Dark breathers in a normal dispersion optical microresonator
Chengying Bao,
1,
∗
Yi Xuan,
1, 2
Daniel E. Leaird,
1
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
Breathers are localized waves, that are periodic in time or space. The concept of breathers is
useful for describing many physical systems including granular lattices, Bose-Einstein condensation,
hydrodynamics, plasmas and optics. Breathers could exist in both the anomalous and the normal
dispersion regime. However, the demonstration of optical breathers in the normal dispersion regime
remains elusive to our knowledge. Kerr comb generation in optical microresonators provides an
array of oscillators that are highly coupled via the Kerr effect, which can be exploited to explore
the breather dynamics. Here, we present, experimentally and numerically, the observation of dark
breathers in a normal dispersion silicon nitride microresonator. By controlling the pump wavelength
and power, we can generate the dark breather, which exhibits an energy exchange between the
central lines and the lines at the wing. The dark breather breathes gently and retains a dark-pulse
waveform. A transition to a chaotic breather state is also observed by increasing the pump power.
These dark breather dynamics are well reproduced by numerical simulations based on the Lugiato-
Lefever equation. The results also reveal the importance of dissipation to dark breather dynamics
and give important insights into instabilities related to high power dark pulse Kerr combs from
normal dispersion microreosnators.
The interaction between nonlinearity and dispersion
leads to various localized solutions to the nonlinear
Schr ̈
o
dinger equation (NLSE). Solitons are one of the
most well-known solution [1] and have advanced various
research arenas such as Bose-Einstein condensates (BEC)
[2] and optics [3]. Notably, soliton mode-locking has en-
abled the coherent Kerr comb generation from optical
microresonators recently [4–6] (a Kerr comb is an array
of equally spaced spectral lines generated by cascaded
four-wave-mixing in high-
Q
microresonators, see ref. [5]
for a review). Breathers are another type of solution
to the NLSE that occur widely in physical systems [7–
11]. Different from solitons which preserve their shape
as they propagate, breathers feature periodic variation
in time or space. Systems in which breathers may exist
include BEC [11], granular lattices [12], hydrodynamics
[13], optics [10, 14]. Breathers are attracting growing
attention recently due to their close relationship to the
growth and decay of the extreme events known as rogue
waves [10, 13–17].
Optical systems are widely used to study breathers,
due to the ease to tailor the linear and nonlinear prop-
erties of the systems.
Optical breathers were first
studied in conservative systems, namely optical fibres
[10, 14, 16, 17]. Recently optical breathers have also
been observed in dissipative systems such as externally
pumped fiber cavities [18] and microresonators [19–22].
Kerr comb generation in optical microresonators [4, 5]
constitutes an excellent experimental platform to explore
breather dynamics, as microresonators provide essen-
tially infinite propagation distance where breathers are
densely sampled every round-trip. Furthermore, the wide
frequency spacing for Kerr combs generated from mi-
croresonator [5] makes it easy to separate individual lines
and study their breathing dynamics [19]. To date, opti-
cal breathers have been mainly explored in the anomalous
dispersion regime (for which nonlinearity and dispersion
have the same sign). Nonlinear dynamics in the normal
dispersion regime (opposite sign for nonlinearity and dis-
persion) are also important; for instance they help to
improve pulse energy of mode-locked lasers [23]. Dark
solitons, which have been reported in optics [24], BEC
[25] and hydrodynamics [26] may exist in this regime.
However, experimental observation of optical breathers
in the normal dispersion regime remains elusive, to our
knowledge. Discrete dark breathers have been shown to
exist in granular lattices (e.g., Fermi-Pasta-Ulam lattice
and Klein-Gordon lattice) [27, 28]. Nevertheless, many
questions about dark breathers, such as their dynamics in
highly driven cases, remain open. Kerr comb generation
in normal dispersion microresonators [29–36] can provide
a unique opportunity to study optical dark breathers.
Numerical simulations [29, 35, 36] and preliminary exper-
iments [29] suggest that optical dark breathers may exist
in normal dispersion microresonators. In this Article, we
present the first clear observation and detailed characteri-
zation of optical dark breathers in the normal dispersion
regime, which we obtain by destabilizing a dark pulse
Kerr comb in a silicon-nitride (SiN) microresonator. The
dark breather breathes relatively gently and features an
energy exchange between comb lines. Furthermore, the
breathing behaviour changes abruptly from line to line.
The observed dark breather dynamics in dissipative cav-
ities is governed by the Lugiato-Lefever equation (LLE),
i.e., driven-damped NLSE [37, 38] and differs substan-
tially from the breathing “double periodic” solution in
the NLSE with normal dispersion [39]. In the highly
driven regime, we observe that the dark breather in our
arXiv:1709.03912v1 [physics.optics] 12 Sep 2017
2
dissipative cavity undergoes a period-tripling bifurcation
and then a transition into a chaotic state. This behavior
is prohibited in systems governed by the NLSE. The ex-
perimental observation of dark breathers is confirmed by
rigorous comparison with numerical simulations.
From a practical perspective, studying the dark
breather gives insights into dark pulse Kerr comb gener-
ation in normal dispersion microresonators. The soliton
Kerr combs generated in anomalous dispersion microres-
onators have been used in many applications, including
communications [40], dual-comb spectroscopies [41], and
low noise microwave generation [42]. However, the con-
version efficiency from the pump into the generated comb
lines is usually less than a few percent for single-soliton
combs [43, 44]. In contrast, dark pulse Kerr combs can
reach much higher conversion efficiency (above 30%) [45],
providing an important tool for high power Kerr comb
generation. Dark breathers represent an intrinsic source
of instability for dark pulses; characterization and mod-
eling of such breathers may offer better opportunities for
their control, and trigger the search for dark breathers in
other physical systems.
RESULTS
Dark breather generation
The dark breather is generated in a SiN microresonator
with a radius of 100
μ
m, a loaded
Q
-factor of 0.8
×
10
6
,
geometry of 2000 nm
×
600 nm, and a normal dispersion
of
β
2
=190 ps
2
/km. The same microresonator pumped
at the same resonance was previously used to generate
a mode-locked dark pulse [29]. By tuning the cw pump
laser into resonance from blue to red, we are able to ini-
tialize the comb generation via mode-interaction [29, 30]
at the pump power of 1.7 W (off-chip power, the fiber-
to-chip coupling loss is
∼
2 dB per facet). At the pump
wavelength of 1549.19 nm, we get some coarsely spaced
comb lines as shown in Fig. 1a, which is dominated by the
mode-interaction assisted modulation instability dynam-
ics. The RF spectrum of this comb shows some narrow
sidebands at low frequencies (60 MHz) (Fig. 1b). These
RF sidebands in the modulation instability regime can
be attributed to the beat note between the main comb
lines and the subcomb lines (see the inset of Fig. 1b) as
reported in ref. [46]. Hence, the sidebands are accom-
modated within the halfwidth-half-maximum (HWHM)
of the cavity resonance (125 MHz). Thus, the comb in
Figs. 1a, b is not a dark breather, despite the narrow RF
sidebands. When further tuning the pump into the reso-
nance, we can get a broadband, single free-spectral-range
(FSR) comb at pump wavelength of 1549.26 nm (Fig.
1c). The comb in this state also has narrow RF sidebands
as shown in Fig. 1d. However, the breathing frequency
(740 MHz,
>
12 times of the frequency in Fig. 1b) is
outside the HWHM of the resonance. Thus, these side-
bands are attributed to a different generation mechanism
from the main-sub comb beat. Supplemented with fur-
ther evidence described below, we confirm that the comb
in Figs. 1c, d corresponds to a dark breather and at-
tribute the sidebands to periodic breathing (modulation)
of the comb lines. The averaged simulated optical spec-
trum (see the section on numerical simulations and Meth-
ods below) of the dark breather is in close agreement with
the measured optical spectrum. An important feature of
the dark breather is that it only breathes weakly. We de-
fine the breathing depth as (
P
max
−
P
min
)
/
(
P
max
+
P
min
),
where
P
max
(
min
)
is the maximum (minimum) power. The
comb power including (excluding) the pump breathes
with a depth of
∼
3% (
∼
5%). This is significantly dif-
ferent from bright soliton breathers, which can breathe
with a depth of
∼
50% (excluding the pump) [19]. We
can get the coherent dark pulse comb by reducing the
pump power to 1.6 W at the same wavelength. Now the
comb shows low-noise operation (Figs. 1e, f). Again,
the simulated dark pulse has an optical spectrum which
closely matches the measurement. There is no signifi-
cant difference between the dark breather and the mode-
locked dark pulse in the optical spectrum analyzer aver-
aged spectra, as the dark breather only breathes weakly;
whereas the bright soliton and bright soliton breather
combs show distinct averaged spectra [19, 21].
Characterization of dark breathers
The high frequency RF sidebands, which were also re-
ported in [29], are not sufficient by themselves to confirm
the observation of dark breathers. For example, a comb
with a constant spectrum shape but varying power can
show similar peaks. Hence, we first need to confirm there
is internal energy exchange between the comb lines. To
do that, we use a pulse-shaper [47] to programmably se-
lect groups of comb line and record their fast evolution
[19] (see Methods and Supplementary Information Sec.
1). We first choose the comb lines whose mode number
with respect to the pump are within [
−
3
,
3] (pump line
mode 0 included and larger number means higher fre-
quency) as the center group, while the comb lines within
[
−
10
,
−
4]
⋃
[4
,
10] as the wing group. The oscillation of
the recorded power change of the center group and the
wing group show a phase difference of 0.21
π
in Fig. 2ai.
When further suppressing the comb line within [
−
2
,
1] for
the center group (i.e., only keeping lines
−
3, 2 and 3), the
center group breathes out of phase (0.93
π
) with the wing
group as shown in Fig. 2aii. Thus, it is confirmed that
there is energy exchange within the Kerr comb in Fig.
1c. To gain more insight into the breathing dynamics of
the dark breather, the pulse shaper is programmed to se-
lect single comb lines to record their fast breathing. The
absolute breathing amplitude of the central comb lines
3
1450 1500 1550 1600 1650
1700
Wavelength (nm)
Intensity (20 dB/div)
Intensity (20 dB/div)
Frequency (MHz)
0
50
100
150
HWHM
resonance
main
sub
Intensity (20 dB/div)
Wavelength (nm)
0
500
1000
1500
HWHM
Frequency (MHz)
Intensity (20 dB/div)
20 ns
2
X
3%
Intensity (20 dB/div)
0
500
1000
1500
Frequency (MHz)
Intensity (20 dB/div)
a
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
f
mode-interaction
avg.
FIG. 1.
Generation dynamics of the dark breather.
When tuning the laser into cavity resonance, the comb is initiated
by mode-interaction.
a, b,
In the modulation instability regime, some coarsely spaced comb lines are generated and there
are narrow RF peaks, whose frequencies are within the half-width-half-maximum (HWHM) of the cavity resonance.
c, d
,
With further tuning into resonance, we can get a broadband comb and a dark breather state. The simulated optical spectrum
is in close agreement with the measured spectrum. The dark breather state has narrow RF peaks on the RF spectrum,
whose frequencies are outside the HWHM of the resonance. The inset in
d
shows the measured comb power change of the
dark breather. The dark breather only breathes weakly with a depth
∼
3%.
e, f,
By lowering the pump power at the same
wavelength, we can get a coherent dark pulse comb with low RF noise. The simulated optical spectrum also closely matches
the measured spectrum. There is no significant difference in the optical spectra in the dark breather state and the dark pulse
state.
within [
−
2
,
2] is much higher compared to other comb
lines. However, the average power is also highest in the
central comb lines. The modulation depth of individ-
ual lines tends to be largest in the wing of the spectrum
(Fig. 2b). Moreover, different lines breathe with differ-
ent phases as shown in Fig. 2c, which differs from the
breathers in the NLSE with normal dispersion [39] (Sup-
plementary Section 4). Abrupt changes in the phase and
amplitude of the breathing can be seen between some
adjacent lines, whereas bright soliton breathers have rel-
atively smooth changes in breathing phase [19]. We also
note that the breathing phase is not symmetric with re-
spect to the pump (e.g., see the modes
±
1, illustrated by
the arrows). Mode-interactions may be responsible for
this asymmetry, as the dominant mode-interaction only
exists on one side of the pump, breaking the symmetry
(see numerical simulations below).
To confirm the observation of dark breathers, we need
to verify that the comb in Figs. 1(c, d) has a dark-pulse
like waveform. We use line-by-line pulse shaping (see
Methods) to probe the corresponding waveform [29, 47,
48]. We first adjust the pump power to reach the mode-
locked dark pulse state and apply spectral phase shaping
to obtain a transform-limited pulse at the output (Fig.
2d). We then adjust the pump power to transition into
the dark breather state, while keeping the same phase
profile on the pulse-shaper. The output is still shaped
into a transform-limited pulse. This comparison suggests
that the dark breather has a spectral phase profile close
to that of the mode-locked dark pulse. Since the dark
breather exhibits only a gentle modulation and the power
spectrum remains close to that of the dark-pulse comb,
we deduce that the dark breather retains a dark-pulse-
like waveform (Supplementary Section 2). These findings
confirm that we are observing a dark breather in our
normal dispersion microresonator.
Transition to chaotic breathing
Fig. 1 shows the breathing instability can be elim-
inated by decreasing the pump power. On the other
hand, the absolute breathing depth increases slightly
when increasing the pump power (Fig. 3a). At the
same time, the breathing rate decreases slightly (Fig.
3b). The breathing rate of bright soliton breathers in
the anomalous dispersion regime also decreases with in-
creasing pump power [21]. The decrease in the breathing
rate is an indication of approaching critical transitions
for dynamical systems [21, 49]. When increasing the
pump power further to 1.9 W, the breathing of the dark
breather goes through a period-tripling bifurcation (Fig.
3c). The period-doubling bifurcation occurs more com-
monly in dynamical systems [50] including bright soliton
breathers in microresonators [20, 21]; however, it is not
observed for the studied dark breather. The perturbation
4
-0.2
0.2
a
0
1
2
3
4
5
-0.2
0.2
Time (ns)
P (a. u.)
P (a. u.)
c
i)
ii)
Mode number
Amplitude (a. u.)
b
-10
-5
0
5
10
0
1
2
3
4
5
Mode number
Slow time (ns)
-1
1
Delay time (ps)
AC intensity (a. u.)
d
[4, 10]
[-3, 3]
[-10, -4]
[4, 10]
-3
[-10, -4]
wing
center
wing
center
Breather
Coherent
2, 3
-2
-1
0
1
2
0
0.2
0.4
0.6
0.8
1
Breather
Coherent
Calculated
Calculated
280 fs
-10
-5
0
5
10
0
1
2
0
1
Depth
FIG. 2.
Characterization of the dark breather. a,
The dark breather exhibits energy exchange between different comb
lines. When choosing the comb lines with mode numbers in [
−
3
,
3] as the center group and comb lines in [
−
10
,
−
4]
∪
[4
,
10] as the
wing group, we observe a phase difference in the fast evolving spectrum, but not an anti-phase exchange in (i). When eliminating
modes in [
−
2
,
1] for the center group (keeping mode
−
3, 2, 3 only), there is a nearly out of phase energy exchange between the
wing group and the modified center group in (ii).
b,
The absolute and fractional breathing amplitudes of individual comb lines.
The absolute breathing amplitude is stronger at the center, while the breathing depth is larger at the wing.
c,
Different comb
lines breathe with different phase; the change of the breathing phase varies abruptly and asymmetric with respect to the pump.
The colour density shows the normalized breathing of each individual line (normalized to the breathing amplitude of the same
individual lines) and the circles on line mark the peaks of the oscillation for different lines, illustrating the breathing phase.
The arrows illustrate the peaks for the modes
±
1.
d,
The dark breather retains a dark-pulse-like waveform. By line-by-line
pulse shaping, dark pulses can be shaped into transform-limited pulses. Using the same phase on the pulse-shaper, the dark
breather comb can also be shaped into a transform-limited pulse.
from mode-interaction is found to be important to this
period-tripling, without which it does not occur (see Sup-
plementary Section 5). Such a direct period-tripling tran-
sition was also demonstrated in passively mode-locked
fiber lasers [51].
By increasing the pump power further to 2.1 W, the
breathing becomes chaotic. The measured RF spectrum
shows broadband noise (Fig. 3d), and the phase dia-
gram (plotting first-order time derivative of comb power
d
P
comb
/d
t
versus
P
comb
) shows the comb power changes
chaotically (Fig. 3e). Chaotic breathing was predicted
and observed (in fibre cavities) to exist in the LLE with
anomalous dispersion [18, 52, 53]; it has also been pre-
dicted to exist for dark breathers but not demonstrated
yet [35]. Our results constitute the first observation of
such a chaotic transition for dark breathers in all physi-
cal systems, to our knowledge. Note this chaotic breath-
ing state is distinct from the chaotic state prior to the
mode-locking transition [6, 29] (see Supplementary Sec-
tions 1, 3). The comb in the chaotic breathing state re-
tains the optical spectral features of the dark pulse (Fig.
3f), and the averaged spectrum in simulations is also in
good agreement with the measurement. When we ap-
ply the same phase profile used in Fig. 2d on the pulse
shaper, the auto-correlation (AC) trace obtained under
the chaotic breathing state still shows a clean pulse which
is close to the transform-limit (as calculated from the
spectrometer averaged spectrum). The visibility of the
AC trace exceeds 97% of the expected value; this dif-
fers dramatically from what is observed with the chaotic
comb in the absence of a mode-locking transition [48].
The AC results suggest the chaotic breathing state has
a dark-pulse-like waveform (see also the simulation re-
sults in Fig. 5). This is consistent with the fact that
excitation of the chaotic breathing is reversible, i.e., the
dark breather and the coherent comb can be obtained
by simply decreasing the pump power from this chaotic
state.
Numerical simulations
Kerr comb generation dynamics can be modeled by
the LLE, i.e., the driven-damped NLSE [18, 19, 37]. For
the NLSE, breather-like solutions also exist with nor-
mal dispersion [39] (Supplementary Section 4). However,
dissipative effects in externally pumped microresonators
change the dark breather dynamics dramatically. Here,
we use the LLE to model the dark breather dynamics.
When choosing the pump power as 700 mW and detuning
as 0.0629, a dark breather can be generated (see Meth-
ods and Supplementary Section 3). As shown above, the
simulated spectrum (averaged over slow time) is in close
agreement of the measured spectrum (Fig. 1c). The in-
clusion of the mode-interaction gives the long wavelength
peak around 1657 nm. From Fig. 4a, we can see the pe-
riodic variation of the spectral and temporal shape of the
dark breather (see the dashed boxes). From the tempo-
ral breathing, we can see the waveform-background (high
power part of the dark-pulse-like waveform) stays nearly
5
0
500
1000
1500
0
500
1000
1500
238 MHz
715 MHz
Frequency (MHz)
Frequency (MHz)
Intensity (10 dB/div)
Intensity (10 dB/div)
Power (W)
Breathing rate (MHz)
238 MHz
1.72
1.76
1.8
725
730
735
740
745
750
755
760
b
Power (W)
1450
1550
1650
Intensity (20 dB/div)
Exp
Sim
AC (a. u.)
Cal.
Exp
Wavelength (nm)
Delay (ps)
Measured RF
Background
Measured RF
Background
c
d
e
f
g
-2
0
2
0
1
-1
0
1
-1
0
1
P
comb
(a. u.)
d
P
comb
/d
t
(a. u.)
1.72
1.76
1.8
2.8
3
3.2
3.4
3.6
Breathing depth (%)
a
FIG. 3.
Power dependence of the breathing properties. a,
The breathing depth of the comb power increases with
increasing pump.
b,
The breathing rate decreases slightly with pump power.
c,
When the pump power is increased to 1.9 W,
the system transitions to a period-tripling state, as evidenced by the additional sidebands spaced by
∼
238 MHz.
d,
Broadband
RF noise is observed when the pump power is further increased to 2.1 W.
e,
The phase diagram of the output comb power
(pump included), which is obtained by plotting the first-order time derivative of the comb power (d
P
comb
/d
t
) versus
P
comb
, also
shows chaotic variation.
f,
The optical spectrum in this chaotic breathing state is similar to the dark pulse state; the averaged
simulated spectrum in the chaotic breathing state is in close agreement with the measured spectrum.
g,
The autocorrelation
of the Kerr comb in the chaotic breathing state after pulse shaping (the same amplitude and phase adjustment used in Fig.
2d) is close to the calculated autocorrelation trace of the transform limited pulse. The auto-correlation of the Kerr comb in
the chaotic breathing state after pulse shaping (the same amplitude and phase adjustment used in Fig. 2d) is close to the
calculated auto-correlation trace of the transform limited pulse.
unchanged in the breathing and the waveform-hole (low
power part of the dark-pulse-like waveform) breathes
more strongly (Supplementary Animation 1). This helps
to elucidate the low breathing depth of the comb power,
as the waveform-top carries most of the power. With the
same detuning, a coherent dark pulse comb can be ob-
tained by decreasing the pump power to 628 mW (Fig.
4b), consistent with experiments. As shown in Fig. 4c,
the simulated dark breather has a breathing rate of 782
MHz, and the breathing depth of the comb power includ-
ing (excluding) the pump is 3% (6%), both in agreement
with the experiments. The simulation retains the dark-
pulse-like waveform in the breathing state; the simulated
AC (averaged over slow time, see Methods) remains close
to the AC of the coherent comb after numerical pulse
shaping (Fig. 4d).
The simulated dark breather also exhibits energy ex-
change between comb lines. When choosing the comb
lines in the same manner as used in Fig. 2a, the simu-
lated power change is shown in Fig. 4e. The center group
in [
−
3
,
3] has a different breathing phase with the wing
group in [
−
10
,
−
4]
∪
[4
,
10], but is not completely out of
phase (0.70
π
). Similar to experiments, when eliminating
modes in [
−
2
,
1] from the center group, the center group
and the wing group exhibit nearly out of phase energy
exchange (1.11
π
). The breathing behaviour also differs
greatly from line to line. The absolute breathing am-
plitude is stronger near the spectrum center, while the
breathing depth tends to be stronger at the wing (Fig.
4f). The discrepancy of the absolute breathing amplitude
from experiments (Fig. 2b) may be attributed to the dif-
ficulty in modeling the mode-interaction accurately. The
simulated breathing amplitude is also not symmetric with
respect to the pump. The breathing phase of different
lines changes asymmetrically with respect to the pump
(see especially modes
±
1 in Fig. 4(g)). Note that the
breathing becomes symmetric with respect to the pump
when mode-interaction is excluded (Supplementary Sec-
tion 5). This simulation together with the experimental
result in Fig. 2c suggests that the comb lines in the dark
breathers are highly coupled and the dark pulse instabil-
ity can be affected by a perturbation over 12 THz away.
Furthermore, the breathing phase can change abruptly
between adjacent lines, in agreement with experiments.
The simulated dark dynamics also depend strongly on
the pump power. In agreement with experiments, the
simulated breathing depth increases slightly while the
breathing rate decreases slightly with increasing pump
power, as shown in Fig. 5a. Moreover, when the pump
power is further increased, the simulated dark breather
undergoes a transition into chaotic breathing via period-
tripling. When increasing the pump power to 752 mW,
the simulated RF spectrum shows a period-tripling fea-
ture (Fig. 5b). Note that this period-tripling does not