of 5
Forced Oscillatory Motion of Trapped Counter-Propagating Solitons
Chengying Bao,
1
Boqiang Shen,
1
Myoung-Gyun Suh,
1
Heming Wang,
1
Kemal S ̧afak,
2
Anan Dai,
2
Andrey B. Matsko,
3
Franz X. K ̈artner,
4, 5
and Kerry J. Vahala
1,
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Cycle GmbH, Hamburg 22607, Germany
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
4
Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron, Hamburg 22607, Germany
5
Department of Physics and the Hamburg Center for Ultrafast Imaging,
University of Hamburg, Hamburg 22761, German
Both the group velocity and phase velocity of two solitons can be synchronized by a Kerr-effect
mediated interaction, causing what is known as soliton trapping. Trapping can occur when soli-
tons travel through single-pass optical fibers or when circulating in optical resonators. Here, we
demonstrate and theoretically explain a new manifestation of soliton trapping that occurs between
counter-propagating solitons in microresonators. When counter-pumping a microresonator using
slightly detuned pump frequencies and in the presence of backscattering, the group velocities of
clockwise and counter-clockwise solitons undergo periodic modulation instead of being locked to a
constant velocity. Upon emission from the microcavity, the solitons feature a relative oscillatory
motion having 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 counter-propagating solitons.
Our results highlight the significance of coherent pumping in determining soliton dynamics within
microresonators and add a new dimension to the physics of soliton trapping.
The mutual interaction of two optical soliton pulses
by way of the optical Kerr effect is known to impact
their relative phase and group velocity leading to the phe-
nomenon of optical trapping. Solitons with close group
velocities can be mutually trapped and travel at the same
group velocity [1–6]. Moreover, their pulse phase veloc-
ities can also be locked (i.e., are equal, see Fig. 1(a))
[3] so that both the envelope and the carrier of the soli-
tons travel at the same velocities. Recently, there has
been considerable interest in a coherently pumped opti-
cal soliton [7] that has been realized in mode locked fiber
systems [8] and high-Q microresonators [9–12]. Besides
their practical importance for realization of compact mi-
crocomb systems [13], the coherent pumping of these soli-
tons, via a background field, introduces new physics into
the system. For example, a modulated background field,
induced by dispersive waves [14, 15] or through electro-
optical modulation [16], introduces a trapping potential
and traps the relative positions of solitons and the back-
ground field. Moreover, a modulated background can en-
able the formation of regular arrays called soliton crystals
[17].
In all of these cases, however, the solitons are co-
propagating and the relative positions of interacting soli-
tons will be fixed. Here, we consider a trapping mech-
anism that occurs between coherently pumped counter-
propagating (CP) solitons [18, 19]. It is shown experi-
mentally and theoretically that a new manifestation of
soliton trapping arises for non-degenerately pumped CP
solitons in the presence of optical backscattering. Specif-
ically, the solitons are trapped on average, but exhibit a
periodic relative motion at the pump detuning frequency,
δν
P
. This forced oscillatory motion is observed after the
CP solitons are coupled outside the cavity and features
an amplitude that is larger than the soliton pulse width.
It also introduces an observable fine structure into the
comb lines in the form of sidebands separated by
δν
P
.
Backscattering within a microresonator allows the
fields of both the CP solitons and their respective pumps
to couple into the opposing propagation directions of the
resonator [18]. This provides a way for each soliton to in-
teract with a co-propagating replica of the other soliton.
Consider first the case of degenerate pumping frequen-
cies. Here, a soliton propagating along say the clockwise
(CW) direction will be able to interact with the replica
counter-clockwise (CCW) soliton. The CCW and CW
solitons share the same longitudincal mode family and
similar pump conditions (same pump frequency and sim-
ilar pump power). Moreover, a feature of coherent pump-
ing is that the pump frequency coincides with a soliton
comb frequency. For these reasons both the envelope and
the carrier of the CP solitons will have closely matched
velocities. Adding to this situation, the Kerr effect me-
diated interaction allows the two solitons to readily trap
one another via the backscattered replica fields. This oc-
curs in a way very similar to trapping of conventional co-
propagating solitons [1–3]. Thus, even while the solitons
are propagating in opposite directions, their group and
phase velocities become locked. A caveat here concerns
the nature of the replica solitons. Generally, backscatter-
ing would not be expected to occur at a single point, but
instead to exhibit a complex spatial profile. As a result,
the trapping itself would be expected to reflect the com-
plexities of this scattering process. Indeed, evidence of
cases where counter-propagating solitons are only weakly
locked (i.e., spatially broad replica solitons) are observ-
arXiv:2003.00573v1 [physics.optics] 1 Mar 2020
2
Conventional soliton trapping
ν
f
r2
= f
r1
f
02
=f
01
f
r1
BOC
Oscilloscope
Pump
PS+EDFA
AOM
EDFA
Relative motion
ν
f
r2
= f
r1
f
02
f
01
sidebands
f
r1
coherent pump 1
coherent pump 2
...
...
L/v
g
L/v
p
...
...
1
/
δν
0
1
/
δν
P
...
(a)
(b)
(c)
Moment 1
Moment 2
Moment n
1
/
δν
P
Moment 3
rel. motion
AOM
...
PS+EDFA
Trapping with forced motion
FIG. 1:
Trapping with forced motion in counter-propagating solitons.
(a) In conventional soliton trapping
without coherent pumps, both phase velocity and group velocity will be locked. In the frequency domain, both the
repetition rate
f
r
and 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) In coherently pumped systems, phase velocity and
f
0
cannot be locked when using
non-degenerate pumps. In this case, the group velocity of the solitons will be locked in an averaged way and there
is relative motion in the trapping. This relative motion repeats with a period of 1/
δν
P
. In the frequency domain,
the motion will induce sidebands around the main comb lines. (c) Illustration showing CP solitons at various mo-
ments in time. In the measurements, a balanced optical cross correlator (BOC) is used to measure relative soliton
motion (blue line). The counter-pumping frequencies are controlled by two acousto-optical modulators (AOMs).
The green line illustrates the expected BOC output signal for conventional (non oscillatory) trapping. EDFA:
erbium-doped fiber amplifier, PS: pulse shaper.
able in the measurements.
Next consider the impact of detuning the CW and
CCW pumping frequencies on this process. When the
pump detuning frequency (
δν
P
) is small in comparison
to the cavity free-spectral-range (FSR), 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 will be shifted by
δν
P
relative to the
replica of the other soliton comb. This frequency shift is
set by the external CW and CCW pumps and cannot be
pulled towards zero as in conventional soliton trapping.
In other words, the soliton carriers and phase velocities
will not be synchronized. Thus, each of the backscat-
tered CW (CCW) comb lines can be regarded as a side-
band that modulates the CCW (CW) comb lines. As
shown below, this causes each soliton to experience a pe-
riodically time-varying spectral center frequency change,
δν
s
. And, in turn, their group velocity will vary with
the spectral center as
δ
(1
/v
g
) = 2
πβ
2
δν
s
, where
β
2
is the
group velocity dispersion (negative for cavities that sup-
port solitons) [20]. This velocity modulation causes peri-
odic relative motion of the CP solitons with a period of
1/
δν
P
within their traps. Finally, since the soliton group
velocity and round-trip time are modulated periodically
with a considerable amplitude, sidebands (spacing
δν
P
)
emerge around the main comb lines (Fig. 1(b)).
Observation of the relative motion requires high tem-
poral resolution. We use a balanced optical cross corre-
3
0
0.1
0.2
0.3
0.4
0.5
Time (ms)
0.1 ms
0
10
20
30
40
50
Time (
μ
s)
Relative delay (100 fs/div)
10
μ
s
250 fs
Relative delay (200 fs/div)
1510
1530
1550
1570
1590
Exp.
Sim.
Power (20 dB/div)
Wavelength (nm)
(a)
(b)
(c)
0
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
0
10
20
30
40
50
Time (
μ
s)
Relative delay (ps)
-10
-5
0
5
Relative frequency (GHz)
0
10
20
30
40
50
Time (
μ
s)
Exp. 10 kHz
Exp. 100 kHz
(d)
(e)
(f)
3 dB:1.4 THz
10
μ
s
Region I
Region II
Sim. 100 kHz
760 fs
0
0.2
0.4
0.6
730 fs
0
0.1
0.2
0.3
0.4
FIG. 2:
Observation of oscillatory motion in CP soliton trapping.
(a) Optical spectrum of the CP solitons
with a 3 dB bandwidth of 1.4 THz. The red line is the simulated spectral envelope and is in agreement with exper-
imental measurement. (b, c) BOC measured relative soliton temporal motion when the pump frequency detuning
is 10 kHz and 100 kHz. The motion frequency is measured to be equal to the counter-pumping frequency detuning.
The red and blue dashed lines indicate the center of motion for 10 kHz and 100 kHz detuning, respectively; and
they are shifted by about 200 fs as shown in panel b. (d, e) Simulations of relative soliton temporal motion when
the two pumps are detuned by 10 kHz and 100 kHz, respectively. The red dashed lines indicate the zero delay. (f)
Numerically calculated relative spectral center frequency between two CP solitons for
δν
P
=100 kHz, showing peri-
odic variation. Depending upon the sign of the relative frequency, the motion can be separated into two regions as
indicated by the green dashed lines.
lator (BOC) to record this motion (experimental setup
shown in Fig. 1(c)). The BOC converts temporal mo-
tion into a voltage signal with a steep discriminator slope
[21]. A single laser pump is used and distinct pumping
frequencies for CW and CCW directions are produced
by two acousto-optical modulators (AOMs). The CP
solitons are generated in a 22 GHz high-Q silica wedge
micoresonator [10, 18, 22]. The corresponding optical
spectrum of one of the solitons is shown in Fig. 2(a)
and has a 3 dB bandwidth of 1.4 THz so that the soli-
ton duration is deduced to be 125 fs (equivalently, 220 fs
for full-width-half-maximum, FWHM, pulse width). The
soliton streams (with pumps suppressed) are dispersion
compensated by pulse shapers [23] and amplified to feed
into the BOC.
The output of the BOC for pump detuning frequen-
cies of 10 kHz or 100 kHz is shown in Figs. 2(b), (c).
The ability to reliably observe a signal implies that the
repetition rates of the two solitons are locked on average,
since otherwise the two inputs would temporally walk-off
due to non-synchronized repetition rates [21]. The peak-
to-peak oscillation amplitude is observed to be as large
as 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). Furthermore, the motion is not sinusoidal
but is asymmetric (sawtooth-like for the 10 kHz detuning
case). Significantly, the oscillation frequency is equal to
the pump frequency detuning (
δν
P
). The center of the
relative motion trajectory is also plotted in Figs. 2(b),
(c). It suggests that the CP solitons will oscillate around
different centers in the trap when the pump detuning fre-
quency varies. This oscillatory motion was also observed
to exist for small detunings,
δν
P
<
1 Hz.
To further confirm the experimental observations,
numerical simulations based on coupled generalized
Lugiato-Lefever equations (LLEs) were performed [18,
4
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. 3:
Measured electrical spectra of the beat
between the CP solitons.
(a) Electrical spectrum
when beating two CP soltions with
δν
P
= 10 kHz. (b)
Electrical spectrum when beating two CP soltions with
δν
P
= 100 kHz.
24, 25]. Details on the simulation are provided in the
Supplementary Materials. The simulated soliton spec-
trum is in excellent agreement with experiments (see Fig.
2(a)). Moreover, upon detuning the counter-pumping
frequencies, the CP solitons undergo periodic relative
motion in simulations. Representative plots of the rela-
tive soliton motion are shown in Figs. 2(d), (e). The mo-
tional frequency equals
δν
P
, and both the trajectory and
amplitude are reasonably consistent with experimental
measurements. For example, the sawtooth-like behavior
is numerically reproduced (Fig. 2(d)).
To test the hypothesis that the relative motion is
driven by the detuned-pump-induced soliton spectral-
center-shift, we numerically calculated the relative spec-
tral center frequency between the two solitons Fig. 2(f).
It exhibits periodic oscillation around 0 Hz. The posi-
tive and negative relative frequency regions correspond
to forced motion where the derivative of the relative de-
lay (i.e., relative group velocity) is positive or negative,
respectively (see the green dashed vertical lines in Fig.
2(e), (f)). Accordingly, the experimental and numerical
observations validate the existence of the coherent-pump
forced oscillatory motion between CP solitons in the pres-
ence of backscattering.
Finally, we experimentally verify that the forced mo-
tion introduces fine structure sidebands into the comb
lines. For this measurement, the two CP soliton mi-
crocombs are heterodyned on a balanced photodetector.
The recorded electrical spectra, shown in Fig. 3, contain
multiple RF tones for both pump detuning of 10 kHz and
100 kHz. The lack of any tone other than those at integer
multiples of
δν
P
shows that only fine structure sidebands
spaced by
δν
P
are present in the optical comb spectra.
The range of detuning frequencies over which this trap-
ping occurs depends upon the backscattering level. A
prior study of CP solitons shows that beyond pump de-
tuning frequencies of several 100 kHz, the CP solitons
unlock and feature independently controllable repetition
rates [18]. For small
ν
P
, a locking zone with the same
repetition rate for CP solions was observed in that work,
which was not explored there but can be understood
to result from the average locking described here. In
the regime of independent repetition rates control, CP
solitons can also exhibit interactions via backscattering
which can cause the solitons to experience optical phase
locking at specific pump detuning frequencies (e.g., sev-
eral MHz) [18].
In summary, we observe a new manifestation of soli-
ton trapping in high-Q microresonators as a result of
the coherently pumped nature of the solitons. This phe-
nomenon results from slightly detuned counter-pumping
of CP solitons in the presence of optical backscattering.
The detuned counter pumps cause a periodic variation
of both the spectral center frequencies of the CP solitons
and their relative positions. Our measurements show
that this trapping does not require the solitons to be
constrained within a small temporal range. The oscilla-
tory motion also inserts fine structure sidebands into the
soliton microcomb spectrum that may affect some comb
applications. This mechanism may be engineered to pro-
gram and control the pulse timing of microresonator soli-
tons and could also be important for the development of
microcomb-based gyroscopes.
This work is supported by the Air Force Office of
Scientific Research (FA9550-18-1-0353) and the Kavli
Nanoscience Institute. C.B. gratefully acknowledges a
postdoctoral fellowship from the Resnick Institute at Cal-
tech.
vahala@caltech.edu
[1] Curtis R Menyuk, “Stability of solitons in birefringent
optical fibers. I: equal propagation amplitudes,” Opt.
Lett.
12
, 614–616 (1987).
[2] MN Islam, CD Poole, and JP Gordon, “Soliton trapping
in birefringent optical fibers,” Opt. Lett.
14
, 1011–1013
(1989).
[3] Steven T Cundiff, BC Collings, NN Akhmediev, Jose M
Soto-Crespo, K Bergman, and WH Knox, “Observation
of polarization-locked vector solitons in an optical fiber,”
Phys. Rev. Lett.
82
, 3988 (1999).
[4] William H Renninger and Frank W Wise, “Optical soli-
tons in graded-index multimode fibres,” Nature Commu-
nications
4
, 1719 (2013).
5
[5] Qi-Fan Yang, Xu Yi, Ki Youl Yang, and Kerry Vahala,
“Stokes solitons in optical microcavities,” Nature Physics
13
, 53 (2017).
[6] Jae K Jang, Alexander Klenner, Xingchen Ji, Yoshitomo
Okawachi, Michal Lipson, and Alexander L Gaeta, “Syn-
chronization of coupled optical microresonators,” Nature
Photonics
12
, 688 (2018).
[7] Stefan Wabnitz, “Suppression of interactions in a phase-
locked soliton optical memory,” Opt. Lett.
18
, 601–603
(1993).
[8] Fran ̧cois Leo, St ́ephane Coen, Pascal Kockaert, Simon-
Pierre Gorza, Philippe Emplit, and Marc Haelterman,
“Temporal cavity solitons in one-dimensional Kerr media
as bits in an all-optical buffer,” Nature Photonics
4
, 471
(2010).
[9] Tobias Herr, Victor Brasch, John D Jost, Christine Y
Wang, Nikita M Kondratiev, Michael L Gorodetsky, and
Tobias J Kippenberg, “Temporal solitons in optical mi-
croresonators,” Nature Photonics
8
, 145 (2014).
[10] Xu Yi, Qi-Fan Yang, Ki Youl Yang, Myoung-Gyun Suh,
and Kerry Vahala, “Soliton frequency comb at microwave
rates in a high-Q silica microresonator,” Optica
2
, 1078–
1085 (2015).
[11] Pei-Hsun Wang, Jose A Jaramillo-Villegas, Yi Xuan, Xi-
aoxiao Xue, Chengying Bao, Daniel E Leaird, Minghao
Qi, and Andrew M Weiner, “Intracavity characterization
of micro-comb generation in the single-soliton regime,”
Opt. Express
24
, 10890–10897 (2016).
[12] Chaitanya Joshi, Jae K Jang, Kevin Luke, Xingchen
Ji, Steven A Miller, Alexander Klenner, Yoshitomo
Okawachi, Michal Lipson,
and Alexander L Gaeta,
“Thermally controlled comb generation and soliton mod-
elocking in microresonators,” Opt. Lett.
41
, 2565–2568
(2016).
[13] Tobias J Kippenberg, Alexander L Gaeta, Michal Lipson,
and Michael L Gorodetsky, “Dissipative Kerr solitons in
optical microresonators,” Science
361
, eaan8083 (2018).
[14] Yadong Wang, Fran ̧cois Leo, Julien Fatome, Miro Erkin-
talo, Stuart G Murdoch, and St ́ephane Coen, “Universal
mechanism for the binding of temporal cavity solitons,”
Optica
4
, 855–863 (2017).
[15] Hossein Taheri, Andrey B Matsko, and Lute Maleki,
“Optical lattice trap for Kerr solitons,” Eur. Phys. J. D
71
, 153 (2017).
[16] Jae K Jang, Miro Erkintalo, St ́ephane Coen, and Stu-
art G Murdoch, “Temporal tweezing of light through the
trapping and manipulation of temporal cavity solitons,”
Nature Communications
6
, 7370 (2015).
[17] Daniel C Cole, Erin S Lamb, Pascal DelHaye, Scott A
Diddams, and Scott B Papp, “Soliton crystals in Kerr
resonators,” Nature Photonics
11
, 671 (2017).
[18] Qi-Fan Yang, Xu Yi, Ki Youl Yang, and Kerry Vahala,
“Counter-propagating solitons in microresonators,” Na-
ture Photonics
11
, 560–564 (2017).
[19] Chaitanya Joshi,
Alexander Klenner,
Yoshitomo
Okawachi, Mengjie Yu, Kevin Luke, Xingchen Ji, Michal
Lipson,
and Alexander L Gaeta, “Counter-rotating
cavity solitons in a silicon nitride microresonator,” Opt.
Lett.
43
, 547–550 (2018).
[20] James P Gordon and Hermann A Haus, “Random walk
of coherently amplified solitons in optical fiber transmis-
sion,” Opt. Lett.
11
, 665–667 (1986).
[21] Jungwon Kim and Franz X Kaertner, “Attosecond-
precision ultrafast photonics,” Laser & Photonics Re-
views
4
, 432–456 (2010).
[22] Hansuek Lee, Tong Chen, Jiang Li, Ki Youl Yang, Seok-
min Jeon, Oskar Painter, and Kerry J Vahala, “Chem-
ically etched ultrahigh-Q wedge-resonator on a silicon
chip,” Nature Photonics
6
, 369 (2012).
[23] Andrew M Weiner, “Femtosecond pulse shaping using
spatial light modulators,” Rev. Sci. Instr.
71
, 1929–1960
(2000).
[24] St ́ephane Coen, Hamish G Randle, Thibaut Sylvestre,
and Miro Erkintalo, “Modeling of octave-spanning Kerr
frequency combs using a generalized mean-field Lugiato–
Lefever model,” Opt. Lett.
38
, 37–39 (2013).
[25] Yanne K Chembo and Curtis R Menyuk, “Spatiotem-
poral Lugiato-Lefever formalism for Kerr-comb genera-
tion in whispering-gallery-mode resonators,” Phys. Rev.
A
87
, 053852 (2013).