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arXiv:1610.08145v1 [physics.optics] 26 Oct 2016
Single-mode dispersive waves and soliton microcomb dynami
cs
Xu Yi
1
,
, Qi-Fan Yang
1
,
, Xueyue Zhang
1
,
2
,
, Ki Youl Yang
1
, and Kerry Vahala
1
,
1
T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA.
2
Department of Microelectronics and Nanoelectronics, Tsinghua Un
iversity, Beijing 100084, P. R. China
These authors contributed equally to this work.
Corresponding author: vahala@caltech.edu
(Dated: October 27, 2016)
Dissipative Kerr solitons are self-sustaining optical wav
epackets in resonators. They use the Kerr
nonlinearity to both compensate dispersion and to offset opt
ical loss. Besides providing insights
into nonlinear resonator physics, they can be applied in fre
quency metrology, precision clocks, and
spectroscopy. Like other optical solitons, the dissipativ
e Kerr soliton can radiate power in the
form of a dispersive wave through a process that is the optica
l analogue of Cherenkov radiation.
Dispersive waves typically consist of an ensemble of optica
l modes. A limiting case is demonstrated
in which the dispersive wave is concentrated into a single ca
vity mode. In this limit, its interaction
with the soliton is shown to induce bistable behavior in the s
pectral and temporal properties of the
soliton. Also, an operating point of enhanced repetition-r
ate stability is predicted and observed.
The single-mode dispersive wave can therefore provide quie
t states of soliton comb operation useful
in many applications.
A new dissipative soliton
1
has recently been observed
in optical resonators. These dissipative Kerr solitons
(DKS) have been demonstrated in fiber resonators
2
and in various microcavity systems
3–7
. In microcomb
research
8,9
soliton formation produces phase-locked spec-
tra with reproducible envelopes, as required in frequency
comb applications
10–14
. Moreover, their unusual proper-
ties and interactions create a rich landscape for research
in nonlinear optical phenomena
5,15–24
. Two such phe-
nomena, the Raman-induced soliton-self-frequency-shift
(SSFS) and dispersive-wave generation, are important to
this work.
The Raman SSFS causes a spectral red shift of the
soliton. In optical fiber systems, this shift continuously
increases with propagation distance
25,26
, however, in mi-
croresonators the shift is fixed and depends upon soli-
ton power
17,18,27,28
. Dispersive waves also occur in op-
tical fiber systems
29
. They are formed when a soliton
radiates into a spectral region of normal dispersion
5,30
.
This radiation, which can also be engineered to occur
through spatial mode interactions
19,31
, can be under-
stood as the optical analog of Cherenkov radiation
30
.
Dispersive waves provide a powerful way to spectrally
broaden a soliton within a microresonator as a precursor
to self referencing
12,32
. Their formation also induces soli-
ton recoil
30
which, similar to SSFS, causes a frequency
shift in the spectral center of the soliton
5,17
.
In microcavities, the Raman SSFS enables controlled
tuning of the emission wavelength of a dispersive wave
by varying the cavity-pump detuning frequency
19
(
δω
ω
0
ω
p
where
ω
p
is the pump frequency and
ω
0
is the fre-
quency of the cavity mode that is being pumped). This
happens because the DKS repetition rate is coupled to
δω
by the Raman SSFS
18,19
. As the repetition rate varies
with
δω
, the phase matching condition of the soliton to
the dispersive wave is also varied, thereby tuning the dis-
persive wave. Measurements of this coupling have been
used to determine the dispersion of the soliton-forming
mode family
19
.
Dispersive waves normally consist of an ensemble of
modes. However, a single cavity mode can exhibit behav-
ior similar to that of a dispersive wave as a result of an
avoided-mode crossing
33
. In this work, an avoided-mode
crossing is used to excite a dispersive wave consisting of a
single cavity mode. The coupling of the
single-mode dis-
persive wave
to the soliton is strongly influenced by the
total soliton frequency shift produced by the combined
Raman-induced SSFS and the dispersive-wave recoil.
The combination is shown to induce hysteresis behavior
in soliton properties. Included in this behavior, there
is an operating point of improved pulse-rate stability
wherein reduced coupling of repetition rate and cavity-
pump detuning occurs. Pulse-rate stability is centrally
important in many frequency comb applications
10,13,34
and the fundamental contributions to phase noise in the
pulse train have been considered theoretically
35
. Techni-
cal noise mechanisms are also present. For example, DKS
generation using on-chip silica resonators exhibits phase
noise that tracks in spectral profile the phase noise of
the optical pump
4
. The quiet operation point is shown
to reduce these noise contributions in the soliton pulse
repetition rate.
In what follows both the hysteresis and the regime of
improved stability (quiet point) are measured and then
modeled theoretically.
Experiment and model.
A silica whispering-gallery
resonator
36
is used for soliton generation. The devices
featured a free-spectral-range (FSR) of approximately
22 GHz (3 mm diameter resonator), intrinsic Q-factors
around 250 million and they support multiple, trans-
verse mode families. In order to characterize the fre-
quency spectrum of the soliton-forming mode family,
mode frequencies were measured from 190.95 THz (1570
nm) to 195.94 THz (1530 nm) using an external-cavity
diode laser (ECDL) calibrated by a fiber Mach-Zehnder
2
Dispersive wave power (
μ
W)
Total frequency shift (THz)
Recoil (THz)
Dispersive wave power (
μ
W)
Mode number
μ
190
192
194
196
0
10
20
30
40
Power (
μ
W)
Frequency (THz)
(ω−ω
0
−μ
D
1
)/2
π
(MHz)
-100
0
100
0
100
200
(a)
(b)
(c)
(d)
-100
0
100
-100
μ
=r
Operating Point I
Operating Point II
Comb frequencies
∆ω
μ
,comb
Measurement
Analytical model
II
II
I
I
15
20
25
30
35
40
45
50
0
20
40
Detuning (MHz)
Detuning (MHz)
0.0
-0.2
-0.4
-0.6
-0.8
Measurement
Analytical model
Raman
Recoil+Raman
0.0
-0.2
-0.1
0
10
20
30
40
50
Soliton mode
∆ω
μ
A
Single-mode
dispersive waves
15
20
25
30
35
40
45
50
Pump
10
30
50
D
1
=22 GHz
D
2
=17 kHz
Crossing mode
∆ω
μ
B
Measurement
Eq.(5)
FIG. 1:
Soliton hysteretic behavior induced by mode-interaction.
a,
Measured relative mode frequencies
(blue points)
4
. The green and yellow dashed lines represent the fitted unperturb
ed soliton-forming mode family A
and crossing mode family B, respectively. The red line illustrates the f
requencies of a hypothetical soliton frequency
comb. A non-zero slope on this line arises from the repetition rate ch
ange relative to the FSR at mode
μ
= 0.
b,
Measured soliton optical spectra at two operating points corresp
onding to closely matched cavity-pump detuning
frequencies,
δω
. A strong single-mode dispersive wave at
μ
= 72 is observed for operating point II, resulting in a
recoil of the soliton central frequency beyond that resulting fro
m the Raman-induced SSFS.
c,d,
Dispersive-wave
power and soliton spectral center frequency shift versus cavity
-pump detuning. Inset in 1c: Measured (blue dots)
and theoretical (red line) recoil frequency versus the spur powe
r.
interferometer (MZI)
4
. The measured
relative
-
mode
-
f requency
, ∆
ω
μ
ω
μ
ω
o
μD
1
versus mode index,
μ
, is
presented in fig. 1a. In this expression,
ω
μ
is the resonant
frequency of the
μ
-th mode,
D
1
is the FSR of the soliton-
forming mode family at
μ
= 0 (assumed to be the pump-
ing mode index), and
ω
0
is the frequency of mode
μ
= 0.
The dispersion of a mode family can be characterized by
expanding the mode frequencies using Taylor series at
the pumped mode (
μ
= 0),
ω
μ
=
ω
0
+
D
1
μ
+
1
2
D
2
μ
2
+
...
,
where
D
2
is the second order dispersion at
μ
= 0.
DKSs form when the pump laser is red-detuned rela-
tive to the pump mode frequency and details on their
generation in these resonators are given elsewhere
4,37
.
The soliton-forming mode family (family A in fig. 1a)
must have anomalous dispersion near
μ
= 0 and be
relatively free of avoided crossings with other trans-
verse mode families
38
. In fig. 1a, the former condi-
tion is met by the relative mode frequency spectrum
being parabolic with postive curvature near
μ
= 0. A
green, dashed parabolic curve (∆
ω
μ
=
1
2
D
2
μ
2
) is pro-
vided in the figure to verify this requirement. Concern-
ing the latter condition, a second mode family (family B
in fig. 1a) causes an avoided mode crossing near
μ
= 72
and hybridization of the mode families occurs near the
crossing
19,39
. The relative-mode-frequency of the unper-
turbed soliton-forming mode family and crossing mode
family are denoted as ∆
ω
μA
and ∆
ω
μB
. The lower
branch of the hybrid mode family formed by the avoided
mode crossing is denoted ∆
ω
μ
. Avoided mode crossing
behavior has been intensively studied in the context of
DKS solitons. Generally they can interfere with soliton
formation
38,40,41
, however, avoided crossings also provide
a wave to induce dark soliton formation
16
. In the present
system the avoided-mode-crossing induces only minimal
distortion in the otherwise parabolic shape of the soliton-
forming mode family. Soliton spectra produced on this
mode family by pumping at
μ
= 0 are shown in fig. 1b
along with theoretical sech
2
spectral envelopes predicted
for DKSs. As an aside, the horizontal scales in fig. 1a
and fig. 1b are the same and the location of the
μ
= 0
pumping mode is indicated by a vertical dashed line in
fig. 1b.
Also shown in fig. 1a are the comb frequencies associ-
ated with a fictitious soliton spectrum. This line is given
by,
ω
μ,
comb
= (
ω
rep
D
1
)
μ
δω
(1)
where
ω
rep
is the soliton repetition frequency
19
. (The fre-
quency components of the soliton comb are red-detuned
relative to the “cold-cavity” mode frequencies by the
Kerr nonlinearity.) In the vicinity of the avoided cross-
3
i
ii
iii
iv
i
ii
iii
iv
i
i
ii
ii
iii
iii
iv
iv
(a)
(b)
(c)
(d)
Total frequency shift (THz)
20
30
40
50
Detuning
δω
(normalized to
κ
A
/2)
40
30
20
Detuning
δω
(normalized to
κ
A
/2)
-0.2
-0.4
-0.6
-0.8
0.08
0.00
0.02
0.04
0.06
Dispersive wave power |h
r
|
2
(normalized to soliton power)
Numerical simulation
Analytical model
0
0.04 0.08
Recoil (THz)
0
-0.1
-0.2
-0.3
|h
r
|
2
Optical power
(20 dB / div)
mode number
μ
0
100
-100
mode number
μ
0
100
-100
mode number
μ
0
100
-100
mode number
μ
0
100
-100
Intracavity
power (a.u.)
0
1
2
Time (T
R
)
0
0.1
-0.1
Time (T
R
)
0
0.1
-0.1
Time (T
R
)
0
0.1
-0.1
Time (T
R
)
0
0.1
-0.1
Numerical
simulation
Analytical
model
FIG. 2:
Numerical simulation and analytical model of single-mode d
ispersive wave generation and re-
coil. a,
Numerical (blue dots) and analytical (red solid line) soliton total fre
quency shift versus cavity-pump de-
tuning.
b,
Numerical (blue dots) and analytical (red solid line) dispersive wave p
ower (normalized to total soliton
power) versus cavity-pump detuning. Inset: recoil frequency v
ersus the dispersive wave power.
c,
Optical spectra in
the two mode families (blue: soliton forming mode family A; red: crossin
g mode family B).
d,
Time domain intra-
cavity power.
T
R
is the cavity round trip time.
ing it is possible for a spectral component of the soliton,
μ
= r with relative mode frequency ∆
ω
r
,
comb
, to couple
resonantly with a mode in the lower branch (relative fre-
quency ∆
ω
r
). The hybrid mode obeys the simple driven
oscillator equation,
dh
r
dt
= [
i
ω
r
κ
r
2
]
h
r
+
f
r
e
i
ω
r
,
comb
t
(2)
where
h
r
is the intracavity field amplitude of the mode,
κ
r
is its loss rate and
f
r
is an effective pumping term
associated with the soliton comb line at relative fre-
quency ∆
ω
r
,
comb
. The pumping term is given by
f
r
=
i
Γ(∆
ω
r
A
ω
r
,
comb
)
a
r
, where
a
r
is the field of the unper-
turbed soliton hyperbolic solution at
μ
= r (see Methods
section). Also, the Kerr effect self-frequency shift of
h
r
is of order 10 kHz and is therefore negligible in compari-
son to
κ
r
.
Because the lower-branch mode at
μ
= r has a high
optical Q factor, slight shifts in the slope of the comb
frequency line (equivalently, shifts of ∆
ω
r
,
comb
relative to
ω
r
) will cause large changes in the power coupled to
the mode. These changes are observable in fig. 1b where
a strong spectral line appears in the case of the blue soli-
ton spectrum. Note that scattering from the soliton into
the spectral line is strong enough so that the power in the
line is greater than the comb line power near the spec-
tral center of the soliton, itself. The strong spectral line
can be understood as a
single-mode dispersive wave
and
it induces a recoil in the spectral center of the soliton.
This recoil contribution is indicated for the blue soliton
spectrum in the figure. In the case of the red soliton
spectrum, the operating point was changed and the res-
onance between the soliton and the mode is diminished.
Accordingly, most of the spectral shift in this case results
from the Raman SSFS.
A change in the slope of the soliton comb line will occur
when the soliton repetition frequency,
ω
rep
, is changed
(see eq. (1)). On account of second order dispersion
ω
rep
4
10
1
10
2
10
3
10
4
10
5
10
6
10
7
-120
-80
-40
0
40
Phase noise (dBc / Hz)
Offset frequency (Hz)
(a)
(b)
(c)
I
20
30
40
50
0
-200
-400
-600
RF - offset (kHz)
Detuning (MHz)
II
III
Measurement
Analytical model
Phase noise at
25 kHz offset (dBc/Hz)
20
30
40
50
-140
-120
-100
-80
Detuning (MHz)
-60
I
II
III
δω
δω
noise floor
15
25
35
45
FIG. 3:
Soliton repetition frequency and phase noise measurement.
a,
Measured and theoretical soliton
repetition frequency versus pump-cavity detuning. The offset fr
equency is 22.0167 GHz. Operating points I, II and
III refer to 3b. Point III is near the quiet operation point.
b,
Phase noise spectra of detected soliton pulse stream
at three operating points shown in 3a.
c,
Phase noise of soliton repetition rates at 25 kHz offset frequency p
lotted
versus the cavity-pump detuning. The blue and red dots (lines) den
ote the experimental (theoretical) phase noise of
the upper and lower branch operating points, respectively.
depends linearly on the frequency offset of the soliton
spectral maximum relative to the pump frequency
19,35
.
This frequency offset has contributions from both the
Raman SSFS, Ω
Raman
, and the dispersive-wave recoil,
Recoil
(i.e., Ω = Ω
Raman
+ Ω
Recoil
). Accordingly, the
soliton repetition rate is given by,
ω
rep
=
D
1
+
D
2
D
1
(Ω
Raman
+ Ω
Recoil
)
(3)
where
D
2
is the second order dispersion of soliton-
forming mode family at
μ
= 0 and from fig. 1a is mea-
sured to be 17 kHz. Substituting for the repetition rate
in the comb line expression (eq. (1)) gives,
ω
μ,
comb
=
μD
2
D
1
(Ω
Raman
+ Ω
Recoil
)
δω
(4)
The recoil frequency has a linear dependence on the
power of the hybrid mode (see eq.(
25
) in Methods),
Recoil
=
γ
|
h
r
|
2
=
r
κ
B
D
1
κ
A
E
|
h
r
|
2
,
(5)
where
κ
A
and
κ
B
denote the loss rates of the family A
and family B modes, respectively, and
E
is the circulating
soliton energy. Solving eq. (2) for the steady-state power
in the crossing mode at the soliton comb line frequency
and using eq. (4)-(5) gives the following,
|
h
r
|
2
=
|
f
r
|
2
(∆
ω
r
+
δω
r
D
2
D
1
[Ω
Raman
+
γ
|
h
r
|
2
])
2
+
κ
2
r
4
.
(6)
Eq. (6) suggests that a hysteresis in the dispersive-
wave power is possible when varying the soliton operat-
ing point. Consistent with this possibility, it is noted
that the two soliton spectra in fig. 1b (blue and red),
which show very different dispersive wave powers, were
produced at nearly identical detuning frequencies,
δω
.
A more detailed survey of the dispersive wave power
behavior is provided in fig. 1c and is again consis-
tent with a hysteretic behavior versus detuning. Also,
since the total spectral shift of the soliton is given by
Ω = Ω
Raman
+Ω
Recoil
= Ω
Raman
+
γ
|
h
r
|
2
a corresponding
hysteresis is observed in the overall soliton spectral shift
(fig. 1d). Theoretical fits are provided in fig. 1c and fig.
1d using eq. (6) (see Methods for the fitting procedure
and parameter values). In plotting the data, the determi-
nation of the detuning frequency,
δω/
2
π
, was made from
the measured total soliton spectral shift (Ω) and pulse
width (
τ
s
) using the relation
δω
=
D
2
/
2
D
2
1
(1
2
s
+ Ω
2
)
(eq.(
33
) in Methods)
18
.
The recoil frequency, Ω
Recoil
, can also be extracted
from the data to verify its linear dependence upon dis-
persive wave power predicted in eq.(
5
). To do this, the
Raman SSFS, Ω
Raman
, is determined from the soliton
5
pulse width using eq.(
26
) in Methods and substracted
from the measured total soliton frequency shift. A plot
of the recoil shift versus the spur power is given as the
inset in fig. 1c and verifies the linear dependence. Eq.(
5
)
is also plotted for comparison using parameters given in
the Methods Section.
While the present results are produced using a dis-
persive wave that is blue-detuned relative to the soliton
spectral maximum, the hysteresis behavior also occurs
for a red-detuned dispersive wave. However, in the red-
detuned case, the orientation of the curve in fig. 1c is
reversed with respect to the detuning frequency. The es-
sential feature for appearance of the hysteresis is that the
recoil advances and retreats versus detuning. During a
portion of this behavior, it is therefore possible to com-
pensate the Raman SSFS. The requirements imposed on
the device and mode crossing for this to occur are dis-
cussed below (see section on Existence of quiet point).
Numerical simulation.
To attribute the observed
frequency shift hysteresis to the single-mode dispersive
wave, and to further validate the analytical model, we
perform numerical simulations based on the coupled
Lugiato-Lefever equations
31,42–45
involving the soliton-
forming mode family (family A) and the crossing-mode
family (family B). Further information including param-
eter values is provided in Methods, but is outlined here.
The two mode families are coupled using a model stud-
ied elsewhere
19
. The coupling is characterized by a rate
constant
G
and is designed to induce an avoided-mode-
crossing around mode index
μ
= 72, similar to the ex-
perimental mode family dispersion. Fig. 2 shows the re-
sults of the numerical simulation including 2048 modes.
The hysteresis in the soliton total frequency shift and the
dispersive wave power resembles the experimental obser-
vation and is also in good agreement with the analytical
model (see fig. 2a,b). As predicted by the eq. (
5
) (and
observed in the fig. 1c inset), the recoil is numerically
predicted to vary linearly with the dispersive wave power
(fig. 2b inset).
Frequency and time domain features of the soliton
(blue) and dispersive wave (red) are also studied in fig.
2c,d. They show that the dispersive wave emerges on the
crossing-mode family (family B) and consists primarily
of a single mode. The single-mode dispersive wave leads
to a modulated background field in the resonator with
a period determined by the beating between the pump
and the dispersive wave. This modulation is observable
in fig. 2d. Spectral recoil of the soliton is also observable
in the numerical spectra.
Soliton repetition rate quiet point.
The nonlinear
behavior associated with soliton coupling to the single
mode dispersive wave can be used to suppress soliton
repetition rate noise produced by coupling of pump laser
noise. This noise source is suspected to be a significant
contributor to repetition rate noise in certain frequency-
offset regimes
4
. From eq. (
3
) the repetition frequency
depends linearly on the total soliton spectral-center fre-
1.4
1.2
1
0.8
0.6
0.4
Quiet Point Existence
Quiet Point Absence
|Recoil slope/Raman slope|max
1
2
3
4
5
10
20
30
Crossing mode loss
κ
B
(normalized to
κ
A
)
Coupling rate G (normalized to
κ
A
)
FIG. 4:
Existence study for the quiet point.
The
maximum ratios of
|
Recoil
/∂δω
|
to
|
Raman
/∂δω
|
at
varying normalized modal coupling rate
G
(see Meth-
ods) and normalized crossing-mode damping rate
κ
B
.
The quiet point exists when this ratio is greater than
unity (red region).
quency shift. However, this total shift frequency versus
cavity-pump detuning has a stationary point on the up-
per hysteresis branch (see fig. 1d). As expected from the
simple dependence in eq. (
3
), this same stationary point
is observed in measurements of the repetition frequency
versus detuning (fig. 3a). To measure the repetition fre-
quency the soliton pulse train is directly detected and an
electrical spectrum analyzer is used to observe the pulse
train spectrum. The theoretical prediction using analysis
from the Methods is also provided for comparison.
The coupling of pump laser frequency noise into the
soliton repetition rate is expected to be minimal at the
stationary point. To verify this prediction, the phase
noise of the detected soliton pulse train is measured at
different soliton operating points on the upper and lower
branches in fig. 3a using a phase noise analyzer. Phase
noise spectra corresponding to operating points I, II and
III in fig. 3a are plotted in fig. 3b. Operating points I
and II correspond to nearly identical cavity-pump detun-
ing, but lie on different branches. As expected, operating
point II in the upper branch has a lower phase noise level
compared to operating point I on account of its reduced
slope. Operating point III is close to the zero-slope de-
tuning point in the upper branch. This
quiet point
has
the lowest phase noise among the recorded phase noise
spectra.
For comparison, the phase noise associated with the
detuning frequency
δω
was also measured. For this mea-
surement, the error signal of a Pound-Drever-Hall feed-
back control system is operated open-loop and recorded
using an oscilloscope. Its power spectral density is con-
6
verted into phase noise in fig. 3b (see Supplementary
Material). The relatively high noise floor in this measure-
ment is caused by the oscilloscope sensitivity. Nonethe-
less, a noise bump at 25 kHz offset frequency originates
from the laser and provides a reference point against
which comparison to the soliton phase noise is possible.
The soliton phase noise at 25 kHz offset frequency noise
is plotted versus detuning in fig. 3c. The calculated soli-
ton phase noise is also presented for comparison using
the cavity-pump detunining noise level at 25 kHz offset.
The dip of the phase noise occurs at the quiet point. For
lower offset frequencies, the contributions to noise are be-
lieved to originate from thermal contributions within the
resonator and are under investigation. Nonetheless, the
measured noise contributions at these frequencies show a
trend of reduction for operation at the quiet point.
An analytical study comparing the detuning response
of the Raman and recoil effects was performed to deter-
mine conditions required to observe the quiet point. The
quiet point occurs when the retreating soliton recoil bal-
ances the always advancing SSFS. Accordingly, fig. 4 is
a contour plot of the maximum ratio of
|
Recoil
/∂δω
|
to
|
Raman
/∂δω
|
while varying the coupling strength be-
tween the soliton-mode and crossing-mode families (see
Supplementary Material) and the damping rate of the
crossing mode. The existence regime for observation of
the quiet point corresponds to the ratio
>
1 shown in
red. Stronger mode interaction and weaker dissipation
are required to operate in this regime.
Discussion
In summary, soliton coupling to a dispersive wave con-
sisting of a single mode was studied experimentally and
theoretically and shown to produce a hysteretic depen-
dence of soliton properties upon cavity-pump detuning.
These properties include the frequency shift of soliton
spectral center relative to the pumping frequency, the
soliton repetition frequency, and the optical power in the
single-mode dispersive wave. The accompanying nonlin-
ear response of repetition rate with detuning frequency
was shown to create a condition (quiet operating point)
where coupling of laser pump frequency noise into the
soliton repetition rate is greatly reduced. This reduction
was measured by characterizing the soliton pulse-stream
phase noise upon photo detection. The requirements for
quiet point existence were also studied. Stronger mode
interactions and reduced crossing-mode damping are pre-
ferrable. The operating point for quiet soliton operation
holds potential for ultra-low-noise microwave generation.
Methods
Dynamical equation of hybrid mode.
Equation (2) can
be derived from coupled mode equations that include disper-
sion, mode interaction and the Kerr nonlinearity. The intra
cav-
ity field of mode
μ
in the soliton-forming mode family A can
be represented by
A
μ
(
t
)
e
μA
t
+
iμφ
, where
A
μ
(
t
) is the slowly
varying amplitude,
t
is the time and
φ
is the azimuthal angle
along the resonator. In the rotation frame of comb frequenci
es
ω
0
δω
+
μω
rep
for
μ
, the intracavity field can be expressed as
a
μ
(
t
) =
A
μ
(
t
)
e
i
(
ω
μA
ω
0
+
δω
μω
rep
)
t
. We denote the intracavity
field in the crossing-mode family B as
b
μ
and express it in the same
reference frame as the soliton-forming mode
a
μ
. The intracavity
fields can be calculated using the equations of motion with Ke
rr
nonlinearity terms
44,46
and modal-coupling terms
39
,
da
μ
dt
=
[
κ
A
2
+
i
(
ω
μA
ω
0
+
δω
μω
rep
)
]
a
μ
+
iGb
μ
+
ig
j,k
a
j
a
k
a
j
+
k
μ
+
F δ
(
μ
)
(7)
db
μ
dt
=
[
κ
B
2
+
i
(
ω
μB
ω
0
+
δω
μω
rep
)
]
b
μ
+
iGa
μ
+
ig
B
j,k
b
j
b
k
b
j
+
k
μ
(8)
where
κ
A,B
=
ω
0
/Q
A,B
is the dissipation rate;
g
=
~
ω
2
0
n
2
D
1
/
2
πn
0
A
eff
represents the normalized Kerr nonlinear co-
efficient where
A
eff
is the effective nonlinear mode area and
g
B
is defined in the same way.
G
is the linear coupling coefficient be-
tween the two mode families
19
and
F
is the coupled laser pump
field. Also, to calculate eq. (2) it is not necessary to includ
e Ra-
man coupling terms in eq.(
7
) and eq.(
8
) since the leading-order
contribution to the forcing term,
f
r
, is from the Kerr nonlinearity.
Modal coupling forms two branches of hybrid modes measured
in the mode spectrum (fig.1a). The frequency of the hybrid mod
es
in the upper (+) and lower (-) branches is given by
39,47,48
ω
μ
±
=
ω
μA
+
ω
μB
2
±
G
2
+
1
4
(
ω
μA
ω
μB
)
2
(9)
where the corresponding field amplitude of the hybrid modes i
s a
linear combination of
a
μ
and
b
μ
. In the far-detuned regime where
ω
μA
ω
μB
G
, the field amplitude of the lower branch hybrid
mode is approximately given by,
̃
h
μ
=
Ga
μ
+ (
ω
μA
ω
μB
)
b
μ
G
2
+ (
ω
μA
ω
μB
)
2
.
(10)
In this experiment, only one mode was observed to be near reso
-
nance with the soliton comb and that mode is assigned mode ind
ex
μ
=
r
. Also, as the amplitude of
b
μ
with
μ
6
=
r
is small, the
Kerr interaction summation term can be neglected in eq.(
8
) in this
calculation.
By taking the time derivative of eq. (
10
) and then substituting
using (
7
) and (
8
) the following dyanamical equation results for
̃
h
μ
±
,
d
̃
h
r
dt
=
[
κ
r
2
+
i
(
ω
r
ω
0
+
δω
r
ω
rep
)
]
̃
h
r
+
f
r
(11)
where
f
r
is the pumping term given by,
f
r
=
i
Γ
g
j,k
a
j
a
k
a
j
+
k
r
.
(12)
and where Γ =
G/
|
G
|
2
+
|
ω
μA
ω
μB
|
2
is the weight of the fam-
ily A mode in
̃
h
μ
and
κ
r
κ
B
for
r
when Γ
1. Also,
consistent with fig. 1a, the hybridization of mode r is assume
d
weak (i.e.,
|
ω
rA
ω
rB
| ≫ |
G
|
and
|
ω
rA
| ≫ |
ω
rB
|
) so that
b
r
is
the dominant contribution to
̃
h
r
. When converting eq. (
11
) into
the rotation frame of (
ω
0
+
μD
1
) with
̃
h
r
=
h
r
e
i
ω
r,
comb
t
, the
following expression results,
dh
r
dt
= [
i
ω
r
κ
r
2
]
h
r
+
f
r
e
i
ω
r,
comb
t
(13)
where ∆
ω
r
=
ω
r
ω
o
μD
1
is the relative-mode-frequency of
hybrid mode
h
r
.
Equation (
13
) is identical to eq.(2) in the main
text.
Effective pumping term.
The pumping term in eq.(
11
) can be
expressed in parameters of the resonator and soliton. The so
liton
field envelope takes the form
3,18
A
(
φ, t
) =
B
s
sech[(
φ
φ
c
)
/D
1
τ
s
]
e
i
Ω(
φ
φ
c
)
/D
1
+
(14)
7
where soliton properties are: amplitude
B
s
, angular position
φ
c
,
temporal pulse width
τ
s
, spectral-center frequency shift (relative
to pump) Ω and phase relative to the pump laser
φ
. Also, this
solution assumes
δω
κ
A
. By applying the Fourier transform to
A
(
φ, t
),
a
μ
can be expressed in terms of the soliton properties,
A
(
φ, t
) =
μ
a
μ
(
t
)
e
(
φ
φ
c
)
,
(15)
a
μ
=
B
s
τ
s
D
1
2
sech(
πτ
s
2
(
D
1
μ
Ω))
e
.
(16)
The pump
f
r
can therefore be derived by inserting eq.(
16
) into
eq.(
12
). The following expression results from simplification of t
he
summation,
f
r
=
i
Γ
D
2
4
D
2
1
[(
D
1
r
Ω)
2
+
1
τ
2
s
]
B
s
τ
s
D
1
sech(
πτ
s
2
(
D
1
r
Ω))
e
(17)
where
g
has been replaced using equation
B
2
s
τ
2
s
=
D
2
/gD
2
1
, which
holds for DKSs
18,35
and is also verified in a section below. Finally,
by using
18
δω
=
D
2
2
D
2
1
(
1
τ
2
s
+ Ω
2
) (see derivation below),
f
r
can be
further reduced to
f
r
=
i
Γ(∆
ω
rA
ω
r,
comb
)
a
r
.
(18)
Recoil and Soliton Self Frequency Shift.
In addition to the
Raman SSFS
17,18
, the spectral center of the DKS can also be
shifted by the single line dispersive wave recoil. The effect
of the
recoil and Raman shift can be calculated using the moment ana
ly-
sis method
17,49
. Using the Fourier transform, eq.(
7
) is transformed
into the perturbed Lugiato-Lefever equation (LLE)
44
∂A
(
φ, t
)
∂t
=
(
κ
A
2
+
iδω
)
A
+
i
D
2
2
2
A
∂φ
2
+
F
+
ig
|
A
|
2
A
+
igτ
R
D
1
A
|
A
|
2
∂φ
+
iGB,
(19)
where the Raman shock term has been added
17,18
and
τ
R
is the
Raman time constant. The moment analysis method treats the
soliton as a particle. The energy E and the spectral center mo
de
number
μ
c
are given by,
E
=
μ
|
a
μ
|
2
=
1
2
π
+
π
π
|
A
|
2
d
φ
=
B
2
s
τ
s
D
1
(20)
μ
c
=
μ
μ
|
a
μ
|
2
E
=
i
4
πE
+
π
π
(
A
∂A
∂φ
A
∂A
∂φ
)d
φ.
(21)
Taking the time derivative of eq.(
21
) and substituting
∂A/∂t
using
eq.(
19
), the following equation of motion for
μ
c
is obtained,
∂μ
c
∂t
=
κ
A
μ
c
R
D
1
2
πE
+
π
π
(
|
A
|
2
∂φ
)
2
d
φ
1
2
πE
+
π
π
(
G
B
∂A
∂φ
GA
∂B
∂φ
)d
φ.
(22)
The second term on the right-hand-side corresponds to the Ra
man-
induced frequency shift and the third term is the frequency s
hift
caused by recoil.
The Raman term can be calculated by substituting eq.(
14
) into
the integral. When calculating the recoil term,
B
is simplified to
B
b
r
e
ir
(
φ
φ
c
)
as the power in mode
B
is dominated by the near
resonance mode
r
. In addition, because the integral of
φ
is over 2
π
,
only
a
r
e
ir
(
φ
φ
c
)
has nonzero contribution. Furthermore, equation
(
8
) is used to relate
Ga
r
to
b
r
and finally leads to,
∂μ
c
∂t
=
8
τ
R
D
2
15
D
3
1
τ
4
s
B
E
|
b
r
|
2
κ
A
μ
c
,
(23)
The steady-state spectral center mode number is therefore g
iven
by,
μ
c
=
8
τ
R
D
2
15
κ
A
D
3
1
τ
4
s
B
κ
A
E
(1
Γ
2
)
|
h
r
|
2
=
1
D
1
(Ω
Raman
+ Ω
Recoil
)
,
(24)
where
|
ω
μA
ω
μB
| ≫
κ
B
,
ω
r
(equivalent to
|
b
r
| ≫ |
a
r
|
) is
assumed and the recoil and Raman shifts are,
Recoil
=
γ
|
h
r
|
2
=
B
D
1
κ
A
E
(1
Γ
2
)
|
h
r
|
2
,
(25)
Raman
=
8
τ
R
D
2
15
κ
A
D
3
1
τ
4
s
(26)
where in the main text, Γ
2
1 is assumed.
Eq.(
25
) is eq.(
5
) in the
main text.
The form for the Raman SSFS, Ω
Raman
, is identical to
the form previously derived in the absence of the dispersive
-wave
coupling
18
.
Soliton parameters with Raman and mode-coupling ef-
fects.
In the presence of recoil and Raman, the relations between
soliton parameters in eq.(
14
) can be derived from the Lagrangian
approach
3,18,35
. In addition, the Lagrangian approach verifies the
expression for Ω
Recoil
obtained above as well as providing a path
for calculation of the repetition-rate phase noise
35
. As detailed
in previous literature
18,35
, the perturbation Lagrangian method is
applied to the LLE equation of
A
(eq.
19
). However, now an addi-
tional perturbation term is added to account for the mode cou
pling
to the crossing-mode family. Taking
B
b
r
e
ir
(
φ
φ
c
)
, produces the
following equations of motion,
D
1
∂φ
c
∂t
∂φ
∂t
δω
D
2
2
2
D
2
1
D
2
6
τ
2
s
D
2
1
+
2
3
gB
2
s
= 0
(27)
D
1
∂φ
c
∂t
∂φ
∂t
δω
D
2
2
2
D
2
1
+
D
2
6
τ
2
s
D
2
1
+
1
3
gB
2
s
= 0
(28)
(
B
2
s
τ
s
Ω)
∂t
=
κ
A
B
2
s
τ
s
8
R
B
4
s
15
τ
s
κ
B
πr
|
b
r
|
2
(29)
∂φ
c
∂t
=
D
2
D
1
(30)
(
B
2
s
τ
s
)
∂t
=
κ
A
B
2
s
τ
s
+
f
cos
φB
s
τ
s
π
sech(Ω
τ
s
π
2
) (31)
where we have assumed the mode r is far from the mode center
μ
c
= Ω
/D
1
and the coupling coefficient
G
is smaller than or around
the same order of magnitude with
δω
. Also, higher order terms
are neglected (see Supplement). Subtracting eq.(
28
) from eq.(
27
)
yields
B
s
τ
s
=
D
2
gD
2
1
(32)
This equation was previously verified in the presence of Rama
n-
only interactions
18
.
An additional relation between
δω
,
τ
s
and Ω is derived for steady
state by substituting eq.(
30
) and (
32
) into eq.(
27
)
δω
=
D
2
2
D
2
1
(
1
τ
2
s
+ Ω
2
)
.
(33)
where Ω can be obtained from (
29
) and (
32
),
Ω = Ω
Raman
+Ω
Recoil
=
8
D
2
τ
R
15
κ
A
D
2
1
τ
4
s
B
D
1
κ
A
E
(1
Γ
2
)
|
h
r
|
2
(34)
which provides an independent confirmation of eq.(
24
). Also, eq.
(
33
) is identical in form to an expression which included only th
e
Raman SSFS
18
. Significantly, however, eq. (
33
) is more general
since Ω is the total spectral center shift provided by the com
bined
effects of Raman SSFS and dispersive-wave recoil.
Analytical model fitting and parameters.
Measurements are