Vernier spectrometer using counter-propagating soliton microcombs
Qi-Fan Yang
1
,
∗
, Boqiang Shen
1
,
∗
, Heming Wang
1
,
∗
, Minh Tran
2
, Zhewei Zhang
1
, Ki Youl
Yang
1
, Lue Wu
1
, Chengying Bao
1
, John Bowers
2
, Amnon Yariv
1
and Kerry Vahala
1
,
†
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
2
University of California, Santa Barbara, Department of Electrical
and Computer Engineering, Santa Barbara, CA 93106, USA.
∗
These authors contributed equally to this work.
†
Corresponding author: vahala@caltech.edu
(Dated: December 3, 2018)
Acquisition of laser frequency with high resolu-
tion under continuous and abrupt tuning condi-
tions is important for sensing, spectroscopy and
communications. Here, a single microresonator
provides rapid and broad-band measurement of
frequencies across the optical C-band with a rel-
ative frequency precision comparable to conven-
tional dual frequency comb systems. Dual-locked
counter-propagating solitons having slightly dif-
ferent repetition rates are used to implement a
Vernier spectrometer. Laser tuning rates as high
as 10 THz/s, broadly step-tuned lasers, multi-
line laser spectra and also molecular absorption
lines are characterized using the device. Besides
providing a considerable technical simplification
through the dual-locked solitons and enhanced
capability for measurement of arbitrarily tuned
sources, this work reveals possibilities for chip-
scale spectrometers that greatly exceed the per-
formance of table-top grating and interferometer-
based devices.
Frequency-agile lasers are ubiquitous in sensing, spec-
troscopy and optical communications
1–3
and measure-
ment of their optical frequency for tuning and control
is traditionally performed by grating and interferometer-
based spectrometers, but more recently these measure-
ments can make use of optical frequency combs
4–6
. Fre-
quency combs provide a remarkably stable measurement
grid against which optical signal frequencies can be de-
termined subject to the ambiguity introduced by their
equally spaced comb lines. The ambiguity can be re-
solved for continuously frequency swept signals by count-
ing comb teeth
7
relative to a known comb tooth; and
this method has enabled measurement of remarkably
high chirp rates
8
. However, signal sources can operate
with abrupt frequency jumps so as to quickly access a
new spectral region or for switching purposes, and this
requires a different approach. In this case, a second
frequency comb with a different comb line spacing can
provide a Vernier scale
9
for comparison with the first
comb to resolve the ambiguity under quite general tun-
ing conditions
9–11
. This Vernier concept is also used in
dual comb spectroscopy
12,13
, but in measuring active sig-
nals the method can be significantly enhanced to quickly
identify signal frequencies through a signal correlation
technique
9
. The power of the Vernier-based method re-
lies upon mapping of optical comb frequencies into a
radio-frequency grid of frequencies, the precision of which
is set by the relative line-by-line frequency stability of the
two frequency combs. This stability can be guaranteed by
self-referencing each comb using a common high-stability
radio-frequency source or through optical locking of each
comb to reference lasers whose relative stability is en-
sured by mutual locking to a common optical cavity.
Here, a broad-band, high-resolution Vernier soliton
microcomb spectrometer is demonstrated using a sin-
gle miniature comb device that generates two mutually-
phase-locked combs. The principle of operation relies
upon an optical phase locking effect observed in the
generation of counter-propagating solitons within high-
Q whispering gallery resonators
14
. Soliton generation in
microcavities is being studied for miniaturization to the
chip-scale of complete comb systems
15
and these so-called
soliton microcombs have now been demonstrated in a
wide range of microcavity systems
16–21
. In the counter-
propagating soliton system, it is found that the clock-
wise (cw) and counter-clockwise (ccw) comb frequencies
can be readily phase locked with distinct repetition rates
that are also locked. This mutual double-locking creates
line-by-line relative frequency stability for the underly-
ing microcomb spectra that is more characteristic of fully
self-referenced dual comb systems. The resulting Vernier
of comb frequencies in the optical domain maps to an
exceptionally stable radio frequency grid. Application of
the signal correlation method
9
to this system, then en-
ables a microresonator soliton spectrometer (MSS) for
rapid and high accuracy measurement of frequency.
To establish its performance and for comparison with
dual fiber-mode-locked-laser spectrometers
9
the MSS is
applied to measure a 10 THz/s laser frequency chirp-
ing rate, step tuning of a laser, as well as acquisition of
high-resolution molecular vibronic spectra over the op-
tical C-band. Moreover, a method for signal frequency
extraction is developed that uses the high relative sta-
bility of the cw and ccw combs to unambiguously deter-
mine frequencies in complex spectra containing 100s of
frequencies.
The measurement concept in the frequency domain is
depicted in Fig. 1A where comb spectra from doubled-
locked cw and ccw solitons are shown. The solitons are
arXiv:1811.12925v1 [physics.optics] 30 Nov 2018
2
CCW soliton
CW soliton
Power (20 dB/div)
Wavelength (nm)
1500
1550
1600
CW soliton
CCW soliton
Pump
Pump
E
C
1 mm
B
c.w. pump
50/50
resonator
CIRC
CIRC
Signal laser
Ref laser
50/50
50/50
50/50
PD
50/50
50/50
Gas cell
PD
PD
PD
PD
Optical frequency
∆ν=
N
∆
f
r
c.w. laser
Pump
∆
f
r
μ
=0
μ
=
N
Chemical
absorption
A
Signal processing
Wavelength (pm
+1553.93305 nm)
Phase locking
Residuals (pm)
Frequency (MHz)
Power (arb. unit)
D
n
=54
F
G
Time (1 ms/div)
AOM
AOM
Time (1 ms/div)
1 mm
2.5
3.0
3.5
Wavelength (nm)
1545
1550
1555
1560
-0.1
0.1
0
-0.02
0
0.02
0
1
Frequency (MHz)
0
4
Correlation (arb.unit)
RBW
200 Hz
0
1
Oscilloscope
V
2
V
1
V
3
μ
=
n
f
L
∆
f
n1
∆
f
n2
∆
f
n1
-
∆
f
n2
=
n
∆
f
r
FIG. 1:
Spectrometer concept, experimental setup and static measurement.
(
A
) Counter propagating
soliton frequency combs (red and blue) feature repetition rates that differ by ∆
f
r
. Their propagation in the res-
onator causes phase-locking at the comb line with index
μ
= 0. Also, the comb teeth separated by ∆
ν
=
N
∆
f
r
at
μ
=
N
are derived from a single pump laser and therefore also are effectively locked. This dual locking of the
vernier-like comb frequencies enables precise measurement of a laser (green) at frequency
f
L
when combined with
electrical correlation of the comb signals to determine
μ
=
n
. Once calibrated, the tunable laser can resolve chem-
ical absorption lines (grey) with high precision. (
B
) Experimental setup. AOM: acousto-optic modulator; CIRC:
circulator; PD: photodetector. Supplement includes more detail. Inset: scanning electron microscope image of a sil-
ica resonator. (
C
) Typical measured spectrum of
V
1
V
2
used to determine order
n
. For this spectrum: ∆
f
n
1
−
∆
f
n
2
= 2.8052 MHz and ∆
f
r
= 52 kHz giving
n
= 54. (
D
) The spectrograph of the dual soliton interferogram (pseudo
color). Line spacing gives ∆
f
r
= 52 kHz. White squares correspond to the index
n
= 54 in panel C. (
E
) Optical
spectra of counter-propagating solitons. Pumps are filtered and denoted by dashed lines.(
F
) Measured wavelength
of an external cavity diode laser operated in steady state. (
G
) Residual deviations between ECDL laser frequency
measurement as given by the MSS and a wavemeter. Error bars give the systematic uncertainty as limited by the
reference laser in panel B.
pumped from a single laser source that is modulated as
shown in figure 1B to produce the two mutually-coherent
pump lines at order
μ
=
N
with frequency separation
∆
ν
. The difference in pumping frequencies (MHz range)
causes the soliton repetition rates to differ by ∆
f
r
which
sets up a vernier effect in the respective soliton comb fre-
quencies. As detailed elsewhere, the cw and ccw combs
will experience frequency locking at order
μ
= 0 for cer-
tain pumping frequencies
14
. This locking requires that
∆
ν
=
N
∆
f
r
. Also, because the two pump frequen-
cies are derived from a single laser source and have a
high relative frequency stability (∆
ν
is very stable), the
two combs are also effectively locked at order
μ
=
N
.
The order
N
can readily increased or decreased by ad-
3
Step-tuned laser
Scanning laser
Wavelength (nm
+1553 nm)
A
B
C
Index
n
60
65
0.5
0
-0.5
Time (2 ms/div)
D
Absorbance (dB)
-12.4 THz/s
Index
n
30
60
Wavelength (nm
+1555 nm)
5
0
-5
High-resolution Spectroscopy
Wavenumber (cm
-1
+6433.2090 cm
-1
)
-0.1
0
0.1
Measurement
Fitting
H
12
CN
2v
1
J=25
J=26
J=27
Ground
J=24
J=25
J=26
M: 6433.2090 cm
-1
R: 6433.2095 cm
-1
0
0.02
0.10
0.12
M: 6429.2194 cm
-1
R: 6429.2214 cm
-1
M: 6425.1897 cm
-1
R: 6425.1931 cm
-1
2.6 GHz
0.04
0.06
0.08
Time (0.2 ms/div)
6.186 nm
Time (2 ms/div)
1557.613 nm
Wavelength
(5 pm/div)
Residuals
(pm)
5
0
-5
Time (2 ms/div)
-12.39 THz/s
FIG. 2:
Laser tuning and spectroscopy measurements.
(
A
) Measurement of a rapidly tuning laser showing
index
n
(upper), instantaneous frequency (middle), and higher resolution plot of wavelength relative to average lin-
ear rate (lower), all plotted versus time. (
B
) Measurement of a broadband step-tuned laser as for laser in panel
A
.
Lower panel is a zoom-in to illustrate resolution of the measurement. (
C
) Spectroscopy of H
12
C
14
N gas. A vibronic
level of H
12
C
14
N gas at 5 Torr is resolved using the laser in panel
A
. (
D
) Energy level diagram showing transitions
between ground state and 2
ν
1
levels. The measured (reference) transition wavenumbers are noted in red (blue).
justing ∆
ν
. The line-by-line relative frequency stability
caused by this double locking is comparable to an ex-
cellent radio-frequency source. Moreover, the frequency
spacings between comb tooth pairs occur at precise in-
teger multiples of ∆
f
r
(the stability of which is ensured
through the relation ∆
ν
=
N
∆
f
r
), and thereby creates
an extremely stable optical frequency vernier for mapping
of the comb spectra into a radio frequency grid spectrum.
The spectrometer operates as follows. A test laser fre-
quency
f
L
is measured using either of the following ex-
pressions:
f
L
=
nf
r
1
,
2
+ ∆
f
n
1
,
2
+
f
0
where
n
is the comb
order nearest to the laser frequency,
f
r
1
,
2
are the comb
repetition rates, ∆
f
n
1
,
2
are the heterodyne beat frequen-
cies of the test laser with the two frequency comb teeth at
order
μ
=
n
, and
f
0
is the frequency at
μ
= 0. Comb rep-
etition rates
f
r
1
,
2
and the beats ∆
f
n
1
,
2
are measured by
co-detection of the combs and the test laser to produce
the electrical signals
V
1
,
2
in Fig. 1B. The correlation
method
9
is used to determine
n
. This method can be
understood as a calculation of the frequency difference
∆
f
n
2
−
∆
f
n
1
=
n
∆
f
r
by formation of
V
1
V
2
followed by
fast Fourier transform (FFT). A typical FFT spectrum
of
V
1
V
2
is shown in Fig. 1C and gives a spectral line at
n
∆
f
r
. To determine
n
requires ∆
f
r
=
f
r
2
−
f
r
1
which is
measured by heterodyne of the solitons to produce elec-
trical signal
V
3
. Figure 1D is a narrow frequency span
of the FFT of
V
3
and shows how the optical frequency
vernier is mapped into a stable radio-frequency grid with
line spacing ∆
f
r
. The order corresponding to the FFT
of the
V
1
V
2
signal (Fig. 1C spectrum) is also indicated.
These steps are performed automatically to provide a real
time measurement of
f
L
relative to
f
0
. To determine
f
0
the order of a comb tooth nearest a reference laser (with
known and stable frequency) is determined. This can be
done, for example, by application of the correlation pro-
cedure to the reference laser. Then, as illustrated in Fig.
1B, the beat of the reference laser with this comb order
is monitored for real time measurement of
f
0
during op-
eration of the MSS. In the current system the reference
laser is stabilized using an internal molecular reference.
The counter-propagating solitons are generated in a
high-
Q
silica microresonator with 3 mm diameter and
corresponding 22 GHz soliton repetition rate
22
.
De-
tails of the soliton generation process can be found
elsewhere
14,17,23
. Typical optical spectra of cw and ccw
solitons are plotted in Fig. 1E and span the telecommuni-
cation C-band. The distinct pumping frequencies enable
repetition rate tuning to control ∆
f
r
through the Raman
4
1500
1600
Power (20 dB/div)
A
B
C
Power (arb. unit)
1
48
49
Frequency (GHz+192.79 THz)
2
3
26
27
D
Residuals (MHz)
OSA
MSS
4 ns
OSA
MSS
Mode-locked
laser
Wavelength (nm)
92 X
249.7 MHz
82 X
249.7 MHz
Relative index m
100
0
100
Frequency (GHz
+192.79 THz)
0
650
550
200
2300
2400
2500
E
0
-0.5
0.5
Relative index m
0
100
200
2300
2400
2500
Measurement
Fitting
FIG. 3:
Measurement of a fiber mode-locked laser
(
A
) Pulse trains generated from a fiber mode-locked laser
(FMLL) are sent into an optical spectral analyzer (OSA) and the MSS. (
B
) Optical spectrum of the FMLL mea-
sured by the OSA. (
C
) Optical spectrum of the FMLL measured using the MSS. Only a 60-GHz wavelength range
is selected. (
D
) Measured (blue) and fitted (red) FMLL mode frequencies versus index. The slope of the fitted line
is set to 249.7 MHz, the measured FMLL repetition rate. (
E
) Residual MSS deviation between measurement and
fitted value.
self-frequency shift
14,24–27
. For example, a repetition rate
difference of ∆
f
r
= 52 kHz as seen in Fig. 1D results
from a pumping frequency difference of ∆
ν
= 4
.
000 MHz
(
N
= 77).
As a preliminary test, the frequency of an external-
cavity-diode-laser is measured and compared against a
wavemeter. Fig. 1C and 1D (
n
= 54) are from this
measurement. The real-time measured wavelength of the
laser is presented in Fig. 1F and fluctuates within
±
0
.
02
pm over a 5 ms time interval. The measurement is re-
peated from 1545 to 1560 nm and the acquired wave-
lengths are plotted in Fig. 1G. The data show residual
deviations less than 0.1 pm versus a wavemeter measure-
ment, which is believed to be limited primarily by the
wavemeter resolution (
±
0
.
1 pm). The systematic un-
certainty of the absolute wavelength measurement in the
current setup is around
±
4 MHz (
±
0
.
03 pm) and is dom-
inated by stability of the reference laser.
The large, microwave-rate, free-spectral range of the
MSS enables tracking of fast-chirping lasers in real time
and discontinuous broadband tuning. Although correla-
tion is performed with a time interval
T
W
= 1
/
∆
f
r
, the
instantaneous frequency of the laser relative to the combs
can be acquired at a much faster rate set by the desired
time-bandwidth-limited resolution. To avoid aliasing of
correlation measurement (i.e., to determine
n
uniquely),
the amount of frequency-chirping should not exceed the
repetition rate
f
r
within the measurement window
T
W
,
which imposes a maximum resolvable chirping-rate of
f
r
×
∆
f
r
. This theoretical limit is 1 PHz/s for the MSS
and represents a boost of 100
×
compared with previous
Vernier spectrometers
9
.
To test the MSS dynamically, it is first used to measure
rapid continuous-tuning of an external cavity diode laser.
As shown in the upper panel of Fig. 2A, the correlation
measurement evolves as the laser is tuned over multiple
FSRs of the comb and thereby determines the index
n
as a function of time. The frequency of the scanning
laser is displayed at low resolution in the middle panel of
Fig. 2A and shows a linear chirping-rate of
−
12
.
4 THz/s.
Finally, the lower panel in Fig. 2A shows the measured
frequency versus time at higher resolution by removing
the average linear frequency ramp. As discussed in the
Methods Section, the discontinuities in the measurement
are caused by electrical frequency dividers used to reduce
the detected signal frequency for processing by a low-
bandwidth oscilloscope. These dividers can be eliminated
by using a faster oscilloscope. In Fig. 2B the MSS is
used to resolve broadband step tuning (mode hopping) of
an integrated ring resonator based tunable III-V/Silicon
laser diode
28
. Fast step tuning between 1551.427 nm and
1557.613 nm every 1 ms with the corresponding index
n
stepping between
n
= 64 and
n
= 29 is observed.
The lower panel in Fig. 2B gives a higher resolution
zoom-in of one of the step regions. The data points in
these measurements are each acquired over 1
μ
s so the
resolution is approximately 1 MHz.
This combination of speed and precision is also use-
ful for spectroscopic measurements of gas-phase chemi-
cals using tunable, single-frequency lasers. Figure 2C is
5
an absorption line of H
12
C
14
N at 5 Torr obtained by a
scanning laser calibrated by the MSS. The linewidth is
around 2.6 GHz and the absorbance is as weak as 0.12 dB.
Separate measurements on vibronic transitions between
the ground state and 2
ν
1
states were performed. Fig.
2D summarizes the corresponding pseudo-Voigt fitting
for the transition wavenumbers, which are in excellent
agreement with the HITRAN database
29
.
To illustrate a measurement of more complex multi-
line spectra, a fiber mode-locked laser (FMLL) is char-
acterized as shown in Fig. 3A. For this measurement,
the FMLL was first sent through a bandpass filter to
prevent detector saturation. Also, the frequency extrac-
tion procedure differs and is modified to enable unique
identification of many frequencies (see Supplement). The
FMLL line spacing of 249.7 MHz (measured by photode-
tection) is not resolved in the Fig. 3B spectrum mea-
sured using a grating spectrometer. On the other hand,
the reconstructed FMLL spectrum measured using the
MSS is plotted in Fig. 3C; here, the comb lines are re-
solved and their frequency separations closely match the
value measured by photo detection. Further details on
this measurement are provided in the Supplemental sec-
tion. In a second study of the FMLL, the MSS is used
to measure 6 closely-spaced-in-frequency groups of lines
located at various spectral locations spanning 2500 free-
spectral-range’s of the mode locked laser. The measured
frequencies are plotted in Fig. 3D. A linear fitting defined
as
f
m
=
f
o
+
mf
rep
is plotted for comparison by using
the measured FMLL repetition rate
f
rep
= 249
.
7 MHz
where
m
and
f
o
represents the relative comb index and
fitted offset frequency at
m
= 0, respectively. The resid-
ual deviation between the measurement and linear fitting
is shown in Fig. 3E and gives excellent agreement. The
slight tilt observed in Fig. 3E is believed to be related to
drifting of soliton repetition rates which were not mon-
itored real-time. Also, variance of residuals within each
group comes from the 300 kHz linewidth of each FMLL
line. Drifting of the reference laser and FMLL carrier-
envelope offset also contributes to the observed residuals
across different measurements.
In conclusion, a soliton spectrometer has been demon-
strated using dual-locked counter-propagating soliton mi-
crocombs. The device provides high resolution measure-
ment of rapid continuously and step tuned lasers as well
as complex multi-line spectra. In combination with a
tunable laser, precise measurement of absorption spec-
tra including random spectral access (as opposed to only
continuous spectral scanning) can be performed. Fur-
ther optimization of this system could include generation
of solitons from distinct mode families thereby allowing
tens-of-MegaHertz repetition rate offset to be possible
30
.
If such solitons can be dual-locked, the increased acquisi-
tion speed would enable measurement of chirping-rates
close to 1 EHz/s. Operation beyond the telecommu-
nications band would also clearly be useful and could
employ soliton broadening either internally
18
or using
on-chip broadeners
31
. Besides the performance enhance-
ment realized with the soliton microcombs, the use of
dual-locked counter-propagating solitons provides a con-
siderable technical simplification by eliminating the need
for a second mutually phase locked comb. Also, it is
interesting to note that the counter-propagating dual-
locked solitons are potentially useful in a different ap-
plication wherein dual-comb down conversion is used
to perform TeraHertz spectroscopy
32
. Finally, chip in-
tegrable versions of the current device employing sili-
con nitride waveguides are possible
33
. These and other
recently demonstrated compact and low-power soliton
systems
34,35
point towards the possibility of compact mi-
croresonator soliton spectrometers.
Methods
Experimental details.
The bandwidth limit of the oscilloscope
used in this experiment is 2.5 GHz and in order to measure fre-
quencies ∆
f
n
1
,
2
up to 11 GHz, microwave frequency dividers were
used that function between 0.5 GHz to 10 GHz and provide an 8
×
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. Mean-
while, the repetition rate difference corresponding to the divided
signals will also decrease 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 oscillo-
scope, which eliminates the above unresolvable bands and allows
chirp-rate measurements approaching the theoretical limit.
The pump is a fiber laser with free-running linewidth less than
2 kHz over 100 ms
36
. The long term stability of the soliton is
maintained by introducing a feed back loop control
17,23
.
Acknowledgment
The authors gratefully acknowledge the Defense Ad-
vanced Research Projects Agency (DARPA) under the
SCOUT (W911NF-16-1-0548) and DODOS (HR0011-15-
C-055) programs; the Air Force Office of Scientific Re-
search (FA9550-18-1-0353) and the Kavli Nanoscience In-
stitute.
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Supplementary Information to “Vernier spectrometer using counter-propagating
soliton microcombs”
Qi-Fan Yang
1
,
∗
, Boqiang Shen
1
,
∗
, Heming Wang
1
,
∗
, Minh Tran
2
, Zhewei Zhang
1
, Ki Youl
Yang
1
, Lue Wu
1
, Chengying Bao
1
, John Bowers
2
, Amnon Yariv
1
and Kerry Vahala
1
,
†
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
2
University of California, Santa Barbara, Department of Electrical
and Computer Engineering, Santa Barbara, CA 93106, USA.
∗
These authors contributed equally to this work.
†
Corresponding author: vahala@caltech.edu
(Dated: December 3, 2018)
I. SAMPLE PREPARATION AND SOLITON GENERATION
The silica microresonators are fabricated on a 4-inch silicon wafer with a 8-
μ
m-thick thermally-grown silica layer.
The detailed fabrication process can be found elsewhere
1
. 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 waveguide
2
.
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-clockwise (ccw) soliton generation. Two acousto-optic modulators (AOMs) are used to independently
control the pump frequency and power in both directions. The pump power in each direction 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
3–5
. A fast power modulation is first
applied to extend the existence range of the solitons. This is followed by activation of a servo control loop to stabilize
the solitons at a selected power by feedback to the pump laser frequency
4,6
. Using this approach, the solitons in both
directions can exist indefinitely.
II. CHARACTERIZATION OF SOLITON PHASE LOCKING
The underlying mechanism leading to phase locking of counter-propagating (CP) solitons has been described in
detail elsewhere
5
. Once the solitons are phase-locked, their relative frequency becomes very stable and their baseband
inteferogram features sharp spectral lines (linewidths well below 1 Hz) in the frequency domain (Fig. S2A and S2B).
To ensure that the solitons are locked during the measurement, the spectrogram of the CP soliton beatnotes is
fiber laser
50/50
resonator
CIRC
CIRC
AOM
AOM
EDFA
FBG
FBG
CCW soliton
90/10
PD
Servo
AWG
~
~
PC
PC
feedback loop
CW soliton
FIG. S1:
Detailed experimental setup for soliton generation.
AWG: arbitrary waveform generator; EDFA:
erbium-doped fiber amplifier; AOM: acousto-optic modulator; PC: polarization controller; CIRC: circulator; FBG:
fiber Bragg grating; PD: photodetector.
arXiv:1811.12925v1 [physics.optics] 30 Nov 2018
2
Power (20 dB/div)
Power (20 dB/div)
Frequency (Hz + 1.000000MHz)
-100
0
100
Frequency (MHz)
0
0.5
1
1.5
Time (1 ms/div)
Frequency (MHz)
0
1
2
3
4
Time (1 ms/div)
Power (arb. u.)
0
1
A
C
D
Resolution
bandwidth
1 Hz
B
FIG. S2:
Interferograms of cw and ccw solitons.
(
A
) A typical inteferogram in frequency domain. (
B
) A
zoom-in of line 48 centered at 1
.
000000 MHz (shaded region in panel A). (
C
) Spectrogram of Fig. 1d in the main
text showing more lines. (
D
) Same as in (
C
) but the frequency spacing has been deliberately chosen so that arti-
facts appear in the scan (see discussion in section IIIA).
monitored as shown in Fig. 1D in the main text. In the locked case, sharp, horizontal spectral lines persist over the
measurement time (Fig. S2C). The acquisition time window
T
W
should be chosen to be integer multiples of 1
/
∆
f
r
,
where ∆
f
r
is the repetition rate difference, so that the frequency of the beatnotes can be accurately resolved (details
can be found in section IIIA). If this is not the case, artifacts will appear in the spectrogram due to misalignment of
the frequency grids (Fig. S2D).
III. SIGNAL PROCESSING
A. 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
V
1
,
2
by
V
1
,
2
∝
cos(
ψ
−
2
πν
n
1
,
2
t
)
,
(S1)
where
ν
n
1
,
2
represent the frequencies of nearest comb teeth and have order
n
. We also have
ν
n
2
−
ν
n
1
=
n
∆
f
r
as a
result of the CP soliton locking. A Hilbert transform is used to extract the time-dependent phase
ψ
−
2
πν
n
1
,
2
t
from
V
1
,
2
which thereby gives the heterodyne frequencies via
∆
f
n
1
,
2
=
̇
ψ/
2
π
−
ν
n
1
,
2
,
(S2)
Each data point of ∆
f
n
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