of 8
ARTICLE
Imaging soliton dynamics in optical microcavities
Xu Yi
1
, Qi-Fan Yang
1
, Ki Youl Yang
1
& Kerry Vahala
1
Solitons are self-sustained wavepackets that occur in many physical systems. Their recent
demonstration in optical microresonators has provided a new platform for the study of
nonlinear optical physics with practical implications for miniaturization of time standards,
spectroscopy tools, and frequency metrology systems. However, despite its importance to
the understanding of soliton physics, as well as development of new applications, imaging the
rich dynamical behavior of solitons in microcavities has not been possible. These phenomena
require a dif
fi
cult combination of high-temporal-resolution and long-record-length in order to
capture the evolving trajectories of closely spaced microcavity solitons. Here, an imaging
method is demonstrated that visualizes soliton motion with sub-picosecond resolution over
arbitrary time spans. A wide range of complex soliton transient behavior are characterized in
the temporal or spectral domain, including soliton formation, collisions, spectral breathing,
and soliton decay. This method can serve as a visualization tool for developing new soliton
applications and understanding complex soliton physics in microcavities.
DOI: 10.1038/s41467-018-06031-5
OPEN
1
T.J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA. These authors contributed equally: Xu Yi, Q
i-Fan
Yang. Correspondence and requests for materials should be addressed to K.V. (email:
vahala@caltech.edu
)
NATURE COMMUNICATIONS
| (2018) 9:3565 | DOI: 10.1038/s41467-018-06031-5 | www.nature.com/naturecommunications
1
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T
emporal solitons are indispensable in optical
fi
ber systems
1
and exhibit remarkable nonlinear phenomena
2
. The
potential application of solitons to buffers and mem-
ories
3
,
4
, as well as interest in soliton physics has stimulated
approaches for experimental visualization of multi-soliton tra-
jectories. Along these lines, the display of soliton trajectories in a
co-moving frame
5
allows an observer to move with the solitons
and is being used to monitor soliton control and interactions of
all types in
fi
ber systems
5
12
. However, this useful data visuali-
zation method relies upon soliton pulse measurements that are
either limited in bandwidth (pulse resolution) or record length. It
is therefore challenging to temporally resolve solitons over the
periods often required to observe their complete evolution. For
example, the time-lens method
13
can provide the required fem-
tosecond-resolution, but has a limited record length set by the
pump pulse. Also, while the relative position of closely spaced
soliton complexes
11
can be inferred over time from their com-
posite dispersive Fourier transform (DFT) spectra
14
, Fourier
inversion requires the constituent solitons to have similar wave-
forms which restricts the generality of the technique. Efforts that
combine these two methods were also reported very recently
15
,
16
.
These limitations are placed in sharp focus by recent demon-
strations of soliton generation in microcavities
17
23
. This new
type of dissipative soliton
24
was long considered a theoretical
possibility
3
and was
fi
rst observed in optical
fi
ber resonators
4
.
Their microcavity embodiment poses severe challenges for ima-
ging of dynamical phenomena by conventional methods, because
multi-soliton states feature inherently closely spaced solitons.
Preliminary real-time measurements using time lens
25
, and direct
detection
26
have been explored, but were limited in either
recording length or pulse resolution. Nonetheless, the compact-
ness of microcavity-based soliton systems has practical impor-
tance for miniaturization of frequency comb technology
27
through chip-based microcombs
28
,
29
. Indeed, spectroscopy sys-
tems
30
,
31
, coherent communication
32
, ranging
33
,
34
, and fre-
quency synthesis
35
demonstrations using the new miniature
platform have already been reported. Moreover, the unique
physics of the new soliton microcavity system has led to obser-
vation of many unforeseen physical phenomena involving com-
pound soliton states, such as Stokes solitons
36
, soliton number
switching
37
and soliton crystals
38
.
In this work, we report imaging of a wide range of soliton
phenomena in microcavities. Soliton formation, collisions
10
,
breathing
6
,
39
41
, Raman shifting
42
,
43
, as well as soliton decay are
observed. Signi
fi
cantly, femtosecond-time-scale resolution over
arbitrary time spans (distances) is demonstrated (and required) in
these measurements. Also, real-time spectrograms are produced
along-side high-resolution soliton trajectories. These features are
derived by adapting coherent linear optical sampling
44
and
electric-
fi
eld cross-correlation
45
to the problem of microcavity
soliton imaging. Beyond the necessity to employ a new method
for imaging soliton motion in microcavities, the high-repetition
rate of microcavity solitons (tens of gigahertz and higher) is
advantageous in sampling-based recording of motion.
Results
Coherent sampling of soliton motion
. To image the soliton
trajectories, a separate optical probe pulse stream is generated at a
pulse rate that is close to the rate of the solitons to be imaged in
the microcavity. The small difference in these rates causes a pulse-
to-pulse temporal shift of the probe pulses relative to the
microcavity signal pulses as illustrated in Fig.
1
a. By heterodyne
detection of the combined streams, the probe pulses
coherently sample the microcavity signal producing a temporal
interferogram
46
,
47
shown in Fig.
1
a. Ultimately, the time shift per
pulse accumulates so that the sampling repeats in the inter-
ferogram at the frame rate which is described below, and is close
in value to the difference of sampling and signal rates. Probe
pulses have a sub-picosecond temporal resolution that enables
precise monitoring of the temporal location of the soliton pulses.
Moreover, the coherent mixing of probe and soliton pulses allows
extraction of each soliton
s spectral evolution by fast Fourier
transform of the interferogram. In principle, the probe pulses can
be generated by a second microcavity soliton source operating in
steady state. However, in the present measurement, an electro-
optic (EO) comb is used
45
,
48
,
49
. The EO comb pulse rate is
conveniently adjusted electronically to match the rates of various
phenomena being probed within the microcavity.
The soliton signal is produced by a 3 mm diameter silica wedge
resonator with a free-spectral-range (FSR) of 22 GHz and
intrinsic quality factor above 200 million
18
,
50
. The device
generates femtosecond soliton pulses when pumped at frequen-
cies slightly lower than a cavity resonant frequency
18
. To sample
the 22 GHz soliton signal the EO comb was formed by
modulation of a tunable continuous-wave (CW) laser. The EO
comb features ~1.3 THz optical bandwidth (within 1 dB power
variation) and an 800 fs full-width-at-half-maximum (FWHM)
pulse width is measured by frequency-resolved optical gating
(FROG) and autocorrelation as shown in Fig.
1
b. Further details
on the experimental setup are provided in the Methods section. In
all presented measurements, the pump laser of the resonator
scans linearly from higher to lower frequency to initiate
parametric oscillation
51
in the microcavity followed by chaotic
dynamics. Ultimately, step-like features are observable in the
resonator transmitted power (Fig.
1
c) indicating the formation of
soliton states
17
. The typical pump power and laser scan speed are
~70 mW and ~1 MHz/
μ
s, respectively.
Measuring multiple soliton trajectories
. As described above,
heterodyne detection of the soliton signal and the EO-comb pulse
produces the electrical interferogram. The period of the signals in
the interogram is adjusted by tuning the EO-comb repetition rate.
In the initial measurements, it is set to ~10 MHz lower than the
rate of the microcavity signal so that the nominal period in the
interferogram is ~100 ns. To display the interferogram signal a
co-rotating frame is applied. First, a frame period
T
is chosen that
is close to the period of signals of interest in the interferogram.
Integer steps (i.e.,
mT
) are plotted along the
x
-axis while the
interferogram is plotted along the
y
-axis, but offset in time by the
x
-axis time step (i.e.,
t-mT
). To make connection to the physical
time scale of the solitons, the
y
-axis time scale is compressed by
the same bandwidth compression factor (
T
×FSR) that accom-
panies the sampling process (see Discussion). The
y
-axis scale is
accordingly set to span one microcavity round-trip time. A typical
measurement plotted in this manner is given in Fig.
1
d. Because
this way of plotting the data creates a co-rotating reference frame,
a hypothetical soliton pulse with an interferogram period equal to
the frame rate
T
would appear as a horizontal line in Fig.
1
d. On
the other hand, slower (higher) rate solitons would appear as lines
tilted upward (downward) in the plot. In creating the imaging
plot, a Hilbert transformation is applied to the interferogram
followed by taking the square of its amplitude to produce a pulse
envelope intensity pro
fi
le. The vertical co-rotating time axis can
be readily mapped into the soliton angular position within the
circular microcavity as shown in Fig.
1
d (right vertical axis).
Imaging of soliton formation and multi-soliton trajectories is
observable in Fig.
1
d. For comparison with the transmitted
power, the time-axis scale is identical in Fig.
1
c, d. As the pump
laser frequency initially scans towards the microcavity resonant
frequency its coupled power increases. At ~8
μ
s the resonator
ARTICLE
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2
NATURE COMMUNICATIONS
| (2018) 9:3565 | DOI: 10.1038/s41467-018-06031-5 | www.nature.com/naturecommunications
enters the modulation instability regime
3
,
4
,
17
. Initially, a periodic
temporal pattern is observable in Fig.
1
d corresponding to
parametric oscillation
51
. Soon after, the cavity enters a regime of
non-periodic oscillation. At ~31
μ
s, this regime suddenly
transitions into four soliton pulses. The soliton positions evolve
with scan time and disappear one-by-one. All solitons have
upward curved trajectories, showing that the soliton repetition
rate decreases as the scan progresses. This soliton rate shift is
caused by the combination of the Raman self-frequency shift
effect and anomalous dispersion in the silica resonator
42
,
52
and a
similar effect on soliton trajectory is observed in optical
fi
ber
resonators
43
. Supplementary Movie 1 provides the corresponding
multi-soliton motion around the microcavity. Finally, the cavity
states at four moments in time are plotted within the circular
microcavity in Fig.
1
e. These correspond to parametric oscillation,
non-periodic modulational instability, four soliton, and single
soliton states.
Observation of soliton collisions
. A variety of non-repetitive
multi and single soliton phenomena were measured in both tem-
poral and spectral domains. To enable more rapid imaging the
repetition rate of EO comb was adjusted to produce an inter-
ferogram at a rate of approximately 50 MHz. The frame period,
T
,
was then reduced accordingly to ~20 ns. Figure
2
presents obser-
vations of two solitons interacting. Soliton annihilation is observed
in Fig.
2
a, wherein two solitons move toward each other, collide,
create an intense peak upon collision and then disappear. A new
phenomena, a wave splash, is observed immediately following the
collision. Though not discussed, this feature appears in recently
reported simulations
12
.InFig.
2
b, two solitons collide but quickly
recover and then collide again, after which point one soliton is
annihilated. Figure
2
c shows a third example in which solitons
merge and a single soliton emerges. In a fourth case shown in
Fig.
2
d, soliton hopping accompanies annihilation of a soliton.
Interestingly, all soliton collisions are observed at the beginning of
solitonformation(inthesolitonbreathingregime).Afterthis
regime, the soliton relative motion quickly stabilizes preventing
collisions. This stabilization process is investigated in a later section.
Also, as noted earlier, the observation of these complex motions
requires measurement of events in close temporal proximity over
long time spans. Finally, numerical simulations of soliton collisions
are shown as inset panels in Fig.
2
. The collisional features observed
in experiments, including the wave splash in Fig.
2
a, are reproduced
in the simulations.
Breather soliton spectrograms
. Figure
3
shows measurement of
a breathing soliton
39
in both the temporal and frequency
domains. The intensity of an individual breather soliton is imaged
in Fig.
3
a. Spectral breathing was explored in
fi
ber-ring resonators
using the DFT method
6
. In the current work, the spectral
breathing is observed by applying a Fourier transform to the
interferogram signal
47
. Figure.
3
b shows the resulting spectro-
gram plotted over the same time interval as Fig.
3
a wherein the
spectrum is widest when the breather soliton has its maximum
Time (
μ
s)
0
20304050
10
Transmission
(a.u.)
Co-rotating time (ps)
10
20
30
40
0
c
d
e
MI: parametric oscillation
MI: non-periodic
Single soliton
a
Multiple solitons
0
20304050
10
Intensity (a.u.)
0
1
Time (ns)
100
200
300
400
Interferogram intensity (a.u.)
0
b
Interferogram
Envelope
Angular position
2
π
0
Soliton rep rate
Probe rep rate
Frame period
Temporal
resolution
Frame period
Probe
PD
μ
Disk
1540
1545
1550
1555
1560
Power (10 dB/div)
Wavelength (nm)
780
770
02040
775
Delay (ps)
Wavelength (nm)
46 ps
–2 –1
0
1
2
Intensity
~800 fs
SHG Intensity
0
1
1
1
2
2
3
33
4
4
Fig. 1
Coherent sampling of dissipative Kerr soliton dynamics.
a
Conceptual schematic showing microcavity signal (red) combined with the probe sampling
pulse train (blue) using a bidirectional coupler. The probe pulse train repetition rate is offset slightly from the microcavity signal. It temporall
y samples the
signal upon photo detection to produce an interferogram signal shown in the lower panel. The measured interferogram shows several frame periods duri
ng
which two solitons appear with one of the solitons experiencing decay. PD: photodetector.
b
Left panel is the optical spectrum and right panel is the second
harmonic generation (SHG) intensity of the probe electro-optic comb measured with FROG (pulse repetition period is shown as 46 ps). An intensity
autocorrelation in the inset shows a full-width-at-half-maximum pulse width of 800 fs.
c
Microresonator pump power transmission when the pump laser
frequency scans from higher to lower frequency. Multiple steps indicate the formation of solitons.
d
Imaging of soliton formation corresponding to the scan
in
c
. The
x
-axis is time and the
y
-axis is time in a frame that rotates with the solitons (full scale is one round-trip time). The right vertical axis is scaled in
radians around the microcavity. Four soliton trajectories are labeled and fold-back into the cavity coordinate system. The color bar gives their sig
nal
intensity.
e
Soliton intensity patterns measured at four moments in time are projected onto the microcavity coordinate frame. The patterns correspond to
initial parametric oscillation
51
in the modulation instability (MI) regime
3
,
4
, non-periodic behavior (MI regime), four soliton and single soliton states
4
,
17
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ARTICLE
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3
peak power. This spectrum also reveals the changing breather
period with frequency scan, which has previously been observed
by measurement of soliton power
26
,
41
. A zoom-in of the soliton
temporal breathing is shown in Fig.
3
c. The combined high frame
rate and sub-ps temporal resolution enable the corresponding
amplitude and pulse width of the breather to be extracted and
these are plotted in Fig.
3
d. As an observation unrelated to the
breathing action, the soliton spectral envelope in Fig.
3
b is con-
tinuously red shifted in frequency by the Raman self-frequency
shift
18
,
42
as its average power increases (increasing time in the
plot).
Tracking relative soliton motion
. Monitoring relative soliton
position in real time is important for study of soliton optical
memories
3
,
4
, their interaction and control
5
,
9
as well as in soliton
crystals
38
. Previously, microcavity soliton relative positions have
been measured by autocorrelation
53
, frequency-resolved optical
gating
17
and synchronized cross-correlation
38
. However, with an
update rate limited by a mechanical delay line, these methods are
only useful for measurement of steady-state phenomena. In this
work, relative soliton positions can be measured in real time from
the interferogram thereby enabling study of their relative motion
dynamics. To plot soliton relative position, one soliton is selected
to be the reference (i.e., zero point of the angular position) and
the angular position relative to the reference soliton is de
fi
ned
from
π
to
π
. Two representative measurements are shown in
Fig.
4
a, b wherein the laser frequency is scanned from high to low
frequency. Even though the reference soliton round-trip rate is
20
ac
d
b
10
Fast time (ps)
Fast time (ps)
Fast time (ps)
Fast time (ps)
0
036
036
03
Slow time (
μ
s)
Slow time (
μ
s)
6
036
20
10
0
20
01
Intensity (a.u.)
10
0
20
10
0
Fig. 2
Measurements of non-repetitive soliton events.
a
Two solitons collide and annihilate. A wave splash appears in the collision.
b
Two solitons survive a
collision, but collide again and one soliton is annihilated.
c
Two solitons collide and merge into a single soliton.
d
A soliton hops in location when another
soliton is annihilated. The measurement frame rate is 50 MHz in all panels. Inset panels show similar collision events from numerical simulation, inc
luding
the appearance of the wave splash (inset in
a
)
30
a
b
c
d
20
10
0
048
0
1.0
0.8
0.6
0.4
0.2
Soliton amplitude (a.u.)
0
1
0
0
5
0.2
0.4
0.6
0.8
0
0.2
Resolution limit
0.4
Time (
μ
s)
0.6
0.8
2.5
Pulse width (ps)
2.0
1.5
1.0
04
Time (
μ
s)
8
20
Mode number
Co-rotating time (ps)
0
–20
Intensity (a.u.)
Co-rotating time (ps)
0
30
Intensity (dB)
Fig. 3
Temporal and spectral measurements of breather solitons.
a
Motion of a single soliton state showing peak power breathing along its trajectory.
b
Spectral dynamics corresponding to
a
. The
y
-axis is the relative longitudinal mode number corresponding to speci
fi
c spectral lines of the soliton. Mode
zero is the pumped microcavity mode. The soliton spectral width breaths as the soliton peak power modulates. The spectrum is widest when peak power is
maximum.
c
Zoom-in view of the white rectangular region in
a
.
d
Soliton amplitude and pulse width breathing corresponding to
c
. The frame rate is 50 MHz
for all panels
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