arXiv:1704.08409v2 [physics.optics] 2 May 2017
Counter-propagating solitons in microresonators
Qi-Fan Yang
∗
, Xu Yi
∗
, Ki Youl Yang, and Kerry Vahala
†
T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA.
∗
These authors contributed equally to this work.
†
Corresponding author: vahala@caltech.edu
(Dated: May 4, 2017)
Solitons occur in many physical systems when a
nonlinearity compensates wave dispersion. Their
recent formation in microresonators opens a new
research direction for nonlinear optical physics
and provides a platform for miniaturization of
spectroscopy and frequency metrology systems.
These microresonator solitons orbit around a
closed waveguide path and produce a repetitive
output pulse stream at a rate set by the round-
trip time. In this work counter-propagating soli-
tons that simultaneously orbit in an opposing
sense (clockwise/counter-clockwise) are studied.
Despite sharing the same spatial mode family,
their round-trip times can be precisely and in-
dependently controlled. Furthermore, a state is
possible in which both the relative optical phase
and relative repetition rates of the distinct soli-
ton streams are locked. This state allows a sin-
gle resonator to produce dual-soliton frequency-
comb streams having different repetition rates,
but with high relative coherence useful in both
spectroscopy and laser ranging systems.
The recent demonstration of optical solitons in mi-
croresonators has opened a new chapter in nonlinear op-
tical phenomena
1–5
. These dissipative solitons
6
use the
Kerr nonlinearity to balance wave dispersion and to com-
pensate cavity loss
1,7
. The resulting dissipative Kerr soli-
tons (DKSs) exhibit Raman-related phenomena
8–11
, op-
tical Cherenkov radiation
3,12,13
and can form ordered ar-
rays called soliton crystals
14
. Soliton mode locking also
creates a new and very stable frequency microcomb with
distinct advantages over earlier microcombs
15
. For ex-
ample, internal broadening of these combs by dispersive-
wave generation
3
enables offset frequency measurement
for comb self referencing
16
. Also, the soliton repeti-
tion rate has an excellent phase noise stability
2,17,18
and
its spectral envelope is stable and reproducible so that
the resulting microcombs are suitable for dual comb
spectroscopy
19–21
.
All whispering-gallery microresonators feature clock-
wise (CW) and counter-clockwise (CCW) optical whis-
pering gallery modes, and this degree of freedom has not
been explored for generation of soliton pulse trains. In
this work counter-propagating (CP) solitons are gener-
ated by counter-pumping on a single microcavity reso-
nance (fig.1a). Because DKSs are phase coherent with
their respective optical pump, the tuning of two counter-
propagating pumps causes an offset in the optical fre-
quency of the two soliton pulse streams. Also, on ac-
a
EDFA
AOM
FBG
CW laser
PD
Feedback Loop
Servo
90/10
50/50
AOM
μ
Disk
Power (20 dB / div)
c
Wavelength (nm)
1500
1550
1600
CW soliton
CCW soliton
b
0
20
40
0
1
Delay (ps)
0
20
40
0
1
Delay (ps)
A-corr
(a.u.)
A-corr
(a.u.)
Pump
Pump
FIG. 1:
Observation of counter-propagating soli-
tons. a,
Rendering showing counter-propagating soli-
tons within a high-Q wedge resonator.
b,
Experimental
setup. A continuous-wave fiber laser is amplified by an
erbium-doped-fiber-amplifier (EDFA) and sent into two
acousto-optic modulators (AOM). The outputs from
the AOMs are counter-coupled into the microresonator
and generate counter-propagating solitons. The opti-
cal power of solitons in one direction is used to Servo
lock the pump laser to a certain frequency detuning
relative to the cavity mode
22
. FBG: fiber-Bragg grat-
ing; PD: photodetector.
c,
Optical spectra of counter-
propagating solitons. The location of the pump line is
indicated by a dashed line. Measured autocorrelation
traces are provided as insets.
count of the Raman-induced soliton self-frequency shift
(SSFS), the repetition rate of a soliton pulse stream de-
pends on the detuning of the pump frequency relative to
the resonant frequency being pumped
10,13
. As a result,
the pulse rate of each CP soliton pulse stream can be
independently controlled.
Besides independent repetition rate control there are
two soliton phase locking effects that are observed. In
both of these locked states, the soliton streams are also
optically phase locked even though the soliton frequen-
2
Frequency (kHz +21.9756 GHz)
RF power (20 dB / div)
60 kHz
CW soliton
CCW soliton
20
0
10
Interferogram
intensity (a.u.)
16.5
μ
s
b
c
0
-500
500
∆ν
=3.9 MHz
∆ν
=77 kHz
CCW soliton
CW soliton
RF power
(20 dB / div)
Pump
Pump
Frequency (MHz +21.9756 GHz)
-5
-10
0
5
10
a
∆ν
=3.9 MHz
d
∆ν=ν
ccw
-
ν
cw
(kHz)
0
60
20
∆
f=f
ccw
-f
cw
(kHz)
0
1000
3000
2000
40
4000
0
-200
400
200
0
10
-5
5
Locked
Measured
Fitted
Time (
μ
s)
FIG. 2:
Counter-propagating solitons with independently tuned re
petition rates. a,
Electrical spectrum
of photo-detected CW and CCW soliton pulse streams with pump freq
uency difference ∆
ν
= 3
.
9 MHz. Strong
central peaks give the repetition rate of each soliton. The weaker
spectral lines occurring over a broader spectral
range are inter-soliton beat frequencies. Beat frequencies prod
uced by the pump line of one soliton beating with
higher and lower frequency comb lines that neighbor the pump line of t
he other soliton are indicated by arrows.
These spectral lines are shifted by ∆
ν
=
±
3.9MHz relative to the two, strong repetition-rate lines.
b,
Upper trace
is gray-banded region from fig. 2a. The pair of strong central pea
ks give the CW and CCW soliton repetition rates.
Lower trace is the same electrical spectrum when the soliton repet
ition rates have locked to the same frequency.
Pump frequencies differ by ∆
ν
= 77 kHz.
c,
Temporal interferogram of the baseband inter-soliton beat signa
l under
unlocked condition in fig. 2a.
d,
Plot of the difference in CW and CCW repetition rates versus versus d
ifference in
pump frequencies. The red line is a fit using the model in Methods. The
inset shows that the two soliton repetition
rates are locked over approximately 150 kHz pump difference frequ
ency range.
cies reside on a distinctly different grid of optical fre-
quencies. In the first, the CW and CCW solitons are
observed to phase lock with identical repetition rates. In
the second locking effect the CP solitons experience rela-
tive rate locking at different repetition rates. As a result,
the microresonator produces two, soliton streams having
different repetition rates but with high relative coher-
ence. This form of locking is potentially useful in dual
comb spectroscopy and in laser ranging systems
23
(LI-
DAR) where it would eliminate the need for independent
and mutually-locked frequency combs.
The counter-propagating solitons are typically several-
hundred femtoseconds in duration and the microcavity
round-trip time is 46 ps. To produce the solitons, a
continuous-wave fiber laser is amplified and split using a
directional coupler so as to pump CW and CCW modes
of a microcavity resonance using a fiber taper coupler
(see experimental setup in fig. 1b). Two acousto-optic
modulators (AOM) are used to control the pump power
and frequencies in each pumping direction. The resid-
ual transmitted pump power is filtered by a fiber Bragg
grating filter (FBG).
The CP solitons are stabilized indefinitely using the
active capture technique
22
. It is found that application
of this locking technique to only one of the soliton pulse
streams automatically locks the other pulse stream. In
figure 1c the measured optical spectra and the autocor-
relation traces (insets) for typical CW and CCW soli-
ton streams are shown. The system can be controllably
triggered and locked with a single or a specified number
of solitons in each propagation direction. The microres-
onator, a high-Q silica wedge design
24
with 3 mm di-
ameter, has anomalous dispersion at the pumping wave-
length near 1.55 microns and is engineered to produce
minimal avoided mode crossings over the optical band of
the solitons
2
.
A feature of the dissipative Kerr soliton when viewed as
a mode-locked frequency comb, is that the pump provides
one of the comb frequencies and is therefore coherent with
the soliton. Non-degenerate counter-pumping therefore
introduces a controlled frequency offset between CW and
CCW solitons. Because the counter-propagating pumps
are derived from a single laser source, the mutual optical
coherence of CW and CCW pump comb teeth is excellent
and determined primarily by the stability of the radio-
frequency signals used to drive the AOMs shown in fig.
1b.
A key parameter that controls the soliton properties
is the cavity-pump detuning frequency
δω
cw
,
ccw
=
ω
0
−
ω
cw
,
ccw
where
ω
0
is the cavity resonant frequency which
is pumped and
ω
cw
,
ccw
are the CW and CCW pump
frequencies. Soliton pulse width
1,2
, average power
1,2
,
3
0
1
3
2
4
Radio frequency (MHz)
Power (10 dB / div)
1.4
1.45
1.55
1.5
1.6
a
c
d
0
1
3
2
4
Radio frequency (MHz)
Power (10 dB / div)
1.4
1.45
1.55
1.5
1.6
-100
0
100
Frequency offset (Hz)
f
g
25 kHz
250 kHz
1475 kHz
Power (10 dB / div)
RBW
1 Hz
RBW
1 Hz
RBW
1 Hz
Beat note number N
Variation of
∆ν
(Hz)
0 20
60
40
-20
-40
-60
-80
e
-200
-100
-200
-100
0
Phase noise (dBc / Hz)
-50
-40
-10
-20
-30
-70
-60
10
100
1000
Radio frequency (kHz)
1 Hz (measured)
10 Hz (measured)
10 Hz (fitted)
1 Hz (fitted)
Unlocked
Locked
pump
pump
Optical power
δ
+
∆
f
δ
∆
f-
δ
μ
=r
∆ν
∆ν
-
∆
f
∆
f+
∆ν
μ
=0
Optical frequency
RF power
Radio frequency
b
∆
f+
δ
δ
2
∆
f-
δ
∆ν
-2
δ
∆
f+
∆ν
-2
δ
∆
f
∆ν
-
∆
f
∆ν
Locked
Unlocked
CCW
CW
2
∆
f
∆ν
∆ν
-
∆
f
∆
f-
δ
Locked
Unlocked
FIG. 3:
Counter-propagating soliton phase locking at different rep
etition rates. a,
Schematic view of the
counter-propagating soliton comb lines. ∆
ν
and ∆
f
denote the pump frequency and repetition rate differences, re-
spectively, and
μ
is mode number relative to the pump mode (
μ
= 0).
b,
Illustration of inter-soliton radio-frequency
(RF) beatnotes produced under locked and unlocked conditions.
c,
Measured RF beatnotes of locked CP solitons
(∆
ν
= 1
.
5MHz, ∆
f
= 25 kHz).
d,
Measured RF beatnotes of unlocked CP solitons (∆
ν
= 1.5 MHz).
e,
Measured
beat-note spacing for locked and unlocked conditions plotted vers
us beatnote number.
f,
High-resolution, zoom-in
spectrum of RF beatnotes in
c
. The corresponding beat note frequency is provided in the legend (
25kHz is the fun-
damental beat note frequency).
g,
Phase noise of the beatnotes at 1 Hz and 10 Hz offset frequencies in
the phase
noise spectrum plotted versus beatnote frequency. The fitting lin
es have an
f
2
dependence.
and self-Raman-shift
8–10
depend upon this detuning. In
cases where the self-Raman shift is strong, the soli-
ton repetition rate also depends upon the cavity-pump
detuning
10,13
and the CW and CCW soliton repetition
rates (
f
cw
and
f
ccw
) can be separately controlled by tun-
ing of the respective pump frequencies.
To measure the CW and CCW soliton repetition rates,
their pulse streams are combined and simultaneously
4
photodetected. The electrical spectrum of the photocur-
rent is shown in fig. 2a when the difference in pumping
frequencies is set to ∆
ν
≡
(
ω
ccw
−
ω
cw
)
/
2
π
= 3.9 MHz and
δω
cw
∼
20 MHz. A zoom-in of the spectrum in the upper
panel of fig. 2b shows that two strong central spectral
peaks differ by 60 kHz. These peaks are the fundamental
repetition rates associated with the CW and CCW soli-
ton streams. The weaker, non-central beats appearing in
fig. 2a and the upper panel in fig. 2b are inter-soliton
beat frequencies between comb teeth belonging to differ-
ent soliton combs. These beat frequencies are equally
separated by the difference in the repetition rates (60
kHz). As an aside, the maxima at the extreme wings of
the spectrum are caused by the mode crossing distortion
in the comb spectra seen in fig. 1c near 1542 nm.
An interferogram showing the electrical time trace of
the co-detected dual-soliton pulse streams is shown in
fig. 2c. This time trace can be understood as a stro-
boscopic interference of the respective soliton pulses on
the detector. The strobing occurs at the rate difference
(∆
f
≡
(
f
ccw
−
f
cw
)) of the two soliton streams giving
the repetitive signal a period of 16.5
μ
s. By varying the
pump detuning, ∆
ν
, it is possible to observe tuning of
the repetition rate difference, ∆
f
, as shown in fig. 2d.
A theoretical fit discussed in Methods is provided in the
figure. Near ∆
ν
= 0 locking of the repetition rates is
observed over a range of ∆
ν
around 150 kHz. The as-
sociated electrical zoom-in spectrum under this locked
condition is shown in the lower panel of fig. 2b. Im-
portantly, nearly all of the weaker peaks that appear
in the unlocked spectrum shown in fig. 2a disappear
as a result of locking. This can be understood to re-
sult from the high relative temporal stability of the two
pulse streams. In particular, under the locking condition,
inter-soliton pulse mixing on the photo-detector, which is
guaranteed under conditions of unequal repetition rates,
now requires strict spatial-temporal alignment of the two
pulse streams at the detector. Consistent with this phys-
ical picture, the interferogram trace is observed to show
no periodic strobing behavior. This locking behavior is
believed to occur when pump light from a given pump-
ing direction is backscattered into the opposing direction
where the two pump signals can mix by the Kerr-effect.
This induces four-wave mixing sidebands on the soliton
comb lines that subsequently induce locking.
In addition to locking at identical repetition rates (de-
generate locking), the soliton pulse streams are observed
to lock when their repetition rates are different. Fig.
3a illustrates the principle of this locking mechanism.
Therein, soliton spectra for CW and CCW directions are
presented. A zoom-in of the higher frequency portion of
the spectra is shown in which the respective soliton spec-
tral lines are superimposed next to shaded areas repre-
senting the cavity resonances. The mode index
μ
= 0,
which is by convention the optical pump, is also indi-
cated. As required for DKS generation, this pump fre-
quency and the other soliton comb teeth are red-detuned
in frequency relative to their respective cavity resonances.
At
μ
= 0, the two pump lines are separated by the
pump frequency difference, ∆
ν
. Under conditions where
these pump frequencies are well separated so that de-
generate rate locking does not occur (see fig. 2d), the
soliton having the more strongly red-detuned pump will
feature a slightly lower repetition rate on account of the
self-Raman-effect discussed above and in Methods. Ac-
cordingly, the CW and CCW comb lines will shift in fre-
quency so as to become more closely spaced as
μ
de-
creases. For a certain negative value of
μ
the CW and
CCW comb lines will achieve closest spectral separation.
In the illustration, this occurs at comb tooth
μ
=
r
where CW and CCW comb lines have frequency sepa-
ration
δ
= ∆
ν
+
r
∆
f
. Backscattering within the res-
onator will couple power between these nearly resonant
lines. This power coupling is shown in Methods to induce
locking with a corresponding bandwidth.
Because the original comb teeth at
μ
= 0 are derived
from the same laser, the additional locking at
μ
=
r
causes the CW and CCW solitons to be mutually phase
locked. Moreover, the difference in the soliton repeti-
tion rates must be an integer fraction (1/
|
r
|
) of pump
frequency difference,
∆
f
=
−
∆
ν/r,
(1)
This result shows that pulse rates have a relative stability
completely determined by the radio frequency signal used
to set the pump frequency offset. Accordingly, the beat
signal between the CCW and CW solitons exhibits very
high stability when the system is locked in this way. The
above relation also shows that the locked CP solitons play
the role of a frequency divider of the pump frequency
difference into the pulse-rate difference frequency. The
phase noise of the rate difference is therefore
r
2
lower
than the phase noise of the relative pump signal,
S
∆
f
=
1
r
2
S
∆
ν
.
(2)
where
S
∆
f
and
S
∆
ν
are the phase noise spectral density
functions of the inter-soliton fundamental beat signal and
the pump difference signal.
Fig. 3b illustrates the effect of the locking condition
on the electrical spectrum produced by photodetection
of combined CCW and CW soliton streams. Under un-
locked conditions, the electrical spectrum will feature two
distinct spectra with spacing ∆
f
. However, under locked
conditions the difference in the frequency of the comb
teeth at
μ
= 0 (i.e., optical pumps) is an integer multiple
of the difference in the repetition rates. As result, the
two electrical spectra merge to form a single spectrum.
Fig. 3c shows a typical measured RF spectrum in the
locked state. It is obtained by Fourier transforming an
interferogram recorded over 1s. A set of equidistant spec-
tral lines is observed with a 50 dB signal-to-noise ratio
(SNR) at the 1 Hz resolution bandwidth (RBW). In this
measurement, ∆
ν
is set to be 1.5 MHz which is 60 times
∆
f
= 25 kHz. Meanwhile, the unlocked state shown in
5
fig. 3d features relatively noisier spectral lines and lower
SNR. This noise results from fluctuations of the absolute
pump frequencies which induce fluctuations in the two
Raman-shifted repetition rates
18
. The resulting noise is
multiplied with each comb tooth index relative to the
pump comb tooth. Fig. 3e plots the spacing between the
RF comb lines. It shows collapse to a sub-Hz stability
under the locked condition.
In strong contrast to the unlocked case, the spectral
line beatnotes in fig. 3c actually improve in stability with
decreasing order relative to the pump line. This a conse-
quence of the frequency division noted in eq. (1) and eq.
(2). In particular, the lowest frequency inter-soliton beat-
note features the minimum linewidth as shown in fig. 3f.
Furthermore, to confirm the scaling of phase noise with
the frequency division given in eq. (2), fig. 3g shows the
phase noise spectral density versus the spectral beatnote
number measured at two phase-noise offset frequencies
(1 Hz and 10 Hz). The dependence follows the predicted
quadratic form typical of a frequency divider. It is noted
that the inferred linewidth for the lowest order beatnote
is 40
μ
Hz ( assuming that it is limited by white frequency
noise).
Counter-propagating solitons have been demonstrated
in a high-Q optical microresonator. Both the repeti-
tion rates and the spectral location for the clockwise and
counter-clockwise directions are independently tuned by
tuning of the corresponding optical pumping frequencies.
Two distinctly different locking phenomena have been
observed while tuning the soliton repetition frequencies.
In the first, the repetition rates lock to the same value.
The pumping frequencies are different when this locking
occurs so that the two soliton comb spectra are offset
slightly in the optical frequency, but have identical comb
line spacings. The interferogram of the two pulse trains
has no baseband time dependence when this locking oc-
curs. In the second form of locking, the pumps are typi-
cally tuned apart to larger difference frequencies and the
solitons are observed to lock at different repetition rates
with a difference that divides into the pump-frequency
difference. The origin of this locking is associated with
optical locking of two comb teeth, one from each soliton.
Since the two pumps are derived from the same laser,
this additional comb tooth locking effectively results in
the two comb spectra being locked at two different posi-
tions in their spectra. The resulting high level of mutual
soliton coherence is observable in the base-band inter-
soliton beat spectra which features very narrow spectral
lines spaced by the difference in the locked soliton rep-
etition rates. In effect, this second form of locking cre-
ates two frequency combs in the same device with dis-
tinct repetition rates and optical frequencies, but that
are optically locked. It is potentially useful in dual comb
spectroscopy and dual comb LIDAR applications where it
would obviate the need for two separate frequency combs
and the associated inter-comb locking hardware. Finally,
it is noted that while single clockwise and counter-clock-
wise solitons have been generated, it is also possible to
create states containing multiple solitons.
Methods
Repetition rate control of CP solitons.
The Raman SSFS,
Ω
R
, is dependent on the pump-cavity detuning,
δω
, by
10
Ω
R
=
−
32
D
2
1
τ
R
15
κD
2
δω
2
(3)
where
τ
R
is the Raman shock time,
κ
is the cavity decay rate,
D
1
(
D
2
) is the free-spectral-range (second-order dispersion) at
mode
μ
= 0 (the pumping mode). The soliton repetition rate,
f
, is
coupled to the SSFS as
2
πf
=
D
1
+
Ω
R
D
2
D
1
(4)
Therefore the interferogram between the counter-propagat
ing soli-
tons with cavity-pump detuning
δω
cw
and
δω
ccw
has a repetition
rate difference
f
ccw
−
f
cw
=
−
16
D
1
τ
R
15
πκ
(
δω
2
ccw
−
δω
2
cw
)
=
−
16
D
1
τ
R
15
πκ
(2
δω
ccw
∆
ω
−
∆
ω
2
)
(5)
The second form of this equation uses ∆
ω
=
ω
ccw
−
ω
cw
= 2
π
∆
ν
and is applied for the theoretical plot in fig. 2d.
Locking of CP solitons
The dissipative Kerr solitons are gov-
erned by the Lugiato-Lefever equation augmented by the Rama
n
term
9,10,25
. The presence of scattering centers can induce coupling
between the CP solitons as follows,
∂A
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
A
)
A
+
i
D
2
2
∂
2
A
∂φ
2
+
F
+
ig
|
A
|
2
A
+
igτ
R
D
1
A
∂
|
A
|
2
∂φ
+
i
∫
2
π
0
Γ(
θ
)
B
(
φ
−
2
θ,t
)
e
−
i
∆
ωt
d
θ
(6)
∂B
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
B
)
B
+
i
D
2
2
∂
2
B
∂φ
2
+
F
+
ig
|
B
|
2
B
+
igτ
R
D
1
B
∂
|
B
|
2
∂φ
+
i
∫
2
π
0
Γ(
θ
)
A
(
φ
+ 2
θ,t
)
e
i
∆
ωt
d
θ
(7)
Here
A
and
B
denote the slowly varying field envelopes of the
CW and CCW solitons, respectively.
φ
is the angular coordi-
nate in the rotational frame
1
.
g
is the normalized Kerr nonlinear
coefficient
1,10
,
F
denotes the normalized continuous-wave pump
term and Γ(
θ
) represents the backscattering coefficient in the lab
frame
θ
.
Considering the spectral misalignment of CP soliton comb li
nes
presented in fig. 3a, it is assumed that only the
r
-th comb lines will
induce inter-soliton coupling. Accordingly, the equation
of motion
for the soliton field amplitude
A
, eq.
6
, is reduced to the following,
∂A
(
φ,t
)
∂t
=
−
(
κ
2
+
iδω
A
)
A
+
i
D
2
2
∂
2
A
∂φ
2
+
F
+
ig
|
A
|
2
A
+
igτ
R
D
1
A
∂
|
A
|
2
∂φ
+
iGb
r
e
irφ
(8)
where the expansion
B
(
φ,t
)
e
i
∆
ωt
=
∑
μ
b
μ
e
iμφ
is used to extract
the
r
-th comb line from soliton field
B
. A similar equation of
motion to eq.(
8
) holds for the amplitude
B
(with corresponding
expansion
A
(
φ,t
) =
∑
μ
a
μ
e
iμφ
). The coupling coefficient
G
=
∫
Γ(
θ
) exp(
−
2
irθ
)d
θ
.
The soliton field amplitude in the presence of the soliton sel
f-
frequency shift can be expressed as,
10,26
A
=
B
s
sech[(
φ
−
φ
Ac
)
/D
1
τ
s
]
e
iμ
A
(
φ
−
φ
Ac
)+
iψ
A
(9)
where
B
s
and
τ
s
are the pulse amplitude and duration, respectively.
μ
A
is the mode number of the soliton spectral maximum (
μ
= 0
is the mode number of the pump mode). This mode number is
related to the soliton self-frequency shift by Ω
R
=
μ
A
D
1
.
ψ
A
is
6
a constant phase determined by the pump
1,10
.
φ
Ac
is the peak
position of the CW soliton, which is coupled to
μ
A
by
10
∂φ
Ac
∂t
=
μ
A
D
2
.
(10)
The soliton energy
E
A
and the spectral maximum mode number
μ
A
are given by
E
A
=
∑
μ
|
a
μ
|
2
=
1
2
π
∫
+
π
−
π
|
A
|
2
d
φ
=
B
2
s
τ
s
D
1
/π
(11)
μ
A
=
∑
μ
μ
|
a
μ
|
2
E
A
=
−
i
4
πE
A
∫
+
π
−
π
(
A
∗
∂A
∂φ
−
A
∂A
∗
∂φ
)d
φ
(12)
Taking the time derivative of eq.(
12
) and substituting
∂A/∂t
using
eq.(
8
), the equation of motion for
μ
A
is obtained as
∂μ
A
∂t
=
−
κμ
A
−
gτ
R
D
1
2
πE
A
∫
+
π
−
π
(
∂
|
A
|
2
∂φ
)
2
d
φ
−
1
2
πE
A
∫
+
π
−
π
(
G
∗
b
∗
r
e
−
irφ
∂A
∂φ
−
irGA
∗
b
r
e
irφ
)d
φ
(13)
The second term on the right-hand-side corresponds to the st
eady-
state Raman-induced center shift
9,10
and is denoted by
κR
A
. The
third term is the soliton spectral shift caused by coupling t
o the
opposing CP soliton through its comb tooth
b
r
. By using
A
=
∑
μ
a
μ
e
iμφ
, eq. (
13
) yields,
∂μ
A
∂t
=
−
κμ
A
+
κR
A
−
ir
E
A
(
a
r
b
∗
r
G
∗
−
c
.
c
.
)
=
−
κμ
A
+
κR
A
+
2
r
E
A
|
a
r
b
r
G
|
sin Θ
.
(14)
where Θ = (
ψ
r
A
−
ψ
r
B
−
ψ
G
) with the phases,
ψ
rA
and
ψ
rB
, of the
comb lines
a
r
and
b
r
given by the following expression,
ψ
rA
=
ψ
A
−
rφ
Ac
.
(15)
ψ
rB
=
ψ
B
−
rφ
Bc
+ ∆
ωt.
(16)
Also, Ψ
G
is the phase of the backscatter coefficient
G
. The time
dependence of
ψ
rA
can be derived from eq.
10
as,
∂ψ
rA
∂t
=
−
r
∂φ
Ac
∂t
=
−
rμ
A
D
2
.
(17)
Similarly, the derivative of the phase of
b
r
is given by,
∂ψ
rB
∂t
=
−
rμ
B
D
2
+ ∆
ω.
(18)
Therefore the time derivative of the phase term Θ = (
ψ
r
A
−
ψ
r
B
−
ψ
G
) is given by,
∂
Θ
∂t
= ∆
ω
+
rD
2
(
μ
B
−
μ
A
) = 2
π
(∆
ν
+
r
∆
f
) = 2
πδ.
(19)
Similar to eq.(
14
), a parallel equation exists for the soliton
B
and
is given by,
∂μ
B
∂t
=
−
κμ
B
+
κR
B
−
2
r
E
B
|
a
r
b
r
G
|
sin Θ
.
(20)
Taking a time derivative of eq.(
19
) and using eq.(
14
) and eq.(
20
)
gives the following equation of motion for the relative phas
e Θ,
∂
2
Θ
∂t
2
+
κ
∂
Θ
∂t
=
−
2
r
2
D
2
(
1
E
A
+
1
E
B
)
|
a
r
b
r
G
|
sin Θ + 2
πκδ
′
,
(21)
where 2
πδ
′
= ∆
ω
+
rD
2
(
R
B
−
R
A
) is the frequency difference be-
tween the
r
th
comb lines induced by the shifted pumps and Raman
SSFS when the CP solitons have no interaction. The above equa
-
tion is similar to the Alder equation of injection locking
27
, only
with an additional second order time-derivative term. Sett
ing the
time derivatives of Θ equal to zero gives the locking bandwid
th,
ω
L
, of
δ
′
as
ω
L
= 4
π
|
δ
′
max
|
=
4
r
2
D
2
κ
(
1
E
A
+
1
E
B
)
|
a
r
b
r
G
|
.
(22)
Moreoever, eq.(
19
) gives
δ
= 0 so that the pump frequency differ-
ence ∆
ν
is divided by the repetition rate difference as follows,
∆
f
=
−
∆
ν
r
,
(23)
which is eq. (1) in the main text.
Parameters.
In the measurement, the loss rate is
κ/
2
π
= 1
.
5
MHz.
D
2
/
2
π
= 16 kHz and
r
=
−
60. For a soliton with
τ
s
= 150
fs, the mode number of the Raman SSFS is
μ
R
∼ −
20 and the ratio
|
a
r
|
2
/E
A
=
D
1
τ
s
sech
2
[
π
(
r
−
μ
R
)
D
1
τ
s
/
2]
/
8
∼
7
×
10
−
4
. As the CP
solitons have similar powers, the locking bandwidth is esti
mated as
ω
L
∼ |
G
|
/
4. In this case a backscattering rate of 4 kHz can provide
a 1 kHz locking bandwidth.
Acknowledgment
The authors gratefully acknowledge the Defense Ad-
vanced Research Projects Agency under the PULSE and
DODOS programs, NASA, the Kavli Nanoscience Insti-
tute.
1
Herr, T.
et al.
Temporal solitons in optical microres-
onators.
Nat. Photon.
8
, 145–152 (2014).
2
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).
3
Brasch, V.
et al.
Photonic chip–based optical frequency
comb using soliton cherenkov radiation.
Science
351
, 357–
360 (2016).
4
Wang, P.-H.
et al.
Intracavity characterization of micro-
comb generation in the single-soliton regime.
Opt. Express
24
, 10890–10897 (2016).
5
Joshi, C.
et al.
Thermally controlled comb generation and
soliton modelocking in microresonators.
Opt. Lett.
41
,
2565–2568 (2016).
6
Ankiewicz, A. & Akhmediev, N.
Dissipative Solitons:
From Optics to Biology and Medicine
(Springer, 2008).
7
Leo, F.
et al.
Temporal cavity solitons in one-dimensional
kerr media as bits in an all-optical buffer.
Nat. Photon.
4
,
471–476 (2010).
8
Mili ́an, C., Gorbach, A. V., Taki, M., Yulin, A. V. &
Skryabin, D. V. Solitons and frequency combs in silica
microring resonators: Interplay of the raman and higher-
order dispersion effects.
Phys. Rev. A
92
, 033851 (2015).
9
Karpov, M.
et al.
Raman self-frequency shift of dissipative
kerr solitons in an optical microresonator.
Phys. Rev. Lett.
116
, 103902 (2016).
10
Yi, X., Yang, Q.-F., Yang, K. Y. & Vahala, K. Theory
and measurement of the soliton self-frequency shift and
efficiency in optical microcavities.
Opt. Lett.
41
, 3419–
3422 (2016).
11
Yang, Q.-F., Yi, X., Yang, K. Y. & Vahala, K. Stokes soli-
tons in optical microcavities.
Nat. Phys.
13
, 53–57 (2017).
12
Matsko, A. B., Liang, W., Savchenkov, A. A., Eliyahu, D.
& Maleki, L. Optical cherenkov radiation in overmoded
7
microresonators.
Opt. Lett.
41
, 2907–2910 (2016).
13
Yang, Q.-F., Yi, X., Yang, K. Y. & Vahala, K. Spatial-
mode-interaction-induced dispersive-waves and their ac-
tive tuning in microresonators.
Optica
3
, 1132–1135
(2016).
14
Cole, D. C., Lamb, E. S., Del’Haye, P., Diddams, S. A.
& Papp, S. B. Soliton crystals in kerr resonators.
arXiv
preprint arXiv:1610.00080
(2016).
15
Kippenberg, T. J., Holzwarth, R. & Diddams, S.
Microresonator-based optical frequency combs.
Science
332
, 555–559 (2011).
16
Brasch, V., Lucas, E., Jost, J. D., Geiselmann, M. & Kip-
penberg, T. J. Self-referenced photonic chip soliton kerr
frequency comb.
Light Sci Appl.
6
, e16202 (2017).
17
Liang, W.
et al.
High spectral purity kerr frequency comb
radio frequency photonic oscillator.
Nat. Commun.
6
, 7957
(2015).
18
Yi, X., Yang, Q.-F., Zhang, X., Yang, K. Y. & Vahala,
K. Single-mode dispersive waves and soliton microcomb
dynamics.
arXiv preprint arXiv:1610.08145
(2016).
19
Suh, M.-G., Yang, Q.-F., Yang, K. Y., Yi, X. & Vahala,
K. J. Microresonator soliton dual-comb spectroscopy.
Sci-
ence
354
, 600–603 (2016).
20
Dutt, A.
et al.
On-chip dual comb source for spectroscopy.
arXiv preprint arXiv:1611.07673
(2016).
21
Pavlov, N.
et al.
Soliton dual frequency combs in crys-
talline microresonators.
Opt. Lett.
42
, 514–517 (2017).
22
Yi, X., Yang, Q.-F., Youl, K. & Vahala, K. Active capture
and stabilization of temporal solitons in microresonators
.
Opt. Lett.
41
, 2037–2040 (2016).
23
Coddington, I., Swann, W., Nenadovic, L. & Newbury, N.
Rapid and precise absolute distance measurements at long
range.
Nat. Photon.
3
, 351–356 (2009).
24
Lee, H.
et al.
Chemically etched ultrahigh-q wedge-
resonator on a silicon chip.
Nat. Photon.
6
, 369–373 (2012).
25
Lugiato, L. A. & Lefever, R. Spatial dissipative struc-
tures in passive optical systems.
Phys. Rev. Lett.
58
, 2209
(1987).
26
Matsko, A. B. & Maleki, L. On timing jitter of mode
locked kerr frequency combs.
Opt. Express
21
, 28862–
28876 (2013).
27
Adler, R. A study of locking phenomena in oscillators.
Proc. IEEE
34
, 351–357 (1946).