of 15
1
Orbital
Angular Momentu
m
-
based Spatial Division Multiplexing for High
-
capacity
Underwater
Optical
Communications
Yongxiong Ren
1
*,
Long Li
1
*
,
Zhe Wang
1
,
Seyedeh
Mahsa Kamali
2
, Ehsan Arbabi
2
, Amir
Arbabi
2
,
Zhe Zhao
1
, Guodong Xie
1
, Yinwen Cao
1
, Nisar Ah
me
d
1
, Yan Yan
1
,
Cong Liu
1
, Asher J.
Willner
1
, Solyman Ashrafi
3
,
Moshe Tur
4
,
Andrei Faraon
2
, and Alan E. Willner
1
1
Department of Electrical Engineering, University of Southern California, Los
Angeles, CA
90089, USA.
2
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA
91125, USA
3
NxGen Partners, Dallas, TX75219, USA
4
School of Electrical Engineering, Tel Aviv University, Ramat Aviv 69978
,
I
srael
.
*These authors contributed equally to this work.
1
.
OAM beams underwater propagation:
M
easurement results under different scattering
levels
and
thermal gradient
s
We show below measurement
s
of OAM beams propagation
under
a different scattering
level and
thermal gradient
.
Supplementary Figs.
S
1(a1
a2) show BER fluctuations over 36 seconds
at a
fixed transmitted power of
-
26 dBm
when different amounts of
0.5% diluted
Maalox solution are
added into the water.
These measurements are performed
every
2
second
s
and
are
repeated
18
times
before the Maalox particles are evenly distributed in the water (
before
circulation
over 1
minute
)
.
We see that
the case of 0
.
5
-
millilitre
of
Maalox solution has a larger BER fluctuation
range
due to a more uneven suspension of Maalox particles along the link path
. After uniform
2
scattering suspension is obtained, the
0
.
5
-
and
1.
5
-
millilitre
Maalox solutions introduce power
losses of 2.2 dB and 4.5 dB to the link, respectively.
Fig.
S
1(c)
presents
the
statistics of beam
displacement
with respect to the propagation axis after
propagation through thermal gradient
-
induced turbulence, in which the
room
temperature
water
and
the
heated water
have a
temperature difference
of 0.
3
o
C
. We see that th
e
received beams’
maximal
displacement
is
estimated to
be
1
.
4
mm
, which is larger than that shown in Fig. 2(b) of the main manuscript
.
The
captured intensity profiles of OAM beam
=+3
under
3
different turbulence reali
s
ations are also
shown
to illustrate the time
-
varying distortions caused by dynamic turbulence.
Fig.
S
1(d) show
s
the
BERs and received power for
the
OAM
=+3
channel
under various turbulence reali
s
ations.
A wider range of power fluctuation is observed tha
n
that shown in Fig.
5
(a) of the main
manuscript.
2.
Implementation details of the
4 Gbit/s
OAM multiplex
ed
underwater
optical
link
using
directly modulated green laser diodes
In this section, we
describe
t
he
experimental
implementation of the 4 Gbit/s four OAM
multiplexed
underwater link
, in which laser diodes at 520 nm are directly modulated and act as
light signal sources.
Supplementary Fig.
S
2
presents the link schematic
. Two 1
-
Gbit/s OOK
signal beams at 520 nm are generated by directly modulating each of the
two 520
-
nm
green
laser
diodes
with a binary sequence. The two modulated green light beams
are
launched on
to two
liquid crystal
-
based spatial light modulators
(SLMs)
to
create
two different
OAM beam
s
with
ℓ=+1 and
+3.
The SLM (Santec Inc.)
has a pixel resolution of 14
40×1050 and an operating
wavelength range of 500
-
1650nm.
The
se
two OAM beams ℓ=+1 and +3 are
coaxially
combined
and then split into two identical copies, one of which is reflected
three
times
using mirrors
arranged to introduce a ~
50
n
s delay
for data
sequence
decorrelation
between the two copies
.
3
A
nother two OAM beams with opposite
values
of
-
1
and
-
3 are obtained
due to the odd number
of reflections
,
and these new beams
are
then
combined
with
the original OAM beams ℓ =
+1
and
+3
by a beam splitter
. T
he resulting
four
multiplexed OAM beams propagate through
the
underwater channel
.
Table
1
.
The power transfer
between the four multiplexed OAM
channel
s (a) at the
transmitter and (b
) after propagating through tap
water.
We
c
haracteri
z
e
the power
leakage and crosstalk between
all OAM channels.
The power leakage
is measured in the following way:
W
e first transmit a
520
-
nm
signal
over OAM
channel ℓ=
-
3
while all the other channels (OAM
beams
ℓ=
-
1, +1, and
+3 are off. Then we record the
power
leaked
in
to
other
OAM mode
s
(OAM channels
ℓ =
±
1 and
±3
)
.
T
he above measurement
s
are
repeated
for all transmitt
ed
OAM
channels until
a
full 4
×
4 power transfer matrix
is obtained
.
Table 1
shows the
power transfer
matrices
between
all
the four channels
at the transmitter and
after propagating the tap water channel
. The crosstalk of a specific channel can be calculated
from the
4×4
power transfer matrix
by adding the received power from all other channels divided
dBm
=
-
3
=
-
1
= +1
= +3
=
-
3
-
23.72
-
34.12
-
40.82
-
44.53
=
-
1
-
35.08
-
21.42
-
36.52
-
39.46
= +1
-
40.60
-
32.89
-
21.78
-
33.06
= +3
-
40.13
-
38.43
-
32.58
-
20.91
dBm
=
-
3
=
-
1
= +1
= +3
=
-
3
-
23.39
-
34.01
-
39.90
-
42.87
=
-
1
-
36.80
-
21.89
-
32.45
-
39.23
= +1
-
39.20
-
34.12
-
22.67
-
34.50
= +3
-
42.80
-
39.26
-
34.89
-
22.35
(b) After propagating through
tap
water
(a) Multiplexed OAM beams at the transmitter
4
by the received power of this channel.
We
see that the crosstalk values for all four multiplexed
channels are below
-
10 dB
in
both cases, and the degradation introduced by tap water is
negligible.
Three different underwater channel conditions are emulated
in a 1.2
-
met
re
-
long rectangular tank
filled with tap water
, as
described
in the main manuscript. T
he w
ater
current
and scattering are
created by distributed circula
tion pumps and the
Maalox solution,
respectively. The thermal
gradient
-
induced
turbulence is
produced by mixing the room
temperature and heated water
.
After
propagating through the various water conditions,
the four
multiplexed
OAM channels are
sequentially
demultiplexed and detected
at the receiver
. To recover the data channel carried by
OAM beam
+
, the SLM
used for demult
iplexing
is loaded with a inverse
spiral phase pattern
of
-
. As a result, only OAM beam with
+
is converted into a Gaussian
-
like beam
(
=
0
)
and all
the other OAM beams maintain ring
-
shaped profiles, which can be efficiently filtered out by a
spatial filt
er (simply a
pin hole
). The Gaussian
-
like beam is subsequently focused onto a high
-
sensitivity silicon avalanche photodiode
detector (APD
)
with a 3
-
dB bandwidth of
1
GHz
.
The
APD has a
spectral
responsivity of
~
1
5
/
at 520 nm and a
low
noise equivalent power
(NEP)
of 0.4
/
.
The signal after detection is
amplified,
filtered
and
sent to a 1
-
Gbit/s receiver
for bit
-
error rate (BER) measurements.
3.
Implementation details of
the
40
Gbit/s
OAM multiplex
ed
underwater optical
link
using
PPLN
-
based frequency doubling
We present below the implementation details of the 40 Gbit/s
four OAM multiplexed
underwater
link
, in which each 532
-
nm green OAM beam carries
a 10 Gbit/s OOK signal
generated from
PPLN
-
based frequency doubling
.
The expe
rimental setup is depicted in Supplementary Fig
.
S
3
.
5
A 1064
-
nm
single
-
mode laser
wi
th a linewidth of less than 100
MHz is sent to a L
ithium
N
iobate
modulator
to produce a 10
Gbit/s on
-
off keying (OOK) signal.
The
transmitted RF signal is a
pseudorandom bin
ary sequence
of a
length 2
15
-
1
. The 10
Gbit/s
OOK signal at 1064 nm is
amplified by a y
tterbium
-
d
oped fibre
a
mplifier
(YDFA)
and fed into a collimat
or
to convert the
single
-
mode fibre output to a collimated Gaussian beam with a diameter of 2.6 mm. This
Gaussian
beam is then focused
into the
centre of a
p
eriodically poled lithium niobate
(PPLN)
crystal for frequency doubling.
A half
-
wave plate is inserted after the collimator to align the
polari
s
ation of
incident
light with the polari
s
ation
orientation
of
the PPLN crystal
to maximi
s
e
the conversion efficiency
.
The
PPLN
crystal
is
z
-
cut
to
20
mm
×
1
mm
×
1
mm in dimension.
Both
its
input and output
facets
are
dual
-
coated
with reflectivity of <1%
at 532
nm and
1064
nm
to reduce
the
Fresnel reflection loss. The crystal oven
together
with a
temperature controller
offers a stability of ±0.01
o
C
.
The power of the green beam
generated from frequency doubling
depends upon both
the
temperature and
the
input pump power.
For a
34 dBm
1064
-
nm i
nput
beam, the power of
the
output green beam
at 532
nm is around 22 dBm with
the
oven
temperature
being
79.5
o
C
.
The beam after
the
PPLN crystal is collimated and passes through a
dichroic mirror to reflect the unconverted 1064
-
nm beam. A bandpass filter
with a centre
wavelength of 532
nm is foll
owed to further separate the
gree
n light from the remaining 1064
-
nm beam.
As a result, a 532
-
nm green beam carrying a 10
Gbit/s
OOK signal is produced.
We
note that t
his frequency
-
doubling process is transparent to intensity
-
based modulation format.
The resulting
signal
beam at 532
nm is split into two copies, which
then
pass through two
transmissive metasurface
phase masks to
generate
two OAM beams with ℓ
=+1 and +3
,
respectively.
The
metasurface
phase masks
have high efficiency (power loss of ~3dB at 532
nm
)
and the generated
OAM modes are of high quality
(
s
ee Supplementary Note
4
for more details
6
about the design
).
These two OAM beams are spatially combined using
a
beam splitter
.
Using a
similar approach as in the previous section, a
nother two OAM beams with opposite
values
(
ℓ =
-
1
and
-
3)
can be
obtained
,
which are
then
multiplexed
with
the original OAM beams ℓ =
+1
and
+3.
T
he beam sizes
for OAM beams
ℓ =
-
3,
-
1, +1
and
+3 are
2.
1
, 1.5
, 1.
4
and
2
.
0
mm,
respectively. T
he
resulting four
multiplexed OAM beams (
ℓ = ±
1
and ±
3)
are sent
through a
1.2
-
metre
under
water
channel
.
The
power transfer
matrices
between the four channels
at the
transmitter and after propagating the tap water channel are
shown in Table.
2
.
We see that
the
crosstalk values for all four channels are below
-
1
1
dB
in
both cases.
Table
2
.
The power transfer
between the four multiplexed OAM
channel
s (a) at th
e
transmitter and (b) after propagating through the
tap
water channel.
After propagation through water, a phase mask with an inverse phase pattern of the desired OAM
channel will be used to convert the chosen OAM beam into a Gaussian like beam.
The other
beams maintain their ring
-
shaped profiles and helical phases
after passing through the phase
mask. The
Gaussian like
beam has a bright high
intensity at the beam centre and is thus separable
dBm
=
-
3
=
-
1
= +1
= +3
=
-
3
-
17.3
-
29.8
-
31.2
-
39.6
=
-
1
-
31.5
-
17.4
-
28.3
-
38.74
= +1
-
38.7
-
30.9
-
17.5
-
28.3
= +3
-
17.7
-
32.3
-
29.2
-
17.7
dBm
=
-
3
=
-
1
= +1
= +3
=
-
3
-
19.99
-
31.8
-
33.65
-
40.18
=
-
1
-
32.07
-
20.1
-
31.66
-
37.42
= +1
-
35.49
-
30.1
-
19.89
-
33.12
= +3
-
40.18
-
35.1
-
30.6
-
19.83
(b) After propagating through
tap
water
(a) Multiplexed OAM beams at the transmitter
7
from the other beams through spatial filtering
(simply a pin hole)
. This beam is then
focused
onto a
9
-
GHz
-
bandwidth
silicon
APD
, which has
a
spectral
responsivity of
~
0.2
A
/
W
at 5
32
nm
and a
NEP
of
<45
pW
/
HZ
.
The signal after
the
APD is
amplified,
filtered
and
sent
to a
10
Gbit/s
OOK
receiver for BER measurements. Given that four OAM beams each
bearing
a 10
Gbit/s
data
stream
are transmitted, a total capacity of 40 Gbit/s
is
achieved.
Supplementary Fig
s.
S
4
(a1
a4)
show
the
captured
intensity profiles of
the generated
OAM
beams
of
=
±
1 and
=
±
3
at the transmitter
.
Figures
.
S
4(b1
b2) presents
t
he
measured
interferograms
for
OAM beams
=
+
1
and +3
, in which t
he state number of the two OAM
beams can be deduced from the number of rotating arms
. Each
interferogram
can be obtained
interfering
an
OAM beam
(either
=
+
1
or +3)
with an expanded Gaussian beam
.
Fig
s.
S
4
(
c
1
c4
)
show the measured intensity profiles (c1
-
c2) and interferograms (c3
-
c4) of the received OAM
beams
=
+
1,
and
=
+
3
after propagating through tap water.
Figs.
S
4
(d1
d4) depict
t
he
intensity profile
s
of demultiplex
ed
beams
at the receiver
when
only the
OAM channel
=
+3
is
transmitted
. We
see
that
only when the phase mask is of inverse spiral phase of
=
-
3
, can
the
OAM
beam
with
=
+3
be
converted into a
Gaussi
an like
beam
with a high
intensity at the beam
cent
re.
4
. High
-
efficiency dielectric metasurface OAM generator for visible wavelengths
In this section, we describe the design, fabrication and characterization of dielectric metasurface
phase masks for OAM generation.
Dielectric metasurface phase masks
made of square cross
-
section SiN
x
nano
-
posts on fused silica are designed, fabricated and characteri
s
ed for the
generation an
d detection of OAM beams at 532
nm.
Polari
s
ation
-
insensitive ph
ase ma
s
ks are
composed of 630
-
nm
-
tall SiN
x
n
ano
-
posts on a square lattice w
ith the lattice constant of 348
nm.
8
By changing the nano
-
posts width in the range of 60
nm to 258
nm, the transmission phase can
be changed from 0 to 2π
while
maintaining
high trans
missivity
at the wavelength of 532 nm.
Therefore
,
any arbitrary
phase profile
can be designed using this metasurface platform. Here, a
blazed
grating ‘
fork
phase pattern, which is a combination of the helical phase
structure
of the
desired OAM mode and a
small linear phase ramp
(~
2
degrees)
,
is designed. Such a combination
pattern can help separate the generated OAM beam from the residual unmodulated Gaussian
beam, providing high
-
quality OAM generation and detection. Supplementary Fig.
S
5
(
a
)
shows a
schema
tic illustration of the blazed
grating ‘
fork
phase mask generating
the
OAM mode with
ℓ=+3.
The metasurfaces
are
fa
bricated by depositing a 630
-
nm
thick layer of SiN
x
on a fused
silica substrate. The pattern is defined and transferred to the SiN
x
layer usi
ng e
-
beam lithography
followed by the lift
-
off process and dry etching. Two different 1.5
-
mm
diameter metasurface
phase masks with ℓ=
+
1 and ℓ=
+
3
are
designed and
fabricated.
Figs. S
5
(b1
b2) show the
scanning electron micrographs of the fabricated phase mas
ks of
=+1
and
=+
3, respectively.
To
quantify
the quality of OAM beam generation, we measure the OAM power spectrum of
generated OAM beams
ℓ =
+1 and ℓ =
+3, as shown in Fig
.
S
5
(
c
)
.
We see that the majority of the
power resides in the desired modes and the power leakage onto the adjacent modes
is
-
13 dB less
than the
desired
modes.
5
.
Implementations details of
the
CMA
-
based
multi
-
channel equalisation algorithm
In general, f
or a
n
underwater optical link using
multiplexed OAM beams, a
multiple
-
input
multiple
-
output
(MIMO)
channel processing with a dimension
of
×
would
be need
ed
to
reduce the inter
-
channel crosstalk caused by underwater propagation (mainly thermal gradient
-
in
duced turbulence in
the
experiment
s
). Mathematically speaking, the received signal vector for
all
OAM channels
=
(
1
,
2
,
...
,
)
T
can be express
ed
as
9
=
+
(1)
where
=
1
,
2
,
...
is the received
signal
of
OAM
channel
and
=
(
1
,
2
,
...
,
)
with
being
the
transmitted signal
for
OAM
channel
.
is
the
channel matrix, which can be written
as
=
1
,
1
1
,
2
2
,
1
2
,
2
1
,
2
,
M
,
1
,
2
,
×
,
(2)
where
,
depict
s
the
transfer
function
between
OAM channel
to
OAM channel
j
.
=
(
1,
2,
...,
)
and
is the noise for
i
-
th
OAM
channel
.
is
determined
by the power loss of
each OAM channel and crosstalk between all
OAM channels,
which are
directly related
to
the
underwater
channel
conditions and system design (e.g. aperture sizes
and link distance
)
.
To
recover the OAM data streams,
the received signals of all OAM channels could then be
multiplied with the inverse
of
channel matrix
theoretically
.
A variety of implementation approa
ches for MIMO processing based on
have been proposed
,
including joint maximum likelihood
sequence
estimation of the data symbols in different streams,
minimum mean
-
square error
detection combined with serial interference cancellation,
zero
forcing
detection
, and p
ilot
-
aided channel equali
s
ation
.
For
the
experiment
using
two
multiplexed OAM modes,
a
2
×
2 CMA
-
based
equalisation
algorithm
is implemented
to
equalise
crosstalk between the two OAM channels, thus allowing data
recover
y
.
After demultiplexing
,
each of the two received OAM channels is converted into a Gaussian
-
like beam and detected by
a 1
-
GHz bandwidth APD. The two signals are amplified, sampled by a real
-
time scope and
recorded for offline DSP.
Given that we use the OOK modulation format for
the signal
10
generation, which does not have a
constant modulus
in the signal constellation, the DC
components of the two received channels are subtracted to apply the CMA algorithm.
The
CMA
-
based
equalization
algorith
m
utilises
a linear equaliser for each channel
.
For a
2
×
2 equalisation,
t
he equali
s
er includes
4
adaptive finite
-
impulse
-
response (FIR) filters
, each with a tap number of
.
Specifically, t
he output of the equali
s
er corresponding to each channel can be expressed as
:
=
2
=
1
,
=
1
,
2
(3)
where
(
=1,2) is the coefficient vector of the FIR filter with a vector length of
(tap
number)
.
represents the inner product operation between two vectors
and
is the output
of the FIR filter. All the FIR coefficients are initiali
s
ed as
zero with only the
centre
weight be
ing
1
and then updated until the coefficients converge
based on
CM
A
:
+
1
=
+
(4)
where
is the step size,
=
2
is the error signal of the adaptive estimation and
is the normali
s
ed reference power.
The main idea of
CMA
-
based MIMO equalisation
is to update
filter
weights such that each channel output can have
a
cle
ar amplitude
.
The tap number
in
each
FIR
filter is set to
be 11, which is
suf
f
icient
to cover the differential time delays among each
data sequence and mitigate temporal ISI effects.
The obtained FIR
filter coefficients are used to
equali
s
e the crosstalk among
two
OAM channels
of
Eq
uation
(
3
)
. After equali
s
ation,
the bit
-
error
rate
s
(BER
s
)
are evaluated
for both channels
.