nature physics
https://doi.org/10.1038/s41567-024-02420-4
Artic�e
Topological temporally mode-locked laser
In the format provided by the
authors and unedited
Contents
1 Experimental Setup
2
2 Pulse Characterization
4
3 Data Analysis
5
3.1
Positioning the Experimental Traces . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
Plotting the Experimental Traces . . . . . . . . . . . . . . . . . . . . . . . . .
7
4 Master Equation Description of Temporal Mode-Locking
8
4.1
Derivation of the Tight-Binding Model . . . . . . . . . . . . . . . . . . . . . .
8
4.2
Steady-State Behavior of the Tight-Binding Model . . . . . . . . . . . . . . .
11
4.3
Master Equation Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
5 Lumped Element Simulations
14
5.1
Elements of the Lumped Element Simulations . . . . . . . . . . . . . . . . . .
14
5.1.1
Passive Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
5.1.2
Erbium-Doped Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5.1.3
Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5.1.4
Intensity Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5.1.5
Intracavity Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
5.2
Lumped Element Simulations at Lower Power . . . . . . . . . . . . . . . . . .
17
5.3
Lumped Element Simulations at Higher Power . . . . . . . . . . . . . . . . .
17
6 Robustness of the Hatano-Nelson Model
18
6.1
Additional Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
6.2
Alternative Metric of Localization and Finite Size Effects . . . . . . . . . . . .
20
7 Potential for Robust Frequency Combs
25
8 Potential for Sensing Applications
34
9 Comparison with Earlier Work
36
9.1
Comparison with Conventional Harmonically Mode-Locked Lasers . . . . . .
36
9.2
Comparison with Other Temporally Mode-Locked Lasers . . . . . . . . . . . .
37
9.3
Comparison with Continuous-Wave Topological Lasers . . . . . . . . . . . . .
38
9.4
Comparison with Two-Loop Architecture . . . . . . . . . . . . . . . . . . . .
39
1
1 Experimental Setup
We present a detailed schematic of our topological temporally mode-locked laser in Fig. 1. This
figure shows the various components involved in calibrating, operating, and stabilizing the laser.
In this section, we summarize these components and briefly mention their functions.
At the core of our topological temporally mode-locked laser is the “main cavity,” which is
the fiber laser cavity colored blue in Fig. 1. In our system, the main cavity consists of individual
patch cables connected with fiber mating sleeves. All fiber in the experiment is polarization
maintaining, and the patch cables are terminated with FC/APC connectors to reduce back re-
flections. The other optical components in the main cavity are also terminated with FC/APC
connectors.
The main cavity contains two 90:10 splitters. The first splitter acts as an input to the cavity,
and, when we calibrate the length of the main cavity, we couple in the pulses of an auxiliary
mode-locked laser with this splitter (see Methods). This splitter recirculates 90% of the light in
the cavity and couples in 10% of the light from the input port.
The second 90:10 splitter is the output port. It couples out 10% of the light from the main
cavity and recirculates the remaining 90%. The light coupled out of the cavity is sent to a 50:50
splitter, and one arm of this splitter goes to a fast 5 GHz detector. The other arm goes to a
second 50:50 splitter, where the power is divided between a slow (kHz) detector and a power
meter. The slow detector is used to stabilize the delay lines, and the power meter enables us to
monitor the power in the laser.
After the first 90:10 splitter, there is an erbium-doped fiber amplifier (EDFA), which is the
source of slow laser gain in the cavity. The EDFA is followed by a 200 GHz, Channel 34 dense
wavelength division multiplexing (DWDM) filter, which removes amplified spontaneous emis-
sion (ASE) when we operate the cavity below threshold and inject auxiliary pulses to calibrate
the system.
After the filter, the light in the main cavity is divided at a 50:50 splitter. Half of the light
continues in the main cavity (still colored blue), and the remaining light enters a common delay
line path (colored brown). Another 50:50 splitter divides the light in the common delay line path
between the
±
T
R
delay lines. Each delay line contains three elements: an intensity modulator,
which enables us to modify the pulse-to-pulse couplings produced by the delay line; a free space
delay, which enables us to coarsely tune the length of the delay line; and a fiber phase shifter,
which is used to stabilize the delay line. The delay lines recombine into a common path at a
50:50 splitter before finally recombining with the main cavity at another 50:50 splitter.
The remaining elements in the main cavity are a fiber stretcher, a fiber phase shifter, and
an intensity modulator. The fiber stretcher and the fiber phase shifter are used to stabilize the
length of the main cavity during the calibration stage, but they are unused when we operate our
topological temporally mode-locked laser above threshold. We drive the intensity modulator
with a
∼
250
MHz sinusoid from an RF function generator. We use a 10 MHz reference from
the auxiliary laser as a clock signal for the RF function generator.
The calibration portion of our system begins at the auxiliary mode-locked laser in Fig. 1.
2
To Oscilloscope
fs 1.55 μm
Laser
EDFA
FPS
Fiber
Stretcher
Tunable
Free
-
Space Delay
IM
+1
IM
-
1
FPS
FPS
-
1T
R
+1T
R
Filter
Main
Cavity
5 GHz
Detector
IM
C
IM
00
IM
01
Acronyms and Color Scheme:
IM: Intensity Modulator
PM: Phase Modulator
FPGA
: Field
-
Programmable Gate Array
EDFA: Erbium
-
Doped Fiber Amplifier
FPS: Fiber Phase Shifter
PID: Proportional
Integral Derivative Controller
FPGA
Data Acquisition
Module
Amps
Low Pass
Filter
PID +
Dither
PID +
Dither
PID +
Dither
250 MHz
Reference
From PC
From PC
From IM
Photodiodes
To IM Bias
Ports
Amp
Amp
Amp
90:10
Function
Generator
To
PID+Dither
kHz
Detector
Power
Meter
50:50
To Oscilloscope
1.6 GHz
Detector
250 MHz
Sinusoid
Attached for the calibration, but disconnected for the experiment
+T
R
Delay Line
-
T
R
Delay Line
Delay Line Common Path
Main Cavity
Signals To and From Modulators
Signal For FPGA Clocking
Signal For Stabilization Circuits
Signals To the Oscilloscope
Oscilloscope
From Detectors
RF Phase
Shifter
Figure 1:
Detailed Schematic of the Topological Temporally Mode-Locked Laser.
3
This mode-locked laser produces femtosecond pulses at a repetition rate of
∼
250
MHz. We
first send these pulses through a Channel 34 DWDM filter to stretch the pulses to
∼
5
ps. Then,
we divide the stretched pulses at a 90:10 splitter.
We send the output of 10% port directly to a 600 MHz detector. The RF output of this
detector passes through a 300 MHz low pass filter to generate a
∼
250
MHz clock signal for the
FPGA that drives our laser’s delay line intensity modulators. In this way, we lock the electronic
driving signals from the FPGA to the repetition rate of our auxiliary mode-locked laser.
We send the output of the 90% port through two consecutive intensity modulators, which
control the pulse pattern that we transmit into the main cavity. We use two intensity modulators
to achieve a greater extinction ratio [1, 2].
After these intensity modulators, there is another 50:50 splitter in Fig. 1. This splitter is not
actually present during the calibration, but it is used during our experiments. When this splitter
is present, it sends 50% of the light from the auxiliary mode-locked laser to a 1.6 GHz detector,
and we compare this signal to the output of our laser to determine the positions of the pulses in
our synthetic lattice. We discuss the details of this positioning procedure in Sec. 3.1.
During calibration, when this 50:50 splitter is absent, the pulses from the auxiliary mode-
locked laser are sent directly to the main cavity. We emphasize that this light is only injected
into the main cavity during the calibration. During our experiments, we remove this external
source.
To conclude this section, we summarize the electronics used to stabilize our delay lines,
which are stabilized using a Pound-Drever-Hall (PDH) locking scheme [3]. The output of the
slow detector in Fig. 1 is sent to several Red Pitaya STEMlabs (shown as the PID+Dither ele-
ments in Fig. 1). These devices feature built-in PDH locking capabilities, and they generate the
dither and feedback signals for both delay lines. We amplify and add these signals with com-
mercial piezo-drivers and custom PCBs, and we apply these signals to the fiber phase shifters
in the delay lines. A similar procedure is used to lock the main cavity during the calibration
procedure. However, there we do not sum the dither and feedback signals. Instead, we apply
the dither signal to a faster fiber phase shifter, and we apply the feedback signal to the fiber
stretcher, which can produce a larger phase shift.
2 Pulse Characterization
We characterize the pulses in our temporally mode-locked laser by reconstructing the pulses
with a fast photodetector and by measuring the spectrum with a spectrum analyzer. For these
measurements, we program the couplings of our temporally mode-locked laser to implement
Hermitian nearest-neighbor couplings.
To reconstruct the pulses in the time domain, we replace the 5 GHz detector in Fig. 1 with
a 30 GHz detector. This detector has a nominal impulse response of 15 ps at 1560 nm. Using
this detector, we record the output of our laser on a 20 GHz oscilloscope with a 80 GSa/s
sampling rate. We use the samples from the recorded trace to reconstruct the pulse envelope,
4
(a)
(b)
-500
0
500
Time (ps)
0
0.5
1
Normalized Intensity
1549.8
1549.9
Wavlength (nm)
0
0.5
1
Normalized PSD
Figure 2:
Reconstruction of the Laser Pulses. (a)
Temporal reconstruction of the laser pulses
in a temporally mode-locked laser with Hermitian nearest-neighbor coupling.
(b)
Power spec-
tral density corresponding to the pulses measured in (a).
and this reconstruction is shown in Fig. 2(a). As shown by the red dashed curve, our pulse
reconstruction is well fit by a Gaussian (
r
2
= 0
.
9675
), which reveals the width of the pulse
reconstruction to be
∼
100
ps.
We show the power spectral density of the pulse train produced by our temporally mode-
locked laser in Fig. 2(b). The power spectral density is also well fit by a Gaussian (
r
2
= 0
.
9967
).
3 Data Analysis
3.1 Positioning the Experimental Traces
An important part of analyzing the mode-locked pulse patterns in the main text is verifying that
they appear where we expect them to occur in our laser’s temporal synthetic lattice. This step is
important because certain factors, such drift in the delay line couplings, can cause the observed
pulse patterns to appear in unexpected locations. This could lead us to misidentify certain pulse
patterns if the shape of the pulse pattern resembles the correct shape, but its location is incorrect.
We present an example of this situation in Fig. 3, in which we show a pulse pattern that
appeared in the presence of a Hatano-Nelson domain wall in our topological temporally mode-
locked laser. Theoretically, we expect that a skin mode should localize at the domain wall and,
based on shape alone [Fig. 3(a)], we might conclude that observed pulse pattern does just that.
However, using the positioning protocol outlined in this section, we find that the observed pulse
pattern localizes to the left of the domain wall [Fig. 3(b)], likely due to an error in the applied
5
0
50
100
150
200
250
Time (ns)
0
0.5
1
Normalized
Power
Theory
Data
0
50
100
150
200
250
Time (ns)
0
0.5
1
Normalized
Power
Theory
Data
(a)
(b)
Figure 3:
Catching an Erroneously Positioned Trace. (a)
The observed trace is theoretically
predicted to occur at a Hatano-Nelson domain wall, and, based on its shape, one might mis-
takenly conclude that it is indeed localized there.
(b)
The positioning protocol reveals that the
trace is localized to one side of the domain wall. The disagreement between the observed trace
and the theory likely arises due to an error in the applied couplings.
couplings.
Our positioning procedure relies on the auxiliary mode-locked laser introduced in Sec. 1.
During an experiment, we program one of the intensity modulators outside of the main cavity
in to transmit one of every 64 pulses to the
1
.
6
GHz detector shown in Fig. 1. Because our
topological temporally mode-locked laser supports 64 pulses and has the same repetition rate as
the auxiliary mode-locked laser, the reference pulses detected on the
1
.
6
GHz detector provide
a reference from which we can determine the positions of the pulses in the lattice [see Fig. 4].
To use these reference pulses, we determine the position of the reference pulses relative to
the 32nd timeslot defined by the coupling waveform, which is the pulse directly to the left of
the domain wall in Fig. 3. We do this by configuring the 50:50 splitter outside of the cavity in
Fig. 1 to transmit light both through the main cavity and to the
1
.
6
GHz detector. We launch
a pulse through the splitter in what was defined to be the 32nd timeslot in our calibration, and
we observe both the signal directly from the 50:50 splitter and the signal that traveled through
the cavity on our oscilloscope. We measure the difference between the arrival times of the
pulses from each path and account for the delay through the 50:50 splitter
1
to estimate the delay
between the reference pulses and the 32nd timeslot of the main cavity. With this procedure, we
estimate the delay between the reference pulses and the 32nd timeslot to be
∼
104
.
7
ns.
In our experiments, we observe some discrepancies between this estimated delay and the
delay we measure between the reference pulses and the mode-locked pulses generated in the
1
As we mentioned in the Sec. 1, this 50:50 splitter is not present during our calibration (when we define the
32nd time slot in the main cavity), but it is present during the experiment.
6
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Time (ns)
0
0.5
1
Normalized Power
Reference
Experiment
Figure 4:
Reference Signal for Positioning Traces.
We determine the position of the mode
measured in Fig. 3 using a reference pulse from our auxiliary mode-locked laser, which has the
same repetition rate as our topological temporally mode-locked laser. On each roundtrip of our
topological temporally mode-locked laser, we view one pulse from the auxiliary mode-locked
laser on our oscilloscope, and this pulse acts as a reference to determine the positions of the
pulses in our synthetic temporal lattice.
32nd timeslot of our synthetic lattice. We believe that these discrepancies arise because, during
our calibration procedure, we tune the phase of the RF drive applied to the intensity modulator
in the main cavity (see Methods). Changing the phase of this drive changes the positions of the
mode-locked pulses in the main cavity.
Due to these discrepancies, we use the following procedure to compare our measured traces
with the theoretically predicted states: First, we use the estimated position of the 32nd timeslot
to ballpark the position of the 32nd pulse in our synthetic lattice. Then, we use the shape of the
observed pulse pattern to resolve any residual ambiguity about where the 32nd pulse occurs.
Using this procedure, we measure the delay between the reference pulse and the 32nd pulse in
the Hatano-Nelson nascent skin mode to be
∼
105
.
2
ns, and we measure the delay between the
reference pulse and the 32nd pulse in the NH-SSH skin mode to be
∼
107
.
2
ns.
While the measured delay in the case of the NH-SSH skin mode differs from the estimated
delay by
∼
2
.
5
ns, we recall that the pulse repetition period in our mode-locked laser is
4
ns.
Therefore, depending on where the RF driving signals overlap with the optical pulses in the
delay line intensity modulators, there can be up to a roughly
4
ns window in which the 32nd
pulse sees the correct couplings. Given the excellent agreement between the observed data and
the theoretically predicted state, we have strong reason to believe that the pulse we identified as
the 32nd pulse sees the correct couplings for the 32nd time window.
3.2 Plotting the Experimental Traces
Here, we summarize the procedures used to generate the averaged heat maps and data traces
presented in the main text.
To produce the heat maps shown in Main Text Fig. 2, Main Text Fig. 4, and Extended Data
7
Fig. 1, we first apply the positioning procedure described above. Then we compute the maxi-
mum value in each timeslot. We divide the maxima into different roundtrips and normalize the
power in each roundtrip by dividing each maximum by the largest maximum on that roundtrip.
Next we subtract an offset value to account for the presence of noise in our data. Following this
we assign each maximum to be the amplitude of a Gaussian pulse whose width is chosen to be
easily visible on the final heat map. We arrange the Gaussian pulses that correspond to the pulse
maxima on a single roundtrip into a single row of the heat map, and we construct the full heat
map by assembling the rows for all the roundtrips under study.
The averaged data traces in Main Text Fig. 2 and Main Text Fig. 3 illustrate the correspon-
dence between the observed pulse patterns and the theoretically predicted states. To generate
these plots, we divide the data trace under study into roundtrips, and we normalize the data from
each roundtrip. Normalizing the traces in this manner removes any fluctuations in the power and
enables us to more directly compare the pulse patterns observed on different roundtrips. Next,
we average over the normalized roundtrip traces. Finally, we normalize the averaged trace so
that we can compare our result with the theoretically predicted pulse pattern.
For the NH-SSH skin mode plotted in Main Text Fig. 2, the theoretical state is the lowest-
loss right eigenstate corresponding to the domain wall described in the main text. On the other
hand, the theoretical state for Main Text Fig. 3(d) is generated using a fitting procedure because
it is difficult to accurately calibrate the boundary conditions of our Hatano-Nelson lattice. To
perform this fitting procedure, we first assume that we correctly implement the desired cou-
pling ratio
w/v
=
√
2
for the Hatano-Nelson model under consideration. We then compute
the lowest-loss right eigenstate of the Hatano-Nelson lattice as we sweep the coupling between
the first and final sites of the Hatano-Nelson lattice from open boundary conditions to periodic
boundary conditions. At each point in the sweep, we compute the 2-norm of the difference
between the theoretical lowest-loss state and the measured pulse amplitudes, and we consider
the state that corresponds to the smallest 2-norm to be our best fit state. With this procedure,
we find that the best fit occurs when the coupling between the first and final lattice sites is sup-
pressed by a factor of
∼
11
.
6
, which is reasonable given that our delay line intensity modulators
have a nominal DC power extinction ratio of
22
dB.
4 Master Equation Description of Temporal Mode-Locking
4.1 Derivation of the Tight-Binding Model
In this section, we derive the tight-binding model presented in the main text. As our starting
point, we propose a modified master equation for active mode-locking. This master equation
has the form,
8