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Applications of orbital angular momentum in high-capacity free-space optical communications

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Applications of orbital angular momentum in high-capacity free-space optical communications

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APPLICATIONS OF ORBITAL ANGULAR MOMENTUM IN HIGH-CAPACITY
FREE-SPACE OPTICAL COMMUNICATIONS

by
Long Li

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)

May 2019

Copyright 2019 Long Li
ii

Dedication

This dissertation is dedicated
to my loving wife Di Lu
for her whole-hearted support and everlasting love,
to my parents Youqing Cao and Zhigang Li
who have always loved me unconditionally,
and to my respectful Prof. Alan E. Willner
who has always been the greatest mentor.

iii

Acknowledgements
I would like to sincerely appreciate my Ph.D. adviser and dissertation committee
chair, Prof. Alan E. Willner, for providing me the opportunity to conduct research in
this exciting field, for mentoring me not only technical knowledge and skills but also
ethics and life philosophy, for giving me guidance for my Ph.D. and beyond-Ph.D.
careers, and for always encouraging me for greater goals.
I would like to thank Prof. Stephan W. Haas and Prof. Andreas F. Molisch for
being my dissertation committee members and providing me valuable comments. I
would like to express my deepest gratitude to Prof. Moshe Tur from Tel Aviv
University for always enlightening and encouraging me during my Ph.D. study.
I also want to extend my appreciation to Dr. Solyman Ashrafi (NxGen Partners),
Dr. Robert Bock (R-DEX Systems), Prof. Todd Brun (University of Southern
California), Prof. Andrei Faraon (California Institute of Technology), Dr. Brittany
Lynn (Space and Naval Warfare Systems Command), Dr. Dmitry Starobudov
(Univeristy of Southern California), and Prof. Wei Wu (University of Southern
California), for their valuable advices and comments during my doctoral study.
I sincerely acknowledge my excellent colleagues at the Optical Communications
Laboratory (OCLab) for their endless help, advice, and excellent ideas. I feel
extremely lucky for being in the same team with them.
Last but not least, I appreciate the tremendous help and support by the EE staff
beloved Tim Boston, Diane Demetras, Gerrielyn Ramos, Corine Wong, and Susan
Wiedem.
iv

Table of Contents
Dedication ................................................................................................................ ii
Acknowledgements .................................................................................................... iii
List of Figures ............................................................................................................ vi
Abstract .............................................................................................................. xiii
Chapter 1 Introduction .......................................................................................... 1
1.1 Background of Orbital Angular Momentum .......................................... 1
1.2 Challenges for OAM-Based FSO Systems ............................................. 4
1.3 References .............................................................................................. 8
Chapter 2 High-Capacity Free-Space Optical Communications between a
Ground Transmitter and Receiver via a UAV Using Orbital
Angular Momentum Multiplexing .................................................... 10
2.1 Introduction .......................................................................................... 10
2.2 Link Demonstration .............................................................................. 12
2.3 Limited Aperture Effects ...................................................................... 23
2.4 Discussion ............................................................................................. 31
2.5 Reference .............................................................................................. 36
Chapter 3 Turbulence Effects Mitigation for Orbital-Angular-Momentum-
Multiplexed Free-Space Optical Communications between a
Ground Station and a UAV Using MIMO Equalization ................ 38
3.1 Introduction .......................................................................................... 38
3.2 Link Demonstration and Turbulence Effects Mitigation ...................... 40
3.3 Reference .............................................................................................. 48
Chapter 4 Beaconless Beam Displacement Tracking and Correction in
Orbital-Angular-Momentum-Multiplexed Free-Space Optical
Links .................................................................................................... 50
4.1 Introduction .......................................................................................... 50
4.2 Beaconless OAM Beam Displacement Tracking ................................. 51
4.3 Reference .............................................................................................. 59
Chapter 5 Orbital-Angular-Momentum-Multiplexed Free-Space Optical Link
Using Transmitter Lenses .................................................................. 62
5.1 Introduction .......................................................................................... 62
5.2 Concept and Simulation ....................................................................... 63
5.3 Experimental Demonstration ................................................................ 69
5.4 Conclusion ............................................................................................ 74
v

5.5 Reference .............................................................................................. 75
Chapter 6 Power Loss Mitigation of Orbital-Angular-Momentum-
Multiplexed Free-Space Optical Links Using Non-Zero Radial
Index Laguerre-Gaussian Beams ...................................................... 78
6.1 Introduction .......................................................................................... 78
6.2 OAM Beams with Non-Zero Radial Indices ........................................ 79
6.3 Link Demonstration .............................................................................. 85
6.4 Conclusion ............................................................................................ 90
6.5 Reference .............................................................................................. 91
Chapter 7 Mode and Space Diversity in Free-Space Optical Link to Increase
System Tolerance to Turbulence ....................................................... 94
7.1 Introduction .......................................................................................... 94
7.2 Concept and Experimental Setup ......................................................... 95
7.3 Link Demonstration and Analysis ........................................................ 98
7.4 Reference ............................................................................................ 101
vi

List of Figures
Figure 1.1 (a) The intensity and wavefront profiles of a Gaussian beam and OAM
beams ℓ = +1 and ℓ = +2; (b) Multiple data channels, each carried by a
different OAM beam, can be multiplexed together to produce a high
capacity free-space data transmission [13]. ............................................................ 2
Figure 1.2 Normalized intensity and phase profile of Gaussian, LG 0,5, LG 1,5, and
LG 3,5 beams. ............................................................................................................ 3
Figure 1.3 OAM beam generation and detection. ............................................................. 4
Figure 1.4 Divergence of regular Gaussian and OAM-carrying LG beams. .................... 5
Figure 1.5 Effects of misalignment on system performance. (a)-(d) Illustration of
OAM-based communications link with perfect alignment, displacement,
tip/tilt error at Rx, and tip/tilt error at Tx, respectively. ......................................... 6
Figure 1.6 Turbulence effects for OAM beams. ............................................................... 7
Figure 2.1 Concept of using OAM multiplexing in high-capacity FSO
communications between a UAV and a ground station [12]. ............................... 13
Figure 2.2 Experimental setup of of an FSO communication link between a UAV
and a ground station using OAM multiplexing [12]. ............................................ 14
Figure 2.3 Received power on different OAM modes under horizontal, vertical,
and simulated displacement when OAM -1 is transmitted. The roundtrip
transmission distance is ~100 m and the tracking system is off. The
displacement refers to the distance between the OAM beam center and the
center of the receiver. Tx:-1, Rx:+3: received power on OAM +3 mode
when OAM -1 beam is transmitted [12]. .............................................................. 16
Figure 2.4 Effects of propeller-induced airflow on system performance. (a) and (b)
Measured power distribution when the UAV’s propellers are turned off or
on, respectively. The power is normalized to its mean during the
measurement. (c) Measured OAM spectrum when the UAV’s propellers are
turned off and on, when OAM +1 is transmitted over ~100 m roundtrip [12]. .... 18
Figure 2.5 Beam jitter, power, and crosstalk measurements in flight environment.
(a)-(c) Measured statistics of beam displacement with respect to the
receiver center when the UAV is static on the ground, hovering, and moving
at a speed of ~0.1 m/s. OAM +3 is transmitted and the received beam
vii

diameter is ~6.2 cm. (d) Schematic diagram of different locations where the
UAV hovers. Location 1 is ~10 m above ground, ~50 m away from the
transceiver with an angular position of 0°; location 2 is ~20 m above
ground, ~40 m away from the transceiver with an angular position of 15°;
location 3 is ~5 m above ground, ~10 m away from the transceiver with an
angular position of -5°. (e) The received power on different modes in a 60-
second period, when OAM -1 is transmitted and the UAV is hovering. (f)-(i)
OAM spectrum in a 60-second period when OAM +1 is transmitted and the
UAV is hovering at different locations or moving at a speed of ~0.1 m/s,
respectively [12]. .................................................................................................. 20
Figure 2.6 Bit-error-rate (BER) measurements in flight environment. Each channel
transmits a 40-Gbit/s quadrature phase shift keying (QPSK) signal. (a)
BERs in a 60-second period and (b) BERs measurements for OAM +3 and -
1, when OAM +3 and -1 beams are transmitted. At each transmitted power
level, 10 data frames (each has 4096 bits) over ~30-second period are
measured. The number at the bottom represents the number of error-free
frames during the measurement period at a certain transmitted power level.
(c) BER measurements when two OAM beams with different mode spacing
are transmitted. The UAV is hovering ~10 m above the ground and ~50 m
away from the ground station. (d) BER measurements for OAM +3 when
OAM +3 and -1 are transmitted. The UAV is hovering at different distances
away from the ground station. FEC: forward-error-correction [12]. .................... 22
Figure 2.7 Concept of limited-size aperture effects on an OAM-based link with: (a)
full receiver aperture and perfect alignment; (b) limited-size receiver
aperture and perfect alignment; (c) full receiver aperture and misalignment;
(d) limited-size receiver aperture and misalignment. ........................................... 24
Figure 2.8 Measured received power on different OAM modes when only OAM
+3 is transmitted with 10-dBm power, (a) without and (b) with intentionally-
added displacement, respectively. (c) Crosstalk for OAM +3 when OAM +3
and -1 are transmitted under various intentionally-added displacement. (d)
Measured crosstalk for OAM+3 when various modes are transmitted under
intentionally-added 1.4-mm displacement. The UAV is static on the ground
~50-m away from the ground station, and the tracking system is off. Rx,
receiver; Tx, transmitter. (e) and (f) Simulated received power on different
modes as functions of lateral displacement when OAM +3 is transmitted,
with a receiver aperture diameter of 3 cm and 7 cm, respectively. ...................... 25
Figure 2.9 (a) Measured OAM spectrum of OAM -1 when OAM -1 is transmitted
and the UAV is on the ground. (b) Measured received power on different
modes in a 60-s time period when OAM -1 is transmitted and the UAV is
hovering ~50-m away. In (a) and (b), the receiver aperture diameter is 6.5
cm. (c) Measured OAM spectrum of OAM -1 when OAM -1 is transmitted
and the UAV is hovering ~50-m away with various receiver aperture size.
The color bar represents the fluctuation range of the received power. (c)
viii

Measured crosstalk for OAM+3 when OAM+3 and -1 are transmitted and
the UAV is on the ground, hovering, or moving ~50-m away, with various
receiver aperture sizes. Rx:6.5 cm, receiver diameter is 6.5 cm. ......................... 29
Figure 2.10 Experimental bit-error-rates (BERs) for each 8192-symbol data frame
when OAM+3 and -1 are transmitted and the UAV is hovering. (a) BER
measurements in a 60-s time period for both channels when transmitted
power is 10 dBm for each channel. (b) BER for both channels as functions
of transmitted power. (c) BER for OAM+3 as a function of transmitted
power with various receiver aperture sizes. For (a) and (b), the receiver
aperture diameter is 6.5 cm. The values at the bottom of (b) and (c)
represent the number of error-free frames out of the 10 measured frames for
each transmitted power level. FEC: forward error correction. ............................. 30
Figure 3.1 Concept of (a) an orbital-angular-momentum (OAM)-multiplexed free-
space optical (FSO) communication link between a ground station and a
retro-reflecting unmanned-aerial-vehicle (UAV) through atmospheric
turbulence; (b) an OAM beam distorted by turbulence; and (c) mitigation for
turbulence effect in an OAM-multiplexed link using multiple-input-
multiple-output (MIMO) equalization. ................................................................. 39
Figure 3.2 Experimental setup. BS: beamsplitter; DeMUX: demultiplexer; EDFA:
erbium-doped fiber amplifier; FSM: fast steering mirror; LO: local
oscillator; MUX: multiplexer; PD: photodetector; PSD: position sensitive
detector; QPSK: quadrature phase-shift keying. .................................................. 42
Figure 3.3 Measured power distribution when the turbulence emulator is (a) placed
in the link and rotating at 40 round/minute and (b) not placed in the link. A
~5-cm diameter 1550-nm Gaussian probe beam is transmitted over ~100 m
roundtrip, and a ~1-mm diameter point detector is used and the receiver. .......... 43
Figure 3.4 Measured OAM spectrum when OAM +1 beam is transmitted and the
UAV is static on the ground (a) with and (b) without the turbulence phase
plate. Measured OAM spectrum in a 60-second period with and without the
turbulence phase plate when OAM +1 beam is transmitted and the UAV is
(a) hovering and (b) moving at a maximum speed of 0.1 m/s. In all cases,
the UAV is ~50-m away, ~5-m above the ground. ............................................... 45
Figure 3.5 Measured received power and crosstalk for OAM+1 and -3 when both
beams are transmitted under 12 different turbulence realizations. The UAV
is hovering ~50-m away, ~5-m above the ground. ............................................... 45
Figure 3.6 Experimental measurement of bit-error-rates (BERs) as a functions of
transmitted power when OAM +1 and OAM -3 are simultaneously
transmitted, each carrying a 20-Gbit/s quadrature phase-shift keying (QPSK)
signal: (a) for the OAM -3 channel without MIMO equalization; (b) for both
channels with and without MIMO equalization. (c) Recovered QPSK
ix

constellations at transmitted power of 10 dBm for all channels with and
without MIMO equalization when OAM +1 and -3 are transmitted. The
UAV is hovering ~ 50-m away, ~5-m above the ground. .................................... 46
Figure 3.7 BERs for both OAM +1 and OAM -3 channels with and without MIMO
equalization under 12 different turbulence realizations. The UAV is
hovering ~ 50-m away, ~5-m above the ground. .................................................. 48
Figure 4.1 (a) Scheme of displacement tracking for multiple OAM beams using
OAM-beams-based position detection. (b) Intensity profiles of an OAM
beam and a Gaussian beam. FSM: fast steering mirror; PSD: position
sensitive detector. ................................................................................................. 52
Figure 4.2 Experimental setup of an OAM-multiplexed FSO communication link
with displacement tracking executed in a beaconless manner by the OAM
beams themselves (i.e., the 1530 nm laser is off) or with the help of a
Gaussian beacon (i.e., the 1530 nm laser is on). BS: beamsplitter; Col.:
collimator; EDFA: erbium-doped fiber amplifier; PC: polarization
controller; QPSK: quadrature phase-shift keying; Rx: receiver; SLM: spatial
light modulator; Tx: transmitter [18]. ................................................................... 53
Figure 4.3 Simulation and experimental results for normalized output of PSD as
functions of spot-centroid displacement when (a) OAM ℓ = +1 beam is
transmitted with various spot sizes (measured in diameter) on PSD plane,
and (b) Gaussian, ℓ = +1, ℓ = +3 beams are transmitted one at a time with
beam diameter of 1.5 mm on PSD plane. ............................................................. 54
Figure 4.4 Measurement of beam centroid on the SLM-3 plane when displacement
range is (a) ±5 mm, without tracking; (b) ±5 mm, with Gaussian-beacon-
based tracking; (c) ±5 mm, with OAM-beams-based tracking when ℓ = +1 is
transmitted with a spot size of 1.2 mm on PSD [18]. ........................................... 55
Figure 4.5 Received signal power and (b) crosstalk from other three channels for
receiving OAM ℓ = +3 over 60 seconds. Displacement range is ±5 mm and
spot size is 1.2 mm [18]. ....................................................................................... 56
Figure 4.6 Received signal power and crosstalk for all four channels when OAM ℓ
= ±3 and ±1 are transmitted (a) without displacement, and (b) with
displacement up to ±5 mm and OAM-beams-based tracking. ............................ 57
Figure 4.7 Measurement of bit-error-rates as a function of optical signal-to-noise
ratio (OSNR) when ℓ = ±3 and ±1 are transmitted: (a) for all four channels;
(b) for ℓ = +3 with Gaussian-beacon-based and OAM-beams-based tracking;
(c) for ℓ = +3 with various displacement levels; and (d) for ℓ = +3 with
various beam sizes on the PSD. In (a), (b) and (c), beam diameter on PSD is
1.2 mm; In (a), (b) and (d), displacement is up to ±5 mm. FEC: forward
error correction [18]. ............................................................................................. 59
x

Figure 5.1 Concept of an OAM-multiplexed free-space optical communication link
using a pair of transmitter lenses. Tx: transmitter; Rx: receiver; f 0: equivalent
focal length; d: center-to-center spacing between the two transmitter lenses;
Δ: spacing offset between two transmitter lenses. ................................................ 64
Figure 5.2 (a) Intensity profiles of OAM beams with limited-size apertures, with
and without transmitter lenses at the receiver. (b) Simulated power loss as a
function of transmission distance of different orders of OAM beams; (c) and
(d): Simulated power loss as a function of spacing offset between two
transmitter lenses in 1 km and 10 km link, respectively. In (b), both
transmitted beam size and aperture size are 10 cm, and the equivalent focal
length of transmitter lenses is 1 km. In (c) and (d): both transmitted beam
sizes and aperture sizes are 10 cm in (c), and 30 cm in (d); focal lengths of
transmitter lenses are 0.5 m in (c) and 1 m in (d). Tx: transmitter; f 0:
equivalent focal length of transmitter lenses; z: transmission distance. ............... 66
Figure 5.3 Alignment between transmitter and receiver as well as received OAM
spectrum in: (a) Perfectly aligned link; (b) Link with only angular error; (c)
Link with only displacement (Tx: Transmitter; Rx: Receiver); (d) and (e): At
the receiver, OAM beams with and without transmitter lenses under angular
error and displacement, respectively. ................................................................... 67
Figure 5.4 Simulated OAM power distribution of 1 km free-space optical link: (a)
With angular error and transmitter lenses; (b) With angular error but without
transmitter lenses; (c) With displacement and transmitter lenses; and (d)
With displacement but without transmitter lenses. Transmitted beam size is
10 cm, and only OAM+3 beam is transmitted. Rx OAM = ℓ: power coupled
into the receiver for mode OAM ℓ. ....................................................................... 68
Figure 5.5 Experimental setup for a 1 m OAM-multiplexed free-space optical link
with transmitter lenses. Col: Collimator; SLM: spatial light modulator; BS:
beam splitter; Tx: transmitter; Rx: receiver. ......................................................... 70
Figure 5.6 (a) and (b) Simulated and experimental power loss as a function of
receiver aperture size when only OAM +3 or only OAM +7 is transmitted
with perfect alignment; (c) and (d) Experimental results of OAM power
distribution when only OAM +3 is transmitted under angular errors with and
without transmitter lenses, respectively; (e) and (f) Experimental results of
OAM power distribution when only OAM +3 is transmitted under
displacement with and without transmitter lenses, respectively. .......................... 71
Figure 5.7 (a) and (b): Experimental results of crosstalk for channel OAM +3
when OAM ±1 and ±3 are transmitted under angular error and displacement,
respectively; (c) and (d): Experimental measurement of bit-error-rate as a
function of optical signal-to-noise ratio (OSNR) for channel OAM +3 when
OAM ±1 and ±3 are transmitted under angular error and displacement,
xi

respectively. w/: with transmitter lenses; w/o: without transmitter lenses; x
µrad: with x µrad angular error; y mm: with y mm displacement. ...................... 73
Figure 6.1 Normalized intensity and phase profile of Gaussian, LG 0,5, LG 1,5, and
LG 3,5 beams. .......................................................................................................... 80
Figure 6.2 Concept of free-space optical link transmitting LG beams with non-zero
radial indices. Tx: transmitter; Rx: receiver. ........................................................ 81
Figure 6.3 Simulated power loss of LG beams with radial indices of p = 0 and p >
0 as functions of transmission distance, with transmitted beam size of 8 cm
and receiver aperture diameter of (a) 4 cm, (b) 6 cm, (c) 8 cm, and (d) 10
cm. ........................................................................................................................ 83
Figure 6.4 Simulated power loss of LG beams with radial indices of p = 0 and p >
0 as functions of receiver aperture diameter, with transmitted beam size of 8
cm and transmission distance of (a) 500 m and (b) 1 km. .................................... 84
Figure 6.5 Simulated power loss of LG beams with different radial indices as
functions of (a) transmission distance, with receiver aperture diameter of 4
cm, and (b) receiver aperture diameter, with transmission distance of 500 m.
In both cases, the transmitted beam size is 8 cm. ................................................. 85
Figure 6.6 Experimental setup for a ~1-m OAM-multiplexed free-space optical
link transmitting OAM beams with non-zero radial indices. Col.: collimator;
SLM: spatial light modulator; BS: beam splitter; EDFA: Erbium-doped fiber
amplifier; PC: polarization controller; f 1, f 2: lenses. ............................................. 86
Figure 6.7 Experimental results of (a) measured intensity profile and
interferrogram of LG 1,1 and LG 1,3 beams, and (b) measured power loss due to
limited-size receiver as a function of aperture diameter, with all transmitted
beam sizes of 2.0 mm in diameter. ....................................................................... 87
Figure 6.8 (a) and (b) Experimental results of OAM power distribution when only
LG 0,3 or LG 1,3 beam is transmitted with a receiver aperture of 3 mm,
respectively. (c) and (d) Experimental results of received power on designed
and neighboring modes with different receiver aperture sizes when only
LG 0,3 beam or LG 1,3 beam is transmitted. Tx: LG p,ℓ: LG p,ℓ beam is
transmitted; Rx: LG p,ℓ: received power on LG p,ℓ mode. ...................................... 88
Figure 6.9 Experimental measurement of (a) power distribution of all four OAM
channels (i.e., LG 1,-3, LG 1,-1, LG 1,1, and LG 1,3), and (b) bit-error-rates as
functions of OSNR of the back-to-back case as well as all different channels
when all four OAM channels are transmitted simultaneously. OSNR: optical
signal-to-noise ratio; b2b: back-to-back. .............................................................. 90
xii

Figure 7.1 Concept of a free-space optical (FSO) communication link using both
orbital-angular-momentum (OAM) mode and space diversity for increased
system tolerance to atmospheric turbulence. Rx: receiver; Tx: transmitter. ........ 96
Figure 7.2 Experimental setup. BS: beamsplitter; EDFA: erbium-doped fiber
amplifier; LD: laser diode; QPSK: quadrature phase-shift keying; Rx:
receiver; S: center-to-center aperture spacing; SLM: spatial light modulator;
Tx: transmitter. ..................................................................................................... 97
Figure 7.3 Experimental results on (a) received OAM spectrum of transmitted
Gaussian beam when only one or both aperture(s) are transmitted. Each
color bar represents the fluctuation range; (b) Received power distribution
and (c) link outage percentage as a function of required received power with
no diversity, only mode diversity, only space diversity, and both mode and
space diversity. All measurements are taken under 50 random turbulence
realizations by rotating the turbulence phase plate. For (b) and (c), total
transmit power is 10 dBm. .................................................................................... 99
Figure 7.4 Experimental results on (a) link outage percentage when mode diversity
is not used, only at the transmitter, only at the receiver, and both, with a total
transmit power of 10 dBm; Link operating percentage as a function of total
transmit power with (b) different OAM modes and (c) different aperture
spacing when mode and space diversity is implemented. For (b) and (c), the
required received power is -40 dBm. .................................................................. 100
Figure 7.5 (a) Experimental bit-error-rates (BERs) of a 100-Gbit/s QPSK link with
and without using mode and space diversity under 50 random turbulence
realizations. Total transmit power is 10 dBm. (b) Experimental BER as a
function of transmitted power for different diversity schemes under a
random turbulence realization. (c) Simulation results for required
transmitted when different numbers of apertures are implemented with a
system reliability requirement of 99% under various turbulence strengths. In
the simulation, transmitted beam diameter D is 3 mm and the required
received power is -40 dBm. FEC: forward error correction. .............................. 101
xiii

Abstract
FSO communication links can potentially benefit from the simultaneous
transmission of multiple independent data-carrying beams, which is known as space-
division-multiplexing (SDM). Mode-division-multiplexing (MDM) is a subset of
SDM, where each of the multiple beams is a unique mode from an orthogonal modal
basis set. Orthogonality minimizes crosstalk among the modes and enables efficient
multiplexing at the transmitter, co-propagation of overlapping beams, and low-
crosstalk demultiplexing at the receiver. One possibility of MDM is to use orbital-
angular-momentum (OAM) modes that are circularly symmetric.
OAM modes are characterized by a phase front having an angular dependence
of the form exp(iℓφ), where φ is the azimuthal angle and ℓ is the OAM order and
counts the number of 2π phase shifts in the azimuthal direction. Beams with different
OAM orders are mutually orthogonal, thus each beam can carry an independent data
stream, and the total information capacity of the spatially overlapped beams equals the
data rate of one beam multiplied by the total number of independent beams.
Due to the unique phase and intensity structures of OAM beams, OAM-based
FSO communication systems may present unique challenges, including: beam
divergence, link alignment, and atmospheric turbulence effects. My dissertation will
introduce my research on these limitations and challenges, as well as techniques to
improve the system performance, including: high-capacity OAM-multiplexed FSO
link between a ground transmitter and a ground receiver via a hovering retroreflecting
unmanned-aerial-vehicle (UAV); turbulence effect mitigation using multiple-input-
multiple-output (MIMO) equalization; beaconless beam displacement tracking and
correction in OAM-multiplexed FSO links; power loss mitigation in OAM-
multiplexed FSO links using transmitter lenses or using LG beams with nonzero radial
indices; and turbulence effect mitigation using mode and space diversity.
1

Chapter 1 Introduction
This chapter will introduce the background and basic concept of orbital angular
momentum (OAM), followed by the key challenges of using OAM in free-space
optical (FSO) communication systems.
1.1 Background of Orbital Angular Momentum
FSO communication links can potentially benefit from the simultaneous
transmission of multiple independent data-carrying beams, which is known as space-
division-multiplexing (SDM) [1,2]. Mode-division-multiplexing (MDM) is a subset
of SDM, where each of the multiple beams is a unique mode from an orthogonal modal
basis set [3,4]. Orthogonality minimizes crosstalk among the modes and enables
efficient multiplexing at the transmitter, co-propagation of overlapping beams, and
low-crosstalk demultiplexing at the receiver [5-8]. Although different orthogonal
modal basis sets could be used, one possibility is to use orbital-angular-momentum
(OAM) modes that are conveniently circularly symmetric [9].
OAM modes are characterized by a phase front having an angular dependence
of the form exp(iℓφ), where φ is the azimuthal angle and ℓ is the OAM order and
counts the number of 2π phase shifts in the azimuthal direction [10]. ℓ is an integer
which can assume a positive, negative, or zero value corresponding to a clockwise
phase helicity, counter-clockwise phase helicity, or no helicity (i.e., a conventional
Gaussian beam), respectively [11]. These helical beams “twist” as they propagate. The
intensity of an OAM beam with non-zero order (e.g., Laguerre-Gaussian (LG) beam
with non-zero azimuthal order and zero radial order) is circularly symmetric and has
a ring shape with little power in the center, as shown in Figure 1.1 (a) [12]. Beams
with different OAM values are mutually orthogonal to one another, thus each of these
2

beams can carry an independent data stream, and the total information capacity of the
spatially overlapped beams equals the data rate of one beam multiplied by the total
number of independent beams, as shown in Figure 1.1(b) [13].

Figure 1.1 (a) The intensity and wavefront profiles of a Gaussian beam and OAM beams ℓ = +1 and ℓ
= +2; (b) Multiple data channels, each carried by a different OAM beam, can be multiplexed together
to produce a high capacity free-space data transmission [13].

Among many OAM-carrying helically phased light beams, LG beams are a
special subset. An LG beam has two indices, ℓ and p, in which the azimuthal index ℓ
represents the beam’s OAM order, and p refers to its radial structure in the intensity
distribution, as shown in Figure 1.2. An LG beam carrying OAM has two important
characteristics: (1) a beam with ℓ ≠ 0 and p = 0 has a “doughnut” shape intensity profile
with a single ring annulus and little power near the beam center; (2) a beam with ℓ ≠ 0
and p > 0 has a multi-ringed intensity profile of p + 1 intensity rings. LG beams form
a complete and orthogonal mode basis in the spatial domain, and are the paraxial eigen
solutions of the wave equation in the cylindrical coordinates of homogeneous media,
e.g., free space. In this dissertation, we refer a general “OAM beam” as all helically
phased beams only defined by its azimuthal index ℓ and regardless of its radial index
3

p. Therefore, such a beam could be expanded into a group of LG beams, each with the
same ℓ but a different p index.

Figure 1.2 Normalized intensity and phase profile of Gaussian, LG0,5, LG1,5, and LG3,5 beams.

To generate an OAM beam, we can pass a fundamental Gaussian beam through
a spiral phase plate that has multiple 2π phase change in the azimuthal direction, such
that the phase change can be impinged onto the incoming beam and convert it into a
corresponding OAM beam. To detect an OAM beam, we can pass the OAM beam
through an inverse spiral phase plate, such that the helical phase change could be
canceled, and the beam will be converted back into a Gaussian-like beam, as shown in
Figure 1.3. These spiral phase plate could be spatial light modulators (SLMs), wave
plates, integrated devices, etc. Recently, there are reports on other methods for OAM
generation and detection, such as multi-plane conversion.

4

Figure 1.3 OAM beam generation and detection.

1.2 Challenges for OAM-Based FSO Systems
Due to the unique phase and intensity structures of OAM beams, OAM-based
FSO communication systems may present unique challenges, including: beam
divergence, link alignment, and turbulence effects. My research focuses on investigate
these limitations and challenges, and explore techniques to improve the system
performance under various conditions.

Incoming
fundamental
Gaussian beam
Incoming OAM
beam
Spatially de-multiplexed
Converted into
an OAM beam
Holographic
phase filter
Integrated devices, wave
plates, spatial light
modulator (SLM) …
Converted into a
fundamental
Gaussian beam
Generation
Detection
5

Figure 1.4 Divergence of regular Gaussian and OAM-carrying LG beams.

One challenge for an OAM-multiplexed FSO communication system is OAM
beam divergence. In general, the receiver should capture the complete phase change
of each of the different OAM beams in order to maintain orthogonality among the
beams. Given that the most phase change occurs near the beam center in the azimuthal
direction, the receiver would typically need to be placed on axis [14]. Unfortunately,
OAM beams have a fairly large area near the center for which the signal power is low.
Therefore, although orthogonality is maintained, the recovered signal power can
become fairly low when the aperture size is smaller than the beam’s annulus. This
limited-receiver-aperture scenario is common since an optical beam diverges with
propagation, and divergence increases with higher OAM values, as shown in Figure
1.4 [15].

!
"
ℓ +%
!
&
ℓ +%
Transmitted
beam radius
Received beam
radius
Transmission distance z
!
"
ℓ +'(+%
!
&
ℓ +'(+%
Transmitted
beam radius
Received beam
radius
Transmission distance z
!
"
!
&
=!
"
%+
&*
+!
"
'
'
Transmitted
beam radius Received beam radius
Transmission distance z
Gaussian beam divergence
LG
ℓ,0
beam divergence
LG
ℓ,p
beam divergence
6

Figure 1.5 Effects of misalignment on system performance. (a)-(d) Illustration of OAM-based
communications link with perfect alignment, displacement, tip/tilt error at Rx, and tip/tilt error at Tx,
respectively.

Another technical challenge is link alignment. In general, beam tracking is
considered important for single-beam non-OAM FSO links due to the relatively small
beam diameters. This issue is even more pronounced for OAM-multiplexed systems,
since any deviation from coaxial detection of the uniquely-structured beams can
produce both additional power-coupling loss of the desired mode and crosstalk from
one mode to other unwanted modes, as shown in Figures 1.5(e) and 1.5(f). In ideal
cases, the Tx and Rx are perfectly aligned, as shown in Figure 1.5(a), where the
receiver center coincides with the transmitted beam center, and the beam axis is
perpendicular to the receiver plane. In such a scenario, if one OAM beam is
transmitted, power will be received only on the desired mode due to the orthogonality
among OAM beams. However, when there is lateral displacement, where the receiver
center does not overlap with the transmitted beam center, and/or tip/tilt error, where
the beam axis is not perpendicular to the receiver plane, the power on the transmitted
OAM mode will be coupled into its neighboring modes, thereby causing crosstalk
between OAM channels [16].
7

A third technical challenge for even single Gaussian beam systems is
atmospheric turbulence. For an OAM-multiplexed link, such turbulence would
dynamically distort the structured phase front of the beams. Since the power coupling
is mode specific and the orthogonality of multiple co-propagating OAM beams
depends on their unique helical phase-fronts, this distortion would cause received
power fluctuations of the desired mode and increased channel crosstalk from the
unwanted modes [17].

Figure 1.6 Turbulence effects for OAM beams.

This dissertation is organized with the following structure: Chapter 2 presents
a high-capacity OAM-multiplexed FSO link between a ground transmitter and a
ground receiver via a hovering retroreflecting unmanned-aerial-vehicle (UAV).
Chapter 3 presents turbulence effect mitigation for OAM-multiplexed FSO link
between ground station and hovering UAV using multiple-input-multiple-output
(MIMO) equalization. Chapter 4 presents beaconless beam displacement tracking and
correction in OAM-multiplexed FSO links. Chapter 5 and 6 present two the studies in
Atmospheric Turbulence
Transmitted OAM Distorted OAM
Power
!
1
OAM Modes
Power
!
1
OAM Modes
!
4
!
2
!
3
!
5
8

power loss mitigation in OAM-multiplexed FSO links, with one using transmitter
lenses and the other one using LG beams with nonzero radial indices. Chapters 7
presents turbulence mitigation in FSO links using OAM mode and space diversity.
1.3 References
[1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, "Orbital angular
momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys.
Rrv. A 45, 8185 (1992).
[2] D. J. Richardson, J. M. Fini, and L. E. Nelson, "Space-division multiplexing in
optical fibres," Nat. Photonics 7, 354–362 (2013).
[3] S. Berdagué and P. Facq, "Mode division multiplexing in optical fibers," Appl. Opt.
21, 1950–1955 (1982).
[4] P. J. Winzer, "Making spatial multiplexing a reality," Nat. Photonics 8, 345–348
(2014).
[5] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S.
Franke-Arnold, "Free-space information transfer using light beams carrying orbital
angular momentum," Opt. Express, 12, 5448–5456 (2004).
[6] A. M. Yao, and M. J. Padgett, "Orbital angular momentum: origins, behavior and
applications," Adv. Opt. Photonics, 3, 161–204 (2011).
[7] G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys., 3,
305–310 (2007).
[8] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Turbulence-induced channel
crosstalk in an orbital angular momentum-multiplexed free-space optical link," Appl.
Opt. 47, 2414–2429 (2008).
[9] J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S.
Dolinar, M. Tur, and A. E. Willner, "Terabit free-space data transmission employing
orbital angular momentum multiplexing, " Nat. Photonics 6, 488–496 (2012).
[10] H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J.
Willner, B. I Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett,
M. Tur, and A. E. Willner, "100 Tbit/s free-space data link enabled by three-
9

dimensional multiplexing of orbital angular momentum, polarization, and
wavelength," Opt. Lett. 39, 197–200 (2014).
[11] G. A. Tyler, and R. W. Boyd, "Influence of atmospheric turbulence on the
propagation of quantum states of light carrying orbital angular momentum," Opt. Lett.
34, 142–144 (2009).
[12] Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao,
L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, "High-capacity
milimetre-wave communications with orbital angular momentum multiplexing," Nat.
Commun. 5,1–9 (2014).
[13] A. E. Willner, J. Wang, and H. Huang, "A different angle on light
communications." Science 337, 655-656 (2012).
[14] P. Martelli, A. Gatto, P. Boffi, and M. Martinelli, "Free-space optical transmission
with orbital angular momentum division multiplexing," Electron. Lett. 47, 972–973
(2011).
[15] R. L. Phillips, and L. C. Andrews, "Spot size and divergence for Laguerre
Gaussian beams of any order," Appl. Opt. 22, 643–644 (1983).
[16] G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery,
N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner,
"Performance metrics and design considerations for a free-space optical orbital-
angular-momentum multiplexed communication link," Optica 2, 357-365 (2015).
[17] C. Paterson, "Atmospheric turbulence and orbital angular momentum of single
photons for optical communication," Phys. Rev. Lett. 94, 153901 (2005).

10

Chapter 2 High-Capacity Free-Space Optical
Communications between a Ground
Transmitter and Receiver via a UAV
Using Orbital Angular Momentum
Multiplexing
2.1 Introduction
The data communications capacity needs of manned and unmanned aerial
platforms have been increasing dramatically over the past several years, thereby
driving the need for higher-capacity links between these platforms and their ground
stations [1]. One example of an aerial platform is a UAV, such as flying drones that
are proliferating for numerous applications [2]. In addition to the need for high-speed
communications, there is also the desire to minimize the probability of possible
interception of the data exchange in order to achieve enhanced privacy and security
[3]. Due to the higher carrier frequency of the lightwave, FSO communications
generally holds the promise of having both higher capacity and lower probability of
intercept (LPI) than RF and millimetre-wave techniques [4]. Specifically, there have
been several reports of FSO communication links with moving aerial platforms [5].
Importantly, an approach for significantly increasing capacity for fixed ground-
based FSO links has gained interest over the past few years. This technique, known as
SDM, is based on the simultaneous transmission of multiple independent data-carrying
beams. MDM is a subset of SDM, where each of the multiple beams is a unique mode
from an orthogonal modal basis set. Orthogonality minimizes crosstalk among the
11

modes and enables efficient multiplexing at the transmitter, co-propagation of
overlapping beams, and low-crosstalk demultiplexing at the receiver. Although
different orthogonal modal basis sets could be used, one possibility is to use OAM
modes that are conveniently circularly symmetric [6].
Prior art in FSO communications with multiple OAM beams between two fixed
ground stations includes [7-10]: (a) 2.5 Tbit/s over ~1 m using 32 OAM modes; (b)
100 Tbit/s over ~1 m using 1008 OAM modes comprised of 12 OAM modes on each
of 2 polarizations and 42 wavelengths; (c) 80 Gbit/s over 260 m using 2 OAM modes
and 400 Gbit/s over 120 m using 4 modes; (d) <1-kbit/s single-beam transmission of
OAM superposition between two 143-km-spaced islands.
To date, there has been little reported on the use of OAM multiplexing in FSO
communications between a ground station and a moving platform. Importantly, such
a scenario is likely to face a few major challenges arising from the special structured
nature of the OAM beams themselves. Challenge include the following: (a)
Alignment: Low inherent crosstalk and power-coupling loss generally relies on
accurate on-axis detection of the multiple OAM beams, thereby necessitating more
tracking sophistication for an OAM-multiplexed link over a single conventional
Gaussian-based link; (b) Turbulence: Turbulence resulting from the atmosphere or
from a UAV’s propellers could significantly distort the OAM beam’s phase front, thus
resulting in increased received power fluctuations and intermodal crosstalk, as
compared to recovering a single conventional Gaussian beam [11].
In this chapter, we explore the use of OAM multiplexing to increase the capacity
and decrease the probability of intercept of data transmission to moving platforms. We
experimentally investigate the performance of an FSO communication link between a
ground station and a flying UAV transmitting two multiplexed OAM beams up to
~100 m roundtrip distance. For ease of demonstration, our UAV carries a retro-
reflector, but does not carry a transmitter or receiver as in a real ground-to-UAV
12

communication system. Instead, the ground station transmits the beams to the UAV,
whereupon the beams are reflected back to a receiver on the ground station that is co-
located with the transmitter. Each OAM beam carries a 40-Gbit/s QPSK signal,
thereby a total capacity of 200 Gbit/s at a single carrier wavelength of 1550 nm is
achieved. We measure the impact of channel impairments, including tracking errors
and propeller-induced airflow on beam quality and system performance, in terms of
received signal power, intermodal crosstalk among channels, and BERs. We find that:
(a) when the UAV hovers in the air, the power on the desired mode fluctuates by 2.1
dB over a 60-second period, whereas the crosstalk to the other mode is -19 dB below
the power on the desired mode, and the crosstalk fluctuates within a 7.8-dB range; and
(b) when the UAV moves in the air at a speed of ~0.1 m/s, the power fluctuation on
the desired mode increases to 4.3 dB and the crosstalk to the other mode increases to
-10 dB below the power on the desired mode. Furthermore, the channel crosstalk
decreases with an increase in OAM mode spacing, and the number of error-free
transmitted data frames increases when channel OAM mode spacing increases from 2
to 3.

Moreover, we investigated the limited-aperture effects on system performance.
Results indicate that in such an OAM-multiplexed airborne FSO link, limited-size
receiver apertures would lead to both power loss and increased channel crosstalk [12].
2.2 Link Demonstration
Figure 2.1 illustrates our prospective application for using OAM multiplexing
in high-capacity FSO communications between a UAV and a ground station [12]. The
ground station contains an OAM transmitter (Tx), an OAM receiver (Rx), and a beam
tracking system. A retro-reflector carried by the UAV is flown up to ~50 m away (i.e.,
~100 m round trip) from the ground station to efficiently reflect the OAM beams
coming from the transmitter back to the receiver with little distortion. During the
experiment, the octocopter UAV moves and hovers at different locations up to ~20 m
above the ground and up to ~50 m away [12].
13

Figure 2.1 Concept of using OAM multiplexing in high-capacity FSO communications between a UAV
and a ground station [12].

The OAM transmitter optics are shown in Figure 2.2 [12]. We use a custom-
designed OAM (de)multiplexer pair based on multi-plane mode conversion for: (a)
generating and multiplexing multiple OAM beams at the Tx, and (b) demultiplexing
and receiving them at the Rx. The OAM multiplexer has seven SMF pig-tailed input
ports, which are connected to a fiber array followed by a microlens array, such that
seven collimated Gaussian beams could be generated and propagate in free space.
These Gaussian beams are then sent to a multipass cavity where the beams are
reflected 15 times at different locations on a reflective phase plate. In each reflection,
the wavefront of the beams are shaped by different transverse phase profiles. The
succession of these transverse phase profiles forms a spatial unitary transform that
14

converts the seven Gaussian beams into seven OAM beams of ℓ = −3, −2, −1, 0, +1,
+2, +3. By using the multiplexer reversely, multiple OAM beams could be transformed
back to multiple Gaussian beams which are then sorted to corresponding SMF outputs.
In a back-to-back measurement of an OAM (de)multiplexer pair, the highest power-
coupling loss for a desired mode is ~11.8 dB, and the highest crosstalk is ~-19.1 dB
from a mode to its nearest mode (i.e., mode spacing of 1), ~-23.9 dB from a mode to
its second nearest mode (i.e., mode spacing of 2), and ~-26.1 dB from a mode to its
third nearest mode (i.e., mode spacing of 3). Such crosstalk level could be sufficient
to enable error-less high speed communications [12].

Figure 2.2 Experimental setup of of an FSO communication link between a UAV and a ground station
using OAM multiplexing [12].

A 50-Gbaud (100-Gbit/s) QPSK data signal at 1550 nm is amplified by an
Erbium-doped fibre amplifier (EDFA) and split into two beams, one of which is
delayed using a ~10-m SMF to decorrelate the data sequence. These two beams are
sent to two of the seven input ports of the OAM multiplexer, generating two
FSM
Ground station Transmitted
beam
Reflected
beam
Fiber
connection
Feedback
control
50/50
coupler
Ground
gimbal
Mirror
PSD
OAM
Demux
Coherent
detection
Mirror
Controller 1530 nm
filter
Lens
50/50 BS
90/10 BS
Iris
UA V
Gimbal
Retro-
reflector
Mirror
Probe beam
at 1530 nm
EDFA
100 Gbit/s
QPSK
signal
generator
at 1550 nm
EDFA
OAM
Mux
15

multiplexed OAM beams. Simultaneously, a 1530-nm probe beam used for tracking
is sent to the ℓ = 0 input of the OAM multiplexer. These co-axially propagating beams
are transmitted through a 1:10 beam expander to enlarge the beam sizes, after which
they propagate in free-space to the gimbal-mounted retro-reflector on the UAV. The
diameter of the transmitted beams are ~3 cm for Gaussian, ~4.2 cm for OAM ±1, ~5.2
cm for OAM ±2, and ~6 cm for OAM ±3 beams. The retro-reflector reverses the order
of an OAM beam between +ℓ and -ℓ, and the relative purity of the OAM beam itself
is not significantly altered. At the receiver, the reflected beams pass through an on-
axis tunable iris acting as the receiver aperture and its size could be adjusted. After
beam reduction, the beams are coupled into the OAM demultiplexer and received by
coherent detection [12].
In general, beam tracking is considered important for single-beam non-OAM
FSO links due to the relatively small beam diameters. This issue is even more
pronounced for OAM-multiplexed systems, since any deviation from coaxial detection
of the uniquely-structured beams can produce both additional power-coupling loss of
the desired mode and crosstalk from one mode to other unwanted modes. In ideal cases,
the Tx and Rx are perfectly aligned. However, in a dynamic ground-to-UAV FSO link,
the Tx and Rx may have residual tracking accuracy limitations. Such limitations that
may result in residual tracking errors could lead to various misalignment problems,
including lateral displacement and tip/tilt error at the Tx and Rx. As only one example
in our experiment, the retro-reflector itself can produce two misalignment issues: (a)
It reflects the beams back in their original direction within an error of <1 arcsecond
due to fabrication imperfections, which produces a ~0.2 mm offset and is a relatively
minor issue at a ~100-m-roundtrip distance; and (b) Unless the beams are exactly in
the center of the retro-reflector (which was not perfectly the case in our system), there
is a natural lateral displacement such that the Rx centre does not completely overlap
with the reflected beams’ center [12].
16

Figure 2.3 Received power on different OAM modes under horizontal, vertical, and simulated
displacement when OAM -1 is transmitted. The roundtrip transmission distance is ~100 m and the
tracking system is off. The displacement refers to the distance between the OAM beam center and the
center of the receiver. Tx:-1, Rx:+3: received power on OAM +3 mode when OAM -1 beam is
transmitted [12].

In order to determine the sensitivity of our system to misalignment, we transmit
one mode and measure the received power on different OAM modes for various
displacements between the Rx centre and the axis of the received beam. For these
initial results: (a) the UAV and the tracking system are turned off, (b) the retro-
reflector on the UAV is placed on the ground ~50 m away from the transceiver, and
(c) the lateral deviation of the beam away from the centre of the retro-reflector that
produces a lateral displacement between the centre of the receiver and the axis of the
received beam is varied by manually changing the beam location. Figures 2.3(a) and
2.3(b) show the measured received power on different OAM modes under various
horizontal and vertical displacements. Furthermore, simulation results for our system
are shown in Figure 2.3(c), which show similar trends of the dependence of the
received power on displacement. However, the magnitude of the crosstalk to the wrong
mode is experimentally higher partially due to the non-ideal performance of the OAM
(de)multiplexer. Moreover, our system is more tolerant to vertical as compared to
horizontal displacement, which is likely due to the fact that the OAM demultiplexer
has an OAM-dependent transfer function along the horizontal direction. Our
0 2 4 6 8
Displacement (mm)
-60
-50
-40
-30
-20
-10
0
Received power (dBm)
Tx:-1, Rx:+3
Tx:-1, Rx:+1
Tx:-1, Rx:-1
Tx:-1, Rx:-3
Tx:0, Rx: 0
0 2 4 6 8
Displacement (mm)
-60
-50
-40
-30
-20
-10
0
Received power (dBm)
Tx:-1, Rx:+3
Tx:-1, Rx:+1
Tx:-1, Rx:-1
Tx:-1, Rx:-3
Tx:0, Rx: 0
0 2 4 6 8
Displacement (mm)
-60
-50
-40
-30
-20
-10
0
Received power (dBm)
Tx:-1, Rx:+3
Tx:-1, Rx:+1
Tx:-1, Rx:-1
Tx:-1, Rx:-3
Tx:0, Rx: 0
Simulation Experiment
Experiment
(a) (b) (c)
17

measurement indicates that a horizontal displacement of >3 mm would lead to a
power-coupling loss of >1.5 dB for a desired mode and a crosstalk of >-13 dB between
two OAM modes with a mode spacing of 4, when the received beam diameter is ~4.3
cm; this crosstalk might lead to high BERs for >40-Gbit/s QPSK signal transmissions;
we note that such a displacement (i.e., 30-μrad tracking error at 100 m) is readily
mitigated by tracking systems. Finally, Figure 2.3 also shows that smaller OAM
spacing would lead to higher crosstalk under similar displacement conditions.
Misalignment arising from mechanical UAV vibrations, should also be mitigated by
the tracking system [12].
Importantly, the tight alignment tolerance for low crosstalk and low power loss
in an OAM-multiplexed link can be viewed as a potential benefit of increasing the
difficulty of eavesdropping (i.e., LPI) by any off-axis receiver. As shown in Figure
2.3(a) when comparing a single Gaussian beam to 2 multiplexed OAM beams, a
displacement of 4 mm of an off-axis eavesdropper when transmitting OAM -1 and -3
simultaneously would result in: (a) an increased power loss of ~2.5 dB, and (b) a
crosstalk level between -1 and -3 modes is ~ -4.2 dB, whereas crosstalk is not an issue
for eavesdropping a Gaussian beam link [12].
Another technical challenge for even single Gaussian beam systems is
atmospheric turbulence. For an OAM-multiplexed link, such turbulence would
dynamically distort the structured phase front of the beams. Since the power coupling
is mode specific and the orthogonality of multiple co-propagating OAM beams
depends on their unique helical phase-fronts, this distortion would cause received
power fluctuations of the desired mode and increased channel crosstalk from the
unwanted modes. Additionally, the airflow induced by the UAV propellers may distort
the OAM beam’s phase front and result in increased received power fluctuation and
channel crosstalk. To estimate this effect, we measure the Rytov variance (σ
2
) using a
transmitted 1550-nm Gaussian probe beam and a ~1-mm diameter point detector over
a 10-minute period. In order to separate mechanical vibrations from airflow turbulence,
18

the retro-reflector is detached from and placed underneath the UAV. The retro-
reflector is fixed ~50 m away from the transceiver. The ~5-cm diameter probe beam
propagates from the transmitter to the retro-reflector and back to the receiver. We
compare the measured power distributions when the UAV’s propellers are off or on as
shown in Figures 2.4(a) and 2.4(b), respectively. By fitting the power distribution into
a lognormal distribution, we find that the σ
2
value is ~0.0028 and the refractive-index
structure parameter (Cn
2
) is ~9.4×10
-15
m
-2/3
when the propeller is turned off. Due to
the increased airflow when the UAV propellers are turned on, σ
2
increases to ~0.0080
[12].
In our communication system, the aperture is large enough to capture the entire
beam and the received power may benefit from the aperture averaging effect. To verify
this, we measure the received OAM spectrum when OAM +1 beam is transmitted for
the scenarios of the propellers off and on, as shown in Figure 2.4(c). We observe that
the overall impact of the propellers on the received OAM spectrum does not appear to
be significant in this measurement [12].

Figure 2.4 Effects of propeller-induced airflow on system performance. (a) and (b) Measured power
distribution when the UAV’s propellers are turned off or on, respectively. The power is normalized to
its mean during the measurement. (c) Measured OAM spectrum when the UAV’s propellers are turned
off and on, when OAM +1 is transmitted over ~100 m roundtrip [12].

19

The system measurements for the OAM-multiplexed FSO link between the
flying (i.e., hovering or moving) UAV and the ground transceiver are performed under
clear weather conditions in the daytime. The wind varies but is typically ~8 km/h from
west to east, and the UAV is located ~50 m northwest of the ground transceiver [12].
In order to evaluate the effects of beam jitter caused by various issues (including
those outlined in the previous section), we measure the statistics of the received beam
centroid before beam reduction. Figures 2.5(a)-2.5(c) show the distributions of relative
position of beam centroid when the UAV is static on the ground with tracking system
off, hovers in the air ~10 m above ground with tracking system on, moves horizontally
in the air at a speed of ~0.1 m/s with tracking system on, respectively. OAM +3 beam
is transmitted during the measurements. The statistics of each scenario is obtained by
continuously capture 1000 intensity profiles of the beam over a 120-second period
using an infrared camera. The beam jitter variance is ~0.0218 mm
2
when the UAV is
grounded, and it increases to ~0.0877 mm
2
and ~0.4604 mm
2
when the UAV is
hovering and moving, respectively. We believe that these increases are caused by the
accuracy and speed limitations of our self-made tracking system, which can be
significantly improved by advanced tracking systems [12].
Figure 2.5(d) shows different UAV positions, and Figures 2.5(e)-2.5(i) show the
coupled signal power and modal crosstalk under hovering and moving conditions.
Figure 5e shows the received power on different modes during a continuous 60-second
period when OAM +1 beam is transmitted and the UAV is hovering ~10 m above
ground and ~50 m away from the ground station, as shown in location 1 in Figure
2.5(d). The power on the desired mode (i.e., OAM +1) fluctuates within a 2.1-dB range.
The crosstalk on the OAM +3 mode is <-19 dB below the power coupled into the
desired mode, and the crosstalk fluctuates within a 7.8-dB range. Figures 2.5(f)-2.5(i)
show the measured OAM spectrum under different flight conditions when OAM +1 is
transmitted, and the shaded portion of each bar shows the power fluctuation range. We
observe that both the power on the desired mode and the crosstalk into other modes
20

experience more fluctuations when the UAV is moving than when it is hovering, which
agrees with our beam jitter measurements of Figs. 2.5(a)-2.5(c) [12].

Figure 2.5 Beam jitter, power, and crosstalk measurements in flight environment. (a)-(c) Measured
statistics of beam displacement with respect to the receiver center when the UAV is static on the ground,
hovering, and moving at a speed of ~0.1 m/s. OAM +3 is transmitted and the received beam diameter
is ~6.2 cm. (d) Schematic diagram of different locations where the UAV hovers. Location 1 is ~10 m
above ground, ~50 m away from the transceiver with an angular position of 0°; location 2 is ~20 m
above ground, ~40 m away from the transceiver with an angular position of 15°; location 3 is ~5 m
21

above ground, ~10 m away from the transceiver with an angular position of -5°. (e) The received power
on different modes in a 60-second period, when OAM -1 is transmitted and the UAV is hovering. (f)-
(i) OAM spectrum in a 60-second period when OAM +1 is transmitted and the UAV is hovering at
different locations or moving at a speed of ~0.1 m/s, respectively [12].

To verify link performance under flying conditions, Figure 2.6 shows
measurements of the BER for each 4096-symbol data frame. We note that these are
raw BERs and no error correction codes have been used. OAM +3 and OAM -1 beams
are multiplexed and transmitted simultaneously, each carrying a 40-Gbit/s QPSK
signal. Figure 2.6(a) shows a 60-second time sequence of BER measurements of the
OAM +3 channel when the UAV is hovering in location 1 of Figure 2.5(d). The
transmitted power of each channel is fixed at 10 dBm. We observe that the BERs
fluctuate, mostly being below (or even error free) but sometimes above the 7%
forward-error correction (FEC) limit, we again note that a better tracking system
would significantly improve link performance [12].
Figure 2.6(b) shows BER measurements as a function of transmitted channel
power, such that there is a range of measurement values at each power level due to the
non-ideal beam tracking. For each transmitted channel power level, 10 BER
measurements (i.e., 10 data frames) are taking within a ~30-second measurement time
period. The value at the bottom of the figure represents the number of error-free frames
out of the 10 measured frames for each power level [12].
Since crosstalk tends to be higher for neighbouring modes than for those far
away, larger channel mode spacing could produce lower crosstalk and potentially
better link performance. Figure 2.6(c) shows the BER measurements for different
transmitted channel mode spacing when the UAV is hovering at location 1 of Fig.
2.5(d). The results indicate that the performance is better for mode spacing of 3 than
for 2. As an example, the number of error-free frames increases from 4 to 7 when mode
22

spacing increases from 2 to 3 when transmitted power is 7 dBm. Figure 2.6(d) shows
the BER for OAM +3 when OAM +3 and -1 are transmitted for different link distances
and when the UAV is hovering. A slight decrease on the number of error-free frames
is observed when the transmission distance increases from 40 m to 100 m, which may
indicate a minor BER performance degradation [12].

Figure 2.6 Bit-error-rate (BER) measurements in flight environment. Each channel transmits a 40-Gbit/s
quadrature phase shift keying (QPSK) signal. (a) BERs in a 60-second period and (b) BERs
measurements for OAM +3 and -1, when OAM +3 and -1 beams are transmitted. At each transmitted
power level, 10 data frames (each has 4096 bits) over ~30-second period are measured. The number at
the bottom represents the number of error-free frames during the measurement period at a certain
transmitted power level. (c) BER measurements when two OAM beams with different mode spacing
are transmitted. The UAV is hovering ~10 m above the ground and ~50 m away from the ground station.
(d) BER measurements for OAM +3 when OAM +3 and -1 are transmitted. The UAV is hovering at
different distances away from the ground station. FEC: forward-error-correction [12].
23

2.3 Limited Aperture Effects
When an OAM-multiplexed link is well aligned, i.e., the center of the receiver
overlaps with the center of the received beams and the receiver is perpendicular to the
beam axis, limited-size receiver apertures would cause power loss on the desired mode
but little crosstalk to other modes, as shown in Figures 2.7(a) and 2.7(b) [13]. When
there is link misalignment, power on OAM modes would be coupled into their
neighboring modes, thereby causing intermodal crosstalk, as shown in Figure 2.7(c)
[13]. This crosstalk might increase when the receiver aperture is limited, as shown in
Figure 2.7(d) [13]. Generally in an airborne FSO link, misalignment (e.g., beam jitter)
would still exist even when a beam tracking system is used. Therefore, the limited-
size aperture effects may result in increased system degradation for airborne OAM-
multiplexed FSO data links as compared to a fixed data link.
In this section, we experimentally explore the limited-size receiver aperture
effects in an OAM-multiplexed FSO link between a ground station and a retro-
reflecting UAV [13]. Two 100-Gbit/s OAM beams are transmitted from a ground
station, propagated to and retro-reflected from a UAV ~50-m away, and detected by
the receiver co-located on the ground station, achieving a total capacity of 200-Gbit/s.
Data is measured when the UAV is on the ground, hovering in the air, and flying at a
24

maximum speed of 0.1 m/s. Measured BERs are mostly below 3.8×10
-3
when receiver
aperture diameter is >5 cm, with a maximum received beam diameter of ~6.2 cm.
Results indicate that in such an OAM-multiplexed airborne FSO link, limited-size
receiver apertures would lead to both power loss and increased channel crosstalk.

Figure 2.7 Concept of limited-size aperture effects on an OAM-based link with: (a) full receiver
aperture and perfect alignment; (b) limited-size receiver aperture and perfect alignment; (c) full receiver
aperture and misalignment; (d) limited-size receiver aperture and misalignment.

OAM order
Power
Tx
Rx Full aperture
Aligned
(a)
OAM
spectrum

OAM order
Power
Tx
Rx
Limited-size
aperture
Aligned
(b)
OAM
spectrum

OAM order
Power
Tx
Rx Full aperture
Misaligned
(c)
OAM
spectrum

OAM order
Power
Tx Rx
Misaligned
(d)
OAM
spectrum
Power loss
Crosstalk
Limited-size
aperture
Higher
crosstalk
25

Figure 2.8 Measured received power on different OAM modes when only OAM +3 is transmitted with
10-dBm power, (a) without and (b) with intentionally-added displacement, respectively. (c) Crosstalk
for OAM +3 when OAM +3 and -1 are transmitted under various intentionally-added displacement. (d)
Measured crosstalk for OAM+3 when various modes are transmitted under intentionally-added 1.4-mm
displacement. The UAV is static on the ground ~50-m away from the ground station, and the tracking
system is off. Rx, receiver; Tx, transmitter. (e) and (f) Simulated received power on different modes as
3 4 5 6 7
Aperture diameter (cm)
-60
-50
-40
-30
-20
Crosstalk (dB)
Rx:OAM+3
Rx:OAM+1
Gaussian
Rx:OAM-1
Rx:OAM-3
No displacement added
19.4 dB
17.0 dB
(a)
Tx: OAM+3
3 4 5 6 7
Aperture diameter (cm)
-25
-20
-15
-10
-5
0
Crosstalk (dB)
Aligned
1.4mm displacement
2.8mm displacement
4.2mm displacement
(c)
Tx: OAM+3/-1
3 4 5 6 7
Aperture diameter (cm)
-25
-20
-15
-10
-5
0
Crosstalk (dB)
Tx:+3/-2,Rx:+3
Tx:+3/-1,Rx:+3
Tx:+3/0,Rx:+3
Tx:+3/+1,Rx:+3
1.4mm
displacement
(d)
3 4 5 6 7
Aperture diameter (cm)
-60
-50
-40
-30
-20
Crosstalk (dB)
Rx:OAM+3
Rx:OAM+1
Gaussian
Rx:OAM-1
Rx:OAM-3
(b)
1.4 mm displacement added
Tx: OAM+3
17.7 dB
11.2 dB
Rx:OAM+3
Power (dBm)
Power (dBm)
(f) (e)
Simulation
Simulation
Tx: OAM+3
Tx: OAM+3
26

functions of lateral displacement when OAM +3 is transmitted, with a receiver aperture diameter of 3
cm and 7 cm, respectively.

Figures 2.8(a) and 2.8(b) show the measured power and crosstalk when the UAV
is static on the ground, 50-m away from the ground station. The tracking system is
turned off. We transmit one OAM mode (OAM +3, with 10-dBm transmitted power)
and measure the received power on different modes for different aperture sizes. The
results without and with intentionally-added displacement between the received beams’
center and the receiver aperture’s center are shown in Figures 2.8(a) and 2.8(b). We
observe that when the receiver aperture size decreases, power on the desired mode (i.e.,
OAM +3) decreases, while power on the other modes remains almost unchanged,
indicating increasing crosstalk.
Figure 2.8(c) shows the intermodal crosstalk for OAM +3 when both OAM +3
and OAM -1 beams are transmitted, under various intentionally-added displacement.
Here crosstalk from OAM -1 to OAM +3 is the power coupled from OAM -1 to OAM
+3, over the power received from OAM +3 to OAM +3. We observe that crosstalk
increases as the aperture size decreases, and this effect is more significant under larger
displacement, which agrees with the results shown in Figures 2.8(a) and 2.8(b). Figure
28(d) shows the crosstalk for OAM +3 when two OAM modes with different mode
spacings (i.e., order difference between two OAM modes) are transmitted. Results
show that crosstalk for a mode spacing of 3, 4, and 5 are similar, and are all lower than
27

for mode spacing of 2. Figures 2.8(e) and 2.8(f) show the simulation results of received
power on different modes as functions of lateral displacement when OAM +3 is
transmitted, with a receiver aperture diameter of 3 cm and 7 cm, respectively. Results
indicate that crosstalk level would be higher when receiver aperture size is smaller.
Figure 2.9(a) shows the measured OAM spectrum when OAM -1 is transmitted,
and the UAV is static on ground, ~100-m away from the ground station. Due to the
beam jittering when the UAV is hovering or moving, the received power on different
OAM modes fluctuates. Figure 2.9(b) measures the received power on different OAM
modes during a continuous 60-second period when only OAM -1 beam is transmitted
and the UAV is hovering ~10 m above ground and ~50 m away from the ground station.
The power on the desired mode (i.e., OAM -1) fluctuates within a 3.8-dB range. The
crosstalk on the OAM +3 mode is <−17 dB below the power coupled into the desired
mode, and the crosstalk fluctuates within a 9.4-dB range. In Figures 2.9(a) and 2.9(b),
the receiver diameters are both 6.5 cm. Figure 2.9(b) shows the measured OAM
spectrum under different flight conditions and receiver aperture sizes when OAM -1
is transmitted. The shaded portion of each bar shows the power fluctuation range.
Figure 2.9(c) shows the intermodal crosstalk for OAM +3 when OAM +3 and
OAM -1 are transmitted, and this crosstalk also varies with time. Each bar in Figure
2.9(c) represents the crosstalk fluctuation range. Results show that in most cases, the
crosstalk and its fluctuation range increase when the UAV is moving rather than
hovering, which may be due to the relatively larger beam jittering, as shown in Figure
28

2.5. It is also observed that this effect is more significant when the aperture size
decreases from 6.5 cm to 4.5 cm.
Figure 2.10 shows the BER for each 8192-symbol data frame when both OAM
+3 and OAM -1 are transmitted, each carrying a 100 Gbit/s QPSK signal. We note that
other OAM modes might also be used, such as OAM +3 and OAM -3, or OAM +3
and OAM -2. The UAV is hovering ~ 5-m above the ground, ~50-m away from the
ground station. Figure 2.10(a) shows a 60-s time sequence of BERs for both channels.
For each channel, the transmitted power is fixed at 10 dBm, and the aperture diameter
is 6.5 cm. We observe that the BERs vary with time, mostly below the 7% overhead
forward-error correction (FEC) limit of 3.8×10
-3
. The two measurements in Figure
2.10(a) are conducted in two different test runs due to the limitations of our
measurement system. Therefore, the BER values with the same time label do not
correspond to each other and cannot be directly compared.
29

Figure 2.9 (a) Measured OAM spectrum of OAM -1 when OAM -1 is transmitted and the UAV is on
the ground. (b) Measured received power on different modes in a 60-s time period when OAM -1 is
transmitted and the UAV is hovering ~50-m away. In (a) and (b), the receiver aperture diameter is 6.5
cm. (c) Measured OAM spectrum of OAM -1 when OAM -1 is transmitted and the UAV is hovering
~50-m away with various receiver aperture size. The color bar represents the fluctuation range of the
received power. (c) Measured crosstalk for OAM+3 when OAM+3 and -1 are transmitted and the UAV
is on the ground, hovering, or moving ~50-m away, with various receiver aperture sizes. Rx:6.5 cm,
receiver diameter is 6.5 cm.
(a) (b)
Tx: OAM-1
Hovering
Tx: OAM-1
3 4 5 6 7
Receiver aperture diameter (cm)
-25
-20
-15
-10
-5
0
Crosstalk (dB)
On ground
Hovering
Moving
Tx: OAM +3 and -1
Rx: OAM -1
(c)
(d)
Fluctuation range
30

Figure 2.10 Experimental bit-error-rates (BERs) for each 8192-symbol data frame when OAM+3 and
-1 are transmitted and the UAV is hovering. (a) BER measurements in a 60-s time period for both
channels when transmitted power is 10 dBm for each channel. (b) BER for both channels as functions
of transmitted power. (c) BER for OAM+3 as a function of transmitted power with various receiver
aperture sizes. For (a) and (b), the receiver aperture diameter is 6.5 cm. The values at the bottom of (b)
0 2 4 6 8 10
Transmitted power (dBm)
1e-4
1e-3
1e-2
Bit error rate
Rx:OAM+3, Hovering
Rx:OAM-1, Hovering
Rx:OAM+3, Static
0 10 20 30 40 50 60
Time (s)
1e-4
1e-3
1e-2
1e-1
Bit error rate
Rx:OAM-1
0 10 20 30 40 50 60
Time (s)
1e-4
1e-3
1e-2
1e-1
Bit error rate
Rx:OAM+3
7% overhead FEC limit
Error
free
Error
free
7% overhead FEC limit
(a)
7% overhead FEC limit
7 5 4 7 4 5 2 1
1 2 0 1
Error
free
(c)
7% overhead
FEC limit
7 5 4 6 4 4 2 2
Error
free
(b)
Back-to-back
31

and (c) represent the number of error-free frames out of the 10 measured frames for each transmitted
power level. FEC: forward error correction.

Figures 2.10(b) and 2.10(c) show BER as a function of transmitted channel
power. For each transmitted channel power, BER measurements for 10 data frames
are taken within a ~30-s time period, such that there is a range of BER values at each
power level due to the non-ideal beam tracking. The values at the bottom of the figure
represent the number of error-free (below 10
-5
) data frames out of the 10 measured
data frames at each power level. As a comparison, Figure 2.10(b) also shows BER
measurements for the OAM +3 channel when the UAV is static on the ground, as well
as the back-to-back case in which the UAV is static on the ground and a single
Gaussian beam is transmitted. Figure 2.10(c) shows the BER measurements for the
OAM +3 channel with different receiver aperture sizes when the UAV is hovering.
We observe that the BERs increase as the receiver aperture decreases from 6.5 cm to
5 cm, but could still be mostly below 3.8×10
-3
when the transmitted power is >8 dBm.
We note that further decreasing the aperture size would lead to BERs mostly above
the FEC limit even when the transmitted power is >10 dBm.
2.4 Discussion
This chapter describes our demonstration of up to 200-Gbit/s FSO
communication link multiplexing 2 OAM modes between a ground transmitter and a
32

ground receiver via a UAV up to a 100 m roundtrip distance. Although our UAV only
carries a reflector and acts as a simple relay for the ground transceiver, we believe that
our results show the potential that a communication link with an aerial platform that
carries transmitter/receiver equipment could be possible [12].
In general for a well-aligned FSO link in which the transmitter and the receiver
are on opposite sides of the link, the received beam’s propagation direction would be
perpendicular to the receiver aperture plane, and the beam center and the receiver
aperture center would coincide with each other. However, there could be misalignment
issues (e.g., increased crosstalk) in a non-ideal system, which arise from: (a) lateral
displacement, such that the beam center does not overlap with the receiver aperture
center; and (b) angular tip/tilt, such that the beam’s propagation direction is not
perpendicular to the receiver aperture plane [12].
In our system, however, the transmitter and the receiver are on the same side of
the link such that the beam is retro-reflected from the UAV. The retro-reflector can
produce both types of misalignment issues mentioned above but caused by different
mechanisms:
(a) Due to an optical-path-length differential inside the retro-reflector, there
could be a lateral displacement between the reflected beam’s centre and the receiver
aperture centre when the transmitted beam is not incident on the centre of the retro-
reflector, and in our case up to an 8-mm displacement is measured.
33

(b) For an ideal system, there should be negligible tip/tilt. However, due to non-
ideal fabrication, the retro-reflector may not perfectly reflect the beam back in its
original direction but may have an angular error. For the retro-reflector in our case,
tip/tilt is <1 arcsecond between the received beam and the receiver aperture.
We believe that such a tip/tilt level would have minor issues in our system,
whereas displacement could be a more important performance factor that affects
crosstalk. We note that for the case where the transmitter and the receiver are on
different sides of a link, both displacement and tip/tilt could be significant

and should
be explored further.
Moreover, with the addressed challenges properly met, we believe that future
high-capacity communication links to moving platforms enabled by OAM
multiplexing has the potential to achieve multi-Tbit/s capacities over multi-km
distances [12]. To extend the transmission distance to multi-km, the following issues
should be considered:
(a) Divergence: OAM beams have a vortex intensity profile and a beam
divergence that grows with ℓ. To capture sufficient signal power at the receiver or limit
the amount of beam divergence, larger receiver aperture sizes or larger transmitted
beam sizes than in our experiment would be required at longer distances, respectively.
For example, a transmitted OAM +3 beam with a diameter of 20 cm would be ~20 cm
at 1 km, ~75 cm at 10 km, and ~7.1 m at 100 km. With a 20-cm-diameter receiver
aperture, the link loss due to beam divergence would be 1.7 dB at 1 km, 27 dB at 10 km,
34

120 dB at 100 km. In order to achieve a < 20-dB power loss due solely to divergence,
the receiver aperture diameter should be >23 cm at 10 km, and >2.2 m at 100 km. In
general, larger aperture size would indicate larger-size optical elements, which could
increase the size and weight of the system. This would cause design challenges
especially for an aerial platform. Potential approaches to overcome these challenges
could be the utilization of optical elements with light-weight materials, and/or the
utilization of aerial platforms that can carry more weight.
(b) Turbulence: The effect of atmospheric turbulence would be more significant
as transmission distances increase. For a similar atmospheric condition as in our
demonstration (i.e., a Cn
2
of ~9.4 × 10
−15
m
−2/3
), the σ
2
would increase to ~0.19 at 1 km
and even higher at 10 km. Although in our experiment we did not use any
compensation techniques, adaptive optics (AO), multiple-input-multiple-output
(MIMO)-based channel equalization, and other digital-signal-processing (DSP)
techniques have been demonstrated to help mitigate turbulence effects in stationary
OAM-multiplexed FSO links. As an example, promising turbulence mitigation
techniques in OAM links have been experimentally demonstrated for relatively weak
turbulence

and theoretically analyzed for moderate-to-strong turbulence. Moreover,
turbulence compensation could possibly be performed on the ground station as pre-
and/or post-mitigation, instead of being performed on the aerial platform itself.
(c) Tracking: The unique structure of OAM beams places a premium on accurate
tracking. In general, there have been advances both commercially and experimentally
35

that has the potential to meet the system requirements. Furthermore, the vortex
amplitude profile has sharp gradients that may actually help the tracking system
performance under certain conditions.
To increase the link capacity to beyond Tbit/s, there exist a number of potential
approaches, including the following:
(a) Mode Spacing: More OAM modes can be accommodated in a given link by
reducing the mode spacing and ensuring low inter-modal crosstalk, and we believe
that a mode spacing of 2 for a UAV link is achievable. Of course, optical components
(e.g., (de)multiplexer) with still higher performance would be helpful. Furthermore,
MIMO-based channel equalization techniques can be used to mitigate crosstalk;
however, the total number of modes may be limited due to the increased signal
processing complexity.
(b) Mode Order: More modes that are located at higher orders can be achieved
by utilizing larger optical elements since beam size and beam divergence increase with
larger OAM value. For example, with the same transmitted beam size of 20 cm and a
distance of 1 km, an OAM beam of +3 and +20 has a diameter at the receiver of ~20 cm
and ~47 cm, respectively
Beyond using only OAM multiplexing, there might well be key advantages to
dramatically increasing capacity by employing channel multiplexing in multiple
domains, such as wavelength- and polarization-division-multiplexing (WDM and
36

PDM). Indeed, fixed FSO links have used OAM +WDM + PDM in the lab to achieve
100 Tbit/s. However, the various optical components should maintain their high
performance across the link’s wavelength spectrum, and the integrity of the
polarization states should be preserved.
Finally, to further enhance the privacy/security offered by the use of OAM
beams, mode hopping a data channel’s location among different modes can be
employed, in analogy with frequency hopping in radio-based communication links
[12].
2.5 Reference
[1] A. K. Majumdar, Advanced Free Space Optics (FSO) Springer, 203-225 (2015).
[2] S. S. Muhammad, T. Plank, E. Leitgeb, A. Friedl, K. Zettl, T. Javornik, and N.
Schmitt, "Challenges in establishing free space optical communications between
flying vehicles," In 2008 6th international symposium on communication systems,
networks and digital signal processing, 82-86. (2008).
[3] A. Kaadan, H. H. Refai, and P. G. LoPresti, "Multielement FSO transceivers
alignment for inter-UAV communications," J. Lightw. Technol. 32, 4183-4193 (2014).
[4] A. Glenn, "Low probability of intercept," IEEE Communications Magazine 21, 26-
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[5] W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro,
J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, "Free-
space optical communications research and demonstrations at the U.S. Naval Research
Laboratory," Appl. Opt. 54, F189-F200 (2015).
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[6] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, "Orbital angular
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Dolinar, M. Tur, and A. E. Willner, "Terabit free-space data transmission employing
orbital angular momentum multiplexing, " Nat. Photonics 6, 488–496 (2012).
[8] H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner,
B. I Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur,
and A. E. Willner, "100 Tbit/s free-space data link enabled by three-dimensional
multiplexing of orbital angular momentum, polarization, and wavelength," Opt. Lett.
39, 197–200 (2014).
[9] Y. Zhao, J. Liu, J. Du, S. Li, Y. Luo, A. Wang, L. Zhu, and J. Wang, "Experimental
demonstration of 260-meter security free-space optical data transmission using 16-
QAM carrying orbital angular momentum (OAM) beams multiplexing," Optical Fiber
Communications Conference and Exhibition, 1-3 (2016).
[10] Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed,
A. J. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic,
M. A. Neifeld, and A. E. Willner, "Experimental characterization of a 400 Gbit/s
orbital angular momentum multiplexed free-space optical link over 120 m," Opt. Lett.
41, 622-625 (2016).
[11] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Turbulence-induced channel
crosstalk in an orbital angular momentum-multiplexed free-space optical link," Appl.
Opt. 47, 2414–2429 (2008).
[12] L. Li, R. Zhang, Z. Zhao, G. Xie, P. Liao, K. Pang, H. Song, C. Liu, Y. Ren, G.
Labroille, P. Jian, D. Starodubov, B. Lynn, R. Bock, M. Tur, and A. E. Willner, "High-
capacity free-space optical communications between a ground transmitter and a
ground receiver via a UAV using multiplexing of multiple orbital-angular-momentum
beams," Sci. Rep. 7, 17427 (2017).
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Pang, G. Labroille, P. Jian, D. Starodubov, B. Lynn, R. Bock, M. Tur, and A. E.
Willner, "Effect of limited aperture size on a retro-reflected communication link
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momentum," Optical Fiber Communications Conference and Exhibition, 1-3 (2018).
38

Chapter 3 Turbulence Effects Mitigation for
Orbital-Angular-Momentum-Multiplexed
Free-Space Optical Communications
between a Ground Station and a UAV
Using MIMO Equalization
3.1 Introduction
In general, atmospheric turbulence is one of the key challenges for FSO
communications that degrade link performance [1]. This issue is of greater concern for
OAM-multiplexed links, since turbulence can cause phase distortions to OAM beams
and increased crosstalk between different OAM channels. There have been reports of
turbulence effect mitigation for OAM-multiplexed FSO links between fixed
transmitters and receivers using: (i) adaptive optics with deformable mirrors or spatial-
light-modulators [2,3], and (ii) signal processing algorithms to improve single mode
performance at any given time [4,5]. Moreover, there have been reports of using MIMO
equalization to mitigate turbulence effects in an OAM-multiplexed link in a lab
environment either in free-space or underwater over roughly a 1-m distance [6,7].
Recently, a 100-m round-trip OAM-multiplexed link between a ground station and a
flying UAV has been reported, but the turbulence issue has not been addressed in that
experiment [8].
39

Figure 3.1 Concept of (a) an orbital-angular-momentum (OAM)-multiplexed free-space optical (FSO)
communication link between a ground station and a retro-reflecting unmanned-aerial-vehicle (UAV)
through atmospheric turbulence; (b) an OAM beam distorted by turbulence; and (c) mitigation for
turbulence effect in an OAM-multiplexed link using multiple-input-multiple-output (MIMO)
equalization.

In this chapter, we experimentally demonstrate MIMO equalization to mitigate
turbulence in a 40-Gbit/s retro-reflected FSO link multiplexing 2 OAM modes between
a ground transmitter and a ground receiver, connected via a flying retro-reflecting UAV
over 100-m round-trip distance [9]. The receiver is co-located with the transmitter on
the ground station. A rotatable phase plate with a pseudo-random phase distribution,
Ground station
Tracking
system
OAM
transmitter
OAM
receiver
UA V carrying a
retro-reflector
Transmitted and reflected
OAM beams
…
Atmospheric turbulence
Multiplexed data-carrying
OAM beams
(a)
ℓ
1
Power
OAM
order
Transmitted
OAM beam
Distorted
OAM beam
Power
OAM
order
ℓ
1
Crosstalk to
other channels
(b)
Transmitted power
ℓ
1 Power
OAM ℓ
1
OAM order
OAM ℓ
2
OAM ℓ
n
…
(c)
Received power
ℓ
1 Power
OAM ℓ
1
OAM order
OAM ℓ
2
OAM ℓ
n
…
Power
Distorted
OAM ℓ
1
OAM order
Distorted
OAM ℓ
2
Distorted
OAM ℓ
n
…
MIMO
equalization
40

obeying Kolmogorov spectrum statistics is used at the transmitter to emulate weak-to-
moderate atmospheric turbulence over a 1-km distance [2]. Results indicate that MIMO
equalization could help mitigate the crosstalk caused by turbulence, and improve both
error vector magnitude (EVM) and BER of the signal in an OAM-multiplexed link for
flying platforms. In our experiment, MIMO equalization helps achieve BER values
mostly below 3.8×10
-3
under the emulated turbulence conditions [9].
Figure 3.1(a) shows the concept of an OAM-multiplexed FSO link between a ground
station and a retro-reflecting UAV. Multiple independent data-carrying OAM beams
are multiplexed and transmitted from the ground station to the UAV, and retro-reflected
back to the same ground station. Due to atmospheric turbulence, the OAM beams may
be distorted when propagating in free-space, such that signal power on each particular
OAM mode may be coupled to its neighbouring modes, as shown in Figure 3.1(b).
Therefore, the received signal at a particular mode may also contain crosstalk from its
neighbours. MIMO equalization could help reduce crosstalk among channels by
applying the inverse of the channel matrix to the received signals, thus mitigating
system performance degradation, as shown in Figure 3.1(c) [6,7].
3.2 Link Demonstration and Turbulence Effects
Mitigation
The experimental setup is shown in Figure 3.2. During the measurement, the
UAV is either on the ground, hovering, or moving at a maximum speed of ~0.1 m/s,
~50-m away from the ground station. A 20-Gbit/s quadrature phase-shift keying
(QPSK) signal at 1550 nm is generated and split into two branches. One branch is
relatively delayed using a ~10-m single-mode fiber to decorrelate the data sequences.
The two branches are fed to two input ports of a custom-designed OAM
41

generator/multiplexer, generating multiplexed OAM beams [10]. Another 1530-nm
beacon for beam tracking is sent to the ℓ = 0 input port of the OAM multiplexer. These
co-axially propagating beams then pass through a thin phase plate mounted on a
rotation stage. This phase plate is designed to generate a pseudo-random phase
distribution obeying the Kolmogorov spectrum statistics with an effective r0 of 1 mm,
which represents weak-to-moderate turbulence over a 1-km distance [2]. Then the
beams are expanded and propagate to the gimbal-mounted retro-reflector carried by
the UAV. The beam diameters after expansion are ~ 6.0 cm and ~ 4.2 cm for OAM -
3 and +1 beams. The retro-reflector reverses the beam’s OAM order from +ℓ to -ℓ. At
the receiver, the beams after beam reduction are coupled into the OAM demultiplexer
for heterodyne detection and MIMO equalization based on a constant modulus
algorithm that utilizes a linear equalizer for each channel [2]. After equalization,
frequency offset estimation and carrier phase recovery are applied to recover the
signals, and the BERs are evaluated for all channels [2]. A two-stage beam tracking
system is used [8]. A coarse tracking system controls the ground gimbal to ensure that
the OAM beams are pointing to the UAV, while a fine tracking system keeps the
reflected OAM beams hit the centre of the OAM demultiplexer [8].
First, we measure the the Rytov variance σ
2
with and without placing the
turbulence phase plate in the link. The retro-reflector carried by the UAV is placed on
the ground ~50 m away from the transmitter/ receiver. A ~5-cm diameter 1550-nm
Gaussian probe beam propagates from the transmitter to the retro-reflector and back
42

to the receiver. At the receiver, a ~1-mm diameter point detector is used to record the
received power over a 10-minute period [8]. Figures 3.3(a) and 3.3(b) show the
received power distributions when the turbulence plate is rotating at 40 round/minute
or not presented in the link, and σ
2
is found to be 0.11 and 0.003, respectively. The
D/r0 for OAM -3 and +1 beams are ~6.0 and ~4.2, where D is the beam diameter.

Figure 3.2 Experimental setup. BS: beamsplitter; DeMUX: demultiplexer; EDFA: erbium-doped fiber
amplifier; FSM: fast steering mirror; LO: local oscillator; MUX: multiplexer; PD: photodetector; PSD:
position sensitive detector; QPSK: quadrature phase-shift keying.
Mirror
FSM
Ground station
20 Gbit/s
QPSK signal
generator at
1550 nm
50/50
coupler
OAM
MUX
Probe beam at 1530 nm
PSD
OAM
DeMUX
Filter
50/50
BS
90/10 BS
Ground
gimbal
Mirror
Controller
EDFA
UA V
Gimbal
Retro-
reflector
Turbulence
emulator
1:10 beam
expansion
Beam
reduction
EDFA
LO
PD
PD
Heterodyne
demodulation
MIMO
equalization
Frequency offset
estimation
Carrier phase
recovery
Decision & BER
Offline signal processing
EDFA
Ground
station
Transmitted
beams
Reflected
beams
Fiber
connection
Feedback
control
43

Figure 3.3 Measured power distribution when the turbulence emulator is (a) placed in the link and
rotating at 40 round/minute and (b) not placed in the link. A ~5-cm diameter 1550-nm Gaussian probe
beam is transmitted over ~100 m roundtrip, and a ~1-mm diameter point detector is used and the
receiver.

Figure 3.4 shows the measured OAM spectrum when only the OAM +1 beam is
transmitted under various flight conditions with and without placing the turbulence
phase plate. In Figures 3.4(a) and 3.4(b), the UAV is static on the ground, ~50-m away
from the ground station; In Figures. 3.4(c) and 3.4(d), the UAV is hovering or moving
at a maximum speed of ~0.1 m/s, ~50-m away from the ground station and ~5-m above
the ground. When the the UAV is hovering or moving, the received power on different
modes fluctuates, which may be due to the beam jitter caused by both imperfect beam
tracking and turbulence effect [8]. The shaded portion of each bar in Figures 3.4(c)
and 3.4(d) represents the fluctuation range of received power. In this measurement,
the turbulence phase plate is fixed at a random angle when placed in the link, emulating
0 1 2
Normalized power
0
0.5
1
1.5
2
Probablity density function
Measured
Normalized
!
2
= 0.11
0 1 2
Normalized power
0
2
4
6
8
Probablity density function
Measured
Normalized
w/o turbulence plate
!
2
= 0.003
w/ turbulence plate (a) (b)
44

a random turbulence realization. Results show that turbulence increases the crosstalk
to and from unwanted modes under all flight conditions.
Figure 3.5 shows the instantaneous power and crosstalk for OAM +1 and -3
channels when both beams are simultaneously transmitted under 12 different
turbulence realizations. These realizations are selected when the turbulence phase
plate randomly rotates to different angles. Here, crosstalk of a specific channels is
defined as the power received from other unwanted modes over the power received
from the desired mode. The UAV is hovering ~50-m away from the ground station
and ~5-m above the ground. Results show that the OAM -3 channel generally has
lower power and suffers from higher crosstalk compared with the OAM +1 channel.
This may be due to the relatively larger D/r0 of the OAM -3 beam.
45

Figure 3.4 Measured OAM spectrum when OAM +1 beam is transmitted and the UAV is static on the
ground (a) with and (b) without the turbulence phase plate. Measured OAM spectrum in a 60-second
period with and without the turbulence phase plate when OAM +1 beam is transmitted and the UAV is
(a) hovering and (b) moving at a maximum speed of 0.1 m/s. In all cases, the UAV is ~50-m away, ~5-
m above the ground.

Figure 3.5 Measured received power and crosstalk for OAM+1 and -3 when both beams are transmitted
under 12 different turbulence realizations. The UAV is hovering ~50-m away, ~5-m above the ground.
(b)
On ground,
w/o turbulence plate
(a)
On ground,
w/ turbulence plate
(c)
Hovering
Fluctuation
range
Moving
Fluctuation
range
(d)
OAM+1 crosstalk
OAM-3 crosstalk
46

Figure 3.6 Experimental measurement of bit-error-rates (BERs) as a functions of transmitted power
when OAM +1 and OAM -3 are simultaneously transmitted, each carrying a 20-Gbit/s quadrature
phase-shift keying (QPSK) signal: (a) for the OAM -3 channel without MIMO equalization; (b) for both
channels with and without MIMO equalization. (c) Recovered QPSK constellations at transmitted
power of 10 dBm for all channels with and without MIMO equalization when OAM +1 and -3 are
transmitted. The UAV is hovering ~ 50-m away, ~5-m above the ground.

7% overhead FEC limit
7% overhead FEC limit
EVM=0.54
OAM -3 OAM +1
EVM=0.32
w/o equalization
EVM=0.26 EVM=0.27
OAM -3 OAM +1
w/ equalization
(c)
(b)
(a)
47

To verify the link performance, Figure 3.6 shows the BER measurements when
both OAM -3 and +1 channels are transmitted, each carrying a 20- Gbit/s QPSK signal.
Figure 3.6(a) shows BERs for the OAM -3 channel as functions of transmitted power
when the UAV is static on the ground or hovering ~50-m away, with and without the
turbulence phase plate. No MIMO equalization has been used. It is shown that the
measured BER curve of OAM -3 without MIMO equalization exhibits a severe error
floor due to the inter-channel crosstalk. Figure 3.6(b) shows BERs for both channels
as functions of transmitted power when the UAV is hovering ~50-m away with phase
plate fixed at a random angle. We observe that the BERs dramatically decrease to
below the 7% overhead FEC limit of 3.8×10
-3
for all channels after MIMO
equalization. The received QPSK constellation diagrams and corresponding EVMs for
both channels are shown in Figure 3.6(c). The transmitted power for both channels are
10 dBm.
We then rotate the phase plate randomly to different angles to test the system
under various turbulence realizations. Figure 3.7 shows the measured BERs for both
channels under 12 different turbulence realizations. Note that the results in Figure 3.5
and Figure 3.7 are measured in two different runs, and the turbulence realizations do
not correspond to each other. For each turbulence realization, the transmitted power
for each channel is 10 dBm. We observe that the BER improvement of using MIMO
equalization varies for different realizations, but the BERs could mostly be kept below
3.8×10
-3
with MIMO equalization for both channels.
48

Figure 3.7 BERs for both OAM +1 and OAM -3 channels with and without MIMO equalization under
12 different turbulence realizations. The UAV is hovering ~ 50-m away, ~5-m above the ground.

3.3 Reference
[1] Z. Qu, and I. Djordjevic, "500 Gb/s free-space optical transmission over strong
atmospheric turbulence channels," Opt. Lett. 41, 3285 (2016).
[2] B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L.
Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, "Simulating thick
atmospheric turbulence in the lab with application to orbital angular momentum
communication," New J. Phys. 16, 033020 (2014).
[3] Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. J. Lavery, B. I.
Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H.
Shapiro, and A. E. Willner, "Adaptive optics compensation of multiple orbital angular
momentum beams propagating through emulated atmospheric turbulence," Opt. Lett. 39,
2845 (2014).
7% overhead
FEC limit
49

[4] G. Xie, Y. Ren, H. Huang, M. P. J. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z.
Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H Shapiro, and
A. E. Willner, "Phase correction for a distorted orbital angular momentum beam using a
Zernike polynomials-based stochastic-parallel-gradient-descent algorithm," Opt. Lett.
40, 1197 (2015).
[5] M. Krenn, J. handsteiner, M. Fink, R. Fickler, and A. Zeilinger, "Twisted photon
entanglement through turbulent air across Vienna," PNAS 112, 14197 (2015).
[6] H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S.
J. Dolinar, and A. E. Willner, "Crosstalk mitigation in a free-space orbital angular
momentum multiplexed communication link using 4x4 MIMO equalization," Opt. Lett.
39, 4360 (2014).
[7] Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y.
Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and
A. E. Willner, "Orbital angular momentum-based space division multiplexing for
high-capacity underwater optical communications," Sci. Rep. 6, 33306 (2016).
[8] L. Li, R. Zhang, Z. Zhao, G. Xie, P. Liao, K. Pang, H. Song, C. Liu, Y. Ren, G.
Labroille, P. Jian, D. Starodubov, R. Bock, M. Tur, and A. E. Willner, "80-Gbit/s
100-m Free-space optical data transmission link via a flying UAV using
multiplexing of orbital-angular-momentum beams," Sci. Rep. 7, 17427 (2017).
[9] L. Li, R. Zhang, P. Liao, Y. Cao, H. Song, Y. Zhao, J. Du, Z. Zhao, C. Liu, K.
Pang, H. Song, D. Starodubov, B. Lynn, R. Bock, M. Tur, A. F. Molisch, and A.
E. Willner, "MIMO equalization to mitigate turbulence in a 2-channel 40-Gbit/s
QPSK free-space optical 100-m round-trip orbital-angular-momentum-
multiplexed link between a ground station and a retro-reflecting UAV," In 2018
European Conference on Optical Communication (ECOC), Th2.33 (2018).
[10] G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur,
"Efficient and mode selective spatial mode multiplexer based on multi-plane light
conversion," Opt. Express 22, 15599 (2014).
50

Chapter 4 Beaconless Beam Displacement
Tracking and Correction in Orbital-
Angular-Momentum-Multiplexed Free-
Space Optical Links
4.1 Introduction
Beam tracking is considered important for a single-beam non-OAM FSO link.
In such a link, potential misalignment between the transmitter and the receiver might
occur, which gives rise to increase in power coupling loss and bit error rates (BERs)
[1,2]. For OAM-multiplexed FSO links, beam tracking might be even more important
[3-15]. This is because any deviation away from coaxial detection of the multiple
OAM beams could lead to power coupling into unwanted modes, thereby increasing
the channel crosstalk and power penalty [4]. Beam tracking has been demonstrated in
single-beam non-OAM FSO links with or without using a beacon beam at a separate
wavelength [1,2], as well as in an OAM-multiplexed FSO link using a Gaussian
beacon [15]. However, there has been little reported work on beacon-less beam
tracking for data-carrying OAM beams [16].
In this chapter, we experimentally demonstrate beacon-less beam displacement
tracking for multiple data-carrying OAM beams in an FSO link using OAM-beams-
based position detection. Four OAM beams are multiplexed and transmitted over a
51

distance of ~ 1 m. Each beam carries a 100-Gbit/s quadrature phase-shift keying
(QPSK) signal, and arrives at the receiver suffering from emulated pseudo-random
beam displacements. A beam displacement tracking system is designed and
implemented utilizing the OAM beams themselves for position detection.
Experimental results show that this beam tracking system could help reduce channel
crosstalk to below -18 dB and achieve power penalties less than 3 dB for all channels,
with displacement of up to ±10 mm and transmitted beam diameters of ~6 mm. In
addition, we compare system performance with and without using a Gaussian beacon.
In our demonstration, using OAM-beams-based and Gaussian-beacon-based position
detections show slightly different position detection responses, but similar crosstalk
and power penalties.
4.2 Beaconless OAM Beam Displacement Tracking
Our scheme of displacement tracking and correction using OAM-beams-based
position detection is shown in Figure 4.1(a). Note that similar schemes have been used
for beam tracking in single-beam non-OAM FSO links [13,14]. At the receiver, a lens
is used to convert the lateral displacements of the OAM beams’ center from that of the
lens’ center, into different angular errors at the lens’ focal plane. A fast steering mirror
(FSM) is placed at the lens’ focal plane to adaptively correct these angular errors. Then
the beams are split into two branches. One branch is sent to a position sensitive
detector (PSD) and the other one is sent to the OAM receiver. The PSD and the OAM
receiver are at the same distance from the beamsplitter. The PSD we used has a quad-
cell structure, which detects beam position by measuring beam power on each of the
four cell quadrants (Q1 to Q4), as shown in Figure 4.1(b). The PSD output voltages
52

Vx and Vy serve as feedbacks to the proportional-integral-derivative (PID) controller,
which controls the angle of FSM to keep the OAM beams close to the centers of both
the PSD and the OAM receiver. These voltages are evaluated by [13]:
𝑉
$
=
('
(
) '
*
),('
-
) '
.
)
'
-
) '
(
) '
.
) '
*
𝑉
/
=
('
-
) '
(
),('
.
) '
*
)
'
-
) '
(
) '
.
) '
*
(1)

Figure 4.1 (a) Scheme of displacement tracking for multiple OAM beams using OAM-beams-based
position detection. (b) Intensity profiles of an OAM beam and a Gaussian beam. FSM: fast steering
mirror; PSD: position sensitive detector.

In our system, the OAM mode numbers and beam tracking ranges would be
related to multiple design parameters, including transmission distance, transmitted
beam size, and transmitter and receiver aperture sizes [16]. Moreover, to
simultaneously track lateral displacements and angular errors, e.g., tracking beams in
an FSO link for aerial platforms where both misalignment may exist, a possible
OAM
receiver
aperture
OAM
beams with
displacement
OAM beam
intensity profile
Q
1
Gaussian beam
intensity profile
OAM-beam-based
tracking loop
Lens
FSM at lens
focal plane
Beamsplitter
FSM
controller
PSD
Q
2
Q
3
Q
4
PSD
outputs
(a) (b)
53

approach could be cascading multiple FSMs to provide more degrees of freedom for
beam steering [17].

Figure 4.2 Experimental setup of an OAM-multiplexed FSO communication link with displacement
tracking executed in a beaconless manner by the OAM beams themselves (i.e., the 1530 nm laser is off)
or with the help of a Gaussian beacon (i.e., the 1530 nm laser is on). BS: beamsplitter; Col.: collimator;
EDFA: erbium-doped fiber amplifier; PC: polarization controller; QPSK: quadrature phase-shift keying;
Rx: receiver; SLM: spatial light modulator; Tx: transmitter [18].

The experimental setup is shown in Figure 4.2 [18]. A 100-Gbit/s QPSK signal
at 1550 nm is generated, amplified and split into two branches. One branch is relatively
delayed using a ~10-m single-mode fiber for signal decorrelation. The two signals are
then fed into two collimators which emit collimated Gaussian beams with beam
diameters of 3 mm. These two Gaussian beams are converted to OAM ℓ = +1 and +3
beams using spatial light modulator (SLM)-1 and SLM-2, respectively. After being
Col.
50/50
coupler
EDFA
100 Gbit/s
QPSK signal
generator
@ 1550 nm
SLM-1
Coherent
receiver
Delay
PC
Col.
SLM-2
SLM-3
Mirror
BS-1
BS-2
BS-3
Tx
1-m free space transmission
PC
Col.
PSD
FSM
controller
FSM
BS-5
Lens-1
Lens-2 Lens-3
Col.
Mirror
Motored
linear stage
Telescope
Rx
1530 nm
laser
BS-4
Filter
Telescope
54

combined with a beamsplitter, these two beams are split into two identical copies. One
copy is reflected by three mirrors to generate ℓ = -1 and -3, and are then multiplexed
with the other copy (i.e., ℓ = +1 and +3). Afterwards, the resulting four OAM beams
pass through a telescope and are coaxially multiplexed with a 1530 nm Gaussian beam.
This Gaussian beam is turned on only for the beacon-assisted performance evaluation.

Figure 4.3 Simulation and experimental results for normalized output of PSD as functions of spot-
centroid displacement when (a) OAM ℓ = +1 beam is transmitted with various spot sizes (measured in
diameter) on PSD plane, and (b) Gaussian, ℓ = +1, ℓ = +3 beams are transmitted one at a time with beam
diameter of 1.5 mm on PSD plane.

Lateral displacements are emulated by a mirror mounted on a motorized linear
stage. The linear stage implements bounded random walk, with a minimum step of 0.1
mm and a maximum speed of 5 cm/s. After passing through a lens with a focal length
of 500 mm, the beams are reflected by an FSM placed at the focal plane of the lens.
The FSM has an angular tuning range of ±1.5° with a resolution of <1 μrad. Our
tracking system has a tracking speed of ~ 550 Hz. We note that a faster tracking system
would benefit high speed FSO communications. The OAM beams are then split into
two branches by a beamsplitter, where one part is sent to a PSD with a diameter of 3
0 0.2 0.4 0.6 0.8 1 1.2
Spot-centroid displacement (mm)
0
0.2
0.4
0.6
0.8
1
Normalized PSD output (V)
OAM+3
OAM+1
Gaussian
0 0.2 0.4 0.6 0.8 1 1.2
Spot-centroid displacement (mm)
0
0.2
0.4
0.6
0.8
1
Normalized PSD output (V)
Spot size 1.5mm
Spot size 1.0mm
Spot size 0.6mm
(a)
OAM ℓ = +1
Line: simulation
Marker: experimen t
(b)
Line: simulation
Marker: experiment
Beam diameter
on PSD: 1.5 mm
55

mm. The other branch is launched onto SLM-3 which converts the desired OAM mode
to a Gaussian-like beam, while other undesired modes remaining their ring shapes.
The Gaussian-like beam is then coupled into fiber for coherent detection. The PSD
and SLM-3 are placed at the same optical distance of ~10 cm from the FSM. A 1530-
nm filter is placed before the PSD only when the 1530 nm Gaussian beacon is used.

Figure 4.4 Measurement of beam centroid on the SLM-3 plane when displacement range is (a) ±5 mm,
without tracking; (b) ±5 mm, with Gaussian-beacon-based tracking; (c) ±5 mm, with OAM-beams-
based tracking when ℓ = +1 is transmitted with a spot size of 1.2 mm on PSD [18].

Figure 4.3 shows PSD output as beam moves off the PSD center. In Figure 4.3(a),
ℓ = +1 beam is transmitted with various beam diameters. The beam sizes are adjusted
by the transmitter telescope. We observe that OAM beams with larger beam sizes on
-1 -0.5 0 0.5 1
X positon (mm)
-1
-0.5
0
0.5
1
Y positon (mm)
◇ displacement up to ±5 mm
w/ Gaussian tracking
-1 -0.5 0 0.5 1
X positon (mm)
-1
-0.5
0
0.5
1
Y positon (mm)
+ displacement up to ±5 mm
w/o tracking
(a) (b)
-1 -0.5 0 0.5 1
X positon (mm)
-1
-0.5
0
0.5
1
Y positon (mm)
◦ displacement up to ±5 mm
w/ OAM tracking
(c)
56

the PSD plane could help increase the maximum tracking range of the system. In
Figure 4.3(b), Gaussian, ℓ = +1 and ℓ = +3 beams are transmitted one at a time, with a
beam diameter of 1.5 mm on the PSD. In view of their individual intensity distributions,
OAM and Gaussian beams produce slightly different responses of the PSD.

Figure 4.5 Received signal power and (b) crosstalk from other three channels for receiving OAM ℓ =
+3 over 60 seconds. Displacement range is ±5 mm and spot size is 1.2 mm [18].

Figures 4.4(a)-(c) show the statistics of displacement of OAM ℓ = +1 beam
respect to the center of OAM receiver aperture (i.e., SLM-3) without tracking, with
Gaussian-beacon-based tracking, and with OAM-beams-based tracking, respectively
[18]. Each marker represents the instantaneous location of the beam centroid. The
OAM beam is aligned to the center of SLM-3 when the linear stage is at its original
(a)
(b)
±5 mm displacement range
Spot size 1.2 mm on PSD
Rx: OAM ℓ = +3
±5 mm displacement range
Spot size 1.2 mm on PSD
Rx: OAM ℓ = +3
57

position. The linear stage randomly travels in a ±5 mm range. Results show that the
FSM and PSD together could help the beam come closer to the center of SLM-3, and
OAM-beams-based tracking and Gaussian-beacon-based tracking show similar
performance. When the linear stage travels in a ±12.5 mm range.
Figure 4.5 shows the received power and crosstalk for channel ℓ = +3 when four
OAM beams of ℓ = ±3 and ±1 are transmitted simultaneously, each carrying a 100-
Gbit/s QPSK signal [18]. The linear stage travel range is ±5 mm, and OAM-beams-
based position detection is used. Beam diameter of the multiplexed beams on the PSD
plane is 1.2 mm. It is observed that both power and crosstalk fluctuate for more than
20 dB, and remain steady when displacement tracking is implemented. Results show
that this beam tracking system is capable of reducing channel crosstalk to below -18
dB under such a displacement level. Note that Figure 6 is result of six different test
runs. The received power and crosstalk without displacement are shown in Figure
4.6(a), with crosstalk <-18 dB for all four channels. The received power and crosstalk
when displacement is up to ±5 mm and OAM-beams-based tracking is used are shown
in Figure 4.6(b), which indicate similar power and crosstalk level compared with the
well-aligned case in Figure 4.6(a).

Figure 4.6 Received signal power and crosstalk for all four channels when OAM ℓ = ±3 and ±1 are
transmitted (a) without displacement, and (b) with displacement up to ±5 mm and OAM-beams-based
tracking.

Tx\Rx +3 +1 -1 -3
+3 -22.8 -43.5 -47.1 -54.0
+1 -44.2 -21.2 -43.7 -46.0
-1 -43.8 -41.1 -21.1 -44.6
-3 -58.4 -46.8 -42.8 -22.5
(a) (b)
Tx\Rx +3 +1 -1 -3
+3 -22.4 -44.1 -43.1 -46.6
+1 -46.7 -22.1 -42.9 -45.4
-1 -42.5 -42.1 -21.9 -42.3
-3 -46.7 -48.4 -43.1 -22.6
Without displacement With displacement and OAM tracking
58

Figure 4.7(a) shows measured BERs for all four channels as a function of the
optical signal-to-noise ratio (OSNR), with beam diameters of 1.2 mm on PSD plane
[18]. Displacement is up to ±5 mm, and OAM-beams-based position detection is used.
All four channels have power penalties < 3 dB compared with the back-to-back case
(i.e., a single Gaussian beam is transmitted without displacement). We note that
without beam tracking, the BERs could not be measured due to severe channel
crosstalk. Figure 8(b) shows BERs for ℓ = +3 channel when using OAM-beams-based
and Gaussian-beacon-based displacement tracking and displacement range is up to ±5
mm, as well as without displacement. Beam diameter is 1.2 mm on PSD plane. It is
observed that with and without using a Gaussian beacon show similar BER
performance. In Figure 4.7(c), BERs for ℓ = +3 channel is measured under different
displacement levels with beam diameter of 1.2 mm on PSD plane. We observe that
BERs increases when maximum displacement is > 10 mm, which is out of the tracking
range of our displacement tracking system. BERs of channel ℓ = +3 with various beam
sizes are shown in Figure 4.7(d), when displacement is up to ±5 mm and OAM-beams-
based position detection is used. Results indicate that beams with a diameter <0.6 mm
(i.e., 20% of PSD diameter) on the PSD plane may decrease the tracking range of our
tracking system.

59

Figure 4.7 Measurement of bit-error-rates as a function of optical signal-to-noise ratio (OSNR) when ℓ
= ±3 and ±1 are transmitted: (a) for all four channels; (b) for ℓ = +3 with Gaussian-beacon-based and
OAM-beams-based tracking; (c) for ℓ = +3 with various displacement levels; and (d) for ℓ = +3 with
various beam sizes on the PSD. In (a), (b) and (c), beam diameter on PSD is 1.2 mm; In (a), (b) and (d),
displacement is up to ±5 mm. FEC: forward error correction [18].

4.3 Reference
[1] K. Kazaura, K. Omae, T. Suzuki, and M. Matsumoto, "Enhancing performance of next
generation FSO communication systems using soft computing-based predictions," Opt.
Express 14, 4958 (2006).
[2] E. Ciaramella, Y. Arimoto, G. Contestabile, M. Presi, A. D’Errico, V. Guarino, and M.
Matsumoto, "1.28 Terabit/s (32x40 Gbit/s) WDM transmission system for free space
optical communications," IEEE J. Selected Areas in Commun. 27, 1639 (2009).
10 12 14 16 18
OSNR (dB)
10
-4
10
-3
10
-2
Bit Error Rate
5mm OAM track
5mm Gaussian track
w/o displacement
10 12 14 16 18
OSNR (dB)
10
-4
10
-3
10
-2
Bit Error Rate
OAM+3
OAM+1
OAM-1
OAM-3
Back to back
(a)
(b)
FEC limit
FEC limit
10 12 14 16 18
OSNR (dB)
10
-4
10
-3
10
-2
Bit Error Rate
5mm OAM track
10mm OAM track
12.5mm OAM track.
(c)
10 12 14 16 18
OSNR (dB)
10
-4
10
-3
10
-2
Bit Error Rate
Spot size 1.2mm on PSD
Spot size 1.0mm on PSD
Spot size 0.6mm on PSD
Spot size 0.5mm on PSD
(d)
FEC limit
FEC limit
Rx: OAM +3
Rx: OAM +3
Rx: OAM +3
60

[3] A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and
A. Forbes, "Optical communication beyond orbital angular momentum," Sci. Rep. 6,
27674 (2016).
[4] A. Yao, M. Padgett, "Orbital angular momentum: origins, behavior and applications,"
Adv. Opt. Photon. 3, 161 (2011).
[5] J. Wang, J.-Y. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar,
M. Tur, and A. E. Willner, "Terabit free-space data transmission employing orbital
angular momentum multiplexing," Nat. Photon. 6, 488 (2012).
[6] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-
Arnold, "Free-space information transfer using light beams carrying orbital angular
momentum," Opt. Express 12, 5448 (2004).
[7] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Turbulence-induced channel crosstalk
in an orbital angular momentum-multiplexed free-space optical link," Appl. Opt. 47, 2414
(2008).
[8] I. Djordjevic, "Deep-space and near-Earth optical communications by coded orbital
angular momentum (OAM) modulation," Opt. Express 19, 14277 (2011).
[9] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular
momentum states of photons," Nature 412, 313 (2001).
[10] B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher,
N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, "Simulating thick atmospheric turbulence in
the lab with application to orbital angular momentum communication," New J. Phys. 16,
033020 (2014).
[11] M. Li, Y. Takashima, X. Sun, Z. Yu, and M. Cvijetic, "Enhancement of channel
capacity of OAM-based FSO link by correction of distorted wave-front under strong
turbulence," In 2014 Frontier in Optics, FTh3B-6 (2014).
[12] D. Richardson, J. Fini, and L. E. Nelson, "Space-division multiplexing in optical
fibres," Nat. Photon. 7, 354 (2013).
[13] H. Hao, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I.
Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E.
61

Willner, "100 Tbit/s free-space data link enabled by threedimensional multiplexing of
orbital angular momentum, polarization, and wavelength," Opt. Lett. 39, 197-200 (2014).
[14] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular
momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys.
Rev. A 45, 8185 (1992).
[15] L. Li, R. Zhang, Z. Zhao, G. Xie, P. Liao, K. Pang, H. Song, C. Liu, Y. Ren, G.
Labroille, P. Jian, D. Starodubov, R. Bock, M. Tur, and A. E. Willner, "80-Gbit/s 100-m
Free-space optical data transmission link via a flying UAV using multiplexing of orbital-
angular-momentum beams," Sci. Rep. 7, 17427 (2017).
[16] L. Li, R. Zhang, G. Xie, Y. Ren, Z. Zhao, Z. Wang, C. Liu, H. Song, K. Pang, R. Bock,
M. Tur, and A. E. Willner, "Experimental beam displacement tracking and correction of
data-carrying orbital-angular-momentum beams in a free-space optical link," In 2017
Optical Fiber Communication Conference, Tu2F.6 (2017).
[17] G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N.
Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, "Performance
metrics and design considerations for a free-space optical orbital-angular-momentum–
multiplexed communication link," Optica 2, 357-365 (2015).
[18] L. Li, R. Zhang, G. Xie, Y. Ren, Z. Zhao, Z. Wang, C. Liu, H. Song, K. Pang, R. Bock,
M. Tur, and A. E. Willner, "Experimental demonstration of beacon-less beam
displacement tracking for an ortbial-angular-momentum-multiplexed free-space optical
link," Opt. Lett. 43, 2392-2395 (2018).

62

Chapter 5 Orbital-Angular-Momentum-
Multiplexed Free-Space Optical Link
Using Transmitter Lenses
5.1 Introduction
Due to the beams’ phase and amplitude structure, OAM multiplexed FSO
communication links present unique challenges. An OAM beam has two important
characteristics: (1) It has a “doughnut-shape” intensity profile, with little power at the
center but more power at the annulus; (2) It has a rich phase-change at the center but
less around the annulus [1-13]. To correctly recover an OAM beam, the rapid phase
change that occurs in the center needs to be collected in order to ensure orthogonality
among the OAM beams, while on the other hand, sufficient optical power should also
be received to satisfy the system requirement for optical signal-to-noise-ratio (OSNR)
[14-15]. Therefore, sufficient capture of an OAM beam is critical for an operational
communication system and will thus limit the transmission distance and the number
of modes that can be supported [16]. One approach to achieve sufficient capture is to
increase the transmitter and/or receiver aperture sizes, although this might increase
total cost and the weight of the FSO link, and degrade system flexibility. Another
approach is to use lenses at the transmitter to focus OAM beams, thus achieving
smaller beam sizes at the receiver [17]. For an OAM-multiplexed FSO link, it would
be worth investigating the effect of using transmitter lenses, on both the performance
and robustness of the system.
63

In this chapter, we show the performance effects of using transmitter lenses in
an OAM-multiplexed FSO communication link, through both simulation and
experiment, exploring its potential benefits and limitations for system performance
and robustness. Simulation results indicate that in 1 km and 10 km OAM-based FSO
links, using transmitter lenses to focus OAM beams could reduce power loss; and that
the use of transmitter lenses may reduce the power loss for higher-order OAM beams
by more than 15 dB in both links. For system robustness, we investigate the effect of
using transmitter lenses under conditions of link misalignment (i.e., angular error or
displacement between the transmitter and receiver). Both simulation and experimental
results show that transmitter lenses could also reduce angular-error-induced channel
crosstalk by ~10 dB, while on the other hand channel crosstalk resulting from
displacement might increase. A 1-m FSO communication link multiplexing four
different OAM beams each carrying 100 Gbit/s quadrature phase-shift keying (QPSK)
signal was established in a laboratory environment. The results show that power loss
could be reduced by >20 dB and that channel crosstalk could be reduced by 10~20 dB
under angular errors, by using transmitter lenses.
5.2 Concept and Simulation
The concept and model simulating the use of transmitter lenses in an OAM-
multiplexed FSO communication link are shown in Figure 5.1. Independent data
streams are carried by collimated Gaussian beams at the wavelength of 1550 nm. Each
beam pass through a spiral phase plate (SPP) with a specific OAM order, to convert
the data-carrying Gaussian beam into a corresponding data-carrying OAM beam.
Multiple OAM beams are then multiplexed and passed through a pair of transmitter
lenses (𝑓
1
and 𝑓
2
) before transmitting in free space. The equivalent focal length of the
lens pair 𝑓
3
satisfies: 1 𝑓
3
=1 𝑓
1
+1 𝑓
2
−𝑑 𝑓
1
𝑓
2
⁄ ⁄ ⁄ ⁄ , where the lens-spacing offset Δ
is defined as 𝑓
1
+𝑓
2
−𝑑 , with 𝑑 representing the center-to-center spacing between
64

these two lenses. Note that such a structure is widely used in traditional FSO systems
as a telescope where the output beam is collimated (i.e., Δ=0) [18]. In our simulation,
we use these transmitter lenses to focus OAM beams at the receiver by tuning Δ, as
shown in Figure 5.1. In the simulation, OAM beams with or without transmitter lenses
are numerically propagated by using Kirchhoff–Fresnel diffraction to the receiver
aperture placed at a certain transmission distance. At the receiver, for a specific
incoming OAM beam, its power distribution in different OAM modes is analyzed via
modal decomposition, which is based on the assumption that the power of a desired
OAM beam could be perfectly collected by its own receiver, and that there is no power
loss during the demultiplexing process [16].

Figure 5.1 Concept of an OAM-multiplexed free-space optical communication link using a pair of
transmitter lenses. Tx: transmitter; Rx: receiver; f0: equivalent focal length; d: center-to-center spacing
between the two transmitter lenses; Δ: spacing offset between two transmitter lenses.

Since an OAM beam diverges when it propagates in free-space, its spot size
might be larger than the hard-truncation receiver aperture. This results in a power loss
that grows with the propagation distance and OAM order, as Figures 5.2(a) and 5.2(b)
shows. Transmitter lenses which focus the beams could be a potential solution up to a
65

given propagation distance. Figure 5.2(b) shows that in an OAM-based FSO
communication link, using transmitter lenses reduces power loss caused by beam
divergence and limited size apertures. In this simulation, the transmitted beam
diameter, which is defined as twice of the beam waist, is 10 cm; both transmitter and
receiver aperture diameters are 10 cm; the equivalent focal length of the transmitter
lenses, 𝑓
3
, is 1 km. Figure 5.2(c) shows that in a 1 km FSO communication link with
a transmitted beam diameter of 10 cm and both transmitter and receiver aperture
diameters of 10 cm and 𝑓
1
=𝑓
2
=0.5 m, power loss decreases when the equivalent
focal length 𝑓
3
of transmitter lenses is adjusted to be around the transmission distance.
Moreover, both Figures 5.2(b) and 5.2(c) show that higher-order OAM beams (e.g.,
OAM +7 in this simulation) would show greater benefit, which might result from their
divergence characteristics. In Figure 5.2(d), we also show the use of beam transmitter
lenses to reduce power loss in a 10 km link that has a transmitted beam diameter of 30
cm and aperture diameters of 30 cm, and 𝑓
1
=𝑓
2
=1 m. Note that for FSO links of
longer distances, more accurate adjustment of Δ would be required.
We next analyze the effect of transmitter lenses on the robustness of FSO
communication links. Ideally, transmitter and receiver are perfectly aligned, i.e., the
centers of the receiver and the transmitted beam overlap, and both transmitter and
receiver are perpendicular to the line connecting their centers (see Figure 5.3(a)).
When transmitter and receiver are perfectly aligned, all power of the transmitted OAM
beam is received at the desired OAM mode, with no power leakage to neighboring
modes. However, in a practical system, transmitter and receiver may have angular
error; i.e., the center of the receiver still overlap that of the transmitter, but the receiver
may have an angular shift (see Figure 5.3(b)), and/or displacement; i.e., the receiver is
still perpendicular to the transmission direction but its center has lateral shift (see
Figure 5.3(c)) [16]. Under displacement and/or angular error, some power of the
transmitted OAM beam will spread into neighboring modes, thus causing channel
crosstalk, as shown in Figures 5.3(b) and 5.3(c). Here, channel crosstalk of a specific
66

channel is defined as the power received on a desired channel when all other channels
are transmitted and this desired channel is off (Pothers), over the received power on this
desired channel when only this specific channel is transmitted (Pself): ηxtalk (dB) =
10log10(Pothers/Pself).

Figure 5.2 (a) Intensity profiles of OAM beams with limited-size apertures, with and without transmitter
lenses at the receiver. (b) Simulated power loss as a function of transmission distance of different orders
of OAM beams; (c) and (d): Simulated power loss as a function of spacing offset between two
transmitter lenses in 1 km and 10 km link, respectively. In (b), both transmitted beam size and aperture
size are 10 cm, and the equivalent focal length of transmitter lenses is 1 km. In (c) and (d): both
transmitted beam sizes and aperture sizes are 10 cm in (c), and 30 cm in (d); focal lengths of transmitter
lenses are 0.5 m in (c) and 1 m in (d). Tx: transmitter; f0: equivalent focal length of transmitter lenses;
z: transmission distance.

67

Figure 5.3 Alignment between transmitter and receiver as well as received OAM spectrum in: (a)
Perfectly aligned link; (b) Link with only angular error; (c) Link with only displacement (Tx:
Transmitter; Rx: Receiver); (d) and (e): At the receiver, OAM beams with and without transmitter lenses
under angular error and displacement, respectively.

Careful design of the transmitter lenses could help reduce power loss, and but it
is also useful to know its effect on the robustness of an OAM-multiplexed FSO link
under angular error or displacement. This is because angular error would introduce
phase mismatch between the incoming OAM beam and the receiver plane due to
different additional phase-shift increasing linearly along the radial direction (see
Figure 5.3(d)). Therefore, using transmitter lenses might reduce phase mismatch and,
consequently, power leakage. Conversely, under the same displacement, beams with
larger spot size would experience smaller mismatch, as Figure 5.3(e) illustrates.
Therefore, using transmitter lenses may increase channel crosstalk due to lateral
displacement. Angular error and displacement could exist simultaneously in a practical
68

FSO link, necessitating a trade-off in designing how the OAM beams are focused when
using transmitter lenses at a certain level of angular error and displacement.

Figure 5.4 Simulated OAM power distribution of 1 km free-space optical link: (a) With angular error
and transmitter lenses; (b) With angular error but without transmitter lenses; (c) With displacement and
transmitter lenses; and (d) With displacement but without transmitter lenses. Transmitted beam size is
10 cm, and only OAM+3 beam is transmitted. Rx OAM = ℓ: power coupled into the receiver for mode
OAM ℓ.

Simulated power distributions of OAM beam as a function of angular error with
and without transmitter lenses are depicted in Figures 5.4(a) and 5.4(b), respectively.
Power distributions as a function of displacement with and without transmitter lenses
are shown in Figures 5.4(c) and 5.4(d), respectively. In these simulations, OAM beam
of ℓ = +3 is transmitted to a distance of 1 km, with transmitted beam size and aperture
sizes diameter 30 cm diameter, and focal length of transmitter lenses equivalent to 1
69

km. Figure 5.4(a) shows less power leakage to neighboring modes compared with
Figure 5.4(b) under the same angular error, indicating that the use of transmitter lenses
could enhance system robustness under angular error, through possible decrease in
channel crosstalk. Conversely, using transmitter lenses might reduce the link tolerance
of displacement, as Figures 5.4(c) and 5.4(d) indicate.
5.3 Experimental Demonstration
An experiment is conducted in a laboratory environment, over a transmission
distance of ~1 m, in which spatial light modulators (SLMs) are functioned as SPP in
simulation, (see Figure 5.5). A narrow linewidth laser at 1550 nm is sent to a Mach–
Zehnder modulator to produce a 100 Gbit/s QPSK signal. After amplification, this
signal is split into two copies, one of which is delayed using a ~10 m length of single-
mode-fiber (SMF) to decorrelate the data sequence. The two fiber branches are
coupled to collimators, each of which emits a collimated Gaussian beam of diameter
2.2 mm. One of the Gaussian beams is converted to OAM+1 by SLM-1, while the
other is converted to OAM+3 by SLM-2. After being combined via a beam splitter
(BS-1), the multiplexed OAM beams are split into two identical copies by another
beam splitter (BS-2). One of the copies is reflected by three mirrors to generate OAM
-1 and OAM -3 beams, which are then multiplexed with the other copy (i.e., OAM +1
and +3) by a third beam splitter (BS-3).

70

Figure 5.5 Experimental setup for a 1 m OAM-multiplexed free-space optical link with transmitter
lenses. Col: Collimator; SLM: spatial light modulator; BS: beam splitter; Tx: transmitter; Rx: receiver.

The four resulting multiplexed OAM beams pass through two transmitter lenses,
both of focal length 10 cm. The spacing between these two lenses is adjustable such
that the equivalent focal length is tunable. After free-space transmission of
approximately 1 m, the beams are sent to SLM-3 loaded with an inverse-spiral phase
pattern of the particular OAM channel to be detected. Such an OAM beam is converted
to a Gaussian-like beam and then coupled into an SMF and sent for power
measurement and/or coherent detection. First, we measured the power loss of the
OAM beam when transmitted with/without transmitter lenses. Figures 5.6(a) and 5.6(b)
show simulated and experimental power losses of OAM +3 and OAM +7 as a function
of receiver aperture size when only OAM +3 or OAM +7 respectively is transmitted
with perfect alignment. Limited size receiver apertures are implemented by adding a
truncated pattern onto SLM-3. The results show that an FSO link of ~1 m using
transmitter lenses shows >20dB less power loss than that without transmitter lenses.
Measured and simulated power losses show similar trends.

71

Figure 5.6 (a) and (b) Simulated and experimental power loss as a function of receiver aperture size
when only OAM +3 or only OAM +7 is transmitted with perfect alignment; (c) and (d) Experimental
results of OAM power distribution when only OAM +3 is transmitted under angular errors with and
without transmitter lenses, respectively; (e) and (f) Experimental results of OAM power distribution
when only OAM +3 is transmitted under displacement with and without transmitter lenses, respectively.

Next, we measure power distribution when OAM +3 is transmitted under
angular errors with and without transmitter lenses (Figures 5.6(c) and 5.6(d)
respectively). The results indicate that when using transmitter lenses, the link is more
72

tolerant of angular errors. For example, with an angular error of 200 µrad in the ~1 m
FSO link, the power received by the designed mode (i.e., OAM +3) is still >20 dB
more than the power leaked into channel OAM +1 and OAM +5; at the same angular
error without using transmitter lenses, the power received by OAM +3 is almost the
same as that leaked into OAM +1 and OAM +5. The power distributions when OAM
+3 is transmitted under displacement with and without transmitter lenses are shown in
Figures 5.6(e) and 5.6(f), which indicate that using transmitter lenses might increase
channel crosstalk under displacement in such an FSO link. Note that in Figures 5.6(c)–
5.6(f), some power is still leaks into neighboring modes even when angular error or
displacement is zero, which might be caused by imperfections in the alignment and
free-space-to-fiber coupling.
Next, all four channels (i.e., OAM -3, OAM-1, OAM +1 and OAM+3) were
transmitted simultaneously. Figures 5.7(a) and 5.7(b) show the crosstalk of OAM +3
channel under angular error and displacement, respectively. The results show that with
transmitter lenses, crosstalk of channel OAM +3 from all three of the other channels
decreases by 10~20 dB under angular errors ranging from 100 µrad to 500 µrad. On
the other hand, with transmitter lenses, crosstalk of channel OAM +3 increases by
5~10 dB under displacement ranging from 0.1 mm to 0.5 mm. Measured crosstalk is
in good agreement with the simulation results.

73

Figure 5.7 (a) and (b): Experimental results of crosstalk for channel OAM +3 when OAM ±1 and ±3
are transmitted under angular error and displacement, respectively; (c) and (d): Experimental
measurement of bit-error-rate as a function of optical signal-to-noise ratio (OSNR) for channel OAM
+3 when OAM ±1 and ±3 are transmitted under angular error and displacement, respectively. w/: with
transmitter lenses; w/o: without transmitter lenses; x µrad: with x µrad angular error; y mm: with y mm
displacement.

Figures 5.7(c) and 5.7 (d) show bit error rate (BER) curves of OAM +3 when all
four channels are transmitted with or without transmitter lenses under angular errors
and displacements, respectively (each beam carries a 50 Gbaud QPSK signal). Figure
5.7(c) shows that when the link has an angular error of 100 μrad, without transmitter
lenses, this ~1 m FSO link could not achieve the 7% overhead forward error correlation
(FEC) limit of 3.8×10
-3
, whereas a link with transmitter lenses could. Moreover, with
transmitter lenses, this FSO link could still achieve the FEC limit even under 300 μrad
or 400 μrad angular error, with little power penalty compared with the 100 μrad case,
74

demonstrating improved system robustness under angular errors. On the other hand,
such a link would have higher power penalty under lateral displacement. Figure 5.7(d)
shows that at 0.20 mm displacement, the use of transmitter lenses imposes a power
penalty of ~8 dB compared with an FSO link without using transmitter lenses, and that
higher power penalty might occur under larger displacement.
5.4 Conclusion
The potential benefits and limitations of using transmitter lenses with FSO
communication link multiplexing multiple OAM beams are investigated through
simulation and experiments [21]. We find that, within certain transmission distances,
the use of transmitter lenses to focus OAM beams could reduce power losses in OAM-
based FSO links with limited-size apertures; and could enhance system robustness
under angular error but degrade tolerance of displacement. In practical cases, angular
error and displacement are generally time-varying stochastic processes. Our approach
may help analyze the upper and lower bounds of system robustness with or without
transmitter lenses. This work only considers using continuous wave (CW) OAM
beams. For OAM-carrying pulsed beams, the intensity profiles would be more
complicated due to the broader bandwidth. This may result in unique effects on OAM
multiplexing in a lens system as well as different power loss and channel crosstalk
compared with CW beams, which needs to be further studied [19-20]. Additionally,
atmospheric turbulence might also result in distortion in OAM-based FSO links, the
effects of which could be considered as angular error and displacement discussed
above. However, severe signal fading might occur due to the intensity and phase
fluctuation of the OAM beams. Such effects are not considered in this work and need
further exploration.

75

5.5 Reference
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Arnold, “Free-space information transfer using light beams carrying orbital angular
momentum,” Opt. Express 12, 5448-5456 (2004).
[2] I. Djordjevic, "Deep-space and near-Earth optical communications by coded orbital
angular momentum (OAM) modulation," Opt. Express 19, 14277-14289(2011).
[3] J. Wang, J.-Y. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar,
M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital
angular momentum multiplexing,” Nat. Photon. 6, 488-496 (2012).
[4] D. Richardson, J. Fini, and L. E. Nelson, "Space-division multiplexing in optical fibres,"
Nat. Photon. 7, 354-362 (2013).
[5] A. M. Yao, and M. J. Padgett, "Orbital angular momentum: origins, behavior and
applications," Adv. Opt. and Photon. 3, 161-204 (2011).
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momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys.
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momentum states of photons,” Nature 412, 313-316 (2001).
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in an orbital angular momentum-multiplexed free-space optical link," Appl. Opt. 47,
2414-2429 (2008).
76

[11] H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Carcia-Vidal,
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satellite-borne, free-space optical communication systems," Opt. Engineering 24,
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S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, "Performance metrics and
design considerations for a free-space optical orbital-angular-momentum–multiplexed
communication link," Optica 2, 357-365 (2015).
[17] L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan,
Z. Wang, N. Ashrafi, S. Ashrafi, R. D. Linquist, M. Tur, and A. E. Willner, "Performance
enhancement of an orbital-angular-momentum-based free-space optical communication
link through beam divergence controlling," In 2015 Optical Fiber Communication
Conference, M2F-6. (2015).
[18] Y. Arimoto, M. Presi, V. Guarino, A. D’Errico, G. Contestabile, M. Matsumoto, and E.
Ciaramella, "320 Gbits/s (8× 40Gbits/s) double-pass terrestrial free-space optical link
transparently connected to optical fibre lines," In Proc. ECOC, pp. 1-2. (2008).
[19] S. Feng, and H. G. Winful, "Higher-order transverse modes of ultrashort
isodiffracting pulses," Phys. Rev. E 63, 046602 (2001).
[20] Bliokh, Konstantin Y., and Franco Nori. "Spatiotemporal vortex beams and
angular momentum," Phys. Rev. A 86, no. 3 (2012): 033824.
77

[21] L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery,
Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, A. E. Willner,
"Orbital-angular-momentum-multiplexed free-space optical communication link
using transmitter lenses," Appl. Opt. 55, 2098-2103 (2016).

78

Chapter 6 Power Loss Mitigation of Orbital-
Angular-Momentum-Multiplexed Free-
Space Optical Links Using Non-Zero
Radial Index Laguerre-Gaussian Beams
6.1 Introduction
Previous publications describing the use of OAM-multiplexed data channels have
typically shown data transmission using p = 0 beams of different ℓ values [1-14]. Previously,
an OAM-multiplexed FSO link using both the radial and azimuthal degrees of freedom is
demonstrated [15]. In general, the receiver should capture the complete phase change of each
of the different OAM beams in order to maintain orthogonality among the beams. Given that
the most phase change occurs near the beam center in the azimuthal direction, the receiver
would typically need to be placed on axis [16]. Unfortunately, LG beams have a fairly large
area near the center for which the signal power is low [17]. Therefore, although orthogonality
is maintained, the recovered signal power can become fairly low when the aperture size is
smaller than the beam’s annulus. This limited-receiver-aperture scenario is common since an
optical beam diverges with propagation, and divergence increases with higher OAM values
[17].
The above issue of small power in the beam center is most sever for LG beams with p
= 0 radial indices. However, the beam’s intensity structure changes for higher p values [18].
Indeed, for limited transmission distances, the size of the innermost ring of a p > 0 beam can
be smaller than that of the single ring of the p = 0 beam with the same ℓ value. This indicates
79

that p > 0 beams may have more signal power near the beam center for such transmission
distances [18]. This observation can potentially provide higher received signal-to-noise ratio
under limited-size receiver apertures for OAM-multiplexed systems [18-19].
In this paper, we explore the use of LG beams with p > 0 in OAM-multiplexed
FSO communication links. Through simulation and experiments, we investigate the
power loss incurred by the beam divergence and limited-size receiver apertures. Our
results show that with limited-size receiver apertures, the received signal power of a p
> 0 beam is higher as compared to a p = 0 beam that has the same ℓ value under certain
short transmission distances. We also show that such advantage of using p > 0 beams
may be negated when using larger size receiver apertures, and/or at longer
transmission distances. Moreover, we experimentally demonstrate a ~1-m FSO link
by multiplexing four LG beams with the same non-zero p value but different ℓ values
(i.e., LG1,-3, LG1,-1, LG1,1, and LG1,3). The measured signal power loss of a p = 1 beam
is about 6 dB less than using a p = 0 beam with the same ℓ value. The worst-case
intermodal crosstalk among these four modes is below -15 dB, and a total capacity of
400-Gbit/s with power penalties less than 6 dB for all channels is achieved.
6.2 OAM Beams with Non-Zero Radial Indices
At a transmission distance of z, the beam radius of an LGp,ℓ beam could be
characterized as [20]:
𝑤(𝑧)=𝑤
3
?2𝑝+ℓ+1B1+C
DE
FG
H
(
I
2
(2)
where w(z) is the beam radius at the transmission distance of z, w0 is the beam
waist of the transmitted beam at z = 0, and λ is the wavelength of the beam. A set of
LG beams with different ℓ values and different p values, each of which has a
transmitted beam waist of w0 = C/(2p+|ℓ|+1)
1/2
(where C is a constant), have the same
80

beam size (i.e., the same beam radius) at z = 0. Here, the beam size is defined by the
second moment of the intensity of a beam [20]. Among these beams, a p > 0 beam
would have more power near the beam center than a p = 0 beam under limited
transmission distances, as Figure 6.1 shows. Note that between each ring of a p > 0
beam, there is a “dark gap” where little power resides.

Figure 6.1 Normalized intensity and phase profile of Gaussian, LG0,5, LG1,5, and LG3,5 beams.

To investigate the potential advantage of using LG beams with p > 0, we
simulate signal power loss of LG beams under limited-size receiver apertures at
different transmission distances. The concept is illustrated in Figure 6.2, where all LG
beams have the same transmitted power and the same transmitted beam size at z = 0,
i.e., all LG beam have the same w(0) as illustrated in Eq. 1 and the single ring of a p =
0 beam has a power equal to the total power of all rings of a p > 0 beam. These
multiplexed beams coaxially pass through the transmitter aperture, and numerically
propagate using Kirchhoff-Fresnel diffraction to the receiver aperture placed at a
certain transmission distance [16]. At the receiver, each specific incoming LG beam
is down-converted into a Gaussian-like beam by passing through an inverse spiral
81

phase plate (SPP). Note that such SPPs only contain azimuthal phase change
regardless of the p values of the incoming LG beam. We assume perfect alignment
between the transmitter and the receiver in our simulation.

Figure 6.2 Concept of free-space optical link transmitting LG beams with non-zero radial indices. Tx:
transmitter; Rx: receiver.

Figure 6.3(a) shows the power loss for LG beams of different ℓ and p values as
functions of transmission distance, when the transmitted beam diameter is 8 cm and
the receiver aperture diameter is 4 cm. We observe that for the same ℓ value, p > 0
beams have less power loss than p = 0 beams. This is because the receiver aperture
covers only part of the single ring of the p = 0 beams as well as only part of the inner
ring of p > 0 beams, and p > 0 beams show advantage due to the relatively smaller size
of the inner ring. Since a p > 0 beam diverges faster compared with a p = 0 beam of
the same ℓ value and transmitted beam size, the size of the innermost ring of the p > 0
beam may also grow faster than the single ring of the p = 0 beam as the transmission
distance increases, thus negating the advantage of using p > 0 beams at relatively
longer transmission distances. As an example, the advantage of using LG1,5 decreases
from 14 dB at 100 m to 6 dB at 700 m.
82

The advantage of using p > 0 beams may also diminish under lager size receiver
apertures. We simulate power loss for LG beams of different ℓ and p values as
functions of transmission distance when the receiver aperture diameter is 6 cm, 8 cm
and 10 cm, as shown in Figures 6.3(b), 6.3(c) and 6.3(d), respectively. In all cases, the
transmitted beam diameter is 8 cm. It can be seen that: (1) the power loss of LG0,1 is
smaller than LG1,1 at most transmission distances, which is because though the inner
ring of LG1,1 is smaller than the single ring of LG0,1, the receiver aperture here could
cover most of the single ring of LG0,1 but not large enough for the outer ring of LG1,1;
(2) LG1,5 has higher power loss than LG0,5 at relatively short transmission distances,
i.e., < 300 m in Figure 6.3(b), < 650 m in Fig. 3(c) and < 900 m in Figure 6.3(d), which
is because the receiver aperture covers most of the single ring of LG0,5 but only part
of the outer ring of LG1,5. As the beam diverges, the power loss of LG0,5 continuously
increases as the transmission distance grows, because the part of the single ring that
the receiver aperture could cover continuously decreases. While for LG1,5, as the
transmission distance increases, the power loss: (1) at first slightly increases because
the receiver aperture covers part of the outer ring and this part decreases as the beam
diverges; (2) then, i.e., from 200 to 450 m in Figure 6.3(b) and from 400 to 650 m in
Figure 6.3(c), remains almost unchanged, because the edge of the receiver aperture is
within the “dark gap” between the outer ring and the inner ring, such that there is little
additional power loss though the beam diverges; (3) continues to increase since the
receiver aperture only covers part of the inner ring, and this is where the p > 0 beam
shows its advantage.
In this paper, we explore the use of LG beams with p > 0 in OAM-multiplexed
FSO communication links. Through simulation and experiments, we investigate the
power loss incurred by the beam divergence and limited-size receiver apertures. Our
results show that with limited-size receiver apertures, the received signal power of a p
> 0 beam is higher as compared to a p = 0 beam that has the same ℓ value under certain
short transmission distances. We also show that such advantage of using p > 0 beams
83

may be negated when using larger size receiver apertures, and/or at longer
transmission distances. Moreover, we experimentally demonstrate a ~1-m FSO link
by multiplexing four LG beams with the same non-zero p value but different ℓ values
(i.e., LG1,-3, LG1,-1, LG1,1, and LG1,3). The measured signal power loss of a p = 1 beam
is about 6 dB less than using a p = 0 beam with the same ℓ value. The worst-case
intermodal crosstalk among these four modes is below -15 dB, and a total capacity of
400-Gbit/s with power penalties less than 6 dB for all channels is achieved.

Figure 6.3 Simulated power loss of LG beams with radial indices of p = 0 and p > 0 as functions of
transmission distance, with transmitted beam size of 8 cm and receiver aperture diameter of (a) 4 cm,
(b) 6 cm, (c) 8 cm, and (d) 10 cm.

84

Figure 6.4(a) and 6.4(b) show the power loss of LG beams as functions of
aperture diameter when the transmission distances are 500 m and 1 km, respectively.
In both cases, the transmitted beam diameter is 8 cm. In Fig. 4, the intercept points of
the curves for LG beams with p = 0 and p = 1 with the same ℓ values are shown,
indicating that the potential advantage of using LG beams with p > 0 might be more
significant when the aperture size is small. The flat region of a p = 1 curve is where
the edge of the receiver aperture is in the “dark gap” between the inner ring and the
outer ring. It is also observed that the intercept points for LG beams with different |ℓ|
values shift to the right as the |ℓ| value increases, indicating that the aperture diameter
range where p > 0 beams may have more received power is larger when the |ℓ| value
is higher.

Figure 6.4 Simulated power loss of LG beams with radial indices of p = 0 and p > 0 as functions of
receiver aperture diameter, with transmitted beam size of 8 cm and transmission distance of (a) 500 m
and (b) 1 km.

85

The simulation results of LG beams with the same ℓ value but different p values
are shown in Figure 6.5, in which the receiver aperture size is fixed to 4 cm in Figure
6.5(a), and the transmission distance is fixed to 500 m in Figure 6.5(b). We observe
that for the same ℓ value, LG beams with higher p values has less power loss than LG
beams with lower p values. As p value increases, the extra advantage of a higher p
value decreases, e.g., the power loss between LG3,5 and LG1,5 is smaller than that
between LG1,5 and LG0,5.

Figure 6.5 Simulated power loss of LG beams with different radial indices as functions of (a)
transmission distance, with receiver aperture diameter of 4 cm, and (b) receiver aperture diameter, with
transmission distance of 500 m. In both cases, the transmitted beam size is 8 cm.

6.3 Link Demonstration
An ~1-m FSO communication link in a lab environment is demonstrated, where
spatial light modulators (SLMs) serve as the phase holograms in our simulation, shown
in Figure 6.6. A narrow line-width laser beam at 1550 nm is sent to a Mach-Zehnder
86

modulator. After amplification, this signal is split into two copies, one of which is
delayed using a ~10-m length of single-mode-fiber (SMF) to decorrelate the data
sequence. The two fiber branches are coupled to collimators, each of which emits a
collimated Gaussian beam with a beam diameter of 2.2 mm. One of the Gaussian
beams is converted to LG1,1 by SLM-1, while the other one is converted to LG1,3 by
SLM-2. Both the generated LG1,1 and LG1,3 beams right after the SLMs have beam
diameters of ~4 mm. After being combined by using a beam splitter (BS-1), the
multiplexed OAM beams are split into two identical copies by another beam splitter
(BS-2). One of the copies is reflected by three mirrors to generate LG1,-1 and LG1,-3
beams, which are then multiplexed with the other copy (i.e., LG1,1 and LG1,3) by the
third beam splitter (BS-3). After free-space transmission, the beams are sent to SLM-
3 loaded with an inverse spiral phase pattern of the particular channel to be detected.
Such spiral phase pattern only contains phase change in the azimuthal direction
regardless of the p values. The LG beam is converted to a Gaussian-like beam and then
coupled into an SMF and sent for power measurement and coherent detection. The
measured intensity profile and interferogram of LG1,1 and LG1,3 beams are depicted in
Figure 6.7(a).

Figure 6.6 Experimental setup for a ~1-m OAM-multiplexed free-space optical link transmitting OAM
beams with non-zero radial indices. Col.: collimator; SLM: spatial light modulator; BS: beam splitter;
EDFA: Erbium-doped fiber amplifier; PC: polarization controller; f1, f2: lenses.
50/50
coupler
EDFA
50 Gbaud
QPSK
signal
generator
SLM-1
Coherent
receiver
Delay
PC
Col.
SLM-2
SLM-3
Mirror
BS-1
BS-2 BS-3
f
1
f
2
Tx
1-m free space transmission
Rx
87

We measure the signal power of LG0,1 and LG1,1, LG0,3 and LG1,3, and LG0,5 and
LG1,5 with different receiver aperture sizes as shown in Figure 6.7(b). In this
measurement, different holograms are loaded on the SLM to generate different LG
beams. The holograms are designed such that LG0,1 and LG1,1, as well as LG0,3 and
LG1,3 beams have the same beam size, respectively. Our measurement shows that
when the receiver aperture diameter is < 3 mm, LG1,1, LG1,3 and LG1,5 beams have a
received signal power 6 dB higher than LG0,1, LG0,3, and LG0,5 beams, respectively;
when the receiver aperture is > 4 mm, the power loss between the p = 0 beam and the
p = 1 beam with the same ℓ value shows little difference, which indicates that p > 0
beams may allow more signal power being received compared with p = 0 beams with
limited size receiver apertures. Note that the receiver aperture size in our experiment
is relatively small, such that LG1,1 has less power loss compared to LG0,1.

Figure 6.7 Experimental results of (a) measured intensity profile and interferrogram of LG1,1 and LG1,3
beams, and (b) measured power loss due to limited-size receiver as a function of aperture diameter, with
all transmitted beam sizes of 2.0 mm in diameter.

88

Figure 6.8 (a) and (b) Experimental results of OAM power distribution when only LG0,3 or LG1,3 beam
is transmitted with a receiver aperture of 3 mm, respectively. (c) and (d) Experimental results of received
power on designed and neighboring modes with different receiver aperture sizes when only LG0,3 beam
or LG1,3 beam is transmitted. Tx: LGp,ℓ: LGp,ℓ beam is transmitted; Rx: LGp,ℓ: received power on LGp,ℓ
mode.

Figure 6.8(a) and Fig. 6.8(b) show the power distribution for different ℓ values
when only LG0,3 or LG1,3 is transmitted with a receiver aperture diameter of 3 mm,
respectively. Here, 3 mm is chosen where p > 0 beams would show less signal power
loss than p = 0 beams. It is observed that when LG0,3 or LG1,3 are individually
transmitted, LG1,3 has higher signal power as compared with LG0,3, and their received
OAM power spectra are similar. Figure 6.8(c) and 6.8(d) show the received power on
different ℓ values when only LG0,3 or LG1,3 is transmitted with different receiver
apertures, respectively. Note that in ideal cases, no power leaks into neighboring
modes for both transmitting LG0,3 and LG1,3 beams, and the crosstalk here might due
89

to imperfect devices and alignment. In this experiment, the above measured results
indicate that using LG beams with p > 0 does not seem to introduce appreciable
additional crosstalk compared with using p = 0 beams under similar limited-aperture
and well-aligned conditions.
We also transmit all four LG beams (LG1,-3, LG1,-1, LG1,1, and LG1,3)
simultaneously, each of which carries a 100-Gbit/s quadrature-phase-shift-keying
(QPSK) signal. Figure 6.9(a) shows the measured received power on all four channels.
The crosstalk of channel LG1,-3, LG1,-1, LG1,1, and LG1,3 are -17.0 dB, -21.0 dB, -15.8
dB, and -17.4 dB respectively. Figure 6.9(b) shows the bit-error-rate (BER) as a
function of optical signal-to-noise ratio (OSNR) of all four channels as well as the
back-to-back case (i.e., using LG0,0 beam without multiplexing), and the power
penalties of all four channels are less than 6 dB. We observe in Figure 6.9(b) that the
LG1,-1 channel has the lowest power penalty and LG1,1 has the highest power penalty,
which might be caused by imperfect alignment, such that the LG1,-1 channel has the
lowest crosstalk from all other channels while the LG1,1 channel has the highest. This
is consistent with the channel crosstalk we measured in Figure 6.9(a). The results show
that this system could achieve the 7% overhead forward error correction (FEC) limit
of 3.8×10
-3
for all four channels. Note that previous publications have shown 1-m FSO
links transmitting 400 G-bit/s QPSK signal using LG0,-3, LG0,-1, LG0,1, and LG0,3
beams, and similar power penalties have been observed [21].

90

Figure 6.9 Experimental measurement of (a) power distribution of all four OAM channels (i.e., LG1,-3,
LG1,-1, LG1,1, and LG1,3), and (b) bit-error-rates as functions of OSNR of the back-to-back case as well
as all different channels when all four OAM channels are transmitted simultaneously. OSNR: optical
signal-to-noise ratio; b2b: back-to-back.

6.4 Conclusion
In this work, we explore the use of p > 0 LG beams in OAM-multiplexed FSO
communication links. Its potential advantage and impact on system performance are
investigated. Our simulation and experiment results show that LG beams with non-
zero radial indices could help in receiving higher signal power in an FSO link under
certain short transmission distances and limited-size receiver apertures. We emphasize
that the design decision of either using p > 0 or p = 0 beams will depend on many
system factors, such that certain aperture sizes and/or transmission distances may favor
one approach over the other.
Our experiment primarily focuses on LG beams with p = 1, and could also be
applied to LG beams with other p > 0 values. In our experiment, we transmit four LG
beams with different ℓ values but the same p value. It is also possible to transmit
multiple LG beams with: (i) the same ℓ value but with different p values by detecting
91

the beam’s radial phase change [22], and (ii) different ℓ values and different p values
by detecting the beam’s phase change both in the radial direction as well as in the
azimuthal direction [15]. It is also worth noting that: (1) when fully received, LG
beams with different ℓ or p values are orthogonal to each other; (2) when the
transmitter and receiver aperture sizes are limited and only part of the beams are
received: (i) LG beams with different ℓ values might still be separated with low
inherent crosstalk, regardless of their p values [16], whereas (ii) LG beams with the
same ℓ value but different p values may suffer intermodal crosstalk [22].
Further studies may investigate the effect of OAM beams with different p values,
such that for each OAM beam with a specific ℓ value, we may design a specific p value
accordingly to further improve the system performance. Moreover, the crosstalk
tolerance for p > 0 beams under transmitter-receiver misalignment, where the ring
structure of the device for de-multiplexing does not match the cross structure of the
received beam, also needs further studies [23].
6.5 Reference
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R. D. Linquist, M. Tur, and A. E. Willner, "Performance metrics for a free-space
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94

Chapter 7 Mode and Space Diversity in Free-
Space Optical Link to Increase System
Tolerance to Turbulence
7.1 Introduction
Free-space optical (FSO) communication links are of potential importance due
to their increased data capacity and reduced probability of eavesdropping, as compared
to radio-frequency (RF) links [1,2]. However, FSO links are more sensitive to various
degrading effects, such as atmospheric turbulence, rain, fog, and small obstructions
[3,4]. A common approach in wireless communication systems for increasing the link
reliability and performance is to use spatial diversity transmission and reception, such
that multiple copies of the same data stream are simultaneously transmitted and
received by multiple spatial apertures [5,6].
In FSO links, two different approaches to utilizing spatial diversity were
reported. The first approach is similar to RF systems, in which multiple data
transmitter and receiver apertures are used [7,8]. A second approach is to use multiple
orthogonal spatial modes employing a single transmitter/receiver aperture pair [9,10].
Such spatial mode diversity could use one Laguerre-Gaussian (LG) mode and one
Hermite-Gaussian mode [9], or multiple orbital-angular-momentum (OAM) modes
[10]. OAM beams, which is a subset of LG beams, have phase fronts “twist” in a
helical fashion as the beams propagate. A beam’s OAM mode order ℓ corresponds to
the number of 2π phase shifts in the azimuthal direction. OAM beams with different
orders are mutually orthogonal [11].
95

In this chapter, we experimentally explore using both OAM mode and space
(i.e., aperture) diversity in an FSO communication link to further increase system
tolerance to turbulence. We demonstrate a ~1-m, 100-Gbit/s quadrature phase-shift
keying (QPSK) FSO link using 2 transmitter/receiver aperture pairs, each transmitting
and receiving a Gaussian beam and an OAM beam, with all beams carrying the same
data stream. A rotatable phase plate with a pseudo-random phase distribution, obeying
Kolmogorov spectrum statistics is used to emulate atmospheric turbulence [8]. Results
indicate that using mode and space diversity could help improve system reliability
under turbulence. In our experiment, the two-prong diversity approach helps to
achieve bit-error-rates (BERs) mostly below 3.8×10
-3
under emulated turbulence
conditions with a Fried parameter r0 of 0.4 mm and transmitted beam diameters of 3
mm.
7.2 Concept and Experimental Setup
In this chapter, we experimentally explore using both OAM mode and space
(i.e., aperture) diversity in an FSO communication link to further increase system
tolerance to turbulence. We demonstrate a ~1-m, 100-Gbit/s quadrature phase-shift
keying (QPSK) FSO link using 2 transmitter/receiver aperture pairs, each transmitting
and receiving a Gaussian beam and an OAM beam, with all beams carrying the same
data stream. A rotatable phase plate with a pseudo-random phase distribution, obeying
Kolmogorov spectrum statistics is used to emulate atmospheric turbulence [8]. Results
indicate that using mode and space diversity could help improve system reliability
under turbulence. In our experiment, the two-prong diversity approach helps to
achieve bit-error-rates (BERs) mostly below 3.8×10
-3
under emulated turbulence
conditions with a Fried parameter r0 of 0.4 mm and transmitted beam diameters of 3
mm.
96

The concept of an FSO link using both OAM mode and space diversity is
illustrated in Figure 7.1. The link consists of N transmitter/receiver aperture pairs
arranged in a linear uniform structure. Each of the transmitter apertures coaxially
transmits M multiple coaxial OAM beams (ℓ = 0 is the Gaussian beam). All beams
carry the same data stream. In comparison to a single-transmitter-aperture, single-
transmitted-mode case, the transmit power of each beam is scaled by 1/NM to conserve
the overall transmit power. N receiver apertures are used to individually capture the
beams emitted from the corresponding transmitter aperture. For each receiver aperture,
the beams are OAM-demultiplexed and received as different modes. In our
demonstration, selection combining is employed to combine all modes from all
receiver apertures, from which the received mode with maximal received power is
determined and used for signal detection [12]. In such a mode and space diversity
scheme, OAM beams from different apertures are affected by the atmospheric
turbulence in a statistically independent way so that appropriate processing of the
multiple outputs can potentially achieve an overall link outage probability lower than
a single-channel case [7-10].

Figure 7.1 Concept of a free-space optical (FSO) communication link using both orbital-angular-
momentum (OAM) mode and space diversity for increased system tolerance to atmospheric turbulence.
Rx: receiver; Tx: transmitter.

Tx aperture
array
Rx aperture
array
OAM mode
separation
Aperture #1
Aperture #2
Aperture #N
Aperture #1
Aperture #2
Aperture #N
…
…
…
Transmit same
data stream
OAM generation
& coaxial combine
Distorted
Gaussian and
OAM beams
Multiple coaxial
Gaussian and
OAM beams
Signal detection
OAM ℓ
0
OAM ℓ
M
…
…
Channel selection
OAM ℓ
0
OAM ℓ
M
…
OAM ℓ
0
OAM ℓ
M
…
Atmospheric
turbulence
…
OAM generation
& coaxial combine
OAM ℓ
0
OAM ℓ
1
OAM ℓ
M
…
Tx
Rx
OAM generation
& coaxial combine
97

Figure 7.2 shows the experimental setup. Two laser diodes at 1550 and 1552 nm
are individually modulated with identical 100-Gbit/s QPSK signal, amplified and each
split into two branches. All branches are fed to collimators which emit 3-mm-diameter
Gaussian beams. The two 1550-nm beams are converted to OAM beams using two
spatial light modulators SLM-1 and SLM-2, and then each coaxially combined with a
1552-nm beam. Therefore, two transmitter apertures, each transmitting a Gaussian
beam and an OAM beam, are built with a center-to-center spacing S of 12 mm. The
resulting beams from the two apertures pass through a thin rotatable phase plate
emulating atmospheric turbulence with a Fried parameter r0 of 0.4 mm. Since S >> r0,
we assume that beams from the two apertures would experience independent
turbulence-induced distortions [9]. When the phase plate is rotated, the beams pass
through time-varying random turbulence realizations. Two receiver apertures are
located at a distance of ~1 m and aligned with the corresponding transmitter apertures.
At each receiver, the multiplexed beams are received as different OAM modes using
an OAM mode separator based on multiplane conversion [13]. The channel with the
maximal received power is then selected and sent for coherent detection. We note that
wavelengths of the two laser diodes are close to each other, and the effects of
wavelength diversity are not considered in our demonstration.

Figure 7.2 Experimental setup. BS: beamsplitter; EDFA: erbium-doped fiber amplifier; LD: laser diode;
QPSK: quadrature phase-shift keying; Rx: receiver; S: center-to-center aperture spacing; SLM: spatial
light modulator; Tx: transmitter.

Transmitter
EDFA
SLM 2
100 Gbit/s
QPSK signal
50/50
coupler
Collimator SLM 1
Rotatable
turbulence plate
Tx aperture 1
Tx aperture 2
OAM mode
separator
OAM mode
separator
Rx aperture 1
Rx aperture 2
Receiver
Channel selection
EDFA
Coherent detection
1-m free space transmission
50/50 BS
Modulator
Modulator
LD
LD
ℓ
0
ℓ
0
ℓ
1
ℓ
1
ℓ
0
ℓ
0
ℓ
1
ℓ
0
ℓ
0
ℓ
1
ℓ
1
ℓ
1
S
98

7.3 Link Demonstration and Analysis
First we measure the received OAM spectrum when Gaussian beams are
transmitted under 50 random turbulence realizations through only one or both
apertures with selection combining, as shown in Figure 7.3(a). We observe that the
OAM spectra for both apertures fluctuate significantly as turbulence varies, and using
aperture diversity reduces the fluctuation range. Figure 7.3(a) also indicates that a fair
amount of power from the transmitted mode is coupled into its neighboring modes.
The received power distribution and link outage percentage are measured under the
different diversity schemes as shown in Figures 7.3(b) and 3(c). For no diversity, only
aperture 1 is transmitting/receiving a single Gaussian beam; for mode diversity, only
aperture 1 is transmitting/receiving a Gaussian beam and an OAM +1 beam; for space
diversity, both aperture 1 and 2 are transmitting/receiving Gaussian beams; for mode
and space diversity, both aperture 1 and 2 are transmitting/receiving Gaussian beams
and OAM +1 beams. We note that the total transmit power is 10 dBm for all diversity
and no diversity schemes. Results show that mode and space diversity helps further
improve the received power distribution and reduce link outage percentage under
turbulence.
99

Figure 7.3 Experimental results on (a) received OAM spectrum of transmitted Gaussian beam when
only one or both aperture(s) are transmitted. Each color bar represents the fluctuation range; (b)
Received power distribution and (c) link outage percentage as a function of required received power
with no diversity, only mode diversity, only space diversity, and both mode and space diversity. All
measurements are taken under 50 random turbulence realizations by rotating the turbulence phase plate.
For (b) and (c), total transmit power is 10 dBm.

Figure 7.4(a) shows the link outage percentage when different mode diversity
schemes are implemented. For no mode diversity, only Gaussian beams are
transmitted and Gaussian modes are received; for Tx mode diversity, both Gaussian
beams and OAM +1 beams are transmitted and only Gaussian modes are received; for
Rx mode diversity, only Gaussian beams are transmitted and both Gaussian and OAM
+1 modes are received; for Rx/Tx mode diversity, both Gaussian and OAM +1 beams
are transmitted and both modes are received. Space diversity is implemented in all
cases. The Rx and Rx/Tx mode diversity schemes show lower outage percentages
among these schemes. Figure 7.4(b) shows the link performance when mode and space
diversity is implemented using different OAM modes. In our measurement, the use of
Gaussian and OAM +1 modes shows the highest operating percentage (i.e. lowest
outage percentage), which may result from the higher intermodal coupling between
(a) (b) (c)
95%
system
operates
-44 dBm
-38 dBm
Gaussian beam
transmitted
Fluctuation
range
100

these two modes. Figure 7.4(c) shows that using mode and space diversity with
different aperture spacing has similar performance in our demonstration.

Figure 7.4 Experimental results on (a) link outage percentage when mode diversity is not used, only at
the transmitter, only at the receiver, and both, with a total transmit power of 10 dBm; Link operating
percentage as a function of total transmit power with (b) different OAM modes and (c) different aperture
spacing when mode and space diversity is implemented. For (b) and (c), the required received power is
-40 dBm.

In the BER measurements, all beams carry the same 100-Gbit/s QPSK signal.
Figure 7.5(a) shows BERs with and without using mode and space diversity under 50
random turbulence realizations with a total transmit power of 10 dBm. We observe
that mode and space diversity helps reduce BERs to mostly below the 7% forward-
error correction (FEC) limit of 3.8×10
-3
. Figure 7.5(b) shows BER as a function of
total transmit power under one of the turbulence realizations. In Figure 7.5(c), we
simulate the required transmit power when more apertures are implemented with a
system reliability requirement of 99% under various turbulence strengths.

(a) (b) (c)
Overlap
Separate
Space diversity
is used in all cases
S is center-to-center
spacing between two
apertures
OAM 0 is
Gaussian beam
101

Figure 7.5 (a) Experimental bit-error-rates (BERs) of a 100-Gbit/s QPSK link with and without using
mode and space diversity under 50 random turbulence realizations. Total transmit power is 10 dBm. (b)
Experimental BER as a function of transmitted power for different diversity schemes under a random
turbulence realization. (c) Simulation results for required transmitted when different numbers of
apertures are implemented with a system reliability requirement of 99% under various turbulence
strengths. In the simulation, transmitted beam diameter D is 3 mm and the required received power is -
40 dBm. FEC: forward error correction.

7.4 Reference
[1] R. Fields, C. Lunde, R. Wong, J. Wicker, D. Kozlowski, J. Jordan, B. Hansen, G.
Muehlnikel, W. Scheel, U. Sterr, and R. Kahle, "NFIRE-to-TerraSAR-X laser
communication results: satellite pointing, disturbances, and other attributes consistent
with successful performance," In Sensors and Systems for Space Applications III 7330,
73300Q (2009).
[2] A. Kaadan, H. H. Refai, and P. G. LoPresti, "Multielement FSO transceivers
alignment for inter-UAV communications," J. Lightw. Technol. 32, 4183-4193 (2014).
[3] R. L. Phillips, and L. C. Andrews, "Spot size and divergence for Laguerre Gaussian
beams of any order," Appl. Optics. 22, 643-644 (1983).
[4] L. Zhu, and J. Wang, "Demonstration of obstruction-free data-carrying N-fold
Bessel modes multicasting from a single Gaussian mode," Opt. Lett. 40, 5463-5466
(2014).
(a) (b) (c)
7% FEC limit
7% FEC limit
Mode diversity
(Gaussian
and OAM +1) is
used in all cases
Experiment
Experiment Simulation
Mostly below FEC limit
Intermittently above FEC limit
102

[5] S. M. Navidpour, M. Uysal, and M. Kavehrad, "BER performance of free-space
optical transmission with spatial diversity," IEEE Trans. Wireless Commun. 6,
2813 (2017)
[6] S. Sanayei, and A. Nosratinia, "Antenna selection in MIMO systems," IEEE
Commun. Magazine 42, 68-73 (2004).
[7] T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, "Optical
wireless links with spatial diversity over strong atmospheric turbulence
channels," IEEE Trans. Wireless Commun. 8, 951-957 (2007).
[8] Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed,
N. Ashrafi, and S. Ashrafi, "Atmospheric turbulence mitigation in an OAM-based
MIMO free-space optical link using spatial diversity combined with MIMO
equalization," Opt. Lett. 41, pp.2406-2409 (2016).
[19] M. A. Cox, L. Cheng, C. Rosales-Guzmán, and A. Forbes, "Modal Diversity for
Robust Free-Space Optical Communications," Phys. Rev. A 10, 024020 (2018).
[10] S. Huang, G. R. Mehrpoor, and M. Safari, "Spatial-mode diversity and
multiplexing for FSO communication with direct detection," IEEE Trans.
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[11] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital
angular momentum of light and the transformation of Laguerre-Gaussian laser
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[12] W. O. Popoola, Z. Ghassemlooy, and V. Ahmadi, "Performance of sub-carrier
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[13] G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J. F. Morizur,
"Efficient and mode selective spatial mode multiplexer based on multi-plane light
conversion," Opt. Express 22, 15599-15607 (2014).
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103

both mode and space diversity in a 100-Gbit/s QPSK free-space optical link to
increase system tolerance to turbulence," In 2019 Optical Fiber Communication
Conference, 1-3 (2019).

Abstract (if available)

Abstract FSO communication links can potentially benefit from the simultaneous transmission of multiple independent data-carrying beams, which is known as space-division-multiplexing (SDM). Mode-division-multiplexing (MDM) is a subset of SDM, where each of the multiple beams is a unique mode from an orthogonal modal basis set. Orthogonality minimizes crosstalk among the modes and enables efficient multiplexing at the transmitter, co-propagation of overlapping beams, and low-crosstalk demultiplexing at the receiver. One possibility of MDM is to use orbital-angular-momentum (OAM) modes that are circularly symmetric. ❧ OAM modes are characterized by a phase front having an angular dependence of the form exp(iℓφ), where φ is the azimuthal angle and ℓ is the OAM order and counts the number of 2π phase shifts in the azimuthal direction. Beams with different OAM orders are mutually orthogonal, thus each beam can carry an independent data stream, and the total information capacity of the spatially overlapped beams equals the data rate of one beam multiplied by the total number of independent beams. ❧ Due to the unique phase and intensity structures of OAM beams, OAM-based FSO communication systems may present unique challenges, including: beam divergence, link alignment, and atmospheric turbulence effects. My dissertation will introduce my research on these limitations and challenges, as well as techniques to improve the system performance, including: high-capacity OAM-multiplexed FSO link between a ground transmitter and a ground receiver via a hovering retroreflecting unmanned-aerial-vehicle (UAV)

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Asset Metadata

Creator Li, Long (author)

Core Title Applications of orbital angular momentum in high-capacity free-space optical communications

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 04/29/2019

Defense Date 02/07/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag free-space optical communications,multiplexing,OAI-PMH Harvest,optical vortices,orbital angular momentum,space diversity,turbulence

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Wilner, Alan (committee chair)

Creator Email longl@usc.edu,longlirent@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-158373

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