ARTICLE
Received 24 Oct 2016
|
Accepted 6 Feb 2017
|
Published 23 Mar 2017
Single-mode dispersive waves and soliton
microcomb dynamics
Xu Yi
1,
*, Qi-Fan Yang
1,
*, Xueyue Zhang
1,2,
*, Ki Youl Yang
1
, Xinbai Li
1,3
& Kerry Vahala
1
Dissipative Kerr solitons are self-sustaining optical wavepackets in resonators. They use the
Kerr nonlinearity to both compensate dispersion and offset optical loss. Besides providing
insights into nonlinear resonator physics, they can be applied in frequency metrology,
precision clocks, and spectroscopy. Like other optical solitons, the dissipative Kerr soliton
can radiate power as a dispersive wave through a process that is the optical analogue of
Cherenkov radiation. Dispersive waves typically consist of an ensemble of optical modes.
Here, a limiting case is studied in which the dispersive wave is concentrated into a single
cavity mode. In this limit, its interaction with the soliton induces hysteresis behaviour
in the soliton’s spectral and temporal properties. Also, an operating point of enhanced
repetition-rate stability occurs through balance of dispersive-wave recoil and Raman-induced
soliton-self-frequency shift. The single-mode dispersive wave can therefore provide quiet
states of soliton comb operation useful in many applications.
DOI: 10.1038/ncomms14869
OPEN
1
T.J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
2
Department of Microelectronics and
Nanoelectronics, Tsinghua University, Beijing 100084, China.
3
State Key Laboratory of Advanced Optical Communication Systems and Networks, School of
Electronics Engineering and Computer Science, Peking University, Beijing 100871, China. * These authors contributed equally to this work. Corresp
ondence
and requests for materials should be addressed to K.V. (email: vahala@caltech.edu).
NATURE COMMUNICATIONS
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1
A
new dissipative soliton
1
has recently been observed in
optical resonators. These dissipative Kerr solitons (DKS)
have been demonstrated in fibre resonators
2
and in
various microcavity systems
3–7
. In microcomb research
8,9
soliton
formation produces phase-locked spectra with reproducible
envelopes, as required in frequency comb applications
10–14
.
Moreover, their unusual properties and interactions create a rich
landscape for research in nonlinear optical phenomena
5,15–24
.
Two such phenomena, 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 fibre systems, this shift continuously increases with
propagation distance
25,26
, however, in microresonators the shift
is fixed and depends upon soliton power
17,18,27,28
. Dispersive
waves also occur in optical fibre systems
29
where they are an
important process in continuum generation
30
and have been used
to study general nonlinear phenomena
31
. They are formed when
a soliton radiates into a spectral region of normal dispersion and
can be understood as the optical analog of Cherenkov radiation
32
.
In microcavities, dispersive waves provide a powerful way to
spectrally broaden a soliton within a microresonator as a
precursor to self referencing
12,33
. Their formation also induces
soliton recoil
32
which, similar to SSFS, causes a frequency shift in
the spectral centre of the soliton
5,17
. Dispersive waves normally
consist of an ensemble of modes that are phase matched to a
soliton. This phase matching can be assisted by avoided mode
crossings in microcavities
19,34
. Avoided mode crossings can also
produce zero group velocity effects
35,36
, enable microcombs to
form in regions of normal dispersion
37
, and provide a way to
induce dark solitons
16
.
In this work, an avoided-mode crossing is used to excite a
dispersive wave consisting of a single cavity mode. The coupling
of this single-mode dispersive 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 behaviour in
soliton properties. Included in this behaviour, there is an
operating point of improved pulse-rate stability (a quiet point)
where the coupling of repetition rate and cavity-pump detuning is
greatly reduced. Pulse-rate stability is centrally important in
many frequency comb applications
10,13,38
. Coupling of pulse rate
and cavity-pump detuning through avoided-mode-crossing recoil
effects has been observed in crystalline resonators
39
. Also, the
fundamental contributions to phase noise in the pulse train have
been considered theoretically
40
. However, technical 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 technical noise contributions
to the soliton pulse repetition rate. Both this regime of operation
and the hysteresis behaviour are measured and modelled
theoretically.
Results
Mode family dispersion
. A silica whispering-gallery resonator
41
is used for soliton generation. The devices feature a free-spectral-
range (FSR) of
B
22 GHz (3 mm diameter resonator) and have
intrinsic
Q
-factors around 250 million. Specific details on soliton
formation in these resonators are given elsewhere
4,42
. The
resonators support multiple, transverse mode families. It is
essential that the soliton-forming mode family feature dispersion
that is primarily second-order and anomalous
43
. To characterize
the frequency spectrum of the resonator, mode frequencies were
measured from 190.95 THz (1,570 nm) to 195.94 THz (1,530 nm)
using an external-cavity diode laser calibrated by a fibre Mach–
Zehnder interferometer
4
. This provides a set of mode frequencies
{
o
m
,
s
} for each spatial mode family ‘
s
’ with
m
as the mode index.
The mode family frequency data are presented in Fig. 1a by
plotting the relative-mode-frequency,
D
o
m
,
s
o
m
,
s
o
0
m
D
1
versus mode index
m
where
o
0
and
D
1
are specific to the
soliton-forming mode family.
o
0
is the frequency of the mode
(set to have index
m
¼
0) that is optically pumped to produce the
soliton, and
D
1
is the FSR of the soliton-forming mode family at
m
¼
0 (note:
m
is a relative and not an absolute mode index). By
plotting the data in this way the second- and higher-order
dispersion of the soliton-forming mode family become manifest.
To illustrate, the relative-mode-frequency of the soliton-mode
family is fit with a green, dashed parabolic curve of positive
curvature in Fig. 1a showing that it features anomalous second-
order dispersion over a wide range of mode numbers.
A second mode family also appears in Fig. 1a and causes an
avoided-mode-crossing near
m
¼
72. Hybridization of this ‘cross-
ing-mode’ family with the soliton-mode family occurs near the
avoided crossing
19,44
. The relative-mode-frequencies of the
unperturbed soliton-forming mode family and crossing-mode
family are denoted as
D
o
m
A
and
D
o
m
B
. Over the range of mode
indices measured
D
o
m
A
¼
1
2
D
2
m
2
where
D
2
is the second-order
dispersion at
m
¼
0. The lower (upper) branch of the hybrid mode
family is denoted by
D
o
m
(
D
o
m
þ
). The spatial modes
associated with the soliton and crossing mode families are
identified in the Supplementary Note 3. Avoided mode crossing
behaviour has been intensively studied in the context of DKS
formation and can interfere with soliton generation by creation of
distortions in the dispersion spectrum
43,45,46
. 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
m
¼
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,b are identical and the
location of the
m
¼
0 pumping mode is indicated by a vertical
dashed line in Fig. 1b.
Single-mode dispersive-wave formation
. Also shown in Fig. 1a
are the comb frequencies associated with a hypothetical soliton
spectrum plotted in the relative frequency frame. This comb line
is given by,
D
o
m
;
comb
¼
o
m
;
comb
o
0
D
1
m
¼
o
rep
D
1
m
do
;
ð
1
Þ
where
o
m
,comb
¼
mo
rep
þ
o
p
is the frequency of
m
th comb line,
o
rep
is the soliton repetition frequency,
o
p
is the pump frequency,
and
do
o
0
o
p
is the cavity-pump detuning frequency. It is
necessary to distinguish between relative frequencies for the soliton
comb and the resonator modes because the frequency components
of the soliton comb are strongly red-detuned relative to the cold-
cavity mode frequencies by the Kerr nonlinearity. Indeed, dis-
persive waves typically form when a set of modes break this rule
and become resonant with a set of comb lines. A limiting case of
this condition is shown in Fig. 1a, where the occurrence of an
isolated resonance between a hybrid mode with relative frequency
D
o
r
and a comb line at
D
o
r,comb
is illustrated. The equation of
motion for the hybrid mode field amplitude
h
r
is shown in the
Methods to have the following form,
d
h
r
d
t
¼
i
D
o
r
k
r
2
hi
h
r
þ
f
r
e
i
D
o
r
;
comb
t
ð
2
Þ
where
k
r
is its loss rate and
f
r
is an effective pumping term
associated with the soliton comb line. The pumping term is given
by
f
r
¼
i
G
(
D
o
rA
D
o
r,comb
)
a
r
, where
G
is the fraction of the family
ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14869
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NATURE COMMUNICATIONS
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A mode in the hybrid mode, and
a
r
is the field amplitude of the
unperturbed soliton hyperbolic solution at
m
¼
r. Also, the Kerr-
effect shift of
h
r
is of order 10 kHz and is therefore negligible in
comparison to
k
r
.
Because the damping rate
k
r
is low (that is, the mode has a
high optical
Q
-factor) slight shifts in the slope of the comb
frequency line (equivalently, shifts of
D
o
r,comb
relative to
D
o
r
)
will cause large changes in the power coupled to the hybrid
mode. These changes are observable in Fig. 1b where a
strong spectral line appears in the case of the blue soliton
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 spectral centre 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 centre 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 resonance
between the soliton and the mode is diminished. Accordingly,
most of the spectral shift in this case results from the Raman
SSFS.
Soliton recoil and hysteresis
. A change in the slope of the soliton
comb line will occur when the soliton repetition frequency,
o
rep
,
is changed (equation (1)). On account of second-order dispersion
o
rep
depends linearly on the frequency offset,
O
, of the
soliton spectral maximum relative to the pump frequency
19,40
.
This frequency offset has contributions from both the Raman
SSFS,
O
Raman
, and the dispersive-wave recoil,
O
Recoil
(that is,
O
¼
O
Raman
þ
O
Recoil
). Accordingly, the soliton repetition
rate is given by,
o
rep
¼
D
1
þ
D
2
D
1
O
Raman
þ
O
Recoil
ðÞð
3
Þ
where
D
2
(the second-order dispersion of soliton-forming mode
family at
m
¼
0) is measured to be 17 kHz from Fig. 1a.
Substituting for the repetition rate in the comb line expression
(equation (1)) gives,
D
o
m
;
comb
¼
m
D
2
D
1
O
Raman
þ
O
Recoil
ðÞ
do
ð
4
Þ
It is shown in the Methods (equation (25)) that the soliton recoil
frequency has a linear dependence on the hybrid mode power,
O
Recoil
¼
g
h
r
jj
2
¼
r
k
B
D
1
k
A
E
h
r
jj
2
ð
5
Þ
where
k
A
and
k
B
denote the power loss rates of the family A and
family B modes, respectively; and
E
is the circulating soliton
energy.
Solving equation (2) for the steady-state power in the hybrid
mode at the soliton comb line frequency and using equations
(4 and 5) gives the following result,
h
r
jj
2
¼
f
r
jj
2
D
o
r
þ
do
r
D
2
D
1
O
Raman
þ
g
h
r
jj
2
2
þ
k
2
r
4
ð
6
Þ
Equation (6) suggests that a bistable state and hysteresis
behaviour in the dispersive-wave power is possible when varying
the soliton operating 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
Dispersive wave power (
μ
W)
Total frequency shift (THz)
Recoil (THz)
Dispersive wave power (
μ
W)
Mode number
190
192
194
196
0
10
20
30
40
Power (
μ
W)
Frequenc
y
(THz)
(
–
0
–
D
1
) / 2
(MHz)
–100
0
100
0
100
200
–100
0
100
–100
=r
Operating Point I
Operating Point II
Comb frequencies
Δ
, 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 1020304050
Soliton mode
Δ
A
Single-mode
dispersive wave
15
20
25
30
35
40
45
50
Pump
10
30
50
D
1
=22 GHz
D
2
=17 kHz
Crossing mode
Δ
B
Measurement
Eq.(5)
Δ
–
Δ
+
ac
bd
Figure 1 | Soliton hysteretic behaviour induced by mode interaction.
(
a
) Measured relative mode frequencies are shown as blue points
4
. The green and
yellow dashed lines represent the fitted relative mode frequencies (
D
o
m
A
and
D
o
m
B
) of the unperturbed soliton-forming mode family A and crossing mode
family B, respectively. Relative mode frequencies for upper and lower branch hybrid-modes are
D
o
m
þ
and
D
o
m
. The red line illustrates the frequencies of
a hypothetical soliton frequency comb. A non-zero slope on this line arises from the repetition rate change relative to the FSR at mode
m
¼
0. (
b
) Measured
optical spectra at soliton operating points I and II, corresponding to closely matched cavity-pump detuning frequencies,
do
. A strong single-mode
dispersive wave at
m
¼
72 is observed for operating point II and causes a soliton recoil frequency shift. This frequency shift adds to the shift resulting from
the Raman-induced SSFS. (
c
,
d
) Dispersive-wave power and soliton spectral centre frequency shift versus cavity-pump detuning. Operating points I and II of
b
are indicated. Inset in
c
: Measured (blue dots) and theoretical (red line) recoil frequency versus the dispersive wave power.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14869
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3
nearly identical detuning frequencies,
do
. A more detailed survey
of the dispersive-wave power behaviour is provided in Fig. 1c and
is again consistent with a hysteresis behaviour versus detuning.
Moreover, since the total spectral shift of the soliton is given by
O
¼
O
Raman
þ
O
Recoil
¼
O
Raman
þ
g
|
h
r
|
2
, a corresponding beha-
viour is observed in the overall soliton spectral shift (Fig. 1d).
Theoretical fits are provided in Fig. 1c,d using equation (6).
The fitting procedure and parameter values are provided in the
Methods.
In plotting the data, the detuning frequency,
do
/2
p
, was
determined from the measured total soliton spectral shift (
O
) and
pulse width (
t
s
) using the relation
do
¼
(
D
2
/2
D
2
1
)(1/
t
2
s
þ
O
2
).
This expression is a generalization of a relationship derived
elsewhere
18
. The generalization extends the shift
O
to include
both the SSFS and the recoil and is derived as equation (33) in the
Methods. As an aside, the pulse width is determined by fitting the
soliton optical spectrum
4
.
Likewise, the recoil frequency,
O
Recoil
, can also be extracted
from the data as
O
O
Raman
by first using the soliton pulse width
to determine the Raman shift using
O
Raman
¼
8
t
R
D
2
/15
k
A
D
2
1
t
4
s
.
A plot of the recoil shift determined this way versus the
dispersive-wave power is given as the inset in Fig. 1c and verifies
the linear dependence (equation (5)). Equation (5) is also plotted
for comparison using parameters given in the Methods. As an
aside, the Raman shift formula noted above is also a general-
ization of a result proven elsewhere
18
. Curiously, as shown in the
Methods, this formula maintains its previous form in the
presence of the dispersive wave.
Within narrow detuning frequency bands in the vicinity of the
hysteresis both measurements and calculations show that the total
cavity power (soliton and dispersive-wave contributions) can
decrease with increasing cavity-pump detuning as opposed to
increasing with detuning as is typical for a soliton. Under these
special conditions, the pump-cavity detuning will no longer be
dynamically stable on account of the thermal nonlinearity
47
.
Evidence of this was observable in the current work as it was not
possible to completely map out the theoretically predicted
hysteresis curves.
While the present results are produced using a dispersive wave
that is blue-detuned relative to the soliton spectral maximum, the
hysteresis behaviour is also predicted to occur for a red-detuned
dispersive wave. However, in the red-detuned case, the orienta-
tion of the curve in Fig. 1c is reversed with respect to the detuning
frequency. The essential feature for appearance of the hysteresis is
that the recoil advances and retreats versus detuning. As a result,
the existence of hysteresis behaviour predicted in equation (6) is
not limited to microresonator materials having a strong Raman
SSFS. It is also predicted to occur, for example, in crystalline
resonators given an appropriate avoided-mode crossing. The
requirements imposed on the device and mode crossing for this
to occur are discussed further below.
Numerical simulation
. To further investigate the single-mode
dispersive-wave phenomena, we perform numerical simulations
based on the coupled Lugiato-Lefever equations
34,48–51
involving
the soliton-forming mode family (family A) and the crossing-
mode family (family B). Additional information including
parameter values is provided in the Methods, but is outlined
here. The two mode families are coupled using a model studied
elsewhere
19
. The coupling is characterized by a rate constant
G
and is designed to induce an avoided-mode-crossing around
mode index
m
¼
72, similar to the experimental mode family
dispersion. Figure 2 shows the results of the numerical simulation
including 2,048 modes. The hysteresis behaviour in the soliton
total frequency shift and the dispersive-wave power resembles the
experimental observation and is also in agreement with the
analytical model (Fig. 2a,b). As predicted by equation (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 in units of
intracavity power. They show that the dispersive-wave emerges
on mode 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. The combined power of
mode A and B spectra in Fig. 2c is the total intracavity power.
Soliton repetition rate quiet point
. The nonlinear behaviour
associated with soliton coupling to the single-mode dispersive
wave can be used to suppress soliton repetition rate noise pro-
duced by coupling of pump-laser frequency noise. This noise
source is suspected to be a significant contributor to repetition-
rate noise in certain frequency-offset regimes
4
. From equation (3)
the repetition frequency depends linearly on the total soliton
spectral-centre frequency shift,
O
. However, this total shift
frequency versus cavity-pump detuning has a stationary point
on the upper hysteresis branch (Fig. 1d). As expected from the
simple dependence in equation (3), this same stationary point is
observed in measurements of the repetition frequency versus
detuning (Fig. 3a). To measure the repetition frequency the
soliton pulse train is directly detected and an electrical spectrum
analyser 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
analyser. 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 detuning, 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 detuning point in the upper branch. This quiet
point has the lowest phase noise among the recorded phase noise
spectra. At higher offset frequencies, the phase noise is shot noise
limited, while at lower offset frequencies the phase noise indicates
4
0 dBc Hz
1
and is mainly contributed by frequency drift of the
repetition rate.
For comparison, the phase noise associated with the detuning
frequency
do
was also measured. For this measurement, the error
signal of a Pound–Drever–Hall feedback control system is
operated open-loop and recorded using an oscilloscope. Its
power spectral density is converted into phase noise in Fig. 3b
(Supplementary Note 1). The relatively high noise floor in this
measurement is caused by the oscilloscope sensitivity. None-
theless, a noise bump at 25 kHz offset frequency originates from
the laser and provides a laser-noise 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 soliton phase noise is calculated in the
Methods and the results are presented for comparison using
the cavity-pump detuning noise level at 25 kHz offset. The dip of
the phase noise occurs at the quiet point. One outlier point (red
branch) is believed to have resulted from loss of lock of the phase
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i
ii
iii
iv
i
ii
iii
iv
i
i
ii
ii
iii
iii
iv
iv
ab
c
d
Total frequency shift (THz)
20
30
40
50
Detuning
(normalized to
A
/2)
40
30
20
Detuning
(normalized to
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 per division)
Mode number
0
100
–100
Mode number
0
100
–100
Mode number
0
100
–100
Mode number
0
100
–100
Intracavity
power (a.u.)
0
1
2
Time (
T
R
)
Time (
T
R
)
Time (
T
R
)
Time (
T
R
)
0
0.1
–0.1
0
0.1
–0.1
0
0.1
–0.1
0
0.1
–0.1
Numerical
simulation
Analytical
model
Mode A
Mode B
Figure 2 | Numerical simulation and analytical model of single-mode dispersive-wave generation and recoil.
(
a
) Numerical (blue dots) and analytical
(red solid line) soliton total frequency shift versus cavity-pump detuning. Points i, ii, iii and iv correspond to specific soliton operating points n
oted in other
figure panels. (
b
) Numerical (blue dots) and analytical (red solid line) dispersive-wave power (normalized to total soliton power) versus cavity-pump
detuning. Inset: recoil frequency versus the dispersive-wave power. (
c
) Comb spectra contributions from the two mode families (blue: soliton forming mode
family A; red: crossing mode family B). (
d
) Time domain intracavity power.
T
R
is the cavity round trip time.
10
1
10
2
10
3
10
4
10
5
10
6
10
7
–120
–80
–40
0
40
Phase noise (dBc Hz
–1
)
Offset frequency (Hz)
I
20
30
40
50
0
–200
–400
–600
RF-offset (kHz)
Detuning (MHz)
II
III
Measurement
Theory
Phase noise at
25 kHz offset (dBc Hz
–1
)
20
30
40
50
–140
–120
–100
–80
Detuning (MHz)
–60
I
II
III
measure floor
Phase noise floor
15
25
35
45
–160
ab
c
Figure 3 | Soliton repetition frequency and phase noise measurement.
(
a
) Measured (blue dots) and theoretical (red) soliton repetition frequency versus
pump-cavity detuning. The offset frequency is 22.0167 GHz. The distinct soliton operating points I, II and III refer to phase noise measurments in 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 and also the noise of the
cavity-pump detuning (green) with its noise floor (brown). The black line connecting the square dots is the measurement floor of the phase noise analyse
r.
(
c
) Phase noise of soliton repetition rates at 25 kHz offset frequency plotted versus the cavity-pump detuning. The blue and red dots (lines) denote the
experimental (theoretical) phase noise of the upper (blue) and lower (red) branch operating points, respectively.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14869
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5
noise analyser. For lower offset frequencies, the contributions to
noise are believed 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 determine conditions
required to observe the quiet point. The quiet point occurs when
the retreating soliton recoil balances the always advancing SSFS.
Accordingly, Fig. 4 is a contour plot of the maximum ratio of
|
@
O
Recoil
/
@
do
|to|
@
O
Raman
/
@
do
| while varying the coupling
strength between the soliton-mode and crossing-mode families
and the damping rate of the crossing mode (see Methods). The
existence regime for observation of the quiet point corresponds to
the ratio
4
1 shown in red. Stronger mode interaction and weaker
dissipation are required to operate in this regime. Also, the
impact of these parameters on the detuning range of the
hysteresis is studied in the Supplementary Note 4.
Discussion
Microfabrication control of resonator diameter, oxide thickness
and wedge angle all impact the spectral placement of mode
families. Numerical simulation of these families based on
scanning electron micrograph measurement of resonator cross
sections provides reasonably accurate dispersion maps for
prediction of resonator properties. Also, process control of the
resonator fabrication is sufficient to guarantee fabrication of
mode families exhibiting the features shown in Fig. 1a within the
1,530–1,570 nm band.
In summary, coupling of a dissipative Kerr soliton to a single-
mode dispersive wave has been shown to produce hysteresis
behaviour in both the dispersive-wave power and in the soliton
properties. These properties include the frequency shift of the
soliton spectral centre relative to the pumping frequency and
the soliton repetition frequency. The hysteresis results from the
dependence of the dispersive-wave phase matching condition
upon the dispersive-wave power. The hysteresis behaviour of the
dispersive wave also leads to an operating point wherein coupling
of laser pump frequency noise into the soliton repetition rate is
greatly reduced. This reduction was modelled and measured, and
the requirements for quiet point existence were also studied. The
operating point for quiet soliton operation is of potential use for
ultra-low-noise microwave generation.
Methods
Dynamical equation of hybrid mode
.
Equation (2) can be derived from coupled
mode equations that include dispersion, mode interaction and the Kerr non-
linearity. The intracavity field of mode
m
in the soliton-forming mode family A can
be represented by
A
m
ð
t
Þ
e
i
o
m
A
t
þ
i
mf
, where
A
m
(
t
) is the slowly varying amplitude,
t
is the time and
f
is the azimuthal angle along the resonator. In the rotation frame
of comb frequencies
o
m
,c
omb
¼
o
0
do
þ
mo
rep
, the intracavity field can be
expressed as
a
m
ð
t
Þ¼
A
m
ð
t
Þ
e
i
o
m
A
o
0
þ
do
mo
rep
ðÞ
t
. We denote the intracavity field
in the crossing-mode family B as
b
m
and express it in the same reference frame as
the soliton-forming mode
a
m
. It should be noted that the relative mode number
m
is
referenced to the mode that is being optically pumped, and does not represent the
actual azimuthal index. The intracavity fields can be calculated using the equations
of motion with Kerr nonlinearity terms
50,52
and modal-coupling terms
44
,
d
a
m
d
t
¼
k
A
2
þ
i
o
m
A
o
0
þ
do
mo
rep
hi
a
m
þ
iGb
m
þ
ig
X
j
;
k
a
j
a
k
a
j
þ
k
m
þ
F
d
ð
m
Þ
ð
7
Þ
d
b
m
d
t
¼
k
B
2
þ
i
o
m
B
o
0
þ
do
mo
rep
hi
b
m
þ
iGa
m
þ
ig
B
X
j
;
k
b
j
b
k
b
j
þ
k
m
ð
8
Þ
where
k
A,B
¼
o
0
/
Q
A,B
is the dissipation rate.
g
¼
:
o
2
0
n
2
D
1
/2
p
n
0
A
eff
represents the
normalized Kerr nonlinear coefficient with
A
eff
the effective nonlinear mode area.
g
B
is defined similarly.
G
is the linear coupling coefficient between the two mode
families
19
and
F
is the normalized coupled laser pump field. Also, to calculate
equation (2) it is not necessary to include Raman coupling terms in equations
(7 and 8) since the leading-order contribution to the forcing term,
f
r
, is from the
Kerr nonlinearity.
Modal coupling causes two branches of hybrid modes to form as shown in
Fig. 1a. The frequencies of the hybrid modes in the upper (
þ
) and lower (
)
branches are given by (refs 44,53,54),
o
m
¼
o
m
A
þ
o
m
B
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
2
þ
1
4
o
m
A
o
m
B
2
r
ð
9
Þ
where the corresponding field amplitude of the hybrid modes is a linear
combination of
a
m
and
b
m
. In the far-detuned regime where
o
m
A
o
m
B
G
,the
field amplitude of the lower branch hybrid mode is approximately given by,
~
h
m
¼
Ga
m
þ
o
m
A
o
m
B
b
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
2
þ
o
m
A
o
m
B
2
q
ð
10
Þ
In this experiment, only one mode was observed to be near resonance with the
soliton comb and that mode is assigned mode index
m
¼
r. Consistent with Fig. 1a,
the hybridization of mode r is assumed weak (that is,
o
rA
o
rB
jj
G
jj
and
D
o
rA
jj
D
o
rB
jj
)sothat
b
r
is the dominant contribution to
~
h
r
. Also, since the
amplitude of
b
m
with
m
a
r is small, the Kerr interaction summation term can be
neglected in equation (8) in this calculation.
By taking the time derivative of equation (10) and then substituting using (7)
and (8) the following dynamical equation results for
~
h
m
,
d
~
h
r
d
t
¼
k
r
2
þ
i
o
r
o
0
þ
do
r
o
rep
hi
~
h
r
þ
f
r
ð
11
Þ
where
f
r
is the pumping term given by,
f
r
¼
i
G
g
X
j
;
k
a
j
a
k
a
j
þ
k
r
ð
12
Þ
and where
G
¼
G
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
jj
2
þ
o
m
A
o
m
B
2
q
is the fraction of the family A mode in
~
h
m
and
k
r
E
k
B
is assumed for r when
G
1. When converting equation (11)
into the rotation frame of (
o
0
þ
m
D
1
) with
~
h
r
¼
h
r
e
i
D
o
r
;
comb
t
, the following
expression results,
dh
r
dt
¼
i
D
o
r
k
r
2
hi
h
r
þ
f
r
e
i
D
o
r
;
comb
t
ð
13
Þ
where
D
o
r
¼
o
r
o
o
m
D
1
is the relative-mode-frequency of hybrid mode
h
r
. Equation (13) is identical to equation (2) in the main text.
Effective pumping term
.
The pumping term in equation (11) can be expressed in
parameters of the resonator and soliton. The soliton field envelope takes the
form
3,18
A
ð
f
;
t
Þ¼
B
s
sech
f
f
c
ðÞ
=
D
1
t
s
½
e
i
O
f
f
c
ðÞ
=
D
1
þ
i
j
ð
14
Þ
1.4
1.2
1
0.8
0.6
0.4
Quiet point existence
Quiet point absence
|
Recoil slope/raman slope
|
max
1
234
5
10
20
30
Crossing mode loss
B
(normalized to
A
)
Coupling rate G (normalized to
A
)
Figure 4 | Existence study for the quiet point.
The maximum ratios of
|
@
O
Recoil
/
@
do
|to|
@
O
Raman
/
@
do
| at varying normalized modal-coupling rate
G
(see Methods) and normalized crossing-mode damping rate
k
B
(dashed
curve is unity ratio). The quiet point exists when this ratio is greater than
unity (red region). Parameters correspond to a silica resonator.
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NATURE COMMUNICATIONS
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where soliton properties are: amplitude
B
s
, angular position
f
c
, temporal pulse
width
t
s
, spectral-centre frequency shift (relative to pump)
O
and phase relative to
the pump laser
j
. Also, this solution assumes
do
k
A
. By applying the Fourier
transform to
A
(
f
,
t
),
a
m
can be expressed in terms of the soliton properties,
A
ð
f
;
t
Þ¼
X
m
a
m
ð
t
Þ
e
i
mf
f
c
ðÞ
ð
15
Þ
a
m
¼
B
s
t
s
D
1
2
sech
pt
s
2
D
1
m
O
ðÞ
e
i
j
ð
16
Þ
The pump
f
r
can therefore be derived by inserting equation (16) into equation (12).
The following expression results from simplification of the summation,
f
r
¼
i
G
D
2
4
D
2
1
D
1
r
O
ðÞ
2
þ
1
t
2
s
B
s
t
s
D
1
sech
pt
s
2
D
1
r
O
ðÞ
e
i
j
ð
17
Þ
where
g
has been replaced using equation
B
2
s
t
2
s
¼
D
2
/
gD
2
1
, which holds
for DKSs
18,40
and is also verified in a section below. Finally, by using
18
do
¼
D
2
2
D
2
1
1
t
2
s
þ
O
2
(see derivation below),
f
r
can be further reduced to
f
r
¼
i
GD
o
rA
D
o
r
;
comb
a
r
ð
18
Þ
Recoil and soliton self frequency shift
.
In addition to the Raman SSFS
17,18
, the
spectral centre of the DKS is also shifted by the single-mode dispersive-wave recoil.
The effect of the recoil and Raman shift can be calculated using the moment
analysis method
17,55
. Using the Fourier transform, equation (7) is transformed into
the perturbed Lugiato-Lefever equation (LLE)
50
@
A
ð
f
;
t
Þ
@
t
¼
k
A
2
þ
i
do
A
þ
i
D
2
2
@
2
A
@
f
2
þ
F
þ
ig A
jj
2
A
þ
ig
t
R
D
1
A
@
A
jj
2
@
f
þ
iGB
ð
19
Þ
where the Raman shock term has been added
17,18
and
t
R
is the Raman time
constant. The moment analysis method treats the soliton as a particle. The energy
E
and the spectral centre mode number
m
c
are given by,
E
¼
X
m
a
m
2
¼
1
2
p
Z
þ
p
p
A
jj
2
d
f
¼
B
2
s
t
s
D
1
=
p
ð
20
Þ
m
c
¼
P
m
m
a
m
2
E
¼
i
4
p
E
Z
þ
p
p
A
@
A
@
f
A
@
A
@
f
d
f
ð
21
Þ
Taking the time derivative of equation (21) and substituting
@
A
/
@
t
using
equation (19), the following equation of motion for
m
c
is obtained,
@
m
c
@
t
¼
k
A
m
c
g
t
R
D
1
2
p
E
Z
þ
p
p
@
A
jj
2
@
f
2
d
f
1
2
p
E
Z
þ
p
p
G
B
@
A
@
f
GA
@
B
@
f
d
f
ð
22
Þ
The second term on the right-hand-side corresponds to the Raman-induced
frequency shift and the third term is the frequency shift caused by recoil.
The Raman term can be calculated by substituting equation (14) into the
integral. When calculating the recoil term,
B
is simplified to
B
b
r
e
i
r
f
f
c
ðÞ
as the
power in mode
B
is dominated by the near resonance mode
r
. In addition, because
the integral of
f
is over 2
p
, only
a
r
e
i
r
f
f
c
ðÞ
has nonzero contribution.
Furthermore, equation (8) is used to relate
Ga
r
to
b
r
and finally leads to,
@
m
c
@
t
¼
8
t
R
D
2
15
D
3
1
t
4
s
r
k
B
E
b
r
jj
2
k
A
m
c
ð
23
Þ
The steady-state spectral centre mode number is therefore given by,
m
c
¼
8
t
R
D
2
15
k
A
D
3
1
t
4
s
r
k
B
k
A
E
1
G
2
h
r
jj
2
¼
1
D
1
O
Raman
þ
O
Recoil
ðÞ
ð
24
Þ
where
o
m
A
o
m
B
k
B
;
D
o
r
(equivalent to
b
r
jj
a
r
jj
) is assumed and the
recoil and Raman shifts are,
O
Recoil
¼
g
h
r
jj
2
¼
r
k
B
D
1
k
A
E
1
G
2
h
r
jj
2
;
ð
25
Þ
O
Raman
¼
8
t
R
D
2
15
k
A
D
2
1
t
4
s
ð
26
Þ
In the main text,
G
2
1 is assumed. Equation (25) is equation (5) in the main
text. The form for the Raman SSFS,
O
Raman
, is identical to the form previously
derived in the absence of the dispersive-wave coupling
18
.
Soliton parameters with Raman and mode-coupling effects
.
In the presence of
recoil and Raman, the relations between soliton parameters in equation (14) can be
derived from the Lagrangian approach
3,18,40
. In addition, the Lagrangian approach
verifies the expression for
O
Recoil
obtained above as well as providing a path for
calculation of the repetition-rate phase noise
40
. As detailed in previous
literature
18,40
, the perturbation Lagrangian method is applied to the LLE equation
of
A
(equation 19). However, now an additional perturbation term is added to
account for the mode coupling to the crossing-mode family. Taking
B
b
r
e
i
r
f
f
c
ðÞ
, produces the following equations of motion,
O
D
1
@
f
c
@
t
@
j
@
t
do
D
2
O
2
2
D
2
1
D
2
6
t
2
s
D
2
1
þ
2
3
gB
2
s
¼
0
ð
27
Þ
O
D
1
@
f
c
@
t
@
j
@
t
do
D
2
O
2
2
D
2
1
þ
D
2
6
t
2
s
D
2
1
þ
1
3
gB
2
s
¼
0
ð
28
Þ
@
B
2
s
t
s
O
@
t
¼
k
A
B
2
s
t
s
O
8
g
t
R
B
4
s
15
t
s
k
B
p
r
b
r
jj
2
ð
29
Þ
@
f
c
@
t
¼
D
2
D
1
O
ð
30
Þ
@
B
2
s
t
s
@
t
¼
k
A
B
2
s
t
s
þ
f
cos
j
B
s
t
s
p
sech
O
t
s
p
2
ð
31
Þ
where we have assumed the mode r is far from the mode centre
m
c
¼
O
/
D
1
and the
coupling coefficient
G
is smaller than or around the same order of magnitude with
do
. Also, higher order terms are neglected (Supplementary Note 2). Subtracting
equation (28) from equation (27) yields
B
s
t
s
¼
ffiffiffiffiffiffiffiffi
D
2
gD
2
1
s
ð
32
Þ
This equation was previously verified in the presence of Raman-only interactions
18
.
An additional relation between
do
,
t
s
and
O
is derived for steady state by
substituting equations (30) and (32) into equation (27)
do
¼
D
2
2
D
2
1
1
t
2
s
þ
O
2
:
ð
33
Þ
where
O
can be obtained from (29) and (32),
O
¼
O
Raman
þ
O
Recoil
¼
8
D
2
t
R
15
k
A
D
2
1
t
4
s
r
k
B
D
1
k
A
E
1
G
2
h
r
jj
2
ð
34
Þ
This result provides an independent confirmation of equation (24). Also,
equation (33) is identical in form to an expression, which included only the Raman
SSFS
18
. Significantly, however, equation (33) is more general since
O
is the total
spectral centre shift provided by the combined effects of Raman SSFS and
dispersive-wave recoil.
Phase noise transfer function
.
The repetition rate of the soliton can be expressed
as follows
19
,
o
rep
¼
D
1
þ
@
f
c
@
t
¼
D
1
þ
D
2
D
1
O
:
ð
35
Þ
The variation in both
D
1
and
O
contribute to fluctuations in the repetition rate.
While
D
1
is subject to thermorefractive noise and fluctuations from the
environment, a significant contributor to fluctuations in
O
results from fluctuations
in the pump-laser frequency detuning frequency,
do
. The noise conversion from
cavity-pump detuning to repetition rate can be calculated by linearizing equations
(27)–(31) using the small-signal approximation
40
. Accordingly, all soliton
parameters (
X
) can be expressed as
X
¼
X
0
þ
D
X
, where
X
0
is the steady-state value
and
D
X
is a small-signal fluctuation. For simplicity, we further denote the Raman
and recoil terms in equation (29) as
8
g
t
R
B
4
s
/15
t
s
k
B
p
r|
b
r
|
2
k
A
B
2
s
t
s
F
(
do
)so
that
O
¼
F
(
do
) is the function of detuning measured in Fig. 1d (that is, steady-state
O
versus
do
). For simplicity, we assume this steady-state holds in the dynamical
equations below. This can be shown to be true when offset frequencies (see
definition below) are small compared to the cavity decay rate.
In the following derivation,
t
s
in equations (27)–(31) is eliminated using
equation (32). Equation (29) can therefore be expressed as
@
B
s
O
@
t
¼
k
A
B
s
O
F
ð
do
Þ
½
:
ð
36
Þ
Applying the small-signal approximation and Fourier transform to equation (36)
gives the result,
1
þ
i
o
=
k
A
ðÞ
D
~
O
ð
o
Þ¼
@
F
@
do
D
f
do
ð
o
Þ
i
o
O
0
k
A
B
s0
D
e
B
s
ð
o
Þ
;
ð
37
Þ
where
D
~
X
ð
o
Þ
is the Fourier transform of
D
X
,
o
is the Fourier frequency (that is,
offset frequency in the phase or frequency-noise spectrum) and where the Fourier
transform of
@
D
X
/
@
t
equals
i
o
D
~
X
ð
o
Þ
.
D
f
do
ð
o
Þ
represents the cavity-pump
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