of 5
Supplementary Material for “Observation of Breathing Dark Pulses in Normal
Dispersion Optical Microresonators”
Chengying Bao
1
,
,
Yi Xuan
1
,
2
, Cong Wang
1
, Attila F ̈ul ̈op
3
, Daniel E.
Leaird
1
, Victor Torres-Company
3
, Minghao Qi
1
,
2
, and Andrew M. Weiner
1
,
2
1
School of Electrical and Computer Engineering, Purdue University,
465 Northwestern Avenue, West Lafayette, Indiana 47907-2035, USA
2
Birck Nanotechnology Center, Purdue University,
1205 West State Street, West Lafayette, Indiana 47907, USA
3
Photonics Laboratory, Department of Microtechnology and Nanoscience,
Chalmers University of Technology, SE-412 96 Gothenburg, Sweden and
Present address: T. J. Watson Laboratory of Applied Physics,
California Institute of Technology, Pasadena 91125, USA
1. Experimental methods
Line-by-line pulse shaping.
The pulse shaper is used for time domain characterization. The method for shaping
the output comb into transform-limited pulses is described in ref. [S1]. Comb lines with mode number in the range
[
7
,
11] (relative to the pump) are used for pulse shaping. The intensity of the comb lines is adjusted to prevent
the strongest comb lines from dominating the pulse shaping. In this intensity adjustment, the pump line (mode 0) is
attenuated by 20 dB; modes
±
1 are attenuated by 10 dB; modes
±
2 are attenuated by 5 dB, and the other modes are
not attenuated. The autocorrelation (AC) is measured by a noncollinear second-harmonic generation based intensity
autocorrelator.
1530
1540
1550
1560
-2
0
2
4
6
-2
-1
0
1
2
0
1
Thru-port
estimated intracavity
Wavelength (nm)
Optical spectral phase (rad)
(a)
(b)
Fast time (ps)
Intensity (a. u.)
Breathing
Stable
Cavity position (
L
)
-0.5
-0.25
0
0.25
0.5
FIG. S1:
Intracavity waveform retrieval.
(a) The retrieved comb phase via line-by-line pulse shaping. The intracavity
pump phase is estimated based on the through-port data. (b) Using the averaged power spectra of the stable dark pulse and the
breathing dark pulse together with the retrieved phase, the intracavity waveform can be reconstructed, showing a dark-localized
waveform. Note that points on the fast time axis are equivalent to spatial positions within the cavity (
L
is the cavity round
trip length).
Intracavity waveform retrieval.
Using the comb phase retrieved in line-by-line pulse shaping (the mode-locked
state and the breathing state share a nearly identical phase profile as shown in Fig. 3(a) of the main text), the
intracavity waveform for the device characterized in the main text can be reconstructed [S2]. Note that the pump
line in the through-port contains both the transmitted uncoupled pump and the coupled-out component from the
cavity. Hence, the intensity and phase in the through-port are corrected to yield the pump line in the cavity, based
on the method described in ref. [S2] and the Supplementary Information therein. By applying the retrieved phase in
Electronic address:
cbao@caltech.edu
Electronic address:
amw@purdue.edu
2
Fig. S1(a) to the spectra of the breathing dark pulse and the stable dark pulse, we obtain dark-localized waveforms
(Fig. S1(b)). Since the breathing dark pulse breathes weakly, this confirms that the breathing dark pulse retains a
dark-localized waveform.
Fast comb breathing recording.
The method to record the fast evolution of the spectral breathing is the same
as that used in ref. [S3]. A pulse shaper is used to select individual comb lines, and the output is recorded by a fast
photodetector and amplified by a low noise amplifier. A portion of the output prior to the pulse shaper is used to
trigger the oscilloscope in order to provide a timing reference for measurements performed on different comb lines. The
recorded traces are Fourier transformed and numerically filtered by a 20 MHz bandpass filter to isolate the strongest
breathing tone; the filtered spectra are inverse Fourier transformed to yield the spectral breathing traces analyzed in
Fig. 3.
2. Numerical simulations details
0
1
2
3
4
50
100
150
50
100
150
50
100
150
Intensity (20 dB/div)
1450
1550
1650
Wavelength (nm)
(b)
(c)
Fast time (ps)
Power (W)
@ 250 ns
@ 150 ns
@ 50 ns
@ 50 ns
@ 150 ns
@ 250 ns
1450
1550
1650
Wavelength (nm)
(a)
50
100
150
200
250
Slow time (ns)
20 dB/div
δ
0
= 0.0102;
Δ
φ
= -0.815
δ
0
= 0.0272;
Δ
φ
= -0.815
δ
0
= 0.0629;
Δ
φ
= -0.326
FIG. S2:
Tuning the detuning to generate the breathing dark pulse.
(a) The spectral dynamics while the detuning
and mode interaction parameter ∆
φ
are varied. (b) Typical spectra at 50 ns, 150 ns and 250 ns; at 250 ns, spectrum with
characteristic dark pulse Kerr comb is obtained. (c) Typical waveforms at 50 ns, 150 ns and 250 ns; the waveform at 250 ns
confirms that the breathing state retains the dark-localized waveform.
Simulated comb generation dynamics.
To initiate the comb generation from noise, mode interaction is included
in simulations. To include mode interaction, we add an identical phase shift ∆
φ
per round trip to 4 modes within the
range [
57
,
54] relative to the pump. In Fig. S2(a), we show the simulated breathing dark pulse generation dynamics.
During the first 100 ns, we set
δ
0
=0.0102 (frequency detuning 377.5 MHz), ∆
φ
=
0.815; the comb is initiated by
the mode interaction and coarsely spaced. During the second 100 ns, we set
δ
0
=0.0272 (frequency detuning 1006.7
MHz), ∆
φ
=
0.815; the comb grows and becomes chaotic. During the third 100 ns,
δ
0
=0.0629 (frequency detuning
2328.1 MHz), ∆
φ
=
0.326, a breathing dark pulse can be obtained. The change in the mode interaction strength
in this step can be attributed to thermal effects and change of intracavity power when tuning the laser further into
resonance [S2]. Since the full evolution over 200
300 ns are plotted in Fig. S2(a), it is hard to resolve the breathing
dynamics. For visualization of the breathing on a smaller time scale, see Fig. 1 in the main text. In Figs. S2(b, c),
we show the typical spectra and waveforms (at 50 ns, 150 ns, 250 ns) in each step. Note that the chaotic comb during
the period 100
200 ns (prior to breathing dark pulse formation) is distinct from the chaotic breathing in Fig. 4 of
the main text, and the time-domain waveform and spectrum do not resemble those in the stable dark pulse state.
3
Numerical pulse shaping and AC in Figs. 3, 4.
For the AC traces presented in Fig. 3(b) and Fig. 4(d) of
the main text, we extract the simulated phase of the stable dark pulse, which we denote as
φ
dp
(
ω
), and then add
φ
dp
(
ω
) to compensate the phase and obtain a transform-limited pulse. The AC is then calculated. For the periodic
breathing and chaotic breathing comb, the same
φ
dp
(
ω
) is added to the complex spectral amplitude. Then a series
of AC traces are computed based on snapshots of the intensity profile. 10
4
individual AC traces are calculated in a
1
μ
s time window (equivalent to
782 periods for the regular breathing dark pulse) and averaged to yield the traces
in Fig. 3(b) and Fig. 4(d). All the simulated comb modes are used in the simulated pulse shaping; hence, the width
of the AC trace is shorter than the experiments, where only a subset of the lines are accommodated by our pulse
shaper. The intensity of the comb lines within [
2
,
2] (relative to the pump) are adjusted in the same way used in
the line-by-line pulse shaping in experiments.
3. Breathing dark pulses characterized by an optical sampling oscilloscope
Breathing dark pulses generated in ‘Device 2’ are characterized by an optical sampling oscilloscope. This microres-
onator also has a free-spectral-range of 231 GHz and the loaded
Q
-factor is
1 million. The breathing dark pulse
is generated when we increase the pump power slightly after generating the mode-locked dark pulse. In contrast to
‘Device 1’ discussed in the main text, the optical and RF spectra are measured from the drop-port (shown in Figs.
S3(b, c)), which provides a direct replica of the intracavity field without the complication of a strong superimposed
pump field [S4]. The breathing frequency is 635 MHz, which as before is well outside the half-width-half-magnitude
(HWHM) of the cavity resonance. This breathing frequency is slightly lower than that reported breathing rate in
the main text, which can be attributed to the slightly higher loaded-
Q
factor and smaller resonance linewidth. The
breathing dark pulses are measured in the time domain by an equivalent-time optical sampling oscilloscope (OSO)
with 500 GHz bandwidth. The OSO is based on four-wave-mixing optical sampling in a nonlinear fiber [S5]. The
sampling gate is provided by a mode-locked laser whose repetition rate is asynchronous to the input signal. Hence,
the delay between the pulse under test and the sampling gate is automatically swept. In Fig. S3(d), the waveform is
reconstructed by using the breathing dark pulses as trigger, which further supports the observation that the breathing
dark pulse retains a dark-localized waveform. Since the breathing frequeny and the repetition rate of the breathing
dark pulses Kerr comb are incommensurate, it is not possible to resolve the breathing dynamics within a full breathing
period with an equivalent-time sampling oscilloscope. However, the OSO can be triggered instead by the 635 MHz
breather frequency, as indicated in the schematics in Fig. S3(a). In this case, the variation of the power levels within
the dark pulses over the breathing period can be clearly observed but their structure is washed out (see Fig. S3(e)).
We further analyzed the occurrence probability for different power levels for the OSO measurement. The histogram
in Fig. S3(f) (analyzed from the measurement in Fig. S3(d)) shows two peaks, which reflects the low-power-level
(waveform hole) and high-power-level (waveform background) of the breathing dark pulse. When further analyzing
the power-level occurrence probability within different time slots using the measurement shown in Fig. S3(e), we can
clearly observe the change of the most probable low-power-level and high-power-level. This shows the dark-localized
waveform is varying in the time domain. Furthermore, the change in the probability of the low-power-level peak and
high-power-level peak suggests slight changes in the duty cycle of the dark pulse.
4. Breathing dark pulses in the absence of mode interaction
The mode interaction is an important perturbation to Kerr comb generation in microresonators. It has recently
been found that mode interaction can induce breathing of bright solitons in anomalous dispersion microresonators
[S6]. Here, we show that mode interaction is not required for the breathing of the dark pulse studied in our device. In
experiments, we first generate the breathing dark pulse at a relatively low chip temperature, similar to experiments in
Figs. 1, 2 of the main text. Then, we increase the chip temperature which shifts the resonance (the pump wavelength
is tuned accordingly to follow the shifted resonance), while maintaining the breathing comb. At a relatively high
temperature, the spectral spike around 1657 nm vanishes (Fig. S4(a)), indicating the mode interaction is relatively
weak. The change in the strength of the mode interaction may be understood as arising from different temperature
dependent frequency shifts for resonances corresponding to different spatial modes. The comb is observed to retain
the breathing (Fig. S4(b)). With the same parameters as used in Fig. 1 of the main text, numerical simulations still
show the breathing of the dark pulse (see Figs. S4(c, d)) when turning off the mode interaction term (i.e., setting
φ
=0). Furthermore, the simulated spectrum (averaged over slow time
t
) when turning off the mode interaction
term also closely matches the experiments. These experimental and numerical results show that the breathing of dark
pulse observed here is not caused by the mode interaction.
4
Time (ps)
Power (a. u.)
1450
1500
1550
1600
1650
Intensity (10 dB/div)
Wavelength (nm)
0
500
1000
1500
Intensity (10 dB/div)
HWHM 100 MHz
635 MHz
(a)
(b)
(c)
(d)
Frequency (MHz)
Optical Sampling Oscilloscope
Optical Spectrum
Pump laser
Microresonator
RF Spectrum
Breathing trigger
Drop-Port
Through-Port
0
5
10
15
20
0
5
10
15
20
25
30
PD
Pulse trigger
Probability (a. u.)
Breathing trigger
0
1
2
3
4
5
Time (ns)
Breathing period
slot 1
slot 2
(f)
(e)
pulse trigger
slot 1
slot 2
0
1
FIG. S3:
Characterization of breathing dark pulses by an optical sampling oscilloscope.
(a) Experimental setup of
the generation and characterization of breathing pulses in another chip (Device 2). A commercially available equivalent-time
optical sampling oscilloscope (OSO) is used for the time domain characterization. A breathing dark pulse is obtained when
increasing the pump power slightly from the stable dark pulse state. The optical spectrum and the RF spectrum are shown
in (b) and (c) respectively. Note that the measurements are performed at the drop-port as it allows direct characterization
of the intracavity field without the superimposed pump. (d) Intracavity waveform of the breathing state as measured by the
OSO when it is triggered by the optical pulses themselves; a dark-localized waveform is observed. (e) When the OSO is instead
triggered by the 635 MHz breathing frequency, the equivalent-time nature of the oscilloscope washes out the individual pulses,
as the breathing period does not correspond to an integer multiple of the pulse round trip time. (f) Occurrence probability for
different certain power-levels. The histogram is analyzed from the measurement triggered by the input pulses shown in (d),
while green and red lines are analyzed from measurement within different time slots (corresponds to the shaded regions in (e))
when the OSO is triggered by the breathing frequency.
5. Illustration of animations
Animation 1,
spectral and temporal breathing of the periodic breathing dark pulse (corresponds to Fig. 1 in the
main text). The breather repeats itself around 1.2
1.3 ns and
2.5 ns (end of the animation), as the period is 1.27
ns.
Animation 2,
spectral and temporal breathing of the chaotic breathing dark pulse (corresponds to Fig. 4 in the
5
Intensity (20 dB/div)
Wavelength (nm)
(a)
Exp
Sim
1450 1500 1550 1600 1650 1700
w/o strong
mode interaction
Measured RF
Background
Intensity (10 dB/div)
HWHM
0
500
1000
1500
(b)
Frequency (MHz)
w/o mode interaction
(d)
Slow time (ns)
0
1
2
3
4
5
(c)
Wavelength (nm)
1400
1500
1600
1700
Slow time (ns)
20 dB/div
Fast time (ps)
0
1
2
3
4
5
0
1
2
3
4
20
120
PWR (W)
FIG. S4:
Experimentally observed (a, b) and simulated (c, d) breathing dark pulses in the absence of strong
mode interaction.
(a) By changing the chip temperature, the mode interaction strength can be reduced such that there is
no significant spike in the spectrum around 1657 nm. The simulated (averaged over slow time) spectrum when turning off
the mode interaction term closely matches the experimentally measured spectrum. (b) Breathing is also observed without the
strong mode interaction. (c) Spectral breathing is also observed in simulations in the absence of the strong mode interaction,
using the other parameters which are the same as in Fig. 3 of the main text. (d) The corresponding temporal breathing.
main text). The chaotic breather does not repeat itself in the full 2.5 ns.
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