of 4
Localization of seismic waves with
submarine fiber optics using
polarization-only measurements:
Supplemental Document
SUPPLEMENTARY NOTE 1
Data processing: Eigenvalue method
The following processing was used to recover local earthquake data using the eigenvalue method:
For every pulse launched into the fiber,
M
reflections are recorded corresponding to the
number of repeaters in the cable. An acquisition (at time
t
) comprises every set of 3 consecutive
measurements (obtained from probing the fiber with three different input SOP). By storing the
recovered normalized stokes vectors from each
m
-th repeater as the columns of a 3x3 matrix, we
are able to construct the matrices that encode the matrices
A
m
(
t
)
After constructing the
A
m
(
t
)
matrices from measurements, we perform the singular value
decomposition of each to compute the closest unitary matrix to
A
m
(
t
)
,
ˆ
A
m
(
t
) =
UV
T
as a
denoising step.
The local birefringence matrices (
X
m
(
t
)
) are then calculated using the measured (and denoised)
cumulative matrices. The eigenvalues of this matrix are then calculated, which are of the form
1, exp
(+
i
θ
)
, exp
(
i
θ
)
. We keep the one with the positive argument (
exp
(+
i
θ
)
) and measure its
difference with the first acquisition.
The final output is a matrix of
M
columns and
T
rows, where
T
is the total number of acquisi-
tions. The processing steps are depicted in figure S1.
Fig. S1.
Eigenvalue processing stack.
Visual depiction on how to recover the local time series
using the eigenvalue method.
t
is an index referring to each acquisition and
m
is an index corre-
sponding to each span or repeater.
Data processing: Direct SOP method
The direct SOP method consists of probing the fiber repeatedly with the same input SOP, and
recording changes to each of the three Stokes components of the output normalized Stokes vectors.
As such, this method produces three output time-series, one for each of the output S components.
For the input SOP, the input SOP must remain stationary, which is not the case in our experi-
ments (the SOP of probe pulses is being cycled through a set of three Stokes vectors). As such,
we split the data into three datasets to be processed independently each corresponding to one of
the three possible input SOP. Each of these datasets produces itself three output S components,
resulting in a total of nine independent measurements.
When performing the direct SOP method we may or may not calculate the difference with the
previous repeater. In both cases, the measurements are not localized. In our experiments we did
not calculate the difference with the previous repeater. For the 2D plot of Figure 3 in the main text,
the nine measurements were processed independently to generate a 2D plot, and all 9 resulting
2D plots were combined by averaging.
SUPPLEMENTARY NOTE 2
Acquisition rate
Fig. S2.
Acquisition rate.
Histogram of time interval between acquisitions over 24 hours,
demonstrating the longer delay occurring every third pulse.
In the field demonstration presented in the main text, the pulse repetition rate is adjusted
to align with the fiber length constraint (calculated as 105 ms using equation 6). However, the
PSY-201 polarization synthesizer can only accommodate voltage change commands (V1 and V2)
every 200 ms. This constraint means that the pulse triplet cannot be efficiently generated by
controlling the polarization of each individual pulse, necessitating the selection of probe pulses
with the appropriate modulation.
We generate the input basis according to the following procedure, assuming the initial probe
state is
s
1
:
• Capture
s
1
SOP at t = 75 ms and update V1 at t = 275 ms.
• Capture
s
2
SOP at t = 350 ms and update V2 at t = 550 ms.
Capture
s
3
SOP at t = 625 ms, then sequentially update V1 at t = 825 ms and V2 at t = 1025
ms.
• Capture
s
1
SOP again at t = 1100 ms and update V1 at t = 1300 ms.
Note that the delay between the first and second captures and between the second and third
captures is fixed at 275 ms. However, a longer delay of 475 ms occurs between the third and fourth
captures, resulting in non-uniform sampling. This can be seen in the histogram of acquisition
intervals in Figure S2.
2
SUPPLEMENTARY NOTE 3
Q Measurements
Fig. S3.
Histogram of Qs.
Histogram registered over 24 hours, where Q is shown to be kept
consistently above 0.9.
In principle, the eigenvalue method can operate with any trio of input SOPs that spans the
entire Stokes space. However, it is advantageous to use an orthogonal basis for optical noise
robustness and mitigation of crosstalk.
One possible figure of merit (FOM) for the orthogonality of the selected input basis can be
Q
=
|
det
(
A
0
)
|
,
(S1)
where
A
0
is a 3x3 matrix where each column is the normalized Stokes vector of each of 3
consecutive input SOP.
Q
varies between 0 for a set of three input vectors that do not form a basis
of the full Stokes space, to 1 where the set of vectors forms an orthonormal basis.
We monitored the
Q
of each acquisition by measuring the 3 polarization states launched into
the fiber at every acquisition, and kept the
Q
consistently above 0.9. The histogram of measured
Q
values is displayed in figure S3
SUPPLEMENTARY NOTE 4
Crosstalk and nonlocal effects
We performed a numerical evaluation of the effects of different perturbation parameters and
input triplets on the observed crosstalk. The simulations comprise a simulated cable composed of
N
spans, each including (independent) forward and backward paths. Each path has a random
birefringence vector orientation with a fixed, pre-determined birefringence strength. The fixed
simulation parameters are represented in table S1.
In order to assess the onset of crosstalk and nonlocal effects when using the eigenvalue method,
we added a sinusoidal variation of 0.02 percent to both the forward and backwards paths of the
second span in the cable.
We repeated the measurement for the same cable while varying the input basis set to have
different
Q
(equation S1) and changing the frequency of the sinusoidal perturbation (while
keeping the acquisition rate and amplitude fixed). Altering the perturbation frequency results in a
more significant waveform change over the three acquisitions needed for a complete measurement
of the
A
(
m
)
matrix, thus leading to a stronger breach of the stationarity assumption. For this
simulation, we considered infinite optical SNR.
3
a.)
b.)
Fig. S4.
Numerical simulation results.
Simulated results of crosstalk observed as a function of
input
Q
and maximum perturbation slew rate. Crosstalk is measured as the median variance
measured in spans after the perturbed span, normalized to the variance of the perturbed span.
Parameter
Value
Number of Spans
40
Span length
100 km
Average birefringence strength (
n
)
1
×
10
8
Perturbation amplitude (change in
n
)
0.02% of birefringence
Perturbed span
2
Table S1.
Summary of Simulation Parameters
We define crosstalk as the median variance of the signal observed in all (unperturbed) spans lo-
cated after the perturbed span, normalized to the variance of the signal observed in the perturbed
span (which may change between runs of the simulation, due to the nonlinearity of eigenvalue
measurements). In figure S4a, we plot the crosstalk against the orthogonality figure of merit
(
Q
) and the maximum birefringence change between consecutive acquisitions (as a measure of
non-stationarity).
There is a clearly observable a threshold effect where the crosstalk suddenly increases by
about 10 dB, for combinations of low
Q
and high slew-rate. Two possible ways to improve or
mitigate crosstalk are to ensure consistently high
Q
and orthogonality, as well as maximizing
the acquisition rate of the system. This also implies that the amount of crosstalk to subsequent
spans is frequency dependent. For fixed amplitude, higher frequency perturbations may lead to
increased crosstalk to subsequent positions, which also suggests that simple low-pass filtering of
all channels may be sufficient to reduce crosstalk for some applications.
We also observed the effects of Q and optical SNR on the measurement noise power, as depicted
in figure S4b. Higher Q and higher SNR seem to correlate with lower measurement noise. Our
simulations seem to suggest that there is a minimum value of Q (around 0.3) at which robustness
to low optical SNR increases.
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