1
SUPPLEMENTARY NOTE 1: EXPERIMENTAL SETUP AND DETUNING NOISE MEASUREMENT
(a)
EDFA
AOM
μ
Disk
FBG
PD
PC
Feedback Loop
Servo
CW laser
EOM
PD
OSA
OSC
PD
PNA
LO
Pump
Comb
(b)
0.0
0.5
1.0
-0.1
0.0
0.1
-1
0
1
2
-2
300
0
100
200
Signal voltage (V)
Scanning time (
μ
s)
Transmission
Error signal
MZI
Monitoring point
Supplementary Figure 1
:
Experimental setup and details on detuning-noise measurement
(a) The
experimental setup includes both the soliton generation and characterization setup
1,2
and a Pound-Drever-Hall
(PDH) system operated open loop. The PDH is added to make possible the pump-cavity detuning noise measure-
ment. Components included in the set up are an EOM: electro-optic modulator; EDFA: Erbium-doped fibre am-
plifier; AOM: acousto-optic modulator; PC: polarization controller; FBG: fibre Bragg grating; PD: photodetector;
OSA: optical spectral analyzer; PNA: phase noise analyzer; LO: local oscillator. The OSA and and PNA are shown
for completeness. They are used to measure the soliton spectrum and repetition rate phase noise. They are not in-
volved in measuring the detuning frequency noise. (b) Measurements that illustrate the pump-cavity detuning mea-
surement. The green trace is the measured power transmission when scanning the pump laser frequency across a
cavity resonance. The pump laser is phase modulated, the transmitted signal is detected and the resulting pho-
tocurrent is then mixed with the PDH local oscillator signal to generate the PDH error signal. Upon laser scan
the PDH error signal (as measured on the oscilloscope) is generated as shown in the red trace. The pump laser is
filtered using the fibre Bragg grating. The monitoring point for the detuning frequency measurement is indicated
by the black dot. In order to convert scanning time into laser frequency, a calibrated Mach-Zehnder interferometer
(MZI) records power transmission (blue trace) on an oscilloscope. The free-spectral-range of the MZI is 40 MHz.
Solitons are generated and locked using the active-capture and locking technique
1,2
. In this method a feedback
loop controls the pump laser frequency (fibre laser is used in this work) to maintain soliton power. In order to
measure the detuning noise, an additional Pound-Drever-Hall (PDH) loop is embedded into the setup and operated
open loop (see supplementary figure 1a). The pump frequency is red detuned relative to the cavity resonance in
order to form the soliton pulse train. Moreover, the amount of cavity-laser detuning required to generate solitons
is many cavity linedwidths so that the conventional PDH error signal near the resonance frequency cannot be used
to monitor the detuning frequency. However, the higher-frequency PDH sideband can be tuned to reside close to
the cavity resonance. Path phases in the PDH loop can be adjusted so that a PDH error signal is produced by the
interaction of this sideband with the cavity resonance. When the soliton is formed, we tune the PDH local-oscillator
(LO) frequency to approximately match the cavity-laser detuning. This is accomplished by monitoring the PDH error
signal (see red trace in supplementary figure 1b). For this measurement the transmitted pump light is filtered from
the soliton spectrum using a fibre Bragg filter. By setting this LO frequency to the indicated monitoring point, the
corresponding error-signal output will convert detuning frequency to a voltage output. This output can be recorded
and then analyzed to produce a noise spectrum. The calibration of voltage into frequency is performed by using the
Mach-Zehnder interferometer trace (see blue trace in supplementary figure 1b). This calibration is performed on the
resonator at reduced power levels where solitons do not form and where the Lorentzian lineshape of the resonator is
unaffected by the Kerr nonlinearity.
2
SUPPLEMENTARY NOTE 2: APPROXIMATIONS IN THE EQUATIONS OF MOTION
The coupling with mode family B results in the recoil term in eq.(29) in the Methods. In eq. (27), (28), (30) and
(31) (Methods), higher order terms have been neglected. Here, we list the higher order terms versus the leading order
terms in each equation and therefore establish the validity conditions for these equations.
2
π
|
G
|
2
τ
s
D
1
∆
ω
′
rB
sech
2
(
πτ
s
(r
D
1
−
Ω)
/
2)
gB
2
s
(
κ
2
B
+ 4∆
ω
′
2
rB
)
≤
π
|
G
|
2
|
a
r
|
2
κ
B
δωB
2
s
τ
s
D
1
1
(1)
2
π
|
G
|
2
τ
s
D
1
∆
ω
′
rB
sech
2
(
πτ
s
(r
D
1
−
Ω)
/
2)[2
−
Ω
τ
s
π
tanh(Ω
τ
s
π/
2)]
gB
2
s
(
κ
2
B
+ 4∆
ω
′
2
rB
)
≤
2
π
|
G
|
2
|
a
r
|
2
κ
B
δωB
2
s
τ
s
D
1
1
(2)
|
G
|
2
D
1
sech(
πτ
s
(r
D
1
−
Ω)
/
2)F(Ω
τ
s
)
2
κ
B
gB
2
s
Ω
≈
|
G
|
2
|
a
r
|
2
κ
B
δω
Ω
1
(3)
π
|
G
2
|
B
2
s
τ
2
s
D
1
κ
B
sech
2
(
πτ
s
(r
D
1
−
Ω)
/
2)
κ
A
B
2
s
τ
s
(
κ
2
B
+ 4∆
ω
′
2
rB
)
≤
4
π
|
G
|
2
|
a
r
|
2
κ
A
κ
B
B
2
s
τ
s
D
1
1
(4)
where F(Ω
τ
s
) =
iπ
2
8
[csc
2
(
1+Ω
τ
s
i
4
π
) + csch
2
(
Ω
τ
s
+
i
4
π
)]
∼
O
(1) in supplementary equation (3) and ∆
ω
′
rB
= ∆
ω
rB
−
∆
ω
r
,
comb
. Supplementary equations (1)-(4) hold under the experimental conditions and ignoring these terms gives
the equations of motion eq.(27)-(31) in the Methods.
-100
-50
0
50
100
-10
-5
0
5
10
Mode number
μ
(ω−ω
0
−μ
D
1
)/2
π
(GHz)
-10
-5
0
5
10
-100
-50
0
50
100
Mode number
μ
(ω−ω
0
−μ
D
1
)/2
π
(GHz)
Measurement
soliton mode
crossing mode
Simulation
(a)
(b)
Intensity (a.u.)
1
0
Supplementary Figure 2
:
Mode spectra measurement and simulation
(a) Measured mode spectra (blue
dots) and fitted lines. Green and orange lines represent fitting for soliton forming mode (mode A) and crossing
mode (mode B). Red and pink lines are fitted for another two high-
Q
modes.
D
1
/
2
π
is 22.0167 GHz. (b) Simulated
mode spectra of four transverse electric (TE) modes. Cross-sectional images of the mode spatial intensity distribu-
tions are included as insets.
SUPPLEMENTARY NOTE 3: SPATIAL MODE IDENTIFICATION
The microresonator geometry has a similar design as in our previous work
1,3
, however, the actual resonator is
different. To identify the spatial mode families of the soliton mode (family A) and the crossing mode (family B), a
numerical simulation of the wedge resonator is used as described elsewhere
1,4
. In the simulation, the geometry of the
3
1.5
1.0
0.5
0
Bistable range (normalized to
κ
A
)
Supplementary Figure 3
:
Contour plot of detuning range
Color plot of detuning range of bistability versus
spatial-mode coupling strength,
G
, and dissipation rate of crossing mode,
κ
B
. All quantities are normalized to the
dissipation rate,
κ
A
, of the soliton-forming mode.
wedge resonator was first set to the design parameters: 3 mm diameter, 30
◦
wedge angle and 7.5
μ
m silica thickness.
These parameters were then slightly tuned to maximize the similarity of measured and simulated mode spectra. In
the simulation presented in supplementary figure 2b, the resonator has diameter 2.9826 mm, wedge angle 30
.
7071
◦
and oxide thickness 7.5154
μ
m. The simulated mode spectrum is in reasonable agreement with the measurement.
The soliton mode (family A) is identified as the fundamental transverse electric (TE) mode and the crossing mode
(family B) is the TE21 mode. At the mode-crossing point in the simulation (
μ
= 57), the absolute mode numbers for
the soliton mode and the crossing mode are calculated to be 8683 and 8584, respectively.
SUPPLEMENTARY NOTE 4: DETUNING RANGE OF BISTABILITY
The detuning range of bistability is critical for observation of hysteresis behavior. Analysis shows that this range
can be increased by increasing the spatial-mode coupling strength, optical quality factor or moving the mode crossing
frequency closer to the pump laser frequency (smaller r). In fig. (3), the calculated detuning range of bistability is
presented as a contour plot versus coupling strength,
G
, and the dissipation rate of crossing mode,
κ
B
. It can be
seen that the range increases with
G
and decreases with
κ
B
. This is mainly because larger
G
and smaller
κ
B
enhance
the dispersive wave power and therefore increase the strength of recoil and nonlinearity. In our experiment, we have
measured
G/κ
A
= 20 and fitted
κ
B
/κ
A
= 1
.
7 and the corresponding normalized bistable range is estimated to be
0.67.
SUPPLEMENTARY REFERENCES
1
Yi, X., Yang, Q.-F., Yang, K. Y., Suh, M.-G. & Vahala, K. Soliton frequency comb at microwave rates in a high-Q silica
microresonator.
Optica
2
, 1078–1085 (2015).
2
Yi, X., Yang, Q.-F., Youl, K. & Vahala, K. Active capture and stabilization of temporal solitons in microresonators.
Optics
Letters
41
, 2037–2040 (2016).
3
Yang, Q.-F., Yi, X., Yang, K. Y. & Vahala, K. Spatial-mode-interaction-induced dispersive-waves and their active tuning in
microresonators.
Optica
3
, 1132–1135 (2016).
4
Yang, K. Y.
et al.
Broadband dispersion-engineered microresonator on a chip.
Nature Photonics
10
, 316–320 (2016).