Imaging soliton dynamics in optical microcavities
Xu Yi
∗
, Qi-Fan Yang
∗
, 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 22, 2018)
Solitons are self-sustained wavepackets that occur
in many physical systems. Their recent demon-
stration in optical microresonators has provided
a new platform for study of nonlinear optical
physics with practical implications for miniatur-
ization of time standards, spectroscopy tools and
frequency metrology systems. However, despite
its importance to understanding of soliton physics
as well as development of new applications, imag-
ing the rich dynamical behaviour of solitons in
microcavities has not been possible. These phe-
nomena require a difficult combination of high-
temporal-resolution and long-record-length in or-
der 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 arbi-
trary time spans. A wide range of complex soliton
transient behaviors are characterized in the tem-
poral or spectral domain, including soliton for-
mation, collisions, spectral breathing and soliton
decay. This method can serve as a universal visu-
alization tool for understanding complex soliton
physics in microcavities.
Temporal solitons are indispensable in optical fiber
systems
1
and exhibit remarkable nonlinear phenomena
2
.
The potential application of solitons to buffers and
memories
3,4
as well as interest in soliton physics has
stimulated approaches for experimental visualization of
multi-soliton trajectories. Along these lines, the display
of solitons 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
fiber systems
5–9
. However, this useful data visualization
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
10
can provide the required femtosecond-resolution, but has
a limited record length set by the pump pulse. Also, while
the relative position of closely-spaced soliton complexes
9
can be inferred over time from their composite DFT
spectra
11
, Fourier inversion requires the constituent soli-
tons to have similar waveforms which restricts the gen-
erality of the technique.
These limitations are placed in sharp focus by
recent
demonstrations
of
soliton
generation
in
microcavities
12–18
.
This new type of dissipative
soliton
19
was long considered a theoretical possibility
3
and was first observed in optical fiber resonators
4
.
Their microcavity embodiment poses severe challenges
for imaging of dynamical phenomena by conventional
methods, because multi-soliton states feature inherently
closely spaced solitons. Nonetheless, the compactness of
these systems has tremendous practical importance for
miniaturization of frequency comb technology
20
through
chip-based microcombs
21,22
.
Indeed, spectroscopy
systems
23
, coherent communication
24
, ranging
25,26
, and
frequency synthesis
27
demonstrations using the new
miniature platform have already been reported. More-
over, the unique physics of the new soliton microcavity
system has lead to observation of many unforeseen
physical phenomena involving compound soliton states,
such as Stokes solitons
28
, soliton number switching
29
and soliton crystals
30
.
In this work, we report imaging of a wide range of
soliton phenomena in microcavities. Soliton formation,
collisions
8
, breathing
6,32–34
, Raman shifting
35,36
as well
as soliton decay are observed. Significantly, femtosecond-
time-scale resolution over arbitrary time spans (dis-
tances) is demonstrated (and required) in these mea-
surements. Also, real-time spectrograms are produced
along-side high-resolution soliton trajectories.
These
features are derived by adapting coherent linear opti-
cal sampling
37–39
to the problem of microcavity solition
imaging. To image the soliton trajectories, a separate op-
tical 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 rela-
tive to the microcavity signal pulses as illustrated in fig.
1a. By heterodyne detection of the combined streams,
the probe pulses coherently sample the microcavity signal
producing a temporal interferogram
40,41
shown in fig. 1a.
Ultimately, the time shift per pulse accumulates so that
the sampling repeats in the interferogram 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 soli-
ton 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 sec-
ond microcavity soliton source operating in steady state.
However, in the present measurement, an electro-optical
arXiv:1805.07629v1 [physics.optics] 19 May 2018
2
Time (
μ
s)
0
20
30
40
50
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
20
30
40
50
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
Kerr cavity
Probe
PD
1540
1545
1550
1555
1560
Power (10 dB/div)
Wavelength (nm)
780
770
0
20
40
775
Delay (ps)
Wavelength (nm)
46 ps
-2
-1
0
1
2
Intensity
~800 fs
1
1
2
2
3
3
3
4
4
FIG. 1:
Coherent sampling of dissipative Kerr soliton dynamics. a,
Conceptual schematic showing micro-
cavity 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 temporally samples the signal upon
photo detection to produce an interferogram signal shown in the lower panel. The measured interferogram shows
several frame periods during which two solitons appear with one of the solitons experiencing decay.
b,
Left panel
is the optical spectrum and right panel is the FROG trace of the probe EO comb (pulse repetition period is shown
as 46 ps). An intensity autocorrelation in the inset shows a full-width-half-maximum pulse width of 800 fs.
c,
Mi-
croresonator pump power transmission when the pump laser frequency scans from higher to lower frequency. Mul-
tiple “steps” indicate the formation of solitons.
d,
Imaging of soliton formation corresponding to the scan in panel
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 signal 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
31
in the modulation instability (MI) regime
3,4
, non-periodic behavior (MI regime),
four soliton and single soliton states
4,12
.
(EO) comb is used
38,39,42
. The EO comb pulse rate is
conveniently adjusted electronically to match the rates
of various phenomena being probed within the microcav-
ity.
The soliton signal is produced by a 3 mm diameter sil-
ica wedge resonator with FSR of 22 GHz and intrinsic
quality factor above 200 million
13,43
. The device gener-
ates femtosecond soliton pulses when pumped at frequen-
cies slightly lower than a cavity resonant frequency
13
.
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 opti-
cal bandwidth (within 1 dB power variation) and an 800
fs FWHM pulse width measured by frequency-resolved
optical gating and autocorrelation as shown in fig. 1b. A
detailed schematic of the complete experimental setup is
provided in the supplementary fig. S1. In all presented
measurements, the pump laser of the resonator scans lin-
early from higher to lower frequency to initiate paramet-
ric oscillation in the microcavity followed by chaotic dy-
namics. Ultimately, “step-like” features are observable in
the resonator transmitted power (fig. 1c) indicating the
formation of soliton states
12
. The typical pump power
and laser scan speed are
∼
70 mW and
∼−
1 MHz/
μ
s,
respectively.
As described above, heterodyne-detection of the soli-
ton signal and the EO-comb pulse produces the electrical
interferogram. The period of the signals in the intero-
gram 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 nom-
inal 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 pe-
riod 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 con-
nection to the physical time scale of the solitons, the y-
axis time scale is also compressed by the same bandwidth
3
Time (
μ
s)
0
6
3
Co-rotating time (ps)
10
20
0
0
6
3
Co-rotating time (ps)
10
20
0
c
b
a
d
Intensity (a.u.)
0
1
Time (
μ
s)
0
8
4
Mode number
-20
20
0
Co-rotating time (ps)
10
20
0
30
0
8
4
Intensity (a.u.)
0
1
Intensity (dB)
0
30
0
0.2
0.4
0.6
0.8
5
0
FIG. 2:
Temporal and spectral measurements of non-repetitive soliton events. a,
Two solitons collide
and annihilate.
b,
Two solitons survive a collision, but collide again and one soliton is annihilated.
c,
Motion of
a single soliton state showing peak power breathing along its trajectory. A zoom-in view of the white rectangular
region is shown as the inset.
d,
Spectral dynamics corresponding to panel
c
. The y-axis is the relative longitudinal
mode number corresponding to specific 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. The frame rate is 50 MHz for all panels.
compression factor (
T
×
FSR) that accompanies the sam-
pling process. 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. 1d. 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 hori-
zontal line in fig. 1d. On the other hand, slower (higher)
rate solitons would appear as lines tilted upward (down-
ward) 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 profile. The vertical co-rotating time
axis can be readily mapped into an image of the soliton
angular position within the circular microcavity as shown
in fig. 1d.
Imaging of soliton formation and multi-soliton trajec-
tories is observable in fig. 1d. For comparison with the
transmitted power, the time-axis scale is identical in fig.
1c and fig. 1d. As the pump laser frequency initially
scans towards the microcavity resonant frequency its cou-
pled power increases. At
∼
8
μ
s the resonator enters the
modulation instability regime
3,4,12
. Initially, a periodic
temporal pattern is observable in fig. 1d corresponding
to parametric oscillation
31
. Soon after, the cavity en-
ters 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 disap-
pear one-by-one. All solitons have upward curved trajec-
tories, 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 ef-
fect and anomalous dispersion in the silica resonator
35,44
and a similar effect on soliton trajectory is observed in
optical fiber cavities
36
. The features of soliton formation
and evolution observed in fig. 1d compare well with nu-
merical simulations presented in Supplementary fig. S2.
Moreover, relative soliton positions can be extracted from
the interferogram measurement (Supplementary fig. S3)
and illustrate solitons stabilizing their relative positions.
Movies of the corresponding multi-soliton motion around
the microcavity are also provided in the Supplementary
Section. Finally, the cavity states at four moments in
time are plotted within the circular microcavity in fig. 1e.
These correspond to parametric oscillation, non-periodic
modulational instability, four soliton and single soliton
states.
A variety of non-repetitive multi and single soliton phe-
nomena were measured in both temporal and spectral do-
mains. To enable more rapid imaging the repetation rate
of EO comb was adjusted to produce an interferogram at
a rate of approximately 50 MHz. The frame period,
T
,
was then reduced accordingly to approximately 20 nsec.
Fig. 2a-b present observations of two solitons interact-
ing. Soliton annihilation is observed in fig. 2a, wherein
two solitons move towards each other, collide, create an
intense peak upon collision and then disappear. A new
phenomena, a “wave splash”, is observed immediately
following the collision. In fig. 2b, two solitons collide but
quickly recover and then collide again, after which point
4
a
b
400
600
800
Time (ns)
Resolution limit
0.0
0.2
0.4
0.6
Envelope (a.u.)
0.5
1.0
1.5
2.0
0.0
0.4
0.8
Pu
lse width
τ
s
(ps)
τ
s
A
E
(a.u.)
FIG. 3:
Characterization of soliton decay. a,
Inter-
ferogram envelope showing a single soliton experiencing
decay. An exponential fitting is given as the dashed
black line.
b,
The measured pulse width (blue) is plot-
ted versus time and its resolution limit is set by the EO
comb pulse width. The product of soliton amplitude
and pulse width is plotted in red.
one soliton is annihilated. Significantly, the observation
of these complex motions requires measurement of events
in close temporal proximity over long time spans. Figure
2c-d shows measurement of a breathing soliton
32
in both
the temporal and frequency domains. The spectrogram is
obtained by applying a Fourier transform to the interfer-
ogram signal
41
. The spectrum is widest when the soliton
has maximum peak power. As an observation unrelated
to the breathing action, the soliton spectral envelope in
fig. 2d is continuously red shifted in frequency by the
Raman self-frequency shift
13,35
as its average power in-
creases (increasing time in the plot).
Finally, soliton decay is analyzed using the sampling
method. The measurement results are shown in fig. 3. In
the experiment, the pump laser frequency is continuously
tuned towards lower frequencies. After soliton formation,
at some point the cavity-laser frequency detuning exceeds
the soliton existence range and the soliton decays
12,13
.
Fig. 3a shows the interferogram signal just before and
during the decay. Pulse widths (
τ
s
) are extracted during
the decay process and are plotted in fig. 3b. Also plotted
in fig. 3b is the product of pulse width and soliton peak
amplitude (
A
E
). Curiously, the soliton pulse width and
peak amplitude preserve the same soliton product rela-
tionship as prior to decay. This is an indication that the
decaying soliton pulse in the microcavity is constantly
adapting itself to maintain the soliton waveform. A sim-
ilar behavior is known to occur for conventional solitons
in optical fiber
45
. To the authors knowledge, this is the
first time this behavior has been observed in real time.
In the Method section the amplitude decay of the soliton
in the interferogram trace is analyzed to extract a decay
time and the cavity Q factor.
Coherent sampling induces a large bandwidth com-
pression of the ultrafast signal that is equal to the sam-
pling rate divided by the difference in the signal rate and
the sampling rate. This compression is well known in
the related techniques of dual comb spectroscopy
41
and
dual comb ranging
40
, and is also present in sampling of
optical signals by four-wave mixing in optical fibers
46
.
In order to avoid spectral folding, the compressed signal
bandwidth must lie within half of the EO comb sampling
rate
40,41
(the Nyquist condition for sampling). As shown
in the Method section, this basic condition establishes the
following relationship between temporal resolution (
τ
),
frame rate (
f
) and the sampling rate (approximately the
microcavity free-spectral-range, FSR):
f < τ
FSR
2
/
2.
This condition also reveals the quadratic importance of
high sampling rates (equivalently large FSRs and corre-
spondingly large soliton repetition rates) to create fast
frame rates. In the current system, a temporal resolu-
tion of less than 1 ps combined with a 22 GHz sampling
rate can enable frame rates as high as 200 MHz.
Imaging of nonlinear dynamical phenomena including
complex soliton interactions with high temporal/spatial
resolution over arbitray time/length spans has been
demonstrated. The temporal resolution in the current
experiment is limited to 800 fs, however, resolution at
the 10s of fs level is possible by spectrally broadening
the EO comb
47
used for coherent sampling. It is also
possible to replace the EO-comb with a microcomb that
is closely matched to the FSR of a microcavity to be sam-
pled. Such matching has been recently used to implement
dual soliton microcomb spectroscopy measurements
23
. In
this case, even higher sampling rates would be possible
that would enable GHz-scale frame rates. The coherent
sampling method can serve as a general real-time state
visualization tool to monitor the dynamics of microcav-
ity systems. It would provide an ideal way to monitor
the formation and evolution of soliton complexes such as
Stokes solitons
28
, soliton number switching
29
and soli-
ton crystals
30
. It can also be used to monitor the state
of chip-based optical memories based on microresonator
solitons.
Methods
Time constant in soliton decay.
In the soliton decay process,
the average intracavity energy decays exponentially and its time
constant equals the dissipation rate of the cavity (
κ
=
ω/Q
), where
ω
is the optical frequency and Q is the loaded cavity Q factor. For
large cavity-laser frequency detuning
12,44
, the average intracavity
energy is approximately the soliton energy,
τ
s
A
2
E
, such that
τ
s
(
t
)
A
2
E
(
t
) =
τ
s
(0)
A
2
E
(0)
e
−
κt
.
(1)
When the dissipation rate is relatively small compared to soliton
Kerr nonlinear shift, the dissipation is a perturbation and the pulse
maintains its soliton waveform
45
. The corresponding balance of
dispersion and Kerr nonlinearity requires that the product of soli-
ton amplitude and pulse width be constant. This condition was
also verified experimentally in figure 3b
44,45
,
τ
s
(
t
)
A
E
(
t
) =
τ
s
(0)
A
E
(0)
.
(2)
5
Inserting eq. (2) into eq. (1) gives,
A
E
(
t
) =
A
E
(0)
e
−
κt
,τ
s
(
t
) =
τ
s
(0)
e
κt
,A
2
E
(
t
) =
A
2
E
(0)
e
−
2
κt
.
(3)
In particular, the soliton amplitude decays at the cavity dissipation
rate, the pulse width exponentially grows, and the soliton peak
power decays twice as fast as the cavity dissipation rate. In the
experiment, the fitted decay constant of the soliton amplitude is
133 ns, which corresponds to
κ/
(2
π
) = 1
.
2 MHz giving
Q
= 161
million. This value is in reasonable agreement with the measured
loaded-Q factor of 140 million.
Nyquist condition for sampling.
In the EO comb sampling
process the optical to electrical conversion is accompanied by a
large bandwidth compression of the sampled signal. In effect, sam-
pling stretches the time scale so that, for example, the optical
temporal resolution (
τ
) is stretched to
τ
×
FSR
/f
after conver-
sion to the electrical signal where
f
is the frame rate given by
f
≈
FSR
−
f
comb
. This stretching means that the THz EO comb
resolution bandwidth is compressed to an electrical bandwidth of
f/
(
τ
FSR). To avoid nonsensical signals in the electrical spectrum,
the compressed bandwidth should lie within the Nyquist frequency
set by the FSR
40
. This gives the condition
f/
(
τFSR
)
<
FSR
/
2, or
f < τ
FSR
2
/
2. In practice, when the oscilloscope bandwidth (
f
osc
)
is smaller than the Nyquist frequency, the interferogram signal will
be limited by the oscilloscope instead of the Nyquist frequncy, such
that
f/
(
τ
FSR)
< f
osc
, or
f < τf
osc
FSR. This is, in fact, the case
in the present measurement as the oscilloscope bandwidth is 4 GHz
while the Nyquist frequency is 11 GHz. In addition, the frequency
components of the interferogram signal must be positive to avoid
frequency folding near zero frequency. This requires that the car-
rier frequency of the interferogram signal is larger than half of the
electrical bandwidth. In the present measurement, the carrier fre-
quency is the frequency offset between the EO comb pump laser
and the microcavity pump laser (defined as ∆Ω). As a result, this
condition is expressed as ∆Ω
> f/
(2
τ
FSR).
Data availability.
The data that support the plots
within this paper and other findings of this study are
available from the corresponding author upon reasonable
request.
Acknowledgement
The authors thank Stephane Coen and Yun-Feng Xiao
for helpful comments during the preparation of this
manuscript and gratefully acknowledge the Air Force Of-
fice of Scientific Research (AFOSR), NASA and the Kavli
Nanoscience Institute.
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