of 18
Reviewers' Comments
:
Reviewer #1
:
Remarks to the Author
:
In this manuscript, authors reported a miniature microwave source by means of the soliton microcomb
technique. They carefully characterized the noise spectrum of the repetition frequency of the
microcavity soliton, particularly around the so
-
called “quiet
point” of the laser cavity detuning where
the dispersive wave and Raman coupled noise can be counterbalanced. Given this situation, the phase
noise of the soliton repetition rate is much reduced, to a record low level among chip scale soliton
microcomb pla
tforms, and the system reaches the limitation of thermorefractive noise (TRN). On the
latter topic, authors further studied the coupling of TRN between a higher order mode of the cavity
and the soliton mode family
.
In my view, the results of the presentin
g work are solid, with fruitful details and supporting theories.
Although the high
-
order mode coupled TRN seems not a dominant noise source compared with the
fundamental one, especially at low offset frequencies (<1 kHz), this effect was carefully derived
here
for the first time
.
However, my concern is that overall the novelty of the presenting work is not obvious. This work is the
one after several others reporting on the soliton microcomb used for microwave signal generation. As
such, the methodology inc
luding the quiet point operation regime in the presenting work has already
been tested in previous works, particularly in the work by Lucas et al., “Ultralow
-
noise photonic
microwave synthesis using a soliton microcomb
-
based transfer oscillator,” Nature Co
mmunications,
11, 374, 2020 (In fact this work is highly related to the present one). In microcavities, the TRN
limitation to the microwave signal is also well known, and clearly, with an ultrahigh Q factor, the TRN
has limited impact such that the signal
noise can be improved, as expected. Nevertheless, there
remains a gap in the noise performance between the presenting work and a crystalline cavity
microcomb (Ref. 4). Since the latter still holds the record of the lowest phase noise (ca.
-
110
dBc/Hz@1kHz)
of soliton microcomb and the system was also miniature, the representing claim of a
“record low” phase noise, though better than other chip based platforms, is degraded
.
To conclude, there is no doubt the presenting manuscript is of high quality and is e
ducational for
people who are not familiar with the fundaments of soliton microcombs. But given that a similar work
has just been published ahead of the presenting one and the current results are not clearly
outstanding, I may not suggest the work to be pu
blished in Nature Communications
.
Reviewer #2
:
Remarks to the Author
:
The manuscript entitled ‘Dispersive
-
wave induced noise limits in miniature soliton microwave sources’
reports on a detailed investigation of the noise processes of a microcomb
-
based m
icrowave source
operating at a so
-
called quiet point. This quiet point occurs as a consequence of a delicate balance
between Raman
-
induced soliton self
-
frequency shift, and spectral recoil associated with the emission
of dispersive wave into a non
-
soliton
transverse mode, as nicely demonstrated by the authors in ref.
[20]. As a result, the detuning noise couples minimally to the soliton repetition rate and the
microwave noise is significantly suppressed
.
Using such a quiet point to reduce the noise of the r
esulting microwave signal is not new in itself (ref.
[29] in the main text). However, I believe the novelty of the current manuscript lies in revealing and
characterizing a new noise mechanism, which originates from correlations (or lack of) between the
th
ermal fluctuations of the competing transverse modes. Such effect can arise whenever multiple
transverse modes interact. This noise mechanism is thus likely to be of a broad interest to studies
concerning micro
-
and nano
-
scale multimode systems, especially
as the fundamental noise limit is
approached
.
In my opinion, the manuscript merits publication in Nature Communications. However, I would first
like to invite the authors to consider my comments below
:
a. Since the work concerns photonic
-
chip
-
based microw
ave sources, it would be appropriate to
introduce ref. [33
-
36] earlier to give audience a broader overview of progress that is in parallel to
microcomb technology
.
b. To appeal to broad audience of Nature Communications, it would be pertinent to compare/co
ntrast
dispersive waves phase
-
matched by higher order dispersion (same mode family), with those arising
from intermodal interaction. Ultrafast researchers more often associate dispersive waves with the
former process in non
-
resonant single
-
pass waveguides.
The statement ‘These waves are emitted
when solitons radiate into resonator modes that do not belong to the soliton
-
forming mode family’
with no further elaboration may cause confusion
.
c. In addition to ref. [19,42,43], Jang et al. Opt. Lett. 39, 5503 (2
014) reports on dispersive wave
generation in a driven fiber resonator and deserves to be acknowledged in my opinion
.
d. In discussion, the authors claim the measured phase noise is a record low among photonic
-
chip
-
based microwave sources. While I do not d
oubt the claim, it would be good to compare the
performance of their system with other systems, possibly in a tabular form
.
e. Please clarify how detuning is measured. I see in the methods that the authors use Eq (23) but
there is no clear reference to tha
t expression in the main text
.
f. In Fig. 3e, phase noise reductions initially exceed the expected black dotted curve at 500 Hz and 1
kHz. Why is that? Is that due to the correlation
-
induced reduction in TRN? Please also refer to the next
point
.
g. On a re
lated note, how does the conclusion from the study of intermode TRN relate to the results of
Fig. 3d and e? Specifically, because the correlation R is positive at lower Fourier frequencies, the
intermode TRN is reduced according to Eq (4) and is supported
by Fig. 5b. However, noise suppression
factor saturates more strongly at lower frequencies in Fig. 3d and e, and seem to contradict Eq (4)
.
h. Please clarify what the authors mean by ‘spectral center’ of the comb. Is it the peak of the sech2
envelope fit,
or is it the first
-
order moment of the measured spectrum
?
i. Just a remark. Figure 4b is very nice and illustrative
!
Other minor comments
:
1. Just below Eq (1), strictly D1 is 2*pi*FSR
.
2. Page 2 right column, there is repeated ‘of the of the’
.
3. In main
text and caption of Fig. 3, the authors refer to dashed orange curve in Fig. 3d. It looks like
a dashed red curve
.
4. In section ‘Thermal noise in the dispersive wave’, the first sentence ‘Constant heat exchange
associated with thermal equilibrium’ reads c
ontradictory because at thermal equilibrium, there should
be no heat exchange. Perhaps more appropriate to clarify e.g. ‘fluctuations about thermal equilibrium’
or equivalent
.
5. What does wide tilde in Eq (11) and (16) signify
?
Reviewer #3
:
Remarks to
the Author
:
Please see the attached file for reviewer comments
.
Id:
ncomms
-
20
-
39020
Title:
Dispersive
-
wave induced noise limits in miniature soliton microwave sources
Authors:
Qi
-
Fan Yang
et al
.
In this manuscript the authors use a frequency comb generated by
continuous wave pumping
of a
n ultra
-
high
quality
microresonator
to generate a microwave
frequency of about 15 GHz.
In
particular
,
the authors investigate and present phase fluctuations of
the
microwave beat
frequency
when the system is operated around the so
-
called quiet point, whe
re the
Raman
-
induced
self
-
frequency shift of the soliton
is balanced by the
dispersive
-
wave induced recoil
.
Both are a function of the frequency detuning defined as
the difference between
the frequency
of the cavity mode nearest to the pump frequency and the pump frequency
.
A calibration tone, generated as a side band of the pump frequency, is used to compare the
phase noise measured for different detuning frequencies.
The c
alibration tone is used to
determine the
expected noise suppression for different detuning frequencies.
It is found in the
experiment that the actual
noise suppression is less than that derived from the calibration tone
at and around the quiet point.
From this the authors conclude that
other noise sources are
limiting the phase noise around the quiet point.
The authors subsequently
consider various noise sources, like pump intensity noise,
noise
induced by quantum vacuum fluctuations
and temperature
-
induced
fluctuation
in the free
-
spectral range of the resonator and conclude that these are all too small
to e
xplain the
measured phase noise of the microwave beat frequency.
The authors conclude
, by comparing
theoretical
and simulation results with measurements,
that
dispersive wave induced noise
originating from
fundamental
intermode thermorefractive noise
is th
e limiting
noise source
for
their device
.
Overall, t
he manuscript is well
structured
and written.
The research is novel,
of high interest
,
and
t
he claims are well
formulated
and supported by
appropriate
measurements and
simulations
.
However,
the authors
provide
uncertainty in measured/calculated values
only for
the Allan
deviation
(fig. 2c), while such information is absent for all other results presented.
It
would benefit the reader to have more insight in the accuracy of the measured and calculated
values presented in this manuscript.
Other
minor
issues that need to be clarified are
:
1.
!
is defined as the frequency of the cavity mode pumped by the laser. Is this the cold
-
cavity frequency or the hot
-
cavity
frequency
of the re
sonance
?
2.
Definition
of
Δ
is not clear
.
Is
this parameter the difference between hybridized modes
or non
-
hybridized mo
des
?
The main text
sugge
sts
this to be the difference between
non
-
hybridized modes, while
the
supplementary
information
uses inline formula
"
,
$
%
=
"
,
&
Δ
”,
suggesting that Δω is the frequency difference between a
hybridized and non
-
hybridized mode.
3.
Formula S1:
From the reference given for formula S1 and formula S1 itself, I would
assume
Δ
to be the frequency difference of the non
-
hybridized modes. However, if
true, formula S1 assumes no loss or at least no loss difference between the non
-
hybridized
resonances, although this is an ultra
-
high
-
Q resonator and losses are
extremely low, is the difference still considered significantly small compared to the
coupling term
and/or
Δ
for equation S1 to hold?
4.
“As a bench
mark of the stability
...” The auth
ors
provide
an Allan deviation analys
is of
the soliton repetition rate
and retrieve the minimum deviation. However
,
this value is
not placed in a context.
How does this device compare to other comb
-
based microwave
sources
, in particular
how does it compare to the ele
ctronic microwave source used for
the locking demonstration
?
5.
The calibration tone is visible as a marker in figure
3d, but not in the measured traces of
the SSB phase noise.
Do these represent two different measurements
, i.e., one with
calibration tone and a subsequent measurement without calibration tone
?
6.
“Their Q
factors are also measured, as shown in Fig. 4d
”.
Why is this figure included?
The Q factors seem not to be used elsewhere in the manuscript.
7.
Equation 4 requires a reference
to the methods section
.
8.
Authors mention that the microwave beat signal could
track the external microwave
source over a range of 30 kHz.
The limits of such tracking are not discussed.
What limits
this
tuning range
? Wh
at is the minimum tuning step that this system can realize?
9.
The laser scan is
precisely measured by a radio
-
frequency calibrated Mach
-
Zehnder
interferometer
. The reference
u
sed
provides a similar statement without providing
more
details
.
The reference should point
to a paper
explaining th
is measurement
technique. The
current reference is inapp
ropriate.
10.
averaged
temperature of an optical mode
.
I woul
d not
call this a temp
erature of a
n
optical
mod
e but an
optical mode we
ight
e
d
ave
rage
temperature
.
11.
It is not complet
ely clear what the
de
nsity
(
)
represents.
Does
(
)
in eq.
9
represent
the transverse distribution
of the electric field
of a
single
tran
s
verse mode or the
total
transverse di
stribution if multiple transverse mode
s are present
? Also, as we are dealing
with a
rela
tively
br
oad
frequency com
b,
even for a single transverse mode, the mode
profile w
ill slightly very
for different
resona
nt fr
e
quencies
.
I
s
this
dependency included
or
neglected?
12.
It is unclear what the tilde in eq. 11 represents.
If it has the same meaning
as in eq 16
, it
should represent the Fourier transf
orm, however that is only introduced w
ith eq
. 16.
13.
Comparing eq. 11 with t
he references shows a
discrepancy
by a factor of 2. Is this a
typo?
14.
In eq
uation 12, t
he
in
(
!"##
)
*
is act
ually the
amplitude squared of the entropy injection
!
+
a
ccording to reference
17. Also
,
again
there seem to be a factor of 2 discrepancy
with the references.
15.
In eq
uation 14 the
Four
ier transform is indicated by
while in eq. 16
(and eq. 11) a tild
e
above the variable is used to indicate the Fourier transform
. T
wo different notati
ons are
used
, where only
one
should be
needed.
16.
The de
nsity
(
)
seems to be missing in t
h
e
source term
of
equation 16
.
The source
term s
hould be the Fourier transform
of eq. 10 multiplied by
!
conform equation 7
.
17.
E
quations 17 & 18
only contains a linear coupling term.
Nonlinear coupling (e.g., cross
-
phase modulation) can also be present.
Why has this not been included in the model
,
i.e.,
why is this effect considered weak enough to neglect even for the dispersive wave
(eq. 18)?
18.
It is stated that the detuning noise is correlated to the noise in the error signal of the
PDH, which makes sense. However, it is unclear how the conversion from P
DH
-
error
signal to detuning noise is performed.
The text only mentions that it is “extracted
from
the residual error signal in the locking loop”
Small typos:
(see Fig. 1.
” Missing closing bracket
being pump by optical
”.
S
hould
be
b
eing pumped by optical
“of
the of the”. Repetition
maintext
Should be
main text
.
Supplemental inf
ormation
f
iled
.
Should
b
e
field
. Furthermore,
Fig. s1a
indicate
s th
at the
power
is plotted and not the
fiel
d
and
it is not explained what
cavity angle
means
(horizontal label in Fig. s1a).
R
e
s
p
o
n
s
e
t
o
r
e
v
i
e
w
e
r
#
1
:
In this manuscript, authors reported a miniature microwave source by means of the soliton
microcomb technique. They carefully characterized the noise spectrum of the repetition
frequency of the microcavity soliton, particularly around the so-called “quiet point” of the
laser cavity detuning where the dispersive wave and Raman coupled noise can be
counterbalanced. Given this situation, the phase noise of the soliton repetition rate is much
reduced, to a record low level among chip scale soliton microcomb platforms, and the
system reaches the limitation of thermorefractive noise (TRN). On the latter topic, authors
further studied the coupling of TRN between a higher order mode of the cavity and the
soliton mode family.
In my view, the results of the presenting work are solid, with fruitful details and supporting
theories. Although the high-order mode coupled TRN seems not a dominant noise source
compared with the fundamental one, especially at low offset frequencies (<1 kHz), this effect
was carefully derived here for the first time.
However, my concern is that overall the novelty of the presenting work is not obvious. This
work is the one after several others reporting on the soliton microcomb used for microwave
signal generation. As such, the methodology including the quiet point operation regime in the
presenting work has already been tested in previous works, particularly in the work by Lucas
et al., “Ultralow-noise photonic microwave synthesis using a soliton microcomb-based
transfer oscillator,” Nature Communications, 11, 374, 2020 (In fact this work is highly related
to the present one). In microcavities, the TRN limitation to the microwave signal is also well
known, and clearly, with an ultrahigh Q factor, the TRN has limited impact such that the
signal noise can be improved, as expected. Nevertheless, there remains a gap in the noise
performance between the presenting work and a crystalline cavity microcomb (Ref. 4). Since
the latter still holds the record of the lowest phase noise (ca. -110 dBc/Hz@1kHz) of soliton
microcomb and the system was also miniature, the representing claim of a “record low”
phase noise, though better than other chip-based platforms, is degraded.
To conclude, there is no doubt the presenting manuscript is of high quality and is educational
for people who are not familiar with the fundaments of soliton microcombs. But given that a
similar work has just been published ahead of the presenting one and the current results are
not clearly outstanding, I may not suggest the work to be published in Nature
Communications.
R
e
p
l
y
: We thank the reviewer for his/her appreciation of our work. We agree that the soliton
microcombs generated in crystalline resonators feature outstanding performance due to their
ultrahigh-Q factors and low thermo-optic coefficients. However, it remains a challenge to
transfer these devices to photonic chips. The silica resonator used in our work holds the
record Q factor among all chip-based platforms, and its integration with a photonic
waveguide has also been demonstrated [K. Yang et al, Bridging ultrahigh-Q devices and
photonic circuits, Nature Photonics 12, 297 (2018)]. Indeed, our results surpass all other
chip-based soliton microcombs and optoelectronic oscillators. To highlight this point, in the
revised manuscript a new table (also attached here) is included to compare the phase noise
and Allan deviation of several state-of-the-art optoelectronic oscillators.
Besides the record chip-based performance, a new noise source of soliton microcombs is
revealed in this work. While the dispersive wave induced ‘quiet point’ mechanism has been
used to reduce the impact of pump phase noise [Nature Communications, 8(1), 14869;
Nature Communications, 11(1); Nature Photonics, 14(8), 486–491; Physical Review Letters,
125(15), 153901], limits of this technique have not yet been explored. In this work we have
shown that while quiet point reduction can exceed 36 dB, intermode thermorefractive noise
channels through the dispersive wave involved with the quiet point operation, itself. And,
indeed this limits phase noise suppression by the quiet point. Since researchers have been
striving to generate low-noise soliton microcombs to facilitate a wide range of applications,
understanding this noise mechanisms is important. Hence, we believe the key findings of our
work are novel and are of great interest to the community of optical frequency combs and
microwave photonics.
R
e
s
p
o
n
s
e
t
o
r
e
v
i
e
w
e
r
#
2
:
The manuscript entitled ‘Dispersive-wave induced noise limits in miniature soliton microwave
sources’ reports on a detailed investigation of the noise processes of a microcomb-based
microwave source operating at a so-called quiet point. This quiet point occurs as a
consequence of a delicate balance between Raman-induced soliton self-frequency shift, and
spectral recoil associated with the emission of dispersive wave into a non-soliton transverse
mode, as nicely demonstrated by the authors in ref. [20]. As a result, the detuning noise
couples minimally to the soliton repetition rate and the microwave noise is significantly
suppressed.
Using such a quiet point to reduce the noise of the resulting microwave signal is not new in
itself (ref. [29] in the main text). However, I believe the novelty of the current manuscript lies
in revealing and characterizing a new noise mechanism, which originates from correlations
(or lack of) between the thermal fluctuations of the competing transverse modes. Such effect
can arise whenever multiple transverse modes interact. This noise mechanism is thus likely
to be of a broad interest to studies concerning micro- and nano-scale multimode systems,
especially as the fundamental noise limit is approached.
In my opinion, the manuscript merits publication in Nature Communications. However, I
would first like to invite the authors to consider my comments below:
R
e
p
l
y
: We thank the reviewer for his/her recommendation for publication.
a. Since the work concerns photonic-chip-based microwave sources, it would be appropriate
to introduce ref. [33-36] earlier to give audience a broader overview of progress that is in
parallel to microcomb technology.
R
e
p
l
y
: We thank the reviewer for this suggestion and have included these references in the
introduction in the revised manuscript.
b. To appeal to broad audience of Nature Communications, it would be pertinent to
compare/contrast dispersive waves phase-matched by higher order dispersion (same mode
family), with those arising from intermodal interaction. Ultrafast researchers more often
associate dispersive waves with the former process in non-resonant single-pass
waveguides. The statement ‘These waves are emitted when solitons radiate into resonator
modes that do not belong to the soliton-forming mode family’ with no further elaboration may
cause confusion.
R
e
p
l
y
: we thank the reviewer for this suggestion. We have added the following sentence to
clarify mechanism that can produce dispersive waves: “
Dispersive waves can emerge as a
result of higher-order dispersion [23,24], supermodes [25], or when solitons radiate into
resonator modes that do not belong to the soliton-forming mode family.
” in the revised
manuscript. We also have noted in the discussion a way to reduce the noise through use of
dispersive waves formed in the same mode longitudinal mode family:
First, use of
dispersive waves within the same longitudinal mode family (as formed by higher order
dispersion
23,24,42,43
) could be investigated. In this case, better overlap of the dispersive
wave modal profile with the soliton mode would be expected to reduce the dispersive
wave noise.
c. In addition to ref. [19,42,43], Jang et al. Opt. Lett. 39, 5503 (2014) reports on dispersive
wave generation in a driven fiber resonator and deserves to be acknowledged in my opinion.
R
e
p
l
y
: We thank the reviewer for the comment. We have included this citation in the revised
manuscript.
d. In discussion, the authors claim the measured phase noise is a record low among
photonic-chip-based microwave sources. While I do not doubt the claim, it would be good to
compare the performance of their system with other systems, possibly in a tabular form.
R
e
p
l
y
: We thank the reviewer for this suggestion. We have included a table in the revised
manuscript (also included here). The phase noise values are all scaled to 15.2 GHz. Our
work does show best noise performance among all chip-based platforms.
e. Please clarify how detuning is measured. I see in the methods that the authors use Eq
(23) but there is no clear reference to that expression in the main text.
R
e
p
l
y
: The detuning is calculated based on the fitted spectral center and pulse width of the
soliton with Eq (23). We have added a sentence “
Here the detuning
is calculated based
ω
δ
on equation (23) in Methods
” in the revised manuscript.
f. In Fig. 3e, phase noise reduction initially exceed the expected black dotted curve at 500
Hz and 1 kHz. Why is that? Is that due to the correlation-induced reduction in TRN? Please
also refer to the next point.
R
e
p
l
y
: we thank the reviewer for this helpful comment. We believe that this is an artifact of
using a calibration tone at an offset frequency of 10 kHz and then making noise
measurements at a series of other offset frequencies (500 Hz, 1 kHz and 8 kHz).
Specifically, the artifact does not appear in the 8 kHz data. In the revised manuscript, we
have added a comment as follows: “As an aside, the quiet-point-induced phase noise
reduction is also slightly higher than indicated by the calibration tone for lower noise
suppression levels (when measured at 500 Hz and 1 kHz offsets). This could result from
possible instrument calibration error associated with calibration using a 10 kHz tone.”.
g. On a related note, how does the conclusion from the study of intermode TRN relate to the
results of Fig. 3d and e? Specifically, because the correlation R is positive at lower Fourier
frequencies, the intermode TRN is reduced according to Eq (4) and is supported by Fig. 5b.
However, noise suppression factor saturates more strongly at lower frequencies in Fig. 3d
and e, and seem to contradict Eq (4).
R
e
p
l
y
: Even though there is stronger correlation in the intermode TRN at lower offset
frequencies, the frequency response of the TRN noise noise strongly increases as offset
frequency decreases. And this increase is so strong that it dominates the improved
intermode TRN correlation discussed later. This is indeed a confusing point and we have
added the following discussion in the main text to clarify.
“Notice that despite the improved correlation of the intermode TRN at lower offset
frequencies, it still dominates the microwave phase noise measured in Fig. 3d,e. This
happens because the TRN noise rises very rapidly as offset frequency decreases, even
overcoming the improving correlation of TRN between the dispersive wave mode and soliton
forming mode.”
h. Please clarify what the authors mean by ‘spectral center’ of the comb. Is it the peak of the
sech2 envelope fit, or is it the first-order moment of the measured spectrum?
R
e
p
l
y
: ‘Spectral center’ is the peak of the sech2 envelope fit. We have rephrased the first
appearance of this expression and defined it: “.....center frequency of the soliton spectrum
(which has an overall sech
2
envelope)...”
i. Just a remark. Figure 4b is very nice and illustrative!
R
e
p
l
y
: We thank the reviewer for the remark.
Other minor comments:
Just below Eq (1), strictly D1 is 2piFSR.
R
e
p
l
y
: we have corrected this definition.
Page 2 right column, there is repeated ‘of the of the’.
R
e
p
l
y
: we have corrected the typo.
In main text and caption of Fig. 3, the authors refer to dashed orange curve in Fig. 3d. It
looks like a dashed red curve.
R
e
p
l
y
: We have corrected the typo to “dashed red curve”.
In section ‘Thermal noise in the dispersive wave’, the first sentence ‘Constant heat exchange
associated with thermal equilibrium’ reads contradictory because at thermal equilibrium,
there should be no heat exchange. Perhaps more appropriate to clarify e.g. ‘fluctuations
about thermal equilibrium’ or equivalent.
R
e
p
l
y
: We have changed the expression to “
fluctuations associated with thermal
equilibrium
”.
What does wide tilde in Eq (11) and (16) signify?
R
e
p
l
y
: The tildes signify Fourier transform of the quantity. In the revised manuscript, we
have unified the notations of Fourier transform to be scripted
F
.
R
e
s
p
o
n
s
e
t
o
r
e
v
i
e
w
e
r
#
3
:
In this manuscript the authors use a frequency comb generated by continuous wave
pumping of an ultra-high quality microresonator to generate a microwave frequency of about
15 GHz. In particular, the authors investigate and present phase fluctuations of the
microwave beat frequency when the system is operated around the so-called quiet point,
where the Raman-induced self-frequency shift of the soliton is balanced by the
dispersive-wave induced recoil. Both are a function of the frequency detuning defined as the
difference between the frequency of the cavity mode nearest to the pump frequency and the
pump frequency.
A calibration tone, generated as a side band of the pump frequency, is used to compare the
phase noise measured for different detuning frequencies. The calibration tone is used to
determine the expected noise suppression for different detuning frequencies. It is found in
the experiment that the actual noise suppression is less than that derived from the
calibration tone at and around the quiet point. From this the authors conclude that other
noise sources are limiting the phase noise around the quiet point.
The authors subsequently consider various noise sources, like pump intensity noise, noise
induced by quantum vacuum fluctuations and temperature-induced fluctuation in the free
spectral range of the resonator and conclude that these are all too small to explain the
measured phase noise of the microwave beat frequency. The authors conclude, by
comparing theoretical and simulation results with measurements, that dispersive wave
induced noise originating from fundamental intermode thermorefractive noise is the limiting
noise source for their device.
Overall, the manuscript is well structured and written. The research is novel, of high interest,
and the claims are well formulated and supported by appropriate measurements and
simulations.
However, the authors provide uncertainty in measured/calculated values only for the Allan
deviation (fig. 2c), while such information absents for all other results presented. It would
benefit the reader to have more insight in the accuracy of the measured and calculated
values presented in this manuscript.
R
e
p
l
y
:
We thank the reviewer for his/her appreciations of our work, as well as the expert
comments. We have included a detailed analysis of uncertainty in Methods of the revised
manuscript. Error bars are also provided in the revised Fig3 and Fig 5.
Minor issues that need to be clarified are:
1.
is defined as the frequency of the cavity mode pumped by the laser. Is this the
ω
0
cold-cavity frequency or the hot-cavity frequency of the resonance?
R
e
p
l
y
:
denotes cold-cavity frequency. We have clarified this point in the revised
ω
0
manuscript as “
cold-cavity mode
”.
2. Definition of
is not clear. Is this parameter the difference between hybridized modes
ω
Δ
or non-hybridized modes? The main text suggests this to be the difference between
non-hybridized modes, while the supplementary information uses inline formula “
” suggesting that
is the frequency difference between a hybridized and
ω
ω
r
,
D
=
ω
r
,
S
Δ
ω
Δ
non-hybridized mode.
R
e
p
l
y
: we thank the reviewer for raising this question. In the previous version of the
manuscript, we were assuming that the mode hybridization is weak, and we were not
distinguishing their differences. We have added discussion in the revised manuscript, and it
is shown that the Δω in the previous version should be frequency difference between a
hybridized and non-hybridized mode. To clarify this point, in the revised manuscript, we use
to represent difference between the non-hybridized modes and
to represent the
ω
Δ
ω
Δ
frequency difference between the hybridized dispersive wave mode and non-hybridized
soliton mode.
3. Formula S1: From the reference given for formula S1 and formula S1 itself, I would
assume Δω to be the frequency difference of the non-hybridized modes. However, if true,
formula S1 assumes no loss or at least no loss difference between the non-hybridized
resonances, although this is an ultra-high-Q resonator and losses are extremely low, is the
difference still considered significantly small compared to the coupling term G and/or Δω for
equation S1 to hold?
R
e
p
l
y
: Yes, we are assuming that the difference is small compared to the couplings and
detuning. A complete expression yields
e
a
l
[
]
[
1
(
)
]
.
ω
r
D
=
2
ω
+
ω
r
S
r
D
R
G
2
+
1
6
4
Δ
ω
(
κ
κ
)
+
4
j
(
κ
κ
)
Δ
ω
2
S
D
2
S
D
=
2
ω
+
ω
r
S
r
D
G
2
+
4
Δ
ω
2
+
O
G
+
2
4
Δ
ω
2
(
κ
κ
)
S
D
2
In over case,
=2.17 MHz, while the denominator
=12.2 MHz.
)
/
2
π
(
κ
D
κ
S
/
2
π
G
2
+
4
Δ
ω
2
A comment is added in the Supplement to clarify this point.
4. “As a benchmark of the stability ...” The authors provide an Allan deviation analysis of the
soliton repetition rate and retrieve the minimum deviation. However, this value is not placed
in a context. How does this device compare to other comb-based microwave sources, in
particular how does it compare to the electronic microwave source used for the locking
demonstration?
R
e
p
l
y
: We thank the reviewer for raising this question. We have compared both phase noise
and Allan deviation of our result with those of several other platforms: crystalline microcomb,
SiN microcomb, photonic Brillouin microwave synthesizer and an integrated opto-electrical
oscillator. However, we could not quantify Allan deviation of the table-top electronic
microwave source, as it is the most stable microwave reference in our lab.
As shown in the figure, our result exceeds the performance of SiN microcomb and Brillouin
microwave synthesizer, but is not as good as the Crystalline microcomb.
5. The calibration tone is visible as a marker in figure 3d, but not in the measured traces of
the SSB phase noise. Do these represent two different measurements, i.e., one with
calibration tone and a subsequent measurement without calibration tone?
R
e
p
l
y
: They are the same measurements. We were using a signal source analyzer
(R&S®FSUP Signal Source Analyzer) with spur rejection function. The spurs were
simultaneously detected and do not appear in the plotted phase noise traces.
6. “Their Q factors are also measured, as shown in Fig. 4d”. Why is this figure included? The
Q factors seem not to be used elsewhere in the manuscript.
R
e
p
l
y
: The Q factors are measured for evaluating the loss of the modes κ_S and κ_D, which
is further used for evaluating the noise transduction factor.
7. Equation 4 requires a reference to the methods section.
R
e
p
l
y
: Thanks for the suggestion. We have added the reference in the revised version.
8. Authors mention that the microwave beat signal could track the external microwave
source over a range of 30 kHz. The limits of such tracking are not discussed. What limits this
tuning range? What is the minimum tuning step that this system can realize?
R
e
p
l
y
: We thank the reviewer for the comment. It is limited by the pump power of the soliton,
as the possible detuning range between the pump laser and cavity mode depends primarily
on the pump power [Nature Physics, 13(1), 94–102]. Continuous tuning is feasible. We have
included a sentence “This range is likely determined by the soliton existence range, which
is, in turn, determined by the pump laser power
33
.” in the revised manuscript.
9. “The laser scan is precisely measured by a radio-frequency calibrated Mach-Zehnder
interferometer”. The reference used provides a similar statement without providing more
details. The reference should point to a paper explaining this measurement technique. The
current reference is inappropriate.
R
e
p
l
y
: We thank the reviewer for noticing this. A more detailed description is included in
[Optics Express, 20(24), 26337 (2012)]. We have added this citation.
10. “averaged temperature of an optical mode”. I would not call this a temperature of an
optical mode but an “optical mode weighted average temperature ".
R
e
p
l
y
: We thank the reviewer for suggesting a more accurate description. We have replaced
the phrase “
averaged temperature of an optical mode
” with “optical mode weighted average
temperature”.
11. It is not completely clear what the density q(r) represents. Does q(r) eq. 9 represent the
transverse distribution of the electric field of a single transverse mode or the total transverse
distribution if multiple transverse modes are present?
R
e
p
l
y
:
q(r)
represents the transverse distribution of the electric field of a single transverse
mode.
Also, as we are dealing with a relatively broad frequency comb, even for a single transverse
mode, the mode profile will slightly vary for different resonant frequencies. Is this
dependency included or neglected?
R
e
p
l
y
: We are neglecting this effect. We have run FEM simulations to verify this
approximation over the measured wavelength range (1530-1570 nm).
We have evaluated the intermode TRN between the TM0 modes at frequencies 191 THz
and 195 THz. Both material and geometry dispersion are considered. This noise is
considered to directly modulate the FSR of the resonator and impose a noise limit on its
repetition rate. The results are plotted below, along with the measured phase noise.
12. It is unclear what the tilde in eq. 11 represents. If it has the same meaning as in eq 16, it
should represent the Fourier transform, however that is only introduced with eq. 16.
R
e
p
l
y
: we thank the reviewer for noticing this. This is a typo, and there should not be the
tilde.
13. Comparing eq. 11 with the references shows a discrepancy by a factor of 2. Is this a
typo?
R
e
p
l
y
: We thank the reviewer for noticing this. It is not a typo. We were using
to
δ
T
|
|
represent amplitude of
, which is different from the reference [Physics Letters A, 372(12),
T
δ
1941–1944 (2008)]. In the revised manuscript, to unify the notation where the reference, we
have changing the expression to
, where <> denotes time
Δ
δ
T
)
r
W
d
i
s
s
=
k
T
0
<
(
2
>
d
3
averaging.
14. In equation 12, the
in
is actually the amplitude squared of the entropy injection
f
π
f
W
ˉ
h
d
i
s
s
according to reference 17. Also, again there seems to be a factor of 2 discrepancy with
F
0
2
the references.
R
e
p
l
y
: Here,
is the frequency of the noise.
is normalized as 1 in the calculations, and
f
F
0
2
we have included
in the denominator of equation (12) for classification in the revised
F
0
2
manuscript. Here, we used single side band definition of power spectral density, while the
references are using one-sided (two side band) definition. The “2” factor comes from the
different definition.
15. In equation 14 the Fourier transform is indicated by F while in eq. 16 (and eq. 11) a tilde
above the variable is used to indicate the Fourier transform. Two different notations are
used, where only one should be needed.
R
e
p
l
y
: We have unified the notations in the revised manuscript as fancy
F
.
16. The density
seems to be missing in the source term of equation 16. The source
(
r
)
q
term should be the Fourier transform of eq. 10 multiplied by
form equation 7.
ω
T
i
0
R
e
p
l
y
: We thank the reviewer for pointing out this typo. We have corrected this.