Supplementary Materials for “Oscillatory Motion of a Counter-Propagating Kerr
Soliton Dimer”
Chengying Bao
1
, Boqiang Shen
1
, Myoung-Gyun Suh
1
, Heming Wang
1
, Kemal
S ̧afak
2
, Anan Dai
1
, Andrey B. Matsko
3
, Franz X. K ̈artner
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, 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 and
5
Department of Physics and the Hamburg Center for Ultrafast Imaging,
University of Hamburg, Hamburg 22761, Germany
I. NUMERICAL SIMULATIONS OF THE FORCED MOTION IN CP SOLITON TRAPPING
The numerical simulations are based on two coupled LLEs, which are written as [S1–S3]
∂A
1
∂T
=
−
(
κ
2
+
iδω
1
)
A
1
−
i
β
2
L
2
T
R
∂
2
A
1
∂t
2
+
iκ
b
A
2
e
i
∆
ω
P
T
+
iγLA
1
T
R
∫
+
∞
−∞
R
(
t
′
)
|
A
1
(
T,t
−
t
′
)
|
2
dt
′
+
√
κ
e
T
R
A
in1
(S1)
∂A
2
∂T
=
−
(
κ
2
+
iδω
2
)
A
2
−
i
β
2
L
2
T
R
∂
2
A
2
∂t
2
+
iκ
b
A
1
e
−
i
∆
ω
P
T
+
iγLA
2
T
R
∫
+
∞
−∞
R
(
t
′
)
|
A
2
(
T,t
−
t
′
)
|
2
dt
′
+
√
κ
e
T
R
A
in2
(S2)
where
T,t
are slow time and fast time, respectively;
A
1
and
A
2
are the envelope of the intracavity field in two directions;
γ
is the nonlinear coefficient and
β
2
is the group velocity dispersion;
δω
1(2)
is the resonator-pump frequency detuning,
κ
is the total loss rate,
κ
e
is the coupling loss rate, and
κ
b
is the backscattering rate (we assume a single scattering
point, thus the pulse waveform is retained). Also, the nonlinear response function
R
(
t
) = (1
−
f
R
)
δ
(
t
) +
f
R
h
R
includes
the instantaneous electronic and delayed Raman contributions. The Raman contribution is calculated in the frequency
domain assuming a Lorentzian gain spectrum centered at
−
14 THz and a bandwidth of 5 THz, and
f
R
=0.21 [S4].
Other cavity parameters are
L
=9.4 mm,
T
R
=40.96 ps,
κ
= 2
π
×
2
.
1 MHz,
κ
e
= 2
π
×
0
.
7 MHz,
β
2
=
−
22 ps
2
/km,
γ
=2.8 W
−
1
km
−
1
, and
κ
b
=2
π
×
10 kHz. The pump condition is chosen as
δω
1
=2
π
×
29 MHz (∆
ω
P
=
δω
1
−
δω
2
and is
chosen to be either
−
2
π
×
10 kHz or
−
2
π
×
100 kHz),
|
A
in1
|
2
=153 mW,
|
A
in2
|
2
=156 mW.
II. ASYMMETRY IN THE FORCED MOTION
The experimentally observed and numerically simulated motion is asymmetric. This asymmetry can arise from the
fact that the pulse profiles in the motion directions are slightly different. In Fig. S1, we show the peak powers of
the two simulated solitons at different delays. For the same relative delay, it can be seen that the peak powers of
the solitons in the two motion directions are different. Hence, the interaction of the CP solitons can be different in
the two directions and an asymmetric motional trajectory can result. Also, the comb lines are red-detuned from the
cavity resonance in the soliton operation. The sidebands around the comb lines arising from the soliton motion can
therefore be slightly asymmetric, which can also contribute to the asymmetric motion.
[S1] S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett.
38
, 37 (2013).
[S2] Y. K. Chembo and C. R. Menyuk, Phys. Rev. A
87
, 053852 (2013).
[S3] Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, Nature Photonics
11
, 560 (2017).
[S4] C. Bao, J. A. Jaramillo-Villegas, Y. Xuan, D. E. Leaird, M. Qi, and A. M. Weiner, Phys. Rev. Lett.
117
, 163901 (2016).
∗
Electronic address:
vahala@caltech.edu
2
0
10
20
Time (
μ
s)
0
100
200
300
400
Relative delay (fs)
548
548.2
548.4
548.6
120
280
549.6
550
550.4
550.8
Peak power (W)
Peak power (W)
Upward
Downward
Upward
Downward
(a)
(b)
(c)
Soliton
A
1
Soliton
A
2
Upward
Downward
FIG. S1:
Asymmetry of the simulated motion trajectory.
(a) The simulated relative soliton motion with a counter-pump
frequency detuning of 100 kHz. (b, c) The peak powers of the two solitons (
A
1
,
A
2
in eqns. S1, S2) at different relative delays
(the studied delay span is indicated by the shaded box in panel (a)). The peak powers of the solitons at the same delay are
different for the two relative motion directions.