# Coherent terahertz-to-microwave link using electro-optic-modulated Turing
rolls

Wenle Weng<EMAIL_ADDRESS>Institute of Physics, Swiss Federal Institute
of Technology (EPFL), CH-1015 Lausanne, Switzerland  Miles H. Anderson
Institute of Physics, Swiss Federal Institute of Technology (EPFL), CH-1015
Lausanne, Switzerland  Anat Siddharth  Institute of Physics, Swiss Federal
Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland  Jijun He
Institute of Physics, Swiss Federal Institute of Technology (EPFL), CH-1015
Lausanne, Switzerland  Arslan S. Raja  Institute of Physics, Swiss Federal
Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland  Tobias J.
Kippenberg<EMAIL_ADDRESS>Institute of Physics, Swiss Federal
Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland

###### Abstract

Arising from modulation instability, Turing rolls in optical Kerr
microresonators have been used in the generation of optical frequency combs
and the synthesis of microwave and terahertz frequencies. In this work, by
applying electro-optic modulation on terahertz-frequency Turing rolls, we
implement electro-optic frequency division with a microcomb to synthesize
variable low-noise microwave signals. We also actively stabilize the terahertz
oscillations to a microwave reference via intracavity power modulation,
obtaining fractional frequency instabilities that are better than those of the
free-running situation by up to six orders of magnitude. This study not only
highlights the extraordinary spectral purity of Turing roll oscillations but
also opens the way for bidirectional terahertz-to-microwave links with hybrid
optical frequency comb techniques.

##  I Introduction

As “one of the most ubiquitous types of instabilities in nature” [1],
modulation instability (MI) has been found in a variety of physical systems
ranging from hydrodynamics [2], plasma [3], optics [4, 5], to ultracold gases
[6, 7, 8]. In optical Kerr microresonators pumped by continuous-wave (cw)
lasers, MI gives rise to cavity-enhanced optical frequency generation through
four-wave mixing [9], leading to the discovery of Kerr microcombs [10, 11].
Although in recent years the research focus of the microcomb community has
shifted to the state of dissipative Kerr solitons (DKS) [12, 13] for its
exceptional dynamical stability and the intrinsic broad spectral span, the
state of MI-induced Turing rolls (TRs) is easier to access and can be
advantageous in certain applications such as efficient terahertz (THz)
excitation [14], coupled laser-microresonator comb generation [15], perfect
soliton crystal inception [16], and noise-robust fiber communications [17].
Also known as cnoidal waves, TRs are periodic pulse-like solutions to the
Lugiato-Lefever equation (LLE) – the master equation that describes the cw-
driven Kerr cavity system [18, 19]. Aside from the many applications in optics
and photonics, TRs have been attracting interests for their fundamental
complex properties and particularly their close relation to other nonlinear
dynamical states [20, 21, 22].

Owing to the compactness and the simplicity of the setup, microwave generation
based on TRs in crystalline microresonators have been demonstrated [23, 24].
Yet, compared with the DKS counterparts [23, 25, 26], TR oscillations utilized
in prior studies exhibit substantially higher phase noise. Moreover, in a
microresonator with an ultrahigh quality factor ($Q$), the frequency of the TR
oscillation is usually a high multiple of the cavity free spectral range
(FSR). As a result, the oscillation frequency can fall into the THz range that
cannot be detected and measured with conventional electronics. Although the
photodetection and the optical-to-THz signal conversion of THz-repetition-rate
microcombs have been realized with advanced electronic-photonic solutions [14,
27], they haven’t reached the technical maturity that is required for wide-
spread applications. Consequently, TRs’ usage in low-noise signal synthesis
and metrology at radiofrequency (rf) and THz ranges is severely hindered.

In this work, we overcome the twofold disadvantage of TRs by adopting the
electro-optic frequency division (EOFD) technique [28] to synthesize low-noise
microwaves from a THz TR oscillation. The developed TR-EOFD scheme allows the
spectral quality of a TR oscillation to be faithfully transferred into
variable microwave frequencies with the benefit of a substantial reduction of
phase noise due to the frequency division mechanism. With the synthesized
10-GHz signal reaching phase noise levels of $-60$ and $-110$ dBc$/$Hz at
offset frequencies of 10 Hz and 1 kHz respectively, the spectral purity of the
microwaves is comparable to the best performances ever achieved with DKS [23,
26]. In addition, the system can be easily adapted to transfer the frequency
stability of an rf reference to the TRs, thus enabling a bidirectional
coherent THz-to-microwave link that could potentially find a variety of
applications in sensing, telecommunication and metrology.

Figure 1: Low-noise microwave synthesis with the TR-EOFD technique. (a) The
schematic layout of the TR-EOFD setup. (b) The phase noise spectra of computed
fundamental quantum noises and simulated technical noises of a terahertz TR
oscillation, a single-soliton DKS oscillation, and a microwave synthesized
with the TR-EOFD scheme, respectively. The synthesized microwave has a
frequency that is the same as the DKS repetition rate (12 GHz in the
simulations, details of which are presented in Appendix A and Appendix B). The
black arrows indicate the phase noise reduction due to the frequency division.

##  II Principle

Figure 1 (a) shows the principle of the TR-EOFD scheme. A cw-laser-pumped
microresonator produces a TR oscillation with a frequency in the THz range,
corresponding to two intensive first-order TR sidebands (and multiple higher-
order sidebands with much weaker amplitudes) in the optical spectrum. After
proper filtering, the two TR sidebands pass through cascaded electro-optic
phase modulators (EOMs) that are modulated with a microwave frequency $f_{\rm
mod}$ derived from a voltage-controlled oscillator (VCO). If the modulation is
strong enough, each TR sideband becomes an electro-optic frequency comb (EOC)
[29, 30, 31] whose spectral width can cover the frequency of the THz TR
oscillation ($f_{\rm TR}$). Two closely separated comb teeth, each of which is
generated from a different first-order TR sideband, can be filtered out at the
center of the two TR sidebands to produce a beat frequency $f_{\rm rf}$. When
the VCO is free running, the single-sideband phase noise of the beat frequency
signal $\mathcal{L}_{\rm rf}$ can be expressed as:

$\mathcal{L}_{\rm rf}=\mathcal{L}_{\rm TR}+n^{2}\mathcal{L}_{\rm VCO}$ (1)

where $\mathcal{L}_{\rm TR}$ is the phase noise of the THz TR oscillation. The
second term on the right-hand side represents the additional phase noise
caused by the multiplication of the free-running VCO noise in EOC [30, 31],
and the integer $n\approx\frac{f_{\rm TR}}{f_{\rm mod}}$ is the division
factor. By tuning $f_{\rm mod}$, $f_{\rm rf}$ can be tuned to closely match
the frequency of a reference rf oscillator (e. g., a 10-MHz reference signal).
With a phase-locked loop (PLL), the VCO can be controlled to lock $f_{\rm rf}$
to the rf reference, and then $f_{\rm mod}$ is related to $f_{\rm TR}$ as:

$f_{\rm mod}=\frac{f_{\rm TR}+(-)f_{\rm rf}}{n}$ (2)

where the sign in the second term depends on the relative positions of the two
teeth. One should note that in principle the two teeth can be set to have the
exactly same frequency in order to eliminate the need for the reference rf
oscillator. In practice, however, using a beat frequency in the rf domain for
the servo facilitates the locking because the error signal can be conveniently
amplified by a rf amplifier without being interfered by the relatively high
technical noises (e. g., electronic noise in the photodetector and laser
intensity fluctuations) in the low frequency range. Because normally the phase
noise of the rf reference signal is extremely low, within the PLL locking
bandwidth the phase noise of the controlled VCO is:

$\mathcal{L}^{\prime}_{\rm VCO}=\frac{\mathcal{L}_{\rm TR}}{n^{2}}$ (3)

showing that the same noise characteristics of the THz TR oscillation is
inherited by the controlled VCO, with an additional reduction by a factor of
$\frac{1}{n^{2}}$ in the phase noise magnitude [28].

Figure 2: Experimental realization of the TR-EOFD scheme. (a) An ECDL is
locked to a microresonator mode with PDH technique to generate a TR microcomb.
After amplification using an erbium-doped fiber amplifier (EDFA), the
microcomb is phase modulated by three EOMs. An optical bandpass filter (BPF)
with a passband centered at 1549.5 nm is used to filter out the overlapped
EOCs to generate a beat frequency around 10 MHz. The PLL servo phase-locks
$f_{\mathrm{mod}}$ to the TR oscillation frequency, thus generating low-noise
microwaves whose phase noises are measured with a phase noise analyzer (PNA).
(b) The optical spectra of the original TR microcomb and the combs after
electro-optical modulation at three different modulation frequencies
respectively. The two first-order TR sidebands are in deep red. (c) A photo of
the MgF2 microresonator. (d) The laser-sweeping spectra of the mode resonance
and the generated microcomb power, and the corresponding PDH error signal. The
PDH modulation frequency is 9 MHz. (e) Electrical spectra of the free-running
$f_{\mathrm{rf}}$ and the phase-locked $f_{\mathrm{mod}}$ measured with an
electrical spectrum analyzer (ESA).

Our in-depth analysis detailed in Appendices A, B and C shows that, as long as
the phase noise is not limited by thermal noise, the TR-EOFD scheme could
yield low-noise microwaves with spectral purities outperform those of DKS
oscillations. In Fig. 1 (b) we plot the calculated quantum noises resulting
from fundamental phase diffusion [32, 33, 34] and the simulated technical
noises for the TR oscillation (of frequency $f_{\rm TR}=16\nu_{\rm FSR}$,
where $\nu_{\rm FSR}=$ 12 GHz is the frequency of the microresonator FSR), the
DKS oscillation and the synthesized microwave of the frequency of FSR,
respectively. The resonator-related parameters used in the calculations and
the simulations are similar to those of the crystalline microresonator in our
experiments, and the technical noises include both the amplitude-modulation
(AM) -to-phase-modulation (PM) noise caused by laser relative intensity noise
(RIN) and the PM-to-PM noise induced by laser phase noise (see details of the
simulation process and the discussion on the noise transfer mechanism in
Appendix B). One should note that changing the absolute values of the laser
RIN and phase noise would not change the relative levels of the technical
noises as the noise transfer functions do not depend on the input noise
amplitude. This result shows that TR-EOFD scheme takes advantage of the
frequency division principle and generates microwaves with phase noises that
are lower than those of DKS oscillations in the same microresonator by up to
20 dB.

##  III Variable low-noise microwave synthesis

We carry out experiments using the setup depicted in Fig. 2 (a). With the
Pound-Drever-Hall (PDH) technique a 1550-nm external-cavity diode laser (ECDL)
is frequency-locked to a high-$Q$ ($\sim 1\times 10^{9}$) resonance in a
magnesium fluoride (MgF2) whispering-gallery-mode microresonator. The
microresonator is placed in a metal enclosure without active temperature
control, only to shield it from ambient air flow. Approximately 4 mW of
optical power is coupled into the microresonator via a tapered fiber,
generating two first-order TR sidebands with a frequency gap of 0.372 THz.
After filtering and amplification, the TR sidebands are modulated with three
EOMs. In this proof-of-concept experiment, we employ a commercial microwave
signal generator instead of a VCO to provide $f_{\mathrm{mod}}$. With a 10-MHz
reference signal (whose phase noise is below $-130$ dBc$/$Hz at all relevant
offset frequencies between 10 Hz and 10 MHz) as the local oscillator, a PLL
servo feeds back to the frequency-modulation (FM) port of the microwave signal
generator to execute the lock. Figure 2 (b) shows the TRs spectrum and the
comb spectra that are electro-optically modulated at three different
frequencies, corresponding to a division factor of 22, 36, and 70,
respectively. As shown in Fig. 2 (d), the TRs conversion efficiency is below
$10\%$. We note that the conversion efficiency is intentionally kept low so
the TR oscillation is close to the three-mode hyperparametric oscillation
studied in [32, 34], although it can be raised above $20\%$ without entering
the chaotic regime [9] by increasing the pump power. Figure 2 (e) shows the
spectra of $f_{\mathrm{rf}}$ when the PLL is not activated and the signal of
$f_{\mathrm{mod}}=10.3$ GHz when the PLL is engaged, respectively. The control
bandwidth of approximately 50 kHz is manifested as the servo bumps at the
corresponding offset frequency in the PLL-locked spectrum of
$f_{\mathrm{mod}}$.

The phase noise spectra of the synthesized microwaves are presented in Fig. 3,
along with the phase noise spectrum of the free-running $f_{\mathrm{rf}}$. As
an indirect measurement of the TR oscillation phase noise, the noise spectrum
of $f_{\mathrm{rf}}$ contains the intrinsic noise of the microwave signal
generator compounded with a multiplication effect. Comparing with the
$f_{\mathrm{rf}}$ phase noise at frequencies below 100 Hz where the noise
level is above the microwave multiplication noise floor, the synthesized
microwaves exhibit a phase noise reduction of approximately 27, 31, and 37 dB
respectively, showing excellent agreement with the expected noise reduction
effect. Out of the PLL control bandwidth, the phase noise amplitudes of the
synthesized microwaves are the same as the intrinsic noise levels of the
microwave source. Remarkably, the synthesized 10.3-GHz microwave phase noise
spectrum reaches $-60$, $-90$, and $-110$ dBc$/$Hz at offset frequencies of
10, 100 and 1000 Hz respectively. Besides demonstrating an improvement by
nearly 40 dB over phase noise levels previously achieved with crystalline
microresonator TRs [23, 24], the TR-EOFD scheme is unprecedented in the
flexibility of generating low-noise microwaves of different frequencies from a
single TR source.

Figure 3: Phase noise spectra of (solid lines) synthesized microwave signals
and (dotted lines) the corresponding uncontrolled $f_{\mathrm{mod}}$. As an
indirect measurement of the phase noise of the 0.372-THz TR oscillation, the
phase noise spectrum of $f_{\rm rf}$ when the PLL is deactivated is also
presented, showing that at frequencies above 100 Hz the measurement is limited
by the multiplication noise of the uncontrolled $f_{\rm mod}$. The noise peaks
around 1 kHz is caused by the phase noise of the pump laser.  Figure 4: Phase-
coherent microwave-to-THz link based on actively frequency-stabilized Turing
rolls. (a) The optical spectra of the TRs before (top) and after (bottom) the
electro-optic modulation. (b) Spectrum of $f_{\mathrm{rf2}}$ when the PLL is
activated. (c) Measured frequencies of $f_{\mathrm{rf2}}$ without (top) and
with (bottom) the PLL control. (d) The histogram of the counted frequencies
with the PLL activated. The standard deviation is indicated. (e) Computed
modified Allan deviations of the frequency traces presented in (c). The error
bars are too small to be visible.

##  IV Microwave-to-THz link

Next, we reverse the path of coherence transfer by stabilizing a THz TR
oscillation to a microwave signal. To demonstrate the accessibility of TRs
with different oscillation frequencies and the broad spectral span of the
EOCs, in a different transverse mode family (i. e., modes with dissimilar
polar and/or radial field maxima numbers) we generate TR sidebands that are
apart from the pump laser by 0.68 THz. Without filtering out the pump line,
three intensive sets of EOCs are generated by setting $f_{\mathrm{mod}}$ to
11.5 GHz (i. e., a division factor of 59). We use two optical bandpass filters
(BPF1 and BPF2) to select the overlapped edges of the EOCs to generate two rf
signals ($f_{\mathrm{rf1}}$ and $f_{\mathrm{rf2}}$) that are of the identical
frequencies of $\sim 29$ MHz. One of the signals ($f_{\mathrm{rf1}}$) is used
to generate the error signals for the PLL, while the other
($f_{\mathrm{rf2}}$) is for spectral measurement and frequency counting of the
THz oscillation (see Fig. 4 (a)). Instead of controlling $f_{\mathrm{mod}}$
with the PLL, in this experiment the PLL servo feeds back to the pump power
via an acousto-optic modulator (AOM). Mainly relying on the thermal expansion
and thermo-optic effects with intracavity power modulation, this approach has
a low control bandwidth of a few tens of hertz. Nevertheless, the PLL can be
tightly locked, as shown by the spectrum of $f_{\mathrm{rf2}}$ (see Fig. 4
(b)) whose linewidth is limited by the ESA’s minimum resolution bandwidth
(RBW) of 1 Hz. We use a frequency counter with a gate time of 0.1 s to count
the frequency of $f_{\mathrm{rf2}}$ for more than half an hour in free-running
and PLL-locked conditions respectively. The counting results are presented in
Fig. 4 (c) and (d), showing that the intracavity power control suppresses the
frequency instability of $f_{\mathrm{rf2}}$ to a sub-Hz level. The computed
modified Allan deviations of the fractional frequency instabilities (with a
nominal frequency of 0.68 THz) are displayed in Fig. 4 (e). At averaging times
above 10 s the free-running frequency instabilities are reduced by nearly six
orders of magnitude by the feedback control, reaching relative Allan
deviations below $1\times 10^{-14}$. We have verified that the frequency
instability attributed to the microwave multiplication is below $1\times
10^{-15}$ at all averaging times. Therefore this measurement reliably
evaluates the frequency stability of the THz oscillation.

##  V Discussion

By leveraging the frequency division technique, we unleashed new capabilities
of TRs in synthesizing microwaves with high spectral purity. Compared with the
previous work of using DKS to purify an injected microwave signal [25], the
TR-EOFD scheme not only generates microwaves with variable frequencies but
also has a significantly larger frequency locking range that is only limited
by the maximum frequency deviation of the frequency-modulated microwave
source. The demonstrated microwave phase noise performance is very close to
that obtained with a DKS state in a similar microresonator [26], which, to the
best of our knowledge, is to date the lowest phase noise level achieved with a
microresonator Kerr oscillator. One should note that, in [26] the DKS state
was operated at a “quiet point” – a particular pump-resonance detuning
position where the noise transduction is minimized due to the mutual
cancelation of multiple detuning-to-repetition-rate coupling effects [35].
Tracking down the quiet point is non-trivial, and its operation is delicate
and vulnerable to perturbations in temperature and coupling condition. In
contrast, the TR-EOFD scheme offers great ease in its implementation, and the
flexibility in switching microwave frequencies provides unprecedented
convenience for practical applications.

Additionally, the system is capable of synthesizing ultrastable THz
oscillations from rf sources. Compared with prior endeavors in generating low-
noise THz radiation with DKS microcombs [27, 36, 37], the state of TRs
explored here yields not only inherently low power threshold and high
conversion efficiency but also simplicity in system development. With future
effort in packaging and integration, the technology can be fruitfully utilized
in advanced THz imaging [38] and next-generation communication networks [39].

###### Acknowledgements.

This publication was supported by Contract D18AC00032 (DRINQS) from the
Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office
(DSO) and funding from the Swiss National Science Foundation under grant
agreement No. 192293.

## Appendix A: Quantum noises of TR oscillations

The theory [33] for computing quantum noise contribution to the phase noise of
DKS repetition rate has been verified both numerically [40] and experimentally
[41]. In this section we simulate the quantum-noise-limited phase noise
spectra of the TR oscillations by numerically integrating the modified LLE
that contains the Langevin noise term. The master equation can be written as:

$\frac{\partial{A}}{\partial
t}-i\frac{1}{2}D_{2}\frac{\partial^{2}{A}}{\partial\phi^{2}}-ig{|A|^{2}A}=-\left({\frac{\kappa}{2}+i(\omega_{0}-\omega_{\mathrm{p}})}\right){A}\\\
+\sqrt{\frac{\kappa_{\rm ex}P_{\rm in}}{\hbar\omega_{0}}}+f(t,\phi)$ (A1)

where $A$ is the slowly varying field amplitude, $\phi$ is the angular
coordinate in the co-rotating frame that is related to the round-trip fast
time coordinate $\tau$ by $\phi=\tau\times D_{1}$ (where $D_{1}=2\pi\nu_{\rm
FSR}$), ${\kappa}$ is the cavity decay rate, ${\kappa_{\rm ex}}$ is the
external coupling rate, and $P_{\rm in}$ is the pump power. Here
$g=\frac{\hbar\omega_{0}^{2}cn_{2}}{n_{0}^{2}V_{\rm eff}}$ is the single
photon induced Kerr frequency shift, where $n_{0}$ and $n_{2}$ are the
refractive and nonlinear optical indices, $V_{\rm eff}$ is the effective mode
volume, and $c$ is the speed of light. The Langevin force term $f(t,\phi)$
obeys [34]:

$\langle
f(t,\phi)f^{*}(t^{\prime},\phi^{\prime})\rangle=\frac{\kappa\hbar\omega_{0}\delta(t-t^{\prime},\phi-\phi^{\prime})}{\Delta\tau}$
(A2)

where $\Delta\tau=\frac{\Delta\phi}{D_{1}}$ is the fast time simulation
resolution. In the simulations we set $\Delta\phi=\frac{2\pi}{256}$.

Using step size of a single cavity round trip, we simulate for consecutively
$1\times 10^{6}$ round trips. The phase evolution of the TR oscillation is
computed with [42]:

$\Psi(t)={\rm Arg}[e^{i2\pi(m\nu_{\rm FSR}+\delta
f)t}(a_{-}a_{0}^{*}+a_{0}a_{+}^{*})]$ (A3)

where $a_{0}$, $a_{-}$ and $a_{+}$ are the complex intracavity fields of the
pumped mode, the low-frequency first-order TR sideband and the high-frequency
first-order TR sideband at slow time $t$, respectively. These complex fields
can be derived by applying Fourier transformation to the slowly varying field
amplitude $A$ in Eq. A1. Without loss of generality, $\delta f$ is included in
the term that represents the TR oscillation frequency for minor frequency
deviation from $m\nu_{\rm FSR}$, where $m$ is the number of TRs per round
trip. We note that only the light in the pumped mode and the two first-order
sidebands are taken into consideration as only they are relevant in our
experiments.

Figure 5: The profiles of the intracavity field amplitude of the simulated TRs
with three different $D_{2}$ settings. The values of $D_{2}$ and the numbers
of TRs are shown in the corresponding panels. The insets plot the relative
intracavity powers of the pumping laser and the first-order TR sidebands.
Figure 6: Numerically simulated quantum-noise-limited phase noise spectra for
three different TR oscillation frequencies show that the phase noise levels
are independent of the roll number $m$. The theoretical estimation is also
presented for comparison.

To numerically test the quantum phase noises of TR oscillations with different
roll numbers ($m$), besides using $\frac{D_{2}}{2\pi}=2$ kHz that is close to
the experimental parameter, we also repeat the simulations with a smaller
$D_{2}$ ($=2\pi\times 0.4$ kHz) and a larger $D_{2}$ ($=2\pi\times 10$ kHz),
respectively. The number of rolls (close to MI threshold) can be estimated
according to [9] using:

$m=\sqrt{\frac{2}{\beta}(\alpha-2\rho)}$ (A4)

where $\alpha=\frac{2(\omega_{0}-\omega_{\mathrm{p}})}{\kappa}$,
$\beta=-\frac{4\pi D_{2}}{\kappa}$, and $\rho$ is calculated to be 1.3 with
$F^{2}=(1+(\rho-\alpha)^{2})\rho$ and $F=\sqrt{\frac{8g\kappa_{\rm ex}P_{\rm
in}}{\kappa^{3}\hbar\omega_{0}}}$. We estimate $m$ to be 7.23, 16.17, and
36.15 for $\frac{D_{2}}{2\pi}=$ 10, 2, and 0.4 kHz, respectively, showing
excellent agreement with the simulation results in Fig. 5. One should note
that, in practice the number of rolls might be tunable by temperature tuning
via the mode-interaction-induced avoided mode crossing phenomenon [43, 44].

In Fig. 6 we plot the computed phase noise spectra of the three TR oscillation
frequencies of $f_{\mathrm{TR}}=$ 7$\nu_{\mathrm{FSR}}$,
16$\nu_{\mathrm{FSR}}$, and 36$\nu_{\mathrm{FSR}}$, respectively. One can see
that these phase noise spectra have the same magnitude, despite the fact that
the carrier frequencies of the TR oscillations are quite different. This
confirms that the quantum noise of a TR oscillation does not depend on $m$,
which agrees with the theoretical derivation in [34], which is written as:

$\mathcal{L}_{\rm
QN}(f)=\frac{\kappa^{2}\hbar\omega_{0}}{4\pi^{2}f^{2}P_{\mathrm{out}}}$ (A5)

where $P_{\mathrm{out}}$ is the output power of the first-order sideband for
which we use the simulation result to obtain. The theoretically calculated
quantum phase noise is also presented in Fig. 6, showing excellent agreement
with numerical simulations.

From the analysis presented in this section, one can conclude that, as long as
the TR oscillation frequency is within the frequency span of the EO comb, the
TR-EOFD scheme benefits from a large roll number $m$ in quantum-noise-limited
phase noise performance.

## Appendix B: Technical noises of TR oscillations

Most of the time technical noises are the dominant noises that degrade Kerr
oscillation spectral purity, especially at offset frequencies below 100 kHz.
Technical noises can be either of thermal nature or non-thermal. For thermal
noises such as resonator temperature fluctuations caused by ambient
temperature change and intracavity power absorption due to laser RIN or the
instability of coupling condition, we will briefly discuss their impact
together with the influence of fundamental thermal noise in the next section.
In this section we focus our attention on non-thermal technical noises
including laser-RIN-induced AM-to-PM noise and laser-phase-noise-induced PM-
to-PM noise. The noise conversion is simulated using methods similar to [42].
To simulate the technical phase noise spectrum of the DKS repetition rate
presented in Fig. 1 (b), the pump power is set to 20 mW and the pump-resonance
detuning is $-10\kappa$. All the other parameter values are the same as those
used for TR oscillation simulations. Like the method in [42], a small third-
order dispersion $\frac{D_{3}}{2\pi}=-4$ kHz is included in the simulations in
order to effectively induce the noise transfer mechanism. (In principle, even
if the resonator has no higher-order dispersion, $D_{2}$ is sufficient to
cause noise transfer. However, in reality the higher-order dispersion is
always present, and the noise transfer caused by the higher-order dispersion
is significantly stronger.)

Figure 7 shows the simulated technical phase noises of TR oscillations. The
input noises, including the laser RIN and the optical phase noise, are
synthesized to possess a power spectral density slope of $f^{-2}$ (see Fig. 7
(a) and (b)). This slope of the noise spectra is consistent with most
realistic laser noises. We confirm that the TR oscillation phase noises
converted from laser RIN and phase noises are independent and that the TR
oscillation phase noise resulted from compound noises is equal to the sum of
RIN-only-induced noise and laser-phase-noise-only-induced noise (see Fig. 7
(c)). In addition, the different slopes exhibited by the spectra of RIN- and
laser-phase-noise-induced phase noises are in agreement with the
characteristics of the noise transfer functions revealed in [42].

Figure 7: (a) Spectrum of the laser RIN that is used for technical noise
simulation. (b) Spectrum of the computer-generated laser phase noise that is
for numerical simulation. (c) Phase noise spectra of the TR oscillation
simulated with input noise of (brown) laser RIN only, (pink) laser phase noise
only, and (red) both laser RIN and phase noise.

We also repeat the same noise transfer simulation with different
$\frac{D_{2}}{2\pi}$ values of 10 and 0.4 kHz respectively while $D_{3}$ is
kept the same. Figure 8 presents the simulation results. While the phase noise
amplitudes differ with different $D_{2}$ because of the different recoil
strengths that are related to the change in microcomb spectral span, the
relative levels of the phase noises are very similar. Just like the result
shown in Fig. 1 (b), the microwaves yielded by the TR-EOFD scheme exhibit
phase noise amplitudes that are lower than those of DKS oscillations by 10 to
20 dB. Physically, the smaller technical noise transfer magnitude of TR state
might be related to its narrower optical spectral span. Under the influence of
the same higher order dispersion effect, a wider optical spectral span usually
results in larger spectral asymmetry, thus leading to larger spectral recoil
[45] and more prominent technical noise transfer. Since the optical spectral
span of DKS is normally much wider than that of the TRs, the technical noise
transfer in the repetition rate is also severer for the DKS state.

Figure 8: Simulated technical phase noise spectra of the DKS oscillation, the
TR oscillation, and the TR-EOFD-synthesized microwave with the same frequency
of the DKS oscillation when $\frac{D_{2}}{2\pi}$ is set to 10 kHz (a) and 0.4
kHz (b) respectively.

## Appendix C: Discussion on thermal noises

Since a Kerr oscillator’s fractional frequency change caused by the change of
the resonator temperature, either fundamental or technical, is independent of
the nominal frequency of the Kerr oscillation, in principle the TR-EOFD scheme
would not make any phase noise improvement if the Kerr oscillation (either TRs
or DKS) spectral purity is thermal-noise-limited. Yet, from a practical point
of view, the TR-EOFD scheme may still be preferable. Because the TR state is
excited at near-threshold power level, the coupled-in laser power is
relatively low (i. e., below a few milliwatts for most high- or ultrahigh-$Q$
resonators). On the contrary, in order to generate stable DKS state and to
have soliton steps that are wide enough, the pump power is often set to be
higher than the MI threshold by one to two orders of magnitude. Consequently,
the usage of higher power could result in larger temperature fluctuations with
the same laser RIN amplitude.

Thermo-refractive noise often dictates the fundamental resonance frequency
noise limit in dielectric microresonators [46] at relatively low Fourier
frequencies. It was indicated in [47] that for DKS in on-chip SiO2
microresonators the thermo-refractive noise is higher than the quantum noise
in the DKS oscillation fundamental phase noise at offset frequencies below a
few kHz. While the finite element method is often used to obtain a more
accurate estimation of the fundamental thermo-refractive noise [48, 49], here
we use the analytical expression derived in [46] to compute the thermo-
refractive-noise-limited phase noise of the TR oscillation in the resonator
used in our experiment, and we compare it with the quantum noise. The formula
for computing the thermo-refractive-noise-limited phase noise is:

$\mathcal{L}_{\rm TRN}(f)=\frac{1}{2f^{2}}(\nu_{\rm
FSR}\frac{1}{n_{0}}\frac{dn}{dT})^{2}\frac{k_{\rm B}T^{2}}{\rho C_{\rm
p}V_{\rm eff}}\frac{R^{2}}{12D}\left[1+(\frac{R^{2}}{D}\frac{2\pi
f}{9\sqrt{3}})^{3/2}+\frac{1}{6}(\frac{R^{2}}{D}\frac{2\pi
f}{8l^{1/3}})^{2}\right]^{-1}$ (C1)

where $k_{\rm B}$ is the Boltzmann’s constant, $\rho$ is the density of the
dielectric material, $C_{\rm p}$ is the specific heat capacity of the
material, and the thermal diffusion coefficient $D$ is related to the thermal
conductivity $K$ as $D=\frac{K}{\rho C_{\rm p}}$. The whispering gallery mode
azimuthal number $l$ can be estimated by $l\approx\frac{Rn_{0}\omega_{0}}{C}$.
The thermo-optic coefficient ($\frac{dn}{dT}$) of MgF2 for transverse electric
(TE) mode (o-ray in z-cut disks) is $8.4\times 10^{-7}$K-1 [50].

Figure 9: Computed fundamental phase noises including the thermo-refractive
noise and the quantum noise of a microwave that is synthesized with the TR-
EOFD scheme. The TR oscillation is of a roll number of 16. The frequency of
the synthesized microwave is equal to $\nu_{\rm FSR}$.

In Fig. 9 we plot the thermo-refractive noise limit and the quantum noise
limit for the phase noise of the microwave frequency that is obtained with
EOFD from a TR oscillation with a roll number of 16. This comparison shows
that the thermo-refractive noise dominates the fundamental noise floor only at
offset frequencies below 10 Hz, which means that practically the thermo-
refractive noise is unlikely to limit the phase noise performance since at low
frequencies technical noises would set the noise level. It is only for
microresonators with much smaller effective mode volumes and larger thermo-
optic coefficients (such as those on-chip SiO2 and Si3N4 microresonators in
[47, 49]) that thermo-refractive noise might be the dominant fundamental phase
noise type in certain frequency ranges.

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