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aaw2317
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Supplementary
Material
s for
Vernier spectrometer using counterpropagating soliton microcombs
Qi-Fan
Yang
, Boqiang Shen
, Heming
Wang
, Minh Tran
, Zhewei
Zhang, Ki Youl
Yang
,
Lue
Wu
, Chengying Bao
, John Bowers
, Amnon
Yariv
, Kerry
Vahala*
*Corresponding author. Email:
vahala@caltech.edu
Published
21 February
2019 on
Science
First Release
DOI:
10.1126/science.
aaw2317
This PDF file includes:
Materials and Methods
Figs. S1 to S5
2
Materials and Methods
Acquisition of correlation signal
The correlation method can be understood as a calculation of the frequency difference
Δ
2
Δ
1
=
Δ
by formation of
1
2
followed by fast Fourier transform (FFT). The
FFT spectrum of
1
2
gives the spectral line at
Δ
(Fig. 1D). To determ
ine
requires
Δ
=
2
1
which is measured by heterodyne of the solitons to produce electrical
signal
3
.
Figure 1
E is a narrow frequency span of the FFT of
3
and shows how the optical
frequency Vernier is mapped into a stable radio
-
frequency g
rid with line spacing
Δ
. The
order corresponding to the FFT of the
1
2
signal (Fig. 1D spectrum) is also indicated.
These steps are performed automatically to provide a real time measurement of
relative
to
0
.
Electrical divider discussion
Th
e bandwidth limit of the oscilloscope used in this experiment is 2.5 GHz and in order to
measure frequencies
Δ
f
n1
,
2
up to 11 GHz, microwave frequency dividers were used that
function between 0.5 GHz to 10 GHz and provide an 8X division ratio. The use of
these
dividers created 3 GHz frequency unresolvable bands within one FSR of the optical combs,
which caused the discontinuities in the lower panel in Fig. 2A. Meanwhile, the repetition
rate difference corresponding to the divided signals will also decreas
e proportionally by a
factor of 8, which in turn reduces the maximum resolvable chirping rate to 125 THz/s. The
dividers can be omitted by using a higher
-
bandwidth oscilloscope, which eliminates the
above unresolvable bands and allows chirp
-
rate measuremen
ts approaching the theoretical
limit.
Sample preparation and soliton generation
The silica microresonators are fabricated on a 4
-
inch silicon wafer with a 8
-
μm
-
thick
thermally
-
grown silica layer. The detailed fabrication process can be found elsewhere
(
1
8
).
The intrinsic quality factor of the resonators used in this work ranges between 200 to 300
million. Light is coupled to the resonator via a tapered fiber; however, it is also possible to
use silica resonators having an integrated silicon nitride waveg
uide
(
2
8
)
.
The detailed experimental setup for soliton generation is illustrated in Fig.
S1
. A
continuous
-
wave fiber laser is amplified by an erbium
-
doped fiber amplifier (EDFA), and
split by a 50/50 directional coupler for clockwise (cw) and counter
-
clock
wise (ccw) soliton
generation. The fiber laser has a free
-
running linewidth less than 2 kHz over 100 ms. Two
acousto
-
optic modulators (AOMs) are used to independently control the pump frequency
and power in both directions. The pump power in each directio
n is around 200 mW and is
attenuated after the resonator by a fiber Bragg grating (FBG). The filtered transmitted
power for the cw direction is split by a 90/10 directional coupler and the 10 percent output
port is used in a servo control loop to stabilize
the solitons. By scanning the laser from the
blue side to the red side of the resonance solitons form simultaneously in both directions
with characteristic ``step
-
like'' features in the transmitted power scan
(
12
,
13
,
1
9
)
. A fast
power modulation is first a
pplied to extend the existence range of the solitons. Then the
long
-
term
stability of the solitons is maintained by introducing a feedback loop control
described in detail els
e
where
(
1
9
,
20
)
. This method maintains a constant soliton power on
one of the soli
t
on streams through servo control of the pumping frequency. It has the
desirable effect of forcing the pump laser frequency to track the microresonator pumping
3
mode as its frequency slowly drifts. As a result it was not necessary to provide any
temperature
stabilization to the resonator in the measurement.
Characterization of soliton phase locking
The underlying mechanism leading to phase locking of counter
-
propagating (CP) solitons
has been described in detail elsewhere
(
12
)
. Once the solitons are phase
-
locked, their
relative frequency becomes very stable and their baseband
interferogram
features sharp
spectral lines (linewidths well below 1 Hz) in the frequency domain (Fig. S2A and S2B).
To ensure that the solitons are locke
d during the measurement, the spectrogram of the CP
soliton beatnotes is monitored as shown in Fig. 1
E
in the main text. In the locked case,
sharp, horizontal spectral lines persist over the measurement time (Fig. S2C).
The
acquisition time window
sho
uld be chosen to be integer multiples of
1
/
Δ
, where
Δ
is the repetition rate difference, so that the frequency of the beatnotes can be accurately
resolved
.
If this is not the case, artifacts will appear in the spectrogram due to misalignment
of the
frequency grids (Fig. S2D).
Signal processing: general processing algorithm
Through heterodyne of the test laser with the nearest comb teeth, the phase
ψ
of the test
laser is related to the electrical signals
1
,
2
by
1
,
2
cos
(
ψ
2
π
ν
1
,
2
)
,
(
1
)
where
ν
1
,
2
represent the frequencies of nearest comb teeth and have order
n
. We also have
ν
2
ν
1
=
n
Δ
as a result of the CP soliton locking. A Hilbert transform is used to
extract the time
-
dependent phase
ψ
2
π
ν
1
,
2
t
from
1
,
2
which thereby gives t
he
heterodyne frequencies via
Δ
1
,
2
=
ψ
̇
/
2
π
ν
1
,
2
(
2
)
Each data point of
Δ
1
,
2
is obtained by linear fitting of the phase over a specified time
interval that sets the frequency resolution. Similarly, the heterodyne frequency between the
reference laser and the soliton comb can be retrieved to determine the frequency
0
(see
discuss
ion in main text).
The Fourier transform of the product
1
2
is given by
1
2
̃
(
)
(
ψ
2
π
ν
1
)
+
(
ψ
2
π
ν
1
)
2
0
(
ψ
2
π
ν
2
)
+
(
ψ
2
π
ν
2
)
2
2
π
푖푓푡
d
(
|
|
Δ
)
,
(
S
3)
where sum frequency terms in the integrand are assumed to be filtered out and are therefore
discarded. To accurately extract the above spectral signal the acquisition time window
should be an integer multiple
of
1
/
Δ
, which is also related to the pump frequency offset
Δν
by
=
N
/
Δ
where
is the pump order and
is an integer. Moreover, the
number of sampled points, which equals the product of oscilloscope sampling rate
samp
and
, should also be an integer (i.e.,
samp
/
Δ
is an integer). In this work,
samp
is
usually set to 2.5 or 5 GHz/s and it i
s found that simple adjustment of
Δν
is sufficient to
satisfy this condition.
As a
result,
it is not necessary to synchronize the oscilloscope to
external sources. It is noted that this method is simpler than the asynchronous detection
used in previous wor
k
(
9
).
On account of the limited bandwidth of the oscilloscope used in work, it was necessary
to apply electrical frequency division to the detected signals for
processing
by the
4
oscilloscope. When frequency dividers are used (division ratio
=
8
), the div
ided
electrical signals (indicated by superscript
d
) yield
1
,
2
d
cos
(
(
ψ
2
π
ν
1
,
2
)
/
)
.
(
4
)
As a result, the divided frequencies also satisfy
Δ
1
,
2
d
=
Δ
1
,
2
/
and the correlation
between the divided signals scales proportionally by
Δ
1
d
Δ
2
d
=
Δ
/
.
(
5
)
Therefore,
the required resolution bandwidth to resolve the ambiguity
from the measured
correlation is
Δ
/
which increases the minimal acquisi
tion time to
d
=
.
Signal processing: absorption spectroscopy
To perform the absorption spectroscopy the laser transmission through the H
12
C
14
N gas
cell is recorded while the laser is continuously scanning. A portion of the laser signal is
also measured in the MSS to determine its frequency during the scan. A pseudo
-
Voigt
lineshape (linear combination of Gaussian and Lorentz profile) is fitted t
o the spectrum and
the central frequency is then extracted.
Signal processing: mode
-
locked laser measurement
The algorithm used here to extract a large number of frequencies simultaneously using the
MSS is different from the previous single
-
frequency meas
urements. Rather than
multiplying the signals
1
and
2
followed by Fast Fourier Transform (FFT) in order to
determine the microcomb order, we directly FFT the signals
1
and
2
followed by filtering
and then frequency correlation. This avoids the gene
ration of ambiguities. To explain the
approach, first consider an implementation similar to that reported in the main text. There,
a fiber mode locked laser (FMLL) comb with free
-
spectral
-
range (FSR) of about 250 MHz
was optically filtered to create a narr
ower frequency range of FMLL laser lines extending
over only a few microcomb lines. The signals
1
and
2
upon FFT therefore produce a
large set of frequencies representing the individual beats of each FMLL laser line (index
) with microcomb modes (ind
ex
). Fig. S3A gives a narrow frequency span of a typical
FFT generated this way for both the
1
and
2
signals. A zoom
-
in of one pair of
1
and
2
signals is provided in Figure S3B and a remarkably precise frequency separation between
the beats (in
view of the spectral breadth of each beat) can be determined by correlating
the upper (blue) and lower (red) spectrum (see Fig. S3C). This precision results from the
underlying high relative frequency stability of the cw and ccw microcomb frequencies. As
d
escribed in the main text this frequency separation is a multiple of
Δ
and plot of the
correlation versus the frequency separation (in units of
Δ
) is provided in Fig. S3C where
the peak of the correlation gives the index
=
63
for this pair of beat
frequencies.
Proceeding this way for each pair of peaks in Fig. S3A allows determination of
from
which the frequency of the corresponding FMLL line can be determined. It is interesting
to note that in Fig. 3A, there are two sets of peaks that give
=6
3, 64 and 65. These
correspond to FMLL lines that are higher and lower in frequency relative to the microcomb
modes with indices
=63, 64 and 65. The relative alignment of the blue and red peaks
which switches sign for these sets of beat frequencies allow
s determination of which FMLL
line is lower and higher in frequency relative to the microcomb lines.
5
To provide more rigor to this explanation, the electrical signals consist of multiple
beat components given by,
1
,
2
=
1
,
2
,
1
,
2
cos
(
ψ
2
π
ν
μ
(
)
1
,
2
)
,
(
6
)
where
ψ
and
ν
μ
(
)
1
,
2
represent the phase of the
-
th FMLL mode and the frequencies
of the microcomb order nearest to this FMLL mode, respectively, where
μ
(
)
denotes the
comb order nearest the
-
th FMLL mode. As described in the main text the frequencies
ν
μ
(
)
1
,
2
are related to the repetition rate difference by
ν
μ
(
)
2
ν
μ
(
)
1
=
μ
(
)
Δ
. The
FFT of
1
,
2
is denoted by
1
,
2
̃
and the correlation given in Fig. S3C (and used to determine
the comb order
of each spectral component) is given by,
1
̃
(
)
2
̃
(
+
Δ
)
d
Δ
1
+
κ
/
2
Δ
1
κ
/
2
d
1
(
)
2
π
푖푓푡
d
2
(
)
2
π
(
+
Δ
)
d
=
1
(
)
2
(
)
2
π
푖푛
Δ
d
(
ψ
2
π
ν
μ
1
)
+
(
ψ
2
π
ν
μ
1
)
2
(
ψ
2
π
ν
μ
2
)
+
(
ψ
2
π
ν
μ
2
)
2
2
π
푖푛
Δ
d
δ
(
μ
(
)
)
(
S
7)
where
Δ
1
denotes the peak frequency of the beatnote,
κ
is a predetermined range of
integration to cover
the linewidth of the beatnote (here
κ
=
2
MHz), and where sum
frequency terms in the integrand have been discarded.
Therefore,
for each spectral
component
, its associated microcomb order number
μ
(
)
can be determined by varying
in the above correlation
until it reaches maximum (see Fig. S3C). The
value with the
maximum correlation will be assigned to the peak as the tooth number
μ
(
)
and then the
absolute frequency can be recovered.
The limit of this process to accommodate more FMLL frequencies is muc
h higher
than that given by the filter bandwidth studied in this work. It is instead set by the spectral
density of FMLL
-
microcomb beat lines that can be reasonably resolved within the
microcomb FSR spectral span.
Additional measurements
To validate the reproducibility of results, we conducted multiple measurements with static,
dynamic and mode
-
locked lasers beyond those results presented in the main text. For
example, shown in Fig. S4A is the histogram of residual deviations between an EC
DL laser
frequency measured by the MSS and a wavemeter. This specific data set is compiled from
Fig. 1G of the main text. The
standard deviation
is calculated to be 0.0
4
pm.
To further benchmark the performance of the MSS, we also evaluated the Allan
devi
ation for MSS measurement of a static laser. The extracted Allan deviation is plotted
in Fig. S4B and decreases to
6
×
10
10
(120 kHz in absolute frequency) at 4
μ
s. It is noted
that this Allan deviation includes noise contributions from the reference laser,
the ECDL
as well as the soliton microcomb repetition rate.
Also, additional measurements of fast chirped lasers using the MSS are plotted in Fig.
S5. The laser is an ECDL tuned at its maximum speed, similar to the case presented in Fig.
6
2A of the main tex
t. Both positive and negative chirping rates at different wavelengths are
resolved.
7
Fig. S1
:
Detailed experimental setup for soliton generation.
AWG: arbitrary waveform
generator; EDFA: erbium
-
doped fiber amplifier; AOM: acousto
-
optic modulator; PC:
pol
arization controller; CIRC: circulator; FBG: fiber Bragg grating; PD: photodetector.
8
Fig. S2
:
Interferograms of cw and ccw solitons.
(
A
) A typical inte
r
ferogram in frequency
domain. (
B
) A zoom
-
in of line 48 centered at 1.000000
M
Hz (shaded region in
panel A).
(
C
) Spectrogram of Fig. 1
E
in the main text showing more lines. (
D
) Same as in (
C
) but
the frequency spacing has been deliberately chosen so that artifacts appear in the scan.
9
Fig. S
3
:
Multi
-
frequency measurements.
(
A
)
A
section of
1
,
2
̃
.
Pa
irs
of beatnotes coming
from the same laser are highlighted and the derived
value is marked next to each pair of
beatnotes. (
B
) Zoom
-
in on the highlighted region near 858 MHz in (A). Two beatnotes are
separated by 1.0272 MHz. (
C
) Cross
-
correla
tion of
1
̃
and
2
̃
is calculated for each
and
the maximum can be found at
=
63
.
10
Fig. S
4
:
Statistics and Allan
d
eviation.
(
A
)
Histogram of the residual deviations shown
in Fig. 1G of the main
text. The
standard deviation
is 0.0
4
pm. (
B
) Allan devia
tion of
measured frequency of a static laser as a function of averaging time. The relative frequency
stability reaches
6
×
1
0
10
at 4
μ
s integration time.