of 5
940
Vol. 11, No. 7 / July 2024 /
Optica
Research Article
Observation of interband Kelly sidebands in
coupled-ring soliton microcombs
Maodong Gao,
1
,
Zhiquan Yuan,
1
,
Yan Yu,
1
,
Warren Jin,
2
,
3
Qing-Xin Ji,
1
Jinhao Ge,
1
Avi Feshali,
3
Mario Paniccia,
3
John E. Bowers,
2
AND
Kerry J. Vahala
1
,
*
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Department of Electrical and Computer Engineering, University of California, Santa Barbara, California 93106, USA
3
Anello Photonics, Santa Clara, California 95054, USA
These authors contributed equally to this work.
*vahala@caltech.edu
Received 18 March 2024; revised 7 June 2024; accepted 8 June 2024; published 5 July 2024
Kelly sidebands are a special type of dispersive wave that appear in mode-locked systems and they have recently been
observed by pulsed excitation in integrated microcombs. Here, Kelly sidebands are generated by continuous-wave exci-
tation in a partially coupled racetrack-resonator microcomb. The coupled-racetrack system supports two optical bands
so that, in contrast to earlier studies, the soliton and Kelly sideband reside in distinct bands. The resulting interband
excitation of the Kelly sidebands relaxes power requirements and continuous-wave sideband excitation is demonstrated.
Tuning of sideband spectral position under pulsed excitation is also studied. Numerical simulation and the experiment
showthatthesidebandsrelyuponsymmetrybreakingcausedbypartialcouplingofthetwo-ringsystem.Moregenerally,
multiband systems provide a new way to engineer Kelly sidebands for spectral broadening of microcombs.
© 2024
Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
https://doi.org/10.1364/OPTICA.524074
1. INTRODUCTION
Kelly sidebands (KSs) [1] have been intensively studied in soliton
mode-locked fiber lasers [2–7]. They resemble dispersive waves
(DWs) [8–11] but their phase relative to the mode-locked optical
pulse changes by integer multiples of 2
π
(instead of 0) every cavity
round trip. This round-trip phase mismatch prevents coupling
of the soliton and KS unless a symmetry is broken in the system.
Like DWs, KSs extend the spectral reach of the pulse. However,
generating KSs is challenging in integrated photonic resonators
because of their smaller round-trip path lengths compared to table-
top mode-locked systems. Nonetheless, KSs have recently been
observed in pulsed-pumped soliton microcombs featuring broken
symmetry [12].
Here, interband KSs are observed in soliton microcombs
formed using dual-racetrack Si
3
N
4
microresonators [13]
[Fig. 1(a)]. The Si
3
N
4
waveguides used to fabricate the racetracks
feature normal dispersion; however, partial coupling of racetracks
having slightly different free spectral ranges (FSRs) creates two
optical frequency bands associated with the hybridization of the
individual racetrack modes. The measured integrated dispersion
of these bands is plotted in Fig. 1(b), and shows that the frequency
bands feature spectral windows with anomalous dispersion. For
example, the upper band [soliton mode band in Fig. 1(b)] features
anomalous dispersion in the vicinity of point I, and pumping near
this point has been shown to form bright soliton pulse pairs that
circulate in a mirror-image fashion, as shown in Fig. 1(c) [13]. In
this work, the impact of this multifrequency band structure on
KS formation is studied. Specifically, interband KSs are shown
to form in the orthogonal band in Fig. 1(b). Also, as discussed in
Appendix A, partial coupling of the racetracks creates symmetry
breaking that allows the soliton to couple to these KSs. The spectral
shift provided by interband excitation of the KSs relaxes otherwise
challenging KS excitation requirements on comb bandwidth in
microcombs, and continuous-wave excitation is demonstrated. In
addition, pulsed pumping is studied as a way to tune the spectral
locations of the KSs.
2. RESULTS
The coupled-ring devices are fabricated using the CMOS-
compatible process described in [14]. Continuous-wave laser
pumping around the anomalous dispersion window near point I
produces the optical spectrum shown in Fig. 1(d). This spectrum
corresponds to soliton pulse pair mode locking, as illustrated in
Fig. 1(c) [13]. For comparison, portions of the dispersion spec-
tra for the two frequency bands in Fig. 1(b) are overlaid with the
pulse-pair spectrum. The spectrum in Fig. 1(d) features two DWs
at points II and III. These DWs correspond to modes that phase
match with soliton comb frequencies. Meanwhile, two further
sidebands are observed at points IV and V, which are identified as
KSs. These sidebands are notable because they form on the other
frequency band (interband KS) relative to the frequency band
used to generate the soliton pulse pair. As indicated, the values
of their integrated dispersion differ by one FSR (
D
1
/
2
π
in the
2334-2536/24/070940-05 Journal © 2024 Optica Publishing Group
Research Article
Vol. 11, No. 7 / July 2024 /
Optica
941
Fig. 1.
(a) Optical image of the partially coupled racetrack resonator. (b) Measured integrated dispersion of the two hybrid mode families (orange and
red) is plotted versus the wavelength and relative mode number. When pumped near point I (anomalous dispersion center of the upper band), soliton pulse
pairs form, as illustrated in (c). Also, DWs (interband KSs) appear at points II & III (IV & V). The appearance of KSs at points IV and V is discussed in
Appendix A.1.
ω
μ
, frequency of mode
μ
;
ω
0
, center mode frequency;
D
1
/
2
π
, free-spectral-range (FSR) of the resonator; and
μ
, relative mode number such
that
μ
=
0 corresponds to the pump mode. (c) Schematic showing soliton pulse pair propagation in the coupled racetracks. DW and KS waves are indicated
as oscillatory backgrounds. (d) Measured optical spectrum of soliton pulse pair from continuous-wave laser pumping. DWs and interband KSs are observed
at the predicted location in (b). Dispersion curves are overlaid onto the soliton spectrum for reference.
figure) compared to point I, and as such the sideband phases are
mismatched by 2
π
relative to the pumped mode every round
trip (See Appendix A, Fig. 4). It is important to note that this
mismatch is primarily accumulated because the soliton and KS
waves reside in distinct frequency bands (separated by a frequency
shift comparable to the cavity FSR). As shown in Appendix A, the
accumulation of this shift would normally require a larger number
of mode shifts and hence greater comb bandwidth and power.
While other types of interband sideband generation in coupled-
ring solitons have been studied before [15], the KSs in our study
are fundamentally different from conventional phase-matched
sidebands that do not require symmetry breaking.
Numerical simulation is performed to compare the difference
of KS generation in the partially coupled and fully coupled devices.
The result is shown in Fig. 2. The simulation for the partially
coupled case is based on the coupled mode Lugiato–Lefever equa-
tion (LLE) with a position-dependent coupling term in the lab
coordinate, formalized as [13]
E
R
t
=−
(
κ
2
+
i
δω
R
)
E
R
v
g
E
R
z
i
β
2
v
3
g
2
2
E
R
z
2
+
i g
K
|
E
R
|
2
E
R
+
i g
c
v
g
χ
c
(
z
)
E
L
+
f
p
,
(1)
and
E
L
t
=−
(
κ
2
+
i
δω
L
)
E
L
v
g
E
L
z
i
β
2
v
3
g
2
2
E
L
z
2
+
i g
K
|
E
L
|
2
E
L
+
i g
c
v
g
χ
c
(
z
)
E
R
,
(2)
where
E
L
,
R
denotes the normalized optical field in the left and the
right racetrack,
κ
=
κ
in
+
κ
ex
is the sum of intrinsic and external
loss rate for each racetrack,
δω
L
,
R
is the pump laser detuning,
v
g
is the group velocity,
β
2
is the group velocity dispersion of the
waveguide,
z
is the coordinate of each resonator,
g
c
is the coupling
strength per unit length,
g
K
is the Kerr nonlinear coefficient, and
f
p
is the continuous-wave pumping term. The function
χ
c
(
z
)
is
an indicator function with a value of 1 where the two racetracks
are coupled and a value of 0 where two rings are not coupled. For
the simulation in Fig. 2(a), the two racetracks are partially coupled
together, and this indicator function is
χ
c
(
z
)
=
{
1 z
∈[
0
,
L
c
),
0 elsewhere
,
(3)
where
L
c
is the effective length of the section where two racetracks
are coupled together. For the simulation in Fig. 2(b), the two race-
tracks are fully coupled together (i.e.,
χ
c
is unity at all positions)
so that the coupled LLE in the rotating frame [16] is used to study
soliton dynamics. Parameters used in this simulation are listed in
Appendix A.2.
In Fig. 2, the simulated results of the soliton spectrum are
plotted as the red curves, together with the dispersion as the black
curves. The spectrum in Fig. 2(a) of partial coupling (i.e., sym-
metry broken case) shows a similar structure to our observation in
Fig. 1(d). Specifically, the spectrum contains two DWs at points
II and III, and two interband KSs around points IV and V. For
comparison, the simulated soliton spectrum in the fully coupled
ring is shown in Fig. 2(b). Here, the KSs are absent and only two
DW sidebands appear at points II and III. In this fully coupled
case (i.e., symmetrical case), the frequency-matched interband
modes at points IV and V cannot generate KSs because of a phase
mismatch. As an aside, in comparing the simulated spectrum in
Fig. 2(a) with the measurement in Fig. 1(d), the measured KSs
Research Article
Vol. 11, No. 7 / July 2024 /
Optica
942
Fig. 2.
(a) Simulated soliton spectrum and dispersion profile of the
partially coupled racetrack. The soliton, when pumped at point I, gen-
erates two dispersive waves (point II and III) and two interband Kelly
sidebands (point IV and V). (b) Simulated soliton spectrum and disper-
sion profile of the fully coupled racetrack. In this case, the Kelly sidebands
at point IV and V do not appear, because of phase mismatch. (a) and
(b) Insets: illustration of partially coupled and fully coupled racetrack
configuration.
exhibit an asymmetry in power. This happens for several rea-
sons. First, the optical field of the modes at point IV (V) is mainly
distributed in the right (left) ring, while the measured output is
coupled from the right ring [13]. Accordingly, the KS appears to be
stronger at point IV (blue side) than point V (red side). Second, the
pumping laser is slightly blue-detuned relative to the anomalous
dispersion window center and this contributes to such a difference
in the measured KS power.
Increasing the comb bandwidth would tend to strengthen the
KS. And the comb bandwidth can be increased through control of
the integrated dispersion profile. As discussed in [13], this can be
tuned by controlling
g
c
and the round-trip length ratio between
the rings. A stronger KS is potentially useful for optical frequency
division [17–19]. We also note that the KS strength may be limited
by the strength of the DWs, which tend to destabilize the soliton as
the detuning or pump power is further increased [20].
To further study the tuning properties of the KSs, an optical
pumping pulse is generated by forming an electro-optic comb and
then applying pulse compression methods [21]. This method of
pumping enables higher peak pumping powers and also allows
exploration of the variation of the pumping repetition rate on side-
band formation. The resulting microcomb spectrum is shown in
Fig. 3(a), and the electro-optic comb pump spectral lines are visible
near the spectral center of the comb. Similar to the continuous-
wave pump case, two DWs form around points II & III and the two
KSs appear around points IV & V in the spectrum. In contrast to
the continuous-wave pump case, the repetition rate of the soliton
spectrum is determined by the electro-optic pump pulse [22,23].
The impact of varying this rate on the DW spectral location is
illustrated in the Fig. 3(a) inset, which overlays the dispersion of the
upper frequency band with the equally spaced comb frequencies
as given by a line. The intersection of the dispersion curve with the
soliton line gives the condition for phase matching of the soliton
with the dispersive wave. Tuning of the soliton repetition rate
causes a shift in the frequencies of the DWs as illustrated.
A similar phase-matching condition will also apply for the KSs,
but with respect to the second frequency band shifted by
D
1
/
2
π
.
Here, as the repetition rate increases, both the wavelengths of the
strongest blue and red KSs would be expected to increase. This
Fig.3.
(a) Optical spectrum of soliton pulse pair generated from pulse pumping. The measured mode frequency dispersion for the two bands is also plot-
ted. Inset: zoomed-in view of the dispersion curve of the pumped mode family and its comparison to the 19.97267 GHz comb frequency line (horizontal
magenta line) and 19.97359 GHz comb frequency (tilted blue line). The right vertical axis of the inset is a magnified version of the right vertical axis of the
main panel. (b) and (c) Zoomed-in optical spectra of the two KSs at (b) shorter wavelengths and (c) longer wavelengths for different input pump-pulse rep-
etition rates. Legend gives the repetition rate of the input pump pulse relative to 19.97217 GHz. Vertical dashed grid lines indicate multiples of the comb
repetition rate of 19.97267 GHz.
Research Article
Vol. 11, No. 7 / July 2024 /
Optica
943
is confirmed experimentally in Figs. 3(b) and 3(c), where tuning
by about 1.14 MHz of the repetition rate causes the interband
KS to jump by seven modes (blue side), and by five modes (red
side). For a larger repetition rate, the KS on the red side becomes
indistinguishable from the instrumental noise floor.
3. SUMMARY
In summary, interband Kelly sidebands have been produced
using both continuous-wave and pulsed pumping. The sidebands
reside in a frequency band that is distinct from the soliton pulse
in a coupled-racetrack resonator microcomb. Their interband
nature relaxes excitation requirements. The wavelengths of the
Kelly sidebands agree with predictions based on mode dispersion
measurements. Pulsed-pump operation allowed exploration of the
sideband tuning properties.
APPENDIX A
A.1 Principle of Conventional KS Generation
To illustrate the principle of conventional KS generation [1], con-
sider the spectrum of a frequency comb, as illustrated in Fig. 4(a).
The comb lines (blue) are equally spaced by a separation approx-
imately equal to the resonator FSR (or
D
1
/
2
π
). The comb line
at relative mode number
μ
=
0 is also the optical pump. The
frequencies of the resonator modes are illustrated as red lines. Due
to the dispersion of the resonator, these frequencies are not equally
spaced. Assuming anomalous second-order dispersion (
D
2
), as
required for soliton formation, the frequencies of the resonator
modes and comb lines walk off by
D
2
μ
2
/
2 versus
μ
. The inte-
grated dispersion plot of this mode family in Fig. 4(b) shows that,
as a result of this walk-off, the relative mode number
μ
is aligned
to the
+
1
)
-th comb line when the walk-off is
D
2
μ
2
/
2
=
D
1
.
This mode is orange highlighted in both Figs. 4(a) and 4(b). This
alignment is a necessary condition for generation of the KS at this
frequency, and the large FSR of the microcombs makes excitation
of this KS mode more challenging. In the case of interband exci-
tation, modes from the orthogonal band align with the comb at
a smaller walk-off than the soliton band due to the spectral gap
between the bands, thus relaxing the power requirements for KSs.
A second condition must also be satisfied for KS excitation. The
accumulated round-trip phase of the KS wave in Fig. 4(a) is offset
by 2
π
from the original comb every round trip. This is illustrated
in Fig. 4(c), and the integrated round-trip coupling of the soliton
to the KS wave is zero as a result of phase interference (i.e, phase
mismatch). Generation of the KS wave is thereby prevented in
cases where the resonators are fully coupled together, as shown
in Fig. 2(b). However, if the additional condition of partial race-
track coupling is added [as formalized in Eq. (3)], then the partial
round-trip integral is nonzero and KS generation occurs. In effect,
the fully coupled racetrack system features a reflection symmetry
that is lifted by partial coupling, thereby allowing the soliton and
KS wave to couple when the eigenmode frequency aligns with the
comb frequency [for example, see points IV and V in Fig. 1(b)].
This effect is also discussed in Fig. 2 of the main text. In addition to
KSs in mode-locked soliton combs, similar parametric sidebands
can also be generated in resonators with broken symmetry [24].
A.2 Parameters in Coupled Mode LLE
In Eqs. (1) and (2), the parameter definitions are the same as
those used in [13]. The optical field
E
L
,
R
is normalized to pho-
ton numbers in the length-averaged ring.
κ
in
,
ex
is related to the
resonator
Q
factor by
κ
in
,
ex
=
ω
0
/
Q
in
,
ex
. The laser frequency
detuning
δω
L
=
δω
R
=
12.5
κ
G
,
where
κ
=
κ
in
+
κ
ex
,
G
=
g
c
L
c
D
1
/
2
π
.
G
is the half frequency gap created by cou-
pling.
v
g
=
c
/
n
g
is the group velocity, where
c
is the speed of light
and
n
g
is the waveguide group index.
g
K
=
~
ω
2
0
D
1
n
2
/(
2
π
n
g
A
eff
)
is the nonlinear coefficient with
A
eff
as the effective mode area and
n
2
is the nonlinear coefficient of the waveguide. The group veloc-
ity dispersion is related to
D
1
and
D
2
by
β
2
=−
n
g
D
2
/(
c D
2
1
)
.
f
p
=
κ
ex
P
in
/(
~
ω
0
)
is the pump term, where
P
in
is the on-chip
pump power. The argument
z
is limited to
[
0
,
L
L
,
R
)
, where
L
L.R
is
the respective resonator round-trip length.
The numerical values of the relevant parameters are
Q
in
=
75
×
10
6
;
Q
ex
=
45
×
10
6
;
ω
0
=
2
π
×
193.34 THz;
L
R
=
9.5 mm;
Fig. 4.
(a) Explanation of Kelly sideband (KS) formation. Comb frequencies (blue) walk-off from resonator mode frequencies (red) by
D
2
μ
2
/
2 as a
result of anomalous dispersion. Comb line at
μ
=
0 is also the pump. The comb and mode frequency become aligned (orange highlight) when this walk-
off is equal to an integer multiple of
D
1
(FSR). This frequency is approximately the frequency of the KSs.
μ
, relative mode number;
ω
μ
, frequency of each
mode;
D
1
/
2
π
, FSR of the resonator; and
D
2
, second-order dispersion of the resonator. (b) Dispersion profile of the mode family shown in (a). The first few
corresponding modes in panel (a) are indicated by dashed arrows. When a mode frequency is an integer times
D
1
/
2
π
relative to the pumped mode at
μ
=
0, it becomes possible to generate a KS at this frequency, provided a second condition is satisfied, as described in (c). (c) Illustration of the propagation phase
of the comb and the mode at
μ
. Their relative phase changes by a multiple 2
π
every round trip, leading to phase mismatch of the waves and preventing KS
generation when the resonators are fully coupled.