1)}$ branch creates two zero dispersion
points around $\mu_{*}$ and puts a group of modes into the range of anomalous
dispersion, see Fig. 13(a).

The dressing induced dispersion changes are very large, see Eq. (69), so that
for the nonlinearity to compensate the dispersive pulse spreading it would be
preferential to engage the modes with the bare resonator dispersion around
$\mu=0$, and away from $\mu_{*}$. Thus keeping $\mu_{*}\gg 1$ and increasing
it by tuning the index to make $|\varepsilon_{0}|$ larger, see Eq. (31), is
expected to increase the bandwidth of modes suitable for the bright soliton
modelocking regime.

In other words, the repetition rate difference can be large, but the sum-
frequency matching to the appropriately large sideband number puts the
undesirably large dispersion this difference creates away from the spectral
core of the soliton.

###  XII.2 Optical Pockels and cascaded-Kerr nonlinearities

Since we are expecting to work with the normal dispersion, the bright solitons
would require the negative, i.e., defocusing, nonlinearity to compensate for
it. Derivatives of $\widetilde{\omega}_{\pm\mu\zeta}^{(j)}$ in $|\Omega|$
describe the rate of the nonlinear shifts of the frequencies in the dressed
spectrum with the intra-resonator power,

$\begin{split}\frac{\partial\widetilde{\omega}_{\pm\mu\zeta}^{(1)}}{\partial|\Omega|}&=\frac{1}{2}\frac{\partial\Omega_{\pm\mu}}{\partial|\Omega|}=\frac{|\Omega|}{2\Omega_{\pm\mu}}>0,\\\
\frac{\partial\widetilde{\omega}_{\pm\mu\zeta}^{(2)}}{\partial|\Omega|}&=-\frac{|\Omega|}{2\Omega_{\pm\mu}}<0.\end{split}$
(70)

The signs of the above expressions determine the signs of the effective
nonlinearities experienced by the sidebands. Since the resonator frequencies
are inversely proportional to the refractive index, see Eq. (10), we conclude
that the frequencies $\widetilde{\omega}_{\pm\mu\zeta}^{(2)}$ in the dressed
spectrum experience the net positive (self-focusing) nonlinearity, and
$\widetilde{\omega}_{\pm\mu\zeta}^{(1)}$ \- the net negative (defocusing)
nonlinearity.

Though the branch nonlinearities do not change signs, their dependence on
$|\Omega|$ varies profoundly. Indeed, $\Omega_{\pm\mu}$ admits two
qualitatively different Taylor series expansions, see Table 2,

$\displaystyle\Omega_{\pm\mu}=|\Omega|+\frac{\varDelta_{\pm\mu}^{2}}{2|\Omega|}+\dots,~{}\frac{\varDelta_{\pm\mu}^{2}}{|\Omega|^{2}}\ll
1,$ (71a)
$\displaystyle\Omega_{\pm\mu}=|\varDelta_{\pm\mu}|+\frac{|\Omega|^{2}}{2|\varDelta_{\pm\mu}|}+\dots,~{}\frac{\varDelta_{\pm\mu}^{2}}{|\Omega|^{2}}\gg
1.$ (71b)

Thus, if $\mu$ is near to $\mu_{*}$, then the nonlinear shift of
$\widetilde{\omega}_{\pm\mu\zeta}^{(j)}$ frequency is proportional to the
amplitude of the cw-state, $|\Omega|\sim|\psi_{\text{f}}|$, which corresponds
to the optical Pockels effect, see Eq. (71a). While, for $\mu$ away from
$\mu_{*}$, the nonlinear shifts are proportional to the power,
$|\Omega|^{2}\sim|\psi_{\text{f}}|^{2}$, and hence are the Kerr-like, i.e.,
correspond to the cascaded-Kerr effect, see Eq. (71b).

###  XII.3 CW-state stability and instability

From the above, one should conclude that, for $\varepsilon_{0}>0$, the first
branch of the dressed states should be considered as the one able to form the
bright solitons. Since it provides a combination of the defocusing
nonlinearity and the relatively small normal dispersion, except several
$\mu$’s around $\mu_{*}$, where dispersion is anomalous.

The same condition, as the one just stated, leads to the intra-branch
instability of the cw-state, see and compare Figs. 9(a) for
$\varepsilon_{0}/2\pi=5$GHz, and 14(a) for $\varepsilon_{0}/2\pi=25$GHz.
Recalling the results of Section VII, see Eq. (56), the number, $N$, of the
sidebands that can be simultaneously unstable under these conditions is
clamped by $\mu_{*}$. Therefore, and also in line with the previous
subsection, $\varepsilon_{0}/2\pi=25$GHz brings more of the unstable
sidebands. Fig. 14 shows that there exists the optimal laser power, ${\cal
W}$, achieving the maximal $N$.

$N$ going up and then down with $\delta$ tuned more negative and powers
increasing, see Fig. 14(b), is due to the nonlinear shifts becoming saturated
by the higher order terms in the expansions of the root function in
$\Omega_{\pm\mu}$. The weakened nonlinear shifts gradually bring the
parametric gain below the threshold first for some and finally for all the
sidebands. The $N$ vs $\delta$ dependencies in Fig. 14(b) would be very
different in the Kerr resonators, where the number of the unstable sidebands
tends to infinity with the simultaneous increases of $\delta$ and ${\cal W}$
[50].

It is now important to note that, for $\delta<0$, the above discussed
instabilities happen to the upper state of the cw bistability loop, while the
low branch is either exclusively or largely stable, see Fig. 14(a). The stable
low-amplitude cw-state makes the background for the bright solitons reported
in the next section.

##  XIII Bright soliton frequency combs

The results of the previous Section let us to conclude that, for the normal
dispersion of the bare resonator modes, $D_{2\zeta}<0$, the bright solitons
are expected providing one arranges the index/frequency matching parameter
between the $M$ and $2M$ resonator modes to be positive,

$\varepsilon_{0}\sim(n_{2M}-n_{M})>0,$ (72)

and sufficiently large, so that the sum-frequency matching or near-matching,

$\omega_{0\text{f}}+\omega_{-\mu\text{f}}-\omega_{-\mu\text{s}}=0,$ (73)

happens for

$\mu=\mu_{*}\gg 1.$ (74)

The detuning should than be tuned to $\delta<0$, i.e., ’blue’ detuning.
$\mu_{*}$ is well approximated by the ratio between $|\varepsilon_{0}|$, and
the repetition rate difference, $|D_{1\text{f}}-D_{1\text{s}}|$, see Eq. (31).

We now do as prescribed and select the laser power, ${\cal W}$, and detuning,
$\delta$, corresponding to a large number of the simultaneously unstable
sidebands, see Fig. 14. Initialising Eq. (75) with the cw-state we immediately
observe its instability and the subsequent formation of the stable two-colour
soliton, see Fig. 16.

For $D_{2\zeta}>0$ (anomalous dispersion), the bright solitons would require
$\varepsilon_{0}<0$, adjusting
$\omega_{0\text{f}}+\omega_{\mu\text{f}}-\omega_{\mu\text{s}}=0$ for
$\mu=\mu_{*}\gg 1$, and, then, tuning to $\delta>0$, i.e., ’red’ detuning. The
mixed dispersion case, $D_{2\text{f}}D_{2\text{s}}<0$, requires a separate
analysis.

To trace the soliton families in the parameter space we solve the comb
equations in Section X. Since the teeth of the single soliton combs follow
with step one, i.e., their spatial period is $2\pi$, we set $\nu=1$ in Eqs.
(64), (66). Families of the bright solitons traced in $\delta$ for the laser
power ${\cal W}=333\mu$W and for a range of frequency mismatch parameters,
$\varepsilon_{0}/2\pi\in[21,29]$GHz are shown in Fig. 15. The unstable soliton
branch (dashed red and green lines) split from the middle branch of cw state
(dashed magenta lines) at the point of its $\mu=1$ instability. The stable
soliton branches extend outside the cw-bistability towards more negative
$\delta$’s.

The soliton profiles along the resonator circumference shown in Figs. 15, 17
are characterised by the tails oscillating with the period $2\pi/\mu_{*}$. The
corresponding spectra have pronounced peaks at $\mu=-\mu_{*}$, where the
powers of the fundamental and 2nd harmonic sidebands are balanced due to the
sum-frequency matching. The background of the soliton in the fundamental field
is set primarily by the $\mu=0$ sideband, see spectra in Fig. 15(d), where
$\mu=0$ is stronger than $\mu=-\mu_{*}$. Therefore, the fundamental background
drops with $\delta$ becoming more negative, cf., the correlated changes of the
full CW line in Fig. 15(a) and of the soliton background in Fig. 15(c). The
background of the soliton in the 2nd harmonic is, however, set primarily by
the $\mu=-\mu_{*}$ sideband, see spectra in Fig. 15(f), where $\mu=-\mu_{*}$
is the strongest and its power correlates with the soliton peak power. This
explains why the 2nd harmonic soliton background goes up with $\delta$
becoming more negative, cf., the dropping full CW line in Fig. 15(b) and the
increasing soliton background in Fig. 15(e).

The oscillatory soliton tail should be interpreted as due to the inability of
the nonlinear effects to compensate for the sharp rise and the sign change of
the dressed state dispersion around $\mu_{*}$, see Fig. 13. Therefore, the
nature of the tails here is similar to the soliton Cherenkov radiation in the
resonators [41, 63, 64], fibers [65], and bulk crystals [66]. We note, that,
in the present case, the dispersion can be significantly altered by the pump
power dependent state dressing, which provides a more flexible tool to control
the radiation frequency.

Figure 17 shows how the soliton families change with the tuning of
$\varepsilon_{0}$. Here the structure of the families appears to be more
complex. The period of the oscillations of the soliton tail, $2\pi/\mu_{*}$,
can change only discretely while $|\varepsilon_{0}|$ is tuned continuously, so
that the solution is forced to accommodate itself, as much as it can, to the
rigid period of its tails, which leads to the discrete set of the soliton
families. Each family is centred around the value of
$|\varepsilon_{0}|=\mu_{*}|D_{1\text{f}}-D_{1\text{s}}|$, we recall that
$|D_{1\text{f}}-D_{1\text{s}}|/2\pi=1$GHz. The tail oscillations of the Kerr
solitons in microresonators with the large higher-order dispersions also lead
to somewhat similar ’quantized’ behaviour of the soliton parameters [46].

##  XIV Discussion

There are numerous open problems left for a researcher tempted to understand
the multimode dynamics of the high-Q $\chi^{(2)}$ microresonators by looking
into properties of the individual modes, in line with the present-day
experimental capabilities. The extension of our results to the case of the
exact index matching $\varepsilon_{0}=0$ while keeping
$D_{1\text{f}}-D_{1\text{s}}$ large requires separate consideration. Extending
the bandwidth of the soliton combs by taking the shorter resonators with the
higher repetition rates and lower quality factors, and, perhaps, larger
$\mu_{*}$ needs to be investigated.

The power induced dispersion engineering of the dressed states offers a new
method to control the comb widths and shapes. Observations of the solitons,
the Rabi splitting and the sum-frequency matching associated spectral features
predicted here are of course important and can be attempted with the available
resonators and the index matching control tools. We note, the relatively high
conversion efficiency from the pump to the soliton-comb spectra seen in Fig.
17, which could be an important practical aspect requiring further
exploration.

We should recall here the prior theoretical work on the spatial [58, 56] and
temporal [39, 31] resonator solitons achieved via the 2nd harmonic generation
arrangements requiring the exactly or near matched phase, $\varepsilon_{0}=0$,
and group, $D_{1\text{f}}=D_{1\text{s}}$, velocities. While a combination of
these assumptions appears as the desirable idealisation in the contexts of the
currently available ring microresonators, future studies along these lines are
warranted. The relevance of the strong-coupling and dressed states for the
microresonator half-harmonic generation arrangement remains to be analysed,
including their links to the results on the half-harmonic bright-bright, dark
and dark-bright soliton pulses [16, 32, 28, 59, 67, 69, 70, 68].

##  XV Summary

(1) The theoretical framework, i.e., dressed-resonator method, to study
frequency conversion and solitons is formulated by including the sum-frequency
nonlinearity into the definition of the resonator spectrum.

(2) The Rabi splitting of the dressed frequencies leads to the four distinct
PDC conditions, see Eq. (52), which are used to explain the existence and
generation of the sparse non-soliton, i.e., Turing-pattern-like, frequency
combs.

(3) The effective nonlinearity and dispersion of the dressed states have been
used to demonstrate that the microresonator with the normal dispersion and
naturally large difference of the repetition rates at the fundamental and 2nd
harmonic frequencies, $D_{1\text{f}}-D_{1\text{s}}$, supports a family of the
bright soliton frequency combs, see Figs. 15, 17. Conditions for this are
provided by tuning the index/frequency matching parameter,
$\varepsilon_{0}=2\omega_{0\text{f}}-\omega_{0\text{s}}$, to be positive and
large, so that it exceeds the repetition rate difference by a significant
factor, $\mu_{*}=|\varepsilon_{0}|/|D_{1\text{f}}-D_{1\text{s}}|\gg 1$.
$\mu_{*}$ or $-\mu_{*}$ approximate the mode number associated with the phase-
matched sum-frequency process and set limits on the soliton bandwidth.

##  XVI Acknowledgement

This work was supported by the EU Horizon 2020 Framework Programme (812818,
MICROCOMB).

## Appendices

##  Appendix A Envelope equations

The intra-resonator electric fields of the fundamental and 2nd harmonic are
expressed as per Eq. (2). Envelopes of the fundamental, $\psi_{\text{f}}$, and
second, $\psi_{\text{s}}$, harmonic satisfy the following system of the
partial-differential equations

$\displaystyle i\partial_{t}\psi_{\text{f}}$
$\displaystyle=\delta\psi_{\text{f}}-iD_{1\text{f}}\partial_{\vartheta}\psi_{\text{f}}-\tfrac{1}{2}D_{2\text{f}}\partial^{2}_{\vartheta}\psi_{\text{f}}$
(75a) $\displaystyle-\gamma_{\text{f}}\psi_{\text{s}}\psi_{\text{f}}^{*}-{\cal
N}_{\text{f}}-i\tfrac{1}{2}\kappa_{\text{f}}\big{(}\psi_{\text{f}}-{\cal
H}\big{)},$ $\displaystyle i\partial_{t}\psi_{\text{s}}$
$\displaystyle=(2\delta-\varepsilon_{0})\psi_{s}-iD_{1\text{s}}\partial_{\vartheta}\psi_{\text{s}}-\tfrac{1}{2}D_{2\text{s}}\partial^{2}_{\vartheta}\psi_{\text{s}}$
(75b) $\displaystyle-\gamma_{\text{s}}\psi_{\text{f}}^{2}-{\cal
N}_{\text{s}}-i\tfrac{1}{2}\kappa_{\text{s}}\psi_{\text{s}}.$

Conditions $\psi_{\zeta}(t,\vartheta)=\psi_{\zeta}(t,\vartheta+2\pi)$ make
this system equivalent to a set of the coupled-mode equations for
$\psi_{\mu\zeta}(t)$. The first principle derivation of Eq. (75) is given in
Ref. [30].

All parameters are explained in Section III of the main text, apart from
${\cal H}^{2}$, which characterizes the pump power [30]. If ${\cal
F}=D_{1\text{f}}/\kappa_{\text{f}}=20000$ is the finesse, then ${\cal H}^{2}$
is expressed via the incoming laser power ${\cal W}$ as

${\cal H}^{2}=\frac{\eta}{\pi}{\cal F}{\cal W}.$ (76)

$\eta<1$ is the coupling efficiency. ${\cal N}_{\zeta}$ are the intrinsic
Kerr, i.e. $\chi^{(3)}$, nonlinearity terms,

${\cal
N}_{\text{f},\text{s}}=\gamma_{3\text{f},3\text{s}}\big{(}|\psi_{\text{f},\text{s}}|^{2}+2|\psi_{\text{s},\text{f}}|^{2}\big{)}\psi_{\text{f},\text{s}}.$
(77)

Nonlinear coefficients $\gamma_{\zeta}/2\pi$ and $\gamma_{3\zeta}/2\pi$ have
units of Hz$/\sqrt{\smash[b]{\text{W}}}$ and $\text{Hz}/{\text{W}}$,
respectively [30]. Units and numerical values of other parameters can be found
in Table 1.

##  Appendix B CW-state: $\chi^{(2)}$ vs $\chi^{(3)}$

CW state, i.e., the $\mu=0$ mode in the fundamental and its 2nd harmonic, is a
solution of Eqs. (75) with $\psi_{\zeta}=\psi_{0\zeta}$,
$\partial_{\vartheta}\psi_{0\zeta}=\partial_{t}\psi_{0\zeta}=0$. Let us now
evaluate the relative impact of the $\chi^{(2)}$ and $\chi^{(3)}$ effects on
the cw state. If $\varepsilon_{0}$ dominates over the linewidth and detuning
parameters, then
$\psi_{0\text{s}}\approx-\gamma_{\text{s}}\psi_{0\text{f}}^{2}/\varepsilon_{0}$,
and the net nonlinear frequency shift of the fundamental resonance is

$\left[-\frac{\gamma_{\text{s}}\gamma_{\text{f}}}{\varepsilon_{0}}+\gamma_{3\text{f}}\right]|\psi_{0\text{f}}|^{2}=\gamma_{3\text{f}}\left[-\frac{\varepsilon_{\text{cr}}}{\varepsilon_{0}}+1\right]|\psi_{0\text{f}}|^{2}.$
(78)

Hence, only for $|\varepsilon_{0}|\gtrsim|\varepsilon_{\text{cr}}|$,
$\varepsilon_{\text{cr}}=\gamma_{\text{s}}\gamma_{\text{f}}/\gamma_{3\text{f}}$,
the 2nd harmonic becomes weak enough for the $\chi^{(3)}$ induced shift to
catch up with the $\chi^{(2)}$ one. For the parameters in Table 1 and
$\gamma_{3\zeta}/2\pi\lesssim 1\text{MHz}/{\text{W}}$ [50],
$\varepsilon_{\text{cr}}/2\pi$ is $\gtrsim 100$GHz. Thus, in the range of
$\varepsilon_{0}/2\pi\in[-30,30]$GHz explored in this project and for the
combs with the relatively low powers, the $\chi^{(3)}$ terms can be neglected.
However, most of the numerical data in this work have been calculated with and
without ${\cal N}_{\zeta}$. The differences that we have observed are not
worth mentioning in the context of our study. For all the above reasons, we
set ${\cal N}_{\zeta}=0$. Some results on the interplay of the $\chi^{(2)}$
and $\chi^{(3)}$ effects in the half-harmonic generation setup can be found
in, e.g., Refs. [32, 16].

##  Appendix C CW-state: $\chi^{(2)}$ only

The cw state is sought in the form

$\psi_{0\text{f}}=\frac{\Omega_{~{}}}{\sqrt{2\kappa_{\text{f}}\kappa_{\text{s}}}}{\cal
H}_{*},~{}\psi_{0\text{s}}=\frac{8\gamma_{2\text{s}}\psi_{0\text{f}}^{2}}{\Omega_{\text{s}}}=\frac{\Omega^{2}}{\gamma_{2\text{f}}\Omega_{\text{s}}},$
(79)

where $\Omega$ is the Rabi frequency. $\Omega_{\text{s}}$ is defined in Eq.
(20), and

${\cal
H}^{2}_{*}=\frac{\kappa_{\text{f}}\kappa_{\text{s}}}{4\gamma_{2\text{f}}\gamma_{2\text{s}}}\approx
30\mu\text{W}$ (80)

is the characteristic intra-resonator power. Thus, the explicit relation
between the Rabi frequency and the power in the fundamental field is

$|\psi_{0\text{f}}|^{2}=\frac{|\Omega|^{2}}{8\gamma_{2\text{f}}\gamma_{2\text{s}}}.$
(81)

Using Eqs. (75), one can show that

$\Omega=\sqrt{\frac{\kappa_{\text{f}}\kappa_{\text{s}}}{2}}\sqrt{\frac{{\cal
W}~{}}{{\cal
W}_{*}}}\frac{\kappa_{\text{f}}}{\Omega_{\text{f}}}\left[1-\frac{|\Omega|^{2}}{\Omega_{\text{f}}\Omega_{\text{s}}}\right]^{-1}.$
(82)

Here, $\Omega_{\text{f},\text{s}}$ are defined after Eq. (58), and

${\cal W}_{*}=\frac{\pi{\cal H}_{*}^{2}}{\eta{\cal F}}\approx 10\text{nW}.$
(83)

Taking the modulus squared of Eq. (82) we find the real cubic equation for
$|\Omega|^{2}$,

$\displaystyle\frac{2|\Omega_{\text{f}}|^{2}|\Omega|^{2}}{\kappa_{\text{f}}^{3}\kappa_{\text{s}}}\times$
$\displaystyle\left[1-|\Omega|^{2}\text{Re}\left\\{\frac{2}{\Omega_{\text{f}}\Omega_{\text{s}}}\right\\}+\frac{|\Omega|^{4}}{|\Omega_{\text{f}}|^{2}|\Omega_{\text{s}}|^{2}}\right]$
$\displaystyle=$ $\displaystyle\frac{{\cal W}}{{\cal W}_{*}}=\frac{{\cal
H}^{2}}{{\cal H}_{*}^{2}}.$ (84)

A useful insight into the cw properties is provided by taking the limit when
$|\varepsilon_{0}|$ is large relative to $|\delta|$, and $|\delta|$ is large
relative to $\kappa_{\zeta}$, so that
$\Omega_{\text{s}}\approx-8\varepsilon_{0}$ and
$\Omega_{\text{f}}\approx\delta$. Then, bistability of the cw-state requires
$\delta\varepsilon_{0}<0$ (the square bracket in Eq. (82) can be zero). In
this regime,

$\max_{\delta}|\Omega|^{2}\simeq-\delta\varepsilon_{0},$ (85)

implying that the resonance shifts proportionally to the pump power, i.e., in
the same way as it happens in Kerr effect. It means, that the cw-state behaves
as it would in the Kerr resonator. For $\varepsilon_{0}=0$ or small,
$\Omega_{\text{f}}\approx\delta$, $\Omega_{\text{s}}\approx 16\delta$. Then,
the cw becomes simultaneously bistable for $\delta>0$ and $\delta<0$ [39, 31],
and $\max_{\delta}|\Omega|^{2}\simeq\delta^{2}$, i.e.,

$\max_{\delta}|\Omega|\simeq|\delta|.$ (86)

This is the Pockels regime of the cw-state, when the nonlinear change of the
refractive index is proportional to the first power of the field amplitude,
which we do not consider in this work. Conditions for the $\mu\neq 0$
sidebands to be in either cascaded-Kerr or Pockels regimes are different and
discussed in Section XII B.

##  Appendix D Linearization around the cw-state

In order to develop a theory of the growth of the multi-sideband signal, i.e.,
frequency comb, on top of the cw solution, $\psi_{0\zeta}$, we extend Eqs.
(75) by a pair of the complex-conjugated equations [71], and seek a solution
in the form

$\begin{split}&\begin{bmatrix}\psi_{\text{f}}\\\ \psi_{\text{s}}\\\
\psi^{*}_{\text{f}}\\\
\psi^{*}_{\text{s}}\end{bmatrix}=\begin{bmatrix}\psi_{0\text{f}}\\\
\psi_{0\text{s}}\\\ \psi^{*}_{0\text{f}}\\\
\psi^{*}_{0\text{s}}\end{bmatrix}+\\\ &\sum_{\mu\geq
0}\left\\{\begin{bmatrix}\widetilde{\psi}_{\mu\text{f}}\\\
\widetilde{\psi}_{\mu\text{s}}\\\ \widetilde{\psi}_{-\mu\text{f}}\\\
\widetilde{\psi}_{-\mu\text{s}}\end{bmatrix}e^{i\mu\theta}+\begin{bmatrix}0&0&1&0\\\
0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{bmatrix}\begin{bmatrix}\widetilde{\psi}_{\mu\text{f}}^{*}\\\
\widetilde{\psi}_{\mu\text{s}}^{*}\\\ \widetilde{\psi}_{-\mu\text{f}}^{*}\\\
\widetilde{\psi}_{-\mu\text{s}}^{*}\end{bmatrix}e^{-i\mu\theta}\right\\}.\end{split}$
(87)

If $\vartheta$ is the angle measured along the resonator circumference, then
$\theta=\vartheta-D_{1}t$ is the coordinate in the rotating frame, see Eq.
(7). $\widetilde{\psi}_{\pm\mu\zeta}(t)$ are the sideband amplitudes. Summing
up the first and second lines gives the net signals in the fundamental and 2nd
harmonic, respectively, with the third and fourth lines being their
conjugated, see Eq. (12). Substituting Eq. (87) into the extended Eq. (75), we
assume smallness of the sideband amplitudes, linearise, and then derive
equations for the individual sidebands using the angular momentum matching.
For a given $\mu$ this leads to a coupled system of the four ordinary
differential equations for $\widetilde{\psi}_{\mu\text{f}}$,
$\widetilde{\psi}_{-\mu\text{f}}$, $\widetilde{\psi}_{\mu\text{s}}$, and
$\widetilde{\psi}_{-\mu\text{s}}$, see Eq. (13).

##  Appendix E Approximate PDC conditions

To present the PDC condition in Eq. (54) in a more transparent form, we first
make explicit how $\delta$, $\varepsilon_{0}$, and $\mu_{*}$ are implicated
there,

$\displaystyle|\Omega_{\text{pdc}}|^{2}$
$\displaystyle=\frac{4}{[(\delta-\delta_{\mu\text{f}})+(2\delta-\delta_{\mu\text{s}}-\varepsilon_{0})]^{2}}$
$\displaystyle\times\big{(}\delta-\delta_{\mu\text{f}}\big{)}$
$\displaystyle\times\big{(}2\delta-\delta_{\mu\text{s}}-\varepsilon_{0}\big{)}$
$\displaystyle\times\big{(}3\delta-\delta_{\mu\text{s}}-\delta_{\mu\text{f}}-\varepsilon_{0}(1+\dfrac{\mu}{\mu_{*}})\big{)}$
$\displaystyle\times\big{(}3\delta-\delta_{\mu\text{s}}-\delta_{\mu\text{f}}-\varepsilon_{0}(1-\dfrac{\mu}{\mu_{*}})\big{)},$
(88)

where we have defined

$\delta_{\mu\text{f}}=-\tfrac{1}{2}D_{2\text{f}}\mu^{2},~{}\delta_{\mu\text{s}}=-\tfrac{1}{2}D_{2\text{s}}\mu^{2}.$
(89)

Eq. (22) implies that $|\varepsilon_{0}|$ dominates over all $\delta$’s. The
first case to consider is when $\mu_{*}$ falls between the two nearest
integers, i.e., the sum-frequency matching point, $\mu=\mu_{*}$, has been
missed. Then Eq. (88) simplifies to Eq. (55),

$|\Omega_{\text{pdc}}|^{2}\approx-4\varepsilon_{0}\left(\delta-\delta_{\mu\text{f}}\right)\left[1-\frac{\mu^{2}}{\mu_{*}^{2}}\right].$
(90)

If, the sum-frequency process is either nearly or exactly matched at

$\mu=\widehat{\mu}\approx\mu_{*},~{}\text{where}~{}\widehat{\mu}\in\mathbb{Z},\mu_{*}\in\mathbb{R},$
(91)

then

$|\Omega_{\text{pdc}}^{(\widehat{\mu})}|^{2}\approx
8(\delta-\delta_{\widehat{\mu}\text{f}})\big{(}3\delta-\delta_{\widehat{\mu}\text{s}}-\delta_{\widehat{\mu}\text{f}}\big{)}.$
(92)

Transition from Eq. (90) to Eq. (92) implies transition from the square-root
(Kerr-like) dependence of $|\Omega_{\text{pdc}}|$ vs $\delta$ to the quasi-
linear (Pockels-like) one.

##  Appendix F Laser power at the PDC thresholds

Detunings at the tips of the instability tongues are worked out by imposing
conditions

$\displaystyle|\Omega_{\text{pdc}}|^{2}=|\Omega_{\text{th}}^{(\mu)}|^{2},$
(93a)
$\displaystyle|\Omega_{\text{pdc}}^{(\widehat{\mu})}|^{2}=|\Omega_{\text{th}}^{(\widehat{\mu})}|^{2}.$
(93b)

Eqs. (93a), (55), (61) yield

$\delta_{\text{th}}^{(\mu)}=\delta_{\mu\text{f}}-\frac{\kappa_{\text{f}}~{}\text{sgn}(\varepsilon_{0})}{1-\dfrac{\mu^{2}}{\mu_{*}^{2}}},~{}\mu\neq\widehat{\mu}$
(94)

For $\mu=\widehat{\mu}$, the procedure is the same. In order not to
overcomplicate the answer, we impose a plausible condition
$(D_{2\text{f}}+D_{2\text{s}})=3D_{2\text{f}}$, leading to
$\delta_{\widehat{\mu}\text{s}}+\delta_{\widehat{\mu}\text{f}}=3\delta_{\widehat{\mu}\text{f}}$.
Then, Eqs. (93b), (92), (62) yield

$\delta_{\text{th}}^{(\widehat{\mu})}=\delta_{\widehat{\mu}\text{f}}\pm\sqrt{\frac{|\varepsilon_{0}|}{6}\sqrt{\kappa_{\text{f}}(\kappa_{\text{f}}+\kappa_{\text{s}})}}.$
(95)

Transparent analytic estimates for the laser powers ${\cal W}$ at the tips of
the instability tongues can be worked out after observing that along the tails
of the nonlinear resonances, see Fig. 2, the square bracket in Eq. (82) is
$\approx 1$,

$\frac{{\cal W}}{{\cal
W}_{*}}\approx\frac{2\delta^{2}|\Omega|^{2}}{\kappa_{\text{f}}^{3}\kappa_{\text{s}}},$
(96)

see Eq. (83) for ${\cal W}_{*}$.

The balance of terms in Eq. (95) is such that the root term dominates and
$\delta_{\widehat{\mu}\text{f}}$ can be neglected, which gives the following
estimate for the power

$\frac{{\cal W}_{\text{th}}^{(\widehat{\mu})}}{{\cal
W}_{*}}\approx\frac{4|\varepsilon_{0}|^{2}(\kappa_{\text{f}}+\kappa_{\text{s}})}{3\kappa_{\text{f}}^{2}\kappa_{\text{s}}},~{}\mu=\widehat{\mu}.$
(97)

In Eq. (94), the two terms are balanced for the moderate $\mu$’s leading to a
longer equation not included here, but for $\mu^{2}\gg\mu_{*}^{2}$, the second
term can be neglected, so that

$\frac{{\cal W}_{\text{th}}^{(\mu)}}{{\cal W}_{*}}\approx
2\mu^{4}\frac{D_{2\text{f}}^{2}|\varepsilon_{0}|}{\kappa_{\text{f}}^{2}\kappa_{\text{s}}},~{}\mu\gg\mu_{*}.$
(98)

Thus, the powers to generate the sum-frequency matched sideband,
$\mu=\widehat{\mu}$, scale with $|\varepsilon_{0}|^{2}$, and the ones for
$\mu\neq\widehat{\mu}$ with $|\varepsilon_{0}|$, cf., Eq. (97) and Eq. (98).

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