of 10
ARTICLE
Dispersive-wave induced noise limits in miniature
soliton microwave sources
Qi-Fan Yang
1,2
, Qing-Xin Ji
1,2
, Lue Wu
1,2
, Boqiang Shen
1
, Heming Wang
1
, Chengying Bao
1
,
Zhiquan Yuan
1
& Kerry Vahala
1
Compact, low-noise microwave sources are required throughout a wide range of application
areas including frequency metrology, wireless-communications and airborne radar systems.
And the photonic generation of microwaves using soliton microcombs offers a path towards
integrated, low noise microwave signal sources. In these devices, a so called quiet-point of
operation has been shown to reduce microwave frequency noise. Such operation decouples
pump frequency noise from the soliton
s motion by balancing the Raman self-frequency shift
with dispersive-wave recoil. Here, we explore the limit of this noise suppression approach and
reveal a fundamental noise mechanism associated with
fl
uctuations of the dispersive wave
frequency. At the same time, pump noise reduction by as much as 36 dB is demonstrated.
This fundamental noise mechanism is expected to impact microwave noise (and pulse timing
jitter) whenever solitons radiate into dispersive waves belonging to different spatial mode
families.
https://doi.org/10.1038/s41467-021-21658-7
OPEN
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.
2
These authors contributed equally: Qi-Fan Yang,
Qing-Xin Ji, Lue Wu.
email:
vahala@caltech.edu
NATURE COMMUNICATIONS
| (2021) 12:1442 | https://doi.org/10.1038/s41467-021-21658-7 | www.nature.com/naturecommunications
1
1234567890():,;
S
oliton mode locking in optical microresonators is receiving
intense interest for chip-scale integration of frequency comb
systems
1
. Apart from frequency comb applications, the
microwave signal produced by detection of the microcomb out-
put is, itself, potentially important as a microwave signal source
(see Fig.
1
). However, mode locking of microcombs at microwave
rates is challenging on account of their unfavorable pump power
scaling with repetition rate
2
. Indeed, only ultra-high-Q discrete
silica and crystalline devices were initially able to operate ef
fi
-
ciently at microwave rates
2
5
. Nonetheless, the next generations
of integrated ultra-high-Q resonators are emerging that both
access the microwave-rate realm
6
8
and offer more complete
integrated functionality
8
,
9
. Because of their superior phase noise
performance compared to other miniature photonic microwave
approaches
10
13
, these devices are stimulating interest in minia-
ture stand-alone soliton microwave sources.
While the fundamental limit of phase noise (and equivalently
timing jitter) in the detected soliton pulse stream is induced by
quantum
fl
uctuations
14
,
15
, in practice, phase noise is dominated
by sources of a more technical origin that couple to the soliton
motion in various ways. For example, the Raman self frequency
shift in microcombs
16
,
17
provides a mechanism for transduction
of changes in the detuning frequency (difference in the fre-
quencies of the resonator mode being pumped and the pumping
laser
fi
eld) into the soliton repetition rate
18
. It does this by
causing a frequency shift in the center frequency of the soliton
spectrum (which has an overall sech
2
envelope) as the pump
detuning frequency is varied. Group velocity dispersion then
converts these spectral shifts into changes in the soliton round-
trip propagation time and hence the repetition rate. The Raman
process thereby couples any
fl
uctuation of the resonator fre-
quency (e.g., thermorefractive noise
19
22
) or the pump frequency
into microwave phase noise. Dispersive waves can also induce a
spectral center shift in the Kerr soliton
18
,
23
. 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. And the spectral shift they
induce can offset the Raman self shift. Indeed, when dispersive
wave and Raman shifts are in balance, a
quiet
operating point is
attained whereby coupling of detuning frequency
fl
uctuations
into the soliton repetition rate are greatly reduced
26
.
Here, by investigating possible limits in application of the quiet
operating point, we report the observation of a fundamental noise
source in the soliton repetition rate. Referred to as spatiotemporal
thermal noise, it originates from uncorrelated thermal
fl
uctua-
tions between distinct transverse modes of the microresonator,
and can couple into the soliton repetition rate through the for-
mation of a dispersive wave. Theory and experiment show that
the spatiotemporal thermal noise imposes a considerable limita-
tion on the repetition rate stability of soliton microcombs emit-
ting dispersive waves into spatial mode families, that are distinct
from the soliton-forming mode family. Beyond the study of the
dispersive-wave noise, a convenient way to operate the soliton
microwave source at the quiet point while also disciplining it to
an external reference, such as a clock, is demonstrated.
Results
Soliton generation in silica microresonators
. A silica disk
microresonator with intrinsic
Q
factor exceeding 300 million and
free-spectral-range (FSR) around 15 GHz is used in the study
27
,
28
.
The microresonator is packaged with active temperature stabili-
zation
29
and operated under an acoustic shield to block envir-
onmental perturbations (Fig.
2
a, b). By continuously pumping the
resonator with an ampli
fi
ed
fi
ber laser, bright soliton pulses are
generated, which are further stabilized by servo control of the
pump laser frequency with respect to the average soliton power
30
.
The residual error in the feedback loop is monitored by an
electrical spectrum analyzer. The soliton beatnote is photo-
detected and characterized using a phase noise analyzer and a
frequency counter. The beatnote of the soliton microcomb shows
a 15.2 GHz repetition rate (see Fig.
1
). Its phase noise exhibits a
smooth spectral shape across a wide frequency range as a result of
isolation provided by the package and acoustical shield (Fig.
2
b).
As a benchmark of the stability, the fractional Allan deviation of
the beatnote is plotted in Fig.
2
c and reaches 5.7 × 10
11
at 50 ms
averaging time.
Quiet point operation
. Plotted in Fig.
3
a is a representative
optical spectrum of the soliton microcomb, showing its char-
acteristic sech
2
spectral envelope. Dispersive waves (the spectral
spurs on the envelope) also appear in the spectrum and result
1/
f
rep
PD
a
b
cw laser
microwave (@
f
rep
)
Microresonator
Optical Power
(20 dB per division)
Wavelength (nm)
0
7
5
1
0
3
5
1
Frequency (kHz+15.2 GHz)
RF Power
(20 dB per division)
RBW
10 Hz
1550
Pump
-4
4
0
onator
PD
micr
o
R
1/
f
rep
f
f
Fig. 1 Soliton microcombs as microwave signal sources. a
Apparatus for microwave signal generation using a soliton microcomb. A microresonator
pumped by a continuous-wave (cw) laser emits a repetitive soliton pulse train that is directed into a photodetector (PD) to produce a signal current.
b
Representative optical spectrum of a soliton microcomb with 15.2 GHz repetition rate (left panel). The pump (black dashed line) has been attenuated b
y
an optical notch
fi
lter. The right panel shows the corresponding microwave-rate beat signal with resolution bandwidth (RBW) of 10 Hz.
ARTICLE
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-21658-7
2
NATURE COMMUNICATIONS
| (2021) 12:1442 | https://doi.org/10.1038/s41467-021-21658-7 | www.nature.com/naturecommunications
from frequency degeneracy between comb lines and other
transverse modes, that do not belong to the soliton forming mode
family
3
,
18
,
26
,
31
. It is noted that the spectral envelope center of the
soliton is offset from the pump frequency. This is caused by the
cumulative effect of the Raman-induced soliton-self-frequency-
shift (SSFS)
Ω
Raman
16
18
and dispersive-wave induced spectral
recoil
Ω
recoil
26
,
31
,
32
. The soliton repetition rate,
ω
rep
, is related to
these frequency shifts by
18
,
26
ω
rep
¼
D
1
þ
D
2
D
1
ð
Ω
Raman
þ
Ω
recoil
Þ
;
ð
1
Þ
where
D
1
/2
π
is the FSR and
D
2
is proportional to the group
velocity dispersion (GVD) of the soliton-forming mode
family
2
,
33
. Therefore, through the respective dependence of
Ω
Raman
and
Ω
recoil
on the detuning frequency
δω
=
ω
o
ω
P
(
ω
o
is the frequency of the cold cavity mode being pumped by optical
fi
eld at frequency
ω
P
), the soliton repetition rate becomes a
function of the detuning frequency. As reported in previous lit-
erature, noise in
ω
P
often plays a dominant role in causing
fl
uctuations in
δω
, and subsequently, by way of Eq. (
1
), also in
ω
rep
26
,
34
,
35
. However, it has also been shown that interplay
between Raman SSFS and dispersive-wave induced spectral recoil
can be used to suppress this noise transfer
26
,
35
. Along these lines,
Fig.
3
b is the measured dependence of soliton repetition rate on
detuning
δω
/2
π
, and shows a parabolic-like trend instead of a
monotonic trend. The slope,
β
=
ω
rep
/
δω
, vanishes at around
11.5 MHz detuning, corresponding to the quiet point of operation
where dispersive wave and Raman induced shifts are in balance.
Here, the detuning
δω
is calculated based on Eq. (
23
)in
Methods
section. By operating the soliton microcomb near this
quiet point, the contribution of detuning noise to the soliton
repetition rate noise can be reduced
26
,
35
.
To actively monitor the degree to which the detuning noise
contribution is suppressed through quiet point operation, we
modulate the phase of the pump laser at 10 kHz to create a large
spike in the detuning noise spectrum. This induces calibration
tones in the vicinity of the soliton beatnote
35
, as shown in Fig.
3
c.
Measured phase noise spectra of the detected soliton microwave
signal along with the power of calibration tone (see colored
triangle points) are plotted in Fig.
3
d for different detuning
frequencies. As an aside, the pronounced bump around 20 kHz in
the phase noise spectrum is caused by the piezoelectric tuning
bandwidth of the pump laser. Away from the quiet point, the
phase noise is largest and is found to follow the spectral pro
fi
le of
the detuning noise, which is extracted from the residual error
signal in the locking loop. The contribution of the detuning noise
can be scaled based on the power of calibration tone to determine
its contribution in each measurement. At the quiet point, 36 dB of
noise suppression is measured using the calibration tone. And the
corresponding inferred detuning noise contribution (dashed red
spectrum in Fig.
3
d) is below the actual measured noise spectrum
at the quiet point (purple spectrum in Fig.
3
d). This indicates that
another noise source is limiting the phase noise at the quiet point.
As one possible source of this limit, pump intensity noise could
also couple into the soliton repetition rate through the combined
effect of Kerr and Raman nonlinearity
34
,
35
. However, its
contribution (see dashed gray curve in
fi
gure) is evaluated in
the
Methods
section and appears to be negligible in this
measurement. Figure
3
e gives a comparison of the measured
phase noise reduction (referenced to the highest phase noise
trace) versus the reduction inferred by the calibration tone. A
clear saturation in the measured noise reduction near the quiet
point is shown at several different offset frequencies, suggesting
again that a source of noise is present. The saturation is stronger
at lower offset frequencies indicating that the noise mechanism is
larger at lower frequencies (see Fig.
3
d). 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.
Dispersive-wave induced noise
. Prior analysis of fundamental
sources of repetition rate noise assume that the soliton is formed
and couples solely within a single transverse mode family.
However, the practical need for higher
Q
resonators favors larger
resonator cross-section to minimize the impact of interface and
sidewall roughness
36
. Typically, several transverse modes besides
the soliton forming mode exist in the microresonator. And when
longitudinal modes in these other families experience near
degeneracy with a mode in the soliton, the soliton radiates power
Fractional Allan deviation
Averaging time (s)
10
-9
10
-10
10
-3
10
-2
10
-1
10
0
10
1
PM
EDFA
AOM
c.w. laser
10 kHz
FBG
OSA
Residual error
Microdisk
ESA/Counter
b
a
c
Frequency offset (Hz)
w/ packaging
w/o packaging
SSB phase noise (dBc/Hz)
-20
-40
-60
-80
-100
-120
-140
0
5.7×10
-11
@ 50 ms
10
1
10
2
10
3
10
4
10
5
PD
Acoustic shield
Servo
Fig. 2 Experimental setup and preliminary microwave signal
characterization. a
Experiment setup for soliton generation. PM phase
modulator, EDFA erbium-doped-
fi
ber-ampli
fi
er, AOM acousto-optic
modulator, FBG
fi
ber-Bragg-grating notch
fi
lter, PD photodetector, OSA
optical spectral analyzer, ESA electrical spectral analyzer.
b
Typical single-
sideband (SSB) phase noise spectrum of detected soliton pulse stream
(scaled to 15.2 GHz) obtained using packaged/unpackaged
microresonators. Inset: photo of a packaged microresonator.
c
Fractional
Allan deviation of soliton pulse rate. The errorbar indicates standard
deviation.
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-21658-7
ARTICLE
NATURE COMMUNICATIONS
| (2021) 12:1442 | https://doi.org/10.1038/s41467-021-21658-7 | www.nature.com/naturecommunications
3
creating a dispersive wave (Fig.
4
a)
3
,
18
,
26
,
31
. The radiative power
depends strongly upon the degree of resonance as determined by
Δ
ω
(the frequency difference between the two modes),
Δ
ω
r
(the
frequency difference between the soliton comb line and the
soliton-forming mode with index
r
), and
κ
B
(the optical loss rate
of the dispersive wave mode). The relationship between these
difference frequencies is illustrated in Fig.
4
a. The radiated power
causes a frequency recoil,
Ω
recoil
, in the soliton spectral center
relative to the pump frequency which takes the form
26
Ω
recoil
/
1
ð
Δω
0
Þ
2
½ð
Δω
r

Δω
0
Þ
2
þ
κ
2
B
4

;
ð
2
Þ
where
Δω
0
is the frequency difference between the partially
hybridized crossing mode and the soliton mode, denoted by
Δω
0
¼
Δω
=
2
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δω
2
=
4
þ
G
2
p
(where
G
is the coupling strength
between the soliton and crossing mode).
Δ
ω
r
is determined by
both detuning
δω
and recoil (and thereby
Δ
ω
). And this equation
provides a way for
fl
uctuations in
δω
and
Δ
ω
to impact the
soliton repetition rate. Speci
fi
cally, the resulting
fl
uctuations in
Ω
recoil
cause spectral center
fl
uctuations of the soliton that ran-
domly vary its round trip time as a result of second order dis-
persion. The physical process steps involved in this noise
transduction mechanism are depicted in Fig.
4
b. A transduction
factor
α
ω
rep
/
Δ
ω
relating the repetition rate to changes in
Δ
ω
is de
fi
ned and noted in the
fi
gure. For comparison, the process
steps involved in the transduction of detuning noise into
repetition rate changes (
β
factor de
fi
ned earlier) are also provided.
As noted earlier, detuning noise can be quieted through inter-
ference between the pathways indicated in Fig.
4
b, one of which
uses portions of the dispersive wave recoil process.
To identify the mode families that constitute the soliton
microcomb and the dispersive wave in the experiment, we
perform mode family dispersion spectroscopy using a scanning
external-cavity-diode-laser (calibrated by a separate Mach-
Zehnder interferometer), as shown in Fig.
4
c. Comparing the
measurement with numerical modeling of the modal dispersion,
the mode family that gives rise to the strong dispersive wave in
Fig.
3
a is determined to belong to the TM
4
mode family, while the
soliton is formed on the TM
0
mode family. Their
Q
factors are
also measured, as shown in Fig.
4
d.
Thermal noise in the dispersive wave
. Fluctuations associated
with thermal equilibrium result in spatial and temporal variations
of temperature in the microresonator
19
22
,
37
. Such temperature
fl
uctuations, characterized by a spectral density
S
δ
T
of the modal
temperature
fl
uctuations, induce frequency
fl
uctuations
δ
D
1
in
the resonator FSR through the thermo-optic effect. In turn, this
induces
fl
uctuations in the soliton repetition rate that are char-
acterized by the spectral density
S
δ
D
1
,
S
δ
D
1
¼
n
2
T
n
2
o
D
2
1
S
δ
T
;
ð
3
Þ
10
12
14
Frequency (kHz
+15
219348 GHz)
0
10
20
Quiet point
SSB phase noise (dBc/Hz)
Frequency offset (Hz)
-20
-40
-60
-80
-100
-120
-140
Normalized detuning noise reduction (dB)
010
20
30
40
0
10
20
30
40
Normalized phase noise reduction (dB)
Detuning noise
Fundamental noise
36 dB
Measurement fl
oor
Frequency (kHz+15.219373 GHz)
Calibration
tone
@ 500 Hz
@ 1 kHz
@ 8 kHz
Detuning noise supression
Power (20 dB/div)
RBW
10 Hz
c
b
e
d
From pump
Intensity noise
D
etuni
n
g noise
-160
0
20
a
Power (20 dB/div)
Frequency (THz)
6
9
1
4
9
1
2
9
1
36 dB
Detuning (MHz)
10
1
10
2
10
3
10
4
10
5
Pump
Frequency shift
Ω
Spectral center
Dispersive wave
0
-20
-40
20
40
Fig. 3 Noise spectra near and away from the quiet point. a
Soliton optical spectrum showing spectral envelope (red solid line), the attenuated pump
(black dashed line) and a strong dispersive wave. The spectral center of the soliton (red dashed line) is shifted in frequency relative to the pump freq
uency.
b
Measured soliton repetition rate versus laser-cavity detuning (
δω
/2
π
), where the existence of a quiet point is revealed.
c
Electrical spectrum showing
soliton repetition rate. Two sidebands at 10 kHz offset frequency are induced by phase modulation of the pump and are used to calibrate the contributio
nof
detuning noise.
d
Single-sideband (SSB) soliton microwave phase noise (solid curves) and calibration tone power (triangles) at different detuning
frequencies (indicated by color in accordance with
b
). The optical detuning noise is the blue dotted line. At the quiet operating point, its calibration-inferred
contribution to microwave noise is the dashed red curve. Noise induced by the pump intensity
fl
uctuation (gray dotted line) is also plotted. The phase noise
analyzer instrumental noise
fl
oor is shown as the black line.
e
Plot of actual noise suppression versus calibration tone suppression at several offset
frequencies. The dashed line indicates the expected phase noise suppression if detuning noise is dominant.
ARTICLE
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-21658-7
4
NATURE COMMUNICATIONS
| (2021) 12:1442 | https://doi.org/10.1038/s41467-021-21658-7 | www.nature.com/naturecommunications
with
n
T
the thermo-optic coef
fi
cient and
n
o
the refractive index of
the mode. This noise contribution to the soliton repetition rate,
and that induced by quantum vacuum
fl
uctuations
14
,
15
, are found
to be much smaller than the measured noise in Fig.
3
d. However,
as now shown, thermorefractive noise (TRN) induced in the
modes participating in dispersive-wave emission can be a major
source of repetition rate noise.
From the analysis in the previous section, noise in relative
frequency,
Δ
ω
, will couple to the repetition rate through the
parameter
α
. The TRN induced noise in
Δ
ω
is given by the
following spectral density (see
Methods
section),
S
Δ
ω
¼
n
2
T
n
2
o
ω
2
o
ð
S
δ
T
S
þ
S
δ
T
D

2
R
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
δ
T
S
S
δ
T
D
q
Þ
;
ð
4
Þ
where
δ
T
S
and
δ
T
D
give temperature
fl
uctuations of mode
volumes associated with the soliton and dispersive-wave modes
involved in the de
fi
nition of
Δ
ω
.
R
is a frequency dependent
function discussed in the
Methods
section that accounts for
correlation between the
fl
uctuations
δ
T
S
and
δ
T
D
. This correla-
tion can be modeled using the
fi
nite-element-method (FEM) and
the
fl
uctuation-dissipation theorem (FDT)
21
,
22
,
37
. Simulation
results for different pairs of transverse modes are plotted in
Fig.
5
a. On account of thermal diffusivity, the function
R
decreases rapidly with increasing frequency, so that beyond a
thermal-limited rate the temperature
fl
uctuations of the two
modes become uncorrelated. When this happens, the value of
S
Δω
exceeds
S
δ
D
1
by several orders since it re
fl
ects temperature
fl
uctuations in absolute (as opposed to relative) optical
frequencies.
In order to test the numerical results and parameters used to
simulate these thermally-related quantities
21
,
22
, we measured the
TRN of the soliton mode. The frequency
fl
uctuations of the mode
were tracked by Pound
Drever
Hall (PDH) locking a
fi
ber laser
to a cavity resonance. The locked laser frequency is then
measured using an optical frequency discriminator as described
in the
Methods
section. The measured single-sideband TRN is
plotted in Fig.
5
b, and is in good agreement with the simulation.
The calculated phase noise of the intermode TRN using the
soliton mode and dispersive wave mode is also plotted for
comparison. Suppression of intermode TRN is apparent at low-
offset frequencies relative to the single mode TRN. However, at
higher offset frequencies (above ~1 kHz), the intermode TRN
becomes the summation of TRN contributions belonging to each
mode. The TRN of the FSR is also shown for comparison. Notice
that despite the improved correlation of the intermode TRN at
lower offset frequencies, it still dominates the microwave phase
noise measured in Fig.
3
d, 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.
An additional measurement of soliton microwave phase noise
was performed except using PDH locking of the pump laser to the
resonator as opposed to servo control using soliton power. Under
these conditions, the pump frequency tracks the cavity resonance
thereby suppressing its technical noise contribution to the soliton
Relative mode index
0
150
50
-50
-100
0
(
o
-D
1
)/2
(MHz)
200
c
TRN
TRN
Frequency
Dispersive wave
a
Frequency
Raman self frequency shift
Soliton pulse width
Technical noise
Soliton repetition rate
Source
Channel
b
r
d
-4
0
4
Frequency (MHz)
Normalized Transmission
0
1
Lorentzian fitting
Measurement
Lorentzian fitting
Measurement
-10
0
10
0
1
Q
0
=324 M
Q
C
=737 M
Q
0
=65 M
Q
C
=1152 M
Frequency (MHz)
Soliton mode
Dispersive wave mode
Dispersive wave mode
Soliton mode
TM
0
TM
4
Intermode noise
400
100
-150
Detuning noise (
)
Dispersive wave power
Spectral recoil
Intermode noise (
)
Dispersive wave noise
rep
Fig. 4 Concept of dispersive-wave-induced noise and identi
fi
cation of mode families. a
Spectral relationship of soliton spectrum to dispersive wave
forming mode. Blue lines indicate soliton spectral lines. Red and green shaded regions denote soliton-forming and dispersive-wave resonator modes
,
respectively.
b
Left side: physical steps involved in coupling
fl
uctuations in
Δ
ω
(intermode noise) into soliton repetition rate. Right side: physical steps
involved in coupling
fl
uctuations in
δω
(detuning noise) into soliton repetition rate. Noise sources (top) are transduced (
α
and
β
coupling channels) into the
soliton repetition rate (bottom). Detuning noise results mainly from the pumping laser noise contributing to
δω
and thereby causes technical noise in the
soliton repetition rate. Intermode noise results from fundamental thermo-refractive noise of the dispersive wave and soliton mode frequencies con
tributing
to
Δ
ω
.
c
Measured mode family dispersion of the microresonator. Numerically simulated cross sections of soliton (TM
0
) and dispersive wave (TM
4
)
modes are plotted and identi
fi
ed with the corresponding frequency branches. Orange and green bands (and wavy lines) are suggestive (and highly
magni
fi
ed)
fl
uctuations induced by thermo-refractive noise (TRN).
d
Measured transmission spectra of soliton and dispersive wave resonator modes. The
intrinsic (
Q
0
) and coupling (
Q
C
)
Q
-factors are extracted by
fi
tting the Lorentzian lineshapes and transmission minima.
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-21658-7
ARTICLE
NATURE COMMUNICATIONS
| (2021) 12:1442 | https://doi.org/10.1038/s41467-021-21658-7 | www.nature.com/naturecommunications
5
phase noise. As expected the measured noise spectrum showed a
limitation consistent with the dispersive wave noise (see
Supplementary note II).
A summary of noise contributions to the soliton repetition rate
yields
S
ω
rep
ð
f
Þ¼
α
2
S
Δω
þ
S
δ
D
1
þ
S
Q
þ
β
2
S
δω
þ
S
P
;
ð
5
Þ
where
S
Q
is the quantum noise limit
14
, and
S
P
is noise transferred
from intensity noise of the pump laser. To evaluate the noise
transduction factors, experimental results are
fi
tted with theory
based on the Lugiato
Lefever equation (see
Methods
section).
The Raman frequency shift and dispersive-wave recoil are plotted
in Fig.
5
c, where the error bar is the standard deviation
contributed from
fi
tting of the soliton spectral envelope. The
frequency recoil in Eq. (
2
)is
fi
tted in the same graph to evaluate
mode coupling coef
fi
cients. Noise transduction factors
α
and
β
are then calculated and plotted in Fig.
5
d together with the
measured results. Figure
5
e shows both measured and calculated
phase noise of the soliton repetition rate while operating at the
quiet point. Excellent agreement with the predicted intermode
TRN induced noise is obtained by setting
α
=
24.5 dB
(measurement value), which is close to the theoretical value
α
=
23.6 dB. Other fundamental noise contributions are also
plotted, but are not limiting factors in the current
measurement
14
,
19
22
.
External reference locking at the quiet point
. Most signal
sources provide a feature that allows the oscillator frequency to be
conveniently locked to an external reference such as a clock so as
to provide long term frequency stability
38
. In the present device,
there is a straightforward way to achieve this locking that also
provides
fi
ne tuning control of the microwave frequency near the
quiet point. As a proof of concept, instead of servo controlling
the soliton system by controlling the soliton power
30
, we lock the
soliton repetition rate to a high-performance electrical signal
generator by servo controlling the optical pump frequency. The
resulting soliton beatnote is shown in Fig.
6
a, and can track the
frequency of the microwave source over a 30 kHz range to
achieve
fi
ne tuning control. This range is likely determined by
the soliton existence range, which is, in turn, determined by the
pump laser power
33
. At the same time, the soliton microwave
phase noise, shown in Fig.
6
b, is disciplined to the reference
oscillator within the servo locking bandwidth. The peak around
10 kHz is induced by the servo locking bandwidth. At high-offset
frequencies, the soliton phase noise outperforms the electrical
oscillator (a Keysight PSG) by up to 20 dB. A variation in noise
performance with
fi
ne tuning is apparent with the best perfor-
mance corresponding to operation near the quiet point.
Discussion
. In conclusion, we demonstrated a low-noise 15 GHz
oscillator based on soliton microcombs. The measured phase
noise of
90 dBc Hz
1
at 1 kHz and
140 dBc Hz
1
at 100 kHz
offset frequencies is a record low among existing photonic-chip-
-20
-40
Trasduction factor (dB)
4
1
2
1
0
1
Detuning (MHz)
-30
e
d
Frequency offset (Hz)
-40
-60
-80
-100
-120
-140
SSB phase noise (dBc/Hz)
10
12
14
Detuning (MHz)
Frequency shift (THz)
0
-0.1
-0.2
Recoil
Raman
c
Measurement
Dispersive wave noise
Quantum timing jitter
TRN of
FSR
10
1
10
2
10
3
10
4
10
5
Measurement Theory
Measurement Theory
10
1
Frequency offset (Hz)
10
2
10
3
10
4
10
5
40
0
-40
-80
SSB phase noise (dBc/Hz)
Lo
cking residual
TRN (Measurement)
TRN (Simulation)
b
Intermode TRN (Simulation)
20
-20
-60
-100
TRN of
FSR
(Simulation)
a
10
1
Frequency offset (Hz)
10
2
10
3
10
4
10
5
10
6
1.0
0.4
0.2
0.8
0.6
0
-0.2
Thermal correlation
R
TM
0
& TM
1
TM
0
& TM
2
TM
0
& TM
3
TM
0
& TM
4
1.2
Fig. 5 Intermode thermal noise (between dispersive wave and soliton modes) and its impact on soliton repetition rate. a
Simulated temperature
correlation
R
between transverse mode volumes versus frequency of thermal
fl
uctuation. Speci
fi
c transverse mode pairs are indicated in the legend. Green
region corresponds to
R
< 0.5.
b
Measured and simulated single-sideband (SSB) TRN of a TM
0
mode. The simulated intermode TRN between TM
0
and
TM
4
is also displayed. Green region corresponds to
R
< 0.5 in
a
.
c
Contribution of Raman SSFS and dispersive recoil to total spectral center frequency shift
of the soliton. The error bar indicates standard deviation, and is contributed from
fi
tting of the lineshape.
d
Measured and calculated noise transduction
factors. The error bar indicates standard deviation. The error in detuning is contributed by the lineshape
fi
tting, while the transduction factor error comes
from the signal analyzer.
e
Measured phase noise at maximum quiet point suppression and calculated dispersive-wave induced noise originating from
intermode TRN. Quantum timing jitter and thermorefractive noise (TRN) of the FSR are also plotted for comparison.
ARTICLE
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-21658-7
6
NATURE COMMUNICATIONS
| (2021) 12:1442 | https://doi.org/10.1038/s41467-021-21658-7 | www.nature.com/naturecommunications