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High-capacity optical communications using structured light in random and disturbed media

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High-capacity optical communications using structured light in random and disturbed media

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Content High-Capacity Optical Communications Using Structured Light in Random and
Disturbed Media
by
Runzhou Zhang
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)
August 2022
Copyright 2022 Runzhou Zhang
Dedication
Thisdissertationisdedicatedtomysuppotiveandlovingwife, QiyuChen, mysupportiveparents
and parents-in-law, Xiaomao Zhang, Qiaoyun Zhang, Jianwang Chen, Guihong Zhou, and my
respectful advisor, Professor Alan E. Willner.
ii
Acknowledgements
First of all, I would like to express my greatest gratitude to my Ph.D advisor Prof. Alan E.
Willner. Heissuchagreatmentorandsupervisorthathehasbeenguidingmethroughthewhole
Ph.D study and research from the scratch. I enjoyed very much working with him to try to solve
problems and challenges that may bring a large impact to the community. I would also like to
thank Prof. Moshe Tur from Tel-Aviv University in Israel, who is another great mentor of mine.
His wisdom has been helping me to conduct the cutting-edge research and his advice has been
greatly enhancing the quality of my work. Moreover, I would like to thank Prof. Keith Jenkins
and Prof. Robin Shakeshaft for serving on my dissertation and qualification exams. I would like
to thank Prof. Andreas F. Molisch and Prof. Wei Wu for severing on my qualification exam.
Secondly, I would love to thank my friends and colleagues in Optical Communications Lab at
USC, including Dr. Yan Yan, Dr. Yongxiong Ren, Dr. Changjing Bao, Dr. Ahmed Almaiman,
Dr. Jing Du, Dr. Yinwen Cao, Dr. Zhe Zhao, Dr. Long Li, Dr. Peicheng Liao, Dr. Cong Liu,
Dr. Ahmad Fallahpour, Dr. Kai Pang, Mr. Haoqian Song, Mr. Kaiheng Zou, Mr. Hao Song, Mr.
Huibin Zhou, Mr. Xinzhou Su, Ms. Nanzhe Hu, and Mr. Yuxiang Duan. Without their help, I
couldn’t get through this journey and finish this dissertation.
Thirdly, I would like to acknowledge my collaborators, including Dr. Yiyu Zhou and Prof.
Robert W. Boyd at University of Rochester, Dr. Giovanni Milione at NEC Laboratory, Dr.
Brittany Lynn at Naval Information Warfare Center Pacific, Dr. Ming-jun Li at Corning Inc.,
and Prof. Eric Johnson at Clemson University.
iii
Furthermore, I also would like to acknowledge the tremendous support from the staff of Ming
Hsieh Department of Electrical and Computer Engineering, including but not limited to Ms.
Diane Demetras, Ms. Corine Wong, and Ms. Gerrielyn Ramos.
Last but not the least, I would like to express my deepest love for my family. Their support
is the underlying reason for every bit of my achievements.
iv
Table of Contents
Dedication ii
Acknowledgements iii
List Of Figures vii
Abstract xiv
Chapter 1: Introduction 1
1.1 Structured Light Beams: Orthogonal Spatial Modes . . . . . . . . . . . . . . . . . 2
1.2 Random Media: Air, Fiber, and Turbid Water . . . . . . . . . . . . . . . . . . . . 5
1.3 Data Modulation Formats and Detection Methods . . . . . . . . . . . . . . . . . . 7
1.4 Free-Space Optical Communications . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Fiber Optical Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2: Turbulence-Resilient Free-Space Optical Communications 13
2.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Concept of Pilot-Assisted Optoelectronic Multi-Mode Mixing . . . . . . . . . . . . 16
2.3 Experimental Characterization of Mixing Power Loss . . . . . . . . . . . . . . . . . 22
2.4 Experimental Measurement of LG Modal Spectrum by Off-Axis Holography . . . . 26
2.5 Polarization-Multiplexed 12-Gbit/s 16-QAM Free-Space Optical Data Transmission 28
2.6 Enhancing Spectral Efficiency by Kramers-Kronig Self-Coherent Detection . . . . . 30
2.7 Enhancing Power Efficiency by Differential Self-Coherent Detection . . . . . . . . . 33
Chapter 3: Turbulence-Resilient Mode-Division-Multiplexing in Free-Space Op-
tical Communications 37
3.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 AutomaticCompensationforTurbulence-InducedCrosstalkbyOptoelectronicMix-
ing of Spatial Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Alignment Monitor for Free-Space Optical Links in Presence of Turbulence Effects 41
3.4 Turbulence-Resilient Mode-Division-Multiplexing with Automatic Crosstalk Com-
pensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Chapter 4: Structured Light’s Interaction with Scattering Media and Effects in
Communications 56
4.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Laguerre Gaussian Modal Spectrum by Single-Particle Scattering . . . . . . . . . . 57
4.3 Ballistic and Diffusive Scattering of OAM Modes in Free-Space Optical Communi-
cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
v
4.4 Simultaneous Turbulence Mitigation and Channel Demultiplexing by Adaptive
Wavefront Shaping and Diffusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 5: Optical Mitigation for Intra-Group Modal Power Coupling in Few-
Mode Fiber 81
5.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Spatial Modal Groups in Few-Mode Fibers . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Adaptive Optics to Mitigate Intra-Group Modal Power Coupling . . . . . . . . . . 85
5.4 200-Gbit/s Mode-Division-Multiplexing inside the LP
01
Group . . . . . . . . . . . 87
Chapter 6: Conclusion 92
References 93
vi
List Of Figures
1.1 Spatial intensity and phase profiles of Laguerre-Gaussian spatial modes. . . . . . . 3
1.2 Spatial intensity and phase profiles of orbital-angular-momentum spatial modes. . 3
1.3 LG beam generation and detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Concept of LG decomposition and complex LG spectrum. . . . . . . . . . . . . . . 5
1.5 (a) Concept of random media that distort the spatial profiles of a data-carrying
beam, causing power coupling from the fundamental Gaussian mode to other
higher-order spatial modes; (b) Typical examples of random media in optical com-
munication systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 (a) Concept of amplitude modulation and direction detection; (b) Concept of am-
plitude and phase modulation. For quadrature-amplitude modulation, coherent
detection is typically utilized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Concept of FSO communications. (a) In single-Gaussian-beam FSO links, atmo-
sphericturbulenceeffectscancausepowercouplingfromthefundamentalGaussian
modetomanyhigher-orderLGspatialmodes;(b)Inmulti-spatial-modeFSOlinks,
turbulence can cause inter-channel crosstalk among the transmitted spatial modes.
Tx: transmitter; Rx: receiver; MDM: mode division multiplexing. . . . . . . . . . . 9
1.8 Concept of fiber communications. (a) In conventional single-mode-fiber (SMF)
links, there is little significant spatial distortions; (b) In multi-mode-fiber (MMF)
MDM links, the inhomogeneity of the MMF can cause modal power coupling and
inter-channel crosstalk among the transmitted LG modes. Tx: transmitter; Rx:
receiver; MDM: mode division multiplexing. . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Conceptofturbulence-resilientFSOcommunicationsbypilot-assistedself-coherent
optoelectronic mixing of many spatial modes. I: in-phase; Q: quadrature. . . . . . 14
vii
2.2 Concept of simultaneous amplitude and phase recovery of a QAM data in turbu-
lent FSO links. (a) The performance of coherent detection can be significantly
degraded by the turbulence-induced LG-modal-coupling effects. (b) Pilot-assisted
self-coherent detection can automatically compensate for the turbulence-induced
LG-modal-coupling effects. DC: direct current; SSB: signal-signal beating; SSBI:
signal-signal-beating interference; SLB: signal-LO beating; SPB: signal-pilot beat-
ing. In both cases, the frequency offset f is greater than the data bandwidth B to
avoid the SSBI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Experimental setup for 12 Gbit/s 16-QAM PDM FSO link. Equal copies of the re-
ceivedbeamsaredetectedbythepilot-assistedself-coherentdetectorandsingle-PD
LO-based heterodyne coherent detector. During the detection of the heterodyne
coherent receiver, the pilot is turned off. The same DSP algorithms are applied
to both receivers to retrieve the 16-QAM data. AWG, arbitrary waveform gen-
erator; Mod., modulator; EDFA, erbium-doped fibre amplifier; PC, polarization
controller; PBC, polarization beam combiner; pol., polarization; Col., collimator;
MR, mirror; HWP, half-wave plate; FM, flip mirror; BS, beam splitter; FS-PD,
free-space-coupled photodetector; SMF, single-mode fiber. . . . . . . . . . . . . . . 23
2.4 Measurement of optical and mixing power loss for the pilot-assisted self-coherent
detector under different turbulence strengths. (a) Experimentally measured his-
tograms of optical power loss under two different turbulence distortions (2 w
0
/r
0
≈ 2.2 and 5.5) for X (left) and Y (right) polarizations; (b) Experimentally measured
histograms of mixing power loss (in the electrical domain) under two different tur-
bulence distortions (2w
0
/r
0
≈ 2.2 and 5.5) for X (left) and Y (right) polarizations.
The mixing power loss is measured at the IF of approximately 2.6 GHz in the elec-
trical domain. In (a) and (b), 1,000 different turbulence realizations are measured
for each polarization. (c) Simulated average optical power loss (top) and average
electricalmixingpowerloss(bottom)resultsfordifferentturbulencestrengthsfrom
1 to 7. The average values of experimentally measured data points (including both
X and Y polarizations) are also plotted. . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 The procedures for measuring the LG spectrum of a distorted pilot beam using
the off-axis holography. The spatial amplitude and phase profiles of the distorted
pilot beam are obtained, and subsequently the corresponding LG power spectrum
is calculated. The captured images have 320 × 256 pixels, with a pixel size of 30
µm . FFT: fast Fourier transform; a
l,p
: coupling coefficient for the LG
l,p
mode.
The ratio of optical power coupled to the LG
l,p
mode is given by|a
l,p
|
2
. . . . . . . 27
2.6 Experimental results of turbulence-induced LG modal power coupling and recov-
ered 16-QAM data qualities using the pilot-assisted self-coherent detector. (a) No
turbulence distortion. (b) One example realization (R1) of the weaker turbulence
distortion (2w
0
/r
0
≈ 2.2). (c-d) Two different example realizations (R1 (c) and R2
(d)) of the stronger turbulence distortion (2w
0
/r
0
≈ 5.5). For each of the four real-
izations, we measure the LG modal power spectrum (two indices− 5≤ ℓ≤ +5 and
0≤ p≤ 10) and recover the 16-QAM data constellations. In this demonstration of
PolM FSO data transmission, each polarization carries a 6-Gbit/s 16-QAM signal.
pol., polarization; EVM, error vector magnitude. . . . . . . . . . . . . . . . . . . . 29
viii
2.7 Experimentally measured BER performance of the pilot-assisted self-coherent de-
tector over 200 different emulated turbulence random realizations. (a) Weaker
turbulence effects, X polarization. (b) Weaker turbulence effects, Y polarization.
(c) Stronger turbulence effects, X polarization. (d) Stronger turbulence effects,
Y polarization. To indicate the effects of turbulence-induced modal coupling on
a coherent-detection FSO system with the single-Gaussian-mode LO, the perfor-
manceusinganSMF-coupledLO-basedheterodynecoherentdetectorisalsoshown.
In the PDM FSO data transmission, each polarization carries a 6-Gbit/s 16-QAM
data signal. Note that we measure the BER performance for one polarization at a
time due to limitations of our measurement setup. Therefore, the BER values for
X and Y polarizations with the same realization label may correspond to different
turbulence realizations and are difficult to be compared directly. . . . . . . . . . . 30
2.8 Concept and experimental results of enhancing spectral efficiency of the pilot-
assisted self-coherent detector by using Kramers-Kronig (KK) detection. (a) Con-
cept of employing KK relation to mitigate SSBI for the pilot-assisted self-coherent
detector; (b) The recovered 16-QAM data constellation with and without KK pro-
cessing (X polarization, without turbulence); (c) Measured LG modal spectra and
therecovered16-QAMdataconstellationsunder3cases: (c1)Noturbulencedistor-
tion; (c2) One example realization of the weaker turbulence distortion (2w
0
/r
0
≈ 2.2); (c3)Oneexamplerealizationofthestrongerturbulencedistortions(2w
0
/r
0
≈ 5.5). DC: direct current; SSBI: signal-signal-beating interference; SPB: signal-pilot
beating. EVM: error vector magnitude. . . . . . . . . . . . . . . . . . . . . . . . . 32
2.9 ConceptualcomparisonforKramers-Kronig,gapped-pilot-assisted,anddifferential-
phase-shift-keying detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Concept of a turbulence-resilient DPSK FSO link. A free-space DLI with relaying
imaging combines direct and delayed beams which are similarly distorted. Subse-
quently, free-space-coupled PDs can perform O/E data-data mixing to automati-
cally compensate the turbulence-induced LG modal coupling. . . . . . . . . . . . . 34
2.11 (a) Experimental setup of a 2.25-Gbit/s DPSK FSO link. AWG: arbitrary wave-
form generator; Mod.: I/Q modulator; EDFA: erbium-doped fiber amplifier; PC:
polarization controller; Col.: collimator; FM: flip mirror; M.: mirror; BS: beam
splitter; PD:free-spacephotodetector; SMF:single-modefiber; LO:localoscillator.
(b) and (c): Measured amplitude, phase profiles, and LG spectra of the received
beam without and with turbulence. (d) Measured histogram of mixing power loss
for our approach and SMF-coupled system. (e) Measured Q factor degradation for
our approach and SMF-coupled coherent receiver. . . . . . . . . . . . . . . . . . . . 35
3.1 Concept of turbulence-induced OAM modal power coupling. . . . . . . . . . . . . . 38
3.2 Concept of utilizing optoelectronic OAM beam mixing at the receiver to achieve
OAM modal coupling resilient to the turbulence distortion in an FSO link. (a)
Optoelectronic mixing of a pilot and a data OAM beam using a free-space-coupled
photodiode at the receiver to reduce the turbulence-induced modal coupling in the
electricaldomain. Conj.: conjugate;DC:direct-current;SSB:signal-signalbeating;
(b)OptoelectronicOAMbeammixingcancoupletheturbulence-inducedspreading
power from undesired data channels back to the transmitted data channel in the
electrical domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
ix
3.3 Concept of alignment monitoring in an FSO link in the presence of atmospheric
turbulence effects using the mixing tone between two opposite-order OAM beams
(i.e., ℓ
1
= +1 and ℓ
2
= − 1) on two different wavelengths: a) little mixing tone
in the electrical domain if Tx and Rx are well-aligned; b) resultant mixing tone
reflecting the Tx-Rx alignment condition if Tx and Rx are misaligned. . . . . . . . 42
3.4 (a) Experimental setup for alignment monitoring in an FSO link. EDFA: erbium-
dopedfiberamplifier; PC:polarizationcontroller; MUX:multiplexer; FM:flipmir-
ror; M.: mirror; SLM: spatial light modulator; FS PD: free-space-coupled photode-
tector; FFT: fast Fourier transform; Tx: transmitter; Rx: receiver; (b) Measured
intensity profiles for turbulence-distorted OAM beams. Turb.: turbulence; Real.:
realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Experimental results for the mixing tone of OAM l
1
= +1 and l
2
= − 1 beams:
(a) Normalized tone strength at different horizontal or vertical offsets without tur-
bulence effects; (b) Electrical spectrum with no misalignment under turbulence
realization 2; (c) Electrical spectrum with a horizontal offset of -3.22 mm under
turbulence realization 2. Experimentally measured relations between the normal-
izedtonestrengthandTx-Rxmisalignmentunderdifferentturbulencerealizations:
(d) horizontal displacement and (e) vertical displacement. Rx aperture size 6 mm. 46
3.6 Two-OAM-beam-multiplexed FSO data transmission with inter-channel crosstalk
resilient to turbulence distortion by transmitting extra OAM pilot tones and OAM
beam mixing at the receiver. Conj.: conjugate; DC: direct-current; SSB: signal-
signal beating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Experimentalfortwo-OAM-beam-multiplexeddatatransmissionthroughemulated
turbulence effects. (a) Experimental setup. AWG: arbitrary waveform generator;
MZI: Mech-zen; PC: polarization controller; EDFA: erbium-doped fiber amplifier;
BPF: band-pass filter; MUX: multiplexer; M.: mirror; FM: flip mirror; BS: beam
spliiter; SLM: spatial light modulator; Col.: collimator; FS PD: free-space photo-
diode; Tx: transmitter; Rx: receiver; Scope: real-time oscilloscope; DSP: digital
signal processing. (b) Measured optical spectra carried by the two OAM beams
at the transmitter. Ch.: channel. (c) Measured electrical spectrum for O/E OAM
beam mixing at the receiver. DC: direct current; SSB: signal-signal beating. . . . . 48
3.8 Experimentalresultsofmeasuredinter-channelcrosstalkunderweakerandstronger
turbulencedistortions. (a)MeasuredbeamprofilesforOAMbeams ℓ=− 2,− 1,0,+1,+2;
(b) Measured inter-channel crosstalk using optoelectronic OAM beam mixing; (c)
Measured inter-channel crosstalk using conventional OAM receiver; (d) Measured
signal power loss for OAM beams ℓ=− 2,− 1,0,+1,+2 by using the beam mixing
approach in (d1) and the conventional approach in (d2). . . . . . . . . . . . . . . . 51
x
3.9 Experimental results of 4-Gbit/s OAM-multiplexed data transmission under dif-
ferent strengths of turbulence effects. (a) Comparison of the received 2-Gbaud
QPSK constellation diagram using the O/E OAM beam mixing and conventional
OAM receiver. Different sets of the multiplexed two OAM beams under different
turbulence strengths are evaluated. The transmitted free-space optical powers ap-
proximately 1.6 dBm and 3.7 dBm for OAM beams ℓ
1
and ℓ
2
, respectively. (b-c)
MeasuredEVMvaluesforresilientandconventionalOAMlinksundertwodifferent
turbulencestrength withtheFriedparameters r
0
of1.0 mmand0.4 mmin(b)and
(c), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.10 Experimental results of 8-Gbit/s OAM-multiplexed data transmission under dif-
ferent strengths of turbulence effects. (a-b) Comparison of the received 2-Gbaud
16-QAMconstellationdiagramusingtheO/EOAMbeammixingandconventional
OAM receiver. The Fried parameters r
0
of the emulated turbulence effects are 1.0
mm and 0.4 mm in (a) and (b), respectively. The transmitted free-space optical
powers are approximately 6.4 dBm and 8.6 dBm for OAM beams ℓ
1
= +1 and
ℓ
2
=− 1, respectively. (c-d) Measured BER curves of the data channels carried by
two OAM beams ℓ
1
=+1 and ℓ
2
=− 1 in (c) and (d), respectively. . . . . . . . . . 54
4.1 Conceptualdiagramof LGbeams’scatteringbyasilicasphericalparticle: Asingle
LG beam propagates along the z-axis and the particle is located on the beam waist
plane (z = 0) of the incident beam. After interaction with the particle, the beam
can be decomposed into multiple different LG modes with complex coefficients
(C
1
,C
2
, ...,C
N
).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Effects of different particle size R on the complex field E
x
at the detection plane:
(a) No particle present; (b) R = 2 µm ; (c) R = 5 µm ; (d) R = 8 µm . Each row
represents: (a1-d1) Normalized intensity profile |E
x
|
2
; (a2-d2) Spatialphase profile
̸ E
x
; (a3-d3) Normalized intensity profile |E
x
|
2
at the x = 0 plane, from z =− 10
µm to z =+10 µm . The incident beam is LG
ℓ=+3,p=0
mode. . . . . . . . . . . . . 59
4.3 Effectsoftheparticle’soff-axisdistance ρλ ontheLGmodepowercoupling|C
ℓ=+3,p
|
2
:
(a) Detected intensity profiles when the particle is located on different off-axis dis-
tance ρ/λ =0.26, 0.77; (b) The ratio of power coupling to LG
ℓ,p
modes with the
same OAM order ℓ=+3 and different radial indices p when the particle is located
on different off-axis distance. The incident beam is LG
ℓ=+3,p=0
. . . . . . . . . . . . 60
4.4 (a-b) Effects of the particle size and the choice of incident mode on the determina-
tion of the off-axis distance: (a) Ratio of power coupled to LG
ℓ=+3,p=5
mode (i.e.,
|C
ℓ=+3,p=5
|
2
) with different particle size R=2,4,6,8 µm ; (b) Ratio of power cou-
pledtoLG
ℓ,p=5
mode(i.e.,|C
ℓ,p=5
|
2
)whenusingLGmodes(ℓ=0,+1,+2,+3,p=
0) with different OAM orders ℓ as the sources. (c-d) Phase of different OAM com-
ponent (i.e.,
̸ C
ℓ,p=0
) if the particle is located on different azimuthal locations: (c)
Unwrapped phase of OAM modes ℓ = 0,+1,...,+6; (d) Effects of the particle’s
off-axis distance ( ρ/λ = 1.0,1.5,2.0) on the OAM phase spectra. Radius of the
particle R=5 µm , incident beam LG
ℓ=+3,p=0
. . . . . . . . . . . . . . . . . . . . . 62
xi
4.5 Relation between the slope of OAM phase spectrum and the particle’s azimuthal
location: (a) Calculated slope of the OAM phase spectrum (ℓ = 0,+1,...,+6) as
the particle located on different azimuthal positions; (b) Effects of the particle’s
size R on the OAM phase spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Concept of multiple scattering effects on an OAM beam. When diffusive scattering
dominates, the power of a pure OAM beam may leak to multiple neighboring
modes, degrading the performance of OAM-multiplexed transmission due to the
scattering-induced crosstalk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Experimental setup of multiplexed OAM beams’ transmission through a scattering
medium. QPSK: quadrature phase-shift keying; EDFA: erbium-doped fiber ampli-
fier; Col.: collimator; SLM: spatial light modulator; PC: polarization controller;
BS: beam splitter; FM: flip mirror; IR: infrared; Tx: transmitter; Rx: receiver. . . 66
4.8 IntensityandinterferogramsofanOAMℓ=+3beamaftertraversingthescattering
medium for different values of optical depth γL .. . . . . . . . . . . . . . . . . . . . 67
4.9 ReceivedtotalpowerenteringtheRxandmodalpowerbelongingtothetransmitted
mode when transmitting OAM modes ℓ = +1,+3,+5,+7 separately. Rx aperture
is 12 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.10 Top: Measured OAM normalized power spectrum in the Rx under different values
of γL : (a) Tx: OAM ℓ = +1; (b) Tx: OAM ℓ = +3; Bottom: Measured crosstalk
(XT)fordifferenttransmittedOAMmodesunderdifferentvaluesof γL : (c)second-
neighboring mode (XT-2); (d) third-neighboring mode (XT-3). Rx aperture is 12
mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.11 (a) Measured optical signal-to-interference ratio (OSIR) for transmitting multi-
plexed ℓ=± 1,± 3 beams; (b) Measured crosstalk as a function of Rx aperture size
for transmitting OAM ℓ=+3 beam. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.12 Bit error rate (BER) of different OAM modes when transmitting multiplexed ℓ =
± 1,± 3 beams: (a) Four modes without scattering effects; (b) ℓ = +1 mode; (c)
ℓ=+3 mode for different OD values. Rx aperture is 12 mm. . . . . . . . . . . . . 72
4.13 Concept of using WSD to mitigate the turbulence effects and simultaneously de-
multiplex two data-carrying OAM beams. . . . . . . . . . . . . . . . . . . . . . . . 74
4.14 Experimental setup of using WSD to mitigate turbulence effects in a 200-Gbit/s
OAM-multiplexed link. QPSK: quadrature phase-shift keying; EDFA: erbium-
doped fiber amplifier; SMF: single-mode fiber; Col.: collimator; SLM: spatial light
modulator; PC: polarization controller; BS: beam splitter; FM: flip mirror; MR:
mirror; IR: infrared; PM: power meter; Tx: transmitter; Rx: receiver; Ctrl.: con-
troller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
xii
4.15 (a) Turbulence-distorted OAM ℓ=+1 beam profiles; (b) Received OAM spectrum
of the distorted ℓ = +1 beam; (c) One example of the determined 30 × 30 phase
pattern; (d) Received power of each mode versus the number of iterations during
the wavefront shaping process; (e) Received power of each mode after WSD is
applied to mitigate different turbulence realizations (TS-1: realization 1-5; TS-2:
realization 6-10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.16 Measured XT using OAM and WSD Rx for receiving distorted OAM modes: (a)
ℓ = +1 mode; (b) ℓ =− 1 mode. The XT without turbulence effects using OAM
Rx is also shown for comparison. Tur.: turbulence. . . . . . . . . . . . . . . . . . . 78
4.17 MeasuredBERofOAM-carrieddatachannelasafunctionofOSNRunderdifferent
turbulence effects: (a) ℓ = +1 under TS-1; (b) ℓ = − 1 under TS-1; (c) ℓ = +1
under TS-2; (d) ℓ=− 1 under TS-2. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 (a) Concept of optical communications in single-mode fiber; (b) Concept of mode-
division multiplexing in multi-mode fiber. . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Channel impairment comparison for mode division multiplexing in free-space at-
mosphere and multi-mode fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Concept of different modal groups in few-mode fibers. LP: linearly-polarized; LG:
Laguerre-Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Concept of weaker inter-group modal coupling and stronger intra-group modal
power coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 ConceptdiagramforAOtomitigatemodalXTinsidetheLP
11
groupofaGIFMF.
AO implements an inverse TM to the coupled output modes. . . . . . . . . . . . . 86
5.6 (a) Experimental setup for AO to mitigate modal XT in an FMF-based MDM
system. QPSK: quadrature phase-shift keying; EDFA: erbium-doped fiber ampli-
fier; SMF: single-mode fiber; PC: polarization controller; MUX: multiplexer; Pol.:
polarizer; OL: objective lens; GI: graded index; Col.: collimator; FM: flip mirror;
MR: mirror; IR: infrared; HWP: half-wave plate; SLM: spatial light modulator;
PM: power monitor; Tx: transmitter; Rx: receiver; (b) An example of AO phase
masktomitigatedatachannel1carriedbyOAMℓ=+1bycombiningtwoindivid-
ualdemultiplexingphasepatternswithcoefficients s
T
1
. (c)Anexampleofmeasured
time-varying XT for OAM ℓ=+1 mode. The carried signal is 10-Gbaud QPSK. . 88
5.7 Measured BER performance for multiplexed data channel 1 and 2 with and with-
out AO mitigation: (a) Constellations for QPSK signal; (b) BER as a function
of OSNR. The BER performance for transmitting the corresponding single data
channel is also shown for comparison. Each OAM mode carries an independent
100-Gbit/s QPSK signal.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.8 BER performance with AO mitigation for different data rates (10-, 20-, and 50-
GbaudQPSK):(a)Datachannel1carriedbyOAMℓ=+1mode;(b)Datachannel
2 carried by ℓ=− 1 mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xiii
Abstract
InadditiontothefundamentalGaussianmode,alaserbeamcanbetailoredtooccupyhigher-order
orthogonal spatial modes, such as the Laguerre-Gaussian (LG) modes, thus called ”structured
light”. The unique spatial profile of an LG beam is typically described by two modal indices: (i)
azimuthal index ℓ refers to the number of the 2π phase shift along the azimuthal direction of the
phase profile; and (ii) radial index p+1 leads to the number of intensity ring structure in the
intensityprofile. Moreover,theorthogonalityamongLGbeamsensurethattheycanbeefficiently
multiplexed, co-propagation, and demultiplexed with little inherent crosstalk.
With respect to optical systems in different applications, communicating media are never
perfect and are likely to generate random distortions to the data-carrying beams. From a spatial
modal point of view, the random media can cause power coupling from the transmitted LG mode
to many other LG modes. Depending on system architecture, such distortions can lead to either
modal-coupling loss or inter-channel crosstalk, both of which are detrimental to high-capacity
optical transmission.
This thesis will discuss various optical/electrical compensation techniques to enable high-
capacity and spectral-efficient optical communications in various random media, including (i)
free-space optical communications in atmospheric turbulence; (ii) mode-division-multiplexing
transmission in scattering media; and (iii) mode-division-multiplexing fiber communications in
few-mode fiber.
xiv
Chapter 1
Introduction
Over the last decades, optical communications has been enhancing the data capacity and spectral
efficiency of data communications [1, 2, 3]. As compared to digital or radio-frequency (RF)
systems,onefundamentalbenefitofopticalwavesisthehighcarrierfrequency(attheorderof100
THz), which enables optical waves to carry much higher bandwidth data channel [1]. There have
beenseveralkeytechniquestoenabletheever-growingdataspeeddemand,includingdevelopment
ofsingle-modefibers, erbium-dopedfiberamplifiers, adoptionofwavelength-divisionmultiplexing
(WDM),spectral-efficientmodulationformats,andcoherentdetectionwhichutilizesdigitalsignal
processing(DSP)[3]. Moreover,tofurtherenhancethedatacapacity/spectralefficiency,therehas
beenagrowinginterestinutilizingspace-divisionmultiplexing(SDM)foropticalcommunications
[4], of which a subset is mode-division multiplexing (MDM) [5, 6, 7]. In MDM, the data-carrying
beams are generally higher-order laser spatial modes. As compared to the fundamental Gaussian
laser beam, those spatial modes are also referred as ”structured light beams” because of their
unique spatial amplitude and phase profiles [8, 9].
Optical communications has been the enabling techniques for the long-haul fiber data trans-
mission, which is also the backbone of the Internet [1, 2]. Due to the demand for numerous
emerging platforms, laser communications has found its applications in different scenarios, such
as free-space optical (FSO) communications [10, 11, 12]. As compared with fiber-guided systems,
1
free-space systems generally do not have a confining medium (e.g., optical fiber) to serve a waveg-
uide, but rather rely on laser propagation in free space [10]. The common FSO media include
atmosphere air [13] and underwater environment [14]. Similar to the RF wireless concept, FSO
communications is sometimes also referred as ”optical wireless communications” [15].
Themediaareneverperfectforcommunicatingchannel. Moreover,mostofthecommunication
media may bring dynamic and random distortion to the data-carrying propagation laser beam,
thus often called ”random media” [16]. To enable robust and efficient data transmission through
random media, two sets of compensation techniques can be generally applied: (i) electronic
mitigation: digitalmitigationfortherecordeddatacanbeutilizedbyleveragingDSPalgorithms;
and (ii) optical mitigation: structured light beams themselves can be tailored to counter-act
the degradation effects of the random media.
1.1 Structured Light Beams: Orthogonal Spatial Modes
A fundamental Gaussian laser beam has a Gaussian-like intensity and a flat phase profiles. More-
over, a laser beam can carry spatial structures that exhibit unique spatial intensity and phase
features, such as the Laguerre-Gaussian (LG) spatial modes [17, 18]. In general, the electrical
field of an LG beam can be expressed as [18]:
E
ℓ,p
(r,ϕ,z )=
A
l,p
w(z)
(
r
√
2
w(z)
)
|ℓ|
L
|ℓ|
p
(
2r
2
w
2
(z)
)exp(− r
2
w
2
(z)
− ik
r
2
2R(z)
− iℓϕ − ikz+iψ (z)) (1.1)
where A
l,p
, L
|ℓ|
p
, R(z), w(z), and ψ (z) are the normalization constant, generalized Laguerre poly-
nomials, curvature radius, beam width, and Gouy phase, respectively.
As shown in Fig. 1.1, an LG beam is characterized by two separate and independent modal
indices, ℓ and p: (i) ℓ refers to the number of 2π phase shift along the azimuthal direction for the
phase profiles; and (ii) p+1 is equal to the number of concentric rings in the intensity profiles.
2
Furthermore, the orthogonality among the LG modes ensures that they can be multiplexed, co-
propagation, and demultiplexed efficiently with little inherent crosstalk [7, 19].
-π
π
0
1
LG
5,0
LG
5,3
p ℓ
Normalized
intensity
profile
Phase
profile
LG
5,1
Spatial intensity and phase profiles
LG
0,0
Fundamental
Gaussian
Figure 1.1: Spatial intensity and phase profiles of Laguerre-Gaussian spatial modes.
As shown in Fig. 1.2, a subset of LG modes that received recent interest is orbital-angular-
momentum (OAM) spatial modes [18]. OAM modes are defined by a subset of LG modes with
zero radial indices (p = 0). The intensity profile of an OAM beam has only a single intensity
ring; the phase profile contains twisting wavefront that the number of 2 π phase shift along the
azimuthal direction is equal to the OAM order ℓ.
[m]
[m]
-10 -8 -6 -4 -2 0 2 4 6 8
x 10
-4
-10
-8
-6
-4
-2
0
2
4
6
8
x 10
-4
-10 -8 -6 -4 -2 0 2 4 6 8
x 10
-4
-10
-8
-6
-4
-2
0
2
4
6
8
x 10
-4
=0 =+1
-10 -8 -6 -4 -2 0 2 4 6 8
x 10 -4
-10
-8
-6
-4
-2
0
2
4
6
8
x 10 -4
[m]
[m]
-10 -8 -6 -4 -2 0 2 4 6 8
x 10
-4
-10
-8
-6
-4
-2
0
2
4
6
8
x 10
-4
=+2 =+3
Intensity Phase Intensity Phase Intensity Phase Intensity Phase
Figure 1.2: Spatial intensity and phase profiles of orbital-angular-momentum spatial modes.
3
There are a number of methods to generate and detect an LG beams, including phase plates
[20, 21], spatial light modulators [22, 23], digital micro-mirror device [24], photonic integrated
circuits [25, 26], metasurfaces [27, 28, 29], and etc. Figure 1.4 indicates the generation and
detectionofanLGbeam: (i)anLGbeamcanbegeneratedbyconvertingafundamentalGaussian
beam to an LG beam via passing the Gaussian beam through a spiral phase plate; and (ii) an LG
beam can be detected by converting it back to a fundamental Gaussian beam via passing the LG
beam through a conjugate spiral phase plate.
Incoming fundamental
Gaussian beam
Converted into an
LG
3,0
beam
Holographic
phase filter
Integrated devices, wave
plates, spatial light
modulator …
Incoming LG
3,0
beam
Converted back to a
Gaussian beam
LG beam generation
LG beam detection
Figure 1.3: LG beam generation and detection.
Importantly, the amplitude and phase profiles of a structured beam can be decomposed into
a series of LG modes [9]. The electrical field of an optical beam U(x,y) can be expressed as a
superposition of a series of LG modes [9]:
U(x,y)=
X
l,p
C
l,p
LG
l,p
(x,y), (1.2)
4
where C
l,p
is the complex coefficient (both amplitude and phase) for the LG
l,p
modal component.
TheamountofpowercoupledtoandthephasedelayrelatetotheLG
l,p
modearegivenby|C
l,p
|
2
and
̸ C
l,p
, respectively.
Wavefront of a
structured beam
Amplitude(A)
-3 - n
LG
order
Phase shift ()
+3 +2 +1 0 -1 -2
+n
Complex coefficient
Wavefront of LG
ℓ,p
beam
, =&
ℓ,#

ℓ,#
(,+
ℓ,#
q The wavefront of a structured beam can be decomposed into a set of
LG modes

ℓ,#
=
ℓ,#
exp(
ℓ,#
)
LG mode decomposition
Complex LG spectrum
Figure 1.4: Concept of LG decomposition and complex LG spectrum.
1.2 Random Media: Air, Fiber, and Turbid Water
In different applications and scenarios, data-carrying beams need to propagate through distinct
media that may pose random spatial distortions to the data beams [16]. As shown in Fig. 1.5(a),
boththeamplitudeandphasespatialprofilesofafundamentalGaussianbeamwouldbedistorted
by a random medium. Specifically, the random distortion would cause the optical power coupling
fromthefundamentalGaussianmodetomanyhigher-orderLGmodes[30]. SuchLGmodalpower
coupling is likely to induce severe performance degradation for the data transmission [31].
5
Atmosphere
(temperature variation,
wind speed, etc)
Multi-mode fiber
(bending, temperature
variation, inhomogeneity, etc)
Random media examples
Random media
that distorts the
beam’s spatial
structures
Transmitter
Receiver
!
!"
!
!#
!
$
!
%#
!
%"
#
#
#
"
#
&
!"
!,!
#
$
Optical LG spectrum
!
!
!
"
!
#
!"
!,#
modal coupling
on 2D space
!
$
"
%"
"
%!
"
$
"
&!
"
&"
Optical LG spectrum
Underwater
(particle scattering,
diffusing, water current, etc)
a)
b)
Figure1.5: (a)Conceptofrandommediathatdistortthespatialprofilesofadata-carryingbeam,
causingpowercouplingfromthefundamentalGaussianmodetootherhigher-orderspatialmodes;
(b) Typical examples of random media in optical communication systems.
As shown in Fig. 1.5(b), typical random media for optical communications include: (i) at-
mosphere: free-space optical communications are typically utilized in atmosphere or air for ter-
restrial links or ground-to-space links. One of the fundamental challenge for communicating over
atmosphere is turbulence effects, which is often induced by temperature or wind speed variations
[16, 31, 32, 33, 34]; (ii)multi-mode fiber : multi-mode fibers can support MDM to achieve data
capacity and spectral efficiency enhancement, as compared to single-mode fibers [3]. However,
duetothebending, temperaturevariation, andtheinhomogeneityoffiberitself, multi-modefiber
wouldcausemodalpowercouplingbetweenthetransmittedmultiplespatialmodes(i.e.,crosstalk)
[5, 35, 36]; and (iii) underwater environment: underwater optical communications has been
paid greater attention due to the fact blue/green laser have relatively low loss inside water and
suchvisiblelightcanstillcarryalargebandwidthofdata(atthelevelofGHz)[14]. Onechallenge
6
for underwater communications is turbidity, which includes strong scattering (or diffusing) effects
that may degrade the signal-to-noise ratio of the captured data beam [37, 38, 39].
1.3 Data Modulation Formats and Detection Methods
Data can be encoded in both amplitude and phase of an optical wave [1, 2]. As shown in Fig.
1.6(a), data can be encoded as distinct intensity levels of an optical beam in pulse amplitude
modulation(PAM)[39]. ForPAMsignaldetection,asingle-endedphotodetector(PD)istypically
utilized to directly detect the intensity of received optical signal [40, 41]. The basic version of
PAMis2-PAM,alsowidelyknownason-off-keying(OOK).Toachieveahigherspectralefficiency,
one can utilize a higher-order PAM, such as 8-PAM [41].
To achieve enhancement for spectral efficiency, amplitude and phase modulation formats such
as quadrature amplitude modulation (QAM) are typically utilized in optical communications
[2, 42]. As shown in Fig. 1.6(b), data can be encoded in distinct constellation points in the
2-dimensional in-phase/quadrature (I/Q) space [43]. To recover a QAM data, coherent detection
with a local oscillator (LO) is typically utilized [44]. By mixing the receiver optical signal with
the LO, a coherent receiver is able to efficiently recover the temporal phase of the optical signal
[44]. Moreover, such phase recovery can further enable DSP functions that can mitigate various
temporal distortions [2].
1.4 Free-Space Optical Communications
Free-space optical (FSO) communications, also known as laser communications, utilizes a data-
carrying laser beam to transmit information from one place to another [11, 10]. As shown in Fig.
1.7(a), a data-carrying fundamental Gaussian beam is transmitted through turbulent atmosphere
and turbulence effects can couple the transmitted optical power to many higher-order LG spatial
modes [30, 16]. Such modal power coupling causes significant performance degradation because
7
Direct detection
• Information encoded on both the amplitude
and the phase
• Quadrature amplitude modulation (QAM)
• n bits can create 2
n
phase/amplitude
symbols, e.g., 16-QAM for n=4
• Phase recovery can enable coherent DSP
Amplitude and Phase Modulation
• Information encoded on the amplitude of an
optical wave
• On-off keying (OOK)
• Pulse amplitude modulation (PAM), e.g., 4-
PAM
0110 0011
Amplitude Modulation
4-PAM
16-QAM
Coherent detection
I
Q
I
Q
PD
PD BS
Received
optical
signal
Received
optical signal
Local oscillator
00
01
11
10
00
a)
b)
Figure 1.6: (a) Concept of amplitude modulation and direction detection; (b) Concept of ampli-
tudeandphasemodulation. Forquadrature-amplitudemodulation,coherentdetectionistypically
utilized.
thereceiverislikelytobesingle-modeoperationandasaresult, theavailablesignal-to-noiseratio
(SNR)ofthecapturedataislimitedandthusthereceiverisnotlikelytorecoverthedatachannel
efficiently [31].
Another form of FSO systems that received growing interest is mode-division-multiplexed
(MDM) FSO links [19, 7]. As shown in Fig. 1.7(b), in MDM FSO systems, each LG mode carries
an independent data channel and multiple LG beams are transmitted simultaneously. Because
they are orthogonal to each other, multiple independent data-carrying LG beams can be multi-
plexed, co-axis propagation, and demultiplexed efficiently with little inherent crosstalk [7]. By
multiplexing N LG beams, the data capacity and spectral efficiency of FSO systems can be en-
hanced by N times in a single-transmitter/single-receiver configuration [19]. However, turbulence
can distort the spatial profiles of LG beams and more importantly, such distortion can induce
modal power coupling and inter-channel crosstalk among the multiplexed data channels [45]. One
fundamentalchallengeMDMFSOcommunicationsistoovercomeorundotheturbulence-induced
crosstalk issues.
8
Transmitter
Tx Mode 1
Atmospheric
turbulence
l
1
Mode order
Mode DEMUX
Rx-N
…
Rx-2
Rx-1
Mode order
l
2
…
l
N
l
1
Receiver
Mode 1
Mode 2
Mode N
a)
Atmospheric
turbulence
Mode 1
…
Mode 2
Mode N Mode DEMUX
Rx-N
…
Rx-2
Rx-1
Mode MUX
Tx-1
…
Tx-2
Tx-N
MDM transmitter MDM receiver Crosstalk
Mode order
l
1
l
2
…
l
N
Mode order
l
2
…
l
N
l
1
Mode 1
Mode 2
Mode N
…
b)
Figure1.7: ConceptofFSOcommunications. (a)Insingle-Gaussian-beamFSOlinks,atmospheric
turbulenceeffectscancausepowercouplingfromthefundamentalGaussianmodetomanyhigher-
order LG spatial modes; (b) In multi-spatial-mode FSO links, turbulence can cause inter-channel
crosstalk among the transmitted spatial modes. Tx: transmitter; Rx: receiver; MDM: mode
division multiplexing.
In addition to airbone-based FSO links, FSO links in underwater environment has gained
growing interest due to the low-absorbtion-loss propagation characteristics of blue/green laser in
water [14]. However, underwater environments are complex in nature: (i) water current and/or
temperature variation along the propagation path can lead to turbulence effects that distort the
spatial profiles; and (ii) turbidity of underwater link largely limits the reach of laser beam and
the particle scattering effects can further degrade the captured beam quality [37, 38, 39].
This thesis will include investigation and studies on mitigation of random distortion in both
single-Gaussian-beam and multi-LG-beams FSO systems. Specifically, covered topics include:
(i) turbulence-resilient FSO communications by automatic optoelectronic mixing of many spatial
modes to compensate thet turbulence-induced modal coupling loss; (ii) demonstration of auto-
matic mitigation for turbulence-induced crosstalk in MDM FSO systems; (iii) investigation of
ballistic and diffusive scattering effects for MDM FSO links in turbid water; and (iv) utilization
9
of adaptive wavefront shaping and diffusing to simultaneously mitigate the turbulence-induced
crosstalk and demultiplex LG beams for MDM FSO links.
1.5 Fiber Optical Communications
As shown in Fig. 1.8(a), a conventional fiber communication system consists of a transmitter, a
single-modefiber(SMF),andareceiver. BecausetheSMFitselfcansupportonlythefundamental
Gaussianmode,thereislittlesignificantspatialcoupling/structuredlighteffectsinanSMF-based
system. Thus, this thesis will not discuss SMF-based transmission systems.
In recent years, MDM in multi-mode fiber (MMF) and few-mode fiber (FMF) have gained
growing interest because of their potential to significantly enhance the data capacity and spectral
efficiency of fiber communications [3, 4]. As shown in Fig. 1.8(b), an MDM transmitter consists
of multiple separate single-mode transmitter (similar to the one shown in Fig. 1.8(a)), each
transmittersendsanindependentdatachannelandamodemultiplexercombinesallthebeamsand
feedthemintoanMMF/FMF.AtMDMrecevier,amodedemultiplexerseparatesthetransmitted
spatial modes and multiple data channels are detected and recovered individually by separate
receivers. Because of the orthogonality among the spatial modes, transmitting N modes can
provide N times of capacity and spectral efficiency for fiber communications [4].
One fundamental challenge for MDM fiber systems is the random modal coupling induced by
the MMF and FMF [5, 36]. Due to various ambient variations (e.g., temperature and mechanical
effects) and inhomogeneity of fiber itself, MMF or FMF would induce inter-channel crosstalk
among the transmitted spatial modes, making it difficult for the receivers to efficiently recover
the multiplexed data channels.
This thesis will discuss modal power coupling mitigation approach in MDM FMF systems.
Specifically,anopticalmitigationapproachbasedonopticalmatrixinversionfor200-Gbit/sMDM
transmission will be discussed.
10
Multi-mode fiber
(few-mode fiber)
Mode 1
…
Mode 2
Mode N Mode DEMUX
Rx-N
…
Rx-2
Rx-1
Mode MUX
Tx-1
…
Tx-2
Tx-N
MDM transmitter MDM receiver Crosstalk
Mode order
l
1
l
2
…
l
N
Mode order
l
2
…
l
N
l
1
Mode 1
Mode 2
Mode N
…
Single-mode fiber
Tx Rx
a)
b)
Figure 1.8: Concept of fiber communications. (a) In conventional single-mode-fiber (SMF) links,
there is little significant spatial distortions; (b) In multi-mode-fiber (MMF) MDM links, the
inhomogeneity of the MMF can cause modal power coupling and inter-channel crosstalk among
the transmitted LG modes. Tx: transmitter; Rx: receiver; MDM: mode division multiplexing.
1.6 Thesis Outline
This thesis will mainly discuss experimental demonstrations of high-capacity optical communica-
tions using structured light in different random media. From a system point of view, this thesis
will includes studies on both single-mode and multi-mode systems in air/atmosphere, multi-mode
fibers, andturbid/diffusivemedia. Mostoftheworktobementionedarefocusedonefficientopti-
cal/electrical compensation approaches for the media’s random spatial distortions and to achieve
robust, high-capacity, and spectral-efficient data transmission inside these media.
To be specific, Chapter 2 presents the turbulence-resilient FSO data transmission by mitigat-
ing the turbulence-induced modal coupling loss using optoelectronic mixing of many LG spatial
modes. The optoelectronic mixing techniques are applied to two different data modulations, in-
cluding quadrature-amplitude modulation and differential-phase-shift keying modulation. Chap-
ter3investigatestheutilizationofoptoelectronicmulti-modemixingtoautomaticallycompensate
turbulence-induced inter-channel crosstalk in a multi-mode MDM system. The combination of
polarization, wavelength, and spatial mode multiplexing is also discussed. Chapter 4 studies the
11
interactionbetweenthespatialmodesandscatteringmediumincludingballisticanddiffusivescat-
tering. Furthermore, the utilization of adaptive wavefront shaping and diffusing is also discussed
to achieve simultaneously turbulence mitigation and channel demultiplexing. Last but not the
least,Chapter5presentstheopticalmatrixinversioninafew-modefibertoachievenear-error-free
MDM data transmission.
12
Chapter 2
Turbulence-Resilient Free-Space Optical Communications
In free-space optical (FSO) communications employing both amplitude and phase data modula-
tion (e.g., quadrature-amplitude-modulation (QAM)), the data is typically recovered by mixing a
Gaussian local oscillator (LO) with a received Gaussian data beam [46, 47, 44]. However, atmo-
sphericturbulencecaninducepowercouplingfromthetransmittedGaussianmodetohigher-order
modes, resulting in significantly degraded mixing efficiency and system performance [32, 30, 31].
In this chapter, a pilot-assisted self-coherent detection approach is proposed to overcome this
problem [48]. Specifically, as shown in Fig. 2.1, a Gaussian data beam and a frequency-offset
Gaussianpilottonebeamaretransmittedsimultaneously,suchthatbothbeamsexperiencesimilar
turbulence and modal coupling. Subsequently, a free-space-coupled photodetector (PD) mixes all
corresponding pairs of the beams’ modes. During mixing, a conjugate of the turbulence-induced
modal coupling is generated and compensates the modal coupling experienced by the data, and
thus corresponding modes of the pilot and data efficiently mix. This chapter will first discuss
pilot-assisted self-coherent systems including gapped-pilot-assisted and Kramers-Kronig detec-
tions. Moreover, a differential-detection-based self-coherent system will be discussed to achieve
an optical-power-efficient FSO system.
13
Data
Atmospheric
turbulence
Pilot
Σ
Data Pilot
I
Q
I
Q
Figure 2.1: Concept of turbulence-resilient FSO communications by pilot-assisted self-coherent
optoelectronic mixing of many spatial modes. I: in-phase; Q: quadrature.
2.1 Background and Motivation
Compared to radio/wireless communications, FSO communications has gained significant inter-
est due to higher data capacity and lower probability of interception [15, 11, 49]. In general,
an amplitude-only-modulated (e.g., pulse-amplitude-modulation (PAM)) Gaussian data beam is
transmittedandrecovered[11];sincedataisencodedasdistinctamplitudelevels,thedataconstel-
lation points of PAM lie on a 1-dimensional line in the 2-dimensional in-phase (I) and quadrature
(Q) constellation [43]. Alternatively, FSO systems can benefit from recovering the data beam’s
amplitude and phase to enable complex modulation formats [46, 50] such as QAM [42]. Since
data is encoded as distinct vectors, I/Q constellation points of a QAM data can be arranged in
a 2-dimensional array [43]. In comparison to PAM of the same number of constellation points
(i.e., modulation order) and average power per bit, QAM is generally less demanding in terms of
optical signal-to-noise ratio (OSNR) of the transmitted data due to its larger Euclidean distance
in the 2-dimensional I/Q constellation [43]. This advantage tends to be more pronounced as the
14
modulation order increases [43]. Additionally, phase recovery can enable various digital-signal-
processing (DSP) functions [x], which might benefit future FSO systems [2] (e.g., compensation
for hybrid fiber/FSO system [42] and adaptive-probabilistic-shaped modulations [51]).
Intensitymodulation/directdetection(IM/DD)FSOlinkstypicallyreceiveamplitude-encoded
data by directly detecting the beam’s intensity levels, yet phase information is not readily recov-
ered [11, 46, 47]. Alternatively, FSO systems can recover both amplitude and phase by using
coherent detection, which mixes the data beam with a receiver Gaussian LO beam [46, 47, 44].
However, atmospheric turbulence generally limits coherent detection because it induces power
coupling of the data beam from the Gaussian mode to other Laguerre-Gaussian (LG) spatial
modes [32, 30, 52]. Such turbulence-induced modal coupling can significantly degrade the LO-
data mixing efficiency due to “mode mismatch” between the LO and data beams [32, 52, 31, 53].
Without turbulence, the PD efficiently mixes the data and LO since they typically occupy the
same single Gaussian mode [17], hence “mode matched” in their spatial distributions [54, 55].
With turbulence, however, significant power of the data beam can be coupled into higher-order
LG modes and degrade mixing efficiency by over 20 dB [32, 30, 31], since data power coupled to
orthogonal higher-order modes does not efficiently mix with the Gaussian LO [31, 56].
To enable amplitude and phase recovery in turbulent links, various modal-coupling mitigation
approaches have been demonstrated [57, 58, 59, 60, 61]. One technique employs adaptive optics
to couple the data power back into the Gaussian mode by measuring the distortion using a
wavefront sensor and applying a DSP-calculated conjugate phase to the beam by a wavefront
corrector [57]. Another technique employs multi-mode digital coherent combining [58, 59, 60, 61],
wherein much of the data power in higher-order modes is captured by either a multi-mode fiber
[58, 59, 61] or an array of single-mode-fiber (SMF) apertures [60]. Subsequently, the power from
each of multiple modes is recovered by a separate coherent detector and combined using DSP
[58, 59, 60, 61]. The performance depends on the number of recovered modes, and the detection-
system complexity tends to increase with the number of detected modes [58, 59, 61]. Since
15
turbulence may induce coupling to a large number of modes, a laudable goal towards achieving
simultaneousamplitudeandphaserecoverywouldbetoautomaticallycompensateforsuchpower
coupling without additional data processing, as well as do so in a single element that efficiently
scales to recover all captured modes.
In this work, we experimentally demonstrate near-error-free transmission of a 12-Gbit/s 16-
QAM polarization-multiplexed (PolM) FSO link that is resilient to turbulence-induced LG modal
powercouplingfor200randomturbulencerealizations. Theamplitudeandphaseofthetransmit-
tedQAMdataisretrievedbyusingapilot-assistedself-coherentdetector. WetransmitaGaussian
pilot beam with a frequency offset from the Gaussian data beam, such that both beams expe-
rience similar turbulence-induced LG modal coupling. Subsequently, a single free-space-coupled
PD mixes the received multi-mode data beam with the multi-mode pilot beam in “self-coherent”
detection26. During mixing, a conjugate of the turbulence-induced modal coupling of the pilot
beamisautomaticallygeneratedandusedtocompensateforthemodalcouplinginthedatabeam.
Specifically, each data-pilot LG modal pair efficiently mixes and contributes to the intermediate-
frequency (IF) signal. Since the data and pilot experience similar modal coupling, our approach
can simultaneously mix and recover nearly all the captured data modes using a single PD. Ex-
perimental results for turbulence strength (i.e., ratio of beam size over the Fried parameter)
2w
0
/r
0
≈ 5.5 show an average mixing loss of approximately 3.3 dB.
2.2 Concept of Pilot-Assisted Optoelectronic Multi-Mode
Mixing
In an FSO link, a fundamental Gaussian beam (i.e., LG
l,p
) carrying a data channel (denoted as
S(t,f) with carrier frequency f) is transmitted through turbulent atmosphere. Due to a random
spatial and temporal refractive-index distribution, the turbulence effects can induce transverse,
16
spatially dependent wavefront distortion to the Gaussian beam [16]. Moreover, since such distor-
tion induces modal power coupling, the electrical field of the data beam ( E
data
) at the receiver
aperture can be expressed as a superposition of LG modes [32, 9]:
E
data
(t,f,x,y)=S(t,f)U(x,y)=S(t,f)
X
l
X
p
a
l,p
LG
l,p
(x,y), (2.1)
whereLG
l,p
(x,y)representstheelectricalfieldoftheLGmodes[17]withanazimuthalindex land
aradialindexp;a
l,p
=
RR
U(x,y)LG
∗ l,p
(x,y)dxdyisthecomplexcoefficientofthecorresponding
LG
l,p
component in the wavefront, ∗ denotes the conjugate of the modal electrical field, and the
portionofopticalpowercoupledtotheLG
l,p
modeis|a
l,p
|
2
; andU(x,y)=
P
l
P
p
a
l,p
LG
l,p
(x,y)
represents the turbulence-induced LG modal coupling. Ideally, the complex weights a
l,p
for all
modal components tend to satisfy
P
l
P
p
|a
l,p
|
2
∼ = 1 if the receiver aperture can collect almost
the entire beam [9].
A turbulent IM/DD FSO link (that is, S(t,f) is amplitude-only encoded) may suffer from
turbulence-inducedmodal-couplinglossifanSMF-coupledPDisusedbecausehigher-ordermodes
are not efficiently captured by the SMF [30]. For a free-space-coupled PD, however, an IM/DD
FSO link may not be significantly affected by modal coupling if the receiver aperture can collect
most of the distorted beam [62]. This free-space-coupled PD can utilize the detected optical
intensity(i.e.,|S(t,f|
2
)torecovertheamplitude-encodeddata,butthebeam’sphaseinformation
is not readily recoverable.
As shown in Fig. 2.2(a), coherent-detection FSO links can recover both the amplitude and
phaseofthedataalthoughtheysufferfromperformancedegradationcausedbyturbulence-induced
modal coupling. Here, the transmitted data S(t,f) contain both amplitude- and phase-encoded
data (for example, 16-QAM data). By way of a simple illustrative example, the continuous-wave
LOatthereceiverinasingle-PDheterodynecoherentdetectorhasanopticalfrequencyoffset∆ f
17
Optical Freq.
I
Q
0
1
− − + + , 0
1
, modal
coupling
−
Optical Freq.
Data power from LG modes
, Only the , mode mixes
with the LO efficiently and
can be recovered
I
Q
Conjugate
LG coupling
Atmospheric
turbulence
LO-based heterodyne coherent detection
Data quality degradation
due to mixing loss
Amplitude and
phase recovery
Almost all the LG modes mix
with the LO efficiently and can
be automatically recovered
0
1
Atmospheric
turbulence
(a)
(b)
Multi-LG-
mode data
Data-carrying
Gaussian beam
Optical LG spectrum
− − + + Optical LG spectrum
Distorted
Gaussian beam
− − + + − − + + ( ) � Square-law mixing in PD
�
Optical LG spectrum
0
1
− − + + , Data and pilot
Gaussian beams
Optical LG spectrum
Pilot tone
(to serve like an LO)
∆ Optical Freq.
0
1
− − + + , Optical LG spectrum
QAM transmitter
Data channel ( )
QAM transmitter
Data channel ( )
0
1
, modal
coupling
− − + + Optical LG spectrum
Similarly distorted data
and pilot Gaussian beams
0
1
, modal
coupling
− − + + Optical LG spectrum
Pilot-assisted self-coherent detection using optoelectronic beam mixing
Square-law mixing
in free-space PD
�
Data power from LG modes
− − + + Gaussian
LO at Rx
Multi-LG-
mode data
( ) � Multi-LG-
mode pilot
LG modal coupling
for the data only
Similar LG modal
coupling for the
data and pilot
( ) � ∗
� � �
∗
, ( ) � ∗
� � � ∗
�
, , � = �
�
, �
, , SPB
Electrical Freq.
SSBI
DC
SLB

Optical Freq.
−
Electrical Freq.
DC

SSBI I
Q
I
Q
Figure 2.2: Concept of simultaneous amplitude and phase recovery of a QAM data in turbu-
lent FSO links. (a) The performance of coherent detection can be significantly degraded by the
turbulence-inducedLG-modal-couplingeffects. (b)Pilot-assistedself-coherentdetectioncanauto-
matically compensate for the turbulence-induced LG-modal-coupling effects. DC: direct current;
SSB:signal-signalbeating;SSBI:signal-signal-beatinginterference;SLB:signal-LObeating;SPB:
signal-pilot beating. In both cases, the frequency offset f is greater than the data bandwidth B to
avoid the SSBI.
from the data carrier (denoted as C(f− ∆ f)) and is a Gaussian beam. The square-law mixing
in the PD of the coherent receiver results in a photocurrent [63, 64]:
I ∝
RR
|C(f− ∆ f)LG
0,0
(x,y)+S(t,f)U(x,y)|
2
dxdy
=|C(f− ∆ f)|
2
+|S(t,f)|
2
+2Re[S(t,f)C
∗ (f− ∆ f)]
RR
U(x,y)LG
∗ 0,0
(x,y)dxdy,
(2.2)
where Re[] is the real part of a complex element; I is the generated photocurrent;|C(f− ∆ f)|
2
and |S(t,f)|
2
are the direct current (d.c.) and the signal–signal beating interference (SSBI)
18
photocurrent,respectively;and2Re[S(t,f)C
∗ (f− ∆ f)]generatesthedesiredsignal–LObeating
(SLB) photocurrent. However, the Gaussian-mode LO does not mix efficiently with the multiple-
LG-mode data beam due to the mode mismatch between their LG spectra, which is expressed as
[31]:
RR
U(x,y)LG
∗ 0,0
(x,y)dxdy
=
RRP
l
P
p
a
l,p
LG
l,p
(x,y)LG
∗ 0,0
(x,y)dxdy =a
0,0
,
(2.3)
where orthogonality amongst the LG modes ensures that
RR
LG
0,0
(x,y)LG
∗ 0,0
(x,y)dxdy = 1
and
RR
LG
l,p
(x,y)LG
∗ 0,0
(x,y)dxdy = 0 given that l ̸= 0 or p ̸= 0. Equation (2.2) shows that
only the portion of the transmitted power that remains LG
0,0
after turbulence can be efficiently
mixed with the LO and utilized for recovering the QAM data. Such modal-coupling loss can
result in severe degradation of the mixing IF power and thus the recovered data quality [56].
We note that this mixing-efficiency degradation in coherent detection can occur for a PD that
is: (i) free-space-coupled due to orthogonality between the higher-order modes and the Gaussian
LO [31, 56] and (ii) SMF-coupled due to power in the higher-order modes not being efficiently
coupled into the fiber [30].
Figure 2.2(b) illustrates the simultaneous recovery of the amplitude and phase of QAM
data by utilizing pilot-assisted self-coherent detection, which automatically compensates for the
turbulence-induced modal coupling. In addition to the Gaussian data beam, we transmit a co-
axial Gaussian beam carrying a continuous-wave pilot tone with a frequency offset ∆ f, producing
a frequency gap between the pilot and data beams of roughly the channel bandwidth (B) to
avoid SSBI. The electrical fields of the data and pilot beams are likely to experience similar
turbulence-induced distortion and modal coupling due to their frequency difference being orders
19
of magnitude smaller than their carrier frequencies [16]. This similar distortion produces auto-
matic ‘mode matching’ between the beams, such that the electric field of the pilot tone is [65]:
E
pilot
(f− ∆ f,x,y)=C(f− ∆ f)U(x,y)
=C(f− ∆ f)
P
l
P
p
a
l,p
LG
l,p
(x,y).
(2.4)
Importantly, aturbulence-inducedLG-couplingconjugate U
∗ isautomaticallygeneratedfromthe
pilottocompensateforthemodalcouplingexperiencedbythedistorteddatabeam, andthetotal
generated photocurrent is:
I ∝
RR
|C(f− ∆ f)U(x,y)+S(t,f)U(x,y)|
2
dxdy
=|C(f− ∆ f)|
2
+|S(t,f)|
2
+2Re[S(t,f)C
∗ (f− ∆ f)]
RR
U(x,y)U
∗ (x,y)dxdy,
(2.5)
where S(t,f)C
∗ (f− ∆ f) generates the desired signal–pilot beating (SPB) photocurrent at an IF
of∆ f. Themodalcouplingis(ideally)correctedinanautomaticfashionandthemixingefficiency
is:
Mixingefficiency ∝
RR
U(x,y)U
∗ (x,y)dxdy
=
RRP
l
P
p
a
l,p
LG
l,p
(x,y)
P
l′
P
p′
a
∗ l′,p′
LG
∗ l′,p′
(x,y)dxdy
=
P
l
P
p
P
l′
P
p′
RR
a
l,p
LG
l,p
(x,y)a
∗ l′,p′
LG
∗ l′,p′
(x,y)dxdy
=
P
l
P
p
|a
l,p
|
2
∼ =1,
(2.6)
where each LG
l,p
component of the data beam is efficiently mixed with the corresponding LG
l,p
component of the pilot beam. Consequently, almost all the captured optical power carried by
higher-orderLGspatialmodescancontributetotheIFsignalandcanbeautomaticallyrecovered
using a single square-law free-space PD. The recovered QAM data can thus exhibit resilience
against modal-coupling loss due to the efficient mixing between the data and pilot beams.
Comparison with IM/DD and coherent systems: We note that the pilot-assisted self-
coherent approach shares some similarities with both IM/DD and coherent detection: (1) similar
20
to IM/DD, our approach does not use a receiver-based LO; and (2) similar to coherent detection,
ourapproachrecoverstheamplitudeandphasebymixingan‘LO-like’transmitter-generatedpilot
with the data beam and is often called ‘self-coherent detection’ [63, 66]. Notably, the pilot in our
self-coherent system would experience similar FSO channel loss as the data beam, which may be
noteworthy in longer-distance FSO links, whereas the LO in coherent detection would not [50].
OSNR sensitivity: Generally, the OSNR needed to achieve a desired bit error rate (BER)
depends on both the modulation formats and the detection approaches [43, 63, 67]. When com-
paring our self-coherent detection with heterodyne coherent detection for amplitude- and phase-
encoded data, the transmitted power of self-coherent detection is shared between the pilot and
data beams, resulting in self-coherent detection being more OSNR-demanding compared with
coherent detection (without turbulence effects) [63]. For example, to achieve a given BER for the
same QAM order, our self-coherent approach is likely to require an OSNR of around 3-dB higher
when the carrier (that is, pilot)-to-signal power ratio (CSPR) is 1 compared with heterodyne co-
herent detection [63, 66]. When comparing our amplitude-and-phase self-coherent approach with
amplitude-only IM/DD, the OSNR advantage of self-coherent QAM over IM/DD PAM (with
the same modulation order) becomes more pronounced as the modulation order increases (for
example, conventionally regarded to be many decibels for 16-QAM) [43, 63, 67, 66].
Power sensitivity: In longer-distance FSO links, the required optical power per bit for a
desired BER can be a limiting factor [47, 53]. Since the transmitted power is shared between the
pilotanddatabeams,self-coherentdetectionwillprobablyhavealowersignal-to-noiseratio(SNR)
compared with free-space-coupled IM/DD with the same received optical power and receiver
thermal noise. Moreover, the SNR advantage of QAM over PAM diminishes as the modulation
order decreases [43]. Consequently, IM/DD may have a better BER performance than pilot-
assisted self-coherent detection for low modulation orders, such as 2-PAM [53, 67]. We also
note that IM/DD may have a better performance than self-coherent detection under lower SNR
conditions even at higher modulation formats [43, 63, 67].
21
Compatibilitywithpolarization-division-multiplexing(PDM):Sinceatmospherictur-
bulencetendsnottoinducesignificantdepolarizationeffects[68], ourpilot-assistedsystemshould
be compatible with PDM techniques by transmitting pilot–data pairs on each orthogonal polar-
ization.
2.3 Experimental Characterization of Mixing Power Loss
AsshowninFig. 2.3,wetransmitapairofdata-carryingandpilotGaussianbeamsonbothXand
Y polarizations. A 6-Gbit/s 16-QAM data channel at a wavelength of λ 1
≈ 1.55 µm is generated,
amplified using an erbium-doped fibre amplifier (EDFA) and equally split into two copies. One
copy is delayed using a 15-m SMF to decorrelate the data channels and two independent data
channelsareindividuallycombinedwithanotherpilottoneatawavelengthofλ 2
(withafrequency
offset of 2.6 GHz from λ 1
, λ ≈ 0.02 nm). The polarizations of the signals and pilots are adjusted
andsubsequentlycombinedusingapolarizationbeamcombinertotransmitPolM16-QAMsignals.
The total optical power including the pilot and data beams is around 7 dBm for each of the
polarizations. The optical signal is coupled to free space using an optical collimator (Gaussian
beam size of diameter 2w
0
≈ 2.2 mm), is distorted using a rotatable turbulence emulator (see the
section ‘Experimental emulation of atmospheric turbulence effects’) and then propagates in free
space for 1 m. In this demonstration, we emulate different strengths of atmospheric turbulence
using two separate turbulence emulators with different Fried parameters r
0
of 1.0 mm and 0.4
mm. The emulated turbulence distortion for the transmitted Gaussian beam is characterized by
the ratio of the beam size to the Fried parameter [16], and these are 2w
0
/r
0
≈ 2.2 and 5.5 for the
two emulators.
At the receiver, we demultiplex one polarization at a time using a half-wave plate cascaded
with a polarizer. The receiver has an aperture diameter of approximately 10 mm. We measure
the spatial amplitude and phase profiles of the turbulence-distorted beam and calculate its LG
22
PC
Mod.
EDFA
Decorrelation
delay
Laser
AWG
I Q
Polarization-multiplexed (PolM) QAM transmitter
PC
PBC
PC
Laser
X Pol.
Y Pol.
EDFA
Col.
Rotatable
turbulence
emulator
MR
Polarizer
Col.
Laser
PC
Camera
Off-axis
holography
Oscilloscope
Single-PD LO-based heterodyne
coherent receiver (pilot turned off)
EDFA
Col.
PC
Laser
Pilot-assisted self-coherent detector
MR
Lens
BS
~ 1-meter free-
space propagation
I
Q

, spectrum measurement
(signal on turned off)
or
1.5 Gbaud
16 QAM
X
Y
FM
Z
HWP
MR FS
PD
SMF
SMF
PD Oscilloscope
Off-line DSP
Off-line DSP
Pilot
Data
Figure 2.3: Experimental setup for 12 Gbit/s 16-QAM PDM FSO link. Equal copies of the
received beams are detected by the pilot-assisted self-coherent detector and single-PD LO-based
heterodyne coherent detector. During the detection of the heterodyne coherent receiver, the
pilot is turned off. The same DSP algorithms are applied to both receivers to retrieve the 16-
QAM data. AWG, arbitrary waveform generator; Mod., modulator; EDFA, erbium-doped fibre
amplifier; PC, polarization controller; PBC, polarization beam combiner; pol., polarization; Col.,
collimator; MR, mirror; HWP, half-wave plate; FM, flip mirror; BS, beam splitter; FS-PD, free-
space-coupled photodetector; SMF, single-mode fiber.
decomposition using off-axis holography [69]. After polarization demultiplexing, the distorted
beam is equally split into two copies that are sent to the pilot-assisted self-coherent detector and
a single-PD LO-based heterodyne coherent detector.
In the pilot-assisted self-coherent detector, the entire spatial profiles of the distorted data and
pilotbeamsarefocusedintoafree-space-coupledInGaAsPD(3-dBbandwidthlessthan3.5GHz)
usinganasphericlenswithafocallengthof16mmandanumericalapertureof0.79. Thecoupling
efficiency of the received Gaussian beam, defined as the ratio of the optical power detected by the
PD over the total received optical power by the receiver aperture (without turbulence effects),
is measured to be greater than 92%. The generated photocurrent is recorded using a real-time
23
digital oscilloscope and the I–Q information of the data channel is subsequently retrieved using
off-lineDSPalgorithms. TheNyquist-shaped16-QAMdatachannelhasasymbolrateof1.5GHz
with a roll-off factor of 0.1, expanding the data’s spectrum to around 1.7 GHz. To avoid SSBI
effects,wesettheIF(thatis,thedifferencebetweenthepilotanddatabeams’carrierfrequencies)
at f ≈ 2.6 GHz, which includes a frequency gap of approximately 1.8 GHz between the pilot and
data beams. Thus, the total transmitted pilot-assisted signal spectrum is around 3.5 GHz, which
is roughly twice that of the data spectrum.
0 5 10 15 20 25 30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Measured optical power loss (dB)
X Pol.
LO-based heterodyne
coherent detection
Fraction of measurements
2w
0
/r
0
~2.2, pilot-assisted self-coherent detection
2w
0
/r
0
~5.5, pilot-assisted self-coherent detection
2w
0
/r
0
~2.2, LO-based coherent detection
2w
0
/r
0
~5.5, LO-based coherent detection
Pilot-assisted
self-coherent detection
(a1)
0 5 10 15 20 25 30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
2w
0
/r
0
~2.2, pilot-assisted self-coherent detection
2w
0
/r
0
~5.5, pilot-assisted self-coherent detection
2w
0
/r
0
~2.2, LO-based coherent detection
2w
0
/r
0
~5.5, LO-based coherent detection
(a2)
Measured optical power loss (dB)
Y Pol.
LO-based heterodyne
coherent detection
Fraction of measurements
Pilot-assisted
self-coherent detection
0 5 10 15 20 25 30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(b1) Measured electrical mixing power loss (dB)
X Pol.
LO-based heterodyne
coherent detection
Fraction of measurements
Pilot-assisted
self-coherent detection
0 5 10 15 20 25 30
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(b2) Measured electrical mixing power loss (dB)
Y Pol.
LO-based heterodyne
coherent detection
Fraction of measurements
Pilot-assisted
self-coherent detection
1 2 3 4 5 6 7
0
5
10
15
20
Including both
X and Y Pol.
(c1)
Average optical power loss (dB)
Turbulence strength 2w
0
/r
0
Simulation, LO-based coherent detection
Simulation, pilot-assisted self-coherent detection
Experiment, LO-based coherent detection
Experiment, pilot-assisted self-coherent detection
1 2 3 4 5 6 7
0
5
10
15
20
Including both
X and Y Pol.
(c2)
Average electrical mixing power loss (dB)
Turbulence strength 2w
0
/r
0
Figure 2.4: Measurement of optical and mixing power loss for the pilot-assisted self-coherent
detector under different turbulence strengths. (a) Experimentally measured histograms of optical
power loss under two different turbulence distortions (2 w
0
/r
0
≈ 2.2 and 5.5) for X (left) and
Y (right) polarizations; (b) Experimentally measured histograms of mixing power loss (in the
electrical domain) under two different turbulence distortions (2 w
0
/r
0
≈ 2.2 and 5.5) for X (left)
and Y (right) polarizations. The mixing power loss is measured at the IF of approximately 2.6
GHz in the electrical domain. In (a) and (b), 1,000 different turbulence realizations are measured
foreachpolarization. (c)Simulatedaverageopticalpowerloss(top)andaverageelectricalmixing
power loss (bottom) results for different turbulence strengths from 1 to 7. The average values of
experimentally measured data points (including both X and Y polarizations) are also plotted.
24
At the single-PD LO-based heterodyne coherent detector (the pilot λ 2
is turned off), we set
the same IF value as the pilot-assisted self-coherent receiver. The distorted Gaussian beam is
coupled into an SMF via a collimator (aperture diameter around 3.5 mm), amplified using an
EDFA, and mixed with an LO (at the same wavelength λ 2
as the pilot) at the SMF-coupled PD.
The received optical signal is amplified by the EDFA to meet the power sensitivity requirement
of the SMF-coupled PD. The electrical signal is subsequently recorded using a real-time digital
oscilloscope and processed to retrieve the data channel’s I–Q information using the same off-line
DSP algorithms as the pilot-assisted self-coherent detector. Note that we measure the optical
power loss and electrical mixing power loss of this detector without using the EDFA inside this
receiver. ThemixingpowerlossismeasuredattheIFofaround2.6GHzintheelectricaldomain.
We measure the turbulence-induced optical power loss and electrical mixing power loss of
the pilot-assisted self-coherent detector for each polarization at 1,000 random realizations of the
emulated turbulence. For both X and Y polarizations, Fig. 2.4(a) shows that stronger turbulence
induces <2 dB of optical power loss for self-coherent detection since the free-space-coupled PD
can capture most of the power; we note that free-space-coupled IM/DD systems are likely to have
similar captured power loss. As shown in Fig. 2.4(b), the self-coherent detector has an electrical
mixingpowerlossof<3dBand<6dBfor99%weakerand90%strongerturbulencerealizations
among 1,000 random turbulence realizations, respectively. The relatively low mixing power loss
for self-coherent detection is due to efficient mixing of the pilot and data beams, which is likely
to recover almost all the data power from the captured modes.
As discussed, turbulence-induced modal coupling can result in significant power loss for SMF-
coupled IM/DD or coherent detectors. Figure 2.4(a) shows that the optical power loss for SMF-
coupled systems ranges from 2 to 22 dB and from 7 to 30 dB under 2.2 and 5.5 turbulence
strengths, respectively. Among the 1,000 emulated turbulence realizations, Fig. 2.4(b) shows
that the coherent detector can suffer from a mixing power loss of approximately 28 dB for 99%
25
and 90% of weaker and stronger turbulence, respectively. This mixing loss is due to the SMF-
coupled detector not efficiently capturing the power coupled to higher-order modes [30].
To help further validate our experimental results, we simulate the self-coherent system using
1-random-phase-screen simulation. As shown in Fig. 2.4(c), the simulation results indicate that
self-coherent detection suffers <4 dB of average optical and electrical mixing power loss as the
turbulence strength 2w
0
/r
0
is increased from 1 to 7. Moreover, the plotted experimental results
are generally in agreement with the simulation.
2.4 Experimental Measurement of LG Modal Spectrum by
Off-Axis Holography
In this section, we present the step-by-step digital-image-processing procedures we use to extract
the complex wavefront of a distorted beam using the off-axis holography [69].
As shown in Fig. 2.5, we perform the following steps to extract both the spatial amplitude
and phase profiles of the distorted pilot beam [69]: (i) Record the interferogram between the
distorted pilot beam and another off-axis undistorted reference Gaussian beam using an infrared
camera; (ii) Perform two-dimensional Fourier transform of the interferogram to obtain the spatial
frequency spectrum, filter out the 1st-order diffraction, and shift the 1st-order diffraction to the
center of the spatial frequency spectrum; (iii) Perform two-dimensional inverse Fourier transform
of the shifted spatial frequency spectrum, and subsequently obtain the spatial amplitude and
phase profiles of the distorted beam (i.e., E
rec
(x,y)); and (iv) Decompose the distorted pilot
beam in the two-dimensional LG modal basis using the Eq. 2.7:
a
l,p
=
Z Z
E
rec
(x,y)LG
∗ l,p
(x,y)dxdy, (2.7)
26
where E
rec
(x,y) and LG
l,p
(x,y) are the measured complex fields of the distorted pilot beam and
the theoretical complex field of an LG
l,p
mode, respectively. The ratio of optical power coupling
to the LG
l,p
mode is given by|a
l,p
|
2
.
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Amplitude Phase
0
2
-5 0 +5

0
10

LG spectrum
decomposition
-30
-25
-20
-15
-10
-5
0
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
-30 (dB)
0
Camera-recorded
off-axis interferogram
2-D FFT
Filter out the 1
st
order diffraction
2-D spatial frequency
spectrum
1st order
Filtered 2-D spatial
frequency spectrum
Shifted 2-D spatial
frequency spectrum
Inverse 2-
D FFT
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Reconstructed complex wavefront

!,#
=%
$%&
, *
!,#
∗
,
2
!
/
!
~5.5
Figure 2.5: The procedures for measuring the LG spectrum of a distorted pilot beam using the
off-axis holography. The spatial amplitude and phase profiles of the distorted pilot beam are
obtained, and subsequently the corresponding LG power spectrum is calculated. The captured
images have 320 × 256 pixels, with a pixel size of 30 µm . FFT: fast Fourier transform; a
l,p
:
coupling coefficient for the LG
l,p
mode. The ratio of optical power coupled to the LG
l,p
mode is
given by|a
l,p
|
2
.
27
2.5 Polarization-Multiplexed12-Gbit/s16-QAMFree-Space
Optical Data Transmission
We demonstrate 12-Gbit/s PDM FSO transmission under emulated turbulence effects, with each
polarizationcarrying1.5-Gbaud16-QAMdata. Thetransmittedtotalopticalpowerperpolariza-
tion(includingpilot and databeams)isapproximately 7dBm. ThetransmittedCSPRvaluesare
approximately 1.1 and 1 for X and Y polarizations, respectively. Figure 2.6 shows the recovered
16-QAM constellations using the self-coherent detector under example realizations of the weaker
and stronger turbulence. We measure the turbulence-induced LG spectra for ℓ and p indices of
− 5 to +5 and 0 to 10, respectively. The complex wavefront is measured using off-axis holography
[69].
With no turbulence effects, Fig. 2.6(a) shows that the pilot-assisted self-coherent detector can
achieveanear-error-freeperformanceandrecoveranerrorvectormagnitude(EVM)of 8%forthe
16-QAM data. Under one random realization of weaker turbulence, the measured LG spectrum
ofFig. 2.6(b)showsthatthedatapowerismainlycoupledtotheneighbouringLGmodes. Under
twodifferentrandomrealizationsofstrongerturbulence,Fig. 2.6(c-d)showthatturbulenceeffects
caninduceapowerlossof>25dBandthatpowercanbecoupledtoalargenumberofLGmodes.
The performance of the self-coherent detector is not severely affected by these turbulence effects
andthe16-QAMdatacanberecoveredwithEVMvaluesfromapproximately8%to10%forboth
realizations. This turbulence resiliency is due to the automatic modal-coupling compensation by
the pilot–data mixing, enabling almost all captured LG modes to be efficiently recovered.
To elucidate the effects of turbulence-induced modal coupling on coherent detection, we also
show the recovered 16-QAM data for an SMF-coupled heterodyne coherent detector in Fig. 2.6;
the recovered data quality degrades for both polarizations, from EVM values of approximately
7.5% without turbulence (Fig. 2.6(a)) to >16% for stronger turbulence (Fig. 2.6(c-d)). This
28
X Pol.
~8.1% EVM
0
1
Amplitude Phase
0
2 -5 0 +5
0
10
-30 (dB)
0
LG spectrum ( / ~ )
Pilot-assisted self-coherent detector
Almost all spatial modes recovered
Y Pol.
~8.1% EVM
LO-based Hetero. coherent detector
Only Gaussian mode recovered
X Pol.
~7.5% EVM
Y Pol.
~7.5% EVM
X Pol.
~8.8% EVM
0
1
Amplitude Phase
0
2 -5 0 +5
0
10
LG spectrum ( / ~ . )
Pilot-assisted self-coherent detector
Almost all spatial modes recovered
Y Pol.
~9.3% EVM
LO-based Hetero. coherent detector
Only Gaussian mode recovered
X Pol.
~7.7% EVM
Y Pol.
~7.7% EVM
(a) No turbulence effects (b) Weaker turbulence R1
-30 (dB)
0
X Pol.
~8.2% EVM
0
1
Amplitude Phase
0
2 -5 0 +5
0
10
LG spectrum ( / ~ . )
Y Pol.
~8.6% EVM
LO-based Hetero. coherent detector
Only Gaussian mode recovered
X Pol.
~16.4% EVM
Y Pol.
~15.6% EVM
Pilot-assisted self-coherent detector
Almost all spatial modes recovered
(c) Stronger turbulence R1
-30 (dB)
0
0
1
Amplitude Phase
0
2 -5 0 +5
0
10
LG spectrum ( / ~ . )
(d) Stronger turbulence R2
-30 (dB)
0
X Pol.
~9.1% EVM
Y Pol.
~10.1% EVM
Pilot-assisted self-coherent detector
Almost all spatial modes recovered
LO-based Hetero. coherent detector
Only Gaussian mode recovered
X Pol.
>20% EVM
Y Pol.
>20% EVM
Figure 2.6: Experimental results of turbulence-induced LG modal power coupling and recovered
16-QAM data qualities using the pilot-assisted self-coherent detector. (a) No turbulence dis-
tortion. (b) One example realization (R1) of the weaker turbulence distortion (2w
0
/r
0
≈ 2.2).
(c-d) Two different example realizations (R1 (c) and R2 (d)) of the stronger turbulence distortion
(2w
0
/r
0
≈ 5.5). For each of the four realizations, we measure the LG modal power spectrum
(two indices − 5 ≤ ℓ ≤ +5 and 0 ≤ p ≤ 10) and recover the 16-QAM data constellations. In
this demonstration of PolM FSO data transmission, each polarization carries a 6-Gbit/s 16-QAM
signal. pol., polarization; EVM, error vector magnitude.
degradation is due to data power coupled to higher-order modes that is not efficiently captured
by the SMF [30].
Figure 2.7 shows measured BER values for the pilot-assisted self-coherent detector under
200 random realizations of weaker and stronger turbulence. Results show that the self-coherent
detector can achieve BER values below the 7% forward error correction limit for all realizations.
Since turbulence can cause strong modal-coupling-induced power loss, the performance of the
coherent detector can degrade and does not achieve the 7% forward error correction limit for
some realizations.
29
0 10 20 30 40 50 60 70 80 90 100
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
0 10 20 30 40 50 60 70 80 90 100
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
0 10 20 30 40 50 60 70 80 90 100
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
0 10 20 30 40 50 60 70 80 90 100
-5
10
-4
10
-3
10
-2
10
-1
10
0
7% FEC
(a) Random turbulence realization
X Pol., 2w
0
/r
0
~2.2 (weaker turbulence)
LO-based heterodyne coherent detector, only the Gaussian mode recovered
Pilot-assisted self-coherent detector, almost all the spatial modes recovered
BER
7% FEC
BER
(c) Random turbulence realization
X Pol., 2w
0
/r
0
~5.5 (stronger turbulence)
BER
(b) Random turbulence realization
Y Pol., 2w
0
/r
0
~2.2 (weaker turbulence)
(d)
BER
Y Pol., 2w
0
/r
0
~5.5 (stronger turbulence)
Random turbulence realization
Figure 2.7: Experimentally measured BER performance of the pilot-assisted self-coherent detec-
tor over 200 different emulated turbulence random realizations. (a) Weaker turbulence effects, X
polarization. (b) Weaker turbulence effects, Y polarization. (c) Stronger turbulence effects, X
polarization. (d)Strongerturbulenceeffects,Ypolarization. Toindicatetheeffectsofturbulence-
induced modal coupling on a coherent-detection FSO system with the single-Gaussian-mode LO,
the performance using an SMF-coupled LO-based heterodyne coherent detector is also shown. In
the PDM FSO data transmission, each polarization carries a 6-Gbit/s 16-QAM data signal. Note
that we measure the BER performance for one polarization at a time due to limitations of our
measurement setup. Therefore, the BER values for X and Y polarizations with the same real-
ization label may correspond to different turbulence realizations and are difficult to be compared
directly.
2.6 EnhancingSpectralEfficiencybyKramers-KronigSelf-
Coherent Detection
In our self-coherent approach, a frequency gap between the pilot and data beams is needed to
avoid SSBI. This gap is roughly equal to the data bandwidth, such that our spectrum is around
2× the data bandwidth. However, this frequency gap can be reduced to increase the spectral
efficiency using SSBI mitigation techniques [70, 71] such as Kramers–Kronig (KK) detection [50,
71]. Therefore, As shown in Fig. 2.8(a), we demonstrate a reduction of the data–pilot gap to
approximately 0.1 GHz (IF ≈ 0.9 GHz) using KK detection. Using KK detection, the spectral
30
efficiency of the pilot-assisted approach could be increased by roughly 2 ×. Importantly, the KK
scheme typically utilizes a stronger pilot than the non-KK approach. Hence, it is typically less
power efficient than the non-KK pilot-assisted approach [71], resulting in a trade-off between
power efficiency and spectral efficiency.
The transmitted optical power is approximately 10 dBm for each polarization. The CSPRs
in KK detection are around 13.8 and 14.8 for X and Y polarizations, respectively. The relatively
largerCSPR(thanthenon-KKpilot-assistedself-coherentdetection)istoensurethatthereceived
signal is “minimum-phase conditioned” [71, 63], and the phase of the QAM data (i.e., ϕ (t)) can
be subsequently retrieved by using the Hilbert transform as expressed [71]:
ϕ (t)=
1
2π p.v.
Z
+∞
−∞
log[I(t
′
)]
t− t
′
dt
′
, (2.8)
where p.v. is the principal value; I(t) is the detected photocurrent.
As shown in Fig. 2.8(b), the frequency gap between the transmitted pilot and data is approx-
imately 0.1 GHz and the IF in the electrical spectrum is around 0.9 GHz which is approximately
halfoftheNyquist-shaped16-QAMdatabandwidth(1.5Gbaudsymbolratewitharoll-offfactor
of 0.1); KK detection can effectively mitigate the SSBI and reduce the recovered 16-QAM data’s
EVM from approximately 12.4% to 9.5% (X polarization, without turbulence effects). Moreover,
we experimentally investigate the performance of KK detection under 3 cases: no turbulence
effects, one example realization of the weaker turbulence (2 w
0
/r
0
≈ 2.2), and one example real-
ization of the stronger turbulence (2w
0
/r
0
≈ 5.5). As shown in Fig. 2.8(c), the measured LG
spectra (two indices − 5≤ ℓ≤ +5 and 0≤ p≤ 10) for the two example turbulence realizations
indicate that turbulence can induce power coupling from the fundamental Gaussian mode to a
largenumberofhigher-orderLGmodes. Underthese3cases, therecovered16-QAMdataexhibit
EVMs<12%forbothXandYpolarizations. Thisisbecausethedatapowerfromallthecaptured
LG modes tends to be efficiently recovered by the pilot-assisted self-coherent detector.
31
X Pol.
~9.5% EVM
-5 0 +5

0
10
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
-30 (dB） 0

LG spectrum
Pilot-assisted self-coherent receiver with
Kramers-Kronig (KK) detection
Y Pol.
~9.5% EVM
a)
Pilot-assisted self-coherent receiver using Kramers-Kronig detection
Pilot
tone
Optical Freq.

Data ()
QAM Transmitter
Almost all the LG modes can
be efficiently mixed and
automatically recovered
I
Q
Kramers-Kronig (KK) relation to mitigate SSBI
Electrical Freq.
DC

SSBI
SPB
KK
relation
Electrical Freq.
DC

SPB

c1) No turbulence
Square-law mixing
In free-space PD
"
!
2
!
/
!
~0
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Amplitude
Phase
0
2
X Pol.
~9.9% EVM
-5 0 +5

0
10
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
-30 (dB） 0

LG spectrum
Pilot-assisted self-coherent receiver with
Kramers-Kronig (KK) detection
Y Pol.
~9.8% EVM
c2) Weaker turbulence
2
!
/
!
~2.2
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Amplitude
Phase
0
2
X Pol.
~11.3% EVM
-5 0 +5

0
10
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
-30 (dB） 0

LG spectrum
Pilot-assisted self-coherent receiver with
Kramers-Kronig (KK) detection
Y Pol.
~11.8% EVM
c3) Stronger turbulence
2
!
/
!
~5.5
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Amplitude
Phase
0
2
I
Q
I
Q
0 1 2 3 4
Electrical frequency (GHz)
0
20
40
60
80
100
Normalized power (dB)
~9.5% EVM ~12.4% EVM
w/o KK
(w/o SSBI mitigation)
w/ KK
(w/ SSBI mitigation)
IF~0.9 GHz
b)
Detected electrical spectrum
Figure 2.8: Concept and experimental results of enhancing spectral efficiency of the pilot-assisted
self-coherent detector by using Kramers-Kronig (KK) detection. (a) Concept of employing KK
relation to mitigate SSBI for the pilot-assisted self-coherent detector; (b) The recovered 16-QAM
data constellation with and without KK processing (X polarization, without turbulence); (c)
MeasuredLGmodalspectraandtherecovered16-QAMdataconstellationsunder3cases: (c1)No
turbulencedistortion; (c2)Oneexamplerealizationoftheweakerturbulencedistortion(2w
0
/r
0
≈ 2.2); (c3) One example realization of the stronger turbulence distortions (2w
0
/r
0
≈ 5.5). DC:
direct current; SSBI: signal-signal-beating interference; SPB: signal-pilot beating. EVM: error
vector magnitude.
32
2.7 EnhancingPowerEfficiencybyDifferentialSelf-Coherent
Detection
FSO communications often use an amplitude-modulated data beam and direct detection by a
single PD, and the amplitude modulation can be binary (e.g., on-off keying) or M-ary [11]. As
with general optical communication systems, binary-phase-shift-keying (BPSK) tends to be less
signal-to-noise-ratio (SNR) demanding than amplitude modulation [43]. To recover the BPSK
data,coherentdetectionwithalocaloscillator(LO)canbeutilized[44]. Moreover,inthescenario
of direct detection for BPSK, the data can be differentially encoded (e.g., differential-phase-shift-
keying, DPSK), and the receiver consists of a delay-line interferometer (DLI) and a balanced PD
[72]. A question remains as to whether the automatic optoelectronic multi-mode mixing can be
applied to a power-efficient DPSK FSO link under different turbulence distortions.
Kramers-Kronig
detection
I
Q
Optical Freq. Optical Freq.
I
Q
Gapped pilot-assisted
detection
Optical Freq.
Differential phase shift
keying (DPSK)
I
Q
Optical power efficiency
Single-end
PD
Single-end
PD
Optical
signal
Optical
signal
BS
BS
PD
PD
Delayed
path
Direct
path
Figure 2.9: Conceptual comparison for Kramers-Kronig, gapped-pilot-assisted, and differential-
phase-shift-keying detection.
AsshowninFig. 2.10,weexperimentallydemonstratea2.25-Gbit/sturbulence-resilientDPSK
FSO link using automatic multi-mode optoelectronic beam mixing. A fundamental Gaussian
33
laser beam carrying the DPSK data stream is transmitted in a 1-m FSO link with an emulated
turbulence strength of 2w
0
/r
0
≈ 5.5. At recevier, a free-space DLI with relay imaging setups is
utilized to enable spatial matching and coherently combining the direct beam and its 1-symbol-
delayed copy. After focused onto free-space-coupled PDs, the turbulence-induced modal coupling
is automatically compensated by the square-law data-data O/E mixing. Experimental results
indicate that: (i) our approach exhibits an average mixing loss of 14.6-dB less than that of an
SMF-coupledsystem(basedon200randomturbulencerealizations),and(ii)themulti-modeO/E
mixing achieves less quality (Q) factor variation than an SMF-coupled coherent Rx (based on 50
random turbulence realizations).
DPSK
modulation
Turbulence induces
LG modal coupling
Gaussian
laser beam
I
Q
I
Q
1-symbol
delay
Σ
Free-space delay-line interferometry w/
relay imaging & automatic O/E mixing
O/E
mixing
Turb.-induced
multi-mode beam
Turbulence-resilient
data recovery
(compensation-free)
"#
∗
≈1
Plane
wavefront
Distorted
wavefront
FS-
coupled
PDs
O/E: optoelectronic;
FS: free-space.
Figure 2.10: Concept of a turbulence-resilient DPSK FSO link. A free-space DLI with relay-
ing imaging combines direct and delayed beams which are similarly distorted. Subsequently,
free-space-coupled PDs can perform O/E data-data mixing to automatically compensate the
turbulence-induced LG modal coupling.
A fundamental Gaussian beam carrying a DPSK data is transmitted through atmospheric
turbulence. Due to the random refractive index distribution of turbulence, the recevier would
capture a distorted beam containing many higher-order LG modes [16]. In our approach, a free-
space DLI with relay imaging is used to match and coherently combine the received field and the
1-symbol-delayed copy of itself, and subsequently both beams are focused onto free-space-coupled
34
PDs. The PDs can perform automatic optoelectronic multi-mode data-data differential mixing to
efficiently recover the data stream:
I(t)∝
Z Z
s(t)UU
∗ s(t− T
s
)
∗ dxdy≈ s(t)s(t− T
s
)
∗ (2.9)
where I(t) and s(t) are the generated photocurrent and transmitted DPSK data, respectively;
andtheturbulence-inducedLGmodalcouplingU isautomaticallycompensatedbyoptoelectronic
mixing.
Laser
1.55 µm
Mod.
EDFA Col.

!

!

!
!
BS
BS

"

"

"

"
PD
PD
Off-line DSP
PC
Turbulence
emulation
~ 1-m free-space
propagation FM
O/E multi-mode mixing
w/ 4f relay imaging
PC Col.
SMF
LO
laser
Coherent
detection
Coherent receiver
AWG
(a)
0 5 10 15 20 25 30
Measured mixing power loss (dB)
0
0.1
0.2
0.3
0.4
Fraction of measurements
O/E multi-mode mixing
SMF coherent detection
200 random Turb. realizations
Coupler
Laser
Off-axis
holography
Camera
Q factor ~18.1 dB
(Tx power ~6 dBm)
Amplitude
2
!
/
!
~0
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
2
-5 +5
0
10
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
-30
(dB)
0

Phase
0
LG spectrum

#,#
Measured eye diagram
(b)
Amplitude
2
!
/
!
~5.5
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
2
-5 +5
0
10
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
-30
(dB)
0

Phase
0
LG spectrum (c)
(d)
0 10 20 30 40 50
Random turbulence realization
0
5
10
15
Q factor degradation (dB)
O/E multi-mode mixing
SMF coherent detection
(e)
4
"
−4
!
~
%
M.
Figure 2.11: (a) Experimental setup of a 2.25-Gbit/s DPSK FSO link. AWG: arbitrary wave-
form generator; Mod.: I/Q modulator; EDFA: erbium-doped fiber amplifier; PC: polarization
controller; Col.: collimator; FM: flip mirror; M.: mirror; BS: beam splitter; PD: free-space pho-
todetector; SMF: single-mode fiber; LO: local oscillator. (b) and (c): Measured amplitude, phase
profiles, and LG spectra of the received beam without and with turbulence. (d) Measured his-
togram of mixing power loss for our approach and SMF-coupled system. (e) Measured Q factor
degradation for our approach and SMF-coupled coherent receiver.
As shown in Fig. 2.11, a laser beam is modulated with a 2.25-Gbit/s DPSK data stream
and coupled to free space (beam size 2w
0
≈ 2.2 mm), and normally incident on the turbulence
emulator. The transmitted optical power is approximately 6 dBm. After around 1-m free-space
35
propagation, the distorted beam is captured by receiver and an off-axis holography measures the
wavefront and the corresponding LG spectrum of the distorted beam. At receiver, a free-space
DLI with two 4f relay imaging systems is designed to match the spatial fields between the direct
and delayed beams. After coherent beam combining inside the DLI, two sets of distorted beams
are focused onto separate free-space-coupled PDs (Thorlabs DET08C) and joint off-line digital
signalprocessingisusedtorecovertheDPSKdata. Toillustratetheeffectsofturbulence-induced
modal coupling loss on an SMF-coupled system, we also measure the recovered data quality of
an SMF-coupled coherent receiver. Figure 2.11(b) and (c) show the received beam profiles and
the corresponding LG spectra under no turbulence (2w
0
/r
0
≈ 0) and one random turbulence
realization (2w
0
/r
0
≈ 5.5), respectively. As shown in Fig. 2.11(d), our approach shows an
average mixing loss of 4.6 dB for 200 random realizations while approximately 19.3-dB average
mixinglossfortheSMF-coupledsystem. AsshowninFig. 2.11(e), themulti-modeoptoelectronic
mixing tends to have less Q factor variation (more resilient to turbulence-induced modal coupling
loss) than the SMF-coupled coherent system over 50 random turbulence realizations.
36
Chapter 3
Turbulence-Resilient Mode-Division-Multiplexing in
Free-Space Optical Communications
There has been a great amount of interest in utilizing orthogonal spatial modes to enhance the
performance of free-space optical (FSO) communications. In this chapter, several approaches
involvingspatialmodesandoptoelectronicmodemixingwillbediscussed,including: (i)alignment
monitor for FSO links by mixing two opposite-order modes [73]; and (ii) automatic crosstalk
reduction by mixing spatial pilot and data modes [65].
3.1 Background and Motivation
There has been growing interest in increasing the spectral efficiency of FSO communications
[15, 11]. In general, FSO links transmit a single fundamental Gaussian beam to achieve point-
to-point data transmission [49]. One potential method for spectral efficiency enhancement is
to use space-division multiplexing (SDM), in which multiple data-carrying optical beams are
transmitted simultaneously [3]. A special type of SDM is mode-division multiplexing (MDM),
such that independent data channels are carried by multiple structured light beams and each
optical beam is tailored to be on a different mode of an orthogonal spatial basis set [74, 75]. The
orthogonality among different modes enables efficient multiplexing, co-axial propagation, and
37
demultiplexing with little inherent inter-channel crosstalk. Orbital angular momentum (OAM)
mode is a candidate for MDM FSO systems [19].
Atmospheric
Turbulence
Transmitted OAM
Power

1
OAM mode
Distorted OAM
Power

1
OAM modes

4

2

3

5
Figure 3.1: Concept of turbulence-induced OAM modal power coupling.
AsignificantchallengeforFSOlinksistheirperformancedegradationinducedbyatmospheric
turbulence effects [33, 45, 76]. For multiple-OAM-beam FSO links, atmospheric turbulence can
cause modal coupling among the transmitted OAM modes, resulting in detrimental inter-channel
crosstalk[33,45]. SeveralopticalandelectronicapproachestomitigateturbulenceeffectsinOAM-
based systems have been investigated, including: (i) adaptive optics (AO) wherein the distorted
wavefront is measured and subsequently a conjugate spatial phase modulation is applied to the
distorted beams [77, 78], and (ii) multiple-input-multiple-output (MIMO)-based digital signal
processing (DSP) which reduces the inter-channel crosstalk by using multi-channel equalization
algorithms [79].
38
3.2 AutomaticCompensationforTurbulence-InducedCrosstalk
by Optoelectronic Mixing of Spatial Modes
Figure 3.2 illustrates the concept of utilizing optoelectronic OAM beam mixing to achieve the
OAM modal coupling resilient to turbulence distortions in an FSO link. Originating from the
random nature of the refractive index distribution of the turbulent atmosphere, the transverse,
spatially dependent phase distortions to the wavefront of propagating OAM beams can be rep-
resented by the two dimensional function U(x,y) = exp[jφ(x,y)] [16, 45]. In this approach, a
conjugate phase modulation U
∗ (x,y) is automatically applied to the distorted OAM beams by
the square-law detection and the turbulence-induced modal couplings can also be automatically
compensated. As shown in Fig. 3.2(a), a modulated data and a CW pilot are carried by the
same OAM beam, having an optical frequency difference of ∆ f. After this pair of OAM beams
propagateco-axiallythroughtheturbulentatmosphere,theentirespatialprofilesofthetwoOAM
beams are tightly focused into a single-pixel PD. The electrical spatial fields of the signal OAM
beam (of carrier frequency ν ), S(x,y,ν ), and that of the pilot OAM beam, C(x,y,ν +∆ f), are
likely to experience the same turbulence-induced distortion U(x,y) due to their frequency dif-
ference being orders of magnitudes smaller than their carrier frequencies [80]. At the receiver,
the free-space-coupled PD mixes the similarly distorted pilot OAM and data OAM beams, i.e.,
heterodyne detection to form the photocurrent as expressed in Eq. (3.1):
I =
RR
|CU +SU|
2
dxdy =
RR
(C +S)UU
∗ (C +S)
∗ dxdy
=|C +S|
2
RR
UU
∗ dxdy =|C|
2
+|S|
2
+2Re[SC
∗ ],
(3.1)
where * and Re[] denote the conjugate operation and real part of the complex electrical field, re-
spectively. As shown in Eq. 3.1, the turbulence-induced distortions are (ideally) compensated by
RR
UU
∗ dxdy = 1, so that little crosstalk is induced after the heterodyne detection and the sub-
sequent signal decoding. The first two terms |C|
2
and|S|
2
correspond to the direct current (DC)
39
component and the signal-signal beating (SSB) photocurrent, respectively; the 2Re[SC
∗ ] term
generates the desired signal-pilot-beating photocurrent at intermediate frequency (IF) ∆ f in the
electrical domain. Moreover, if the signal and pilot are carried by different OAM beams ( S
l1
and
C
l2
, with l
1
̸= l
2
), their optoelectronic beam mixing contributes little to the signal-pilot beating
termowingtothespatialorthogonalityofthetwodifferentOAMbeams, i.e.,
RR
S
l1
C
∗ l2
dxdy≈ 0.
Pilot
Optical Freq.
…
×
×
×
∆

!

"

# OAM
Optical Freq.
…
∆

!

"

#
OAM
Signal Signal $
200 400 600 800 1000 1200 1400 1600 1800 2000
200
400
600
800
1000
1200
1400
1600
1800
2000
Pilot $
200 400 600 800 1000 1200 1400 1600 1800 2000
200
400
600
800
1000
1200
1400
1600
1800
2000
×
Square-law
detection
$
!
Single-pixel
photodiode
Lens
Turbulence
distortion
Electrical Freq.
∆
DC
SSB
Reduction of
modal coupling
&$$
∗
$
∗

=&$
∗

Signal-pilot
beating
a)
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
∆

"

!
OAM order in
optical domain

#

$

%
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Pure OAM
Atmospheric
turbulence
Distorted OAM

&

'

"

!

#

$

%

&

'
1 2 3 4 5 6 7
Data channel # in
electrical domain
Resultant inter-data-
channel crosstalk
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Compensated signal
w/o inter-channel
crosstalk
1 2 3 4 5 6 7
Data channel # in
electrical domain
"
!
O/E OAM
beam mixing
OAM order in
optical domain
200 400 600 800 1000 1200 1400 1600 1800 2000
200
400
600
800
1000
1200
1400
1600
1800
2000
b)
Figure 3.2: Concept of utilizing optoelectronic OAM beam mixing at the receiver to achieve
OAM modal coupling resilient to the turbulence distortion in an FSO link. (a) Optoelectronic
mixing of a pilot and a data OAM beam using a free-space-coupled photodiode at the receiver
to reduce the turbulence-induced modal coupling in the electrical domain. Conj.: conjugate; DC:
direct-current; SSB: signal-signal beating; (b) Optoelectronic OAM beam mixing can couple the
turbulence-induced spreading power from undesired data channels back to the transmitted data
channel in the electrical domain.
40
3.3 AlignmentMonitorforFree-SpaceOpticalLinksinPresence
of Turbulence Effects
In almost all FSO links, and especially in links involving moving platforms such as airplanes and
unmanned aerial vehicles, a key challenge is to maintain accurate alignment between the trans-
mitter (Tx) and receiver (Rx) apertures [81]. For OAM-based systems, it is even more important
tokeeptheTx-Rxalignmentbecausemisalignmentcaninducedetrimentalinter-channelcrosstalk
among the transmitted OAM modes [82]. One typical approach for tracking alignment in OAM
links is to: (i) transmit an additional Gaussian beacon beam or utilize the data-carrying OAM
beams themselves as the beacon, and (ii) use a quadrature position-sensitive detector at Rx to
measure the beacon beam’s movement [83, 84]. It has also been investigated using an OAM
beam’s unique intensity gradient for alignment monitoring [85].
However, atmospheric turbulence can corrupt the intensity profiles of both Gaussian and
OAM beams [16, 33]. Furthermore, turbulence distortion may also induce spot wandering of the
received beam [86, 87], possibly making it more difficult to use only the beams’ intensity profiles
for alignment monitoring purposes. A laudable goal would be to monitor the Tx-Rx alignment in
FSO systems using an approach that is tolerant to a turbulent environment.
Figure 3.3 indicates the concept of FSO alignment monitor using the mixing signal of two
opposite-order OAM beams on two different wavelengths. Two co-axial OAM ℓ
1
= +1 and
ℓ
2
=− 1 beams are carried by two different CW optical frequencies f
1
and f
2
(with a frequency
difference ∆ f), denoted as C
l1
(x,y,f
1
) and C
l2
(x,y,f
2
), respectively. These two opposite-order
OAM beams are transmitted, propagate through a turbulent atmosphere, and reach the limited-
size Rx aperture. Since these two OAM beams share the same intensity profiles, they would
likely to experience similar turbulence-induced wavefront distortion e
iφ(x,y)
. Upon arriving at the
Rx aperture, the possibly misaligned aperture can cause truncation effects to the OAM beams,
denoted as a mask function A(x,y) = 0 or 1 for spatial location (x,y). Inside Rx, the entire
41
Atmospheric
turbulence Tx
Rx
Misaligned Rx
aperture (,)

! !
"(,)
"#(%,')

! "
"(,)
"#(%,')
Free-space
coupled PD
Lens
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Electrical Freq.
∆
Mixing
tone
DC
“Error” signal for
monitoring Tx-Rx
alignment
Similar distortion
(,) for two
OAM beams

)
"
*
∗

)
(,)

*
(,)
∆

! !
Opposite-order OAM beams

*
Optical Freq.
∆

)
OAM
!
= +1
Optical Freq.
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
-3
-2
-1
0
1
2
3
+
-
OAM
"
= −1
Atmospheric
turbulence Tx Rx
Well-aligned
Rx aperture

! !
"
"#(%,')

! "
"
"#(%,')
Free-space
coupled PD
Lens
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
Square-law optoelectronic beam mixing
Electrical Freq.
∆
No mixing
tone
DC
Indicator for good
Tx-Rx alignment
Similar distortion
(,) for two
OAM beams
Conjugate
field

)
"
*
∗

)
(,)

*
(,) 20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
-3
-2
-1
0
1
2
3
+
-

! "
∆

! !

*
Optical Freq.
∆

)
OAM
!
= +1
Optical Freq.
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
-3
-2
-1
0
1
2
3
+
-
OAM
"
= −1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
-3
-2
-1
0
1
2
3
+
-

! "
a)
b)
Figure 3.3: Concept of alignment monitoring in an FSO link in the presence of atmospheric
turbulence effects using the mixing tone between two opposite-order OAM beams (i.e., ℓ
1
= +1
and ℓ
2
= − 1) on two different wavelengths: a) little mixing tone in the electrical domain if Tx
and Rx are well-aligned; b) resultant mixing tone reflecting the Tx-Rx alignment condition if Tx
and Rx are misaligned.
collected OAM beams (distorted and possibly misaligned, denoted as E
1
(x,y) = C
l1
Ae
iφ
and
E
2
(x,y)=C
l2
Ae
iφ
) are focused onto a free-space-coupled PD, of which the generated photocur-
rent can be expressed in Eq. (3.2) [54]:
I ∝
Z Z
|E
1
+E
2
|
2
dxdy =
Z Z
(|E
1
|
2
+|E
2
|
2
)dxdy+2Re[E
1
E
∗ 2
]dxdy, (3.2)
where the Re[] denotes the real part of a complex function; the first and second term correspond
to the direct-current (DC) and IF beating tone in the electrical domain, respectively. In this
approach,theIFsignalwithafrequencyof∆ f isusedtoestimatetheTx-Rxalignmentcondition.
As shown in Fig. 3.3(a), if Tx and Rx are well-aligned and the Rx truncation does not occur
(i.e., A(x,y) = 1 across all the transverse beam profiles), the alignment monitor would observe
little mixing IF signal. This is because the square-law mixing is likely to largely compensate
42
the turbulence-induced wavefront distortion by automatically generating a conjugate optical field
(i.e., E
∗ 2
), expressed as in Eq. (3.3):
I
∆ f
∝
RR
E
1
E
∗ 2
dxdy =
RR
C
l1
e
iφ
C
∗ l2
e
− iφ
dxdy
=
RR
C
l1
(x,y)C
∗ l2
(x,y)dxdy
∼ =0,
(3.3)
where C
l1
(x,y) and C
l2
(x,y) are the spatial fields of undistorted OAM ℓ
1
= +1 and ℓ
2
= − 1
beams, respectively; and
RR
C
l1
(x,y)C
∗ l2
(x,y)dxdy
∼ =0 is due to the orthogonality between these
two OAM beams. However, as shown in Fig. 3.3(b), if Tx and Rx are misaligned, the limited-size
Rx aperture can induce truncation effects to the distorted OAM beams and thus the value of the
function A(x,y) is not 1 for all spatial locations of the received beam profiles. Furthermore, this
truncation-induced amplitude distortion can lead to modal power coupling between the OAM
ℓ
1
= +1 and ℓ
2
= − 1 beams [82], which is not likely to be mitigated during the square-law
mixing. Subsequently, the mixing of two OAM beams is likely to result in an IF beating tone
(i.e., I
∆ f
̸=0), as described in Eq. (3.4):
I
∆ f
∝
Z Z
E
1
E
∗ 2
dxdy =
Z Z
A
2
C
l1
e
iφ
C
∗ l2
e
− iφ
dxdy =
Z Z
A
2
(x,y)C
l1
(x,y)C
∗ l2
(x,y)dxdy̸=0.
(3.4)
By measuring the strength of this mixing tone, one can monitor the Tx-Rx alignment in an FSO
link. Furthermore, this OAM mixing tone can be potentially utilized as an “error” signal to
calibrate and align the Tx-Rx aperture pair under atmospheric turbulence distortions.
The experimental setup is shown in Fig. 3.4(a). At Tx, two lasers with different wavelengths
(frequency difference ≈ 2 GHz) around 1.55 µ m are individually amplified by two erbium-doped
fiber amplifiers (EDFA) and fed into an OAM multiplexer [88] to generate co-axial OAM l
1
=+1
and l
2
=− 1 beams (beam sizes ≈ 3.6 mm). Both OAM beams have optical power of ≈ 6 dBm.
Thesetwobeamsarecoupledtothefreespace,passthroughaturbulenceemulator,andpropagate
in free space for 1 m before arriving at the Rx aperture. The turbulence emulator is a rotatable
43
glass plate which is fabricated according to the Kolmogorov spectrum statistics with an effective
Fried parameter r
0
of 1 mm. Different turbulence realizations are emulated by rotating the glass
plate to different random orientations. The misaligned Rx aperture is emulated using a spatial
light modulator (SLM) which displays a circular aperture phase pattern with tunable horizontal
or vertical offsets. Inside Rx, a 4-f system is placed after the SLM to filter out the 1st order
diffracted light and an aspheric lens (focal length ≈ 16 mm and numerical aperture ≈ 0.79) is
used to focus the distorted light beams onto a free-space-coupled PD (detection diameter ≈ 80
µm;bandwidth<5GHz). Weuseareal-timeoscilloscopetorecordthemixingsignalandperform
fast Fourier transform to measure the strength of the mixing tone at the IF of ≈ 2 GHz of the
electrical spectrum. As shown in Fig. 3.4(b), we measure the distorted OAM beam profiles under
5 different random turbulence realizations.
Figure 3.5(a) indicates the relation (in both simulation and experiment) between the normal-
ized OAM mixing tone strength and the horizontal or vertical offsets without turbulence effects.
The orthogonality between two OAM beams ensures that little mixing tone at the IF frequency
is generated if Tx and Rx are well aligned. Moreover, if Tx and Rx are misaligned, the trun-
cation effects would lead to a generated mixing tone of which the strength is dependent on the
alignment condition. We note that the relatively larger deviation from the simulation prediction
for the horizontal measurements is likely due to the imperfect generation of the OAM beams in
this demonstration. Figures 3.5(b) and 3.5(c) show two examples of the measured electrical spec-
tra under the cases of without and with Tx-Rx misalignment, respectively, under the turbulence
realization 2. As described in Eq. (3.3), we measure little OAM mixing tone if Tx and Rx are
well-aligned (Fig. 3.5(b)). However, if Tx and Rx are misaligned, as described in Eq. (3.4), we
measure a mixing tone with a normalized magnitude of approximately 25.5 dB under a horizontal
Tx-Rx offset of -3.22 mm.
WethenmeasurethenormalizedtonestrengthasafunctionoftheTx-Rxmisalignmentunder
different turbulence realizations 1-5. As shown in Fig. 3.5(d-e), under all the measured random
44
Turbulence
emulator
Tx
EDFA
Laser

!

!
→
"
PC
SLM
FS
PD
Scope
Iris
4-f system

"

#

$
Aperture
EDFA
Laser

"

!
→
#
PC
OAM MUX
M.
M.
~1 m free
space
M.
FFT
Rx
FM
Camera

"
=+1
#
=−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
0
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Real. 1 Real. 2 Real. 3 Real. 4 Real. 5 b)
a)
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
No Turb.
Figure3.4: (a)ExperimentalsetupforalignmentmonitoringinanFSOlink. EDFA:erbium-doped
fiber amplifier; PC: polarization controller; MUX: multiplexer; FM: flip mirror; M.: mirror; SLM:
spatial light modulator; FS PD: free-space-coupled photodetector; FFT: fast Fourier transform;
Tx: transmitter; Rx: receiver; (b) Measured intensity profiles for turbulence-distorted OAM
beams. Turb.: turbulence; Real.: realization.
turbulence realizations, the mixing of two OAM beams generates little beating tone IF signal if
Tx and Rx are well aligned. Moreover, it appears that the turbulence distortion does not have
significanteffectsonthemonitoringrelationbetweenthetonestrengthandthehorizontal/vertical
misalignment. Specifically, the alignment monitor measures OAM mixing strength variations of
approximately 21.2 dB and 24 dB as the Rx aperture is misaligned of around 1.8 mm and 2.3 mm
in the horizontal and vertical direction, respectively.
45
0 1 2 3 4 5
Electrical frequency (GHz)
-20
0
20
40
60
Normalized magnitude (dB)
Mixing
tone
Turb. realization 2
Horizontal offset: -3.22 mm
Vertical offset: 0 mm
c)
a)
-6 -4 -2 0 2 4 6
0
10
20
30
40
Normalized tone strength (dB)
Horizontal or vertical offset (mm)
No turbulence effects
Experiment, horizontal
Experiment, vertical
Simulation, horizontal/vertical
0 1 2 3 4 5
Electrical frequency (GHz)
-20
0
20
40
60
Normalized magnitude (dB)
Turb. realization 2
Horizontal offset: 0 mm
Vertical offset: 0 mm
No mixing
tone
b)
-6 -4 -2 0 2 4 6
0
10
20
30
40
-6 -4 -2 0 2 4 6
0
10
20
30
40
No Turb.
Turb. R1
Turb. R2
Turb. R3
Turb. R4
Turb. R5
a)
Normalized tone strength (dB)
Horizontal displacement (mm)
No Turb.
Turb. R1
Turb. R2
Turb. R3
Turb. R4
Turb. R5
b)
Normalized tone strength (dB)
Vertical displacement (mm)
d)
e)
Figure 3.5: Experimental results for the mixing tone of OAM l
1
= +1 and l
2
=− 1 beams: (a)
Normalized tone strength at different horizontal or vertical offsets without turbulence effects; (b)
Electrical spectrum with no misalignment under turbulence realization 2; (c) Electrical spectrum
with a horizontal offset of -3.22 mm under turbulence realization 2. Experimentally measured re-
lations between the normalized tone strength and Tx-Rx misalignment under different turbulence
realizations: (d) horizontal displacement and (e) vertical displacement. Rx aperture size 6 mm.
3.4 Turbulence-ResilientMode-Division-Multiplexingwith
Automatic Crosstalk Compensation
As shown in Fig. 3.6, two independent data channels at the same optical frequency are carried
by two OAM beams with OAM orders ℓ
1
and ℓ
2
varying from -2 to +2. Two additional CW pilot
tones located at the frequency difference ∆ f
1
and ∆ f
2
away from data channel’s frequency are
also carried by the corresponding OAM beams ℓ
1
and ℓ
2
, respectively. All four OAM beams are
spatially combined and co-axially propagate through atmospheric turbulence effects. At the re-
ceiver,thesingle-pixelfree-space-coupledPDautomaticallycancelstheturbulence-inducedmodal
coupling by square-law detection and heterodyne-converting the two carried data channels to the
IF signal at frequencies ∆ f
1
and ∆ f
2
in the electrical domain. We then apply off-line digital
filters to separate the data channels from the signal-signal beating (SSB) and the direct-current
46
(DC) component. The I-Q information of the data channels is retrieved by using off-line DSP
algorithms.
∆
!
Optical Freq.
Data
Ch. 1
Pilot
tone
#1
OAM
!
∆
"
Optical Freq.
Data
Ch. 2
Pilot
tone
#2
Electrical Freq.
∆
!
DC
SSB
∆
"
Data
Ch. 1
Data
Ch. 2
Optoelectronic
beam mixing
$
"
Single-pixel
photodiode
Lens
Turbulence
distortion
Inter-channel
crosstalk
cancellation
#
∗
= 1
Turbulence-resilient
OAM multiplexing
OAM
#
Figure 3.6: Two-OAM-beam-multiplexed FSO data transmission with inter-channel crosstalk
resilient to turbulence distortion by transmitting extra OAM pilot tones and OAM beam mixing
at the receiver. Conj.: conjugate; DC: direct-current; SSB: signal-signal beating.
Figure 3.7 (a) illustrates the experimental setup for the two-OAM-beam multiplexed data
transmission through emulated turbulence effects. At the transmitter, a laser with a wavelength
of λ 0
is modulated with either a 1-Gbaud Nyquist-shaped QPSK or 16-QAM signal by an in-
phase-quadrature (I-Q) optical modulator. The modulated light is subsequently amplified and
equally split to two copies of which one copy is delayed by a 15-m single-mode fiber (SMF) to
decorrelatethedatasequence. ThesetwocopiesarethenindividuallycombinedwithtwoCWpilot
tones at different wavelengths λ 1
and λ 2
, with the frequency differences ∆ f
1
and Deltaf
2
away
from the wavelength λ 0
, respectively. After amplified by erbium-doped fiber amplifiers (EDFA),
these two sets of signal and pilot tones at wavelengths λ 1
and λ 2
are fed to an OAM multiplexer
to generate co-axial OAM beams ℓ
1
and ℓ
2
, respectively. The inset of Fig. ?? (a) indicates
the generated OAM beam profiles with OAM orders ℓ = − 2,− 1,+1,+2 without any emulated
turbulenceeffects. Inthisdemonstration,theOAMordersofthetwomultiplexedOAMbeams, ℓ
1
and ℓ
2
, are assigned from these four OAM beams together with the fundamental Gaussian beam
(i.e., ℓ = 0). After distorted by the turbulence emulator (a thin glass plate), the data-carrying
OAM beams propagate a free-space distance of 1 m and reach the receiver. At the receiver,
the distorted OAM beams are shaped by a spatial light modulator (SLM) and subsequently are
equally split into two copies of which one is sent to optoelectronic (O/E) beam mixing receiver
47
O/E OAM beam mixing Rx to reduce XT
Emulated
turbulence
FS PD
PC
MZI
EDFA
Delay
Laser

!
AWG
I Q
BPF
Tx
Iris
Conventional OAM Rx
PC
EDFA
Laser

"
EDFA
PC
Laser

#
BPF

!
→
"

!
→
#
OAM MUX
M
M
FM
Camera
BS
SLM
4-f system
Col.

#

"
$
PC
PC
EDFA
M
Fiber
PD
Scope
=+2
=−2
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
1000
2000
3000
4000
5000 =+1
=−1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
No turbulence distortion

!
1550.1 1550.2 1550.3 1550.4 1550.5
-60
-50
-40
-30
-20
-10
0
Power (dBm)
Wavelength (nm)
Tx OAM l
1
Tx OAM l
2
Resolution: 0.08 nm

#

"
Pilot
tones
Data Ch. 1
and Ch. 2
0 1 2 3 4 5
0
10
20
30
40
50
Magnitude (dB)
Frequency (GHz)
OAM
"
SSB Ch. 1 Ch. 2
OAM
#
DC
b) c)
Orders
#
,
"
∈
{−2,−1,0,+1,+2}
a)
PC
Laser

$
DSP
Scope
DSP
Figure 3.7: Experimental for two-OAM-beam-multiplexed data transmission through emulated
turbulence effects. (a) Experimental setup. AWG: arbitrary waveform generator; MZI: Mech-
zen; PC: polarization controller; EDFA: erbium-doped fiber amplifier; BPF: band-pass filter;
MUX: multiplexer; M.: mirror; FM: flip mirror; BS: beam spliiter; SLM: spatial light modulator;
Col.: collimator; FS PD: free-space photodiode; Tx: transmitter; Rx: receiver; Scope: real-time
oscilloscope;DSP:digitalsignalprocessing. (b)MeasuredopticalspectracarriedbythetwoOAM
beams at the transmitter. Ch.: channel. (c) Measured electrical spectrum for O/E OAM beam
mixing at the receiver. DC: direct current; SSB: signal-signal beating.
48
by the single-pixel PD and the other copy is sent to a normal OAM receiver for comparison. (i)
With respect to the O/E beam mixing receiver, the SLM is set to be acting as a grating mirror,
and the distorted OAM beams are filtered by a 4-f system ( f
1
= f
2
= 88 mm) and subsequently
focused into a free-space InGaAs PD (Thorlabs DET08C) using an aspheric lens (f
3
= 16 mm,
NA=0.79). (ii)AsfortheconventionalOAMreceiver, theSLMissettobeOAMdemultiplexing
phase pattern which converts one OAM mode of interest to the fundamental Gaussian beam and
couples it to a fiber collimator for signal detection. Another laser at wavelength λ 3
are combined
with this signal and subsequently sent to a fiber-coupled PD. To be fair comparison, the detected
electrical signals from the O/E beam mixing and the conventional OAM receiver are processed by
the same heterodyne DSP algorithm and procedures. Figure 3.7 (b) shows the optical spectra at
the transmitter carried by the multiplexed OAM beams including 1-Gbaud Nyquist-shaped data
channels and the extra CW pilot tones. The resultant electrical spectrum using the O/E OAM
beam mixing at the receiver is illustrated in Fig. 3.7 (c). We observe the DC component, SSB
term, and the transmitted two data channels at two different intermediate IF ∆ f
1
and Deltaf
2
in the electrical domain.
As shown in Fig. 3.8 (a), we transmit the OAM ℓ = − 2,− 1,0,+1,+2 beams through emu-
lated weaker and stronger turbulence with different Fried parameters r
0
of 1.0 mm and 0.4 mm,
respectively. The diameters of OAM beams range from approximately 2.2 mm to 4.6 mm. After
distortion by the glass phase plate, the OAM beams propagate around 1 m in free space and
subsequently their entire spatial profiles are focused onto a free-space PD at the receiver. The
measured inter-channel crosstalk matrix under one turbulence realization for turbulence-resilient
system and conventional OAM system are shown in Fig. 3.8 (b) and Fig. 3.8 (c), respectively.
We note that the power values (in unit of dBm) are measured in the electrical and optical domain
for Fig. 3.8 (b) and Fig. 3.8 (c), respectively. The inter-channel crosstalk of the resilient OAM
systemisnotsignificantlyaffectedbytheturbulencedistortionsandismeasuredlowerthan25dB
under both the weaker and stronger turbulence distortions. However, severe turbulence-induced
49
crosstalk between the OAM beams are measured for the conventional OAM system under turbu-
lence effects. We further measure the mode-dependent loss (MDL) of transmitted OAM beams
by using the optoelectronic beam mixing. As shown in Fig. 3.8 (d1), the MDL under weaker and
stronger turbulence distortions are respectively measured to be <2.5 dB and <5.5 dB. However,
as shown in Fig. 3.8 (d2), the MDL of the conventional OAM system can be >4.2 dB and >10.1
dB for the weaker and stronger turbulence effects, respectively. This is likely due to that the
optoelectronic beam mixing can couple the spread power from different data channels back to the
transmitted data channel in the electrical domain.
We demonstrate a 4-Gbit/s OAM-multiplexed FSO link under different emulated turbulence
strengths, with each OAM beam carrying a 2-Gbit/s Nyquist-shaped QPSK data stream (roll-
off factor of 0.1). Figure 3.9 illustrates the measured distorted spatial profiles of different sets
of the two multiplexed OAM beams ℓ
1
and ℓ
2
under different turbulence realizations and the
corresponding calculated error-vector magnitudes (EVM) using the optoelectronic beam mixing
approach. The EVMs of data channels using a conventional OAM receiver and heterodyne de-
tection method are also shown in Fig. 3.9 for comparison. As shown in Fig. 3.9 (a1), both
the OAM beam mixing and the conventional OAM detection methods can achieve near-error-free
data transmission without turbulence effects (i.e., D/r
0
≈ 0), and the EVMs are measured to be
approximately 13.4% and 14.9% using the optoelectronic OAM beam mixing method and 15.8%
and 13.8% using the conventional OAM receiving method for OAM ℓ
1
andℓ
2
beams, respectively.
The transmitted free-space optical powers are around 1.6 dBm and 3.7 dBm for OAM ℓ
1
and ℓ
2
beams, respectively. Under the spatial distortion by the emulated turbulence effects, the OAM
beams can barely preserve the standard donut-shaped intensity profiles and a large number of
bit errors occur using the conventional OAM receiver. Figures 3.9 (a2-a5) show four examples of
turbulence-induced spatial distortion and inter-channel crosstalk for different OAM beams with
the OAM order ranging through {ℓ
1
,ℓ
2
} = {+1,− 1}, {+2,− 2},{+1,− 2}. When the OAM
beams are distorted by the weaker turbulence, having a Fried parameter of 1 mm (i.e., D/r
0
≈ 50
-65
-55
-45
-35
-25
-15
-65
-55
-45
-35
-25
-15
-65
-55
-45
-35
-25
-15
-2 -1 0 +1 +2
-2
-1
0
+1
+2
Tx
Rx
w/o Turb.
-2 -1 0 +1 +2
-2
-1
0
+1
+2
Weaker Turb.
-2 -1 0 +1 +2
-2
-1
0
+1
+2
Stronger Turb.
-45
-35
-25
-15
-5
-45
-35
-25
-15
-5
-45
-35
-25
-15
-5
-2 -1 0 +1 +2
-2
-1
0
+1
+2
-2 -1 0 +1 +2
-2
-1
0
+1
+2
-2 -1 0 +1 +2
-2
-1
0
+1
+2
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
= −2 = −1 = 0 = +1 = +2
Weaker Turb.

!
= 1 mm
Stronger Turb.

!
= 0.4 mm
w/o Turb. Weaker Turb. Stronger Turb.
a)
b)
c)
Tx
Rx
-2 -1 0 1 2
0
5
10
15
20
-2 -1 0 1 2
0
5
10
15
20
Electrical power loss (dB)
OAM order
Weaker Turb. R1
Weaker Turb. R2
Weaker Turb. R3
Stronger Turb. R1
Stronger Turb. R2
Stronger Turb. R3
Optical power loss (dB)
OAM order d1) d2)
Resilient OAM system using optoelectronic beam mixing
Conventional OAM system
Figure 3.8: Experimental results of measured inter-channel crosstalk under weaker and stronger
turbulence distortions. (a) Measured beam profiles for OAM beams ℓ = − 2,− 1,0,+1,+2; (b)
Measured inter-channel crosstalk using optoelectronic OAM beam mixing; (c) Measured inter-
channel crosstalk using conventional OAM receiver; (d) Measured signal power loss for OAM
beams ℓ = − 2,− 1,0,+1,+2 by using the beam mixing approach in (d1) and the conventional
approach in (d2).
51
3.8 and 4.6), as shown in Fig. 3.9 (a2-a3), both the two multiplexed data channels suffer from the
turbulence-induced crosstalk and the EVMs of data channels increase from <16% to >36% using
the conventional OAM receiver. Under the stronger turbulence of D/r
0
≈ 9.5 and 11.5, as shown
in Fig. 3.9 (a4-a5), the DSP algorithms cannot readily recover the I-Q information of the data
channels due to the higher inter-channel crosstalk (e.g., -4.2 dB in Fig. 3.9 (a4) compared to -7.4
dB in Fig. 3.9 (a2) for OAM ℓ
1
= +1) induced by the stronger turbulence distortion. However,
when the distorted OAM beams are optoelectronically mixed by the single-pixel InGaAs PD, the
two multiplexed data channels are not significantly influenced by the turbulence effects and the
measured EVMs exhibit near-error-free performance similar to the case of no turbulence effects,
albeit with only approximately 2% increased EVM.
Figure 3.9 (b-c) summarizes the EVM comparison for different multiplexed OAM sets under
15 different realizations including the weaker ( r
0
=1 mm in Fig. 3.9 (b)) and the stronger
(r
0
=0.4 mm in Fig. 3.9 (c)) turbulence conditions . It is shown that the OAM beam mixing
method can reduce the EVMs of the two multiplexed data channels by up-to 41.5% and by 6.9%
- 42.1% for the weaker and the stronger turbulence distortions, respectively. Furthermore, the
optoelectronic beam mixing exhibits certain resilience of OAM inter-channel crosstalk to these 15
random turbulence distortions, measured as the data channels’ EVMs variation from 12.9% to
17.4% and from 12.4% to 19.8% for the weaker and stronger turbulence effects, respectively. We
note that the variation of the achieved EVMs is likely due to the difference in coupling efficiencies
for distorted OAM beams upon propagation into the free-space InGaAs PD.
We further demonstrate an 8-Gbit/s two-OAM-multiplexed FSO communications, with two
OAM beams ℓ
1
= +1 and ℓ
2
= − 1 each carrying an independent 1-Gbaud 16-QAM signal.
The transmitted free-space optical powers are approximately 6.4 dBm and 8.6 dBm for OAM
beams ℓ
1
= +1 and ℓ
2
= − 1, respectively. As shown in Figs. 3.10(a-b), compared to the
conventional OAM receiver, the optoelectronic OAM beam mixing method can reduce the 16-
QAM data channels’ EVMs by 13.08% and 13.54% under the weaker turbulence distortion (i.e.,
52
10
20
30
40
50
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
10
20
30
40
50
Stronger turbulence distortion r
0
~ 0.4
w/o Turb., resilent link w/o Turb., conventional link
w/ Turb., resilient link w/ Turb., conventional link
EVM (%)
EVM (%)
Weaker turbulence distortion r
0
~ 1.0
w/o Turb., resilent link w/o Turb., conventional link
w/ Turb., resilient link w/ Turb., conventional link
EVM (%)
Turbulence realization
EVM (%)
Turbulence realization
15.2% EVM
/
!
~3.6 /
!
~3.8

"
=+1
#
=−1
O/E OAM beam mixing
16.0% EVM
Conventional OAM Rx
39.9% EVM 36.6% EVM
14.4% EVM
/
!
~9.0 /
!
~9.5

"
=+1
#
=−1
O/E OAM beam mixing
16.5% EVM
Conventional OAM Rx
49.8% EVM 52.9% EVM
15.8% EVM
/
!
~4.6 /
!
~4.6

"
=+2
#
=−2
O/E OAM beam mixing
15.2% EVM
Conventional OAM Rx
39.5% EVM 52.2% EVM
14.9% EVM
/
!
~11.5 /
!
~9.5

"
=+1
#
=−2
O/E OAM beam mixing
15.4% EVM
Conventional OAM Rx
53.3% EVM 48.0% EVM
13.4% EVM
/
!
~0 /
!
~0

"
=+1
#
=−1
O/E OAM beam mixing
14.9% EVM
Conventional OAM Rx
15.8% EVM 13.8% EVM
b) c)
a)

"
=+1

#
=−1

"
=+1

#
=−1

"
=+2

#
=−2

"
=+2

#
=−2

"
=+1

#
=−2

"
=+1

#
=−2

"
=0

#
=−2

"
=0

#
=−2

"
=0

#
=−1

"
=0

#
=−1
a1) a2) a3) a4) a5)
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data CH.1 Data CH.2
Data CH.1 Data CH.2
Figure 3.9: Experimental results of 4-Gbit/s OAM-multiplexed data transmission under different
strengths of turbulence effects. (a) Comparison of the received 2-Gbaud QPSK constellation di-
agram using the O/E OAM beam mixing and conventional OAM receiver. Different sets of the
multiplexed two OAM beams under different turbulence strengths are evaluated. The transmit-
ted free-space optical powers approximately 1.6 dBm and 3.7 dBm for OAM beams ℓ
1
and ℓ
2
,
respectively. (b-c) Measured EVM values for resilient and conventional OAM links under two
different turbulence strength with the Fried parameters r
0
of 1.0 mm and 0.4 mm in (b) and (c),
respectively.
53
/
!
~3.8

"
=+1
O/E OAM beam mixing
Conventional OAM Rx
21.40% EVM 21.90% EVM
a)
8.32% EVM 8.36% EVM

#
=−1
/
!
~3.6 /
!
~9.5

"
=+1
O/E OAM beam mixing
Conventional OAM Rx
20.55% EVM 19.80% EVM
b)
8.93% EVM 7.68% EVM

#
=−1
/
!
~9.0
c)
d)
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0
1
1 2 3 4 5 6 7 8
10
-4
10
-3
10
-2
1 2 3 4 5 6 7 8
10
-4
10
-3
10
-2
Data CH. 2
OAM l
2
= -1
4-Gbit/s 16-QAM
D / r
0
~ 0 , B 2 B
D / r
0
~ 0 , l
2
= - 1
D / r
0
~ 3 .3 , l
2
= - 1
D / r
0
~ 8 .2 , l
2
= - 1
7% FEC limit
D / r
0
~ 0 , B 2 B
D / r
0
~ 0 , l
1
= + 1
D / r
0
~ 3 .3 , l
1
= + 1
D / r
0
~ 8 .2 , l
1
= + 1
BER
Transmitted power (dBm)
Data CH.1
OAM l
1
= +1
4-Gbit/s 16-QAM
7% FEC
limit
BER
Transmitted power (dBm)
Figure3.10: Experimentalresultsof8-Gbit/sOAM-multiplexeddatatransmissionunderdifferent
strengths of turbulence effects. (a-b) Comparison of the received 2-Gbaud 16-QAM constellation
diagramusingtheO/EOAMbeammixingandconventionalOAMreceiver. TheFriedparameters
r
0
of the emulated turbulence effects are 1.0 mm and 0.4 mm in (a) and (b), respectively. The
transmitted free-space optical powers are approximately 6.4 dBm and 8.6 dBm for OAM beams
ℓ
1
= +1 and ℓ
2
=− 1, respectively. (c-d) Measured BER curves of the data channels carried by
two OAM beams ℓ
1
=+1 and ℓ
2
=− 1 in (c) and (d), respectively.
D/r
0
≈ 3.8), and by 11.6% and 12.1% under the stronger turbulence distortion (i.e., D/r
0
≈ 9.5)
for OAM beams ℓ
1
= +1 and ℓ
2
= − 1, respectively. Figures 3.10(c-d) illustrate the measured
BER performance of these two multiplexed data channels. The back-to-back (B2B) performance
which refers to a single 4-Gbit/s 16-QAM data channel carried by a single fundamental Gaussian
beam without turbulence distortion is also shown for comparison. The BER performance of
both data channels can be readily achieved below the 7% FEC limit of 3.8e-3 under turbulence
distortions. ComparedtotheB2Bperformance,themeasuredBERvaluesexhibitpowerpenalties
of approximately 1.5 dB and 3.6 dB under the weaker distortion; 1.2 dB and 2.9 dB under the
stronger distortion effects, for OAM beams ℓ
1
= +1 and ℓ
2
= − 1, respectively. The inferior
performance of data channel 2 carried by the OAM ℓ
2
=− 1 beam is because the free-space PD
has lower frequency response around 3 GHz (i.e., the electrical signal band for data channel 2).
54
We note that the baud rate of multiplexed data channels is limited by the bandwidth of the
free-space PD used in this demonstration.
55
Chapter 4
Structured Light’s Interaction with Scattering Media and
Effects in Communications
4.1 Background and Motivation
Asthedemandforhigh-bit-ratedatatransmissionincreasesinmanyapplications,coherentoptical
wireless communication (OWC) has received considerable interest. OWC links have the potential
to provide high-capacity data transmission through media of interest, such as the atmosphere
and underwater [89, 15, 37]. To further increase the system’s capacity, one potential approach
could be transmitting multiple optical beams carrying different data streams simultaneously, i.e.,
using space-division-multiplexing (SDM) [3]. Specifically, one subset of SDM is mode division
multiplexing (MDM), where each beam occupies a unique mode from an orthogonal modal basis
set. The spatial orthogonality between different modes in MDM enables efficient multiplexing,
coaxialpropagation,anddemultiplexingwithlittleinherentcrosstalk[74,75]. Onepossiblechoice
for MDM is using multiple beams carrying different amounts of OAM [19, 18, 90].
In addition, there has been a fair amount of interest in utilizing structured optical beams
for various applications, including manipulation, imaging, and sensing [91, 92, 93, 94]. Light
beam interacting with an unknown target, after either passing through, truncation or scattering,
may carry some information regarding the object’s properties and locations. Such a resultant
56
beam structure is typically a spatial representation of the amplitude and phase contribution
from different modes of a larger modal basis set [95]. By investigating different complex spatial
component of a resultant optical field, the object information may be resolved.
In this chapter, interaction between structured light carrying OAM and scattering media, as
wellastheirapplicationinopticalcommunicationsandsensingwillbediscussed. Specifically,this
chapter will include: (i) Laguerre-Gaussian spatial modal spectrum by single-particle scattering
[96]; (ii)ballisticanddiffusivescatteringeffectsonOAM-multiplexedOWClinks[97,98]; and(iii)
simultaneous turbulence mitigation and channel demultiplexing by adaptive wavefront shaping
and diffusing [99, 100].
4.2 Laguerre Gaussian Modal Spectrum by Single-Particle
Scattering
Interaction of a single dielectric particle with LG beams carrying OAM, i.e. Mie scattering of LG
beams, has drawn a lot of attention due to the unique spatial phase and amplitude structure of
LG modes [17, 18, 95, 101, 102]. Specifically, the role of angular momentum of the light beam in
the Mie scattering has been investigated [103, 104]. Moreover, the complex structure of scattered
LG beams can potentially be used to help characterize some geometrical properties of a spherical
particle. Previous report has shown that OAM modes can be used for spiral imaging which uses
theamplitudeofthescatteredmodalspectrumforp=0andmultiplelvalues[105]. Thusitwould
be interesting to investigate the 2-dimensional (varying both azimuthal and radial orders) LG
mode spectrum of the scattered light and explore the potential application of such a complex
spectrum (both amplitude and phase) on determining the location of a single spherical particle.
TheconceptofLGbeam’sscatteringbyasinglesilicaparticleisshowninFig. ??. Withoutloss
ofgenerality,wechoosethebeamwaistw
0
=3µm ,wavelengthλ =1.55µm andsimulatebyfinite-
difference-time-domain (FDTD) calculation the propagation of a linearly-polarized LG
ℓ=+3,p=0
57
Source:

!"#
=
$,& Particle at z=0 plane

Single

$,&
mode
Multiple
$,&

'
×

(
×
…

)
×
+
+
Figure4.1: ConceptualdiagramofLGbeams’scatteringbyasilicasphericalparticle: AsingleLG
beam propagates along the z-axis and the particle is located on the beam waist plane (z = 0) of
theincidentbeam. Afterinteractionwiththeparticle, thebeamcanbedecomposedintomultiple
different LG modes with complex coefficients ( C
1
,C
2
, ...,C
N
)..
beam (x-polarized OAM beam with order ℓ = +3, i.e., E
x
= E
ℓ=+3,p=0
and E
y
= E
z
= 0) from
the emission plane (z = − 10 µm ) to the detection plane (z = +10 µm , 30× 30 µm
2
size). At
the detection plane, the complex field of the scattered light beam E
rec
is recorded and further
analyzed by LG mode complex decomposition. A spherical silica particle (n
silica
= 1.444) is
located on the beam waist plane (z =0) in the free space (n
background
=1.0) and its location can
be described by the off-axis distance ρ and azimuthal angle φ. All the calculation and analysis
are based on the x-polarized component of the scattered electric field.
The incident LG
ℓ=+3,p=0
beam is scattered by a silica sphere with different radius R located
on the z-axis (ρ = 0). As shown in Fig. 4.2, different particle sizes may influence the scattered
complexfield E
x
atthedetectionplane. Thenormalizedintensity|E
x
|
2
andphase
̸ E
x
areshown
in Fig. 4.2 (a1-d1) and Fig. 4.2 (a2-d2), respectively. The intensity pattern |E
x
|
2
on the x = 0
plane from the emission to the detection plane is also shown in Fig. 4.2 (a3-d3) (z-axis along
the vertical direction). For comparison, the detected fields without a particle present are shown
in Fig. 4.2 (a). The radiuses of the spherical particle in Fig. 4.2 (b-d) are R = 2, 5, 8 µm ,
respectively. The inset circles in Fig. 4.2 (b3-d3) demonstrate the geometrical location and size
58
c2
c3
a1
a2
a3

b1
b3
b2
c1
d1
d2
d3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1
-3
-2
-1
0
1
2
3
−
+

!
"
at
z=+10 µm

!
"
at
x = 0

∠
!
at
z=+10 µm
R = 0
R = 2 µm R = 5 µm R = 8 µm
Figure 4.2: Effects of different particle size R on the complex field E
x
at the detection plane:
(a) No particle present; (b) R = 2 µm ; (c) R = 5 µm ; (d) R = 8 µm . Each row represents:
(a1-d1) Normalized intensity profile |E
x
|
2
; (a2-d2) Spatial phase profile
̸ E
x
; (a3-d3) Normalized
intensity profile |E
x
|
2
at the x = 0 plane, from z =− 10 µm to z = +10 µm . The incident beam
is LG
ℓ=+3,p=0
mode.
of particles. If there is no particle, the twisting wavefront of the emitted LG beam (ℓ = +3) is
intactandtheannularintensityringstructureisundistorted. However,withthesphericalparticle
located on the z-axis, the detected intensity profiles may vary with the particle size, as shown
in Fig. 4.2 (b1-d1) while the twisting phase structure is mostly conserved, as shown in Fig. 4.2
(b2-d2). Specifically, for a particle with a relatively smaller radius R = 2 µm (Fig. 4.2 (b)), the
scatteredlighttendstodivergeinthefreespacewhilethescatteredlightislargelyconfinedinthe
near field at the back of the silica sphere with a larger radius R=5 µm (Fig. 4.2 (c)), forming a
‘photonic nanojet’ [102]. As for a particle with a relatively large size, e.g., R = 8 µm in Fig. 4.2
(d), the silica particle tends to act like a convergent lens to ‘refocus’ the light beam.
To quantitatively analyze the scattered light, the detected field E
rec
can be expressed as
superposition of multiple LG
ℓ,p
modes at the detection plane: E
rec
=
P
ℓ
P
p
C
ℓ,p
E
ℓ,p
where the
59
complex coupling coefficients can be calculated as the overlap integral between the scattered field
and the LG
ℓ,p
mode field:
C
l,p
=
R
E
l,p
E
∗ rec
dS
q
R
|E
l,p
|
2
dS
R
|E
rec
|
2
dS
. (4.1)
The intensity |C
l,p
|
2
and phase
̸ C
l,p
of complex coefficients denote the ratio of power coupling
and the phase delay to the LG
ℓ,p
mode, respectively. Since the LG mode forms an orthogonal
modal basis in the free space, the complex coefficient C
l,p
for an arbitrary light field at a prop-
agation distance should follow that
P
l
P
p
|C
l,p
|
2
≈ 1 [9]. By determining the complex coupling
coefficientsofcertainLG
ℓ,p
modes,wemayextractsomeusefulinformationregardingthegeomet-
rical characteristics of the spherical particle. Specifically, the off-axis distance ρ and azimuthal
angle φ are related to power coupling|C
ℓ=+3,p
|
2
and phase distribution
̸ C
ℓ,p=0
, respectively.
Source:
!"#
$"%&
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
1

!
"
= 0.77
a2

!
"
a1
= 0.26
a)
b)
Figure4.3: Effectsoftheparticle’soff-axisdistance ρλ ontheLGmodepowercoupling|C
ℓ=+3,p
|
2
:
(a)Detectedintensityprofileswhentheparticleislocatedondifferentoff-axisdistance ρ/λ =0.26,
0.77; (b) The ratio of power coupling to LG
ℓ,p
modes with the same OAM order ℓ = +3 and
different radial indices p when the particle is located on different off-axis distance. The incident
beam is LG
ℓ=+3,p=0
.
TheLGintensityspectrumofthescatteredlightmaychangesignificantlyastheparticlemoves
off the z-axis. Figure 4.3 (a) shows the detected intensity profiles if the particle is located on the
off-axis positions ρ = 0.26λ and 0.77λ . It is shown that the azimuthal symmetry of the incident
LG
ℓ=+3,p=0
beam is distorted by interaction between light field and silica particle (radius R = 5
60
µm ). Figure 4.3 (b) shows the ratio of power coupling to other LG
ℓ,p
modes with the same OAM
order ℓ=+3 and different radial orders if the particle is located at different off-axis distance ρ/λ
(along x-axis, plotted in dB, i.e., 20log
10
|C
ℓ,p
|). The power coupled to LG
ℓ=+3,p=1,2,3,4,5
mode
varydifferentlyastheparticlemovesoffthez-axis. Forinstance,thepowercoupledtoLG
ℓ=+3,p=1
mode decreases monotonously while the power coupled to LG
ℓ=+3,p=4
and LG
ℓ=+3,p=5
exhibit
oscillation-like variation over the range 0 < ρ/λ < 2.5. Moreover, for certain radial order, the
coupledlightpowerdropsrapidlyiftheparticlemovesoff-axiallywithinasubwavelengthdistance.
Specifically, the coupling power drop |C
ℓ=+3,p=5
|
2
induces a power dip of 25 dB over a detection
range of 0.8λ . By detecting the power drop on a mode of interest, certain LG
ℓ,p
mode with non-
zero radial order may be utilized as a ‘signature’ to potentially achieve sub-wavelength off-axis
localization of single particle.
AsshowninFig. 4.4(a-b),thedeterminationoftheparticle’soff-axisdistancemaybeaffected
bytheparticlesizeRandthechoiceoftheincidentLG
ℓ,p
mode. Wedefinethedetectionrangeas
the distance from the maximum signal power to the first minimum of the signal power; the power
dip as the amount of the decreased power if the particle is located off-axially at the detection
range. These two parameters may give an estimation of the localization sensitivity. As shown in
Fig. 4.4 (a), different particle sizes may influence both the power dip and detection range of the
off-axis detection when the LG
ℓ=+3,p=5
mode component is used as the ‘signature’. We note that
other LG modes with different radial orders (i.e., ℓ=+3,p̸=5) can also be utilized to sense the
dielectric particle’s off-axis displacement. Figure 4.4 (b) shows that the choice of incident beam
may also affect the determination of off-axis displacement. Both the detection range and power
dip may vary when different LG mode ( ℓ = 0,+1,+2,+3, and p = 0) is chosen as the incident
beam. For the silica particle with radius R = 5 µm , the use of the normal Gaussian beam
(LG
ℓ=0,p=0
) as the source may suffer from lower signal power (approximately 4.2-dB lower when
theparticleislocatedonthez-axis)andapproximately17-dBlesspowerdip,comparedtotheuse
of LG
ℓ=+3,p=0
as the source . The use of other LG modes as the source, e.g., ℓ = +1,+2,p = 0,
61
Source:
!"#
$"%&
b) a)

=/8
z=0 plane
Source:
!"#
$"%&
c) d)
Figure 4.4: (a-b) Effects of the particle size and the choice of incident mode on the determination
of the off-axis distance: (a) Ratio of power coupled to LG
ℓ=+3,p=5
mode (i.e.,|C
ℓ=+3,p=5
|
2
) with
differentparticlesize R=2,4,6,8µm ;(b)RatioofpowercoupledtoLG
ℓ,p=5
mode(i.e.,|C
ℓ,p=5
|
2
)
whenusingLGmodes(ℓ=0,+1,+2,+3,p=0)withdifferentOAMorders ℓasthesources. (c-d)
Phase of different OAM component (i.e.,
̸ C
ℓ,p=0
) if the particle is located on different azimuthal
locations: (c) Unwrapped phase of OAM modes ℓ = 0,+1,...,+6; (d) Effects of the particle’s
off-axis distance ( ρ/λ = 1.0,1.5,2.0) on the OAM phase spectra. Radius of the particle R = 5
µm , incident beam LG
ℓ=+3,p=0
.
can achieve similar detection ranges of approximately 0.7λ and power dip of approximately 15.6
dB and 17.4 dB, respectively. This may be because the twisting phase structure of a LG mode is
sensitive to the scattering of the dielectric particle located on the different off-axis positions.
Moreover, the azimuthal location (described by azimuthal angle φ) of the silica sphere can
be inferred from the phase spectra of different OAM components (i.e.
̸ C
ℓ,p=0
). As shown in
Fig. 4.4 (c), the unwrapped phase of different OAM modes ( ℓ̸=+3) vary linearly as the particle
rotatesintheazimuthaldirection(from φ=0toφ=2π ). Thesilicaparticleislocatedatoff-axis
distance ρ = λ/ 2 with size R = 5 µm , probed by a LG
ℓ=+3,p=0
beam. We further observe that
62
the unwrapped phase of each OAM component increases (or decreases) at a rate proportional to
|ℓ− 3| as the particle rotates azimuthally. Figure 4.4 (d) indicates the calculated OAM phase
spectra (ℓ = 0,+1,...,+6) with the particle located at an azimuthal location φ = π/ 8: phase of
different OAM mode component depends linearly on the OAM order ℓ. Moreover, the phases of
different OAM components are almost unaffected if the particle is located at the same azimuthal
positionbutdifferentoff-axisdistance ρ =1.0λ, 1.5λ, 2.0λ . Thisresultshowsthatincertainrange
ofoff-axisdisplacement,theslopeofOAMphasespectraisinsensitivetodifferentoff-axisdistance
of the particle.

z=0 plane
a) b)

=
!"
#
Figure 4.5: Relation between the slope of OAM phase spectrum and the particle’s azimuthal
location: (a) Calculated slope of the OAM phase spectrum (ℓ = 0,+1,...,+6) as the particle
located on different azimuthal positions; (b) Effects of the particle’s size R on the OAM phase
spectrum.
Figure 4.5 (a) indicates the linear dependence of the calculated slope of OAM phase spectra
(ℓ=0,+1,...,+6)ontheparticle’sazimuthalpositionφ(theparticle’soff-axisdistance ρ =λ/ 2).
A good linear relation |slope|∝ φ is observed. This may be due to the fact that the wavefront
of different OAM modes is twisting at a speed proportional to the corresponding OAM order |ℓ|,
and thus the scattered light may induce linear phase delays to different OAM modes. We also
investigate the dependence of OAM phase spectra on the size of the particle size which is placed
on a fixed position ( ρ = 1.0λ , φ = 3π/ 8). As shown in Fig. 4.5 (b), the phases of OAM modes
at the detection plane may differ after scattering by different-size particles ( R =3,5,6 µm ), and
63
subsequently the linear relation between the azimuthal position and the slope of OAM phase
spectra may need to be modified for different particle sizes.
4.3 Ballistic and Diffusive Scattering of OAM Modes in
Free-Space Optical Communications
For OWC links employed in different media, several environmental effects can degrade the link
performance, such as absorption, scattering and turbulence [106, 107, 108]. Absorption of the
medium reduces the signal power arrived at the receiver (Rx) [106]. Turbulence effects may
dynamically distort the wavefront of the optical beams and thus cause power scintillations at the
Rx [107, 108]. Moreover, in an OAM-multiplexed OWC link, turbulence distortion may induce
signal power leakage to neighboring OAM modes, i.e., crosstalk [45].
In addition, scattering is a notable effect, which occurs both in underwater and atmosphere
media [38, 98]. Since proper separation of multiplexed OAM beams critically depends on their
individual unique phase and amplitude structure, it is valuable to understand the effects of a
scattering medium on OAM-multiplexed transmission in terms of mode purity, crosstalk in high
speed OWC.
It has been shown that light propagation in a scattering medium can be broadly categorized
into two regions [109]: (i) In the ballistic, single-scattering region, mostly unscattered (ballistic)
photons reach the receiver in essentially straight trajectories; (ii) In the diffusive region, the
receiver collects photons that may have been scattered multiple times and travelled different
paths. A concept diagram of scattering effects on a pure OAM beam is shown in Fig. 4.6.
Whentheballisticscatteringdominates, apropagatingOAMbeammaymaintainitspuritywhile
suffering scattering-induced power loss. However, as the scattering becomes diffusive, the more
significant power loss would be accompanied by a distorted wavefront, giving rise to loss of mode
purity and crosstalk among co-propagating OAM beams.
64
Pure
OAM mode
OAM mode &
diffusive light
Power

!

"

#

$

%
OAM
order
No
crosstalk
Ballistic
photon
Diffusive
photon
Reflection
Loss
Crosstalk
Power

!

"

#

$

%
OAM
order
Figure 4.6: Concept of multiple scattering effects on an OAM beam. When diffusive scattering
dominates, the power of a pure OAM beam may leak to multiple neighboring modes, degrading
the performance of OAM-multiplexed transmission due to the scattering-induced crosstalk.
PreviousreportsonscatteredOAMbeamshaveshownthefollowing: (i)theeffectsofdifferent
scattering strengths on the transmittance of an OAM beam [110]; (ii) the temporal dispersion of
a modulated OAM beam under different scattering strength [111]. To the best of our knowledge,
there have been few reports of scattering effects on mode purity, crosstalk, and BER performance
of a coherent OAM-multiplexed OWC link.
The transmittance of a laser beam through a scattering medium can be described by the
Beer’s law [38]: P
out
= P
in
exp(− γL ), where γ and L are the extinction ratio and propagation
distance in the scattering medium, respectively. The dimensionless quantity γL is the optical
depth (OD), quantifying the scattering strength of a given-distance medium. As an example
of a scattering medium, diluted commercial antacid solution (Maalox solution, suspensions of
Al(OH)
3
and Mg(OH)
2
) is chosen in this work, which is commonly used to emulate the scattering
properties of seawater [90, 111]. The Maalox solution is contained in an optical glass cuvette
(2-mm light path) and different scattering strengths (i.e. different ODs) are created by varying
the solution’s concentration. In the underwater environment, the extinction ratio γ may range
from approximately 0.08 m
− 1
(pure sea water) to 0.41 m
− 1
(coastal ocean water) for blue-green
65
Multiplexed
OAM beams
Scattering
medium
FM
IR Camera
Laser
@1550 nm
BS
Col.
SLM-1
EDFA
PC
Col.
Col.
Mirror
BS-1
Delay
SLM-2
Mirror
BS-2
Tx
PC
50/50
coupler
20-Gbit/s
QPSK signal
@1550 nm
SLM-3
Lens-1
Lens-2
Col.
Coherent
receiver
Rx
Mirror
EDFA
Figure 4.7: Experimental setup of multiplexed OAM beams’ transmission through a scattering
medium. QPSK: quadrature phase-shift keying; EDFA: erbium-doped fiber amplifier; Col.: col-
limator; SLM: spatial light modulator; PC: polarization controller; BS: beam splitter; FM: flip
mirror; IR: infrared; Tx: transmitter; Rx: receiver.
66
light [90] and in the atmosphere environment, γ may exceed 45 dB/km for near-infrared laser
beam in presence of rain and snow [106].
TheexperimentalsetupisshowninFig. 4.7. Inthetransmitterend(Tx),alightbeamcarrying
20-Gbit/s quadrature-phase-shift-keying (QPSK) signal at 1550 nm is generated, amplified, and
then fed into a 50/50 fiber coupler. One of the two copies is delayed by a 1-m single mode
fiber (SMF) to decorrelate the data sequences, and both copies are then sent to collimators to
generatefree-spaceGaussianbeamsinthefreespace(beamsize 3mm). TwoGaussianbeamsare
launched onto two spatial light modulators (SLM-1 and SLM-2) and converted to OAM beams
ℓ = +1,+3, respectively. A beam splitter-1 (BS-1) combines the OAM ℓ = +1 and ℓ = +3
beams, and OAM ℓ=− 1,− 3 beams are generated by reflecting the combined beams three times.
BS-2 combines the four data-carrying OAM ℓ = ± 1,± 3 beams coaxially, and the multiplexed
OAM beams are normally transmitted through the scattering medium. The emerging beams are
launched onto SLM-3, which is programmed to only extract a single OAM mode of choice and
convert it to a Gaussian-like beam, which is then coupled to a SMF for coherent signal detection.
To investigate the scattering effect on the mode purity and crosstalk of OAM beams, SLM-2 is
blocked and SLM-1 is loaded with different phase patterns to generate OAM beams of different
orders ℓ=+1,+3,+5,+7. An infrared camera is used to record the spatial intensity distribution
of the scattered OAM beams and their interferograms with an unscattered Gaussian beam.
Intensity
Interference
w/ Gaussian
= 0 1.73 7.00 4.33 6.19 8.43
Figure 4.8: Intensity and interferograms of an OAM ℓ=+3 beam after traversing the scattering
medium for different values of optical depth γL .
67
Figure 4.8 shows camera-captured intensity and interferogram (w/ a Gaussian beam) of an
OAM ℓ = +3 beam upon emerging from the scattering medium for different values of ODs.
Generally, the interferogram of an OAM beam could indicate its mode purity. For small values
of ODs, e.g., γL < 6.19, the intensity patterns of the scattered OAM beams keep the ring-like
shapes, similartothatofanunscatteredOAMbeam, andtheinterferogramsshowthemarginally
distorted twisting phase fronts. However, for the cases of large ODs, e.g., γL> 7.00, the annular
ring tends to dissolve in the intensity patterns, and the helical phase structure is ‘blurred’ by the
diffusive light in the interferogram.
Loss of
mode purity
Figure 4.9: Received total power entering the Rx and modal power belonging to the transmitted
mode when transmitting OAM modes ℓ=+1,+3,+5,+7 separately. Rx aperture is 12 mm.
To investigate the OD-dependent degradation of OAM mode purity, we first measure the
total power entering the Rx and then the received power of only the transmitted mode when
transmitting OAM beams ℓ=+1,+3,+5,+7. The beams are transmitted and analyzed one at a
time. As shown in Fig. 4.9, the upper bundle of curves (above the curve of Beer’s law) represents
thetotalpowerofeach receivedbeam, while thelower bundle ofcurves(below thecurveofBeer’s
68
law) presents the modal power, which is the amount of received power when the SLM-3 is set to
only pass the OAM component with the same order of the transmitted OAM mode. It appears
that the different OAM beams tend to interact with the scattering medium in a very similar way.
The transition from ballistic scattering to diffusive scattering region can be inferred from the
deviationsofthetwobundlesfromtheBeer’slawcurve. Intheballisticregion,mostlyunscattered
light arrives at the receiver and, therefore, the received modal power is almost equal to the total
received power for the different modes and also follows the Beer’s law. In the diffusive region,
however, the received total power of OAM beams decay slower with the OD since multiple-
scattered light contributes to total transmittance [110]. The received power on the transmitted
mode (i.e., modal power) decreases much faster than predicted by the Beer’s law because the
twisting phase and amplitude structure of OAM beams are distorted by the multiple scattering
interaction. AmeasureofthedegradationonthemodepurityofeachOAMmodecanbeextracted
from the difference between the received total and modal powers in Fig. 4.9. We observe that,
for each OAM mode, mode purity is almost conserved in the ballistic region while in the diffusive
region, it decreases as the scattering becomes stronger since the received total power is greater
than the modal power, e.g., mode purity may lose > 15 dB at γL =8.43 scattering strength.
Loss of mode purity may also lead to power leakage from the transmitted OAM mode to other
OAM modes. Figure 4.10 (a) and (b) show the received OAM power spectrum for transmitting
OAM ℓ = +1 and ℓ = +3 beams for different values of OD. For both transmitted OAM beams,
the received OAM spectra have very similar relative power distribution for γL< 6.19 since mode
purity remains almost unaffected in the ballistic scattering region. In the diffusive scattering
region, however, the loss of mode purity gives rise to severe power leakage to neighboring OAM
modes (e.g., γL = 7.00, 8.43). Importantly, in an OAM-multiplexed link, this leakage causes
crosstalk between different modes.
Crosstalk (XT) is defined as the ratio of the leakaged power from an unwanted mode to the
received power of the desired mode. The measured second-neighboring crosstalk (XT-2) and
69
a)
b)
c)
d)
Figure 4.10: Top: Measured OAM normalized power spectrum in the Rx under different values of
γL : (a) Tx: OAM ℓ=+1; (b) Tx: OAM ℓ=+3; Bottom: Measured crosstalk (XT) for different
transmitted OAM modes under different values of γL : (c) second-neighboring mode (XT-2); (d)
third-neighboring mode (XT-3). Rx aperture is 12 mm.
third-neighboring crosstalk (XT-3) for different transmitted OAM modes are shown in Figs. 4.10
(c) and (d), respectively. In the ballistic region, the XT-2 and XT-3 are below -21 dB and -33
dB, respectively, and are hardly influenced by scattering effects, while both the XT-2 and XT-3
grow significantly in the diffusive region: a mere increase of the OD by only approximately 2
deteriorates the XT by >10 dB and >20 dB, respectively. It is further observed that both XT-2
and XT-3 are measured to be approximately -10 dB for γL = 8.43. This may be because the
diffusive light tends to distribute power equally among most of the neighboring OAM modes due
to the spatial incoherence nature of the diffusive photons [109].
70
a) b)
Figure4.11: (a)Measuredopticalsignal-to-interferenceratio(OSIR)fortransmittingmultiplexed
ℓ=± 1,± 3beams; (b)MeasuredcrosstalkasafunctionofRxaperturesizefortransmittingOAM
ℓ=+3 beam.
The measured optical signal-to-interference ratio (OSIR) of each mode under different OD
values (Rx aperture is 12 mm), when transmitting multiplexed OAM ℓ=± 1,± 3 beams, is shown
inFig. 4.11(a). OSIRsforall4receivedmodesmaintain >18dBintheballisticregionsincelittle
crosstalkisinducedbytheballisticscattering, whiletheOSIRsdropdrasticallybyapproximately
10 dB from γL = 7.00 to γL = 8.43 in the diffusive region. The aperture size of the Rx may
be optimized to mitigate the increased crosstalk since a limited-size aperture could block some of
the diffusive light entering the Rx. An OAM ℓ = +3 beam (received beam size approximately 4
mm)istransmittedthroughaballistic-scattering(γL =4.33)anddiffusive-scattering( γL =7.00)
media,respectively. AsshowninFig. 4.11(b),boththeXT-1andXT-2remainalmostunaffected
in the ballistic region with different aperture sizes. For the case of the diffusive region, the XT-1
and XT-2 decrease by 3.3 dB and 3.5 dB, respectively, as the Rx aperture shrinks from 12 mm
to 4 mm.
Figure 4.12 (a) indicates the bit error rate (BER) performance of all 4 modes (after demul-
tiplexing by SLM-3) without scattering effects ( γL = 0): all modes can achieve below the 7%
overhead forward error correction (FEC) limit and have power penalties <1 dB at a BER of 1e-4.
The BERs as a function of the optical signal-to-noise ratio (OSNR) of OAM modes ℓ = +1 and
71
ℓ = +3 for different scattering strengths are shown in Figs. 4.12 (b) and 4.12 (c), respectively.
For the cases of γL =1.74 and γL =4.33, the BER performance of both OAM modes are similar
since crosstalk is almost independent of the increased scattering strength in the ballistic region.
As for the case of γL = 7.00, diffusive light contributes to the increased crosstalk among the
different modes and the BER performance of both modes is observed to degrade. We note that
for the case of γL = 8.43, the received power for both OAM modes ℓ = +1 and ℓ = +3 is below
the sensitivity of the EDFA inside the Rx, and thus the BER performance for γL = 8.43 could
not be measured.
a) b)
c)
Figure 4.12: Bit error rate (BER) of different OAM modes when transmitting multiplexed ℓ =
± 1,± 3 beams: (a) Four modes without scattering effects; (b) ℓ=+1 mode; (c) ℓ=+3 mode for
different OD values. Rx aperture is 12 mm.
In general, the overall effects of scattering on a coherent OAM-multiplexed link comprise of
powerlossandincreasedcrosstalkamongdifferentmodes. PowerlossdegradestheOSNR,butcan
be compensated by increasing the transmitted power. Crosstalk, which unfortunately is power-
independent,contributestothedecreaseoftheOSIRofagivenmode. Inthiswork,however,even
forthediffusivescatteringof γL =7.00,theOSIRsformodesℓ=+1andℓ=+3are14.08dBand
18.44 dB, still greater than their received OSNRs 10.47 dB and 12.09 dB, respectively. The BER
performance is dominated by the OSNR and thus does not show large power penalties. In this
scenario, the signal power loss rather than diffusive interference is the dominant impairment to
this link. If we had significantly higher transmitted power, we might have seen OSIR-dominated
72
BERperformanceforγL =8.43. Inpractice,onemayneedtoconsiderbotheffectsanddetermine
the dominant impairment for a specific scattering strength and link parameters.
4.4 SimultaneousTurbulenceMitigationandChannelDemultiplexing
by Adaptive Wavefront Shaping and Diffusing
In general, atmospheric turbulence is a significant challenge for an OAM-multiplexed FSO link
due to the turbulence-induced modal power coupling and resultant inter-channel crosstalk [45].
Mitigation of turbulence effects in OAM links has been demonstrated using: (i) adaptive optics
that uses a distortion sensor and a wavefront corrector [77], and (ii) electronic signal processing
techniques such as multiple-input-multiple-output (MIMO) algorithms [79].
AnadditionalchallengeforOAMFSOlinksistoefficientlydemultiplexmultiplespatialmodes
[88, 112]. Recently, an approach for OAM demultiplexing used adaptive wavefront shaping and
diffusing (WSD) of different modes [113]: multiple OAM modes are shaped to interact differently
with a deterministic multiple-scattering medium, such that the scattered light from different
modes can be separated to different spatial locations.
As shown in Fig. 4.13, two pure orthogonal OAM modes are distorted by atmospheric turbu-
lence, which couples the power of each OAM mode into many neighboring spatial modes [45]. In
anOAM-multiplexedlink,suchpowercouplingleadstointer-channelXTanddegradesthesystem
performance. The atmospheric turbulence is emulated by a single-phase screen, whose random
transfer function U(x,y) = exp(iψ (x,y)) multiplies the incident fields, E
1
(x,y) and E
2
(x,y),
causing them to couple together upon reception if demultiplexed using their original OAM modal
basis. Without the turbulence effects, a WSD system has been utilized to separate multiple pure
OAMmodesbyshapingthebeamswithaspecificphasefront[113]. Onecanmodifysuchaphase
front in WSD so that it can also act as an inverse transfer function U
∗ (x,y) = exp(− iψ (x,y))
73
for that of turbulence [100]. Therefore WSD may perform turbulence mitigation and channel de-
multiplexing simultaneously. In WSD, the distorted wavefront of each OAM beam is shaped by a
phase-only spatial light modulator (SLM), and then focused onto an optical diffuser in which the
lightfieldsarescatteredmultipletimes. Generally, suchamultiple-scatteringprocessmayfurther
degrade the mode purities of the OAM beams [97]. However, by applying the ‘correct’ phase pat-
tern on the SLM, WSD can be mode-selective so that: light fields from different distorted OAM
modes would experience different scattering trajectories inside the diffuser, and then the fields
emerging from the diffuser are refocused to different spatial locations [113, 114]. Single-mode
fibers (SMFs) are used to collect the separated single-OAM-tagged optical fields. These SMFs,
each carrying (ideally) a single data stream, are then connected to individual coherent receivers
for error-free data recovery. Such a look-for phase pattern on the SLM is determined by a genetic
algorithm (GA), that aims to maximize the received power of each distorted OAM mode.
Multiplexed
OAM beams
Diffuser
Lens
SMF
Atmospheric
turbulence
Feedback
ℓ
!
ℓ
"
OAM
order
ℓ
!
ℓ
"
PC
SLM
Power
monitor
Power

1
Power

2
Wavefront
shaping
Optical
Demultiplex
Optical
diffusing
Figure 4.13: Concept of using WSD to mitigate the turbulence effects and simultaneously demul-
tiplex two data-carrying OAM beams.
Figure 4.14 shows the experimental setup. At the Tx, a 100-Gbit/s QPSK signal generated at
1.55 µm using an in phase-quadrature (IQ) modulator. It is then amplified and sent into a 50/50
fibercoupler. Oneofthetwocopiesisdelayedbya1-mSMFtodecorrelatethedatasequences,and
both copies are coupled to free space via SMF collimators. These two optical beams are launched
onto SLM-1 and SLM-2 to convert them to OAM ℓ=+1 and ℓ=− 1 mode (beam size 2.4 mm),
74
respectively. A beam splitter (BS-1) multiplexes the data-carrying OAM ℓ =± 1 beams and the
resultantbeamsaretransmittedthroughtheemulatedturbulence. Theatmosphericturbulenceis
emulatedbya(static)thinphasescreenplatewhichisdesignedtoproduceKolmogorovspectrum
statistics and the turbulence strength is characterized by the effective Fried coherence length
r
0
[45]. Different realizations of emulated turbulence are implemented by rotating the plate to
different orientations. In the experiment, two different phase plates with r
0
= 0.4 mm, 1.0 mm
are employed one at a time. A dimension-less measure for beam distortion of an optical beam is
defined as the ratio of the beam size to the Fried coherence length, i.e.,
D
r0
. Based on the selected
r
0
, we emulate different levels of beam distortion induced by different turbulence strength (TS):
noturbulence(
D
r0
=0),weakerturbulence(
D
r0
=2.4),andstrongerturbulence(
D
r0
=6.0),denoted
as TS-0, TS-1, and TS-2, respectively. At the Rx, the distorted beams are equally split by BS-2.
One branch is sent to a normal OAM Rx [19] for the characterization of the MDL and modal
XT induced by the emulated turbulence. For this purpose, SLM-3 is loaded with a phase pattern
conjugate to the OAM mode of interest. The other branch sends the two modes to the WSD
Rx. For each turbulence realization, the wavefront of the distorted beams are firstly shaped by
SLM-4 and then focused by a lens (f =50 mm) onto the optical diffuser (ground glass with grit
120, Thorlabs). SLM-1, -2, -3, and -4 are Hamamatsu phase-only liquid crystal on silicon (LCoS)
SLMs with a refresh rate of 60 Hz and 792×600 pixels in an effective area of 15.8 × 12.0mm
2
. A
pair of SMFs, spaced apart by 127 µm , is placed approximately 1 mm behind the diffuser. The
collectedpowerforeachSMFportisfedtotheSLM-4controllerinwhichaGA(MATLABglobal
optimizationtoolbox)isappliedtosearchfortheSLM-4-loadedphasepatternthatbestmitigates
the turbulence effects, while demultiplexing the OAM modes. Only simple power measurement is
required to enable the feedback loop.
Figure 4.15 (a) shows the intensity profiles of the received OAM ℓ = +1 beam affected by
the different turbulence strengths. The annual ring structure of the OAM mode is marginally
distorted under the influence of the weaker turbulence (TS-1), while the ‘doughnut’ structure is
75
Multiplexed
OAM beams
IR camera
SLM-1
EDFA
PC
Col.
Col.
MR
BS-1
Delay
SLM-2
MR
OAM Tx
PC
50/50
100-Gb/s
QPSK @
1.55 µm
MR
Emulated
turbulence
FM
BS-2
Diffuser
SLM-4
Coh.
Rx
SMF
WSD Rx
PM
SLM-3
Col.
OAM Rx
Coh.
Rx
MR
Ctrl.
Figure 4.14: Experimental setup of using WSD to mitigate turbulence effects in a 200-Gbit/s
OAM-multiplexed link. QPSK: quadrature phase-shift keying; EDFA: erbium-doped fiber am-
plifier; SMF: single-mode fiber; Col.: collimator; SLM: spatial light modulator; PC: polarization
controller; BS: beam splitter; FM: flip mirror; MR: mirror; IR: infrared; PM: power meter; Tx:
transmitter; Rx: receiver; Ctrl.: controller.
barely present for the stronger turbulence (TS-2). As shown in Fig. 4.15 (b), the ratio of leakage
power to the neighboring OAM modes (ℓ = 0 and ℓ = +2) is as low as approximately -15 dB
without turbulence effects. However, the power of the transmitted OAM ℓ=+1 mode spreads to
many other unwanted OAM modes under turbulence effects. For instance, the ratio of the power
coupledtotheOAM ℓ=− 1modeincreasesfrom-19.04dB(TS-0)to-7.94dBand1.51dBunder
the effects of TS-1 and TS-2, respectively.
OneexampleoftheoptimizedphasepatternisshowninFig. 4.15(c),whichconsistsof30× 30
elements, each of which occupies 20× 20 pixels of the SLM-4. The wavefront-shaping process for
eachturbulencerealizationinWSDconsistsoftwosteps: (i)transmitoneOAMbeamthroughthe
emulated turbulence at a time and independently maximize the received power at the designated
76
SMF.Thustwodifferentphasepatterns θ 1
andθ 2
canbedeterminedforOAMℓ=+1andℓ=− 1
mode, respectively, and then (ii) apply the phase pattern given by θ =arg[exp(iθ
1
)+exp(iθ
2
)] to
simultaneouslymitigatingtheturbulenceeffectanddemultiplexingthe ℓ=± 1modes[115]. Figure
4.15 (d) shows an example of the convergence of this power-based feedback process. The received
powersofthedistortedOAMℓ=+1andℓ=− 1modes(underTS-2)convergetoapproximately-
20.7dBmand-21.9dBm,respectively,afteraround15000measurementiterations. Eachiteration
takes approximately 0.1 s and it is mainly limited by the refresh rate of the SLM.
=+1 =−1
+ =
-3
-2
-1
0
1
2
3
−
+
c)
1
2
3
4
5
6
1
2
3
4
5
6
-3
-2
-1
0
1
2
3
=±1
Simultaneous
turbulence
mitigation and
channel
demultiplexing
0 2 4 6 8 10 12 14
-50
-45
-40
-35
-30
-25
-20
OAM l = +1
OAM l = -1
Received power (dBm)
Number of iterations / 10^3
AO Comp.
d)
TS-2,
!
⁄ =6.0
1 2 3 4 5 6 7 8 9 10
-45
-40
-35
-30
-25
-20
-15
Received power (dBm)
Turbulence realizations
l = +1, OAM Rx
l = +1, WSD Rx
l = -1, OAM Rx
l = -1, WSD Rx
/
!
=2.4
!
⁄ =6.0
Power
enhancement
e)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

!
⁄ = 0
0
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

!
⁄ = 6.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

!
⁄ = 2.4
a)
b)
-3 -2 -1 0 +1 +2 +3 +4 +5
-25
-20
-15
-10
-5
0
-30
Normalized power (dB)
OAM order at the OAM Rx
D /r
0
=0
D /r
0
=2 .4
D /r
0
=6 .0
Turbulence
strength (TS)
-19 dB
Crosstalk Tx: = +1
TS-0
TS-1 TS-2
Figure 4.15: (a) Turbulence-distorted OAM ℓ = +1 beam profiles; (b) Received OAM spectrum
of the distorted ℓ = +1 beam; (c) One example of the determined 30 × 30 phase pattern; (d)
Receivedpowerofeachmodeversusthenumberofiterationsduringthewavefrontshapingprocess;
(e)ReceivedpowerofeachmodeafterWSDisappliedtomitigatedifferentturbulencerealizations
(TS-1: realization 1-5; TS-2: realization 6-10).
77
WSD is also applied to mitigate different realizations of both the weaker (realizations 1-5
with TS-1) and stronger (realizations 6-10 with TS-2) turbulence effects. As shown in Fig. 4.15
(e), different realizations of turbulence effects have influence on the received powers of the OAM
modes. Under the effects of TS-2, the received powers of the ℓ=+1 and ℓ=− 1 modes decrease
by >6.25 dB and >8.41 dB, respectively. Here, the stronger turbulence more severely distorts the
beams’ wavefronts, and consequently induces more power coupling from the transmitted mode
to neighboring ones. After WSD is applied to mitigate realizations 1-5 of the weaker turbulence,
1-6 dB and 3-8 dB MDL are respectively observed for receiving the ℓ = +1 and ℓ =− 1 modes.
However, as for the realizations 6-10 of stronger turbulence, WSD is capable of enhancing the
received powers of both modes. Specifically, approximately 4-19 dB of power improvement is
observed for the two modes. This may be due to the fact that WSD controls a large number
of spatial modes and thus may couple some of the leakage power back to the received mode
[114]. Moreover, as shown in Fig. 4.15 (b), the WSD-mitigated powers for both modes do not
significantly depend on the turbulence realizations. The received powers are approximately -26
dBm and -27 dBm for the ℓ=+1 and ℓ=− 1 modes, respectively, for realizations 1-10.
1 2 3 4 5 6 7 8 9 10
-30
-20
-10
0
+10
Crosstalk for l = -1 (dB)
Realizations
w/ Tur., OAM Rx
w/ Tur., WSD Rx
w/o Tur., OAM Rx
1 2 3 4 5 6 7 8 9 10
-30
-20
-10
0
+10
Crosstalk for l = +1 (dB)
Realizations
w/ Tur., OAM Rx
w/ Tur., WSD Rx
w/o Tur., OAM Rx
a)
/
!
= 2.4
!
⁄ = 6.0
Crosstalk
reduction
b)
Crosstalk
reduction
/
!
= 2.4
!
⁄ = 6.0
Figure 4.16: Measured XT using OAM and WSD Rx for receiving distorted OAM modes: (a)
ℓ=+1 mode; (b) ℓ=− 1 mode. The XT without turbulence effects using OAM Rx is also shown
for comparison. Tur.: turbulence.
78
14 16 18 20 22 24 26
10
-5
10
-4
10
-3
10
-2
10
-1
14 16 18 20 22 24 26
10
-5
10
-4
10
-3
10
-2
10
-1
7% FEC limit
BER

B2B
w/o Tur., OAM Rx
w/ Tur., OAM Rx
w/ Tur., WSD Rx
7% FEC limit
BER
OSNR (dB)
B2B
w/o Tur., OAM Rx
w/ Tur., OAM Rx
w/ Tur., WSD Rx
15 18 21 24 27 30
10
-5
10
-4
10
-3
10
-2
10
-1
7% FEC limit
BER
OSNR (dB)
B2B
w/o Tur., OAM Rx
w/ Tur., OAM Rx
w/ Tur., WSD Rx
15 18 21 24 27 30
10
-5
10
-4
10
-3
10
-2
10
-1
BER
OSNR (dB)
B2B
w/o Tur., OAM Rx
w/ Tur., OAM Rx
w/ Tur., WSD Rx
7% FEC limit
Rx: =+1
Penalty
Rx: =−1
c)
d)
TS-1, /
!
= 2.4
Rx: =+1
TS-1, /
!
= 2.4
Rx: =−1
TS-2, /
!
= 6.0
TS-2, /
!
= 6.0
a) b)
~ 9.3 dB penalty
Figure 4.17: Measured BER of OAM-carried data channel as a function of OSNR under different
turbulence effects: (a) ℓ = +1 under TS-1; (b) ℓ =− 1 under TS-1; (c) ℓ = +1 under TS-2; (d)
ℓ=− 1 under TS-2.
XT improvement for receiving the two OAM modes due to WSD is shown in Fig. 4.16. For
comparison, the XT for receiving both the OAM modes using the OAM Rx is approximately -20
dB (TS-0). Such XT slightly fluctuates over time due to the drift of alignments in the OAM-
multiplexed link. Under the effects of TS-1, the XT for the ℓ = +1 and ℓ = − 1 modes can be
suppressed by >10.0 dB and >5.8 dB, respectively. XTs can also be reduced by >17.7 dB for
stronger-turbulence-distorted OAM modes for realizations 7-10. Furthermore, WSD can suppress
the XT for both modes to less than -15 dB under most of the realizations for both TS-1 and
79
TS-2. It also appears that the mitigation performance of WSD does not largely depend on the
turbulence realizations. We note that the mitigated XT may be even lower than the XT without
turbulence effects. For example, the XT for OAM ℓ =− 1 is decreased from +3.41 dB to -21.17
dB(realization7), whichislowerthantheXTwithoutturbulence, thatis, -17.72dB(usingOAM
Rx). This may be due to the fact that WSD may also correct some misalignment of the optical
link and thus further decrease the XT.
AfterWSDisperformed,eachdemultiplexedsignalissenttoacoherentdetectoroneatatime.
Figure 4.17 shows the measured BER as a function of OSNR under different turbulence effects.
Without any turbulence effects (TS-0), BER values for both modes using the normal OAM Rx
easily fall below the 3.8e-3 FEC limit. However, using the OAM Rx under the influence of TS-1,
a power penalty as large as approximately 9.3 dB at FEC limit is observed for decoding ℓ = +1
mode and the BER performance for receiving ℓ = 1 mode cannot achieve below the FEC limit,
showninFig. 4.17(a)andFig. 4.17(b), respectively. ForthecaseofTS-2, bothOAMmodescan
hardlyachievebelowtheFEClimitduetothepresenceofstrongXTbetweenthesetwochannels,
shown in Fig. 4.17 (c) and Fig. 4.17 (d). After the WSD is applied to mitigate the turbulence
effects, both distorted modes exhibit similar BER performances as the BER performance under
TS-0, albeit with some OSNR penalties at the 7% FEC limit: approximately 0.7 dB and 1.6
dB under TS-1 while 3.2 dB and 1.8 dB under TS-2 for the OAM ℓ = +1 and ℓ = − 1 modes,
respectively.
80
Chapter 5
Optical Mitigation for Intra-Group Modal Power Coupling
in Few-Mode Fiber
5.1 Background and Motivation
Single-mode fiber
Tx Rx
Multi-mode fiber
(few-mode fiber)
Mode 1
…
Mode 2
Mode N Mode DEMUX
Rx-N
…
Rx-2
Rx-1
Mode MUX
Tx-1
…
Tx-2
Tx-N
MDM transmitter MDM receiver Crosstalk
Mode order
l
1
l
2
…
l
N
Mode order
l
2
…
l
N
l
1
Mode 1
Mode 2
Mode N
…
…
…
a)
b)
Figure 5.1: (a) Concept of optical communications in single-mode fiber; (b) Concept of mode-
division multiplexing in multi-mode fiber.
Optical communications has been increasing its capacity by employing various multiplexing
techniques, including polarization- and wavelength-division multiplexing [1, 2, 3]. As shown in
Fig. 5.1 (a), such transmission are typically employed in single-mode fibers (SMFs). SMFs can
81
support only the fundamental Gaussian laser mode and thus there is little spatial modal coupling
in SMF-based systems.
There has been much interest in utilizing space-division-multiplexing (SDM) as a technique to
further increase capacity in optical communication systems [3]. A subset of SDM is the transmis-
sion of multiple orthogonal spatial modes in a few-mode fiber (FMF) or multi-mode fiber (MMF)
to achieve mode-division multiplexing (MDM) [116]. As shown in Fig. 5.1 (b), an can typically
accommodate a limited set of linearly-polarized (LP) modes such that each mode carries an inde-
pendent data channel [5, 117]. The orthogonal spatial profiles of different LP modes ensure that
there is little inherent crosstalk for multiplexing, co-propagation in FMF, and demultiplexing [3].
Rx-1
Rx-2
Rx-N
MDM in free space
MUX
Atmospheric
turbulence
DEMUX
…
Tx-1
Tx-2
Tx-N
…
Rx-1
Rx-2
Rx-N
MDM in multi-mode fiber
MUX
DEMUX
…
Tx-1
Tx-2
Tx-N
…
Divergence
q LG
!,#
beam scales w/ +2+1
Divergence
q No divergence inside fiber
Modal crosstalk
q Mainly by atmospheric turbulence
q Different modes may have mode-
dependent distortion
Modal crosstalk
q Weaker inter-group and stronger intra-
group modal coupling
q Fiber imperfections and bendings
Modal dispersion
q Little modal dispersion for all modes in free
space
Modal dispersion
q Smaller intra-group modal dispersion and
larger inter-group modal dispersion
Modal number
q LG
!,#
modes share the same beam size &
divergence w/ modal index +2+1
Modal group
q LP
!,$
modes divided into groups w/ the
same modal index +2−1
Figure 5.2: Channel impairment comparison for mode division multiplexing in free-space atmo-
sphere and multi-mode fiber.
82
For MDM systems, there are different channel impairments in the free-space atmosphere and
MMF. As shown in Fig. 5.2, the comparison between atmosphere and MMF include: (i) Diver-
gence: in free-space, LG modes diverge as their propagation and the divergence of an LG beam
scales with
p
|ℓ|+2p+1 [118], while there is no divergence effects insides MMFs. (ii) Modal
group: the LG modes in the free-space share the same beam size and divergence with modal
index|ℓ|+2p+1 [18]; the LP modes inside an MMF are divided into the same modal group with
modal index of ℓ+2m− 1 as they share the same group velocity (to be explained later) [119].
(iii) Modal crosstalk: both the atmosphere turbulence and MMF can induce modal crosstalk
for MDM transmission but there are weaker inter-group and stronger intra-group coupling inside
MMF [45, 119]. And (iv) Modal dispersion: there is little modal dispersion in free-space while
significant modal dispersion can occur for long-distance MMF transmission [5].
5.2 Spatial Modal Groups in Few-Mode Fibers
During propagation along the FMF, LP modes tend to experience modal power coupling and
inter-channel crosstalk (XT) [120]. Moreover, as shown in Fig. 5.3, the supported modes of
an FMF can be degenerate (e.g., two-fold degenerate LP
11a
and LP
11b
modes) and divided into
different groups. Different modes from different modal groups (e.g., LP
01
, LP
11
, and LP
21
) tend
tobeweakercoupledbecauseofthelargerdifferenceintheirgroupvelocities. Conversely,different
modes in the same degenerate group would tend to experience stronger power coupling and XT
since they share a similar group velocity [119].
Generally, in addition to LP mode, as shown in Fig. 5.3, orbital-angular-momentum (OAM)
modecanalsobeusedasamodal-basistooltocharacterizeandcontrolthemodalpowercoupling
inanFMF[121]. ThisisduetothefactthatLPmodescanberepresentedbylinearcombinations
of OAM modes [122].
83
LP modes
LP
01
LP
02
LP
11a
LP
12a
LP
21a
LP
31a
LG
0,0
LG
0,1
LG
-1,0
LG
-1,1
LG
-2,0
LG
1,0
LG
1,1
LG
2,0
LP
11b
LP
21b
LP
31b LP
12b
Mode
group 1
LG
3,0
LG
-3,0
0
1
-π
π
Intensity Phase
LG modes
Mode
group 2
Mode
group 3
Mode
group 4
Figure 5.3: Concept of different modal groups in few-mode fibers. LP: linearly-polarized; LG:
Laguerre-Gaussian.
As shown in Fig. 5.4, for the LP
01
and LP
11
modal groups of an FMF, the power coupling
between degenerateLP
11a
andLP
11b
modestends tobestronger thanthecoupling between LP
01
and LP
11a
(or LP
11b
) modes. In this demonstration, such strong coupling within LP
11
group
would correspond to modal XT between output OAM ℓ=+1 and ℓ=− 1 modes. Moreover, if an
approachcanmitigatemodalpowercouplingintheOAMbasis, ittendstoindicatethecapability
to mitigate coupling between the corresponding LP modes [122, 123, 124].
In the case of weaker power coupling among modal groups, an MDM system using FMF can
still operate if each independent data stream is: (i) transmitted on a different modal group, (ii)
transmitted on only one mode within that group, and (iii) is recovered by detecting only the
modes within that group having strong coupling [125, 126].
Alternatively,approachestomitigatethisweakinter-grouppowercouplinginclude: (i)recover
multiplesignalssimultaneouslyfrommultiplemodesandapplyelectronicmultiple-input-multiple-
output (MIMO) equalization algorithms [5, 117]; (ii) measure the distortion of each mode after
FMF propagation and then correct the distortion by an optical SLM. Such an “optical equaliza-
tion”, called digital optical phase conjugation (DOPC), tends to enable the accommodation of
high-bandwidth data channels [127]. With respect to the stronger XT inside modal groups, one
84
x-pol y-pol
v
Stronger modal power coupling inside the LP
11
group
LP
11a
y-pol
LP
11a
x-pol
LP
11b
y-pol
LP
11b
x-pol
Weaker modal power coupling between groups
MIMO DSP is not necessarily
required to enable degenerate
mode-group multiplexing.
LP
01
LP
11
Figure 5.4: Concept of weaker inter-group modal coupling and stronger intra-group modal power
coupling.
approach to limit such XT is to use a specially-designed fiber, such as an elliptical-core FMF
[128, 129]. However, for a circular-graded-index (GI)-FMF-based MDM transmission, electronic
MIMOprocessingistypicallyusedformitigatingtheXTandrecoveringthemultiplexedchannels
within a degenerate modal group [5, 117, 130].
5.3 AdaptiveOpticstoMitigateIntra-GroupModalPower
Coupling
TheconceptofusingAOtomitigateintra-groupmodalXTisshowninFig. 5.5. Twoindependent
datachannelsaretransmittedinsidetheLP
11
groupandpowerofthetransmittedchannelswould
coupletoeachotherduringtheirpropagationinsidetheFMF.Asaresultoftheintra-groupmodal
coupling, the output data channels would contain both the LP
11a
and LP
11b
components. We
can also describe such modal coupling as a complex 2× 2 matrix M at the basis of OAM ℓ=+1
85
and ℓ=− 1. The amplitude and phase of complex elements in matrix M are related to the power
coupling and phase delay between the two OAM modes, respectively. AO then applies an inverse
matrix S to the mixed signals [131]:
S =M
− 1
=




s
11
s
12
s
21
s
22




=




s
T
1
s
T
2




, (5.1)
wheres
T
1
=

s
11
s
12

ands
T
2
=

s
21
s
22

aretherowvectorsofmatrix S andallelementsofS
are complex numbers. The phase mask for the AO mitigation is constructed by using a complex
combination of OAM ℓ = +1 and ℓ = − 1 demultiplexing phase patterns, of which weights are
given by s
T
1
(s
T
2
) for mitigating ℓ = +1 (ℓ = − 1) data channel. Such a phase mask converts
the two OAM modes to Gaussian beams simultaneously with the designed complex conversion
efficiencies (i.e., either s
T
1
or s
T
2
), and the XT component in the received signal would have little
power coupled into the coherent signal detection.
600-m
few-mode fiber
LP
11a
LP
11b
LP
11a
LP
11b
LP
11a
LP
11b
LP
11a
LP
11b
LP
11a
LP
11b
LP
11a
LP
11b
Data 1
Data 2
Opposite
phase by AO
Mode MUX
Complex crosstalk
(amplitude and phase)
LP
11a
LP
11b
LP
11a
LP
11b
Little crosstalk due
to destructive
interference
AO mitigation
Opposite
phase by AO
Data 1
Data 2
Data 1
Data 2
Data 1
Data 2
Figure 5.5: Concept diagram for AO to mitigate modal XT inside the LP
11
group of a GI FMF.
AO implements an inverse TM to the coupled output modes.
86
5.4 200-Gbit/sMode-Division-Multiplexinginsidethe LP
01
Group
The experimental setup is shown in Fig. 5.6 (a). At the transmitter (Tx), an up-to 100-Gbit/s
QPSK data stream is generated at 1.55 µm by an in-phase-quadrature (IQ) modulator. It is
then amplified and sent into a 50/50 fiber coupler. One of the two copies is delayed by a 1-m
SMF to decorrelate the data sequences, and both copies are sent to an OAM mode multiplexer
[88] to generate coaxial ℓ = +1 and ℓ =− 1 modes in free space. The polarizations of these two
data-carrying OAM modes are adjusted by two fiber-based polarization controllers and then a
free-space polarizer ensures that they have the same polarization. An objective lens (20× ) is used
to couple the modes into a approximately 0.6-km GI FMF which supports LP
01
, LP
11
, LP
21
,
and LP
02
degenerate modal groups. The OAM ℓ = +1 and ℓ =− 1 modes are launched into the
FMF to excite linear combinations of degenerate LP
11a
and LP
11b
modes inside the LP
11
mode
group. After propagation in the FMF, the output modes are coupled back to the free space by a
collimator. The intensity profiles of the FMF input and the corresponding output modes are also
shown in Fig. 5.6 (a): the input modes are donut-shape ℓ = +1 and ℓ = − 1 modes with little
power in the beam center; the output modes are observed to contain two intensity lobes, which
are combinations of LP
11
modes. This is consistent with the fact that OAM and LP modes can
berepresentedbylinearsuperpositionsofeachother[122]. Atthereceiver(Rx), thepolarizations
of output modes are tuned by a half-wave plate (HWP) to be aligned with the polarization of the
polarization sensitive SLM. After spatially transformed by the SLM, the reflected light beams are
coupled into an SMF and then equally split to two copies. One copy is sent for coherent signal
detection and the other copy is sent for power and TM measurement.
The process of AO mitigation includes the following steps: (i) Measure the complex 2 × 2
TM (i.e., matrix M) of the FMF using the method in [131, 18]. We send one data-carrying mode
at a time at the Tx and correspondingly load different phase patterns on the SLM at the Rx to
87
EDFA
Delay
50/50
Up to 100-
Gb/s QPSK
@ 1.55 µm
PC
OAM MUX
Pol. OL
0.6-km
GI FMF
Tx
Col.
HWP
SLM
50/50
PM
Rx
Laptop
Optical
switch
Feedback
signal
IR camera
FM
MR
SMF
MR
Col.
Control
signal
Intensity profiles
0 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
50 100 150 200 250 300
50
100
150
200
250
50 100 150 200 250 300
50
100
150
200
250
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200 0
1000
2000
3000
4000
5000
FMF input FMF input
FMF output FMF output
OAM =+1
EDFA
Coherent
detection
SMF
OAM =−1
=+1 =−1

!!
× +
!"
× =

!
"
AO Comp.
−
+
(a)
(b)
(c)
0 200 400 600 800 1000 1200
-12
-10
-8
-6
-4
5.8-dB drop
AO
Threshold
Crosstalk (dB)
Time (second)
Crosstalk for l = +1
AO
AO is on hold
AO turned on AO turned on
10-Gbaud QPSK
Figure5.6: (a)ExperimentalsetupforAOtomitigatemodalXTinanFMF-basedMDMsystem.
QPSK: quadrature phase-shift keying; EDFA: erbium-doped fiber amplifier; SMF: single-mode
fiber; PC: polarization controller; MUX: multiplexer; Pol.: polarizer; OL: objective lens; GI:
graded index; Col.: collimator; FM: flip mirror; MR: mirror; IR: infrared; HWP: half-wave plate;
SLM:spatiallightmodulator; PM:powermonitor; Tx: transmitter; Rx: receiver; (b)Anexample
ofAOphasemasktomitigatedatachannel1carriedbyOAMℓ=+1bycombiningtwoindividual
demultiplexing phase patterns with coefficients s
T
1
. (c) An example of measured time-varying XT
for OAM ℓ=+1 mode. The carried signal is 10-Gbaud QPSK.
88
measure both amplitude and phase of each element of the TM. The amplitude of each element
is obtained by direct power measurement, and its phase is retrieved by sequentially loading four
different phase masks in the SLM and then calculating the phase value from the measured four
power values, as described in [131]; (ii) Calculate the inverse matrix of M by S =M
− 1
; and (iii)
Construct a mitigation phase mask by combining OAM ℓ=+1 and ℓ=− 1 demultiplexing phase
patterns, of which weights are given by the row vector s
T
1
(s
T
2
), to mitigate modal crosstalk for
data channel 1 (2) carried by ℓ=+1 (ℓ=− 1) mode. Figure 5.6 (b) shows an example of the AO
mitigation phase mask for data channel 1 using weights s
T
1
.
Figure 5.6 shows the measured time-varying XT before and after the AO mitigation. The XT
fluctuates from –7 dB to -5 dB for the first approximately 300 s since no AO is applied; the XT
drops by >5.8 dB to <-11 dB when we apply the AO to mitigate modal power coupling. From
approximately 300 s to 880 s, the compensation phase pattern is kept fixed, and subsequently
the XT varies, and the mitigation degrades over time. To re-suppress the XT, we manually set a
threshold of approximately -9 dB and would repeat the AO as the XT reaches the threshold. For
example, the XT drops again after we refresh the AO at approximately 880 s.
We then utilize the AO mitigation to a two-channel 200-Gbit/s MDM link with each OAM
modecarryinganindependent100-Gbit/sQPSKdatastream. AsshowninFig. 5.7(a),measured
XT for ℓ=+1 and ℓ=− 1 mode is decreased from -8.4 dB and -4.6 dB to -14.3 dB and -10.2 dB,
errorvectormagnitude(EVM)isdroppedfrom62.2%and60.6%to22.1%and25.6%,respectively.
Measured BER performance as a function of OSNR for both data channels with AO mitigation
is shown in Fig. 5.7 (b). For comparison, we first measure the BER performance of transmitting
a single data channel carried by ℓ = +1 or ℓ =− 1 mode without AO mitigation: approximately
3.1 dB and 2.1 dB OSNR penalty are observed at the 7% FEC limit, respectively. Since there is
no impact of inter-channel XT, such power penalty is likely to be caused by the residual modal
differential group delay (DGD) of the FMF [5]. We then measure the BER performance for these
two data channels when they are transmitted simultaneously. After AO mitigation, these two
89
b)
OSNR: 28.7 dB
EVM: 22.1% EVM: 62.2%
EVM: 60.6%
EVM: 25.6%
Received data channel 1
w/o AO mitigation w/ AO mitigation
OSNR: 31.0 dB
XT: -8.4 dB XT: -14.3 dB
w/o AO mitigation w/ AO mitigation
OSNR: 30.9 dB
XT: -4.6 dB
OSNR: 27.9 dB
XT: -10.2 dB
Received data channel 2
a)
14 16 18 20 22 24 26
10
-4
10
-3
10
-2
B2B
50-Gbaud QPSK
BER
OSNR (dB)
Data Ch. 1: w/o XT, w/o AO (Tx: l = +1 only, Rx: l = +1)
Data Ch. 1: w/ XT, w/ AO (Tx: l = ±1 MUX, Rx: l = +1)
Data Ch. 2: w/o XT, w/o AO (Tx: l = -1 only, Rx: l = -1)
Data Ch. 2: w/ XT, w/ AO (Tx: l = ±1 MUX, Rx: l = -1)
7% FEC limit
Figure 5.7: Measured BER performance for multiplexed data channel 1 and 2 with and without
AO mitigation: (a) Constellations for QPSK signal; (b) BER as a function of OSNR. The BER
performance for transmitting the corresponding single data channel is also shown for comparison.
Each OAM mode carries an independent 100-Gbit/s QPSK signal.
channels carried by ℓ = +1 and ℓ =− 1 mode can achieve BER under the 3.8e-3 FEC limit and
theOSNRpenaltiesareapproximately0.6dBand3.8dBlargerthanthatofBERperformancefor
transmitting the corresponding single channels, respectively. The inferior performance of ℓ=− 1
mode is consistent with the measured XT: the AO achieves XT only approximately -10.2 dB for
ℓ=− 1 mode as compared with approximately -14.3 dB for ℓ=+1 mode. It may be due to that
ℓ = − 1 mode is coupled to LP
11
group with lower efficiency, possibly limited by the imperfect
mode generation in this demonstration.
WealsoinvestigatetheeffectsofdifferentdataratesontheperformanceofAOmitigation. As
shown in Fig. 5.8, AO can reduce XT and achieve BER below the 7% FEC limit for transmitting
bothmodescarrying10-,20-,and50-GbuadQPSKsignals. Forcomparison,theBERperformance
fortransmittingasingledatachannel(i.e., w/ointer-channelXT)isalsoshowninFig. 5.8. With
AOmitigation,theOSNRpenaltiesattheFEClimitforℓ=+1andℓ=− 1modearerespectively
90
b)
a)
8 10 12 14 16 18 20 22
10
-4
10
-3
10
-2
10 12 14 16 18 20 22 24 26
10
-4
10
-3
10
-2
Data Ch. 1
Rx: l = +1
10 Gbaud
50 Gbaud
20 Gbaud
BER
w/o modal XT, w/o AO w/ modal XT, w/ AO
7% FEC
50 Gbaud
7% FEC
Data Ch. 2
Rx: l = -1
BER
OSNR (dB)
10 Gbaud
20 Gbaud
Figure5.8: BERperformancewithAOmitigationfordifferentdatarates(10-, 20-, and50-Gbaud
QPSK): (a) Data channel 1 carried by OAM ℓ=+1 mode; (b) Data channel 2 carried by ℓ=− 1
mode.
measured as approximately 0.9 dB and 3 dB for 10-Gbaud signal, approximately 1 dB and 1.8
dB for 20-Gbaud signal.
91
Chapter 6
Conclusion
This thesis has discussed various enabling techniques for optical communications in different
disturbing media by controlling the interaction of structured light with the media. By ma-
nipulating Laguerre-Gaussian spatial mode contents inside random distortion, including atmo-
spheric turbulence, scattering media, and multi-mode fibers, one can potentially enhance the
resilience/robustness of optical transmission to the channel impairment.
It is still not clear whether these techniques will be deployed in the practical systems or not.
Theutilizationofstructuredlightinopticalcommunicationshasbeenayoungandemergingfield,
withsomeattractivepotentialadvantagesandalotofchallengesremainingtobeaddressed. Last
but no the least, it would not be surprising to see structured light finds applications in other
promising fields, such as optical sensing and imaging.
92
References
[1] L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems. Artech
House optoelectronics library, Artech House, 1996.
[2] G. Li, “Recent advances in coherent optical communication,” Advances in optics and pho-
tonics, vol. 1, no. 2, pp. 279–307, 2009.
[3] D.Richardson,J.Fini,andL.Nelson,“Space-divisionmultiplexinginopticalfibres,” Nature
Photonics, vol. 7, no. 5, pp. 354–362, 2013.
[4] G.Li,N.Bai,N.Zhao,andC.Xia,“Space-divisionmultiplexing: thenextfrontierinoptical
communication,” Advances in Optics and Photonics, vol. 6, no. 4, pp. 413–487, 2014.
[5] R. Ryf, M. Mestre, S. Randel, C. Schmidt, A. Gnauck, R.-J. Essiambre, P. Winzer, R. Del-
bue, P. Pupalaikis, A. Sureka, et al., “Mode-multiplexed transmission over a 209-km dgd-
compensated hybrid few-mode fiber span,” IEEE Photonics Technology Letters, vol. 24,
no. 21, pp. 1965–1968, 2012.
[6] N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and
S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing
in fibers,” science, vol. 340, no. 6140, pp. 1545–1548, 2013.
[7] A.E.Willner,H.Huang,Y.Yan,Y.Ren,N.Ahmed,G.Xie,C.Bao,L.Li,Y.Cao,Z.Zhao,
et al., “Optical communications using orbital angular momentum beams,” Advances in
Optics and Photonics, vol. 7, no. 1, pp. 66–106, 2015.
[8] H. Rubinsztein-Dunlop, A. Forbes, M. Berry, M. Dennis, D. L. Andrews, M. Mansuripur,
C. Denz, C. Alpmann, P. Banzer, T. Bauer, et al., “Roadmap on structured light,” Journal
of Optics, vol. 19, no. 1, p. 013001, 2016.
[9] J. Rogel-Salazar, J. P. Trevi˜ no, and S. Ch´ avez-Cerda, “Engineering structured light with
optical vortices,” JOSA B, vol. 31, no. 6, pp. A46–A50, 2014.
[10] V.W.Chan,“Free-spaceopticalcommunications,” Journal of Lightwave technology,vol.24,
no. 12, pp. 4750–4762, 2006.
[11] M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A commu-
nication theory perspective,” IEEE communications surveys & tutorials, vol. 16, no. 4,
pp. 2231–2258, 2014.
[12] H. Hemmati, Deep space optical communications. John Wiley & Sons, 2006.
[13] X.ZhuandJ.M.Kahn,“Free-spaceopticalcommunicationthroughatmosphericturbulence
channels,” IEEE Transactions on communications, vol. 50, no. 8, pp. 1293–1300, 2002.
93
[14] Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless
communications,” IEEE communications surveys & tutorials, vol. 19, no. 1, pp. 204–238,
2016.
[15] D. Kedar and S. Arnon, “Urban optical wireless communication networks: the main chal-
lenges and possible solutions,” IEEE Communications Magazine, vol. 42, no. 5, pp. S2–S7,
2004.
[16] L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, vol. 152.
SPIE press Bellingham, WA, 2005.
[17] L.Allen,M.W.Beijersbergen,R.Spreeuw,andJ.Woerdman,“Orbitalangularmomentum
oflightandthetransformationoflaguerre-gaussianlasermodes,” PhysicalReviewA,vol.45,
no. 11, p. 8185, 1992.
[18] A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applica-
tions,” Advances in Optics and Photonics, vol. 3, no. 2, pp. 161–204, 2011.
[19] J.Wang, J.-Y.Yang, I.M.Fazal, N.Ahmed, Y.Yan, H.Huang, Y.Ren, Y.Yue, S.Dolinar,
M. Tur, et al., “Terabit free-space data transmission employing orbital angular momentum
multiplexing,” Nature Photonics, vol. 6, no. 7, pp. 488–496, 2012.
[20] S.Oemrawsingh,J.VanHouwelingen,E.Eliel,J.Woerdman,E.Verstegen,J.Kloosterboer,
et al., “Production and characterization of spiral phase plates for optical wavelengths,”
Applied optics, vol. 43, no. 3, pp. 688–694, 2004.
[21] M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront
laser beams produced with a spiral phaseplate,” Optics Communications, vol. 112, no. 5-6,
pp. 321–327, 1994.
[22] A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with
spatial light modulators,” Advances in Optics and Photonics, vol. 8, no. 2, pp. 200–227,
2016.
[23] N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation
of high-quality higher-order laguerre-gaussian beams using liquid-crystal-on-silicon spatial
light modulators,” JOSA A, vol. 25, no. 7, pp. 1642–1651, 2008.
[24] M. Mirhosseini, O. S. Magana-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd,
“Rapid generation of light beams carrying orbital angular momentum,” Optics express,
vol. 21, no. 25, pp. 30196–30203, 2013.
[25] X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G.
Thompson,andS.Yu,“Integratedcompactopticalvortexbeamemitters,” Science,vol.338,
no. 6105, pp. 363–366, 2012.
[26] T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. Yoo,
“Demonstration of free space coherent optical communication using integrated silicon pho-
tonic orbital angular momentum devices,” Optics express, vol. 20, no. 9, pp. 9396–9402,
2012.
[27] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro,
“Lightpropagationwithphasediscontinuities: generalizedlawsofreflectionandrefraction,”
science, vol. 334, no. 6054, pp. 333–337, 2011.
94
[28] E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating
optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,”
Light: Science & Applications, vol. 3, no. 5, p. e167, 2014.
[29] Z. Zhao, J. Wang, S. Li, and A. E. Willner, “Metamaterials-based broadband generation of
orbital angular momentum carrying vector beams,” Optics letters, vol. 38, no. 6, pp. 932–
934, 2013.
[30] Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical commu-
nication through atmospheric turbulence,” Applied Optics, vol. 44, no. 23, pp. 4946–4952,
2005.
[31] Y. Ren, A. Dang, L. Liu, and H. Guo, “Heterodyne efficiency of a coherent free-space
optical communication model through atmospheric turbulence,” Applied Optics, vol. 51,
no. 30, pp. 7246–7254, 2012.
[32] R.J.Noll,“Zernikepolynomialsandatmosphericturbulence,” JOsA,vol.66,no.3,pp.207–
211, 1976.
[33] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in
an orbital angular momentum-multiplexed free-space optical link,” Applied optics, vol. 47,
no. 13, pp. 2414–2429, 2008.
[34] N.ChandrasekaranandJ.H.Shapiro,“Photoninformationefficientcommunicationthrough
atmosphericturbulence–parti: channelmodelandpropagationstatistics,” Journal of Light-
wave Technology, vol. 32, no. 6, pp. 1075–1087, 2014.
[35] N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. A. Lopez, R. A. Correa, B. Guan,
B. Ercan, R. P. Scott, S. B. Yoo, et al., “30× 30 mimo transmission over 15 spatial modes,”
inOptical Fiber Communication Conference,pp.Th5C–1,OpticalSocietyofAmerica,2015.
[36] R. Ryf, N. K. Fontaine, S. Wittek, K. Choutagunta, M. Mazur, H. Chen, J. C. Alvarado-
Zacarias, R. Amezcua-Correa, M. Capuzzo, R. Kopf, et al., “High-spectral-efficiency mode-
multiplexedtransmissionovergraded-indexmultimodefiber,”in 2018 European Conference
on Optical Communication (ECOC), pp. 1–3, IEEE, 2018.
[37] L. Mullen, A. Laux, and B. Cochenour, “Propagation of modulated light in water: implica-
tions for imaging and communications systems,” Applied optics, vol. 48, no. 14, pp. 2607–
2612, 2009.
[38] B. Cochenour, L. Mullen, and J. Muth, “Effect of scattering albedo on attenuation and
polarization of light underwater,” Optics letters, vol. 35, no. 12, pp. 2088–2090, 2010.
[39] D. Kedar and S. Arnon, “Evaluation of coherence interference in optical wireless commu-
nication through multiscattering channels,” Applied optics, vol. 45, no. 14, pp. 3263–3269,
2006.
[40] K. Szczerba, P. Westbergh, J. Karout, J. S. Gustavsson,
˚A. Haglund, M. Karlsson, P. A.
Andrekson, E. Agrell, and A. Larsson, “4-pam for high-speed short-range optical commu-
nications,” Journal of optical communications and networking, vol. 4, no. 11, pp. 885–894,
2012.
[41] K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. P. T. Lau, “Digital signal processing for
short-reach optical communications: A review of current technologies and future trends,”
Journal of Lightwave Technology, vol. 36, no. 2, pp. 377–400, 2018.
95
[42] R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer,
M. Dreschmann, M. Huebner, C. Koos, et al., “Error vector magnitude as a performance
measure for advanced modulation formats,” IEEE Photonics Technology Letters, vol. 24,
no. 1, pp. 61–63, 2011.
[43] B. Sklar, Digital communications, vol. 2. Prentice Hall Upper Saddle River, 2001.
[44] K. Kikuchi, “Fundamentals of coherent optical fiber communications,” Journal of lightwave
technology, vol. 34, no. 1, pp. 157–179, 2015.
[45] Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P.
Lavery, N. K. Steinhoff, M. Tur, et al., “Atmospheric turbulence effects on the performance
of a free space optical link employing orbital angular momentum multiplexing,” Optics
letters, vol. 38, no. 20, pp. 4062–4065, 2013.
[46] H. Hemmati, Near-earth laser communications. CRC press, 2020.
[47] H. Liu, B. Huang, J. C. A. Zacarias, H. Wen, H. Chen, N. K. Fontaine, R. Ryf, J. E.
Antonio-Lopez, R. A. Correa, and G. Li, “Turbulence-resistant fso communication using a
few-mode pre-amplified receiver,” Scientific Reports , vol. 9, no. 1, pp. 1–7, 2019.
[48] R.Zhang,N.Hu,H.Zhou,K.Zou,X.Su,Y.Zhou,H.Song,K.Pang,H.Song,A.Minoofar,
et al., “Turbulence-resilient pilot-assisted self-coherent free-space optical communications
using automatic optoelectronic mixing of many modes,” Nature Photonics, vol. 15, no. 10,
pp. 743–750, 2021.
[49] H. Kaushal and G. Kaddoum, “Optical communication in space: Challenges and mitigation
techniques,” IEEE communications surveys & tutorials, vol. 19, no. 1, pp. 57–96, 2016.
[50] A.Lorences-Riesgo,F.P.Guiomar,A.N.Sousa,A.L.Teixeira,N.J.Muga,M.C.Medeiros,
andP.P.Monteiro,“200goutdoorfree-space-opticslinkusingasingle-photodiodereceiver,”
Journal of Lightwave Technology, vol. 38, no. 2, pp. 394–400, 2020.
[51] F. P. Guiomar, A. Lorences-Riesgo, D. Ranzal, F. Rocco, A. N. Sousa, M. A. Fernandes,
B. T. Brand˜ ao, A. Carena, A. L. Teixeira, M. C. Medeiros, et al., “Adaptive probabilistic
shaped modulation for high-capacity free-space optical links,” Journal of Lightwave Tech-
nology, vol. 38, no. 23, pp. 6529–6541, 2020.
[52] J. Surof, J. Poliak, and R. M. Calvo, “Demonstration of intradyne bpsk optical free-space
transmission in representative atmospheric turbulence conditions for geostationary uplink
channel,” Optics Letters, vol. 42, no. 11, pp. 2173–2176, 2017.
[53] A. Belmonte and J. M. Kahn, “Efficiency of complex modulation methods in coherent free-
space optical links,” Optics express, vol. 18, no. 4, pp. 3928–3937, 2010.
[54] S. C. Cohen, “Heterodyne detection: phase front alignment, beam spot size, and detector
uniformity,” Applied optics, vol. 14, no. 8, pp. 1953–1959, 1975.
[55] D. M. Chambers, “Modeling heterodyne efficiency for coherent laser radar in the presence
of aberrations,” Optics Express, vol. 1, no. 3, pp. 60–67, 1997.
[56] D. Delautre, S. Breugnot, and V. Laude, “Measurement of the sensitivity of heterodyne
detection to aberrations using a programmable liquid-crystal modulator,” Optics communi-
cations, vol. 160, no. 1-3, pp. 61–65, 1999.
96
[57] M. Chen, C. Liu, D. Rui, and H. Xian, “Experimental results of atmospheric coherent
opticalcommunicationswithadaptiveoptics,” Optics Communications, vol.434, pp.91–96,
2019.
[58] N. K. Fontaine, R. Ryf, Y. Zhang, J. C. Alvarado-Zacarias, S. van der Heide, M. Mazur,
H. Huang, H. Chen, R. Amezcua-Correa, G. Li, et al., “Digital turbulence compensation of
free space optical link with multimode optical amplifier,” 2019.
[59] M.ArikawaandT.Ito, “Performanceofmodediversityreceptionofapolarization-division-
multiplexed signal for free-space optical communication under atmospheric turbulence,”
Optics express, vol. 26, no. 22, pp. 28263–28276, 2018.
[60] D. J. Geisler, T. M. Yarnall, M. L. Stevens, C. M. Schieler, B. S. Robinson, and S. A.
Hamilton, “Multi-aperture digital coherent combining for free-space optical communication
receivers,” Optics Express, vol. 24, no. 12, pp. 12661–12671, 2016.
[61] B.Zhang,R.Yuan,J.Sun,J.Cheng,andM.-S.Alouini,“Free-spaceopticalcommunication
using non-mode-selective photonic lantern-based coherent receiver,” IEEE Transactions on
Communications, vol. 69, no. 8, pp. 5367–5380, 2021.
[62] H. Yuksel, S. Milner, and C. Davis, “Aperture averaging for optimizing receiver design
and system performance on free-space optical communication links,” Journal of Optical
Networking, vol. 4, no. 8, pp. 462–475, 2005.
[63] X. Chen, C. Antonelli, A. Mecozzi, D. Che, and W. Shieh, “High-capacity direct-detection
systems,” in Optical Fiber Telecommunications VII, pp. 419–441, Elsevier, 2020.
[64] T. Takenaka, K. Tanaka, and O. Fukumitsu, “Signal-to-noise ratio in optical heterodyne
detection for gaussian fields,” Applied Optics, vol. 17, no. 21, pp. 3466–3471, 1978.
[65] R.Zhang,N.Hu,K.Zou,H.Zhou,X.Su,Z.Zhao,H.Song,H.Song,A.Almaiman,K.Pang,
et al., “Experimental demonstration of crosstalk reduction to achieve turbulence-resilient
multiple-oam-beam free-space optical communications using pilot tones to mix beams at
the receiver,” in CLEO: Science and Innovations, pp. SW4L–4, Optical Society of America,
2020.
[66] Z.Li, M.S.Erkılın¸ c, K.Shi, E.Sillekens, L.Galdino, T.Xu, B.C.Thomsen, P.Bayvel, and
R. I. Killey, “Digital linearization of direct-detection transceivers for spectrally efficient 100
gb/s/λ wdmmetronetworking,” Journal of Lightwave Technology, vol.36, no.1, pp.27–36,
2017.
[67] J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques
for dwdm systems,” IEEE Journal of selected topics in quantum electronics, vol. 10, no. 2,
pp. 259–272, 2004.
[68] M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Po-
larization measurements through space-to-ground atmospheric propagation paths by using
a highly polarized laser source in space,” Optics express, vol. 17, no. 25, pp. 22333–22340,
2009.
[69] Y. Zhou, B. Braverman, A. Fyffe, R. Zhang, J. Zhao, A. E. Willner, Z. Shi, and R. W.
Boyd, “Vectorial phase conjugation for high-fidelity mode transmission through multimode
fiber,” in Photonic Networks and Devices, pp. NeW2B–5, Optical Society of America, 2020.
97
[70] W.-R.Peng,X.Wu,K.-M.Feng,V.R.Arbab,B.Shamee,J.-Y.Yang,L.C.Christen,A.E.
Willner, and S. Chi, “Spectrally efficient direct-detected ofdm transmission employing an
iterative estimation and cancellation technique,” Optics express, vol. 17, no. 11, pp. 9099–
9111, 2009.
[71] A.Mecozzi,C.Antonelli,andM.Shtaif,“Kramers–kronigcoherentreceiver,” Optica,vol.3,
no. 11, pp. 1220–1227, 2016.
[72] X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Optics Express,
vol. 16, no. 2, pp. 792–803, 2008.
[73] R.Zhang, N.Hu, X.Su, A.Almaiman, H.Song, Z.Zhao, H.Song, K.Pang, C.Liu, M.Tur,
et al.,“Alignmentmonitorforfree-spaceopticallinksinthepresenceofturbulenceusingthe
beating of opposite-order orbital-angular-momentum beams on two different wavelengths,”
in 2020 Optical Fiber Communications Conference and Exhibition (OFC), pp. 1–3, IEEE,
2020.
[74] K. Pang, H. Song, Z. Zhao, R. Zhang, H. Song, G. Xie, L. Li, C. Liu, J. Du, A. F. Molisch,
et al., “400-gbit/s qpsk free-space optical communication link based on four-fold multiplex-
ing of hermite–gaussian or laguerre–gaussian modes by varying both modal indices,” Optics
letters, vol. 43, no. 16, pp. 3889–3892, 2018.
[75] S. Restuccia, D. Giovannini, G. Gibson, and M. Padgett, “Comparing the information
capacityoflaguerre–gaussianandhermite–gaussianmodalsetsinafinite-aperturesystem,”
Optics express, vol. 24, no. 24, pp. 27127–27136, 2016.
[76] B. Rodenburg, M. P. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson,
M.Padgett,andR.W.Boyd,“Influenceofatmosphericturbulenceonstatesoflightcarrying
orbital angular momentum,” Optics letters, vol. 37, no. 17, pp. 3735–3737, 2012.
[77] Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. Lavery, M. Tur, M. A.
Neifeld, et al., “Adaptive-optics-based simultaneous pre-and post-turbulence compensation
of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,”
Optica, vol. 1, no. 6, pp. 376–382, 2014.
[78] S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compen-
sation in orbital angular momentum communications: Advances and perspectives,” Optics
Communications, vol. 408, pp. 68–81, 2018.
[79] H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J.
Dolinar,andA.E.Willner,“Crosstalkmitigationinafree-spaceorbitalangularmomentum
multiplexed communication link using 4× 4 mimo equalization,” Optics letters, vol. 39,
no. 15, pp. 4360–4363, 2014.
[80] Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A.
Neifeld,et al.,“Turbulencecompensationofanorbitalangularmomentumandpolarization-
multiplexed link using a data-carrying beacon on a separate wavelength,” Optics Letters,
vol. 40, no. 10, pp. 2249–2252, 2015.
[81] Y. Kaymak, R. Rojas-Cessa, J. Feng, N. Ansari, M. Zhou, and T. Zhang, “A survey on ac-
quisition,tracking,andpointingmechanismsformobilefree-spaceopticalcommunications,”
IEEE Communications Surveys & Tutorials, vol. 20, no. 2, pp. 1104–1123, 2018.
[82] G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. Lavery, N. Ashrafi,
S. Ashrafi, et al., “Performance metrics and design considerations for a free-space optical
98
orbital-angular-momentum–multiplexedcommunicationlink,” Optica,vol.2,no.4,pp.357–
365, 2015.
[83] L. Li, R. Zhang, G. Xie, Y. Ren, Z. Zhao, Z. Wang, C. Liu, H. Song, K. Pang, R. Bock,
et al.,“Experimentaldemonstrationofbeaconlessbeamdisplacementtrackingforanorbital
angular momentum multiplexed free-space optical link,” Optics Letters, vol. 43, no. 10,
pp. 2392–2395, 2018.
[84] L. Li, R. Zhang, Z. Zhao, G. Xie, P. Liao, K. Pang, H. Song, C. Liu, Y. Ren, G. Labroille,
etal.,“High-capacityfree-spaceopticalcommunicationsbetweenagroundtransmitteranda
groundreceiverviaauavusingmultiplexingofmultipleorbital-angular-momentumbeams,”
Scientific reports , vol. 7, no. 1, pp. 1–12, 2017.
[85] G. Xie, L. Li, Y. Ren, Y. Yan, N. Ahmed, Z. Zhao, Z. Wang, N. Ashrafi, S. Ashrafi, R. D.
Linquist, et al., “Exploiting the unique intensity gradient of an orbital-angular-momentum
beam for accurate receiver alignment monitoring in a free-space communication link,” in
2015 European Conference on Optical Communication (ECOC), pp. 1–3, IEEE, 2015.
[86] J. H. Churnside and R. J. Lataitis, “Wander of an optical beam in the turbulent atmo-
sphere,” Applied Optics, vol. 29, no. 7, pp. 926–930, 1990.
[87] T. Chiba, “Spot dancing of the laser beam propagated through the turbulent atmosphere,”
Applied optics, vol. 10, no. 11, pp. 2456–2461, 1971.
[88] 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,” Optics
express, vol. 22, no. 13, pp. 15599–15607, 2014.
[89] Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. Cheng, “Emerging optical wireless
communications-advances and challenges,” IEEE journal on selected areas in communica-
tions, vol. 33, no. 9, pp. 1738–1749, 2015.
[90] J.Baghdady, K.Miller, K.Morgan, M.Byrd, S.Osler, R.Ragusa, W.Li, B.M.Cochenour,
and E. G. Johnson, “Multi-gigabit/s underwater optical communication link using orbital
angular momentum multiplexing,” Optics express, vol. 24, no. 9, pp. 9794–9805, 2016.
[91] S.W.Hell,“Towardfluorescencenanoscopy,” Naturebiotechnology,vol.21,no.11,pp.1347–
1355, 2003.
[92] M. Padgett and R. Bowman, “Tweezers with a twist,” Nature photonics, vol. 5, no. 6,
pp. 343–348, 2011.
[93] B.K.Singh,H.Nagar,Y.Roichman,andA.Arie,“Particlemanipulationbeyondthediffrac-
tion limit using structured super-oscillating light beams,” Light: Science & Applications,
vol. 6, no. 9, pp. e17050–e17050, 2017.
[94] G. Xie, H. Song, Z. Zhao, G. Milione, Y. Ren, C. Liu, R. Zhang, C. Bao, L. Li, Z. Wang,
et al., “Using a complex optical orbital-angular-momentum spectrum to measure object
parameters,” Optics Letters, vol. 42, no. 21, pp. 4482–4485, 2017.
[95] M.W.Beijersbergen, L.Allen, H.VanderVeen, andJ.Woerdman, “Astigmaticlasermode
converters and transfer of orbital angular momentum,” Optics Communications, vol. 96,
no. 1-3, pp. 123–132, 1993.
99
[96] R. Zhang, H. Song, Z. Zhao, H. Song, J. Du, G. Xie, L. Li, K. Pang, C. Liu, A. Almaiman,
et al., “Scattered complex laguerre-gaussian spectrum to determine the 2-d transverse po-
sition of a spherical silica particle,” in CLEO: QELS Fundamental Science, pp. JTu2A–57,
Optical Society of America, 2019.
[97] R. Zhang, L. Li, Z. Zhao, G. Xie, G. Milione, H. Song, P. Liao, C. Liu, H. Song, K. Pang,
et al., “Coherent optical wireless communication link employing orbital angular momentum
multiplexing in a ballistic and diffusive scattering medium,” Optics Letters, vol. 44, no. 3,
pp. 691–694, 2019.
[98] R. Zhang, L. Li, Z. Zhao, G. Xie, P. Liao, H. Song, C. Liu, H. Song, K. Pang, R. Bock,
et al., “Experimental effect of scattering on an 80-gbit/s qpsk wireless link using 4 orbital-
angular-momentum beams,” in Optical Fiber Communication Conference, pp. Tu2I–5, Op-
tical Society of America, 2018.
[99] R. Zhang, H. Song, Z. Zhao, H. Song, J. Du, C. Liu, K. Pang, L. Li, H. Zhou, A. N.
Willner, et al., “Simultaneous turbulence mitigation and channel demultiplexing for two
100 gbit/s orbital-angular-momentum multiplexed beams by adaptive wavefront shaping
and diffusing,” Optics Letters, vol. 45, no. 3, pp. 702–705, 2020.
[100] R. Zhang, H. Song, Z. Zhao, H. Song, J. Du, C. Liu, K. Pang, L. Li, A. N. Willner, R. W.
Boyd, et al., “Demonstration of independent turbulence mitigation of two 100-gbit/s qpsk
orbital-angular-momentummultiplexedbeamsusingwavefrontshapingandcontrolledscat-
tering,” in 2019 Optical Fiber Communications Conference and Exhibition (OFC), pp. 1–3,
IEEE, 2019.
[101] V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, “Mie scattering
distinguishes the topological charge of an optical vortex: a homage to gustav mie,” New
Journal of Physics, vol. 11, no. 1, p. 013046, 2009.
[102] A. D. Kiselev and D. O. Plutenko, “Mie scattering of laguerre-gaussian beams: Photonic
nanojets and near-field optical vortices,” Physical Review A, vol. 89, no. 4, p. 043803, 2014.
[103] X. Zambrana-Puyalto and G. Molina-Terriza, “The role of the angular momentum of light
in mie scattering. excitation of dielectric spheres with laguerre–gaussian modes,” Journal of
Quantitative Spectroscopy and Radiative Transfer, vol. 126, pp. 50–55, 2013.
[104] X. Zambrana-Puyalto, X. Vidal, P. Wozniak, P. Banzer, and G. Molina-Terriza, “Tailoring
multipolar mie scattering with helicity and angular momentum,” ACS Photonics, vol. 5,
no. 7, pp. 2936–2944, 2018.
[105] D. Petrov, N. Rahuel, G. Molina-Terriza, and L. Torner, “Characterization of dielectric
spheres by spiral imaging,” Optics letters, vol. 37, no. 5, pp. 869–871, 2012.
[106] S. Mori and F. S. Marzano, “Microphysical characterization of free space optical link due
to hydrometeor and fog effects,” Applied optics, vol. 54, no. 22, pp. 6787–6803, 2015.
[107] M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam
scatteringandwanderinginfsocommunications,” Journal of Lightwave Technology, vol.32,
no. 7, pp. 1323–1328, 2014.
[108] S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-
communication systems,” Optics letters, vol. 28, no. 2, pp. 129–131, 2003.
[109] L. Wang, P. Ho, C. Liu, G. Zhang, and R. Alfano, “Ballistic 2-d imaging through scattering
walls using an ultrafast optical kerr gate,” Science, vol. 253, no. 5021, pp. 769–771, 1991.
100
[110] W. Wang, R. Gozali, L. Shi, L. Lindwasser, and R. Alfano, “Deep transmission of laguerre–
gaussian vortex beams through turbid scattering media,” Optics letters, vol. 41, no. 9,
pp. 2069–2072, 2016.
[111] B. Cochenour, K. Morgan, K. Miller, E. Johnson, K. Dunn, and L. Mullen, “Propagation
of modulated optical beams carrying orbital angular momentum in turbid water,” Applied
optics, vol. 55, no. 31, pp. C34–C38, 2016.
[112] N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-
gaussian mode sorter,” Nature communications, vol. 10, no. 1, pp. 1–7, 2019.
[113] R.Fickler,M.Ginoya,andR.W.Boyd,“Custom-tailoredspatialmodesortingbycontrolled
random scattering,” Physical Review B, vol. 95, no. 16, p. 161108, 2017.
[114] I. M. Vellekoop and A. Mosk, “Focusing coherent light through opaque strongly scattering
media,” Optics letters, vol. 32, no. 16, pp. 2309–2311, 2007.
[115] S. R. Huisman, T. J. Huisman, T. A. Wolterink, A. P. Mosk, and P. W. Pinkse, “Pro-
grammablemultiportopticalcircuitsinopaquescatteringmaterials,” Opticsexpress,vol.23,
no. 3, pp. 3102–3116, 2015.
[116] P. Sillard, “Few-mode fibers for space division multiplexing,” in Optical Fiber Communica-
tion Conference, pp. Th1J–1, Optical Society of America, 2016.
[117] S.Randel,R.Ryf,A.Sierra,P.J.Winzer,A.H.Gnauck,C.A.Bolle,R.-J.Essiambre,D.W.
Peckham, A. McCurdy, and R. Lingle, “6× 56-gb/s mode-division multiplexed transmission
over 33-km few-mode fiber enabled by 6 × 6 mimo equalization,” Optics Express, vol. 19,
no. 17, pp. 16697–16707, 2011.
[118] M. J. Padgett, F. M. Miatto, M. P. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of
an orbital-angular-momentum-carrying beam upon propagation,” New Journal of Physics,
vol. 17, no. 2, p. 023011, 2015.
[119] J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division mul-
tiplexing,” Journal of Lightwave Technology, vol. 30, no. 24, pp. 3946–3952, 2012.
[120] K.-i. Kitayama and N.-P. Diamantopoulos, “Few-mode optical fibers: Original motivation
and recent progress,” IEEE Communications Magazine, vol. 55, no. 8, pp. 163–169, 2017.
[121] S. Chen and J. Wang, “Theoretical analyses on orbital angular momentum modes in con-
ventional graded-index multimode fibre,” Scientific reports , vol. 7, no. 1, pp. 1–15, 2017.
[122] H.Huang,G.Milione,M.P.Lavery,G.Xie,Y.Ren,Y.Cao,N.Ahmed,T.AnNguyen,D.A.
Nolan, M.-J. Li, et al., “Mode division multiplexing using an orbital angular momentum
mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” Scientific reports ,
vol. 5, no. 1, pp. 1–7, 2015.
[123] R. Zhang, H. Song, H. Song, Z. Zhao, G. Milione, K. Pang, J. Du, L. Li, K. Zou, H. Zhou,
et al., “Utilizing adaptive optics to mitigate intra-modal-group power coupling of graded-
indexfew-modefiberina200-gbit/smode-division-multiplexedlink,” OpticsLetters,vol.45,
no. 13, pp. 3577–3580, 2020.
[124] J. Carpenter, B. J. Eggleton, and J. Schr¨ oder, “110x110 optical mode transfer matrix in-
version,” Optics express, vol. 22, no. 1, pp. 96–101, 2014.
101
[125] T.Hu, J.Li, D.Ge, Z.Wu, Y.Tian, L.Shen, Y.Liu, S.Chen, Z.Li, Y.He, et al., “Weakly-
coupled 4-mode step-index fmf and demonstration of im/dd mdm transmission,” Optics
express, vol. 26, no. 7, pp. 8356–8363, 2018.
[126] K. Benyahya, C. Simonneau, A. Ghazisaeidi, R. R. Muller, M. Bigot, P. Sillard, P. Jian,
G. Labroille, J. Renaudier, and G. Charlet, “200gb/s transmission over 20km of fmf fiber
using mode group multiplexing and direct detection,” in 2018 European Conference on
Optical Communication (ECOC), pp. 1–3, IEEE, 2018.
[127] S. Bae, Y. Jung, B. G. Kim, and Y. C. Chung, “Compensation of mode crosstalk in mdm
systemusingdigitalopticalphaseconjugation,” IEEE Photonics Technology Letters,vol.31,
no. 10, pp. 739–742, 2019.
[128] E.Ip,G.Milione,M.-J.Li,N.Cvijetic,K.Kanonakis,J.Stone,G.Peng,X.Prieto,C.Mon-
tero, V. Moreno, et al., “Sdm transmission of real-time 10gbe traffic using commercial sfp+
transceivers over 0.5 km elliptical-core few-mode fiber,” Optics express, vol. 23, no. 13,
pp. 17120–17126, 2015.
[129] G. Milione, E. Ip, M.-J. Li, J. Stone, G. Peng, and T. Wang, “Mode crosstalk matrix
measurementofa1kmellipticalcorefew-modeopticalfiber,” Optics letters, vol.41, no.12,
pp. 2755–2758, 2016.
[130] H. Takahashi, D. Soma, S. Beppu, and T. Tsuritani, “Digital signal processing for space-
division multiplexing (sdm) transmission,” in 2019 IEEE Photonics Conference (IPC),
pp. 1–2, IEEE, 2019.
[131] H. Song, H. Song, R. Zhang, K. Manukyan, L. Li, Z. Zhao, K. Pang, C. Liu, A. Al-
maiman, R. Bock, et al., “Experimental mitigation of atmospheric turbulence effect using
pre-signal combining for uni-and bi-directional free-space optical links with two 100-gbit/s
oam-multiplexed channels,” Journal of Lightwave Technology, vol. 38, no. 1, pp. 82–89,
2020.
102

Abstract (if available)

Abstract In addition to the fundamental Gaussian mode, a laser beam can be tailored to occupy higher-order orthogonal spatial modes, such as the Laguerre-Gaussian (LG) modes, thus called ”structured light”. The unique spatial profile of an LG beam is typically described by two modal indices: (i) azimuthal index l refers to the number of the 2π phase shift along the azimuthal direction of the phase profile; and (ii) radial index p + 1 leads to the number of intensity ring structure in the intensity profile. Moreover, the orthogonality among LG beams ensure that they can be efficiently multiplexed, co-propagation, and demultiplexed with little inherent crosstalk.
With respect to optical systems in different applications, communicating media are never perfect and are likely to generate random distortions to the data-carrying beams. From a spatial modal point of view, the random media can cause power coupling from the transmitted LG mode to many other LG modes. Depending on system architecture, such distortions can lead to either modal-coupling loss or inter-channel crosstalk, both of which are detrimental to high-capacity optical transmission.
This thesis will discuss various optical/electrical compensation techniques to enable high- capacity and spectral-efficient optical communications in various random media, including (i) free-space optical communications in atmospheric turbulence; (ii) mode-division-multiplexing transmission in scattering media; and (iii) mode-division-multiplexing fiber communications in few-mode fiber.

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Asset Metadata

Creator Zhang, Runzhou (author)

Core Title High-capacity optical communications using structured light in random and disturbed media

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2022-08

Publication Date 05/10/2022

Defense Date 04/13/2022

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

Tag atmospheric turbulence,coherent communications,multi-mode fiber.,OAI-PMH Harvest,optical communications,random media,structured light

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Willner, Alan (committee chair), Jenkins, Keith (committee member), Shakeshaft, Robin (committee member)

Creator Email runzhou@usc.edu,zhangrunzhou.zhezhe@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111307120

Unique identifier UC111307120

Legacy Identifier etd-ZhangRunzh-10702

Document Type Dissertation

Format application/pdf (imt)

Rights Zhang, Runzhou

Type texts

Source 20220510-usctheses-batch-941 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu