Supplementary
Materials
for
Multimodality integrated microresonators using the Moiré speedup effect
Qing
-
Xin Ji
et al
.
Corresponding author
s
:
John E. Bowers, jbowers@ucsb.edu; Kerry J. Vahala, vahala@caltech.edu
Sci
ence
3
8
3
,
10
80
(
202
4
)
DOI:
10.1126/sci
ence
.
adk9429
The
PDF
file
includes:
Materials and Methods
Figs. S1 to S4
References
Other Supplementary Material for this manuscript includes the following:
Movie S1
2
Materials and Methods
Theory of Moiré speedup of the dispersion spectrum
In this section, we discuss the theory that describes the Moiré
speedup of the dispersion
spectrum when the two rings are differentially tuned. Calculation of the dispersion in the
Vernier
-coupled ring resonators is detailed in (8). Briefly, a transfer matrix, T, is used to
propagate a 2
-component wave function through a round trip.
푇푇
=
푒푒
푖푖휔휔
±
퐿퐿
�
/
푐푐
�
푒푒
−푖푖
휔휔
Δ퐿퐿
/
푐푐
cos
(
푔푔
co
푙푙
co
)
푖푖
sin(
푔푔
co
푙푙
co
)
푖푖
sin(
푔푔
co
푙푙
co
)
푒푒
푖푖
휔휔
Δ퐿퐿
/
푐푐
cos
(
푔푔
co
푙푙
co
)
�
. (S1)
The resulting secular equation gives the eigenfrequencies,
휔휔
푚푚
,±
, of the two mode families
(see plots in Fig. 2B and Fig. 2C in the main text),
휔휔
푚푚
,±
−휔휔
푚푚
= ±
퐷퐷
1
,
푚푚
2휋휋
arccos
[cos
(
푔푔
co
푙푙
co
)cos
(
2πε
m)]
≡퐷퐷
int
,
c
, (S2)
w
here
푚푚
is mode number,
휀휀
=
Δ퐿퐿
/
퐿퐿
�
is the optical path length contrast of the rings (
퐿퐿
�
≡
(n
wg
,
A
푙푙
A
+ n
wg
,
B
푙푙
B
)/2
is the averaged optical path length of the two rings, and
Δ퐿퐿≡
(n
wg
,
A
푙푙
A
−
n
wg
,
B
푙푙
B
)/2
),
푔푔
co
is the coupling strength per unit length,
푙푙
co
is the length of the
coupling section,
휔휔
푚푚
=
2π
mc
/
퐿퐿
�
with
푐푐
the speed of light in vacuum, and
퐷퐷
1
,
푚푚
≡휕휕휔휔
푚푚
/
휕휕
푚푚
.
The coupling strength
푔푔
co
has a weak dependence on m (and wavelength), decreasing by 4.1%
per 1 THz towards higher optical frequencies in the current design. This results from the
wavelength dependence of the evanescent field coupling between the two rings.
F
ree spectral range
퐹퐹퐹퐹푅푅
푚푚
,±
of the two bands is obtained by calculating derivative of
휔휔
푚푚
,±
with respect to
푚푚
:
퐹퐹퐹퐹푅푅
푚푚
,±
=
퐹퐹퐹퐹푅푅
푚푚
±
휀휀
퐹퐹퐹퐹
푅푅
o
cos
(
푔푔
co
푙푙
co
) sin(2
휋휋
휀휀푚푚
)
�
1
−
cos
2
(
푔푔
co
푙푙
co
)
cos
2
(2
휋휋
휀휀푚푚
)
, (S3)
where dispersion of the mode coupling strength
휕휕푔푔
co
/
휕휕
푚푚
is neglected in the calculation, and
where
퐹퐹퐹퐹푅푅
푚푚
=
퐷퐷
1
,
푚푚
/2
휋휋
and
퐹퐹퐹퐹푅푅
o
=
푐푐
/
퐿퐿
�
=
퐷퐷
1
�
�
�
/2
휋휋
. The group velocity dispersion parameter
퐷퐷
2
,
c
(as plotted in Fig. 3C) is obtained by calculating the derivative of
퐹퐹퐹퐹푅푅
푚푚
,±
with respect to
푚푚
,
퐷퐷
2
,
푚푚
,±
−퐷퐷
2
,
o
≈
±
퐷퐷
1
�
�
�
2휋휋휀휀
2
cos
(
푔푔
co
푙푙
co
)
sin
2
(
푔푔
co
푙푙
co
)
cos
(
2휋휋
휀휀푚푚
)
[
1− co
s
2
(
푔푔
co
푙푙
co
)
cos
2
(
2휋휋
휀휀푚푚
)]
3
2
≡퐷퐷
2
,
c
, ( S4
)
w
here again
휕휕푔푔
co
/
휕휕
푚푚
is neglected, and
퐷퐷
2
,
o
/2
휋휋
=
휕휕
퐹퐹퐹퐹
푅푅
푚푚
/
휕휕
푚푚
is the GVD parameter of the
uncoupled resonator waveguide. The r.h.s. of eqn. (S4) vanishes when
2
휀휀
푚푚
=
푁푁
+ 1/
2
where
푁푁
is an integer.
Equation (S3) also predicts the accessible dispersion regime when the two rings are
differentially tuned,
�퐷퐷
2
,
c
�≡�퐷퐷
2
,
푚푚
,±
−퐷퐷
2
,
o
�≤
2
휋휋휀휀
2
퐷퐷
1
�
�
�
cot
(
푔푔
co
푙푙
co
)
. ( S5
)
The r.h.s. of eqn. (S5) corresponds to the shaded orange area in Fig. 3C in the main text.
N
ote that
2
휀휀
푚푚
=
푁푁
(where
푁푁
is an integer) corresponds to the mode numbers
푚푚
where the
ring frequencies overlap and hybridize. The broadband nature of the Moiré
pattern modulation
results from the separation of these mode numbers (
Δ푚푚∼
1/
휀휀
) being large. In the present
system
Δ푚푚
=
200
. Also, note that
2
휀휀
푚푚
=
푁푁
+ 1/
2
corresponds to frequencies where mode
resonances in the two rings interleave (circled dots in Fig. 2A,B in the main text). Differential
temperature tuning varies
휀휀
,
thereby shifting the mode number locations where strong
hybridization occurs (as well as the overall Moiré
pattern). To calculate the corresponding shift
in frequency, consider the change
훿훿푓푓
o
in optical frequency at a given overlap frequency
푓푓
o
(or an
interleave frequency) as given by,
3
훿훿푓푓
o
푓푓
o
=
훿훿푚푚
푚푚
=
−
훿훿휀휀
휀휀
=
−
1
2
휀휀
훿훿
(
퐿퐿
A
−퐿퐿
B
)
퐿퐿
�
≡−푅푅
훿훿
(
퐿퐿
A
−퐿퐿
B
)
퐿퐿
�
, (S6
)
where
푅푅 ≡
1/(
2
휀휀
)
is the Vernier ratio. A simple relation between the differential ring tuning
훿훿
(
퐿퐿
A
−퐿퐿
B
)
and tuning of the dispersion spectrum
훿훿푓푓
o
is thus established. On the other hand, the
mode resonance tuning of a single ring via optical path length change is given by,
훿훿푓푓
푓푓
=
−
훿훿퐿퐿
퐿퐿
, (S7
)
from which it follows that the Vernier tuning scheme enhances the tuning rate by a factor of
푅푅
.
This is also the optical speedup factor in the movement of the Moiré
pattern (10)
.
Heater fabrication and control
A pair of metallic resistance heaters (platinum with titanium as an adhesion layer) are
deposited using ebeam evaporation and liftoff. As shown in Fig. 1C, the metal is offset from the
SiN waveguides by 8
휇휇
m to avoid metal
-induced optical absorption. To suppress cross
-talk
heating is minimized near the coupling section between the two rings as described in Fig. S
1A
.
The heaters are wire
-bonded to a printed circuit board to apply direct
-current voltages. The chip
is attached to a copper submount using thermal
-conductive epoxy. A thermistor and a ceramic
heater are used in combination with a PID control loop to stabil
ize temperature within <
1 mK
error, while ~10 mK temperature variation is generally allowed for
the device operation
. The
metallic heaters are used for differential tuning. The common temperature is varied using the
ceramic heater.
Frequency
response of the integrated heaters
The frequency response of the metallic heater tuners was also measured. The princip
le of
the measurement approach is shown in Fig. S1B
. A single frequency probe laser (red line) is
tuned into a mode resonance of the coupled rings. One of the heaters is then modulated by a
vector network analyzer (VNA) which causes modulation of the mode resonant frequency. The
laser's frequency is fixed
so that the thermal-
induced mode resonance modulation is transferred
as modulation of transmitted laser power on the bus waveguide. The modulated power is
monitored by a photodetector and measured by the VNA to determine the frequency response
|
퐻퐻
�
|
.
The experimental setup is shown in Fig. S1C
. A CW Toptica ECDL is coupled into the
bus waveguide using a lensed fiber, and the power transmission signal is collected by a Newport
1811 photodetector. The laser is loosely stabilized (feedback bandwidth <10 Hz) to the cavity
resonance by locking to a certain transmitted power. The measured
퐻퐻
�
(
푓푓
)
is given Fig. S1
D
where the 6 dB bandwidth is
2
.67
±
0.05
kHz
, when fitted by a Lorentzian response (4
2) .
Stimulated Brillouin lasing in the coupled rings
For the Brillouin laser measurement in Fig. 1D, the III
-V DFB laser is tuned to the mode
resonance of the upper frequency band to induce the self
-injection lock
, and the frequency shift
to the lower frequency band is configured (by differential tuning) to phase match the Brillouin
process. The DFB laser emits ~120 mW of CW light and the on-
chip power is estimated to be
30-
40 mW. Output of the Brillouin laser is m
onitored at the through port of the pumped
waveguide as in Fig. 1D in the main text. Feedback phase of the self
-injection process is
optimized to observe the stable beatnote between the pump and the Brillouin laser, as in the inset
of Fig. 1D. This is done by precise control of the coupling gap between the laser and the
4
resonator chip. The Brillouin laser can operate for minutes under the self
-injection locked mode.
We believe this is limited by slow drift of the feedback phase and relative temperature of the
laser and resonator. The introduction of active control should eliminate this
issue.
For the measurement of the C
-band tunable Brillouin laser in Fig. 3B, an external cavity
diode laser (Toptica DLC pro) is amplified by an Erbium doped fiber amplifier and coupled into
the rings through a lensed fiber. The pump frequency is locked to the cavity by servo control of
the laser frequency (piezo control) to maintain transmitted power. Dispersion is tuned (by
differential temperature tuning) at each pumping wavelength to phase match the Brillouin
process. The nominal threshold of the Brillouin
laser action is 10.4 mW near 1550 nm.
Noise measurement
The enhanced tuning in this system can make possible magnified fluctuations in
dispersion through differential temperature fluctuations. The exact impact of such fluctuations is
still under investigation, however, some preliminary data and discussion is pr
esented here. First,
the optical frequency noise of the pump and SBS laser were characterized by collecting output
from the bus waveguide using a lensed fiber. An Erbium
-doped fiber amplifier was then used to
boost optical power for measurements
, followed
by filtering of the pump (or the Brillouin laser)
using a tunable fiber Bragg grating. The frequency noise spectrum was measured using the
method in the reference (43). Representative frequency noise spectra from the III
-V pump and
the Brillouin laser are shown in Fig. S2.
Second, concerning microcomb repetition rate noise measurements, a similar version of
the integrated microcomb studied here has been characterized in both free
-running operation and
in an optical frequency division (OFD) system. For the latter, the comb OF
D system generated
record
-low noise microwave signals for a microcomb system suggesting that the Moiré
effect
does not significantly impair this critical application (21).
Broadband reconfiguration of bright pulse microcomb and a dark pulse microcomb
In this section, details of the bright/dark pulse microcomb generation are provided. Four
different DFB lasers are exchanged to pump the same coupled
-ring device. The pumping occurs
under self
-injection
-locking conditions (i.e., without any optical amplifi
cation or isolation). The
DFB laser at ~1542 nm is a commercial DFB laser from PhotonX. Inc.; the other three DFB
lasers are from Emcore Corp. Dispersion is tuned (by differential electrical heating) to allow the
generation of bright and dark pulse microcombs at each wavelength. Coherence of the generated
pulses is confirmed by detecting the repetition rate tone using a fast photodetector and
measurement using an electrical signal analyzer as shown in Fig. S3. Repetition rate tones for
microcomb states when pumped at 1548 nm are already included in Fig. 1, and thus not included
here. Microwave spectrum measurement results at the other three pumping wavelengths (see
optical comb spectra in Fig. 3D and Fig. 3E) a
re presented in Fig. S
3. Microwave
phase noise of
the repetition rate tone for the dark pulse mode has been reported elsewhere using a comparable
device (21).
We also measured switching between a bright soliton microcomb and dark pulse
microcomb in real time, as shown in Fig. S
4. In this measurement, a DFB laser (at around 1548
nm) is self
-injection locked to the coupled rings. Switching is induced with an arbitrary
waveform generator (AWG), whose output was amplified by an electrical waveform amplifier
(Amp.) to drive the heater
s for differential thermal tuning. Current applied on the DFB laser is
simultaneously switched to maintain the self
-injection lock. Micr
ocomb states are monitored
5
using an optical spectrum analyzer (OSA), while the repetition rate tone is mixed down by a
local microwave synthesizer near the repetition rate, and then monitored in real time using a
frequency counter (FC). As noted in Fig. 1E and Fig. 1F, the repetition rate frequency of the
bright soliton microcomb is slightly higher than the dark pulse
microcomb
(also noted in Fig. S
3)
The measurement results are shown in Fig. S
4B. In this measurement, the bright soliton state
stays stationary for 200 seconds, at whic
h time the microcomb state is switched. A dark pulse
microcomb state emerges in seconds and remains stationary afterwards for another 200 seconds.
6
Fig. S1.
(B) Details of the heater
routing. The electrodes (gray rectangles) are wire bonded to an external
circuit board to apply the heating current. Heaters (red) are located at the arcs of the racetrack
and are electrically connected with a wider electrical trace (yellow). The wider trace has less
electrical resistance and thus reduces the heating effect at the coupling section of the two rings.
This design improves the differential thermal tuning efficiency. All three parts (red, yellow and
gray) are deposited with the same Pt layer thi
ckness.
(
B
) Principle of integrated heater frequency response characterization.
(
C
), Heater frequency response characterization setup.
(
D
), Measured transfer function of the heater. The dashed line is a Lorentzian fitting where the 6
dB bandwidth is
2
.67
±
0.05
kHz.
7
Fig. S2.
Optical frequency noise spectrum of pump (self
-injection
-locked) and the Brillouin
laser.
8
Fig. S
3. Repetition rate tone of several bright pulse and dark pulse microcombs.
Resolution
bandwidth is 10 kHz for all plots. (
A-C
), Repetition rate tone of bright pulse states at three
pumping wavelengths. (
D-F
), Repetition rate tone of dark pulse microcomb at three pumping
wavelengths.
9
Fig. S
4. Real time switching between a bright pulse microcomb and a dark pulse
microcomb. (
A)
,
Experimental setup.
(
B
),
Measured repetition rate when the microcomb is
switched from a bright pulse microcomb to a dark pulse microcomb.
10
Movie S1
.
Animation of the Vernier-enhanced dispersion tuning process.
(
A
), Formation of the Moiré
pattern and its tuning.
(
B
), Calculated integrated dispersion excluding waveguide dispersion
퐷퐷
int
,
c
as in eqn. (S2)
(
C
), Calculated group velocity dispersion
퐷퐷
2
,
c
as in eqn. (S4). The zero points (interleaved
region) are marked as i, ii and iii (the same as Fig. 2A,B in the main text).
The supplementary movie illustrates the Moiré
speedup effect for dispersion control. In panel A,
modes inside the two isolated rings are partially overlapped to create the Moiré
pattern (similar
to Fig. 2A in the main text). The calculated
퐷퐷
int
,
c
and
퐷퐷
2
,
c
(similar Fig. 2B and Fig. 3B in the
main text) are plotted in
panel B and Fig. panel C. The interleaved region is marked with black
dots in panel C, whose position is extracted for Fig. 2D in the main text. To make the Moiré
pattern easy to visualize, the parameters here are
퐷퐷
1
�
�
�
/2
휋휋
=80 GHz, and
휀휀
= 1/100
as compared
to
퐷퐷
1
�
�
�
/2
휋휋
=20 GHz
, and
휀휀
= 1/400
in the actual device.
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