of 6
PHYSICAL REVIEW A
103
, L011501 (2021)
Letter
Oscillatory motion of a counterpropagating Kerr soliton dimer
Chengying Bao,
1
Boqiang Shen,
1
Myoung-Gyun Suh,
1
,
*
Heming Wang,
1
Kemal ̧
Safak,
2
Anan Dai,
2
Andrey B. Matsko,
3
Franz X. Kärtner,
4
,
5
and Kerry Vahala
1
,
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Cycle GmbH, 22607 Hamburg, Germany
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
4
Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron, 22607 Hamburg, Germany
5
Department of Physics and the Hamburg Center for Ultrafast Imaging, University of Hamburg, 22761 Hamburg, Germany
(Received 25 February 2020; revised 15 December 2020; accepted 18 December 2020; published 6 January 2021)
Counterpropagating (CP) solitons generated in high-
Q
microcavities not only offer useful dual-comb sources,
but also provide a new platform to study soliton interactions. Here, we demonstrate and theoretically explain
a manifestation of soliton trapping that occurs between CP solitons in a silica microcavity introducing a Kerr
soliton dimer. In conventional soliton trapping, the group velocities of two solitons can be synchronized by a
Kerr-effect-mediated interaction. The solitons can then copropagate with a fixed temporal delay. However, as
shown here, when counterpumping a microcavity using slightly detuned pump frequencies and in the presence
of backscattering, the group velocities of clockwise and counterclockwise solitons undergo periodic modulation
instead of being locked to a constant velocity. Upon emission from the microcavity, the solitons feature a relative
oscillatory motion around a locked average relative displacement with an amplitude that can be larger than the
soliton pulse width. This relative motion introduces a sideband fine structure into the optical spectrum of the CP
solitons. Our observation provides insights on coherently pumped soliton dimers in microcavities.
DOI:
10.1103/PhysRevA.103.L011501
Coherently pumped optical solitons were first studied
within the subject of spatial cavity solitons [
1
4
]. Their coun-
terparts, coherently pumped temporal solitons, have also been
demonstrated in passive fiber cavities [
5
] and high-
Q
micro-
cavities [
6
10
]. And the realization of such microcavity soli-
tons represents a significant advance of microresonator-based
Kerr frequency combs (microcombs) [
10
,
11
]. These soliton
microcombs can pave the way towards compact spectrometers
[
12
], Lidar systems [
13
,
14
], optical frequency synthesizers
[
15
], and optical clocks [
16
]. The advent of microcav-
ity solitons and their mathematical relationship to various
multiparticle physical systems also make these systems useful
for improved understanding of intricate physical systems. For
example, they have been utilized to demonstrate soliton crys-
tals [
17
], heteronuclear soliton molecules [
18
], and soliton
formation in photonic dimers [
19
]. Furthermore, they provide
a platform for the study of Cherenkov radiation [
20
24
],
Fermi-Pasta-Ulam recurrence [
25
27
], Feshbach resonances
[
28
], and Bose-Hubbard hopping [
29
33
].
The symmetry of the clockwise (CW) and counterclock-
wise (CCW) propagation directions enables the generation
of counterpropagating (CP) solitons in microcavities [
34
,
35
].
Unlike copropagating solitons, CP solitons can interact via
Rayleigh backscattering of the microcavity [
34
], which adds a
new element for study of soliton physics. For example, the in-
teraction of CP solitons via a single comb line can stabilize the
*
Present address: Physics & Informatics Laboratories, NTT Re-
search, Inc., 940 Stewart Drive, Sunnyvale, California 94085, USA.
vahala@caltech.edu
repetition rate difference (

f
r
) between CP solitons to

f
r
=
P
/
N
, where
P
is the frequency detuning between the
counterpropagating pumping waves (typically several MHz
for silica microcavities) and
N
is an integer [
34
]. A repetition
rate locking regime has also been observed when
P
is small
(e.g., tens or hundreds of kHz) and the pump powers are nearly
balanced [
34
36
]. However, the residual soliton motion in this
regime (hereinafter referred to as the ‘

f
r
=
0 regime’) has
not been revealed due to the insufficient temporal resolution
of the measurement methods.
In this paper, we report the measurement of the relative CP
soliton motion in the

f
r
=
0 regime using a balanced optical
cross-correlator (BOC) [
37
]. This regime can be understood to
arise from the Kerr-mediated soliton trapping effect [
38
,
39
]
between the CW (CCW) soliton and the backscattered field
from the CCW (CW) soliton. Surprisingly, an oscillatory
motion between CP solitons (after coupling out from the mi-
crocavity) at a frequency of
P
is observed. The amplitude
of this motion can be larger than the soliton pulse width. The
motion makes CP soliton trapping different from conventional
soliton trapping, which locks the interacting solitons to travel
at the same group velocity and a fixed time delay. Thus, our
work adds a new insight into the understanding of soliton trap-
ping dynamics. Moreover, it establishes a new type of soliton
dimer molecule [
18
] formed by interacting CP solitons.
CP soliton dimer formation is enabled by the soliton
trapping process. This trapping process can arise from the
cross-phase-modulation induced refractive index trapping po-
tential, which locks the group velocities of two interacting
solitons [
38
41
]. The pulse phase velocities can also be locked
via a four-wave-mixing interaction term [
40
] when the phase
2469-9926/2021/103(1)/L011501(6)
L011501-1
©2021 American Physical Society
CHENGYING BAO
et al.
PHYSICAL REVIEW A
103
, L011501 (2021)
Conventional soliton trapping
f
r2
= f
r1
f
02
=f
01
f
r1
f
r2
= f
r1
f
02
f
01
sidebands
f
r1
coherent pump 1
coherent pump 2
s
...
...
L/v
g
L/v
p
(a)
(b)
Trapping with forced motion
XPM
FWM
CCW soliton
CW soliton
Backscattered
CW soliton
Backscattered
CCW soliton
CW soliton
Backscattered
CCW soliton
1/
P
...
...
...
...
t
FIG. 1. Forced motion in a trapped CP soliton dimer. (a) In conventional soliton trapping without coherent pumps, both the phase velocity
and the group velocity will be locked. In the frequency domain, both the repetition rate
f
r
and the carrier envelope offset frequency
f
0
of
the corresponding combs will be equal. Note that in the illustration the pulse envelopes are plotted to be exactly overlapped and the phase
is identical, which are not necessary in experiments. (b) For CP solitons in coherently pumped systems, backscattering enables a different
trapping outcome. Due to the coherent pumping, the phase velocity and
f
0
cannot be locked when using nondegenerate pumps. In this case,
solitons will be locked in an averaged way, with the spectral center frequency (
ν
s
) and group velocity experiencing periodic modulation (upper
shaded box). As a result, there is relative motion between the CP solitons after being emitted from the microcavity (lower shaded box). This
relative motion repeats with a period of 1
/ν
P
. In the frequency domain, the motion will induce sidebands around the main comb lines. Note
that the power ratio between the solitons and the backscattered counterpart is for illustration and does not reflect the actual ratio.
velocity difference is small enough [see Fig.
1(a)
]. In this
way, both the pulse envelope and the carrier frequency can be
synchronized. And, as a result, the corresponding comb fre-
quencies for each pulse train become identical [see Fig.
1(a)
].
The trapping process (in both the group and the phase velocity
versions) has also been reported for solitons in coherently
pumped fiber and microcavities [
17
,
42
46
].
Unavoidable backscattering within a microcavity couples
the soliton field into the other propagation direction [
34
]. This
provides a way for each soliton to interact with a modified
replica of the other soliton [see Fig.
1(b)
]. Because backscat-
tering is not expected to occur at a single point, but rather over
a complex spatial profile, the backscattered field itself reflects
the complexities of this scattering process and may no longer
be a short solitonlike sech pulse. Let us consider first the case
of degenerate pumping frequencies (
P
=
0). CCW and CW
solitons share the same longitudinal mode family and similar
pump conditions (same pump frequency and similar pump
power). Moreover, a feature of the coherent pumping is that
the pump frequency is one of the soliton comb frequencies.
For these reasons both the envelope and the carrier of CP soli-
tons will have closely matched velocities. Moreover, the Kerr-
effect-mediated interaction allows the two solitons to trap one
another via the backscattered fields in a way very similar to
the trapping of conventional copropagating solitons [
38
40
].
Thus, even while the solitons are propagating in opposite
directions, their group and phase velocities become locked.
When the pump frequency detuning
P
is nonzero but
small compared to the cavity linewidth, we show that the CP
solitons are still bound to each other. However, on account
of the coherent nature of the pumping, the frequency of each
soliton comb line is shifted by
P
relative to the replica
of the other soliton comb. Importantly, this frequency shift
is set by the external pumps and cannot be pulled towards
0 as in conventional soliton trapping [Fig.
1(b)
]. The soliton
carrier and phase velocities are therefore not synchronized.
Thus, each of the backscattered CW (CCW) comb lines can
be regarded as a sideband that modulates the CCW (CW)
comb lines [see upper gray-shaded box in Fig.
1(b)
]. This
modulation causes each soliton to experience a periodic spec-
tral center frequency change (
s
), with the period being
1
/ν
P
. The group velocity of the pulses varies with the
spectral center as

(1
/
v
g
)
=
2
πβ
2
s
, where
β
2
is the group
velocity dispersion (negative for cavities supporting solitons)
[
47
,
48
]. This group velocity modulation causes periodic rel-
ative motion of CP solitons, with a period of 1
/ν
P
within
their traps. Corresponding frequency sidebands spaced by
P
emerge around the main comb lines [see right panel in
Fig.
1(b)
].
We used the experimental setup shown in Fig.
2(a)
to
measure the predicted motion. CP solitons were generated
in a 22-GHz high-
Q
silica wedge microcavity [
7
,
34
,
49
]. A
single pump laser was used for generation of CP solitons and
distinct pumping frequencies for CW and CCW directions
L011501-2
OSCILLATORY MOTION OF A ...
PHYSICAL REVIEW A
103
, L011501 (2021)
0
0.1
0.2
0.3
0.4
0.5
Time (ms)
0.1 ms
010203040
50
Time (
s)
Relative delay (100 fs/div)
10
s
Relative delay (200 fs/div)
1510
1530
1550
1570
1590
Exp.
Sim.
Power (20 dB/div)
Wavelength (nm)
(b)
(c)
(d)
Exp. 10 kHz
Exp. 100 kHz
3 dB:1.4 THz
BOC
Oscilloscope
Pump
PS+EDFA
AOM
EDFA
AOM
PS+EDFA
Delay (5 ps/div)
Voltage (0.2 V/div)
760 fs
(a)
250 fs
FIG. 2. Observation of oscillatory motion in CP soliton trapping.
(a) A balanced optical cross correlator (BOC) is used to experimen-
tally measure relative CP soliton motion (left). The BOC output
signal is shown when the delay is scanned (right). AOM, acousto-
optical modulator; EDFA, erbium-doped fiber amplifier; PS, pulse
shaper. (b) Optical spectrum of the CP solitons with a 3-dB band-
width of 1.4 THz. The red line (spectral envelope) is the simulated
spectrum. (c), (d) BOC-measured relative soliton motion when the
pump frequency detuning is 10 and 100 kHz, respectively. The
motion frequency is measured to be equal to
P
. The dashed red
(upper) and blue (lower) lines indicate the center of motion for 10-
and 100-kHz detuning, respectively; and they are shifted by about
200 fs as shown in (c).
were produced by two acousto-optical modulators [
34
]. The
corresponding optical spectrum of one of the CP solitons is
shown in Fig.
2(b)
and has a 3-dB bandwidth of 1.4 THz so
that the soliton duration is deduced to be 125 fs [equivalently,
220 fs for the full width at half-maximum (FWHM) pulse
width]. We used a BOC to record the soliton motion, which
operates by balanced detection of the sum frequency genera-
tion between two inputs in a PPKTP crystal [
37
]. Before input
into the BOC, the soliton streams (with pumps suppressed by
notch filters) were dispersion compensated by pulse shapers
[
50
] and amplified, so as to enhance the BOC output signal.
Figure
2(a)
also shows an example of the BOC output signal
when the delay line inside the BOC was scanned. The central
portion of the signal is linear. Thus, when setting the delay line
in that region, soliton motion can be converted into a voltage
signal for measurements.
The output of the BOC for pump detuning frequencies of
10 and 100 kHz is shown in Figs.
2(c)
and
2(d)
. These result-
ing temporal modulation rates are much lower than the BOC
detection bandwidth of 4 MHz. The ability to reliably observe
a nonzero signal implies that the repetition rates of the two
solitons are locked on average (i.e., operating in the

f
r
=
0
regime), since otherwise the two inputs would temporally
walk off, resulting in a zero signal due to nonsynchronized
repetition rates [
37
]. A strong oscillation of the BOC signal
is observed, with a peak-to-peak amplitude reaching 0.8 ps,
which is more than twice the soliton FWHM pulse width.
This is also nearly 2% of the round-trip time (46 ps). The
motion is not sinusoidal but is asymmetric (sawtoothlike in
the 10-kHz detuning case). And the oscillation frequency is
equal to the pump frequency detuning (
P
) as expected. The
center of mass of the relative motion trajectory is also plotted
in Figs.
2(c)
and
2(d)
. It suggests that the CP solitons oscillate
around different centers in the trap when the pump detuning
frequency varies. This motion exists for small detunings, e.g.,
P
<
1 Hz, suggesting that the solitons interact on an ul-
tralong time scale. The noisier trace in Fig.
2(d)
compared
to Fig.
2(c)
is under investigation but could result from the
Raman process. Specifically, the Raman-induced soliton fre-
quency shift has a quadratic dependence on pump resonance
detuning [
51
], and hence the relative group velocity of CP
solitons is more susceptible to pump-resonance-detuning fluc-
tuations when
P
is larger.
Numerical simulations based on the coupled Lugiato-
Lefever equations [
1
,
34
,
52
,
53
] confirm the experimental
observations [
54
]. Representative plots of the relative soli-
ton motion are shown in Figs.
3(a)
and
3(b)
. The motion
frequency equals
P
, and both the trajectory and the
amplitude are reasonably consistent with the experimental
measurements. For example, the asymmetric sawtoothlike be-
havior is numerically reproduced. The simulation assumed
a single-point backscatterer, while there are likely multiple
backscatterering centers in the actual microcavity. It is there-
fore expected that some discrepancies exist in the observed
and simulated motion trajectories. More discussion of the
asymmetric motion can be found in [
54
].
To test the hypothesis that the relative motion is driven
by the detuned-pump-induced forced-soliton spectral center
shift, we numerically calculated the relative spectral center
frequency between the two solitons in Fig.
3(c)
. It exhibits
periodic oscillation around zero frequency. The positive and
negative relative frequency regions correspond to forced mo-
tion where the derivative of the relative delay (i.e., relative
group velocity) is positive or negative, respectively [see verti-
cal dashed green lines in Figs.
3(b)
and
3(c)
]. Moreover, the
fact that the relative center frequency oscillates around 0 Hz
indicates that their group velocities are locked on average.
Accordingly, the experimental and numerical observations
validate the existence of oscillatory forced motion of the CP
soliton dimer in the presence of backscattering.
Finally, we experimentally verify that the forced motion
introduces fine-structure sidebands into the comb lines. For
this measurement, the two CP soliton microcombs are hetero-
dyned on a balanced photodetector. The recorded electrical
spectra, shown in Fig.
4
, contain multiple radio-frequency
tones for pump detuning of both 10 and 100 kHz. The lack
L011501-3
CHENGYING BAO
et al.
PHYSICAL REVIEW A
103
, L011501 (2021)
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0.1
0.2
0.3
0.4
0.5
Time (ms)
Relative delay (ps)
Sim. 10 kHz
0.1 ms
Sim. 100 kHz
420 fs
01020304050
Time (
s)
Relative delay (ps)
-10
-5
0
5
Relative frequency (GHz)
01020304050
Time (
s)
(a)
(b)
(c)
10
s
Region I
Region II
Sim. 100 kHz
0
0.2
0.4
0.6
730 fs
0
0.1
0.2
0.3
0.4
FIG. 3. Simulation of forced motion in CP soliton trapping. (a),
(b) Simulated relative soliton temporal motion when the two pumps
are detuned by 10 and 100 kHz, respectively. Dashed red lines indi-
cate the zero delay. (c) Numerically calculated relative spectral center
frequency between two CP solitons for
P
=
100 kHz, showing
periodic variation. Depending upon the sign of the relative frequency,
the motion can be separated into two regions as indicated by the
dashed green lines.
of any tone other than those at integer multiples of
P
shows
that only fine-structure sidebands spaced by
P
are present
in the optical comb spectra. Sidebands due to soliton spectral
bandwidth breathing induced by Rayleigh backscattering in a
microcavity were also simulated in Ref. [
32
]. Different from
that instability, the measured sidebands here arise from the
forced relative soliton motion.
In summary, we have measured the relative oscilla-
tory motion between CP solitons in the

f
r
=
0regime.
This temporal motion results from slightly detuned coherent
0
50
100
150
200
250
300
350
400
450
Frequency (kHz)
Intensity (20 dB/div)
10 kHz
0
100
200
300
400
500
600
700
800
900
1000
Frequency (kHz)
Intensity (20 dB/div)
100 kHz
(a)
(b)
FIG. 4. Measured electrical spectra of the beat between CP soli-
tons. (a)
P
=
10 kHz. (b)
P
=
100 kHz.
counterpumping of CP solitons in the presence of optical
backscattering. The detuned counterpumps cause a periodic
modulation of both the spectral center frequencies of CP
solitons and their relative group velocities. Different from
the

f
r
=
P
/
N
regime, all comb lines participate in the
soliton interaction in the

f
r
=
0 regime. Our measurements
show that it does not require the solitons to be constrained
within a small temporal range for trapping of solitons and
how coherently pumped solitons are different from conserva-
tive solitons. The oscillatory motion also inserts fine-structure
sidebands into the soliton microcomb spectrum that may af-
fect some comb applications. The results provide new insights
and generalize somewhat the concept of a soliton dimer in
microcavities.
This work was supported by the Air Force Office of Scien-
tific Research (Grant No. FA9550-18-1-0353) and the Kavli
Nanoscience Institute. C.B. gratefully acknowledges the post-
doctoral fellowship from the Resnick Sustainability Institute
at Caltech. The research performed by A.M. was carried
out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics
and Space Administration (No. 80NM0018D0004). We thank
Qi-Fan Yang for helpful discussion.
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