of 6
Observation of Breathing Dark Pulses in Normal Dispersion Optical Microresonators
Chengying Bao,
1
,*
Yi Xuan,
1,2
Cong Wang,
1
Attila Fülöp,
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
(Received 8 June 2018; published 21 December 2018)
Breathers are localized waves in nonlinear systems that undergo a periodic variation in time or space.
The concept of breathers is useful for describing many nonlinear physical systems including granular
lattices, Bose-Einstein condensates, hydrodynamics, plasmas, and optics. In optics, breathers can exist in
either the anomalous or the normal dispersion regimes, but they have only been characterized in the former,
to our knowledge. Here, externally pumped optical microresonators are used to characterize the breathing
dynamics of localized waves in the normal dispersion regime. High-
Q
optical microresonators featuring
normal dispersion can yield mode-locked Kerr combs whose time-domain waveform corresponds to
circulating dark pulses in the cavity. We show that with relatively high pump power these Kerr combs can
enter a breathing regime, in which the time-domain waveform remains a dark pulse but experiences a
periodic modulation on a time scale much slower than the microresonator round trip time. The breathing is
observed in the optical frequency domain as a significant difference in the phase and amplitude of the
modulation experienced by different spectral lines. In the highly pumped regime, a transition to a chaotic
breathing state where the waveform remains dark-pulse-like is also observed, for the first time to our
knowledge; such a transition is reversible by reducing the pump power.
DOI:
10.1103/PhysRevLett.121.257401
The interaction between nonlinearity and dispersion
leads to various localized waves. Breathers are a type of
localized wave that occurs in a wide variety of physical
systems
[1
5]
. Unlike solitons which preserve their shape
as they propagate, breathers feature periodic variation in
time or space. Systems in which breathers exist include
Bose-Einstein condensates (BEC)
[5]
, granular lattices
[6]
,
micromechanical oscillator arrays
[7]
, nonlinear electrical
networks
[8]
, pendula arrays
[9]
, hydrodynamics
[2,10]
,
and optics
[4,11]
. Optical systems are widely used to study
breathers due to the ease of tailoring the linear and
nonlinear properties of the systems. Optical breathers were
first studied in conservative systems, namely optical fibers
[4,11]
. Recently bright soliton breathers have also been
observed in dissipative systems such as externally pumped
fiber cavities
[12,13]
and microresonators
[14
17]
.
In optics, the character of the nonlinear dynamics
depends strongly on the sign of the dispersion, i.e.,
derivative of the reciprocal of group velocity with optical
frequency. To date optical breathers have been explored
primarily in the anomalous dispersion regime, under which
group velocity increases with frequency; the joint action of
anomalous dispersion and a self-focusing nonlinearity
gives rise to bright solitons. Here, we consider the nonlinear
dynamics in the opposite, normal dispersion regime, again
with a self-focusing nonlinearity. In principle, equivalent
nonlinear dynamics should also occur for one-dimensional
spatial diffraction with a defocusing nonlinearity
[18]
. Dark
solitons, which have been reported in optics
[18,19]
,
BEC
[20]
, and hydrodynamics
[21]
, exist in this regime.
Although breathing dark pulses (also termed dark soliton
breathers) have been predicted to exist in normal dispersion
microresonators
[22,23]
, experimental study of optical
breathing dark pulses has been limited in conservative as
well as dissipative optical systems, to our knowledge.
In this Letter, we present the first experimental study
focusing on optical breathing dark pulses in the normal
dispersion regime using microresonators. Kerr comb gen-
eration in normal dispersion microresonators
[22
29]
pro-
vides this unique opportunity. We find that although the dark
pulses breathe relatively gently, different spectral lines can
exhibit clear differences in the modulation depth and phase
with which they breathe. In the highly driven regime, we
observe that the breathing dark pulse in our dissipative cavity
can reversibly enter a chaotic state, where the waveform
remains dark localized and similar to that of the stable or the
breathing dark pulse. The breathing dark pulse dynamics in
dissipative cavities is governed by the Lugiato-Lefever
equation (LLE), i.e., driven-damped nonlinear Schrödinger
equation
[30
33]
. Indeed, our experimental observations are
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=
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=
121(25)
=
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© 2018 American Physical Society
consistent with numerical simulations based on the LLE.
Moreover, simulations reveal the importance of modulation
of switching waves (also termed as domain walls or fronts)
[24,34,35]
in the breathing dynamics.
We start our investigation with simulations based on the
LLE
[12,30,33]
, which can be written as

t
R
t
þ
α
2
þ
i
δ
0
þ
i
β
2
L
2
2
τ
2

E
i
γ
L
j
E
j
2
E
¼
ffiffiffi
κ
p
E
in
;
ð
1
Þ
where
E
is the envelope of the intracavity field,
t
and
τ
are
slow time and fast time variables, respectively,
t
R
and
L
are
the round trip time and cavity length,
β
2
and
γ
are the group
velocity dispersion and nonlinearity coefficient,
α
and
κ
are
the total loss and coupling loss,
E
in
is the amplitude of the
pump field, and
δ
0
is the pump phase detuning (angular
frequency detuning to the closest resonance multiplied by
t
R
). Simulation parameters used are
t
R
¼
4
.
3
ps,
β
2
¼
187
ps
2
=
km,
L
¼
628
μ
m,
γ
¼
0
.
9
ð
Wm
Þ
1
,
α
¼
0
.
0068
,
κ
¼
0
.
0054
, which are close to experimental parameters
and enable direct comparison with experiments. The sim-
ulation starts with a noise field in the microresonator, and
the detuning
δ
0
is tuned to generate the comb [see the
Supplemental Material
[36]
for a detailed view of the path to
comb generation]. A breathing dark pulse is obtained with a
pump power of 700 mW (64% of the experimental on-chip
power) and a final detuning of 0.0629 radians (equivalent to
a laser frequency 2.3 GHz to the red of the resonance). We
show the simulated breathing dynamics together with
representative spectra and waveforms in Fig.
1
. Both the
power spectrum and the temporal intensity profile of the
comb vary periodically, repeating with a period of 1.3 ns
(782 MHz),
300
times longer than
t
R
. The formation of
dark pulses can be understood as arising from interlocked
switching waves, which are traveling front solutions con-
necting the bistable homogeneous state of the LLE
[24,34,35]
. The background field (i.e., the high power part
of the dark-localized waveform) stays nearly unchanged
during the breathing cycle, because it follows the homo-
geneous stable state
[22,23,34]
. Because the background
field carries most of the comb power (average power over
one round trip time), the breathing depth of the comb power,
which is defined as
ð
P
max
P
min
Þ
=
ð
P
max
þ
P
min
Þ
with
P
max
ð
min
Þ
being the maximum (minimum) power, is rela-
tively low. The simulated breathing depth of the intracavity
comb power including (excluding) the pump is 3% (6%).
During the breathing, the switching waves are modulated as
illustrated both by Fig.
1(b)
and Animation 1. The corre-
sponding modulation in the tails of the switching waves
results in the stronger breathing in the waveform hole (the
low power part of the dark-localized waveform). Because
the dark pulse has a complex waveform, the breathing within
the waveform hole is also complex.
We have performed experiments in three different
(see Table
I
) optical microresonators which provide data
supporting the simulated dynamics. We focus our
discussion on results from a silicon nitride microresonator
(device 1 in Table
I
) with a radius of
100
μ
m, a loaded
Q
factor of
0
.
8
×
10
6
, geometry of
2000
nm ×
600
nm
(waveguide width vs height), a free spectral range (FSR)
of 231 GHz, and strong normal group velocity dispersion
β
2
¼
190
ps
2
=
km. The experimental setup is sketched in
Fig.
2(a)
. Similar to bright soliton breathers
[12,14
16]
, the
breathing dark pulse state exists in a regime with relatively
strong pump power and small detuning
[22,23]
. Hence, we
generate our comb with a relatively large on-chip pump
power of 1.1 W (obtained by measuring the off-chip power
and adjusted with
2
dB fiber-to-chip coupling loss).
By tuning the continuous wave (cw) pump laser into
resonance from the blue to the red, we are able to generate
a broadband, single FSR comb [Fig.
2(b)
]. In this state the
comb power measured by a fast photodetector exhibits
narrow radio-frequency (rf) tones as shown in Fig.
2(c)
.
The fundamental frequency of the rf tones is 740 MHz, in
close agreement with the simulated breathing frequency
and much larger than the resonance linewidth which has
half-width-half-maximum (HWHM) of 125 MHz. The
other normal dispersion devices listed in Table
I
also show
narrow rf tones that can be attributed to dark pulse
breathing, supporting the generality of our observation.
The observed breathing frequencies decrease with increas-
ing loaded
Q
factor (decreasing resonance linewidth).
(b)
e
0
1
2
3
4
Slow time (ns)
0123
4
20
120
Power (W)
SW
SW
P (W)
SW
SW
2.5 ns
3 ns
1550
1650
0
1
2
3
4
(a)
10 dB/div
1450
50
150
(20 dB/div)
Power
Fast time (ps)
2.5 ns
3 ns
Wavelength (nm)
Slow time (ns)
FIG. 1. (a) Simulated breathing dynamics of optical power
spectrum, showing complex breathing behavior. (b) Simulated
breathing dynamics in the time domain. The switching waves
(SWs) are modulated during breathing and the waveform hole
breathes strongly. Above each panel we show snapshots of the
spectrum and time-domain waveform at moments indicated by
the dashed lines.
TABLE I. Measured breathing frequencies, loaded
Q
factor,
and HWHM resonance linewidths for three microresonators with
similar 231 GHz FSRs.
Frequency
Loaded
Q
Resonance HWHM
Device 1
740
MHz
0
.
8
×
10
6
125
MHz
Device 2
635
MHz
1
×
10
6
100
MHz
Device 3
250
MHz
2
.
4
×
10
6
40
MHz
PHYSICAL REVIEW LETTERS
121,
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In each case, the breathing frequency is approximately six
times the HWHM resonance linewidth. Similar narrow rf
tones were previously observed in a normal dispersion
microresonator, but without experimental evidence to show
the comb corresponds to breathing dark pulse
[24]
. Here,
we confirm that the comb in Figs.
2(b)
and
2(c)
corresponds
to a breathing dark pulse and attribute the rf tones to
periodic modulation of the comb lines.
The measured optical spectrum of the breathing state is
in close agreement with the averaged simulated optical
spectrum [see Fig.
2(b)
]. Furthermore, the overall modu-
lation of the breathing dark pulse is weak, also in agreement
with simulation. The comb power measured at the through
port including (excluding) the pump line breathes with a
depth of
3%
(
5%
). This is significantly different from
bright soliton breathers, for which the modulation of the
comb power excluding the pump can reach
50%
[14]
.
We can transition to a stable dark pulse state by reducing
the pump power to 1.0 W at the same wavelength and can
return to the breathing state by increasing the pump power
back to 1.1 W. At the lower pump power, the comb shows
low-noise operation [Figs.
2(d)
and
2(e)
]. Again, the
simulated stable dark pulse gives an optical spectrum in
agreement with experiment. The time averaged optical
spectra of the breathing and stable dark pulse combs show
no significant difference. This is a consequence of the weak
overall breathing and is a key difference with respect to
bright soliton breathers
[14,16]
. There is a spike in the
spectrum around 1657 nm due to interaction with another
spatial mode, which is reproduced by inclusion of mode
interaction in the simulations [see the Supplemental
Material
[36]
]. Note that the existence of mode interaction
was observed to cause intermode breathing in one recent
report on anomalous dispersion microresonators
[17]
.We
have verified that mode interaction is not responsible for the
observed breathing in our experiments by changing the
chip temperature to a value where the mode interaction
effects vanish [see the Supplemental Material
[36]
].
To establish the observation of a breathing dark pulse, we
first need to verify that the comb in Figs.
2(b)
and
2(c)
has a
dark-localized waveform. We use line-by-line pulse shaping
to probe the corresponding waveform
[24,39,40]
.By
adjusting the phases of the comb lines to compress them
into a transform-limited pulse train, we can retrieve the comb
phase profile and the corresponding waveform [see the
Supplemental Material
[36]
for methods]. In experiments,
we first adjust the pump power to reach the stable dark pulse
state and apply pulse shaping to compress the output into
transform-limited pulses, whose intensity autocorrelations
(ACs) are shown in Fig.
3(a)
. We then adjust the pump power
to transition into the breathing state, while keeping the same
phase profile on the pulse shaper. The output is still com-
pressed into a transform-limited pulse. This comparison
suggests that the breathing dark pulse has a spectral phase
profile very close to that of the stable dark pulse. Because the
breathing dark pulse only exhibits a gentle modulation, we
deduce that the breathing comb retains a dark-localized
waveform [inset of Fig.
3(a)
]. The simulated AC of the
breathing dark pulse after numerical pulse compression and
averaging over slow time [i.e.,
t
in Eq.
(1)
] is close to the AC
of the stable comb (see the Supplemental Material
[36]
for
methods to obtain the numerical AC) [Fig.
3(b)
]. In experi-
ments on another microresonator (device 2), we have con-
firmed that a dark-localized waveform is retained during
breathingby usingan alternative approach based on an optical
sampling oscilloscope [see the Supplemental Material
[36]
].
Next, we confirm that the form of the comb varies
periodically and examine the breathing of individual lines.
To do that, we program the pulse shaper to select individual
comb lines and then measure their time-dependent power
using a photodetector. A sample of the full comb is also
detected; its power modulation under breathing is used as a
Intensity (20 dB/div)
Wavelength (nm)
HWHM
Frequency (MHz)
Intensity (20 dB/div)
20 ns
2
X
3%
Intensity (20 dB/div)
Frequency (MHz)
Intensity (20 dB/div)
HWHM
1450 1500 1550 1600 1650 1700
Exp
Sim
mode interaction
1450 1500 1550 1600 1650 1700
Exp
Sim
mode interaction
Wavelength (nm)
(b)
(c)
(d)
(e)
avg.
0
500
1000
1500
0
500
1000
1500
Pump
Pulse shaper
Autocorrelator
Oscilloscope
PD
PD
Power
Wavelength
stable
breathing
chaotic breathing
Fig. 4
Fig. 2(b,c)
Fig. 2(d,e)
(a)
meas.
floor
meas.
floor
temperature control
FIG. 2. (a) Experimental setup and summary of comb states.
A pulse shaper is used to select individual comb lines; their fast
breathing dynamics are recorded individually using a fast
photodetector (PD) and digital oscilloscope. When recording
the fast breathing dynamics, a portion of the breathing comb is
used to trigger the oscilloscope to provide a timing reference. By
controlling the pump power, both periodic breathing and chaotic
breathing are experimentally studied. The chip is placed on a
temperature controlled stage. (b), (c) Optical and rf spectrum of
the breathing dark pulse state under an on-chip power of 1.1 W.
The simulated optical spectrum is in close agreement with the
measured spectrum. The breathing dark pulse state has narrow rf
tones on the rf spectrum, whose frequencies are outside the half-
width-half-maximum (HWHM) of the resonance. The inset in (c)
is the measured comb power change of the breathing dark pulse
(including pump line) at the through-port, showing a weak
modulation depth of
3%
. (d), (e) Optical and rf spectrum of
the stable dark pulse state under an on-chip power of 1.0 W. There
is no significant difference in the averaged optical spectra
between the breathing and stable state.
PHYSICAL REVIEW LETTERS
121,
257401 (2018)
257401-3
trigger, providing a timing reference to compare the
breathing signals and their rf phase delays for different
comb lines [Fig.
2(a)
]. In experiments, we observe that the
absolute breathing amplitudes of the central comb lines
within
½
2
;
2

relative to the pumped mode are much higher
compared to other comb lines. However, the average
powers are also higher for the central comb lines, and
the modulation depths of individual lines tend to be larger
in the wings of the spectrum [Fig.
3(c)
]. Although the
breathing depth of the total comb power is only 3%, some
individual lines breathe deeply. Furthermore, the phase of
the rf breathing signal varies from line to line [Figs.
3(c)
and
3(d)
]. Consequently, the comb spectrum is changing
during the modulation period, providing definitive proof of
a breathing state. The qualitative trend obtained in simu-
lations is similar; see Fig.
3(d)
. Some of the details of the
simulated breathing do differ from the experiments; these
may be attributed to the difficulty in modeling the mode
interaction accurately and the unconsidered wavelength-
dependent
Q
factor. The phase and amplitude of the
breathing can change abruptly from one line to the next;
this observation is consistent with the simulated complex
breathing dynamics shown in Fig.
1
.
The high
Q
microresonator also enables us to study the
breathing dynamics of dark pulses in the highly driven
regime. By increasing the on-chip pump power up to 1.3 W,
the breathing first exhibits period tripling and then becomes
chaotic. The comb in this chaotic state retains the optical
spectral features of the dark pulse [Fig.
4(a)
], and the
averaged spectrum from simulation is in good agreement
with the measurement. Animation 2 provides a visualization
of the simulated chaotic breathing dynamics. However, the
measured rf spectrum shows broadband noise [Fig.
4(b)
];
this is also observed in simulations when increasing the
pump power to 785 mW under the same detuning as Fig.
1
[Fig.
4(b)
inset]. The modulation depth of the chaotic
breathing dark pulse is at the few percent level in the
simulation, similar to that of the periodic breathing dark
pulse. Chaotic breathing was predicted and observed to exist
in the LLE with anomalous dispersion
[12,13,16,41]
; it has
also been predicted to exist for breathing dark pulses
[22]
but
not demonstrated until now. Our results constitute the first
observation of such a chaotic transition for breathing dark
solitary waves in all physical systems, to our knowledge. It is
also worth noting that by reducing the pump power at a
constant pump frequency, one can move repeatably from the
chaotic breathing state back to the periodic breathing or
stable state. Note that this chaotic breathing state is distinct
from the chaotic state that occurs while tuning the pump
frequency prior to reaching the breathing or mode-locked
dark pulse state
[24]
. When we apply the same phase profile
used in Fig.
3(a)
on the pulse shaper, the AC trace obtained
under the chaotic breathing state still shows a clean peak
pulse which is close to the transform limit (as calculated
from the time averaged spectrum). The visibility of the AC
trace exceeds 97%. This observation differs dramatically
from reports that unstable modulation instability Kerr combs
with broadband noisy rf spectra cannot be compressed into
transform-limited pulses
[40]
. The ability here to compress
using the same phase profile as for the stable dark pulse
suggests that the comb retains a dark-localized waveform
even under chaotic breathing. Consistent with the exper-
imental measurement, the averaged AC trace (in simulation,
after numerical pulse shaping) of the chaotic breathing state
remains close to the AC of the stable state [Fig.
4(d)
].
In summary, optical breathing dark pulses are clearly
observed and comprehensively modeled for the first time,
using normal dispersion microresonators. The breathing
dark pulse features a high frequency modulation and a weak
modulation depth. Different comb lines breathe in distinct
manners, a behavior which is related to a modulation of the
switching waves and their oscillating tails. This behavior is
in sharp contrast to the dynamics observed in breathing
bright solitons in anomalous dispersion microresonators. At
higher pump power, the breathing dark pulse undergoes a
(b)
Breathing
Stable
AC intensity (arb. u.)
Delay time (ps)
0
4
0
150
Power (W)
Time (ps)
Breathing depth
SW
(d)
-1
0
1
2
0
1
SW
-2
-1
0
1
2
0
1
-2
0
2
0
1
Time (ps)
P (arb. u.)
Breathing
Stable
Breathing Cal.
Stable Cal.
Delay time (ps)
(a)
(c)
Experiment
Simulation
AC intensity (arb. u.)
-10
-5
0
5
10
0
1
0
1
2
3
Mode number
-10
-5
0
5
10
0
1
Breathing depth
Mode number
Breathing phase (
)
Breathing phase (
)
0
1
2
3
4
FIG. 3. (a),(b) By line-by-line pulse shaping, stable dark pulses
can be compressed into transform-limited pulses (the shaped
comb lines contain 90% of the comb power). Applying the same
phase on the pulse shaper, the breathing dark pulses can also be
compressed into transform-limited pulses. The intensity auto-
correlations (ACs) of the compressed pulses are plotted in the
figure. Using the phase retrieved in the line-by-line pulse shaping,
the actual intracavity field is deduced to have a dark-localized
waveform for the breathing Kerr comb (inset). The averaged
simulated AC trace of the pulse-shaped breathing dark pulse
comb [see the Supplemental Material
[36]
for numerical meth-
ods] is close to the AC of the pulse-shaped stable dark pulse
comb. The inset is an example of a snapshot of the breathing dark
pulse (SW: switching wave). The AC in simulations is shorter,
because it calculates the full spectrum while only a portion of the
spectrum is shaped in experiments. (c),(d) The breathing depth
(blue) and phase (red) of individual comb lines. The breathing
depths tend to be larger at the wings of the spectrum; different
lines also breathe with different phases. Both the breathing depth
and phase can change abruptly between adjacent lines.
PHYSICAL REVIEW LETTERS
121,
257401 (2018)
257401-4
reversible chaotic transition, where a dark-localized wave-
form is retained despite the broadband intensity noise. The
breathing instability can impair applications which need
stable combs; knowledge of breathing dark pulses can help
us avoid this instability. Our Letter also suggests high
Q
microresonator can be a useful platform to study nonlinear
dynamics, especially in the highly driven regime.
This Letter was supported in part by the Air Force Office
of Scientific Research (Grant No. FA9550-15-1-0211), by
the DARPA PULSE program (Grant No. W31P4Q-13-1-
0018) from AMRDEC, and by the National Science
Foundation (Grant No. ECCS-1509578). A. F. and V. T. C.
acknowledge partial support by the Swedish Research
Council and the European Research Council (DarkComb
Grant Agreement No. 771410). C. B. acknowledge partial
support from a postdoctoral fellowship from the Resnick
Institute, Caltech. We gratefully acknowledge fruitful dis-
cussions with Xiaoxiao Xue, Nail Akhmediev, Zhen Qi,
and Curtis Menyuk.
*
cbao@caltech.edu
Present address: T. J. Watson Laboratory of Applied
Physics, California Institute of Technology, Pasadena
91125, USA.
amw@purdue.edu
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