University of Southern California Dissertations and Theses

Free-space optical communications and probing through turbulent links using structured light

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Free-space optical communications and probing through turbulent links using structured light

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Content Free-Space Optical Communications and Probing through Turbulent Links using
Structured Light
by
Huibin Zhou
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL AND ELECTRONICS ENGINEERING)
May 2024
Copyright 2024 Huibin Zhou

This dissertation is dedicated to my loving girlfriend, Yating Xiang, my supportive parents, Yuanqiao Zhou
and Heguang Wang, and my respectful advisor, Prof. Alan Willner.
ii

Acknowledgements
Firstly, I would like to express my greatest appreciation to my Ph.D. advisor, Prof. Alan E. Willner. I feel
very lucky that I can have the chance to work with Prof. Willner during my Ph.D study. Prof. Willner is
the best mentor in the world and always gives kind support and valuable advice. I have learned so much
from him about how to become a professional researcher and a great person. I believe the lessons and
wisdom from him can be extremely helpful in my future life journey. I would also like to thank Prof. Moshe
Tur at Tel Aviv University for his insightful discussions and extensive support to my research. Moreover,
I would like to thank Prof. Todd A. Brun and Prof. Stephan W. Haas for serving on my dissertation and
qualification exams. I would also like to thank Prof. Andreas F. Molisch and Prof. Wei Wu for serving on
my qualification exam.
Secondly, I want to thank my colleagues in OCLab, including Dr. Ahmed Almaiman, Dr. Long Li, Dr.
Yinwen Bao, Dr. Peicheng Liao, Dr. Zhe Zhao, Dr. Cong Liu, Dr. Ahmad Fallhpour, Dr. Kai Pang, Dr.
Runzhou Zhang, Dr. Haoqian Song, Dr. Hao Song, Dr. Kaiheng Zou, Mr. Xinzhou Su, Mr. Amir Minoofar,
Mr. Narek Karapetyan, Mr. Yuxiang Duan, Mr. Zile Jiang, Mr. Murale Ramakrishnan, Mr. Stanley Ko,
Mr. Abdulrahman Alhaddad, Ms. Yingning Wang, Mr. Ruoyu Zeng, Ms. Yue Zuo, Mr. Hongkun Lian,
and Mr. Zixun Zhao. Working together with them is highly enjoyable. I have learned a lot from them and
I’m proud to be one of the Oclab family.
Thirdly, I would also like to acknowledge my collaborators, including Prof. Robert W. Boyd at University
of Rochester, Prof. Eric Johnson at Clemson University, Prof. Daeyoung Park at Inha University, Dr.
Brittany Lynn at Naval Information Warfare Center Pacific, Dr. Robert Bock at R-DEX Systems inc., Dr.
Hirofumi Sasaki and Dr. Doohwan Lee at NTT, and Dr. Adam Heiniger at TOPTICA Photonics.
iii

Last but not least, I would like to thank my girlfriend Yating Xiang and my parents, Yuanqiao Zhou and
Heguang Wang, for always helping and supporting me in my life during my Ph.D study.
iv

Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List Of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Single- and Multi-Channel Free-Space Optical Links . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Polarization-division multiplexing (PDM) . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Wavelength-division multiplexing (WDM) . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Mode-division multiplexing (MDM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Different structured beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Laguerre-Gaussian (LG) modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Hermite-Gaussian (HG) modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Bessel-Gaussian (BG) modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Degradation effects for beams propagating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Hazy environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.4 Beam divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Digital Modulation Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Data Transceivers in mid-IR FSO Link Demonstrations . . . . . . . . . . . . . . . . . . . . . 8
1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 2: Pilot-Assisted Self-Coherent Turbulence-Resilient Free-Space Optical Communications . . 12
2.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Concept of Turbulence-Resilient Self-Homodyne FSO links . . . . . . . . . . . . . . . . . . . . 14
2.3 Demonstration of Turbulence-Resilient Self-Homodyne FSO links . . . . . . . . . . . . . . . . 18
2.4 Concept of self-coherent turbulence-resilient multi-channel FSO links . . . . . . . . . . . . . . 21
2.5 Demonstration of self-coherent turbulence-resilient multi-channel FSO links . . . . . . . . . . 24
2.6 Concept of self-coherent turbulence-resilient MDM FSO links using a PD array . . . . . . . . 32
2.7 Demonstration of self-coherent turbulence-resilient MDM FSO links using a PD array . . . . 35
Chapter 3: Phase-Conjugation-Based Turbulence Mitigation for Coherent FSO links . . . . . . . . . 40
3.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Concept and Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Experimental Results for Turbulence Mitigation in a Coherent FSO Link . . . . . . . . . . . 44
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Chapter 4: Probing Longitudinal Atmospheric Turbulence Strength using Structured Light . . . . . . 49
4.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Concept of Longitudinal Atmospheric Turbulence Probing . . . . . . . . . . . . . . . . . . . . 51
4.3 Simulation of Longitudinal Atmospheric Turbulence Probing . . . . . . . . . . . . . . . . . . 56
4.4 Experiment of Longitudinal Atmospheric Turbulence Probing . . . . . . . . . . . . . . . . . . 59
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 5: Mid-IR Coherent Free-Space Link through Fog . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Concept of OPO-based Mid-IR data transmitter . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
vi

List Of Figures
1.1 System diagrams for (a) a single-channel FSO communication link and (b) a multi-channel
FSO link, in which multiple data channels are multiplexed and transmitted simultaneously.
(c) Various channel multiplexing schemes in FSO communications links. . . . . . . . . . . . . 2
1.2 Normalized intensity and phase profiles (as shown in insets) for different spatial modes: (a)
LGℓ,p, (b) HGm,n, and (c) BGℓ,kr with various mode indices. . . . . . . . . . . . . . . . . . . 3
1.3 Various degradation effects in FSO communication links. . . . . . . . . . . . . . . . . . . . . . 5
1.4 Atmospheric transmission spectrum. There are two transmission windows with relatively
higher transmission in the mid-IR region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 (a) The data signal can be modulated on the amplitude and/or phase of an optical wave. (e.g.,
on-off keying (OOK), quadrature phase shift keying (QPSK), and 16-quadrature amplitude
modulation (16-QAM)). (b) The constellation and the error vector of a QPSK signal. . . . . 7
1.6 System diagrams for (a) a single-channel FSO communication link and (b) a multi-channel
FSO link, in which multiple data channels are multiplexed and transmitted simultaneously.
(c) Various channel multiplexing schemes in FSO communications links. . . . . . . . . . . . . 9
1.7 System diagrams for (a) a single-channel FSO communication link and (b) a multi-channel
FSO link, in which multiple data channels are multiplexed and transmitted simultaneously.
(c) Various channel multiplexing schemes in FSO communications links. . . . . . . . . . . . . 10
2.1 Overview of the topics related to the pilot-assisted self-coherent turbulence-resilient FSO communications that will covered in this chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 (a) Atmospheric turbulence-induced modal power coupling can significantly decrease the mixing efficiency between the data and LO and cause data quality degradation. (b) Concept of
a previous approach using pilot-assisted self-heterodyne detection to increase the turbulence
resilience of FSO links by automatically mitigating the turbulence-induced modal coupling
effects. There is a frequency offset between the data and pilot for a guard band to avoid
signal-to-signal beating (SSB) interference. As a result, the bandwidth of the detector is not
fully utilized (e.g., the used bandwidth of the PD is around 2x the data channel’s bandwidth).
(c) Concept of turbulence-resilient FSO link using pilot-assisted self-homodyne detection. By
co-transmitting an orthogonally polarized pilot tone together with the data, the pilot-assisted
self-homodyne detection can automatically mitigate the turbulence-induced modal coupling
effects. Since there is no frequency offset between the data and pilot, the resultant data-pilot
mixing term is located at baseband frequency (f = 0) with higher utilization of detector’s
bandwidth compare with the pilot-assisted self-heterodyne approach. . . . . . . . . . . . . . 15
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2.3 Detailed implementation of the pilot-assisted self-homodyne detection at the Rx for recovering
in-phase (I) and quadrature (Q) data information. All components are free space coupled,
which can better support turbulence-distorted multi-mode beam as compared to single-mode
fiber (SMF)-coupled components. HWP: half-wave plate; BS: beam splitter; QWP: quarterwave plate; PBS: polarization beam splitter; FS-PD: free-space coupled photodiode. . . . . . 16
2.4 Experimental setup for a 12-Gbit/s 16-QAM turbulence resilient FSO link using pilot-assisted
self-homodyne detection. We compare the performance with LO-based homodyne or intradyne
detection. AWG: arbitrary waveform generator; EDFA: Erbium-doped fiber amplifier; PC:
polarization controller; FM: flip mirror; HWP: half-wave plate; BS: beam splitter; QWP:
quarter-wave plate; PBS: polarization beam splitter; FS-PD: free-space coupled photodiode;
SMF: single-mode fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 We compare three different detection schemes in our experiments, including (a) pilot-assisted
self-homodyne detection, (b) LO-based coherent homodyne detection, and (c) LO-based coherent intradyne detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Experimentally measured beam profiles and LG modal spectra for the data beam on the X
pol. and the pilot beam on the Y pol. under (a) no turbulence, (b) one weaker turbulence
realization, and (c) one stronger turbulence realization. Recovered 16-QAM data constellation
diagrams and EVM performance for the pilot-assisted self-homodyne, LO-based homodyne,
and LO-based intradyne detection are also shown under these turbulence conditions. . . . . . 22
2.7 (a) Turbulence-induced modal coupling could cause channel crosstalk between transmitted
data channels carried by two OAM modes after mode demultiplexing and LO-based coherent
detection. (b) Concept of a single-channel FSO link using pilot-assisted O/E beam mixing.
Mode components of the data beam could be efficiently mixed with the corresponding component of the pilot beam. (c) Concept of the turbulence-resilient OAM-multiplexed data
transmission link using two OAM beams ℓ1 and ℓ2. Two additional CW pilot tones located
at the frequency difference away from data channel’s frequency are used to automatically
compensate the turbulence-induced modal coupling through O/E mixing. DC: direct current;
SSB: signal-signal beating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Experimental setup for the two-OAM-beam multiplexed data transmission through emulated
turbulence. The inset indicates the generated OAM beam profiles without any emulated
turbulence effects. AWG: Arbitrary Waveform Generator; EDFA: Erbium-doped Fiber Amplifier; PC: Polarization Controller; BPF: Band Pass Filter; M: Mirror; FM: Flip Mirror; BS:
Beam Splitter; HWP: Half-Wave Plate; SLM: Spatial Light Modulator; DSP: Digital Signal
Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.9 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 conventional LO-based heterodyne coherent detection. (c)
Measured inter-channel crosstalk using pilot-assisted O/E beam mixing. (d) Measured signal power loss for OAM beams ℓ = −2, −1, 0, +1, +2 by using the conventional LO-based
heterodyne detection in (d1) and the pilot-assisted O/E beam mixing in (d2). . . . . . . . . . 27
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2.10 Experimental results of 4-Gbit/s OAM-multiplexed data transmission under different strengths
of the emulated turbulence. (a1) 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. (a2) The resultant electrical spectrum using the pilot-assisted O/E OAM beam mixing
at the receiver. (b) Comparison of the received 2-Gbaud QPSK constellation diagram using
the conventional LO-based heterodyne detection and O/E beam mixing. Measured EVM values under two different turbulence strength with the Fried parameters r0 of 1.0 mm and 0.4
mm in (c1) and (c2), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.11 (a) Concept of the turbulence resilient link combining OAM (ℓ1 and ℓ2), polarization (pol. X
and Y) and wavelength (λ1 and λ2) multiplexing. For each OAM mode, two pilot tones are
located at different polarizations with different ∆f. The turbulence-induced distortions are
automatically mitigated by the pilot-assisted O/E mixing. (b) The transmitter side of the
experimental setup for the turbulence resilient link combining two OAM, two polarization,
and two wavelength multiplexing. The receiver side of the experimental setup is the same as
the setup shown in Fig. 2.8. AWG: Arbitrary Waveform Generator; EDFA: Erbium-doped
Fiber Amplifier; PC: Polarization Controller; PBC: Polarization Beam Combiner; BPF: Band
Pass Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.12 Normalized crosstalk (in dB scale) between different data channels for λ1 and λ2 using (a)
the conventional LO-based heterodyne detection and (b) the pilot-assisted O/E beam mixing
without and with weaker or stronger turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.13 (a) Optical spectra of the transmitted pilots and data channels carried by the OAM ℓ1 = +1
and OAM ℓ2 = −2. (b) Electrical spectrum with eight received data channels after the wave
mixing of the pilot tone and data channels. (c) Beam profiles, data channel constellations and
EVM performance (under the corresponding constellations) for the conventional LO-based
heterodyne detection and the pilot-assist O/E beam mixing. . . . . . . . . . . . . . . . . . . . 32
2.14 (a) Conventional LO-based coherent MDM FSO system can be significantly degraded by
turbulence. (b) Our proposed pilot-assisted self-coherent MDM FSO system using a PD
array. Turbulence-induced wavefront distortion is “automatically” mitigated by its conjugate
during the pilot-data beam mixing at the PD array. . . . . . . . . . . . . . . . . . . . . . . . 33
2.15 Extracting the 4 data channels by applying a matrix on the signals detected by 4 sub-PDs
in DSP. (a) An example for a Gaussian pilot beam and 4 multiplexed channels carried by
OAM ℓ = −1, 0, +1, +2 (b) Different relative phase delays detected by different sub-PDs after
the pilot-data mixing. (c) Detailed implementation of the matrix multiplication for channel
demultiplexing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.16 Setup for an MDM FSO link with four 1-Gbuad QPSK channels. Three receivers are compared: (a) a conventional LO-based MDM coherent receiver, (b) a LO-based PD-array Rx,
and (c) our proposed pilot-assisted self-coherent PD-array Rx. AWG: arbitrary waveform
generator, PC: polarization controller, EDFA: erbium-doped fiber amplifier, M: mirror, FM:
flip mirror, BS: beam splitter, Col.: collimator. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.17 Beam profiles, crosstalk (XT), data constellations, and EVM with and without a weaker/stronger
turbulence realization. Pilot-assisted PD-array system is much less degraded due to the “automatic” turbulence mitigation. The transmitted power of the four data channels is ∼0 dBm. 37
2.18 EVMs under 8 different stronger turbulence realizations with r0=1 mm for the three different
receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ix

2.19 BER of data Ch3 (OAM ℓ = +1) with and without multiplexing and turbulence (the stronger
turbulence case in Fig. 2.17). Tur.: turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.20 BER of the four channels with and without turbulence (the stronger turbulence case in Fig.
2.17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 (a) Coherent FSO links can be significantly degraded by turbulence-induced beam distortion
and modal coupling. (b) Our approach using DFWM-based phase conjugation for automatically mitigating turbulence. (c) Detailed process of the DFWM-based phase conjugation.
The distorted probe beam and a Gaussian-like reference beam interfere inside the crystal to
“record” turbulence distortion. A Gaussian data beam is used to “read” the distortion and
generate a phase conjugate data beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Experimental setup for automatic turbulence mitigation using phase conjugation in a 2-Gbit/s
QPSK coherent FSO link. AWG: arbitrary waveform generator; BS: beam splitter; Col.:
collimator; FM: flip mirror; HWP: half-wave plate; Iso.: isolator; M: mirror; PC: polarization
controller; PD: photodiode; SLM: spatial light modulator; SMF: single-mode fiber; YDFA:
Ytterbium-doped fiber amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Experimentally measured intensity profiles of the probe, reference, and data beams under one
turbulence realization (D/r0∼4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Beam profiles, modal spectra, and recovered data constellations of the received data beam. . 46
3.5 (a1-b1) Turbulence-induced LO-data mixing loss, (a2-b2) EVMs, and (a3-b3) data constellations of the received QPSK signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Measured efficiency of the crystal for converting the input Gaussian data beam to the output
phase conjugate data beam. The three turbulence realizations shown in this figure correspond
to the same realizations shown in Fig. 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1 (a)A general scheme of using forward-propagating optical beams for probing turbulence along
a path. At the Rx, beam-turbulence interactions are measured for retrieving turbulence
information. (b) One prior turbulence probing technique by transmitting two probe beams
from two separate sources and detecting them at a multi-element Rx aperture array. (c)
Our proposed approach designs and sequentially transmits multiple longitudinally structured
beams, each having its narrow beam width at a different z along the propagation path, using
a single pair of Tx/Rx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 (a) Three equal-length turbulence regions in simulation with each region having a different turbulence strength of C
2
n,j , j = 1, 2, 3. (b) Simulated beam widths of the three beams designed
for turbulence probing (Di,j for each beam i in each region j). (c) Simulated average modal
spectrum for the three beams at the Rx under 200 turbulence realizations. The bars show
the standard deviation for the simulation results. (d) Normalized average power remained on
the ℓ = 0 for the three beams (e.g., Pi (ℓ = 0) for Beam i) in the simulation. (e) Constructed
equations for retrieving the turbulence distribution. (f) Probed turbulence distribution in
simulation. The simulated relative probing error is ∼4% (∼0.2 dB from the original value)
for this example of probing 3 turbulence regions. . . . . . . . . . . . . . . . . . . . . . . . . . 57
x

4.3 Simulation results for probing a “Gaussian-shaped” turbulence distribution, having the peak
turbulence strength located at different distances, z, in a 10-km path (e.g., z=0, 2.5, 5, 7.5,
10 km). (a) Received beam profiles of Beam 1, 20, and 40 for probing the “Gaussian-shaped”
turbulence distribution with the peak located at the middle of the path (z=5 km) under one
turbulence realization. (b) Average P (ℓ = 0) for the 40 beams under 200 turbulence realizations for different “Gaussian-shaped” turbulence distributions. The bars show the standard
deviation of the simulation results for each beam. (c) The original turbulence strength distribution and its simulated probed values using the 40 probe beams (in both logarithmic and
linear ordinate scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 (a) Simulation results for probing turbulence with various distributions, including (a-b) linear
changing distributions, (c-d) “triangular-shaped” distributions, (e-f) “sin-shaped” distributions, and (d) a distribution based on the Hufnagel-Valley turbulence model describing the
atmospheric turbulence distribution at different altitudes. . . . . . . . . . . . . . . . . . . . . 59
4.5 (a) Experimental setup for probing two emulated turbulence regions using longitudinally structured beams. (b-c) Simulated and experimentally measured intensity profiles of Beam 1 and 2
at different propagation distances. (d) Simulated and experimentally measured beam widths
for Beam 1 and 2. (e) Experimentally measured beam profiles and modal spectrum of Beam
1 and 2 under one turbulence realization for 4 different cases where turbulence regions 1 and
2 have different Fried parameters r0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 (a) Simulated and experimentally measured P (ℓ = 0) for Beam 1 and 2 under 200 turbulence
realizations for the 4 different turbulence distribution cases. (b) Original turbulence and
probed turbulence in simulation and experiment. To directly compare probed turbulence
strengths with the emulated ones, we show the probing results in terms of the Fried parameters
r0 instead of the C
2
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1 (a) Comparison between Near-IR and Mid-IR FSO communication links through harsh fog
effects. (b) Concept of using an OPO-based data transmitter for a coherent Mid-IR FSO link.
Tx: transmitter; Rx: receiver; Mod.: modulator. . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Different types of Mid-IR data transmitters, including native Mid-IR devices, PPLN-based
DFG, and OPO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Channel data rate and link length for Mid-IR FSO link demonstrations of using different
types of Mid-IR transmitters. Our demonstration is a 3-m link with emulated fog effects that
induce up to ∼18-dB Mid-IR power loss (the same fog induces ∼40-dB power loss for ∼1550-
nm). Such a fog-induced attenuation can roughly correspond to a 100-m foggy link with 50-m
visibility or a 4-km link with 1-km visibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 (a) Experimental setup of a coherent Mid-IR communication link through fog using TOPO
to generate a Mid-IR QPSK data channel in the Tx. For comparison, we generate a separate
∼1550-nm QPSK data beam and transmit it through the foggy link along the similar path
as the MIR data beam. AWG: arbitrary waveform generator, Col.: collimator, DM: dichroic
mirror, EDFA: erbium-doped fiber amplifier, FM: flip mirror, HPF: high-pass filter, HWP:
half-wave plate, LO: local oscillator, M: mirror, PC: polarization controller, PPLN: periodically poled lithium niobate, YDFA: Ytterbium-doped fiber amplifier. Optical spectra for (b)
CW laser at 1064 nm as input to TOPO, (c) the Mid-IR CW carrier generated by TOPO,
(d) the Mid-IR 5-Gbaud QPSK data channel generated by TOPO, and (e) the QPSK data
channel that is converted to the C band at the Rx. . . . . . . . . . . . . . . . . . . . . . . . . 71
xi

5.5 (a) Measured fog-induced power loss for Near-IR light at 1550-nm and Mid-IR light at ∼3400
nm. (b) Pictures for a green-light beam propagating through three different fog conditions
measured in (a). The green light serves as a guide light to help to visualize the fog strength. . 72
5.6 Data constellations and error vector magnitudes (EVMs) for (a) 2-Gbaud and (b) 5-Gbaud
Mid-IR QPSK data channel without fog and with different fog conditions. (c) Data constellations and EVMs for 5-Gbaud Near-IR QPSK data channel. To compare the degradations
induced by fog, we intently control the Near-IR and Mid-IR links to have a similar EVM
performance with no fog as shown in (b) and (c). . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.7 BERs of the 5-Gbaud Mid-IR and Near-IR QPSK data channel under different fog. Three
data points for Mid-/Near-IR corresponds to constellations in Fig. 5.6(b)/(c), respectively. . 74
5.8 Data constellations and EVMs for (a) 1-Gbaud and (b) 2-Gbaud Mid-IR 8-PSK data channels
without fog effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
xii

Abstract
As compared to radio systems, free-space optical (FSO) communication systems hold the promise of (i) higher
capacity due to a larger spectrum, and (ii) lower probability of intercept due to the beam’s lower divergence
and higher directionality. Many FSO communication systems transmit data-carrying fundamental Gaussian
beams. However, recent studies have shown various potential benefits of using spatially structured optical
beams for FSO links.
In general, a structured beam refers to the tailoring of the spatial distribution of an optical beam’s
amplitude and phasefront to exhibit unique properties. There are various types of structured beams (e.g.,
Laguerre-Gaussian (LG), Bessel-Gaussian (BG) beams). One important application of structured beams in
FSO links is to utilize their orthogonality to enable the multiplexing of multiple independent data-carrying
beams for enhancing total system capacity; this is often referred to as mode-division multiplexing (MDM).
Data-carrying beams in FSO communications can be affected by different channel impairments. One
key impairment is atmospheric turbulence, which can cause power loss and intermodal power coupling that
can significantly degrade system performance. On the one hand, various approaches have been developed to
mitigate such turbulence effects to increase the performance of the link. On the other hand, such information
on light-turbulence interaction can also be utilized for sensing applications.
This thesis will discuss (i) pilot-assisted self-coherent free-space optical communication systems that
are highly resilient to turbulence degradation, (ii) photorefractive-crystal-based optical phase conjugation to
mitigate turbulence in coherent FSO links, (iii) probing turbulence strength distribution along a propagation
path using longitudinally structured beams, and (iv) optical parametric oscillator (OPO)-based high-power
data transmitter for a mid-IR coherent free-space link through fog.
xiii

Chapter 1
Introduction
1.1 Single- and Multi-Channel Free-Space Optical Links
Free-space optical (FSO) communications has attracted increased attention over the past several years [1, 2].
In general, optical beams tend to support wider spectral bandwidth and have higher propagation directionality when compared to radio-frequency (RF) waves [1]. Therefore, FSO links have the potential for higher
data rates and reduce the probability of detection by eavesdroppers as compared to RF links [1]. In general,
the total data rate of an optical communication system can be enhanced by transmitting multiple independent data channels simultaneously. Different channel multiplexing schemes have been demonstrated in FSO
communication links. Figure 1.1(a) shows a system diagram for a single-channel FSO communication link,
in which a single data channel is transmitted from the transmitter to the receiver. In a multi-channel FSO
link (see Fig. 1.1(b)), multiple data channels can be multiplexed and transmitted simultaneously through a
free-space link. As shown in Fig. 1.1(c), various channel multiplexing schemes have been demonstrated in
FSO links, including the following:
1.1.1 Polarization-division multiplexing (PDM)
In a PDM scheme, two orthogonal polarizations are utilized to carry two independent channels. Therefore,
the system data rate can be doubled by using PDM. Moreover, since the two channels are located at the
same optical wavelength, the spectral efficiency (i.e., bit/s/Hz) of the system can also be doubled [3, 4, 5].
1

1.1.2 Wavelength-division multiplexing (WDM)
WDM can be utilized to further enhance the total data rate of the link. In a WDM link, each data channel
is located at a different optical wavelength. Compared to PDM with a doubled data rate, WDM can scale
the data rate by a factor of the number of transmitted channels [6, 7, 8, 9].
1.1.3 Mode-division multiplexing (MDM)
In a single-channel FSO link, the channel is usually carried on a fundamental Gaussian spatial mode. In
an MDM scheme, multiple data channels are carried on different orthogonal spatial modes (which can be
higher-order modes) and transmitted simultaneously from a transmitter aperture to a receiver aperture
[10, 11, 12, 13]. Similar to PDM, MDM data channels occupy the same optical spectrum band. Therefore,
MDM can increase both the data rate and spectral efficiency of the system by a factor of N with N
multiplexed data channels. We note that MDM is a subset of space-division multiplexing [12, 13].
Figure 1.1: System diagrams for (a) a single-channel FSO communication link and (b) a multi-channel FSO
link, in which multiple data channels are multiplexed and transmitted simultaneously. (c) Various channel
multiplexing schemes in FSO communications links.
2

Each of the above-mentioned multiplexing schemes is compatible and can be combined with others to
further increase the number of multiplexed data channels and enhance the total data rate [14, 15, 16, 17, 18].
For example, in a system combining WDM and MDM, each wavelength can carry multiple data channels
that are multiplexed in different spatial modes [17, 18].
1.2 Different structured beams
Based on Maxwell’s equations, electromagnetic fields of a free-space optical beam can be solved by the
Helmholtz equation [19, 20, 21]. In different coordinates, different types of solutions can be obtained. These
solutions can form the different modal basis for free-space optical beams [20]. Here, we introduce three types
of structured modal beams (LG, HG, and BG beams).
Figure 1.2: Normalized intensity and phase profiles (as shown in insets) for different spatial modes: (a)
LGℓ,p, (b) HGm,n, and (c) BGℓ,kr with various mode indices.
3

1.2.1 Laguerre-Gaussian (LG) modes
In cylindrical coordinates, the solutions of the Helmholtz equation are known as LG modes [19, 20]. Figure
1.2(a) shows beam profiles of some LG beams with different mode indices. LG modes can be characterized by
two modal indices: (1) azimuthal index ℓ, which is the number of 2π phase shifts in the azimuthal direction
(i.e., orbital angular momentum (OAM) values carried by the beam), and (2) radial index p, related to the
number of concentric rings (i.e., p + 1 rings) in the intensity profile [22, 23, 21]. When ℓ = p = 0, the solution
reduces to the fundamental Gaussian mode. In theory, an LG beam is orthogonal to other LG beams with
different ℓ and/or p [24, 25].
1.2.2 Hermite-Gaussian (HG) modes
HG modes are solutions of the Helmholtz equation in Cartesian coordinates [19, 20]. HG modes can be
described by using two spatial indices (m, n) referring to the x and y directions, respectively. Different
from LG modes which have a circular symmetry, HG modes have a rectangular symmetry, as shown in Fig.
1.2(b). The HGm,n mode has m + 1 maxima in the x direction and n + 1 in the y direction [19, 20]. The
lowest-order HG mode, HG0,0, is simply the Gaussian mode. HG modes with different mode indices (m
and/or n) are orthogonal with each other [20, 11].
1.2.3 Bessel-Gaussian (BG) modes
BG modes are another type of solution in cylindrical coordinates [26, 20, 27]. Their transverse spatial
intensity distribution is characterized by the Bessel function with multiple rings [26, 20, 27]. Similar to LG
modes, BG modes also have the azimuthal index (ℓ) which refers to the OAM value carried by the mode [27].
However, instead of the discrete radial index (p) of the LG modes, BG modes have a continuous radial wave
vector, kr, that determines the spacing of the rings in the radial direction [27]. BG modes with different
values of ℓ are orthogonal to each other [28]. However, the orthogonality between BG modes with different
kr is not generally satisfied [28]. Compared to LG and HG modes, there are two unique properties of BG
modes: (i) they are near diffraction-free within a propagation range, and (ii) they can reform their beam
shape after being disrupted by an obstacle in the beam path [27, 29, 30, 31].
4

1.3 Degradation effects for beams propagating
For optical beams propagating through the atmosphere, there can be different degradation effects, such
as atmospheric attenuation, turbulence, hazy environments, and beam divergence, as shown in Fig. 1.3.
Moreover, these effects are wavelength-dependent and may need to be carefully evaluated when choosing the
wavelength for an FSO communication system. In this subsection, we will introduce these effects and discuss
how they affect the system differently for different wavelengths (e.g., mid-IR and near-IR wavelengths).
Figure 1.3: Various degradation effects in FSO communication links.
1.3.1 Attenuation
Atmospheric attenuation is an important factor that can affect the performance of an FSO link. Such
attenuation can be caused by the absorption of water vapor and other molecules [32]. Figure 1.4 shows the
atmospheric transmission spectrum [33]. In the mid-IR regions, there are two transmission windows that have
relatively higher transmission rates. Within the mid-IR transmission windows, some mid-IR wavelengths
have lower atmospheric attenuation than other near-IR wavelengths (e.g., C-band). Therefore, given the
same transmitted optical power and link distance, more mid-IR signal power can be received to help with
better mid-IR link performance compared to near-IR links.
5

Figure 1.4: Atmospheric transmission spectrum. There are two transmission windows with relatively higher
transmission in the mid-IR region.
1.3.2 Turbulence
Atmospheric turbulence can induce dynamic wavefront distortion of the transmitted optical beam [34, 20, 35].
As a result, various degradation effects can be caused, including power loss, scintillation, beam wandering,
and modal power coupling (i.e., power coupling from the transmitted spatial mode to other modes) [34,
20, 35]. In general, these effects can significantly affect the performance of FSO links (both single- and
multi-channel links) [36, 37, 38]. It has been investigated that the turbulence effects are dependent on the
wavelength and tend to be weaker for the mid-IR wavelengths than for C-band wavelengths [34, 18].
1.3.3 Hazy environment
In FSO links, the propagating beam can experience scattering effects caused by various particles in the
atmosphere (e.g., by fog and smoke) [39, 40, 41]. In hazy link conditions, the beam may suffer from significant
power loss caused by particle scattering. It has been demonstrated that mid-IR light, having wavelengths
larger than those of the particles, experiences less scattering and power loss than near-IR beams with shorter
wavelengths [39, 40, 41].
6

1.3.4 Beam divergence
Given the same transmitted beam size, mid-IR beams tend to have larger beam divergence compared to
C-band beams [24]. Therefore, the size of the received mid-IR beam is larger than the C-band beam after
propagating the same distance. As a result, a larger receiver aperture might be required in a mid-IR link
than in a C-band link in order to capture the beam and reduce the power loss caused by beam truncation
[42]. Moreover, the beam divergence is also related to the spatial mode order. A larger-order beam tends
to have larger beam divergence [24]. Therefore, compared to a single Gaussian-channel link, an MDM link
might need a larger receiver aperture since higher-order modes are utilized.
1.4 Digital Modulation Formats
0 1
Amplitude and Phase Modulation
Amplitude Modulation
Phase Modulation
01 00
11 10
1111 1110 1101 1100
OOK 0 1 1 0 0 1
01 10 00 11
QPSK
011
0
001
1 16-QAM
Aref
Aerror
In-phase (I)
Quadrature (Q)
φ
E = A cos (2πft + φ)
(a) (b)
Constellation and error vector
1011 1010 1001 1000
0111 0110 0101 0100
0011 0010 0001 0000
Figure 1.5: (a) The data signal can be modulated on the amplitude and/or phase of an optical wave. (e.g.,
on-off keying (OOK), quadrature phase shift keying (QPSK), and 16-quadrature amplitude modulation (16-
QAM)). (b) The constellation and the error vector of a QPSK signal.
In general, digital data signals can be modulated on the amplitude and phase of the optical wave in
time domain as shown in Fig. 1.5. For amplitude modulation, the bits in the data streams are mapped to
7

multiple amplitude levels of the optical wave. For example, an on-off keying (OOK) signal has two possible
amplitude levels representing the bit ”0” and ”1”. Since only a single photodiode is required at the receiver
to recover the amplitude information (intensity modulated/direct detection (IM/DD)), such modulation is
usually considered to have a low cost and simple implementation. For the phase modulation, the bits are
mapped to the temporal phase, and the amplitude remains constant. As an example, the quadrature phase
shift keying (QPSK) has four possible phase levels, and one QPSK symbol can carry two bits. In addition,
the amplitude and phase modulation can be utilized simultaneously to increase the number of bits carried by
one symbol. As an example, the 16-quadrature-amplitude-modulation (16-QAM) has 16 possibilities mapped
on the in-phase and quadrature amplitude axes encoding four bits of information. Unlike the amplitudemodulated signals, coherent detection is often used at the receiver to extract the phase information when the
temporal phase is also encoded with data. In a coherent receiver, the received signal is sent to a 90◦ hybrid
along with a continuous-wave (CW) local oscillator (LO) laser and detected by balanced photodiodes.
The quality of the received signal can be characterized by the bit error rate (BER), which is defined as
the ratio between the number of bit errors and the total number of the received bits. Alternatively, error
vector magnitude (EVM) can be used to determine the quality of the received signal. As shown in Fig.
1.5(b), the received symbols are represented by the points on the constellation according to their amplitudes
and temporal phases. Due to the noise and distortions during the transmission, the received symbol may
deviate from the transmitted symbol, resulting in an error vector Aerror. The EVM is calculated by the ratio
between the mean root square of the error vectors Aerror and the reference vector Aref as follows [43],
EVM (%) =
q
1
N
PN
i=0 |Aerror,i
|
2
|Aref|
× 100% (1.1)
1.5 Data Transceivers in mid-IR FSO Link Demonstrations
There are generally two types of methods for generating and detecting mid-IR data channels. One method
is to use native mid-IR devices for the transmitter and receiver. In mid-IR FSO systems using native mid-IR
devices as the transceiver, the data channel is generated by (1) direct modulation of a mid-IR laser (Fig.
1.6(a1)) or (2) external modulation with a mid-IR modulator (Fig. 1.6(a2)) at the transmitter. At the
8

receiver, the mid-IR channel is detected by a mid-IR photodetector. Various mid-IR devices have been
demonstrated in such systems, including (i) laser: the quantum cascade laser (QCL) and interband cascade
laser (ICL), (ii) modulator: the Stark modulator, and (iii) detectors: the quantum cascade detector (QCD),
interband cascade infrared photodetector (ICIP), and mercury cadmium telluride (MCT) detector.
Another type of transceiver demonstrated in mid-IR FSO links is based on wavelength conversion (Fig.
1.6(b)). In such systems, the transmitter consists of a laser and a high-speed modulator operated in other
frequency bands, followed by converting the data signal to the mid-IR region through wavelength conversion.
At the receiver, the mid-IR data channel is not directly detected by a mid-IR detector but converted to other
bands for detection.
Figure 1.6: System diagrams for (a) a single-channel FSO communication link and (b) a multi-channel FSO
link, in which multiple data channels are multiplexed and transmitted simultaneously. (c) Various channel
multiplexing schemes in FSO communications links.
9

Figure 1.7: System diagrams for (a) a single-channel FSO communication link and (b) a multi-channel FSO
link, in which multiple data channels are multiplexed and transmitted simultaneously. (c) Various channel
multiplexing schemes in FSO communications links.
1.6 Thesis Outline
This thesis will mainly discuss free-space optical communications and probing using structured light, as
shown in Fig. 1.7. Specifically, Chapter 2 shows concept and demonstrations of pilot-assisted self-coherent
free-space optical communication systems that are highly resilient to turbulence degradation. Chapter 3
discusses principal and demonstrations of using photorefractive-crystal-based optical phase conjugation to
mitigate turbulence in coherent FSO links. Chapter 4 shows an approach for probing turbulence strength
10

distribution along a propagation path using longitudinally structured beams. Simulation and experimental
demonstration of this approach will be discussed. Chapter 5 shows an experimental demonstration of an
optical parametric oscillator (OPO)-based high-power data transmitter for a mid-IR coherent free-space link
through fog.
11

Chapter 2
Pilot-Assisted Self-Coherent Turbulence-Resilient Free-Space
Optical Communications
2.1 Background and Motivation
Free-space optical (FSO) links have the potential for higher bandwidth and lower probability of intercept
as compared to radio links. In FSO systems, the ability to transmit multiple multiplexed data channels
simultaneously can increase the aggregate data rate. One example is mode-division-multiplexing (MDM), in
which each data beam is tailored to have a different orthogonal spatial mode from a modal basis set. One
type of modal set is the orbital-angular-momentum (OAM). Moreover, in many communication systems, it is
desirable to modulate data on the temporal phase of the beam (e.g., quadrature-phase-shift-keying (QPSK)
and quadrature amplitude modulation (QAM)). Such formats can have a higher optical spectral efficiency
and lower optical signal-to-noise ratio requirements than amplitude-only modulation.
Usually, phase-encoded data can be recovered by mixing a data beam with a local oscillator (LO) in
a coherent receiver. However, turbulence presents a key challenge for coherent FSO systems. Specifically,
turbulence can distort the data beam and couple power from the transmitted mode into other modes, such
that the data beam does not efficiently mix with the LO. Moreover, this challenge significantly grows in
complexity for an MDM system, since modal coupling also causes crosstalk among multiple data channels.
12

There have been various “adaptive” approaches to mitigate the above two challenges from turbulence and
recover phase-encoded data in MDM links, including adaptive optics and adaptive multiple-input-multipleoutput (MIMO) digital signal processing (DSP). However, methods have been reported that can mitigate
turbulence in an “automatic fashion” (i.e., the mitigation process does not need to adapt to changing turbulence conditions). One automatic approach is the co-transmitting of a frequency-offset pilot beam with
each data beam. In this method, each pilot beam suffers a similar turbulence distortion as its corresponding data beam, and a distortion conjugate is “automatically” generated during the pilot-data mixing in a
photodetector (PD) for mitigation of modal-coupling-based degradations. Such a pilot-assisted self-coherent
(self-heterodyne) approach has been previously demonstrated for a single-channel turbulence-resilient FSO
link.
Figure 2.1: Overview of the topics related to the pilot-assisted self-coherent turbulence-resilient FSO communications that will covered in this chapter.
In this chapter, three more advanced demonstrations of the pilot-assisted turbulence resilient approach
will be discussed as shown in Fig. 2.1, including (a) using self-homodyne to increase the bandwidth utilization
of the detector, (b) extending this approach to multi-channel multiplexed FSO links, and (c) using a PD
array as the detector to increase the electrical spectral efficiency.
13

2.2 Concept of Turbulence-Resilient Self-Homodyne FSO links
The performance of FSO coherent detection systems can be significantly degraded by turbulence-induced
modal coupling, as shown in Fig. 2.2(a). At the transmitter (Tx), a data channel is carried by a fundamental
Gaussian beam and transmitted through turbulence. Due to the turbulence-induced modal coupling, the
received data beam will contain many higher-order LG modes. In an LO-based coherent detection (e.g.,
homodyne or intradyne), only the Gaussian mode can efficiently mix with the Gaussian LO and be recovered,
resulting in significant power loss and degradation of the recovered data quality.
Figure 2.2(b) shows the concept of a previous demonstration using pilot-assisted self-coherent (selfheterodyne) detection to increase the turbulence resilience of FSO links [38]. At the Tx, a frequency-offset
pilot CW beam is simultaneously transmitted along with the data beam. The pilot beam and the data beam
are generated by two separate lasers. Both beams experience similar turbulence-induced modal coupling and
thus can efficiently mix all modal orders in a single FS-PD. The center frequency of the data-pilot mixing
term is at f=∼1.5B (B is the bandwidth of the data channel). In order to recover both amplitude- and
phase-encoded data, the data-LO mixing term is filtered out, and the frequency offset between the data and
pilot provides a guard band to avoid SSB interference. As a result, the bandwidth of the detector is not
fully utilized (e.g., the used bandwidth of the PD is ∼2X the data channel’s bandwidth).
Figure 2.2(c) shows the concept of our proposed turbulence-resilient FSO link using pilot-assisted selfhomodyne detection. At the Tx, the output from a CW laser is split into two paths, with one path modulated
with a 16-QAM data channel on X polarization (pol.) and another path used as a pilot tone on Y pol. The
data and pilot beams are simultaneously transmitted through turbulence. Since data and pilot beams on
two polarizations experience similar turbulence-induced modal coupling [34, 44], almost all the LG modes
of the data beam can automatically mix with the pilot beam at the receiver (Rx). Because there is no
frequency offset between the data and pilot, the resultant data-pilot mixing term is located at baseband
frequency (f=0). Finally, the amplitude- and phase-encoded data (or in-phase (I) and quadrature (Q) data
information) can be efficiently recovered by homodyne detection.
14

Figure 2.2: (a) Atmospheric turbulence-induced modal power coupling can significantly decrease the mixing efficiency between the data and LO and cause data quality degradation. (b) Concept of a previous
approach using pilot-assisted self-heterodyne detection to increase the turbulence resilience of FSO links by
automatically mitigating the turbulence-induced modal coupling effects. There is a frequency offset between
the data and pilot for a guard band to avoid signal-to-signal beating (SSB) interference. As a result, the
bandwidth of the detector is not fully utilized (e.g., the used bandwidth of the PD is around 2x the data
channel’s bandwidth). (c) Concept of turbulence-resilient FSO link using pilot-assisted self-homodyne detection. By co-transmitting an orthogonally polarized pilot tone together with the data, the pilot-assisted
self-homodyne detection can automatically mitigate the turbulence-induced modal coupling effects. Since
there is no frequency offset between the data and pilot, the resultant data-pilot mixing term is located at
baseband frequency (f = 0) with higher utilization of detector’s bandwidth compare with the pilot-assisted
self-heterodyne approach.
The implementation of the pilot-assisted self-homodyne detection at the Rx is depicted in Fig. 2.3. The
incoming beam consists of the turbulence-distorted data signal EsU on X pol. and a CW pilot EpU on Y
pol., which can be represented by:
E =




Ex
Ey




=




EsU
EpU




=




[As(t)exp(jωt + θs(t))]U
[Apexp(jωt)]U




(2.1)
15

Figure 2.3: Detailed implementation of the pilot-assisted self-homodyne detection at the Rx for recovering
in-phase (I) and quadrature (Q) data information. All components are free space coupled, which can better
support turbulence-distorted multi-mode beam as compared to single-mode fiber (SMF)-coupled components.
HWP: half-wave plate; BS: beam splitter; QWP: quarter-wave plate; PBS: polarization beam splitter; FSPD: free-space coupled photodiode.
where As (t) and θs (t) are amplitude- and phase-encoded data on the data beam, respectively. Ap is the
amplitude of the CW pilot tone. U is the turbulence-induced modal coupling and can be expressed as [38]
U =
X
ℓ
X
p
aℓ,pLGℓ,p (x, y) (2.2)
where aℓ,p are turbulence-dependent coefficients and LGℓ,p (x, y) represent the LG modes with azimuthal
index ℓ and radial index p. The polarization of the incoming beam is first rotated by 45o using a half-wave
plate (HWP) and subsequently split into two copies using a beam splitter (BS) for recovering in-phase (I)
and quadrature (Q) data information, respectively. For each copy, a polarization beam splitter (PBS) is used
to separate the X and Y pol. components, each containing a pair of data and pilot beams. The data and
pilot on each polarization mix in an FS-PD. The I data information is recovered by differentially detecting
16

the data-pilot O/E mixing on two polarizations. For recovering Q information, an additional quarter-wave
plate (QWP) is used to induce 90o phase delay on the pilot. Specifically, optical signals detected by each
FS-PD [45, 46] are as follows:
E1 = 1/2 (EsU + EpU) (2.3)
E2 =
1
2
(EsU − EpU) (2.4)
E3 =
1
2
(EsU + jEpU) (2.5)
E4 =
1
2
(EsU − jEpU) (2.6)
The output photocurrent I1 (t) from FS-PD 1 is then given as [38]
I1 (t) ∝
Z Z (Es + Ep)UU∗
(Es + Ep)
∗
dxdy
= (Es + Ep) (Es + Ep)
∗
Z Z UU∗
dxdy
=
n
|Es|
2 + |Ep|
2 + 2Re [EsEp
∗
]
o Z Z |U|
2
dxdy
(2.7)
where ∗ and Re[ ] denote the conjugate operation and real part of the complex electrical field, respectively.
The integral over |U|
2
represents the mixing efficiency of the pilot and data beam, and, when the detector
collects almost all the modes, its value is
Z Z |U|
2
dxdy =
X
ℓ
X
p
X
ℓ
′
X
p′
Z Z aℓ,pLGl,p (x, y) a
∗
ℓ
′
,p′LG∗
ℓ
′
,p′ (x, y) dxdy =
X
ℓ
X
p
|aℓ,p|
2 ∼= 1 (2.8)
Therefore, each mode component of the data beam is efficiently mixed with the corresponding component
of the pilot beam. As a result, I1 (t) can be simplified to
I1 (t) ∝ |Es|
2 + |Ep|
2 + 2Re [EsEp
∗
] (2.9)
17

Similarly, the pilot and data can also be efficiently mixed in FS-PD 2 to 4 and their corresponding output
photocurrents are [45, 46]
I2 (t) ∝ |Es|
2 + |Ep|
2 − 2Re [EsEp
∗
] (2.10)
I3 (t) ∝ |Es|
2 + |Ep|
2 + 2Im [EsEp
∗
] (2.11)
I4 (t) ∝ |Es|
2 + |Ep|
2 − 2Im [EsEp
∗
] (2.12)
where Im[ ] denotes the imaginary part of the complex electrical field. After the differential detections, the
output photocurrents corresponding to the I and Q data information are
II (t) = I1 (t) − I2 (t) ∝
q
PpPs (t)cos(θs (t)) (2.13)
IQ (t) = I3 (t) − I4 (t) ∝
q
PpPs (t)sin(θs (t)) (2.14)
where Pp = |Ap|
2
/2 and Ps (t) = |As|
2
/2 are the power of the pilot and data signal, respectively. From
II (t) and IQ (t), we can recover both amplitude- and phase-encoded data. We note that the components
we used for the pilot-assisted self-homodyne detection are all free space coupled multi-mode devices, which
can better support multi-mode beams that are distorted by turbulence as compared to single-mode fiber
(SMF)-coupled components.
2.3 Demonstration of Turbulence-Resilient Self-Homodyne FSO
links
Figure 2.4 shows our experimental setup. At the Tx, Laser 1 is separated into two polarizations by using a
PBS. A 3-Gbaud 16-QAM data channel is modulated on X pol. and subsequently combined with the CW
pilot tone on Y pol. by using a polarization beam combiner (PBC). The pair of data and pilot is coupled
to free space by an optical collimator (Gaussian beam size of diameter 2w0 = 3 mm) and propagate in free
space for a distance of 1 m. We experimentally emulate the turbulence effects using glass plates, whose
refractive index distributions are fabricated to emulate Kolmogorov turbulence power spectrum statistics.
18

Figure 2.4: Experimental setup for a 12-Gbit/s 16-QAM turbulence resilient FSO link using pilot-assisted
self-homodyne detection. We compare the performance with LO-based homodyne or intradyne detection.
AWG: arbitrary waveform generator; EDFA: Erbium-doped fiber amplifier; PC: polarization controller; FM:
flip mirror; HWP: half-wave plate; BS: beam splitter; QWP: quarter-wave plate; PBS: polarization beam
splitter; FS-PD: free-space coupled photodiode; SMF: single-mode fiber.
Three rotatable glass plates with different Fried parameters (r0) of 3.0 mm, 1.0 mm, and 0.4 mm are used;
a smaller r0 means stronger turbulence. Different turbulence realizations are emulated by rotating the glass
plate to different orientations. Off-axis holography is used to measure the amplitude and phase profiles of
the distorted beam and its corresponding LG modal spectrum [47].
At Rx, we compare the performance of three different detection schemes under turbulence, as shown in
Fig. 2.5, including (a) the proposed pilot-assisted self-homodyne detection, (b) LO-based coherent homodyne
detection, in which we split the power of Laser 1 and transmit it through an SMF to Rx as an LO, and (c)
LO-based coherent intradyne detection, in which a separated Laser 2 is used as an LO [48]. The optical
power of the transmitted data beam is controlled to be ∼5 dBm for these three detection schemes. For
the pilot-assisted self-homodyne, the received pilot and data beam are mixed using the detection scheme as
shown in Fig. 2.3, in which four free-space-coupled InGaAs PD (3dB bandwidth of 3.5GHz) are used. We
digitally subtract the signals detected from PD 1 and PD 2 to obtain the I information and subtract the
signals from PD 3 and PD 4 for the Q information. For the LO-based coherent detection (homodyne or
intradyne), the transmitted pilot on the Y pol. is turned off and the received data beam on the X pol. is
coupled into an SMF, amplified by an Erbium-doped fiber amplifier (EDFA), and mixed with the LO using
a coherent I/Q receiver.
We note that, since Laser 1 and Laser 2 are two separate lasers and are not phase-locked to each other,
additional digital signal processing (DSP) is used to mitigate laser frequency offset and phase noise for the
19

Figure 2.5: We compare three different detection schemes in our experiments, including (a) pilot-assisted
self-homodyne detection, (b) LO-based coherent homodyne detection, and (c) LO-based coherent intradyne
detection.
LO-based intradyne detection [49, 50]. Such DSP is also required in previous demonstrations of pilot-assisted
self-heterodyne. However, such DSP can be simplified for homodyne detections (either pilot-assisted or LObased), because a single laser (Laser 1) is used for both data and pilot/LO such that the temporal phase
variations of the data and pilot/LO light wave can be automatically canceled during the data-pilot/LO
mixing [51, 52, 53, 54]. In our experiment, we use tunable optical delay lines to reduce the mismatch of the
path length between the data and pilot/LO to optimize the performance of pilot-assisted self-homodyne and
LO-based homodyne systems [53, 55, 56].
As shown in the electrical spectra detected by FS-PDs for the pilot-assisted self-homodyne detection in
Fig. 2.4, there is no frequency guard band for avoiding SSB interference, and 1.5-GHz bandwidth (roughly
half the symbol rate of the data channel) is utilized for each PD, and the occupied PD’s bandwidth is
20

fully utilized for recovering the data. As a comparison, in previous demonstrations using pilot-assisted selfheterodyne detection, the occupied PD’s bandwidth is roughly twice that of the data channel’s symbol rate,
and only half of the occupied PD’s bandwidth is utilized for recovering the data.
Figure 2.6 shows the beam profiles, turbulence-induced LG modal power coupling, and recovered 16-QAM
constellations under two example realizations of the turbulence (one weaker and one stronger). Under the
case of no turbulence, the data/pilot beam is mainly composed of a single Gaussian mode as shown in Fig.
2.6(a). However, with weaker turbulence (r0= 1.0 mm), the measured LG spectrum shows that the power is
coupled from Gaussian mode to the neighboring LG modes. Figure 2.6(c) shows that the stronger turbulence
effect (r0=0.4 mm) can induce a power loss of >20 dB on Gaussian mode and that power can be coupled to
a large number of other LG modes. Our results also show that data and pilot beams on two polarizations
experience similar turbulence-induced beam distortion and modal power coupling, which implies that the
modal coupling experienced by the pilot can be used to automatically mitigate the data beam’s modal
coupling (i.e., UU∗
). For both realizations, the performance of the pilot-assisted self-homodyne detection is
not severely affected by these turbulence effects and the 16-QAM data can be recovered with error vector
magnitude (EVM) values from 8% to 10%. We also show the recovered 16-QAM data for conventional
LO-based homodyne and intradyne detection. The recovered data can be degraded from EVM values of 8%
without turbulence to over 30% for stronger turbulence. This is due to that the power coupled to higher
order modes is not efficiently mixed with the Gaussian LO for the LO-based detections.
2.4 Concept of self-coherent turbulence-resilient multi-channel FSO
links
Figure 2.7 illustrates the concept of utilizing pilot-assisted optoelectronic beam mixing to achieve the modal
coupling resiliency to atmospheric turbulence in an FSO link. Figure 2.7(a) shows that, for an MDM
FSO link, turbulence-induced modal coupling causes channel crosstalk between transmitted data channels
carried by two OAM modes after the mode demultiplexing and conventional LO-based heterodyne detection.
Figure 2.7(b) describes a single-channel FSO link where the pilot-assisted O/E mixing is utilized to mitigate
21

Figure 2.6: Experimentally measured beam profiles and LG modal spectra for the data beam on the X pol.
and the pilot beam on the Y pol. under (a) no turbulence, (b) one weaker turbulence realization, and (c)
one stronger turbulence realization. Recovered 16-QAM data constellation diagrams and EVM performance
for the pilot-assisted self-homodyne, LO-based homodyne, and LO-based intradyne detection are also shown
under these turbulence conditions.
turbulence-induced modal coupling. At the transmitter, a modulated data signal S(t) and a CW pilot of
amplitude C are carried together by the same OAM mode, with an optical frequency difference of ∆f. This
pair of waves propagates co-axially through the turbulent atmosphere. Sharing the same spatial mode and
of close optical frequencies, they can be assumed to experience the same turbulence-induced modal coupling.
Thus, each mode component of the data beam is efficiently mixed with the corresponding component of
the pilot beam and in the absence of losses the integral over |U|
2
approaches unity. Such a pilot-assisted
22

Figure 2.7: (a) Turbulence-induced modal coupling could cause channel crosstalk between transmitted data
channels carried by two OAM modes after mode demultiplexing and LO-based coherent detection. (b)
Concept of a single-channel FSO link using pilot-assisted O/E beam mixing. Mode components of the
data beam could be efficiently mixed with the corresponding component of the pilot beam. (c) Concept
of the turbulence-resilient OAM-multiplexed data transmission link using two OAM beams ℓ1 and ℓ2. Two
additional CW pilot tones located at the frequency difference away from data channel’s frequency are used to
automatically compensate the turbulence-induced modal coupling through O/E mixing. DC: direct current;
SSB: signal-signal beating.
O/E mixing approach could be extended in an OAM-multiplexed communication system that transmits
multiple OAM beams carrying independent data streams. As shown in Fig. 2.7(c), two data channels at
the same optical frequency are carried by two OAM beams with different OAM orders. Two additional
CW pilot tones located at the frequency difference ∆f1 and ∆f2 away from data channel’s frequency are
also carried by the corresponding OAM beams ℓ1 and ℓ2. All OAM beams are spatially combined and coaxially propagate through the same atmospheric turbulence effects. The two modes may experience different
turbulence-induced modal coupling. However, the square-law detection process, outputting the photocurrents
23

of the two modes at distinct IF frequencies, ∆f1 and ∆f2, gives rise to completely independent turbulence
compensation of each mode. DSP-based off-line digital filtering and in-phase-quadrature (I-Q) demodulation
are then used to independently retrieve the two data streams. In this approach, after the O/E mixing, the
detected data channels are located at different IF frequencies in the electrical domain. Thus, the total
electrical bandwidth of the single detector is divided into different parts for different data channels. We note
that this approach may be favorable in scenarios wherein (i) the bandwidth of each data signal is limited
by the transmitter, and the detector has sufficient bandwidth for multiple data channels, or (ii) a sufficient
number of PDs is not readily available, and a simplified receiver architecture is preferred.
2.5 Demonstration of self-coherent turbulence-resilient multi-channel
FSO links
Figure 2.8 illustrates the experimental setup for the two-OAM-beam multiplexed data transmission link
through emulated turbulence. At the transmitter, a laser with a wavelength of λ0 is modulated with a
1-Gbaud Nyquist-shaped (roll-off factor of 0.1) QPSK signal by an I-Q optical modulator. The modulated
light is subsequently amplified and equally split into two copies, one of which is delayed by a ∼15-m singlemode fiber (SMF) to decorrelate the data sequences. These two copies are then individually combined with
two CW pilot tones at different wavelengths λ1 and λ2, with frequency differences ∆f1 and ∆f2 away from
the wavelength λ0, respectively, commensurate with the data bandwidth. After amplification by erbiumdoped fiber amplifiers (EDFA), these two sets of signal and pilot tones at wavelengths λ1 and λ2 are fed
to an OAM multiplexer to generate co-axial OAM beams ℓ1 and ℓ2, respectively. The optical OAM beams
are generated and simultaneously multiplexed using a commercially available multiple-plane light converter
(MPLC) at the wavelength of 1.55 µm [57]. The inset of Fig. 2.8 indicates the generated OAM beam profiles
with OAM orders ℓ = −2, −1, 0, +1, +2 without any emulated turbulence effects. In this demonstration,
the OAM orders of the two multiplexed OAM beams, ℓ1 and ℓ2, are assigned from these four OAM beams
together with the fundamental Gaussian beam (i.e., ℓ = 0). We experimentally emulate the turbulenceinduced distortion by utilizing thin glass plates whose refractive index distributions are fabricated according
24

to Kolmogorov spectrum statistics [58]. Two rotatable glass phase plates are employed in the experiment,
with Fried parameters r0 of 1.0 mm (weaker turbulence effects) and 0.4 mm (stronger turbulence effects).
After being distorted by the turbulence emulator, the OAM beams propagate a free-space distance of 1 m
and reach the receiver. At the receiver, the distorted OAM beams are equally split into two copies, one of
which is sent to the pilot-assisted O/E beam mixing receiver. For the O/E mixing receiver, the beams are
focused on a free-space InGaAs PD using an aspheric lens (f=16 mm, NA = 0.79). The 3-dB bandwidth of
the FS-PD is 3GHz. As for the other copy, a spatial light modulator (SLM) is set to a demultiplexing phase
pattern to convert one OAM mode of interest into the fundamental Gaussian beam, which is then coupled
into a fiber collimator for signal detection. Another laser at a wavelength λ3 is combined with this signal
and subsequently sent to an SMF-coupled PD for heterodyne detection. To ensure a fair comparison, the
detected electrical signals from the O/E beam mixing and the LO-based heterodyne detection are processed
by the same DSP algorithm and procedures.
Figure 2.9(a) shows measured intensity profiles of the transmission of OAM ℓ = −2, −1, 0, +1, +2 beams
through the emulated weaker and stronger turbulence. The measured normalized inter-channel crosstalk
matrices for one turbulence realization for the LO-based heterodyne detection and pilot-assisted O/E beam
mixing are shown in Fig. 2.9(b) and (c), respectively. It is worth noting that the power values are measured
in the optical and electrical domain for Fig. 2.9(b) and (c), respectively. The inter-channel crosstalk for the
pilot-assisted O/E beam mixing approach is not significantly affected by the turbulence distortions and is
measured to be lower than -25.7 dB under both the weaker and stronger turbulence distortions. However, we
observe severe turbulence-induced crosstalk between the OAM beams for the LO-based heterodyne detection
under the same turbulence effects. We further measure the mode-dependent loss (MDL) of transmitted OAM
beams by using the pilot-assisted O/E beam mixing or LO-based heterodyne detection. As shown in Fig.
2.9(d1), the loss for the LO-based heterodyne detection can be 4.2 dB and 10.1 dB for the weaker and
stronger turbulence effects, respectively. However, as shown in Fig. 2.9(d2), the MDL for the O/E beam
mixing under weaker and stronger turbulence distortions are measured to be 2.5 dB and 5.5 dB, respectively.
The smaller power loss might be due to the O/E beam mixing recovering the modal coupling of different
data channels in the electrical domain.
25

Figure 2.8: Experimental setup for the two-OAM-beam multiplexed data transmission through emulated
turbulence. The inset indicates the generated OAM beam profiles without any emulated turbulence effects.
AWG: Arbitrary Waveform Generator; EDFA: Erbium-doped Fiber Amplifier; PC: Polarization Controller;
BPF: Band Pass Filter; M: Mirror; FM: Flip Mirror; BS: Beam Splitter; HWP: Half-Wave Plate; SLM:
Spatial Light Modulator; DSP: Digital Signal Processing.
We subsequently demonstrate a 4-Gbit/s OAM-multiplexed FSO communication link under different emulated turbulence strengths, with each OAM beam carrying a 1-Gbaud Nyquist-shaped QPSK data stream.
Fig. 2.10 (a1) shows the optical spectra at the transmitter carried by the multiplexed OAM beams, including
1-Gbaud Nyquist-shaped data channels and extra CW pilot tones. The carrier (pilot) to signal power ratio
(CSPR) is 0.80 and 0.86 for the data channel carried by OAM ℓ1 and ℓ2, respectively. The resultant electrical spectrum using the O/E OAM beam mixing at the receiver appears in Fig. 2.10(a2). We observe the
DC component, SSB term, and the transmitted two data channels at two different intermediate frequencies
(IF) ∆f1 and ∆f2 in the electrical domain. Figure 2.10(b) shows the measured distorted spatial profiles of
different sets of the two multiplexed OAM beams ℓ1 and ℓ2 under different turbulence realizations and the
corresponding calculated error-vector magnitudes (EVM) using the pilot-assisted beam mixing approach.
26

Figure 2.9: 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 conventional LO-based heterodyne coherent detection. (c) Measured inter-channel crosstalk
using pilot-assisted O/E beam mixing. (d) Measured signal power loss for OAM beams ℓ = −2, −1, 0, +1, +2
by using the conventional LO-based heterodyne detection in (d1) and the pilot-assisted O/E beam mixing
in (d2).
The EVMs of the data channels using a conventional LO-based heterodyne detection are also shown in
Fig. 2.10(b) for comparison. As shown in Fig. 2.10(b1), both approaches can achieve near-error-free data
transmission without turbulence effects, and the EVMs are measured to be ∼13.4% and ∼14.9% using the
pilot-assisted beam mixing method and ∼15.8% and ∼13.8% using the conventional heterodyne detection
27

for OAM ℓ1 and ℓ2 beams, respectively. The transmitted free-space optical powers are ∼1.6 dBm and ∼3.7
dBm for OAM ℓ1 and ℓ2 beams, respectively. Under the spatial distortion by the emulated turbulence effects,
the OAM beams could barely preserve the standard ring-shaped intensity profiles and a large number of bit
errors occur using the conventional LO-based heterodyne detection. Figures 2.10(b2-b5) show four examples
of turbulence-induced spatial distortion and inter-channel crosstalk for different OAM beams with the OAM
order selected from different combinations. When the OAM beams are distorted by the weaker turbulence,
having a Fried parameter of 1 mm (i.e., D/r0∼3.8 and 4.6), as shown in Fig. 2.10(b2-b3), 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 LO-based heterodyne detection. Under the stronger turbulence
of D/r0∼9.5 and 11.5, as shown in Fig. 2.10(b4-b5), the DSP algorithms could not readily recover the I-Q
information of the data channels due to the higher inter-channel crosstalk (e.g., -4.2 dB in Fig. 2.10(b4)
compared to -7.4 dB in Fig. 2.10(b2) for OAM ℓ1 = +1) induced by the stronger turbulence distortion. However, when the distorted OAM beams are O/E mixed by the FS-PD, the two multiplexed data channels are
not significantly influenced by the turbulence effects, and the measured EVMs exhibit similar performance
to the case of no turbulence effects, albeit with only ∼2% increased EVM.
Figure 2.10(c) summarizes the EVM comparison for different multiplexed OAM sets under 15 different
realizations including the weaker (r0=1 mm in Fig. 2.10(c1)) and the stronger (r0=0.4 mm in Fig. 2.10(c2))
turbulence conditions. We observe that the pilot-assisted O/E beam mixing approach can reduce the EVMs
of the two multiplexed channels by up-to 41.5% and by 6.9%-42.1% for the weaker and the stronger turbulence
distortions, respectively. Furthermore, the O/E 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 PD.
Figure 2.11(a) shows the concept of the turbulence resilient link combining OAM, polarization, and
wavelength multiplexing. At the transmitter, for each OAM mode (ℓ1 or ℓ2), independent data channels
located at different wavelengths (λ1 and λ2) and different polarizations (pol. X and pol. Y) are multiplexed.
28

Figure 2.10: Experimental results of 4-Gbit/s OAM-multiplexed data transmission under different strengths
of the emulated turbulence. (a1) 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. (a2) The resultant
electrical spectrum using the pilot-assisted O/E OAM beam mixing at the receiver. (b) Comparison of the
received 2-Gbaud QPSK constellation diagram using the conventional LO-based heterodyne detection and
O/E beam mixing. Measured EVM values under two different turbulence strength with the Fried parameters
r0 of 1.0 mm and 0.4 mm in (c1) and (c2), respectively.
Two pilot tones on the two polarizations are added with a frequency difference ∆f (∆f1 and ∆f2 for OAM ℓ1,
and ∆f3 and ∆f4 for OAM ℓ2) compared with the data channels. After OAM multiplexing, the pilot and data
channels of the same OAM mode (and polarization) are expected to experience similar turbulence-induced
modal coupling. After propagating through the turbulent atmosphere, the beam spatial profiles of the OAM
beam carrying both the pilot and multiplexed data channels are focused on an FS-PD. At the receiver,
the free-space PD would mix the similarly distorted pilot-carrying and data-carrying beams with the same
polarization and same OAM mode and generate signal-pilot beating terms at different frequency ∆f in the
electrical domain. The turbulence-induced modal coupling on each data OAM beam could be compensated
through pilot-assisted beam mixing. By choosing the proper frequency spacing ∆f, data channels can be
down-converted to different frequencies and separated with little turbulence-induced crosstalk.
Figure 2.11(b) shows the transmitter side of the experimental setup for the turbulence resilient link
combining OAM, polarization, and wavelength multiplexing. The receiver side is the same as the setup
shown in Fig. 5. At the transmitter, two different carriers (λ1 and λ2) are modulated with two data
29

Figure 2.11: (a) Concept of the turbulence resilient link combining OAM (ℓ1 and ℓ2), polarization (pol.
X and Y) and wavelength (λ1 and λ2) multiplexing. For each OAM mode, two pilot tones are located at
different polarizations with different ∆f. The turbulence-induced distortions are automatically mitigated
by the pilot-assisted O/E mixing. (b) The transmitter side of the experimental setup for the turbulence
resilient link combining two OAM, two polarization, and two wavelength multiplexing. The receiver side of
the experimental setup is the same as the setup shown in Fig. 2.8. AWG: Arbitrary Waveform Generator;
EDFA: Erbium-doped Fiber Amplifier; PC: Polarization Controller; PBC: Polarization Beam Combiner;
BPF: Band Pass Filter.
channels independently and multiplexed. After pre-amplification, the wavelength-multiplexed data channels
are split into four parts, and their polarizations are intentionally controlled. Four pilot tones are combined
with the data channels. Polarization beam combiners (PBC) are used to achieve polarization multiplexing.
SMFs are used to induce delays to decorrelate different data channels. Two pilot tone and data combinations
are converted into OAM modes ℓ1 and ℓ2. We choose two OAM modes ℓ1 = +1 and ℓ2 = −2 as an example
to demonstrate the turbulence resilience of the system. At the receiver, the proposed turbulence-resilient
receiver using the pilot-assisted O/E beam mixing and conventional LO-based heterodyne detection are
compared. Since we use a single FS-PD (3dB bandwidth of 3GHz) to receive eight data channels at
different IF frequencies in this experiment, the baud rate for each data channel is set to be 0.25 GBaud.
We measured normalized channel crosstalk using conventional LO-based heterodyne detection with and
without turbulence for data channels located at λ1 and λ2, respectively, as shown in Fig. 2.12(a). The
results indicate that, for both λ1 and λ2, the turbulence mainly induces crosstalk between different OAM
modes. OAM ℓ1 = +1 experiences ∼-11dB and ∼-9dB crosstalk, whereas OAM ℓ2 = −2 experiences ∼-10dB
and ∼-2dB crosstalk for weaker turbulence and stronger turbulence, respectively. The pilot tones and data
channels from different polarizations might experience little crosstalk (<-20dB), which maintains the turbulence resilience of the pilot-assisted O/E beam mixing approach when applying polarization multiplexing.
30

Figure 2.12: Normalized crosstalk (in dB scale) between different data channels for λ1 and λ2 using (a) the
conventional LO-based heterodyne detection and (b) the pilot-assisted O/E beam mixing without and with
weaker or stronger turbulence.
Figure 2.12(b) shows the crosstalk matrix using the pilot-assisted O/E beam mixing. The results show that
the inter-channel crosstalk of different data channels carried by different OAM modes and polarization is
not significantly affected by the turbulence distortions. The crosstalk between two OAM modes and two
polarizations are measured lower than ∼-22.6 dB and ∼-28.9 dB, respectively, under both the weaker and
stronger turbulence distortions.
Figure 2.13(a) shows the optical spectrum of the pilot tones and data channels of the OAM ℓ1 = +1
and OAM ℓ2 = −2. The carrier (pilot) to signal power ratio (CSPR) is ∼1.64, ∼0.87, ∼1.06, and ∼1.44
for the pilot 1 to 4, respectively. The corresponding electrical spectrum with eight data channels after pilot
tone-data mixing is shown in Fig. 2.13(b). Figures 2.13(c) show the constellations and EVM performance of
different data channels when using the conventional LO-based heterodyne detection and pilot-assisted beam
mixing with and without turbulence. The results show that the EVM performance for the conventional
31

Figure 2.13: (a) Optical spectra of the transmitted pilots and data channels carried by the OAM ℓ1 = +1
and OAM ℓ2 = −2. (b) Electrical spectrum with eight received data channels after the wave mixing of the
pilot tone and data channels. (c) Beam profiles, data channel constellations and EVM performance (under
the corresponding constellations) for the conventional LO-based heterodyne detection and the pilot-assist
O/E beam mixing.
LO-based heterodyne detection degrades quickly as the turbulence effects become stronger (EVM >32.5%
for stronger turbulence). The pilot-assisted O/E beam mixing suffers much fewer degradations for all eight
data channels (EVM< 22.2% for stronger turbulence).
2.6 Concept of self-coherent turbulence-resilient MDM FSO links
using a PD array
In the above-mentioned work of pilot-assisted self-coherent MDM link, multiple OAM-multiplexed data
beams are transmitted together with their corresponding pilot beams with different frequency offsets (e.g., N
data beams need N pilot beams). At the receiver, a single PD was used to detect all these beams. As a result,
the pilot-data mixing terms for different channels were located at different electrical spectra of the PD and
demultiplexed by electrical filters in DSP. However, the channels need to share the bandwidth of the single
PD, which reduces the electrical spectrum efficiency of the PD. In this work, a single frequency-offset pilot
beam is transmitted together with data beams. At the receiver, the pilot-data mixing terms for different
channels are detected by a PD array with multiple sub-PDs and demultiplexed by applying a matrix on
the detected signals in DSP. Importantly, these pilot-data mixing terms are located at the same electrical
spectrum of the sub-PDs without sharing the bandwidth of the detector.
32

As shown in Fig. 2.14(a), a conventional LO-based MDM coherent FSO link can be significantly degraded by turbulence. At the transmitter (Tx), multiple channels carried by different orthogonal modes are
transmitted simultaneously through turbulence. Due to turbulence-induced wavefront distortion, the beams
experience modal coupling effects, which couple the power from each transmitted mode to other modes [38].
At the receiver (Rx), the data-channel modes are demultiplexed, converted to Gaussian modes, and mixed
with a Gaussian-mode LO to recover the date channels using coherent detection. The modal coupling can
cause two degradation effects [59]: (i) mixing power loss, which is due to that only a single-mode component
of each beam is converted to the Gaussian mode and efficiently mixed with the LO, and (ii) inter-channel
crosstalk, where each channel detector also detects interference from other channels.
Figure 2.14: (a) Conventional LO-based coherent MDM FSO system can be significantly degraded by turbulence. (b) Our proposed pilot-assisted self-coherent MDM FSO system using a PD array. Turbulence-induced
wavefront distortion is “automatically” mitigated by its conjugate during the pilot-data beam mixing at the
PD array.
33

Figure 2.14(b) shows the concept of our proposed pilot-assisted self-coherent MDM FSO system using a
PD array that can “automatically” mitigate turbulence. At the Tx, an additional frequency-offset Gaussianmode pilot beam is transmitted coaxially together with 4 mode-multiplexed channels. The pilot beam
experiences similar turbulence as the data beams and is distorted by the turbulence-induced spatial wavefront
distortion U. At the Rx, the distorted pilot and data beams mix in a PD array consisting of 4 subPDs. During the pilot-data mixing that follows the rule of square-law optoelectrical detection, the terms of
turbulence-induced wavefront distortion (i.e., U(x, y)) and its conjugate are both “automatically” produced
and multiplied together, thereby mitigating the modal coupling effects. As a result, the pilot and data beams
can be efficiently mixed with lower mixing loss compared to the single-mode-LO-based Rx. Specifically, for
each data beam i (i=1, 2, 3, 4), the photocurrent detected by sub-PD j (j= 1, 2, 3, 4) is given by an integral
over the area of the sub-PD [38, 60, 61]:
Ii,j ∝
Z Z |C (f + ∆f)U(x, y) + 1
4
Si(t, f)e
jθi,jU(x, y)|
2
dxdy (2.15)
where f is the optical carrier frequency, C is the pilot with a frequency offset of ∆f, Si
is the data signal
carried by data beam i, and θi,j is the relative phase differences between the spatial profiles of the pilot beam
and the data beam i (see Fig. 2.15(b)). After the heterodyne detection, the data signal of beam i detected
by sub-PD j can be expressed as 1
4
Si
e
jθi,j RR U
∗U dxdy ∼=
1
4
Si
e
jθi,j , given that RR U
∗U dxdy ∼=1.
During the pilot-data mixing, the pilot and data beams are mixed on the 4 sub-PDs and each sub-PD
detects some portion of the mixing results. Figure 2.15 shows an example of 4 channels carried by OAM
ℓ = −1, 0, +1, +2. For the mixing between the pilot and each of the data beams, different sub-PDs would
detect signals with different phase delays. This is because there are azimuthal phase differences between the
spatial profiles of the pilot and data beams. Based on the phase delays, we design and apply a matrix to
the 4 signals detected by 4 sub-PDs in DSP. The detailed implementation of the matrix multiplication is
shown in Fig. 2.15(c). The matrix is designed based on the azimuthal phase differences between the pilot
and data beams. After the matrix multiplication, the 4 signals can be constructively added to obtain one
desired channel at one row of the output [62, 63, 64].
34

Figure 2.15: Extracting the 4 data channels by applying a matrix on the signals detected by 4 sub-PDs in
DSP. (a) An example for a Gaussian pilot beam and 4 multiplexed channels carried by OAM ℓ = −1, 0, +1, +2
(b) Different relative phase delays detected by different sub-PDs after the pilot-data mixing. (c) Detailed
implementation of the matrix multiplication for channel demultiplexing.
2.7 Demonstration of self-coherent turbulence-resilient MDM FSO
links using a PD array
Figure 2.16 shows our experimental setup. At the Tx, we generate a 1-Gbaud QPSK signal. After preamplification by an EDFA, the signal is split into 4 copies for 4 data channels, and their polarizations are
35

Figure 2.16: Setup for an MDM FSO link with four 1-Gbuad QPSK channels. Three receivers are compared:
(a) a conventional LO-based MDM coherent receiver, (b) a LO-based PD-array Rx, and (c) our proposed
pilot-assisted self-coherent PD-array Rx. AWG: arbitrary waveform generator, PC: polarization controller,
EDFA: erbium-doped fiber amplifier, M: mirror, FM: flip mirror, BS: beam splitter, Col.: collimator.
controlled and aligned. Fibers are used to induce delays to de-correlate channels. A pilot tone is generated
by Laser 2 and has a 1.5-GHz frequency offset as compared to the data channels. The frequency offset is used
to provide a guard band for reducing the signal-to-signal beating interference in heterodyne detection. The
4 channels and the pilot tone are converted into different modes and multiplexed together using a multipleplane light converter (MPLC). They are subsequently coupled into free space and propagate together through
a 1-m link. The beam waist of the generated beams is w0=∼2.2 mm. A rotatable phase plate is used to
emulate the turbulence effects with Fried parameter r0=1 mm (stronger turbulence) and 3 mm (weaker
turbulence). We compared three different Rxs: (a) a conventional LO-based MDM coherent Rx, where a
mode demultiplexer is used to demultiplex channels and each channel is mixed with a Gaussian-mode LO
(Laser 3) in a signal-mode-fiber-coupled PD, (b) a LO-based coherent PD-array Rx, where a Gaussian-mode
LO (Laser 4) is mixed with all channels in a free-space-coupled PD array, and (c) our proposed pilot-assisted
self-coherent PD-array Rx. For LO-based approaches (a) and (b), we turn off the pilot and control the
power and wavelength of the LO to be the same as that of the pilot in approach (c). For PD-array-based
approaches (b) and (c), we use the demultiplexing matrix in DSP to extract the 4 channels.
As shown in Fig. 2.17, we measure beam profiles, channel crosstalk, data constellations, and error vector
magnitude (EVMs) with and without turbulence. Even under stronger turbulence, the maximum crosstalk
for the proposed pilot-assisted PD-array Rx is down to -16.3 dB. The crosstalk increases to -0.7 dB and -4.4
dB for the conventional LO-based Rx and LO-based PD-array Rx, respectively. Moreover, the mixing power
36

Figure 2.17: Beam profiles, crosstalk (XT), data constellations, and EVM with and without a
weaker/stronger turbulence realization. Pilot-assisted PD-array system is much less degraded due to the
“automatic” turbulence mitigation. The transmitted power of the four data channels is ∼0 dBm.
loss between the pilot/LO and data beams can be represented as the decrease in the diagonal values in the
crosstalk matrix caused by turbulence. Results show that the mixing loss is <1 dB for the pilot-assisted
PD-array Rx, while the loss is >8 dB and >2.5 dB for the conventional LO-based Rx and LO-based PD-array
Rx, respectively. With the increased crosstalk and mixing loss, the performance of the two LO-based Rxs
degrades quickly and the EVMs of 4 channels increase up to >50%. The proposed pilot-assisted self-coherent
Rx suffers less degradation, and the EVMs are smaller than ∼26%.
We also measure the EVM performance under different turbulence realizations in Fig. 2.18. For the
pilot-assisted PD-array Rx, all channels can have EVMs below 35% under 8 different turbulence realizations
with r0=1 mm. However, for the two LO-based Rxs, the channels have much larger EVMs (>50% for many
realizations).
We measure the BER of Ch3 (OAM ℓ = +1) with and without channel multiplexing and turbulence in Fig.
2.19. The penalty induced by the pilot/LO-data mixing loss can be obtained by comparing the performance
without and with turbulence for a single-channel transmission case (without multiplexing). The channel
crosstalk-induced penalty can be obtained by comparing cases without and with channel multiplexing when
there is turbulence. For the conventional LO-based Rx, the power penalty induced by the mixing loss and
crosstalk is >8 dB and >10 dB, respectively. For the LO-based PD-array Rx, the crosstalk-induced penalty
37

Figure 2.18: EVMs under 8 different stronger turbulence realizations with r0=1 mm for the three different
receivers
Figure 2.19: BER of data Ch3 (OAM ℓ = +1) with and without multiplexing and turbulence (the stronger
turbulence case in Fig. 2.17). Tur.: turbulence
is still large (>10 dB), while the mixing loss is relatively smaller (∼3 dB), which might benefit from the
aperture diversity effects of the multiple sub-PDs [65]. For these two receivers, the BER doesn’t reach below
the 7% FEC limit. For the pilot-assisted PD-array Rx, both LO-data mixing power loss and crosstalk induce
much smaller penalties (<3 dB). Figure 2.20 shows the BER of the 4 multiplexed channels under the stronger
turbulence case shown in Fig. 2.17. All channels can reach the 7% FEC limit and the penalty induced by
turbulence is <3 dB. The performance of Ch4 is relatively worse than other channels for both with and
38

Figure 2.20: BER of the four channels with and without turbulence (the stronger turbulence case in Fig.
2.17).
without turbulence, which might be due to lower mixing efficiency between the pilot (OAM ℓ = 0) and Ch4
data (OAM ℓ = +2) beam.
39

Chapter 3
Phase-Conjugation-Based Turbulence Mitigation for Coherent
FSO links
3.1 Background and Motivation
FSO demonstrations commonly use intensity modulation and direct detection [2]. However, similar to optical
fiber links, there are advantages in increased receiver sensitivity and spectral efficiency by using phase-based
data encoding (e.g., quadrature-phase-shift-keying (QPSK) and quadrature-amplitude-modulation (QAM))
and coherent detection with a local oscillator (LO) in an FSO link [66, 46].
A key challenge in FSO communications is atmospheric turbulence [34, 67]. For direct detection, this can
cause power scintillations and beam wander. The situation can be much worse for coherent detection since
turbulence can cause significant power loss due to the inability to efficiently mix a fundamental Gaussian LO
beam with a distorted data beam in a coherent receiver [38, 58]. This inefficiency comes from the turbulenceinduced coupling of power into higher-order spatial modes [68]. Various approaches to mitigate turbulence
for coherent reception include: (a) adaptive optics by measuring the wavefront distortion and applying a
digital signal processing (DSP)-calculated conjugate phase for correction [69], and (b) multi-mode combining
by collecting multiple modes and recovering their power by multiple coherent detectors and additional DSP
[70, 71, 72]. However, it might be advantageous to “automatically” mitigate turbulence for efficient coherent
detection using a single detector without additional DSP.
40

In Chapter 2, we discussed pilot-assisted self-coherent FSO links that can automatically mitigate turbulence. However, compared to a ”true” coherent-detection system, the transmitted pilot in a self-coherent
system may suffer from link loss and reduce the power efficiency and receiver sensitivity. One approach for
automatic turbulence mitigation for coherent FSO link is to: (i) propagate a probe beam from the receiver
(Rx) to the transmitter (Tx) through the turbulence and experience distortion, (ii) reflect the beam with
a conjugate phase profile by a self-pumped photorefractive crystal [73, 74, 75, 76, 77, 78], (iii) temporally
modulate the data by adding electrical voltages on the crystal [76, 77] or using an external free-space coupled
modulator [78], and (iv) transmit the data beam back through the turbulence to mitigate the distortion.
This approach has been demonstrated with ¡ 1 Mbit/s intensity modulation. However, this approach is
challenged by the dramatically lower bandwidth performance of free-space coupled modulators relative to
that of fiber-coupled modulators (e.g., ¡1 GHz [79, 80] as compared to ¿40 GHz [81], respectively).
In this chapter, we demonstrate automatic turbulence mitigation in a 2-Gbit/s quadrature-phase-shiftkeying (QPSK) coherent FSO link using degenerate four-wave-mixing (DFWM)-based phase conjugation and
fiber-coupled data modulation. Specifically, we counter-propagate a Gaussian probe from the Rx to the Tx
through turbulence. At the Tx, we generate a Gaussian beam carrying QPSK data by a fiber-coupled phase
modulator. Subsequently, we create a phase conjugate data beam through a photorefractive crystal-based
DFWM involving the Gaussian data beam, the turbulence-distorted probe, and a spatially filtered Gaussian
copy of the probe beam. Finally, the phase conjugate beam is transmitted back to the Rx for turbulence
mitigation. Compared to a coherent FSO link without mitigation, our approach shows up to 14-dB higher
LO-data mixing efficiency and achieves error vector magnitude (EVM) performance of ¡16% under various
turbulence realizations.
3.2 Concept and Experimental Setup
Figure 3.1(a) shows the turbulence-induced degradations for a coherent FSO link. The turbulence can distort
the wavefront of the transmitted Gaussian data beam and cause significant modal coupling from the Gaussian
mode (i.e., Laguerre-Gaussian (LG) mode with mode indices ℓ = 0 and p = 0) to other LG modes. At the
41

Rx, the multi-mode data beam cannot efficiently mix with a Gaussian LO beam, thereby inducing mixing
loss and signal quality degradation.
Photorefractive crystal-based self-pumped phase conjugation has been demonstrated to mitigate turbulence. In this approach, a Gaussian probe beam is transmitted from the Rx to the Tx. At the Tx, a
phase-conjugator is used to “reflect” the probe beam and generate a phase-conjugate beam. After phase
conjugation, a free-space data modulator can be used to temporally modulate the data on the beam. When
the data beam propagates back through the same turbulence, turbulence-induced spatial distortion and
modal power coupling are automatically mitigated. However, in this approach, the free-space-based data
modulation may only support a low bandwidth.
Figure 3.1: (a) Coherent FSO links can be significantly degraded by turbulence-induced beam distortion
and modal coupling. (b) Our approach using DFWM-based phase conjugation for automatically mitigating
turbulence. (c) Detailed process of the DFWM-based phase conjugation. The distorted probe beam and a
Gaussian-like reference beam interfere inside the crystal to “record” turbulence distortion. A Gaussian data
beam is used to “read” the distortion and generate a phase conjugate data beam.
As shown in Fig. 3.1(b), we utilize DFWM-based phase conjugation to mitigate turbulence. The
turbulence-distorted probe beam is first divided into two copies at the Tx. One of the copies propagates
through a spatial filter to remove the high-spatial-frequency components and generate a Gaussian-like reference beam [82]. The reference beam and another copy of the distorted probe beam interfere with each other
inside the crystal to form a grating to “record” the spatial turbulence distortion, as shown in Fig. 3.1(c). At
the Tx, a Gaussian data beam is first generated by a wide bandwidth, fiber-coupled modulator and propagates in a direction opposite to that of the reference beam through the crystal to “read” the distortion and
generate a phase conjugate data beam. In order to generate high-fidelity phase conjugation, the Gaussian
42

data beam and the reference probe beam should propagate through the crystal in a counter-propagating
manner [83]. Misalignment between these two beams might induce undesired spatial intensity and phase
variations on the phase conjugate beam, which can affect the performance of turbulence mitigation. The
specific alignment requirements and performance penalties depend on the specific system parameters, and
the performance can be modeled based on various published analyses. The resulting phase conjugate beam
then propagates through the assumingly same turbulence to the Rx.
Figure 3.2: Experimental setup for automatic turbulence mitigation using phase conjugation in a 2-Gbit/s
QPSK coherent FSO link. AWG: arbitrary waveform generator; BS: beam splitter; Col.: collimator; FM:
flip mirror; HWP: half-wave plate; Iso.: isolator; M: mirror; PC: polarization controller; PD: photodiode;
SLM: spatial light modulator; SMF: single-mode fiber; YDFA: Ytterbium-doped fiber amplifier.
Figure 3.2 shows our experimental setup. At the Rx, we generate a CW probe beam at 1064 nm, amplify
it to ∼800 mW by a YDFA, couple it to free space as a Gaussian beam with a waist diameter of ∼2.5 mm,
and propagate it from the Rx to the Tx in a ∼1-m free-space link. The turbulence effect is emulated by
loading phase screens on an SLM placed around the middle of the link. These phase screens are designed
based on Kolmogorov turbulence power spectrum statistics. The strength of the turbulence is characterized
by D/r0, where D is the beam size and r0 is the Fried turbulence parameter [7]. A larger D/r0 results in
stronger turbulence distortion. When the distorted probe beam arrives at the Tx, we use a beam splitter
to create its two copies, with one being spatially filtered to a Gaussian-like reference pump beam. Spatial
filtering is performed by a 4-f system consisting of two lenses with focal lengths of 30 mm and an iris. The
Gaussian-like reference beam and distorted probe beam cross and interfere inside an Rh-reduced KNbO3
crystal to “record” the turbulence distortion. We note that the beam size of the Gaussian data beam and the
probe beam might affect the performance of turbulence mitigation. If the Gaussian data beam is too small
compared to the distorted probe beam, it may not effectively overlap with the probe beam inside the crystal
43

to “read” the turbulence distortion. As a result, the generated phase conjugate beam carries only part of
the spatial conjugation of the distortion, and the turbulence mitigation performance might be degraded. In
our demonstration, we use a lens to reduce the size of the probe beam to ensure good spatial overlap of the
two beams. One may also consider using a beam expander to increase the size of the Gaussian data beam
to achieve an effective beam overlap.
At the Tx, we generate a Gaussian data beam carrying a 2-Gbit/s QPSK signal with a 1.5-GHz frequency
offset through a fiber-coupled phase modulator. The frequency offset is used to provide a guard band for
reducing signal-to-signal beating interference in heterodyne detection. We propagate the Gaussian data beam
in a reverse direction through the crystal to “read” the turbulence distortion and generate a phase-conjugated
data beam, which then propagates back towards the Rx through the same emulated turbulence. At the Rx,
after being coupled into an SMF, the data signal is mixed with an LO in a PD. The power of the LO is 0
dBm. We measure the complex wavefront of the received data beam and perform LG modal decomposition
using off-axis holography to analyze the turbulence-induced wavefront distortions. We note that, since we
only have one laser source at 1064nm, we split the power of the laser for multiple uses, including generating
the probe beam, the data beam, and acting as the LO.
3.3 Experimental Results for Turbulence Mitigation in a Coherent
FSO Link
Figure 3.3 shows the intensity profiles of the probe, reference, and data beams under one turbulence realization. Our results show that the mitigated beam can have a Gaussian-like profile with little turbulence
distortion. In an ideal case, the phase conjugate beam should have a similar intensity profile as the distorted probe beam. However, they show some differences in our results. This might be due to the imperfect
Gaussian reference beam generation.
We measure the beam profiles and LG modal spectra of the received data beam without and with
phase-conjugation mitigation in Fig. 3.4(a) and (b), respectively. We note that, for the phase conjugate
beam generation through the crystal, the conversion efficiency from the input Gaussian data beam to the
44

Figure 3.3: Experimentally measured intensity profiles of the probe, reference, and data beams under one
turbulence realization (D/r0∼4).
output phase conjugate data beam is measured to be ∼-20 dB without turbulence. With turbulence, the
efficiency can be decreased, which might be due to a larger loss during the reference beam generation
through spatial filtering (e.g., ∼5 dB power reduction for the turbulence realization of D/r0∼6). To focus
on the turbulence mitigation performance, we keep the same optical power (∼4 mW) of the transmitted
Gaussian (w/o mitigation) and phase conjugate (w/ mitigation) data beam. Our results show that, without
mitigation, the received data beam becomes more distorted and experiences stronger modal coupling effects
under a stronger turbulence distortion. For the turbulence realization with a D/r0∼6, ∼18-dB power loss is
observed in the fundamental Gaussian mode. Due to the modal power coupling and inefficient mixing with
the LO, the quality of the received QPSK signal is significantly degraded with increased turbulence strength
(e.g., error vector magnitude (EVM) increases from ∼8% to >40%). However, with the phase conjugation
mitigation, the beam distortion, and modal power coupling effect are largely mitigated with up to 14-dB
improvement (e.g., <-4-dB for D/r0∼6). As a result, the EVMs of <12% can be achieved of the signal under
a strong turbulence case.
45

Figure 3.4: Beam profiles, modal spectra, and recovered data constellations of the received data beam.
As shown in Fig. 3.5, with mitigation, the average LO-data mixing loss can be reduced by 5.4 dB and 12.6
dB for the weaker and stronger turbulence, respectively. As a result, with mitigation, EVMs are achieved
under ∼16%, while, without mitigation, the EVMs can be >30% and >40% for the weaker and stronger
turbulence, respectively. The data constellations in Figs. 5(a3-b3) indicate that the quality of recovered
QPSK data can be significantly improved.
To measure the efficiency and response time of the crystal, we turn on the laser at the receiver and
transmit the Gaussian probe beam ( 800 mW) from the receiver to the transmitter (at Time=0 sec). At
the transmitter, we record the power changes of the generated phase conjugate data beam through the
46

Figure 3.5: (a1-b1) Turbulence-induced LO-data mixing loss, (a2-b2) EVMs, and (a3-b3) data constellations
of the received QPSK signal.
crystal at a different time and calculate the efficiency of the conversion from the input Gaussian data beam
to the output phase conjugate beam. As shown in Fig. 3.6, it takes 2 minutes for the crystal to reach
its maximum conversion efficiency of ∼-20 dB without turbulence. The performance of our system can
potentially be improved by using faster crystals. Some crystals have been reported to support a ¡ms refresh
rate in the visible wavelength region, which might enable real-time turbulence mitigation [84, 85, 86]. With
turbulence, the efficiency can be decreased, which might be due to a larger loss during the reference beam
47

generation through spatial filtering (e.g., ∼5 dB power reduction for the turbulence realization of D/r0∼6).
One possible way to mitigate such loss might be to couple the distorted probe beam into an SMF and amplify
it for reference beam generation. This method might need additional control for fiber stabilization to avoid
relative phase drifts between the probe and reference beams.
Figure 3.6: Measured efficiency of the crystal for converting the input Gaussian data beam to the output
phase conjugate data beam. The three turbulence realizations shown in this figure correspond to the same
realizations shown in Fig. 3.4.
48

Chapter 4
Probing Longitudinal Atmospheric Turbulence Strength using
Structured Light
4.1 Background and Motivation
The atmosphere is a medium that affects many applications, such as various forms of air flight and the
transmission of light waves [34, 87, 20, 88]. The atmosphere is considered an inhomogeneous medium with
random fluctuations of varying strengths in both time and space [34, 87]. Moreover, the atmosphere can
seriously affect various applications, including (a) people and aircraft: turbulence can be characterized by
random air motion, which has been identified as a leading cause of serious injury to airline passengers
and damage to various forms of aircraft [20, 88]; and (b) communications and imaging: the temperature
variations of turbulence can cause spatial and temporal changes in the refractive index, thereby causing
wander and distortion to lightwaves and significantly degrading free-space communication links and imaging
systems [34, 87]. In each of these cases, knowledge of the inhomogeneous spatial distribution of atmospheric
turbulence can help in spatially avoiding or actively mitigating the effects of strong-turbulence regions
[88, 89, 90, 91].
In general, optical beams can serve as probes for the detection of atmospheric turbulence due to lightmatter interaction [92, 93, 94, 95, 96, 97]. For example, the total accumulated turbulence can be determined
by measuring the turbulence-induced beam wavefront distortion at the end of its propagation path [98, 69, 99].
However, knowledge of the specific distribution of turbulence strength at various longitudinal z distances
49

could be of immense value for enabling: (a) aerial platforms to avoid local areas of “flight risky” strong
turbulence, and (b) more accurate adaptive optics turbulence compensation in communication links and
imaging systems [99, 100, 101].
Previously, optical approaches have been demonstrated that can measure the distribution of turbulence
strength along a path. One technique is based on lidar, and detects a transmitted pulse that is backscattered
by air; this method requires equipment at only one terminal but is typically limited to a few hundred
meters due to the small power in a reflected pulse [88, 89, 90, 91]; such a short distance may not provide
sufficient warning to avoid turbulence. Another technique that can operate over many kilometers is based on
transmitting two beams that intersect each other along the path. Changing the angle between these beams
results in their overlap at different z locations. The two beams experience similar turbulence where they
overlap, thus enabling an array of receivers to measure the z-dependent turbulence strength [92, 93, 94, 95,
96, 97]. Due to the angular requirement for long distances, however, this multi-aperture approach tends to
either: (i) require a dramatically larger receiver size than transmitter size [102] thus making it far too large
to deploy, or (ii) have relatively poor performance near the transmitter for the case of the transmitter and
receiver being of similar size [103]. An over-arching goal would be to have a relatively accurate z-dependent
turbulence probing technique that operates over long distances and be of reasonable size.
In this chapter, we demonstrate using longitudinally structured beams for probing turbulence strength
(i.e., turbulence structure constant, C
2
n
(z)) along a propagation path using a single pair of Tx/Rx apertures
of similar size. Our longitudinally structured beams are superpositions of multiple low-divergence BesselGaussian (BGℓ=0,kz
) modes with different longitudinal wavenumbers kz and an orbital angular momentum
(OAM) ℓ = 0 order. We tailor the complex coefficients for different kz values to generate z-dependent beam
width, and subsequently measure the resultant turbulence-induced modal power coupling among various ℓ
orders as signatures for retrieving the z-dependent turbulence strength. Since turbulence affects a wider beam
more than a narrower beam, the receiver detects relatively weaker effects where a narrow beam interacts
with turbulence. By sequentially transmitting different beams with the narrow section located at different
z locations, we extract the turbulence-strength distribution when measuring at the receiver the z-dependent
BG modal coupling from ℓ = 0 to ℓ ̸= 0 orders. By simulating a 10-km path with up to 30-dB difference in
50

C
2
n
, our approach shows: (i) relatively uniform probing errors [104, 105] (∼0.1 to ∼0.3 dB); and (ii) a tradeoff between probing resolution and transmitter aperture size. We also experimentally demonstrate probing
of two emulated turbulence regions of ∼15 dB variation and <0.8 dB error. Importantly, our approach
has the potential to support: (i) much longer distances than lidar, and (ii) much smaller size and uniform
performance than crossed overlapping beams.
4.2 Concept of Longitudinal Atmospheric Turbulence Probing
Different turbulence-induced effects on probe beams can be utilized as signatures to help retrieve turbulence
information. In our approach, we measure the turbulence-induced modal power coupling (i.e., the coupling
of power from the original spatial mode of the beam to other spatial modes) [106, 107], and such modal
coupling is related to the z-dependent turbulence strength and beam width. Using a single Tx/Rx aperture
pair, we sequentially transmit bursts of different longitudinally structured beams as z-dependent probes.
Each longitudinally structured beam is a superposition of multiple, low-divergence, ℓ=0 order BGℓ=0,kz
modes with different longitudinal wavenumbers, kz [108]; we note that BG modes can be represented by two
spatial indices: (i) ℓ is the number of 2π phase shifts in the azimuthal direction and is the beam’s OAM
value, and (ii) kz is related to the radial wavenumber kr (kz
2 + kr
2 = (2π/λ)
2
, λ is the wavelength), which
determines the radial ring spacings in the intensity profile [28, 109]. By designing the complex coefficients of
kz of the ℓ=0 order BG modes, the beam width of each beam can be designed to be z-dependent, therefore
resulting in different z- and turbulence-dependent power coupling from the ℓ=0 BG modes to other ℓ orders.
Based on measured ℓ modal coupling and kz designed beam widths, we can extract the inhomogeneously
distributed turbulence strength along the propagation path. Importantly, instead of other modal basis (e.g.,
Laguerre-Gaussian modes), we choose combinations of BG modes for creating the z-dependent probe beams
to achieve a lower beam divergence, thus enabling a smaller size Rx aperture.
Detailed principles of our approach using turbulence-induced l-based power coupling and kz-based longitudinally structured beams are described below.
Optical beams can be distorted when propagating through turbulence, giving rise to power coupling from
the transmitted spatial mode into other modes. In our approach, we consider power coupling among ℓ modes
51

Figure 4.1: (a)A general scheme of using forward-propagating optical beams for probing turbulence along
a path. At the Rx, beam-turbulence interactions are measured for retrieving turbulence information. (b)
One prior turbulence probing technique by transmitting two probe beams from two separate sources and
detecting them at a multi-element Rx aperture array. (c) Our proposed approach designs and sequentially
transmits multiple longitudinally structured beams, each having its narrow beam width at a different z along
the propagation path, using a single pair of Tx/Rx.
52

to characterize such turbulence-induced modal coupling. Applied to a launched fundamental Gaussian beam
(ℓ=0) propagating through a (statistically) longitudinally uniform atmospheric turbulence, the normalized
average power remaining at the end of the link on the ℓ=0 mode, P(ℓ = 0), as well as a parameter β, are:
P(ℓ = 0) ≈ (I0 (β) + I1 (β)) exp(−β) (4.1)
and
β = 1.8025 (D)
5
3 r0
− 5
3 (4.2)
where In ( ) is the modified nth-order Bessel function of the first kind, D is the beam width (2nd order
definition [24]) and β is related to the integration of the turbulence strength along z, as characterized by the
Fried parameter, r0 [110]:
r0 =

0.423k
2
Z L
0
C
2
n
(z) dz!− 3
5
(4.3)
Here, C
2
n
(z) is the z-dependent refractive index turbulence structure parameter, k = 2π/λ and L is the total
propagation distance. The modal coupling increases with increasing turbulence strength (i.e., smaller r0 or
larger C
2
n
(z)) as well as the beam width (i.e., larger D). This can be understood as follows: (i) the Fried
parameter r0 is a measure of the transverse distance scale after which the turbulence refractive index becomes
uncorrelated, and (ii) under a given r0, larger beams experience more uncorrelated refractive index fluctuation
and exhibit stronger turbulence-induced distortion and modal coupling. Heuristically extending Eq. 4.1 to
an inhomogeneous turbulence scenario and comprising M uniform segments of strengths C
2
n,j , j = 1, . . . , M
each of thickness ∆z, we express β of Eq. 4.2 as:
β ≈ 1.8025X
M
j=1
n
0.423k
2C
2
n,j∆z

D
5
3
j
o
(4.4)
where Dj , j = 1, . . . , M are the beam widths of each respective segment and assuming for simplicity that
the widths do not change within a segment. By sequentially transmitting different beams each having its
narrow width at a different longitudinal turbulence segment, we can define a set of equations for solving the
corresponding turbulence strengths, C
2
n,j , j = 1, . . . , M along the link.
53

Longitudinally structured beams can be created by a coherent superposition of multiple co-propagating
BG modes with different longitudinal wavenumbers, kz, and ℓ = 0 orders. The kz values have equal spatial
frequency spacing, thereby forming a comb in the space-frequency domain, as shown in Fig. 4.1(c). Since the
constructive and destructive interference among the BG modes is governed by their complex coefficients, a
longitudinally structured beam can be designed to have a desired on-axis central intensity distribution along
the propagation axis z, from z=0 to L [111]. Equation 4.5 shows the waveform of a longitudinal structured
beam consisting of (2N+1) BG modes all at the same optical frequency ω0 [112]:
Ψ (ρ, z, t) = e
−iω0tG(ρ)
X
N
n=−N
AnJ0 (kr,nρ) e
ikz,nz
(4.5)
where ρ is the radius in the cylindrical coordinate; kρ,n and kz,n are transverse and longitudinal wavenumbers,
respectively, satisfying kρ,n
2 + kz,n
2 = k
2
; G(ρ) is a Gaussian apodization and is chosen to ensure that all
BG modes have relatively low divergence in the relevant longitudinal range; and the coefficients An are given
by
An =
1
L
Z L
0
F (z) e
−i
2π
L nzdz (4.6)
Where the function F (z) defines the desired longitudinal on-axis central intensity distribution. Figure 4.1(c)
shows an example of a longitudinally structured beam that has a rectangular-shaped longitudinal central
intensity distribution with the on-axis intensity higher within zi ≤ z ≤ zj and lower elsewhere, corresponding
to a pattern
F (z) =



1, if zi ≤ z ≤ zj
0, otherwise
(4.7)
Importantly, the location of the intensity-higher region can be almost arbitrarily determined by a proper
choice of the F (z) function. As for the z-dependence of the beam width: In the intensity-higher region
(zi ≤ z ≤ zj ), different BG modes constructively interfere and contribute to a higher intensity at the center
part of the beam, and consequently, a narrow beam width. However, in the other regions, different BG modes
are no longer in-phase and their power is spread over a larger width. In our turbulence probing approach
54

using longitudinally structured beams, such controllable z-dependent beam width can provide z-dependent
signatures that help to extract the turbulence strengths along the z-axis.
Figure 4.1(c) shows the concept of our approach using a single Tx/Rx pair and multiple longitudinally
structured beams, each having its narrow width at a different position along the path. To simplify the
analysis and simulation, the turbulence distribution along z is divided into M equal-length regions, each
having a constant, region-dependent turbulence strength (i.e., C
2
n,j , j = 1, . . . , M for the turbulence region
j). At the Tx, M longitudinally structured beams are sequentially transmitted, in which beam i has its
narrow width in region i. Following Eq. 4.4, the β for beam i can be approximated by:
βi ≈ 1.8025X
M
j=1
n
(Di,j )
5
3

0.423k
2C
2
n,j∆z
o
(4.8)
where Di,j is the width of beam i when it is in turbulence region j, and ∆z =
L
M is the length of each region.
Equation 4.8 can be represented as a set of linear equations:












β1
.
.
.
βM−1
βM












=

1.8025 × 0.423k
2∆z













(D1,1)
5
3
· · · (D1,M−1)
5
3 (D1,M)
5
3
.
.
.
.
.
.
.
.
.
(DM−1,1)
5
3
(DM,1)
5
3
· · ·
(DM−1,M−1)
5
3 (DM−1,M)
5
3
(DM,M−1)
5
3 (DM,M)
5
3
























C
2
n,1
.
.
.
C
2
n,M−1
C
2
n,M












(4.9)
Based on Eqs. 4.1 and 4.2, the normalized average received power remaining on the ℓ = 0 order for beam i
is approximated by:
Pi (ℓ = 0) ≈ (I0 (βi) + I1 (βi)) exp (−βi) (4.10)
Therefore, after measuring Pi (ℓ = 0) for each transmitted beam at the Rx and extracting the corresponding
βi based on Eq. 4.10, we can solve the M equations of Eq. 4.9 for the C
2
n,j , j = 1, . . . , M of the different
regions.
5

4.3 Simulation of Longitudinal Atmospheric Turbulence Probing
To explore the performance of our approach, we simulate the probing of a 10-km path with inhomogeneous
turbulence. As an example in Fig. 4.2(a), we first simulate a case of three regions, each with a different
constant turbulence strength of C
2
n,j , j = 1, 2, 3, respectively. We use a split-step beam propagation method,
in which 40 different phase plates are placed at 250-m spacings. When each beam sequentially propagates
through the phase screens, each screen induces its own spatial phase distortion on the beam. We generate
“random” phase distributions with different turbulence strengths according to the Kolmogorov turbulence
theory. To probe three regions, we transmit three longitudinally structured beams (e.g., Beam 1, 2, and
3) at a wavelength of 1550 nm, in which each beam’s narrower beam width longitudinally overlaps with
its corresponding turbulence region. We set the center of kz longitudinal wavenumbers to be Q = kz,0 =
(1 − 6 × 10−10) × k; N=7; and the size of the Tx/Rx aperture diameters = 1 m.
We calculate the beam width Di,j in Fig. 4.2(b) for each beam (i) in each region (j), and the width
of each beam becomes smaller in its corresponding region (see “Methods” for width calculation). Modal
decomposition is used at the receiver to calculate each beam’s modal spectrum, and Fig. 4.2(c) shows an
average of 200 different turbulence realizations. Subsequently, the normalized average power remaining on
ℓ = 0 for each beam (e.g., Pi (ℓ = 0) for beam i) is compared with the theoretical calculation (Fig. 4.2(d))
and used to calculate βi
, thereby forming the equations relating βi
, Di,j , and C
2
n,j (Fig. 4.2(e)). Finally,
the C
2
n,j values are extracted by solving an inverse problem using optimization algorithms. These simulated
probed turbulence strengths (i.e., C
2
n,j (probed)) are compared to the original values from Fig. 4.2(f). The
relative errors of the turbulence distribution’s L
2 norm of our method can be given by:
Error
C
2
n

=

C
2
n,j − C
2
n,j (probed)

2

C2
n,j (probed)

2
(4.11)
For this example, the relative error is ∼4.0% (∼0.2 dB from the original value).
Next, we explore more complicated scenarios with 40 turbulence regions. In these examples, we simulate
“Gaussian-shaped” turbulence distributions with the peak turbulence strength located at different z (see
5

Figure 4.2: (a) Three equal-length turbulence regions in simulation with each region having a different turbulence strength of C
2
n,j , j = 1, 2, 3. (b) Simulated beam widths of the three beams designed for turbulence
probing (Di,j for each beam i in each region j). (c) Simulated average modal spectrum for the three beams
at the Rx under 200 turbulence realizations. The bars show the standard deviation for the simulation results.
(d) Normalized average power remained on the ℓ = 0 for the three beams (e.g., Pi (ℓ = 0) for Beam i) in
the simulation. (e) Constructed equations for retrieving the turbulence distribution. (f) Probed turbulence
distribution in simulation. The simulated relative probing error is ∼4% (∼0.2 dB from the original value)
for this example of probing 3 turbulence regions.
Fig. 4.3). We design and transmit 40 longitudinally structured probe beams (Beams 1 to 40). For designing
Beam i, we use an on-axis central intensity distribution
Fi (z) =



1, if zi ≤ z ≤ zi + 250m, zi = (i − 1) × 250m
0, otherwise
(4.12)
57

Figure 4.3: Simulation results for probing a “Gaussian-shaped” turbulence distribution, having the peak
turbulence strength located at different distances, z, in a 10-km path (e.g., z=0, 2.5, 5, 7.5, 10 km). (a)
Received beam profiles of Beam 1, 20, and 40 for probing the “Gaussian-shaped” turbulence distribution with
the peak located at the middle of the path (z=5 km) under one turbulence realization. (b) Average P (ℓ = 0)
for the 40 beams under 200 turbulence realizations for different “Gaussian-shaped” turbulence distributions.
The bars show the standard deviation of the simulation results for each beam. (c) The original turbulence
strength distribution and its simulated probed values using the 40 probe beams (in both logarithmic and
linear ordinate scale).
Figure 4.3(a) shows the received beam profiles of Beams 1, 20, and 40 for a turbulence distribution with
its peak strength located at z=5 km. We find that Beam 20 has its minimum size at z=5 km and has the
least relative distortion of the 40 beams. Figure 4.3(b) shows the average P (ℓ = 0) under 200 turbulence
realizations. The results show that the average P (ℓ = 0) is higher for the beams that have narrower widths
in the stronger turbulence regions, and this is because the turbulence causes relatively less distortion and
modal power coupling to narrower beams. Figures 4.3(c1-c5) compare the simulated probed turbulence
distributions with the original values. Our results show relatively uniform performance with errors varying
58

from ∼2% to ∼8% (∼0.1 to ∼0.3 dB from the original value), for various z locations of peak turbulence
strength.
In order to explore the performance of our approach under more general cases, we simulate other turbulence distributions, including linear, “triangular-shaped”, and “sine-shaped” distributions with errors of
<8%, <12% and <32%, respectively. In Fig. 4.4, simulation results show that more complicated distributions with larger longitudinally spatial gradients tend to result in larger errors. In addition, we simulate a
more realistic turbulence profile based on the Hufnagel-Valley (H-V) turbulence model [113], which has been
used to describe turbulence at different altitudes. The probing error for the H-V turbulence profile is <8%.
Figure 4.4: (a) Simulation results for probing turbulence with various distributions, including (a-b) linear
changing distributions, (c-d) “triangular-shaped” distributions, (e-f) “sin-shaped” distributions, and (d) a
distribution based on the Hufnagel-Valley turbulence model describing the atmospheric turbulence distribution at different altitudes.
4.4 Experiment of Longitudinal Atmospheric Turbulence Probing
We conduct a proof-of-concept experiment with emulated turbulence regions under laboratory conditions,
as shown in Fig. 4.5(a). At the Tx, we generate longitudinally structured beams by encoding the desired
pattern into a computer-generated hologram on a programmable SLM. A Gaussian beam with a beam waist
of ∼7 mm is incident on the SLM and acts as the input of the complex amplitude phase modulation for
59

structuring the spatial amplitude and phase of the longitudinally structured beam. Next, a 4-f system and an
iris are used to filter out the desired beam and remove the undesired diffraction orders. We divide the 0.6-m
path into two equal-length regions and emulate turbulence effects by placing a rotatable thin phase plate in
the middle of each region. The phase plates are fabricated to have different Fried parameter r0 (0.4 mm, 1
mm, or 3 mm) to emulate different turbulence strengths (smaller r0 means stronger turbulence) at 1550 nm.
To probe these two turbulence regions, we design and simultaneously transmit two longitudinally structured
beams (i.e., Beam 1 and 2). At the Rx, we use off-axis holography to measure the spatial amplitude and
phase of each beam and numerically calculate its modal spectrum and P (ℓ = 0). To measure the modal
spectrum using this approach, we split the laser light source to provide a coherent reference light beam.
We note that one can also use an SLM at the Rx to perform the inner product between the received beam
and Bessel basis functions for the modal decomposition, which does not require a coherent light beam. The
modal spectrum result for each beam and turbulence distribution is an average of measurements under 200
different turbulence realizations, achieved through the rotation of the phase plates to different orientations.
To directly compare probed turbulence strengths with the emulated ones, we show the probing results in
terms of the Fried parameters r0 instead of the C
2
n
. These two parameters are mutually related by Eq. 4.3.
To probe these two turbulence regions, we generate and transmit Beams 1 and 2 having their narrower
beam widths at different regions, respectively. Figure 4.5(b-c) shows the simulated and experimentally
measured intensity profiles of the two beams. In Fig. 4.5(d), we calculate the beam widths for the two
beams based on the measured intensity profiles. Our results show that the experimentally measured beam
widths are in relative agreement with the simulation results. In Fig. 4.5(e), we show the received beam
profiles and measured modal spectra of Beams 1 and 2 under one turbulence realization for 4 different cases,
where turbulence regions 1 and 2 have different values for r0. From the measured modal spectra, we see that,
if the beam’s narrower-width section is in the stronger turbulence (smaller r0), it suffers less turbulenceinduced mode coupling than the beam wider section. For example, in Figs. 4.5(e2) and (e4), Beam 1 exhibits
a larger measured P (ℓ = 0) than that of Beam 2 for the stronger turbulence region 1 and weaker turbulence
region 2 (see Figs. 4.5(e3) and (e5) for the opposite scenario).
60

Figure 4.5: (a) Experimental setup for probing two emulated turbulence regions using longitudinally structured beams. (b-c) Simulated and experimentally measured intensity profiles of Beam 1 and 2 at different
propagation distances. (d) Simulated and experimentally measured beam widths for Beam 1 and 2. (e) Experimentally measured beam profiles and modal spectrum of Beam 1 and 2 under one turbulence realization
for 4 different cases where turbulence regions 1 and 2 have different Fried parameters r0.
61

Figure 4.6: (a) Simulated and experimentally measured P (ℓ = 0) for Beam 1 and 2 under 200 turbulence
realizations for the 4 different turbulence distribution cases. (b) Original turbulence and probed turbulence
in simulation and experiment. To directly compare probed turbulence strengths with the emulated ones, we
show the probing results in terms of the Fried parameters r0 instead of the C
2
n
.
Figure 4.6(a) shows measurements of the P (ℓ = 0) for Beams 1 and 2 under 200 different turbulence
realizations for the 4 different turbulence distribution cases. Again, the histograms show that the average
P (ℓ = 0) is larger for the beam that has smaller beam widths in the stronger turbulence region. Compared
to Case 1 and 2 with r0=1 mm and 3 mm, Cases 3 and 4 with r0=0.4 mm and 3 mm result in more mode
coupling to ℓ ̸= 0 orders and smaller P (ℓ = 0) due to the stronger turbulence (r0=0.4 mm) region. To probe
the turbulence strengths in the two regions, we first calculate for each case the average P (ℓ = 0) for Beams 1
and 2 for the 200 turbulence realizations. We then utilize the average P (ℓ = 0) and measured beam widths
to extract the r0 for the two regions. As shown in Fig. 4.6(b), the average relative errors in the experiment
for Cases 1 to 4 are around 7.7%, 9.8%, 9.4%, and 15.3% (i.e., all <0.8 dB), respectively. Simulations are
also conducted using the same parameter settings, and these results are in relatively good agreement with
the experimental results.
62

4.5 Discussion
Currently deployed, non-optical turbulence probing techniques are limited, including: (i) radar-based turbulence monitoring has difficulty measuring turbulence in clear air without dense clouds; and (ii) an aircraft
flying through high-turbulence locations can signal other aircraft, but this does not help the original aircraft
avoid the turbulence or if the turbulence changes dynamically.
Alternatively, optical probing techniques do not generally suffer from these drawbacks. However, different
optical approaches can have different advantages and disadvantages when compared to the approach of our
paper:
(a) Back-scattering-based lidar: Lidar relies on detecting relatively low-power backscattered light by air
and is typically limited to a few hundred meters. This distance provides a few seconds of warning time, which
is typically too short for an airplane to actively avoid turbulence. Our approach detects forward-propagating
beams and has a much higher signal power, potentially supporting multi-kilometer-length distances and
providing a longer warrning time for turbulence avoidance. However, since lidar uses equipment at one
terminal and our approach requires separate Tx and Rx terminals, lidar can probe in any direction and does
not need to form a link between two terminals.
(b) Forward-propagating crossed beams: Crossed beams relies on changing the angle and location of beam
overlap. It has been demonstrated for long distances, but: (i) the receiver array can require many elements
and be very large in size (e.g., 4 meters for a 10-km probe), and (ii) the accuracy near the transmitter
is generally poorer. Alternatively, our approach utilizes a single pair of Tx/Rx apertures, the size can be
significantly smaller (e.g., 4x smaller), and the accuracy can be uniform along the entire path. However, the
crossed beams approach uses simple Gaussian beams, which are easier to generate and transmit than our
tunable structured beams.
(c) Focusing Gaussian beams: Given our structured beams with a z-dependent width, we compare our
approach to a simple case of using transmitter lenses to focus a fundamental Gaussian beam at different
distances [114, 115, 56]. We compared a focused Gaussian beam and our longitudinally structured beams for
a 10-km path with a 1-m Tx aperture. Our simulation shows that focused Gaussian beams have: (i) sharper
63

beam width changes and finer probing resolution, but (ii) larger beam divergence resulting in a larger Rx
aperture (e.g., ∼3x larger receiver aperture when the focus is at 1 km).
Another relevant issue to consider is the system architecture. Our approach requires creating a link
between two terminals, which places a limitation on the directions that can be probed. One can envision
several scenarios that can reduce the impact of this limitation and create new capabilities/opportunities,
including the following:
(a) Reconfigurable Path: One aircraft can dynamically and reconfigurably steer the probe beam to many
other aircraft for sequentially probing turbulence in different directions.
(b) Receiver at Touchdown: When an aircraft is landing, the receiver terminal can be located on a ground
station, thereby enabling an aircraft to probe the turbulence of the landing path.
(c) Network of Beams: Multiple probe beams can form a topological grid among different platforms.
For example, a network of aircraft, satellites, and ground stations can each contain many transmitters and
receivers, thereby enabling a mapping network to probe the 3-dimensional distribution of turbulence in a
region of atmosphere.
64

Chapter 5
Mid-IR Coherent Free-Space Link through Fog
5.1 Background and Motivation
There is growing interest in Mid-infrared (Mid-IR) free-space optical (FSO) communications [116, 117, 118].
The Mid-IR has various transmission windows (e.g., 3-5 µm and 8-12 µm) of relatively lower atmospheric
attenuation in comparison to the Near-infrared (Near-IR) [32]. Moreover, Mid-IR tends to be less affected
by aerosol/particle scattering (e.g., fog and smoke) and atmospheric turbulence, thereby offering higher
resilience against severe link conditions [119, 41, 120, 34].
There have been reports of Mid-IR FSO links that have used various native Mid-IR devices to generate
Mid-IR data channels for FSO links. These approaches include (1) direct modulation of a Mid-IR laser (e.g.,
quantum/interband cascade lasers) [121, 122, 123, 124, 125, 126, 127, 128, 129] and (2) external modulation
with a Mid-IR modulator (e.g., Stark modulator) [130, 131]. However, only amplitude modulation has been
demonstrated by using native Mid-IR devices for direct-detection systems [121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 131]. There have also been demonstrations of using high-speed modulators in the Cband followed by converting the data signal to the Mid-IR region with periodically poled LiNbO3 (PPLN)
waveguides-based difference frequency generation (DFG) at the transmitter [132, 133, 134, 18, 9, 135]. This
method can enable both phase and amplitude modulation for coherent-detection systems. Unfortunately,
the relatively low DFG conversion efficiency tends to limit the available transmitted Mid-IR power (<∼5
mW).
65

All these prior reports either did not take into account link degradation effects (e.g., turbulence and fog)
or only achieved a relatively low data rate (<1 Gbit/s) under degradation effects. Furthermore, we are not
aware of prior reports demonstrating phase modulation and coherent detection with degrading effects for
bit rates over 100 Mbit/s [133]. A key limitation of the above demonstrations was to produce a high-power
phase modulated data channel for Mid-IR coherent FSO links. A potential solution might be to utilize a
high-power Mid-IR optical parametric oscillator (OPO) [136, 137]. This OPO has been shown to convert a
continuous wave (CW) 1064-nm beam to generate a >1-W Mid-IR beam over a tunable range of ∼2-4 µm
and has been demonstrated for Mid-IR frequency comb generation and spectroscopy [136, 137]. A laudable
goal might be to utilize such an OPO to generate a high-power phase-modulated Mid-IR data channel and
demonstrate it in a coherent FSO link with harsh degradation effects.
In this chapter, we demonstrate an OPO-based high-power transmitter in a 10-Gbit/s quadrature phase
shift keying (QPSK) Mid-IR free-space link through emulated highly attenuating fog. We first generate a
QPSK data channel at ∼1064 nm and then convert it to ∼3400-nm Mid-IR wavelength using the OPO at
the transmitter. The power of the generated Mid-IR data signal is ∼100 mW. The Mid-IR beam propagates
through a free-space link with emulated fog effects. At the receiver, we convert the Mid-IR signal to the
C-band through DFG and achieve coherent detection of the QPSK data channel. Our results show that
bit-error rates (BERs) of the detected Mid-IR QPSK signal can reach below 20% soft-decision forward error
correction (SD-FEC) limit under a fog with ∼18-dB mid-IR power loss, while the same fog causes ∼40-dB
loss for a ∼1550-nm Near-IR data channel. Our Mid-IR link achieves ∼10X higher data rate compared to
prior demonstrations that have considered link degradation effects.
5.2 Concept of OPO-based Mid-IR data transmitter
Optical beams in the Mid-IR wavelength region tend to have better penetration through various link degradation effects as compared to Near-IR beams. For example, under fog conditions, Mid-IR light, having
wavelengths larger than those of the fog particles, experiences less power loss that Near-IR beams of shorter
wavelengths [39, 40]. Therefore, given the same transmitted optical power for Mid-IR and Near-IR links,
66

more Mid-IR power can be received as shown in Fig. 5.1 (a). This enables Mid-IR FSO communication links
to have higher resilience to foggy environments.
Figure 5.1: (a) Comparison between Near-IR and Mid-IR FSO communication links through harsh fog effects.
(b) Concept of using an OPO-based data transmitter for a coherent Mid-IR FSO link. Tx: transmitter; Rx:
receiver; Mod.: modulator.
In our demonstration, we utilize an OPO (TOPTICA DLC TOPO) to create a QPSK channel at ∼3400
nm, as shown in Fig. 5.1(b). We first modulate a CW laser at ∼1064 nm to generate a data-carrying beam.
This data beam at ∼1064 nm (λin) is focused into a singly-resonant OPO cavity. In this cavity, a nonlinear
crystal is placed at the beam waist and is quasi-phase-matched for parametric oscillation. At the output,
the input light is converted to a Mid-IR wavelength at λout 1 and another wavelength at λout 2. The three
wavelengths are related by [137]:
1
λin
=
1
λout 1
+
1
λout 2
(5.1)
67

Since only the light beam at λout 2 is resonant within the OPO, by energy conservation, the spectral structure
on the input light at λin should be transferred to the non-resonant output at λout 1. Therefore, the phaseencoded data modulated on the input light can be transferred to the Mid-IR wavelength. Moreover, λout 1
and λout 2 can be tuned by tuning the position of the nonlinear crystal. Specifically, λout 1 can be tuned
within 2.19-4 µm while the corresponding λout 2 range is 2.07-1.45 µm. In our demonstration, we focus on
the output Mid-IR wavelength λout 1 at ∼3400 nm, which is mainly limited by the operation wavelength of
a PPLN we used at the receiver [18].
Figure 5.2: Different types of Mid-IR data transmitters, including native Mid-IR devices, PPLN-based DFG,
and OPO.
Figure 5.2 shows a comparison between prior approaches and ours for Mid-IR data transmitters. Compared to a direct-modulated Mid-IR laser or externally modulated Mid-IR amplitude modulator (Fig. 5.2(a)),
our approach can enable high- rate phase modulation and support coherent Mid-IR over lossy links. Compared to PPLN-based DFG (Fig. 5.2(b)), our approach can generate much larger transmitted Mid-IR power
68

due to the higher conversion efficiency of the TOPO implementation. Specifically, the TOPO can convert
10-W 1064-nm input light to >1-W Mid-IR output (∼-10-dB efficiency). However, the conversion efficiency
of PPLN-based DFG is ∼-30 dB and the output Mid-IR power is <5 mW in previous demonstrations. The
higher Mid-IR power generated by the TOPO appears to enable an FSO system with higher tolerance to
link loss.
Figure 5.3: Channel data rate and link length for Mid-IR FSO link demonstrations of using different types
of Mid-IR transmitters. Our demonstration is a 3-m link with emulated fog effects that induce up to ∼18-dB
Mid-IR power loss (the same fog induces ∼40-dB power loss for ∼1550-nm). Such a fog-induced attenuation
can roughly correspond to a 100-m foggy link with 50-m visibility or a 4-km link with 1-km visibility.
Figure 5.3 summarizes the channel data rate and link length of previous Mid-IR FSO link demonstrations
using different types of transmitters. Many prior demonstrations didn’t include link degradation effects. Our
demonstration employs emulated fog effects in a 3-m link that induce up to ∼18-dB Mid-IR power loss (the
same fog induces ∼40-dB power loss for ∼1550-nm). Such an emulated fog-induced attenuation roughly
corresponds to a 100-m foggy link with 50-m visibility or a 4-km link with 1-km visibility [138]. The
visibility is defined as the distance where the optical power of a propagating greenlight beam (wavelength at
550 nm) decreases down to 2% of its original value [138]. Moreover, we also achieve ∼10X higher data rate
compared to prior demonstrations that have considered the link degradation effects.
69

5.3 Experimental Setup
Figure 5.4 shows our experimental setup. At the transmitter, we generate a QPSK data channel at ∼1064 nm
by utilizing a phase modulator to modulate a CW laser. The QPSK signal is first amplified by an Ytterbiumdoped fiber amplifier (YDFA) to a power of 10 W and then fed into a TOPO system for nonlinear wavelength
conversion. The TOPO has three free-space ports for output light at different wavelengths, including (1)
converted Mid-IR QPSK signal at ∼3400 nm, (2) converted light at ∼1550 nm, and (3) remaining part of
the input QPSK signal at ∼1064 nm. We open the port for the Mid-IR signal and block the other two ports.
We note that the typical power of the converted Mid-IR signal can be >1 W. However, we measure that the
Mid-IR power is ∼100 mW in our experiment, which might be due to the imperfect alignment inside our
TOPO system. Subsequently, the generated Mid-IR QPSK data signal is transmitted through a free-space
link with a total length of ∼3 m. In order to emulate fog effects on the link performance, a tube filled with
fog is placed in the link. The diameter and length of the tube are ∼15 cm and ∼0.5 m, respectively. The
fog is generated by using a water-based fog machine and we control the strength of fog effects by changing
the density of the fog inside the tube [41].
At the receiver, the Mid-IR data beam is mixed with a pump beam at ∼1064 nm in a PPLN waveguide
and converted to the C band at ∼1550 nm through DFG. The pump beam is generated from a CW laser
at ∼1064 nm and amplified by a YDFA to ∼3 W. The data and pump beams are combined by a dichroic
mirror (DM) and their polarization states are controlled by two half-wave plates (HWPs). A high-pass
filter (HPF) is placed at the output of the PPLN for the converted C-band signal. We optimize the DFG
conversion efficiency by carefully aligning the beams and tuning the temperature of the PPLN. The power of
the output C-band data channel is ∼0.03 mW. Subsequently, the data channel is coupled into a single-mode
fiber, amplified by an EDFA to ∼1 mW, and detected by a coherent receiver with a ∼10-mW local oscillator.
Subsequentially, the data is processed and recovered through online digital signal processing (DSP), including
channel equalization using a constant modulus algorithm, frequency-offset estimation using a 4th-power-FFT
approach, and phase recovery using blind-phase-searching algorithm [139, 50].
To compare the fog-induced degradations between a Mid-IR and a Near-IR link, we generate a separate
∼1550-nm QPSK data signal by using an I/Q modulator and transmit it through the emulated fog along a
70

Figure 5.4: (a) Experimental setup of a coherent Mid-IR communication link through fog using TOPO
to generate a Mid-IR QPSK data channel in the Tx. For comparison, we generate a separate ∼1550-nm
QPSK data beam and transmit it through the foggy link along the similar path as the MIR data beam.
AWG: arbitrary waveform generator, Col.: collimator, DM: dichroic mirror, EDFA: erbium-doped fiber
amplifier, FM: flip mirror, HPF: high-pass filter, HWP: half-wave plate, LO: local oscillator, M: mirror, PC:
polarization controller, PPLN: periodically poled lithium niobate, YDFA: Ytterbium-doped fiber amplifier.
Optical spectra for (b) CW laser at 1064 nm as input to TOPO, (c) the Mid-IR CW carrier generated by
TOPO, (d) the Mid-IR 5-Gbaud QPSK data channel generated by TOPO, and (e) the QPSK data channel
that is converted to the C band at the Rx.
similar path as the Mid-IR data beam. After propagating through the fog, the received data beam is also
coupled into a single-mode fiber, and the QPSK data is detected by a coherent receiver and recovered by
the same DSP procedures as we use for the Mid-IR link.
5.4 Experimental Results
We first measure optical spectra for different wavelengths without fog effects, as shown in Figs. 5.4 (b)-(e).
Figure 5.4 (b) shows the CW laser at ∼1064 nm we used as the input to TOPO. Figs. 5.4 (c) and (d) show
spectra for the Mid-IR signal at ∼3400 nm generated by TOPO when the phase modulation is off and on,
respectively. The results indicate that the phase-encoded QPSK data modulated on the input pump light
71

is converted to the ∼3400-nm Mid-IR output of TOPO. Figure 5.4(e) shows that the Mid-IR QPSK data
channel is converted to ∼1550 nm at the receiver.
Figure 5.5: (a) Measured fog-induced power loss for Near-IR light at 1550-nm and Mid-IR light at ∼3400
nm. (b) Pictures for a green-light beam propagating through three different fog conditions measured in (a).
The green light serves as a guide light to help to visualize the fog strength.
We subsequently compare the fog-induced power loss for Mid-IR light at ∼3400 nm and Near-IR light at
∼1550 nm under various fog conditions. The Mid-/Near-IR power loss is measured by using a Mid-/Near-IR
free-space power meter with a similar size of detective area at the end of the link path. We plot the relative
losses for each link in dB-scale, as shown in Fig. 5.5. The results show that the Mid-IR link experiences
much smaller power losses under the same fog condition. For example, when the Near-IR link has a power
loss of ∼40 dB, the Mid-IR power is only reduced by ∼18 dB (∼150X smaller loss than Near-IR). We fit
our measured data points with a linear regression algorithm to quantify the loss difference for the Mid-IR
and Near-IR links. In our results, the fog-induced power loss for Near-IR link is ∼2.2X larger in dB-scale
than that for the Mid-IR link, hence showing more resilience of the Mid-IR link to fog. Figure 5.5(b) shows
pictures of a green-light beam propagating through three different fog conditions (Fog 1, 2, and 3 measured
in Fig. 5.5(a)). More scattered light can be visually observed when the fog effects become stronger.
In Fig. 5.6(a) and (b), we measure the data constellations and corresponding error vector magnitudes
(EVMs) for 2- and 5-Gbaud Mid-IR QPSK channels under different fog conditions. We show the link
performance for 3 different strengths of fog effects (the same fog conditions as we have shown in Fig. 5.5).
Under the stronger Fog 3, the EVM of 5-Gbaud QPSK data is degraded from 20.2% to 35.5%, which is
72

Figure 5.6: Data constellations and error vector magnitudes (EVMs) for (a) 2-Gbaud and (b) 5-Gbaud MidIR QPSK data channel without fog and with different fog conditions. (c) Data constellations and EVMs for
5-Gbaud Near-IR QPSK data channel. To compare the degradations induced by fog, we intently control the
Near-IR and Mid-IR links to have a similar EVM performance with no fog as shown in (b) and (c).
caused by the larger fog-induced power loss. We note that there is phase noise/distortion shown in the
resulting data constellations (even without fog effects). There are multiple potential reasons that can cause
such phase noise/distortions, including: (i) the laser linewidth of the two CW lasers at ∼1064 nm (>100-
kHz linewidth) and the LO laser at ∼1550 nm (∼10-kHz linewidth), and (ii) the bandwidth of our phase
73

modulator is limited, which induces more phase distortions for a larger-baud-rate signal. More advanced
DSP algorithms might be used to further mitigate these phase distortion effects.
Figure 5.7: BERs of the 5-Gbaud Mid-IR and Near-IR QPSK data channel under different fog. Three data
points for Mid-/Near-IR corresponds to constellations in Fig. 5.6(b)/(c), respectively.
We compare the BER performance of the 5-Gbaud QPSK Mid-IR and Near-IR links under fog as shown
in Fig. 5.6(b) and (c). To focus on the degradations induced by fog, we control the transmitted power for
both links such that they have a similar back-to-back performance when there is no fog. As shown in Fig.
5.6(b) and (c), both links have an EVM of ∼20% without fog. Our results show that the Near-IR QPSK
data channel cannot be recovered under Fog 2 and 3 and the EVMs are larger than 50% due to the large
power loss (>30-dB loss) induced by fog. In Fig. 5.7, we test the BER performance of the two different
links under different fog conditions. The BER of the Near-IR link can reach below the 20% soft-decision
FEC (SD-FEC) only under the relatively weaker Fog 1. However, the Mid-IR QPSK data channel is more
resilient to fog and can reach below the 7% hard-decision FEC (HD-FEC) under Fog 2 and the 20% SD-FEC
under Fog 3.
In order to test the capability of TOPO in supporting higher-order phase-modulation formats, we generate
an 8-PSK data channel at ∼1064 nm and subsequently convert it to ∼3400 nm using TOPO. We measure
the data constellations and EVMs for 1-Gbuad and 2-Gbaud 8-PSK Mid-IR signals without fog effects,
74

as shown in Fig. 5.8. The results indicate that the 8 different phase levels encoded on ∼1064-nm light
can be converted to Mid-IR signal. Although we only demonstrate phase-encoded data modulation for a
single wavelength in this paper, we believe that the OPO-based Mid-IR data transmitter can potentially
support (i) more complex data modulation formats with both amplitude and phase data encoding, and (ii)
wavelength-division multiplexing (WDM) schemes where multiple WDM data channels at ∼1064 nm can be
simultaneously converted to Mid-IR regimes.
Figure 5.8: Data constellations and EVMs for (a) 1-Gbaud and (b) 2-Gbaud Mid-IR 8-PSK data channels
without fog effects.
75

Chapter 6
Conclusion
This thesis discusses free-space optical communications and probing through turbulent media using structured light. It covers multiple topics including structured light, light-matter interaction, optical communications, and optical sensing, most of which are still young and emerging research fields. The research efforts
focused on these topics could potentially enrich the fields of free-space optics and help with the development
of high-capacity and high-resilient free-space optical communication and sensing systems.
76

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Abstract (if available)

Abstract As compared to radio systems, free-space optical (FSO) communication systems hold the promise of (i) higher capacity due to a larger spectrum, and (ii) lower probability of intercept due to the beam’s lower divergence and higher directionality. Many FSO communication systems transmit data-carrying fundamental Gaussian beams. However, recent studies have shown various potential benefits of using structured beams for FSO links.

In general, a structured beam refers to the tailoring of the spatial distribution of a beam’s amplitude and phasefront to exhibit unique properties. There are various types of structured beams (e.g., Laguerre-Gaussian, Bessel-Gaussian beams). One important application of structured beams in FSO links is to utilize their orthogonality to enable the multiplexing of multiple data-carrying beams for enhancing total system capacity.

Data-carrying beams in FSO communications can be affected by different channel impairments. One key impairment is atmospheric turbulence, which can cause power loss and power coupling that can significantly degrade system performance. On the one hand, various approaches have been developed to mitigate such turbulence effects to increase the performance of the link. On the other hand, such information on light-turbulence interaction can also be utilized for sensing applications.

This thesis will discuss (i) pilot-assisted self-coherent free-space optical communication systems that are highly resilient to turbulence degradation, (ii) photorefractive-crystal-based optical phase conjugation to mitigate turbulence in coherent FSO links, (iii) probing turbulence strength distribution along a propagation path using longitudinally structured beams, and (iv) optical parametric oscillator-based high-power data transmitter for a mid-IR coherent free-space link through fog.

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Creator Zhou, Huibin (author)

Core Title Free-space optical communications and probing through turbulent links using structured light

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2024-05

Publication Date 03/07/2024

Defense Date 02/26/2024

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Tag atmospheric turbulence,OAI-PMH Harvest,optical communications,structured light

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