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Spatial modes for optical communications and spatiotemporal beams

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Spatial modes for optical communications and spatiotemporal beams

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i

SPATIAL MODES FOR OPTICAL COMMUNICATIONS AND SPATIOTEMPORAL
BEAMS

by

Kai Pang

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

May 2022

Copyright 2022 Kai Pang
ii

Dedication

This dissertation is dedicated to my parents, Zaiben Pang and Junlan Sun,

for their love and endless support.
iii

Acknowledgements
The journey towards my PhD is long, and I deeply feel that the PhD period is the most
memorable time in my life up to now. I am sincerely grateful to many people for their support and
encouragement. Without their help and devotion, it would not be possible for me to complete this
dissertation.
First of all, I would like to deeply thank my thesis advisor, Professor Alan E. Willner. It is
one of my most precious experiences to work with him at Optics Communications Laboratory
(OCLab). He provides me with enthusiastic support and guidance for my research. In addition, I
have also learned a lot from him about many personal lessons, which I believe will be greatly
helpful and important for my future life.
In addition, I would like to thank Professor Moshe Tur from Tel-Avis University in Israel. I
have learned so much from his broad knowledge and extensive experience. I would also like to
thank my defense committee Professor Stephan W. Haas and Professor Andreas F. Molisch and
also my qualification exam committee Professor Wei Wu and Professor Andrea M. Armani.
I would also like to acknowledge the continuous support and advice from our collaborators.
Specifically, I want to thank Prof. Robert W. Boyd, Dr. M Zahirul Alam, and Dr. Orad Reshef
from University of Ottawa, Dr. Jiapeng Zhao, and Dr. Yiyu Zhou from University of Rochester
for their help with our research projects. Also, I would like to thank Prof. Ayman Abouraddy and
Murat Yessenov from University of Central Florida, Prof. Tobias J. Kippenberg and Maxim
Karpov from EPFL for helpful discussions.
My grateful thanks are also extended to my colleagues Dr. Yongxiong Ren, Dr. Cong Liu,
and Kaiheng Zou for their insightful discussions and collaborations. Moreover, I am grateful for
other current and past OCLab members, Dr. Morteza Ziyadi, Dr. Guodong Xie, Dr. Ahmed
Almaiman, Dr. Amirhossein Mohajerin-Ariaei, Dr. Yinwen Cao, Dr. Long Li, Dr. Zhe Zhao, Zhe
Wang, Dr. Ahmad Fallahpour, Dr. Fatemeh Alishahi, Haoqian Song, Runzhou Zhang, Hao Song,
Huibin Zhou, Karapet Manukyan, Nanzhe Hu, Amir Minoofar, Xinzhou Su, Yuxiang Duan, and
Narek Karapetyan for all their helpful discussions. I would also like to appreciate the tremendous
support from USC staff, particularly, Ms. Diane Demetras, Ms. Corine Wong, Ms. Gerrielyn
Ramos, Ms. Susan Wiedem, and Mr. Theodore Low.
Finally, I want to thank my warm and loving family for their precious help and support in
my life.
iv

Table of Contents
Dedication ....................................................................................................................................... ii
Acknowledgements ........................................................................................................................ iii
List of Figures ................................................................................................................................ vi
Abstract ..................................................................................................................................... xii
Chapter 1 Introduction .....................................................................................................................1
1.1 Orthogonal Spatial Modes for Optical Communications ...................................................1
1.2 Different Spatiotemporal (ST) Beams by Correlating Spatial Modes and Frequencies ....3
1.3 Thesis Outline .....................................................................................................................4
Chapter 2 400-Gbit/s QPSK Free-space Optical Communication Link Based on Four-fold
Multiplexing of Hermite–Gaussian or Laguerre–Gaussian Modes by Varying Both
Modal Indices .................................................................................................................6
2.1 Introduction ........................................................................................................................6
2.2 Concept and Experimental Setup .......................................................................................7
2.3 Experimental Results ..........................................................................................................9
2.4 Conclusion ........................................................................................................................14
Chapter 3 Experimental Mitigation of the Effects of the Limited Size Aperture or Misalignment
by Singular-value-decomposition-based Beam Orthogonalization in a Free-space
Optical Link using Laguerre–Gaussian Modes ............................................................15
3.1 Introduction ......................................................................................................................15
3.2 Concept and Experimental Setup .....................................................................................16
3.3 Experimental Results ........................................................................................................18
3.4 Conclusion ........................................................................................................................22
Chapter 4 Demonstration of a 10 Mbit/s Quantum Communication Link by Encoding Data on Two
Laguerre–Gaussian Modes with Different Radial Indices ...........................................24
4.1 Introduction ......................................................................................................................24
4.2 Concept and Experimental Setup .....................................................................................25
4.3 Experimental Results ........................................................................................................26
4.5 Conclusion ........................................................................................................................31
Chapter 5 Simulation of Near-diffraction- and Near-dispersion-free OAM Pulses with a
Controllable Group Velocity by Combining Multiple Frequencies, Each Carrying a
Bessel Mode .................................................................................................................32
5.1 Introduction ......................................................................................................................32
5.2 Concept .............................................................................................................................33
5.3 Simulation Results ............................................................................................................35
5.4 Conclusion ........................................................................................................................40
v

Chapter 6 Experimental Generation of Near-Diffraction-Free OAM Pulses Having a Controllable
Group Velocity from 1.0069c-0.9933c by Coherently Combining Different Beams of
Multiple Correlated Bessel Modes and Frequencies ...................................................42
6.1 Introduction ......................................................................................................................42
6.2 Concept and Experimental Setup .....................................................................................43
6.3 Experimental Results ........................................................................................................44
6.4 Conclusion ........................................................................................................................46
Chapter 7 Experimental Demonstration of Dynamic Spatiotemporal Structured Beams that Exhibit
Two Orbital-angular-momenta Simultaneously Using a Kerr Frequency Comb ........47
7.1 Introduction ......................................................................................................................47
7.2 Concept and Experimental Setup .....................................................................................48
7.3 Experimental Results ........................................................................................................49
7.5 Conclusion ........................................................................................................................50
References.. ....................................................................................................................................51
vi

List of Figures
Figure 1.1 Intensity profiles and phase fronts of Laguerre–Gaussian (LG) beams with different
azimuthal indices (ℓ) and radial indices (!). ...................................................................1
Figure 1.2 Conceptual diagram using orthogonal spatial modes for MDM FSO
communication links. ......................................................................................................2
Figure 1.3 Concept of data encoding system using orthogonal spatial modes. ........................3
Figure 1.4 Correlation between frequencies and spatial modes for generation of different
dynamic ST beams. .........................................................................................................4
Figure 2.1 Conceptual diagram using four HG modes or LG modes for MDM FSO
communication links. The simulated intensity (a1) HG beams (HG01, HG10, HG03,
and HG30); (a2) LG beams (LG10, LG−10, LG11, and LG−11). Illustration of an MDM
FSO communication link with (b) perfect alignment and a full Rx aperture, (c) perfect
alignment and a limited-size aperture, (d) lateral misalignment, and (e) rotational
misalignment between the Tx and Rx. Tx, transmitter; Rx, receiver. ............................8
Figure 2.2 (a) Experimental setup of an MDM FSO communication system multiplexing
orthogonal HG or LG modes. QPSK, quadrature-phase-shift-keying; EDFA, erbium-
doped fiber amplifier; PC, polarization controller; Col., collimator; SLM, spatial light
modulator; BS, beam splitter; FM, flip mirror; experimental intensity profiles and
receiver phase patterns of (b) four HG and (c) LG beams ..............................................9
Figure 2.3(a) Normalized received total power and power distribution among the four HG
modes when transmitting (a1) HG01 and (a2) HG03 with the aperture diameter.
Normalized received total power and power distribution among the four LG modes
when transmitting (b1) LG10 and (b2) LG11 with the aperture diameter. ...................10
Figure 2.4 (a1)–(a4) Normalized received power among the four HG modes when only one
HG mode is transmitted with various lateral displacements. Normalized received power
among the four LG modes when transmitting (b1) LG10 or (b2) LG11 with various
receiver lateral displacements. The displacement refers to the distance between the beam
center and the receiver in horizontal direction. .............................................................10
Figure 2.5 Normalized received power among the four HG modes or four LG modes when
transmitting (a1) HG10, (a2) HG30, and (b) LG10 with various receiver rotation angles.
The receiver aperture is circular, with a diameter of 4 mm. .........................................13
Figure 2.6 Crosstalk for each mode when all four HG or LG modes are transmitted with
various (a1) and (a2) receiver lateral displacements or (a3) receiver rotation angles.
Measurement of the BER as a function of the OSNR for (b1) all HG modes, (b2) all LG
modes with well alignment, HG10, HG01 andLG10, LG11 with and without, (b3) a
receiver lateral displacement of 0.2 mm, and (b4) HG10, HG30 and LG10, LG11 with
vii

and without a receiver rotation angle of 10°. All four HG or LG beams are transmitted.
B2B, back to back; Disp., displacement.. .....................................................................13
Figure 3.1 (a) Concept of (a1) limited-size aperture and (a2) misalignment effects on an FSO
link using LG modes. (b) Concept diagram of transmitting each data channel on a
designed beam that is combination of multiple LG modes (multi-mode beam) to mitigate
the effects of the limited-size aperture or misalignments an MDM link. Ch: Channel;
Tx: Transmitter; Rx: Receiver. .....................................................................................18
Figure 3.2 (a) Experimental setup of a four-channel MDM FSO system. QPSK: quadrature
phase-shift keying; EDFA: erbium-doped fiber amplifier; PC: polarization controller;
Col.: collimator; SLM: spatial light modulator; BS: beam splitter. (b) The Tx and Rx
SLM patterns for (b1) single-mode beams, and (b2) designed multi-mode beams. Here,
the radius of receiver aperture is 1.0 mm. .....................................................................19
Figure 3.3 (a1) The complex weights of LG modes carrying data channel 1 and 2 without and
with SVD approach; (a2) Measured intensity profiles of single-mode beams and multi-
mode beams when aperture radius is 1.0 mm. (b) Measured limited-size-aperture
induced (b1) power loss and (b2) crosstalk under various aperture radii when
transmitting data channels on single-mode beams or multi-mode beams (LG10, LG11
and LG12 modes). For each aperture radius, the two multi-mode beams are unique due
to different transmission matrix. ...................................................................................21
Figure 3.4 (a) Measured displacement-induced (a1, b1) power loss and (a2, b2) channel
crosstalk with horizontal displacements when transmitting channels on single-mode
beams or multi-mode beams. ........................................................................................20
Figure 3.5 Measured constellations of four data channels when transmitting each channel on
(a) a single-mode beam or (b) a multi-mode beam. (c) Experimental BERs of four data
channels with OSNR when using multi-mode beams. B2B: back to back; FEC: forward
error correction. .............................................................................................................20
Figure 4.1 (a) Two LGℓp modes with the same ℓ but different p values for data encoding
(Case 1: LG00 and LG01; case 2: LG10 and LG11) (b) Concept of quantum data
encoding based on two LGℓp modes. ...........................................................................26
Figure 4.2 (a) Experimental setup of a quantum communication link based on LGℓp modes.
AWG: arbitrary waveform generator; PC: polarization controller; Col.: collimator; SLM:
spatial light modulator; BS: beam splitter; FM: flip mirror; ATT: attenuator; SPD: single
photon detector; DSP: digital signal processing. ..........................................................27
Figure 4.3 (a) Normalized generated waveforms of the two branches (a1-a2) and their
combination (a3) at the transmitter. (b) Normalized waveforms of the two branches and
their combination received by SPDs using the mode set {LG00 and LG01}. The symbol
period is 100 ns. ............................................................................................................28
Figure 4.4 Crosstalk analysis between LG00 and LG01 modes as a function of the circle radius
(R) of the receiver pattern using the quantum measurement approach. (a1 - a2)
viii

Experimental intensity profiles of the LG00 and LG01 modes in the classical domain.
(a3 - a4) Receiver phase patterns for LG00 and LG01 modes. Tx: LG00, Rx: LG00:
LG00 is transmitted and LG00 is received. The optimized R value is 0.9 mm. ...........30
Figure 4.5 (a1 – a3). Crosstalk matrices between LG00 and LG01 modes for different R values
(0.5, 0.9, and 1.3 mm). (b) Registered photon rates and (c) QSERs with different R (0.5,
0.9, and 1.3 mm) of the receiver phase pattern as a function of average photon number
per pulse (μ). The transmitted data rate is 10 Mbit/s. The optimized R value is 0.9 mm.
.......................................................................................................................................31
Figure 4.6 Crosstalk analysis between LG10 and LG11 modes as a function of the circle radius
(R) of the receiver pattern using the quantum measurement approach. (a1 - a2)
Experimental intensity profiles of the LG10 and LG11 modes in the classical domain.
(a3 - a4) Receiver phase patterns for the LG00 and LG01 modes. The optimized R value
is 0.9 mm.. .....................................................................................................................32
Figure 4.7 (a1 – a3) Crosstalk matrices between LG10 and LG11 modes for different R values
(0.5, 0.9 and 1.3 mm). (b) Registered photon rates and (c) QSERs with different R (0.5,
0.9, and 1.3 mm) of the receiver phase pattern as a function of average photon number
per pulse (μ). The transmitted data rate is 10 Mbit/s. The optimized R value is 0.9 mm...
.......................................................................................................................................32
Figure 5.1 (a) When non-diffractive Bessel beams with the same frequency (f) and different
spatial frequencies kr are coherently combined, a non-diffractive CW OAM beam is
generated; (b) When Bessel beams with different f but the same ℓ and kr value are
coherently combined, the pulse envelop becomes distorted as it propagates due to the
nonlinear relation between f and kz. As a result, a dispersive OAM pulse is generated.
BM: Bessel mode. .........................................................................................................36
Figure 5.2 Each frequency carries a specific Bessel mode with the same ℓ but a unique kr,
leading to a linear relation between f and kz based on "
!
= "
"
+(2'( ) ⁄ − "
"
)/tan2.
As a result, a near-diffraction-free and near-dispersion-free OAM pulse could be
generated. In addition, its group velocity could be tuned by changing θ. .....................37
Figure 5.3 The spatio-temporal spectra for (a1) the dispersive OAM pulse which corresponds
to the cases of Fig. 5.1 (b). (a2) The simulated intensity profiles |4(5,7,8,9)|
#
with
time at z = 0, 1, 10 m for this pulse; (a3) The simulated temporal pulse shapes at the
transverse position where the intensity is maximum in (a2).. .......................................38
Figure 5.4 The spatio-temporal spectra for (a1) the non-dispersive OAM pulse which
corresponds to the cases of Fig. 5.2. (a2) The simulated intensity profiles |4(5,7,8,9)|
#

with time at z = 0, 1, 10 m for this pulse; (a3) The simulated temporal pulse shapes at
the transverse position where the intensity is maximum in (a2). ..................................39
Figure 5.5 Simulated intensity and phase profiles of the non-dispersive OAM pulse at different
times (-0.25, 0, 0.25 ps) when z is 0 m.. .......................................................................39
ix

Figure 5.6 (a) Time averaged intensity profile (I(:,8) = ∫
|4(:,8,9)|
#
dt) at different
propagation distances corresponding to the space-time correlation in Fig. 5.2. (b)
Theoretical and simulated group velocity of the OAM pulse for different values of the
tilt angle θ.. ....................................................................................................................40
Figure 5.7 (a) Temporal pulse duration of the non-dispersive OAM pulse for different
numbers of frequencies. (b) The pulse duration (the blue curve with square markers)
and group velocity (the orange curve with circular markers) of the non-dispersive OAM
pulse in Fig. 5.4 with various ℓ values when the number of frequencies is 20... ..........40
Figure 6.1 (a1) Concept for the generation of near-diffraction-free OAM pulses with a
controllable group velocity. BM: Bessel mode. ............................................................46
Figure 6.2 (a) Experimental setup for generation and detection of near-diffraction-free OAM
pulses with a controllable group velocity. EDFA: erbium-doped fiber amplifier; PC:
polarization controller; Col.: collimator; SLM: spatial light modulator; BS: beam splitter.
The measured intensity profiles |4(5,7 = 0,8 = 0,9)|
#
with time and the transverse
profiles at the reference time delay of 0 for (b) OAM +1 and (c) OAM +3 pulses. .....46
Figure 6.3 The measured intensity profiles |4(5,7 = 0,8 = 0,9)|
#
with time and the
transverse profiles at the reference time delay of 0 for (a) OAM +1 and (b) OAM +3
pulses. ............................................................................................................................47
Figure 6.4 (a) The theoretical and measured group velocities of OAM pulses for different θ
values. ...........................................................................................................................48
Figure 6.5 (a) The time averaged intensity profile (=(5,8) = ∫
|4(5,8,9)|
#
>9) at different z
for OAM pulses based on (a1) LG modes and (a2) our approach. (b) The measured
D0.5Imax values for various transmitter aperture sizes.. ..................................................48
Figure 7.1 (a) When a single frequency line carries a single LG mode, a rotating LG beam at
the central axis is generated; (b) When a single frequency line carries multiple LG
modes, a spatially offset rotating LG beam can be generated; (c) When combining
multiple frequency lines with each carrying a superposition of multiple LG modes
containing one unique ℓ value and multiple p values, a revolving and rotating LG beam
is generated. ..................................................................................................................52
Figure 7.2 (a) Experimental setup for generation and detection of ST beams. EDFA: erbium-
doped fiber amplifier; PC: polarization controller; Col.: collimator; FROG: frequency-
resolved optical gating; SLM: spatial light modulator; BS: beam splitter. (b) Optical
spectrum of the selected nine frequency lines after the waveshaper. ...........................52
Figure 7.3 (a) Simulated amplitude and phase of the complex coefficients of all LG modes
used for superposition. (b) The intensity and phase profiles of the reconstructed
revolving and rotating LG beam in time. ......................................................................53
x

Figure 7.4 (a) Power distribution on light beams with different ℓ values. (b) The mode purity
of the ST beam when using different number of frequency lines for different revolving
radii (R). ........................................................................................................................54
xi

Abstract
“Structured light” is a topic of growing interest in the optics community, not only for its
inherent properties but also for its possible applications in communications, sensing, imaging, or
micromanipulation. In general, structured light refers to an optical beam with a tailored spatial
amplitude/phase distribution and corresponding unique properties. One type of structured light
involves beams carrying orbital angular momentum (OAM). As a spatially orthogonal modal basis
set, OAM modes has been utilized in different applications, such as optical communications,
optical sensing, optical imaging.
The first part of this dissertation will study orthogonal spatial modes for optical
communications including high-speed classical communications and high-dimensional quantum
communications. This part includes: (i) experimental demonstration of free-space mode-division-
multiplexing using Laguerre-Gaussian (LG) modes or Hermite-Gaussian (HG) modes; (ii)
experimental mitigation of the effects of the limited-size aperture or misalignment by transmitting
each of four data channels on multiple LG modes in a free-space optical (FSO) communication
link; (iii) experimental demonstration of a quantum link using data encoding on two LG modes.
The second part of the dissertation discusses some interesting spatiotemporal (ST) beams
which correlate the spatial modes and frequencies. The potential applications of these dynamic ST
beams might include optical communications, optical imaging and sensing. Some simulations are
implemented, and then the experimental generation and detection of these dynamic ST beams are
also explored. By using Kerr frequency comb, two types of dynamic ST beams are experimentally
generated and detected. One is the structured beams that exhibit two orbital-angular-momenta
simultaneously. The other one is the near-diffraction-free OAM pulses having a controllable group
velocity by coherently combining different beams of multiple correlated Bessel modes and
frequencies.
1

Chapter 1 Introduction
This chapter will first introduce the basic concept of the orbital angular momentum (OAM)
modes and Laguerre-Gaussian (LG) modes. In addition, the use of OAM modes and LG in
different communication schemes are also discussed. Then, a brief overview of different
spatiotemporal (ST) beams will be presented, which might have potential applications in optical
communications. Finally, the outline of the rest of the dissertation will be explained.
1.1 Orthogonal Spatial Modes for Optical Communications
Achieving higher data transmission capacity is one of the primary interests of the optical
communications community. This has led to the investigation of using different physical properties
of a lightwave for data transmission. In the recent years, spatially orthogonal modes have been
under intense investigation for improving the transmission capacity of free-space optical (FSO)
communication links. As an example, LGℓp modes is one possible spatial modal basis set with two
modal indices: (1) ℓ represents the number of 2p phase shifts in the azimuthal direction and the
size of the ring grows with ℓ; and (2) !+1 represents the number of concentric amplitude rings, as
shown in Fig. 1.1 [1, 2]. It should be noted that OAM modes are a subset of the LG modal group
[3-5].

Figure 1.1 Intensity profiles and phase fronts of Laguerre–Gaussian (LG) beams with different azimuthal indices (ℓ)
and radial indices (!).
Theoretically, two LG beams with the same beam waist are orthogonal to each other if they
have either a different ! value or a different ℓ value. This orthogonality is of crucial benefit for a
communication engineer. It implies that light beams with different LG modes can be efficiently
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separated. In addition, the possibility of utilizing the orthogonality of LG beams with different !
values along with different ℓ values could provide more communication modes as well as form a
complete 2-dimensional (2D) orthogonal basis.
Generally, there are two different ways to take advantage of the distinction between LG
beams with different ℓ values or ! values in communications. The first approach, employed in
most of the recent demonstrations, is mode-division-multiplexing (MDM), in which each data-
carrying beam occupies a different mode from the LG modal basis set [6]. The orthogonality of
LG beams with different states could enable efficient beam (de)multiplexing and spatially
overlapping transmission with little inherent crosstalk, thereby providing independent data carriers
in addition to wavelength and polarization [7, 8]. A typical scheme of multiplexing of LG beams
is conceptually depicted in Fig. 1.2. In the process, independent data-carrying beams with different
orthogonal modes are multiplexed at the transmitter and then co-propagate in free space. At the
receiver, the received orthogonal beams are demultiplexed and demodulated with little crosstalk.
An obvious benefit of such technique is the improvement in system spectral efficiency, since the
same bandwidth

Figure 1.2 Conceptual diagram using orthogonal spatial modes for MDM FSO communication links.
can be reused for additional data channels.
In the second approach, N different LG modes can be encoded as N different data symbols
representing “0”, “1”, ..., “N−1”. A sequence of LG modes sent by the transmitter therefore
represents data information. At the receiver, the data can be decoded by checking the received LG
modes [9, 10]. In this approach, photon efficiency can be increased if a beam or single photon is
Mode
multiplexing
Mode
demultiplexing
Mode 1
…
Mode
generation
…
…
Mode n
Mode
generation
Mode
detection
Mode
detection
Transmitter
Receiver
Tx aperture Rx aperture
Free-space
propagation
Mode-division-multiplexed system
3

encoded into one of many possible LG modes, thus making a large alphabet for possible data
symbols within a discrete time window. The achievable data bits per photon increases as [?@A
#
(B)]
[11-13].

Figure 1.3 Concept of data encoding system using orthogonal spatial modes.
1.2 Different Spatiotemporal (ST) Beams by Correlating Spatial Modes
and Frequencies
As degrees of freedom for a light beam, both the frequency and spatial modes can potentially
affect some properties of light beams, as shown in Fig. 1.4. In general, when single frequency
carries multiple spatial modes, its spatial distribution at a given propagating distance from a
transmitter can be tailored using coherent interference by using a complex-weighted combination
of multiple spatial modes from a complete modal basis set. As an example, LG modal basis set
with 2D indices (ℓ and p) are one of such modal bases. In this case, it is continuous-wave (CW)
beam and exhibits a nearly any fixed spatial distribution, which is determined by the complex
coefficients of each LG components [14, 15]. However, when combining multiple frequency lines
with each carrying a superposition of multiple spatial modes, some dynamic and interesting
characteristics might occur. In this case, each frequency is correlated with multiple spatial modes,
and thus the interference condition between different frequencies might change with time.
Consequently, some propagation characteristics might be dynamically affected.
Previously, it has been shown that a light beam can be created to exhibit some dynamic
features by correlating frequencies to spatial modes [15-20], including: (i) a Gaussian-like beam
dot that exhibits dynamic circular revolution at a given propagation distance [15-17]; (ii) a
Gaussian-like beam dot that can dynamically move up-and-down in a linear fashion at a given
propagation distance [18]; a light beam that exhibits dynamic rotation around its center at a given
Time
T1 T2 T3 T4
…
…
Data encoding system using spatial modes
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4

distance [19]; a light beam that exhibits dynamic rotation around its center (azimuthal dimension)
as well as in-and-out linear radial movement at a given propagation distance [20]. A laudable goal
is to theoretically and experimentally investigate the generation of different interesting dynamic
ST beams.

Figure 1.4 Correlation between frequencies and spatial modes for generation of different dynamic ST beams.
1.3 Thesis Outline
This dissertation mainly focuses on the orthogonal spatial modes and its applications. The
rest of the dissertation is organized in the following manner. The utilization of spatial modes in
classical and quantum communication links will be discussed from Chapter 2 to 4. Chapters 5 to
7 will present the use of spatial modes in generation of some interesting dynamic ST beams by
using Kerr frequency comb.
To be specific, Chapter 2 presents the experimental demonstration of FSO communication
links based on mode-division-multiplexing of four LG modes or Hermite-Gaussian (HG) modes.
Chapter 3 investigates the utilization of singular-value-decomposition-based beam
orthogonalization to experimentally mitigate the effects of the limited size aperture or
misalignment in an FSO link using LG modes. Chapter 4 shows the demonstration of a 10 Mbit/s
quantum communication link by encoding data on two LG modes with different radial indices.
Chapter 5 presents the simulation of near-diffraction-free and near-dispersion-free OAM pulses
5

with a controllable group velocity. Chapter 6 shows a method to experimentally generate and
detect such near-diffraction-free OAM pulses having a controllable group velocity by coherently
combining different beams of multiple correlated Bessel modes and frequencies. Chapter 7
presents the experimental demonstration of structured beams that exhibit two orbital-angular-
momenta simultaneously.
6

Chapter 2 400-Gbit/s QPSK Free-space Optical
Communication Link Based on Four-fold Multiplexing of
Hermite–Gaussian or Laguerre–Gaussian Modes by
Varying Both Modal Indices
2.1 Introduction
Space-division-multiplexing (SDM) is a potentially significant technique for optical
communication systems, since multiple independent data-carrying beams can simultaneously
propagate over the same spatial medium, thereby increasing aggregate transmission capacity [6].
A subset of SDM is MDM, which could be achieved by different types of orthogonal modal basis
sets. For example, HG modes and LG modes are two types of orthogonal modal basis sets that can
be utilized in MDM communication links. In general, the beam properties of a given orthogonal
modal basis can be fully described by two indices so that these two indices form a 2D set of
orthogonal values, e.g., HG beams can be characterized by (m, n) referring to the x and y directions,
respectively, and LG beams can be characterized by (ℓ, p) by referring to the azimuthal and radial
directions, respectively [1,2, 21–23]. The different mode-multiplexed beams can be chosen from
the different orthogonal values of the modal basis sets.
There have been several reported demonstrations using MDM to achieve higher capacity in
FSO data communication links [24–28]. The modal basis sets used for the FSO transmission of
multiple multiplexed data-carrying beams have included LG modes and vector modes; we note
that orbital-angular-momentum modes are a subset of the LG modal group, and most experimental
demonstrations of MDM links using LG modes focused on OAM modes [3]. To the best of our
knowledge, there have been few FSO reports of MDM for data transmission that (1) use multiple
multiplexed HG modes, or (2) simultaneously utilize and vary both of the two modal indices for
any modal group (HG or LG modes), which could potentially provide a larger two-dimensional
modal space of orthogonal data-carrying channels and increase the transmission capacity for a
communication link.
In this chapter, experimental demonstration of a 400-Gbit/s quadrature phase shift keying (QPSK)
FSO communication link based on the four-fold multiplexing of HG or LG modes is presented.
7

By simultaneously utilizing both modal indices for the HG or LG modes, we transmit four channels,
achieving power penalties of < 4 dB for all channels. Moreover, the performance degradation
dependent on aperture size, as well as the lateral and rotational misalignment between the
transmitter/receiver apertures are also shown. This experimental analysis may help in providing
more insights into the system design, such as minimizing channel crosstalk in systems with a
limited aperture size; many of these insights can also be determined by analyzing the mode profiles.
Various system effects are measured, including (1) power loss and crosstalk for both HG and LG
modes caused by a limited-size aperture, (2) crosstalk increase for LG modes, but small crosstalk
increase for HG0n or HGm0 modes under lateral misalignment between the transmitter and receiver,
and (3) crosstalk increase for HG modes, but small crosstalk increase for LG modes under a
rotational misalignment between the transmitter and receiver.
2.2 Concept and Experimental Setup
Figure 2.1 shows the concept of an MDM FSO link using four orthogonal modes. HGmn
modes (HG01, HG10, HG03, and HG30) or LGℓp modes (LG10, LG−10, LG11, and LG−11) are selected
as examples of the simultaneous utilization of both indices. Specifically, we choose these modes
due to their pairwise symmetry, which helps reduce the mode-dependent penalty (e.g., HG01/HG10,
HG03/HG30, LG10/LG−10, and LG11/LG−11). In the process, independent data-carrying beams with
different indices are multiplexed at the transmitter and then co-propagate in free space. Due to the
orthogonality between the four HG modes or the four LG modes, the received beams can be
demultiplexed and demodulated at the receiver with little crosstalk.
In an ideal case, the receiver could receive the whole transmitted beams and be perfectly
aligned with the transmitter, as shown in Fig. 2.1(b). For a non-ideal case, there might be
misalignment between them, which may cause increased system degradation, when
simultaneously utilizing both modal indices for HG or LG sets [29–31]. Moreover, the receiver
might have a limited aperture size, which would produce power loss and crosstalk, as shown in
Fig. 2.1 (c) [29–31]. The lateral and rotational misalignments between the transmitter and receiver
are shown in Figs. 2.1(d) and 2.1(e), respectively. These issues might have different effects on HG
as opposed to LG modes due to these modes’ different two-dimensional structural characteristics
[29–31].
8

Figure 2.1 Conceptual diagram using four HG modes or LG modes for MDM FSO communication links. The
simulated intensity (a1) HG beams (HG01, HG10, HG03, and HG30); (a2) LG beams (LG10, LG−10, LG11, and LG−11).
Illustration of an MDM FSO communication link with (b) perfect alignment and a full Rx aperture, (c) perfect
alignment and a limited-size aperture, (d) lateral misalignment, and (e) rotational misalignment between the Tx and
Rx. Tx, transmitter; Rx, receiver.
Figure 2.2(a) illustrates the experimental setup. First, a 100 Gbits/s signal is generated by a
bit pattern generator, and it is modulated on the beam from an external laser source at 1550 nm by
a QPSK transmitter. In the transmitter, the incoming beam is split into two branches; then it is
modulated by a Mach–Zehnder interferometer-based modulators at 50 Gbaud and, finally,
combined for output. Subsequently, the signal is amplified by a 15-dB gain EDFA. Then the signal
is split into four branches by a ∼2 dB excess loss coupler. Each branch is sent into a collimator
that generates a collimated Gaussian beam with a diameter of 3 mm at which the intensity values
fall to 1/e
2
of their axial values. The resulting Gaussian beams are sent to two spatial light
modulators (SLMs). The designed phase holograms on each half of the screens are used to generate
the four different HG or four different LG by tailoring the amplitude and phase profile of the
incoming beam. The loss induced by SLMs for HG and LG mode generation is∼8 dB. These four
outputs are multiplexed using three beam splitters. After a coaxial free-space propagation of∼1 m
with minimum excess loss in the lab (without additional turbulence effects), the multiplexed beams
are directed to SLM-3 at the receiver side, which loads the conjugate phase profile of the desired
mode. The experimental intensity profiles and the receiver phase patterns loaded on the SLM-3
for the corresponding HG or LG beams are also shown in Figs. 2.2 (b) and 2.2 (c), respectively.
These phase patterns have circular aperture shapes, whose sizes are defined as aperture sizes in
our experiment. The loss of SLM-3 is ∼2 dB, and the conversion efficiency is ∼−1 dB for the four
LG modes, HG01 and HG10, and ∼−2 dB for HG03 and HG30. Finally, the resulting beam is coupled
into an SMF for coherent detection and bit error rate (BER) measurements.
9

Figure 2.2 (a) Experimental setup of an MDM FSO communication system multiplexing orthogonal HG or LG modes.
QPSK, quadrature-phase-shift-keying; EDFA, erbium-doped fiber amplifier; PC, polarization controller; Col.,
collimator; SLM, spatial light modulator; BS, beam splitter; FM, flip mirror; experimental intensity profiles and
receiver phase patterns of (b) four HG and (c) LG beams.
2.3 Experimental Results
A. Effects of the aperture size, lateral displacement, and receiver rotation on the
system performance
To evaluate the effect of aperture size on the system performance, we measure the received total power and the
power on each of the four HG modes or four LG modes with various aperture sizes, while only one is
transmitted, as shown in Fig. 2.3. In all cases, the power values are normalized to the received total power of
the desired modes. Such total power is tuned to be almost the same value when the aperture diameter is 4 mm.
The received power of the desired mode decreases when decreasing the aperture size of the pattern on SLM-3,
because the received beam is too large to be fully captured. We find that an aperture diameter of 4 mm is large
enough to avoid obvious power loss for all these modes. Moreover, we observe a power coupling for both the
HG and LG modal bases, from the desired mode to other modes when the aperture size is smaller than the
beam diameter of 3 mm. For example, when transmitting the HG01 mode and reducing the aperture
size, a power coupling and crosstalk occur from HG01 into HG03, as shown in Fig. 2.3 (a1). Here,
we only choose two HG and two LG modes, and it is expected that the other modes (HG10 & HG30
and LG−10 & LG−11) show similar trends.
10

Figure 2.3 (a) Normalized received total power and power distribution among the four HG modes when transmitting
(a1) HG01 and (a2) HG03 with the aperture diameter. Normalized received total power and power distribution among
the four LG modes when transmitting (b1) LG10 and (b2) LG11 with the aperture diameter.

Figure 2.4 (a1)–(a4) Normalized received power among the four HG modes when only one HG mode is transmitted
with various lateral displacements. Normalized received power among the four LG modes when transmitting (b1)
11

LG10 or (b2) LG11 with various receiver lateral displacements. The displacement refers to the distance between the
beam center and the receiver in horizontal direction.
Lateral displacement of the receiver from the optimal line of sight is studied in Fig. 2.4,
where only one HG or LG mode is with various horizontal displacements emulated by horizontally
moving the receiver pattern on SLM-3. The receiver aperture diameter is 4 mm, and the beam spot
size is < 4 mm. Figure 2.4 (a) shows that when transmitting the HG01 or HG03 beams, the power
on the desired modes does not change too much with the horizontal displacement. This might be
caused by the invariance of the receiver phase patterns with the horizontal displacement due to
their axial symmetry [30]. However, when transmitting HG10 or HG30 beams, we observe modal
coupling occurs from the desired modes to other modes with the horizontal displacement, which
would affect their crosstalk behaviors. Figure 2.4 (b) shows the cases of LG beams with the lateral
displacement. The results show that as the horizontal displacement increases, it could affect the
power coupled into other modes, while the power on the desired mode decreases. This is due to
the orthogonality between the LG modes degrades with a lateral misalignment. Here, we only
present the case of LG10 and LG11. Furthermore, LG-10 and LG-11 have the similar trends.
The influence of receiver rotation on system performance is also explored. Figure 2.5 (a)
presents the received power on the four HG modes when HG10 or HG30
is transmitted, in which the rotation is emulated by rotating the receiver patterns on SLM-3. We
observe that the power of the desired mode decreases gradually with the rotational angle. Moreover,
a larger rotation angle could also cause a higher power coupling into some other modes. However,
for LG10 modes, the results show that the power change on the four LG modes is negligible when
the receiver rotates, which might be due to the rotational symmetry property of LG modes.
Furthermore, we could expect the performance trends to be similar when transmitting other LG
modes.
12

Figure 2.5 Normalized received power among the four HG modes or four LG modes when transmitting (a1) HG10,
(a2) HG30, and (b) LG10 with various receiver rotation angles. The receiver aperture is circular, with a diameter of 4
mm.
B. Bit error rate (BER) performance of the MDM links
For the BER measurements, we transmit all four HG or all four LG modes simultaneously,
each mode carrying a 100 Gbit/s QPSK signal. First, we measure the crosstalk for all the HG or
LG modes with horizontal displacement and rotation angle, as shown in Figs. 2.5(a1)– 2.5(a3).
Here, the crosstalk for a certain mode refers to the

13

Figure 2.6 Crosstalk for each mode when all four HG or LG modes are transmitted with various (a1) and (a2) receiver
lateral displacements or (a3) receiver rotation angles. Measurement of the BER as a function of the OSNR for (b1) all
HG modes, (b2) all LG modes with well alignment, HG10, HG01 andLG10, LG11 with and without, (b3) a receiver
lateral displacement of 0.2 mm, and (b4) HG10, HG30 and LG10, LG11 with and without a receiver rotation angle of
10°. All four HG or LG beams are transmitted. B2B, back to back; Disp., displacement.
power coupled from all the other three modes over the power on the desired mode. We note that
for all four LG modes, the crosstalk increases with the displacement. However, for HG01 and HG03,
the crosstalk does not change very much with horizontal displacement due to the relative
invariance of their receiver phase patterns with the lateral displacement [30]. In addition, we
observe that there is a crosstalk peak for HG30 at the displacement of 0.3 mm. This might be caused
by the largest mismatch between the HG30 beam and the receiver phase pattern, which causes a
lower received power for the HG30 beam. Furthermore, when the receiver rotation angle increases,
the crosstalk for LG modes remains relatively unchanged, whereas the crosstalk increases rapidly
for the HG modes. Figures 2.5(b1) and 2.5(b2) show the measured BERs of all four HG modes or
four LG modes as a function of optical signal-to-noise ratio (OSNR) with the well-aligned
transmitter and receiver, i.e., without lateral or rotational misalignment. The results show that at a
BER of 3.8 × 10
−3
[the 7% overhead forward error correction (FEC) limit], the power penalties of
the four LG channels are < 2 dB. However, for HG channels, HG01 and HG03 have higher power
penalties than HG10 and HG30 due to their higher crosstalk of ∼3 and ∼2 dB, respectively, which
14

could be caused by the imperfect alignment. This misalignment issue may affect HG and LG
modes differently, which could potentially cause the crosstalk difference between HG and LG
modes. Figures 2.5(b3) and 2.5(b4) show the measured BERs for two HG modes and two LG
modes with a receiver lateral displacement of 0.2 mm and a rotation angle of 10°, respectively.
We note that the higher crosstalk for HG10 over HG01 shown in Fig. 2.5(a1), is reflected in its
higher BER power penalty. Again, LG10 and LG11 modes show little BER power penalties, which
is also consistent with their crosstalk behaviors in Fig. 2.5(a2). With a rotation angle of 10°, higher
power penalties exist for HG modes due to the higher crosstalk. However, for LG modes, the
impact of rotation is relatively small due to the rotational symmetry property. Since HG10 and HG01
behave similarly for a given receiver rotation angle due to their symmetry, we choose non-
symmetric HG10 and HG30 for the BER analysis due to their bigger BER performance difference.
2.4 Conclusion
In conclusion, we experimentally demonstrate the four-fold multiplexing of HGmn or LGlp
modes to achieve a 400-Gbit/s QPSK free-space optical communication link. In this experiment,
both modal indices for the HG and LG modes are simultaneously utilized to achieve a larger
potential orthogonal modal space. Moreover, we also investigate and compare the effects of
aperture size, lateral displacement, and rotation on the system.
Our study uses a circular transmitter/receiver aperture. This was due to that we use circular
lenses and fibers. However, a rectangular aperture might be another choice in an MDM FSO link.
Although a different shape would likely lead to different results for HG or LG modes, we believe
that power loss and crosstalk trends shown in this paper would be similar for rectangular apertures.
15

Chapter 3 Experimental Mitigation of the Effects of the
Limited Size Aperture or Misalignment by Singular-
value-decomposition-based Beam Orthogonalization in a
Free-space Optical Link using Laguerre–Gaussian Modes
3.1 Introduction
As mentioned in previous chapter, an MDM link can multiplex multiple beams, each
composed of a single LG mode [24, 32, 33]. Unfortunately, such MDM FSO links can suffer from
degrading effects that can be induced by a limited-size receiver aperture or misalignments between
the transmitter and receiver apertures. These two issues can cause deleterious signal-power loss
and modal-power-coupling crosstalk in MDM links in which each channel is transmitted on a beam
using a single OAM mode [29, 34-37].
In general, approaches that were utilized to reduce the modal crosstalk caused by a limited-
aperture receiver include the following: (i) Each data channel is transmitted on a beam composed
of one carefully-selected OAM mode such that the received channels on different beams remain
orthogonal [35, 36]; and (ii) Carefully choose specific modes at the transmitter and carefully
choose receiver aperture to reduce the crosstalk such that cross-coupling does not significantly
appear at the other channel [37].
In our approach, each data channel is transmitted on a single beam composed of a unique set
of LG modes. By carefully designing the modal complex weights, all data channels could be
spatially tailored and remain orthogonal at the receiver. This could simultaneously mitigate power
loss and crosstalk induced by limited-size apertures or Tx-Rx misalignments [38].
In this chapter, experimental demonstration of the mitigation of the power loss and crosstalk
induced by the limited-size aperture or misalignments in a 400-Gbit/s four-channel FSO link is
presented. In our approach, we follow these steps: (i) measure the transmission matrix H for the
link that has a limited-size aperture or misalignments (ii) perform SVD on the transmission matrix
to find the U, Σ, and V complex matrices; (iii) transmit each data channel on multi-mode beam
with weights according to the V matrix; and (iv) apply the U matrix to the receiver demultiplexer.
Experimental results when transmitting multi-mode beams instead of single-mode beams show
16

that: (a) with a limited-size aperture, the power loss and crosstalk could be reduced by ~8 and ~23
dB, respectively; and (b) with misalignments, the power loss and crosstalk could be reduced by
~15 and ~40 dB, respectively.
3.2 Concept and Experimental Setup
In a perfectly-aligned LG mode multiplexed FSO link, the aperture of the Rx is large enough
to receive the whole transmitted beam with little power loss and crosstalk. However, if the Rx
aperture size is limited, only a part of the power would be collected. Moreover, this issue could also
cause power coupling from the desired LG mode to other LG modes with the same l value but
different p values, thereby increasing both power loss and crosstalk, as shown in Fig. 3.1 (a1) [31].
In addition, when there is a Tx-Rx misalignment, the power on the desired LG mode would be
coupled to other LG modes with different l or p values. This could induce both power loss and
crosstalk, as shown in Fig. 3.1 (a2) [29].
Figure 3.1 (b) presents the concept of transmitting each data channel on a multi-mode beam
to mitigate effects of the limited-size aperture or misalignments in an MDM link. Generally, the
mode coupling between a set of LG modes in a given link can be described as a complex
transmission matrix H [32]. Our approach is based on SVD of H (D= E∙∑∙H
∗
), which is usually
utilized to find orthogonal basis at the Tx and Rx with no interference between data channels [39-
41]. At the Tx, the multi-mode beams are generated by complex combinations of multiple LG
modes, of which the complex weights are given by the orthogonal column vectors of the V. After
passing through a given link with a limited-size aperture or misalignments, the resulting beams on
different channels are still composed of multiple LG modes, of which the complex weights are the
orthogonal row vectors multiplied by singular values in ∑ matrix. This means that these beams are
still orthogonal to each other, and thus can be demultiplexed with little crosstalk based on the
orthogonal row vectors of the inverse of the U matrix. Besides, intensity profiles of the transmitted
beams would be spatially shaped, which might also reduce the power loss caused by the limited-
size aperture or misalignment.
17

Figure 3.1 (a) Concept of (a1) limited-size aperture and (a2) misalignment effects on an FSO link using LG modes.
(b) Concept diagram of transmitting each data channel on a designed beam that is combination of multiple LG modes
(multi-mode beam) to mitigate the effects of the limited-size aperture or misalignments an MDM link. Ch: Channel;
Tx: Transmitter; Rx: Receiver.
Figure 3.2 shows the experimental setup of a four-channel MDM FSO link. At the Tx side, a
100-Gbit/s QPSK signal at 1550 nm is amplified by an erbium-doped fiber amplifier (EDFA) and
subsequently equally split into four branches by a 1´4 coupler. All of the four copies are delayed
by single mode fibers with different lengths to decorrelate the data sequences. Subsequently, each
branch is sent into a collimator that generates a collimated Gaussian beam with a diameter of 3 mm.
The Gaussian beams are sent to two SLMs loaded with different phase holograms on each half of
the screen to create specific desired beams. These four output beams are multiplexed by three beam
splitters and then coaxially propagate in free space over ~1 m. At the Rx side, SLM 3 is loaded with
a specific phase pattern to convert one of the incoming beams to a Gaussian-like beam. Finally, it
is coupled to an SMF for coherent detection and BER measurements.

(l
2
, p
2
)
LG
lp Power
(l
2
,p
2
)
LG
lp
(l
2
,p
3
) (l
2
,p
1
)
Aperture
Power
Tx
Rx
MDM link by transmitting each data channel on multiple LG modes
Single LG mode
… …
Power coupling into other LG
modes with different p values
Power Power
Power Power
v
11 v
21
+ =
LG
l1p1 + LG
l2p2
…
v
14 v
24
+
=
LG
l1p1 +
LG
l2p2
…
…
…
Limited-size
aperture or
misalignments
Channel 1
Channel 4
Channel 1
Channel 4
(l
2
,p
2
) (l
3
,p
3
) (l
1
,p
1
)
(l
2
,p
2
) (l
3
,p
3
) (l
1
,p
1
)
(l
2
,p
2
) (l
1
,p
1
)
LG
lp
(l
3
,p
3
)
LG
lp
(l
2
,p
2
) (l
1
,p
1
) (l
3
,p
3
)
LG
lp
LG
lp
Limited-size aperture effects
(a1)
(l
2
, p
2
)
LG
lp
Power
(l
2
,p
2
) LG
lp
(l
3
,p
3
) (l
1
,p
1
)
Aperture
Power
Tx
Rx
Single LG mode
…
…
Power coupling into other LG
modes with different l or p values
Misalignment effects
(a2)
Ch 4
Ch 1
Beam Mux
Ch 4
Ch 1
100 200 300 400 500 600 700 800
100
200
300
400
500
600
700
800 100 200 300 400 500 600 700 800
100
200
300
400
500
600
700
800 100 200 300 400 500 600 700 800
100
200
300
400
500
600
700
800
100 200 300 400 500 600 700 800
100
200
300
400
500
600
700
800
Beam Demux
Lateral
displacement
Aligned
(b)
…
…
Ch2
Ch1
Ch2
Ch3
Ch2
Ch1
Ch2
Ch3
Tx
Rx
(l
4
,p
4
)
(l
4
,p
4
)
(l
4
,p
4
)
(l
4
,p
4
)
(b1)
Beam 1
Delay PC
SLM 3
SLM 1
SLM 2
1×4
coupler
Coherent
detection
EDFA
100Gbit/s
QPSK Signal
(a)
BS
BS
BS
Beam 2
(b2)
Single-mode beams
SLM 1
LG
10
LG
11
LG
-10
LG
-11
SLM 1 SLM 2 SLM 2
Beam 3 Beam 4
SLM 1 SLM 1 SLM 2 SLM 2
Multi-mode beams
SLM 3 SLM 3 SLM 3 SLM 3
SLM 3 SLM 3 SLM 3 SLM 3
Tx
Rx
18

Figure 3.2 (a) Experimental setup of a four-channel MDM FSO system. QPSK: quadrature phase-shift keying; EDFA:
erbium-doped fiber amplifier; PC: polarization controller; Col.: collimator; SLM: spatial light modulator; BS: beam
splitter. (b) The Tx and Rx SLM patterns for (b1) single-mode beams, and (b2) designed multi-mode beams. Here,
the radius of receiver aperture is 1.0 mm.
The process of the proposed mitigation method includes the following steps: (i) Measure the
complex transmission matrix H for a set of LG modes using the method in [32]. For an MDM link
with a given limited-size aperture or misalignment, the amplitude and phase of each element in H
represent the power coupling between two LG modes. By transmitting and receiving single LG
modes (i.e., probe beams) separately, the amplitude of each element is obtained by direct power
measurement. In addition, its phase is calculated by sequentially loading four different phase masks
on the SLM at the Tx and measuring four corresponding power values at the Rx with a specific
phase pattern; (ii) Calculate the SVD of H by D= E∙∑∙H
∗
. (iii) At the Tx, utilize different
columns of matrix V to design mutually orthogonal multi-mode beams, each of which is a complex
combination of multiple LG modes and carries one of the four data channels. Subsequently, we
construct the phase patterns on the SLM (1 and 2) for orthogonal multi-mode beams using the
approach in [32]. (iv) At the Rx, utilize different rows of the inverse matrix U to construct the phase
patterns on SLM 3 that can convert the incoming multi-mode beam back to a Gaussian-like beam
[32]. As an example, when the radius of the receiver aperture is 1.0 mm, the Tx and Rx SLM
patterns for single-mode beams and multi-mode beams are shown in Fig. 3.2 (b1) and Fig. 3.2 (b2),
respectively. We note that the Tx and Rx SLM patterns should be different under different aperture
sizes or misalignments due to different transmission matrix.
3.3 Experimental Results
A. SVD Approach for Mitigation of Effects of Limited-size Aperture
First, we evaluate the effect of the Rx aperture size on the MDM link when transmitting data
channels 1 and 2 (Ch 1 and 2) on a single LG10 and LG11 mode, respectively, as shown in Fig.
3.3(a1). It should be noted that the “Ch 1” and “Ch 2” in the following figures carry the same
meaning as the ones in Fig. 3.2. At the Tx, we generate single LG10 or LG11 beam with a beam
radius of 0.7 mm by using the SLM. After propagating through the free space, LG10 and LG11
beams have beam size of ~1.3, and ~1.8 mm at the Rx, respectively. The phase patterns on SLM-
3 have circular aperture shapes, whose radii is defined as aperture sizes in our experiment. As
19

expected, the power loss increases as the aperture size decreases, as shown in Fig. 3.3 (b1). In
addition, we also measured channel crosstalk that refers to the power coupled from the other modes
over the power on the desired mode. We see that the crosstalk for both LG beams become higher
when decreasing the aperture size. This is because a limited-size aperture blocks a part of the LG
beams, and thus degrading the orthogonality between LG modes with different p values [31]. In
our approach, the data channel is transmitted on a multi-mode beam. Since the limited-size aperture
affects the beam profiles in the radial direction, multiple LG modes with different p values are
utilized to tailor the profiles of multi-mode beams in the radial direction, as shown in Fig. 3.3 (a1).
However, we find that there would be a relatively high power loss if we only use the same two LG
modes (LG10, and LG11) as the case of single-mode beams. Therefore, an extra LG mode (LG12)
is utilized in the generation of multi-mode beams. Theoretically, the power loss of the multi-mode
beams is related to the diagonal elements in the ∑ matrix [39]. We see that power loss for both
channels is reduced with an aperture radius of 1.0-1.6 mm. In addition, we also find that when the
aperture radius is less than 0.8 mm, the power loss of the multi-mode beam carrying data channel
2 would become even larger. This might be due to the variation of H with the aperture radius. As
the aperture radius decreases, the power coupling from the desired single LG mode to the other
LG mode gradually increases and becomes similar to the power on the desired LG mode [31]. This
results in a smaller difference between the values of diagonal elements and off-diagonal elements
in H. Consequently, this would induce a decrease in the value of the second diagonal element in
the ∑ matrix after SVD calculation of H, which causes an increase in the power loss for the second
multi-mode beam.
Moreover, the crosstalk between the two channels is also reduced, which is due to that the
orthogonality between the two beams is not degraded by the limited-size aperture. We also note
that the crosstalk for channel 2 increases when the aperture radius is less than 0.8 mm. This is
mainly due to a large power loss of the multi-mode beam carrying channel 2 under a small aperture
size, which will cause a higher crosstalk that refers to the ratio of power coupled from the other
multi-mode beam to the power of desired multi-mode beam. Moreover, it could be expected that
when using additional LG modes with higher p values, the performance of these two channels
20

Figure 3.3 (a1) The complex weights of LG modes carrying data channel 1 and 2 without and with SVD approach;
(a2) Measured intensity profiles of single-mode beams and multi-mode beams when aperture radius is 1.0 mm. (b)
Measured limited-size-aperture induced (b1) power loss and (b2) crosstalk under various aperture radii when
transmitting data channels on single-mode beams or multi-mode beams (LG10, LG11 and LG12 modes). For each
aperture radius, the two multi-mode beams are unique due to different transmission matrix.
might be further improved. This might be due to more sophisticated control of the multi-mode
beam’s spatial profile in the radial direction by using additional LG modes with higher p values
[42]. As an example, the experimental intensity profiles of single-mode beams and multi- mode
beams with an aperture radius of 1.0 mm are shown in Fig. 3.3 (a2). The intensity profiles of multi-
mode beams vary with the aperture radius according to the SVD calculations of transmission
matrices for different aperture radii. This results in different power losses with the aperture radius,
as shown in Fig. 3.3 (b1). We can see that when the aperture radius is 1.0 mm, both multi-mode
beams are smaller than single-mode beams, which results in lower power losses.
B. SVD Approach for Mitigation of Effects of Misalignments
As one kind of the misalignments in an MDM system, the influence of horizontal
displacements on system performance is also investigated. As shown in Fig. 3.4 (a), when
transmitting data channels on single-mode beam with different p values (LG10, LG11), both the
power loss and crosstalk become larger with horizontal displacements. In our approach, the same
two LG modes (LG10, LG11) are utilized to generate the multi-mode beams. Compared with the
case of single-mode beams, the power loss for both channels is reduced within a displacement
Aperture radius (mm) Aperture radius (mm)
Power loss (dB)
Crosstalk (dB)
(b2)
Multi-mode beam
(LG
10
, LG
11
and LG
12
)
carrying Ch 1
Single-mode beam
(LG
10
) carrying Ch 1
Fixed l different
p values
Fixed l different
p values
LG
10
LG
11
Beam 1 Beam 2
(a2)
Single-mode beams
Multi-mode beams
(b1)
Single-mode beam
(LG
11
) carrying Ch 2
Multi-mode beam
(LG
10
, LG
11
and LG
12
)
carrying Ch 2
(a1)
21

range of 0.4-0.7mm. In addition, the crosstalk for both multi-mode beams remains at a relatively
low level (<-27 dB) with the displacement. Furthermore, we also explore the effects of
displacements on LG modes with different l values (LG10, LG-10), as shown in Fig. 3.4 (b). It is
observed that the power loss and crosstalk of LG10, LG-10 modes increase with the displacement.
For both channels, the power loss can be reduced when using the multi-mode beams (composed
of LG10, LG-10 modes) with a displacement range of 0.8-1.1 mm. Moreover, the crosstalk for both
channels stays at a relatively low level (<-17 dB) with the displacement. We also find an increase
and then a decrease of the power loss for the second beam with the displacement in Fig. 3.4. This
might be due to the change of H with the displacement [36]. First, the power coupling from the
desired single LG mode to the other LG mode gradually increases and becomes similar to the
power of the desired LG mode with the displacement. This results in a smaller difference between
the values of diagonal elements and off-diagonal elements in H. Consequently, this would induce
a decrease in the value of the second diagonal element in the ∑ matrix, which causes an increase
in the power loss for the second beam. However, as the power coupling continually increases with
the displacement, it would be higher than the power in the desired LG mode. Therefore, the
difference between the values of diagonal elements and other elements in H will increase. As a
result, the value of the second diagonal element in ∑ matrix would increase, which causes the
power loss of the multi-mode beam to decrease [39].

Figure 3.4 (a) Measured displacement-induced (a1, b1) power loss and (a2, b2) channel crosstalk with horizontal
displacements when transmitting channels on single-mode beams or multi-mode beams.
Horizontal displacement (mm)
Power loss (dB)
Crosstalk (dB)
Horizontal displacement (mm)
(a1)
Power loss (dB)
Crosstalk (dB)
Horizontal displacement (mm) Horizontal displacement (mm)
Fixed l different
p values
Fixed l different
p values
Fixed p different
l values
Fixed p different
l values
(a2)
(b1) (b2)
Multi-mode beam
(LG
10
and LG
11
)
carrying Ch 1
Single-mode beam
(LG
10
) carrying Ch 1
Single-mode beam
(LG
11
) carrying Ch 2
Multi-mode beam
(LG
10
and LG
11
)
carrying Ch 2
Multi-mode beam
(LG
10
, and LG
-10
)
carrying Ch 1
Single-mode beam
(LG
10
) carrying Ch 1
Single-mode beam
(LG
-10
) carrying Ch 2
Multi-mode beam
(LG
10
, and LG
-10
)
carrying Ch 2
22

C. BER Performance for an MAM Link using SVD Approach
We subsequently utilize our approach in a four-channel multiplexed link, each of which
carries a 100-Gbit/s QPSK signal, as shown in Fig. 3.5. Specifically, four single-mode-beams
(LG10, LG11, LG-10 and LG-11) or four multi-mode beams are transmitted and received. Here, the
aperture radius is 1.0 mm and there is no displacement. In the case of single-mode beams, all four
channels have high error vector magnitude (EVM) (>50%). This is due to their higher crosstalk
caused by the limited-size aperture. However, when using the multi-mode beams, the EVM of the
four channels are all at a relatively low level (~22%). Here, first two multi-mode beams are
composed of LG10, LG11 and LG12 modes, while other two multi-mode beams are composed of
LG-10, LG-11 and LG-12 modes. In BER measurements, the power penalties of all four channels are
< 2 dB at the 7%-overhead FEC limit.

Figure 3.5 Measured constellations of four data channels when transmitting each channel on (a) a single-mode beam
or (b) a multi-mode beam. (c) Experimental BERs of four data channels with OSNR when using multi-mode beams.
B2B: back to back; FEC: forward error correction.
3.4 Conclusion
In this chapter, we discuss the SVD-based beam orthogonalization to mitigate the effects of
the limited-size aperture or misalignments. Moreover, we note that the beam size and direction can
be further tailored by adding transfer functions of a lens and a linear grating on the transmitter-
side SLM, respectively [43, 44]. Therefore, this might be an effective complement to our approach
and further improve the system performance. Furthermore, we note that it might also be possible
OSNR (dB)
(c)
B2B
14 16 20 22 24 18
BER 10
-2
10
-3
10
-4
FEC threshold
Ch1 Ch2 Ch3 Ch4
EVM: 55.1% EVM: 51.6%
EVM: 61.2% EVM: 55.6%
(a4) (a3) (a2) (a1)
OSNR=18.3 dB
Single-mode beam
on each channel
EVM: 22.2% EVM: 22.1%
EVM: 22.4% EVM: 22.3%
(b4) (b3) (b2) (b1)
OSNR=18.5 dB
Designed multi-mode
beam on each channel
Ch1 Ch2 Ch3 Ch4
Multi-mode beam (LG
10
, LG
11
and LG
12
) carrying Ch 1
Multi-mode beam (LG
10
, LG
11
and LG
12
) carrying Ch 2
Multi-mode beam (LG
-10
, LG
-11
and LG
-12
) carrying Ch 3
Multi-mode beam (LG
-10
, LG
-11
and LG
-12
) carrying Ch 4
23

to utilize our approach to mitigate other issues in an MDM link, such as atmospheric turbulence
[32].
24

Chapter 4 Demonstration of a 10 Mbit/s Quantum
Communication Link by Encoding Data on Two Laguerre–
Gaussian Modes with Different Radial Indices
4.1 Introduction
Quantum optical communication links have the potential for enhanced system security [45-
49]. Typically, a quantum communication link employs two orthogonal states, such as the
polarization of a photon for data encoding. In such a quantum qubit system, only one bit of
information is encoded on each photon. An increase in the number of orthogonal states in a
quantum communication link could potentially improve both its security and photon efficiency
(e.g., bits/photon) [50-53].
One potential technique to achieve a larger alphabet is to employ a set of multiple orthogonal
spatial modes from a modal basis set for quantum data encoding. The orthogonality would enable
the selection of the desired mode at the receiver with little inherent crosstalk to other quantum
states [54, 55]. In this case, each photon occupies only one of the d orthogonal spatial modes at a
given time slot, where d is the number of possible states that are used for encoding. Compared
with a conventional qubit system where only two orthogonal states are available, this higher
dimensional qudit system might potentially provide a photon efficiency of up to log2(d) bits per
photon [54-58].
One example of a possible spatial basis set is LG modes, which can be characterized by two
indices: the azimuthal index ℓ and the radial index p [1, 2]. LG modes with different ℓ values or p
values are orthogonal with each other. In the classical domain, mode multiplexing (i.e., each mode
carries an independent data stream) and data encoding (i.e., each pulse occupies a given LG mode
state) using different ℓ or p values have been demonstrated [8, 10, 22, 24, 25, 33].
In the quantum domain, a single photon can occupy a given quantum state; for an LG modal
basis set, a given ℓ and p value would represent the state [11-13, 59]. Previously, there have been
several demonstrations of data encoding on the same p and different ℓ values [11, 13, 57, 58, 60].
To the best of our knowledge, there have been few reports of quantum data encoding among LG
modes with different p values [12, 61]. The utilization of LG modes with different p values could
potentially provide more communication modes.
25

In this chapter, we experimentally demonstrate a 10-Mbit/s free-space quantum
communication link by encoding data on orthogonal LG modes with different p indices. By
encoding data on two LGℓp modes (i.e., for ℓ = 0, we encode [“0”, “1”] as [p = 0, p = 1], and for ℓ
= 1, we encode [“0”, “1”] as [p = 0, p = 1]), the quantum symbol error rate (QSER) < 5 % is
achieved at an encoding rate of 10 Mbit/s. Moreover, we also investigate the influence of the circle
radius (R) of the receiver phase pattern on registered photon rates and QSER. The results show
that a receiver phase pattern whose R does not match the beam size of the LG modes would induce
higher crosstalk between the two encoded quantum branches.
4.2 Concept and Experimental Setup
Figure 4.1 shows the concept of quantum data encoding based on two LGℓp modal sets, each
having the same ℓ and different p values (LG00 and LG01 or LG10 and LG11). For quantum data
encoding, photons are encoded by converting each photon into specific LGℓp mode from an LGℓp
modal set with the same ℓ and different p. Within each symbol period, every single photon exists
in only one of the two LGℓp modes. After the free-space transmission followed by the mode
separation and detection, the data stream could be recovered with low inherent crosstalk due to
the orthogonality of the different LGℓp modes.

Figure 4.1 (a) Two LGℓp modes with the same ℓ but different p values for data encoding (Case 1: LG00 and LG01; case
2: LG10 and LG11) (b) Concept of quantum data encoding based on two LGℓp modes.
The experimental setup of quantum encoding using two LG modes with the same ℓ and
different p is shown in Fig. 4.2. Two pseudo-random sequences, generated by an arbitrary
waveform generator (AWG), are first amplified and then used to directly modulate two 850-nm
lasers, respectively. The two branches (branch ① and branch ②) are then coupled into two
collimators, each of which emits a collimated Gaussian beam with a diameter of 3.99 mm. Two
programmable SLMs loaded with different phase holograms on the screens are used to convert the
two incoming beams into the desired LG beams (LG00 and LG01 or LG10 and LG11). A beam splitter
26

(BS) is used to spatially combine the two LGℓp beams. Then the combined beams are attenuated
by an attenuator to the single photon level. The resulting quantum channel propagates in free space
in the lab over ~1 m.
After free-space transmission, the incoming quantum channel is split into two copies by
another BS and then sent to SLM-3 and SLM-4. These SLMs are loaded with the designed phase
patterns to convert the LGℓp photons back into Gaussian-like (LG00) photons simultaneously. Each
of the downconverted photons is coupled into a SMF using two lenses and collimators. The two
branches are simultaneously detected by single photon detectors (SPDs) that have a deadtime of
50 ns and an afterpulsing probability of 0.5 %. When a photon event is detected by the SPD, a 25-
ns-wide pulse would be generated. All the output pulses produced by SPDs are sampled and
recorded by a real-time oscilloscope with a sampling rate of 250 Msample/s. Finally, an offline
digital signal processing (DSP) is used to calculate the QSER and the registered photon rate [11].
In our experiment, the proof-of-concept quantum link is only over ~1 m in the lab. However, when
considering propagation outside over a long distance, atmospheric turbulence might affect the
system performance. In order to reduce the photon loss and QSER, an adaptive optics system might
be utilized to mitigate the turbulence effect.

Figure 4.2 (a) Experimental setup of a quantum communication link based on LGℓp modes. AWG: arbitrary waveform
generator; PC: polarization controller; Col.: collimator; SLM: spatial light modulator; BS: beam splitter; FM: flip
mirror; ATT: attenuator; SPD: single photon detector; DSP: digital signal processing.
4.3 Experimental Results
27

Figure 4.3 (a) Normalized generated waveforms of the two branches (a1-a2) and their combination (a3) at the
transmitter. (b) Normalized waveforms of the two branches and their combination received by SPDs using the mode
set {LG00 and LG01}. The symbol period is 100 ns.
Figure 4.3(a) presents the normalized waveforms of the two branches and their combination
at the transmitter. Here, the transmitted symbol rate is 10 Mbit/s and the signal duty ratio is 25%.
The combined waveform verifies that the photons exist in only one of the two branches in each
100-ns period. At the receiver side, the attenuated signals are detected by the SPDs, as shown in
Fig. 4.3(b). The mode set we use here is LG00 and LG01 and we observe that only one of the two
LG modes is active in each symbol period. This means that the photon only occupies one of the
two LG modes in a symbol period.
Figures 4.4(a1-a2) show the experimental intensity profiles of the LG00 and LG01 beams,
respectively, in the classical domain. Figures 4.4(a3-a4) show the specific phase patterns loaded
on the screens of SLM-3 and SLM-4, respectively, to downconvert the incoming LG00 and
LG01 photons into Gaussian-like (LG00) photons. We note that the receiver phase pattern for the
LG00 mode is an all-zero phase pattern, while the one for the LG01 mode contains a circle inside
with a π phase difference from the outside. The radius of the circle (R) in the receiver phase pattern
for LG01 mode might affect the power loss for LG01 mode and the crosstalk between the LG00 and
LG01 modes. Therefore, in order to achieve an optimized system performance, the R of the receiver
28

phase pattern is required to match the beam size of the received LG01 mode [22]. Here, based on
the quantum measurement approach, different R values are tested, and the normalized registered
photon counts are measured by single photon detectors (SPDs) and an oscilloscope, as shown in
Fig. 4.4(a). Each value is measured as a ratio of the measured photon counts (power) to the
maximum photon counts (power) in this figure in a unit of dB. We note that when receiving LG00
photons, the registered photon counts remain relatively stable. We think this is due to the fact that
the receiver pattern for LG00 photons is an all-zero phase pattern, and may not be sensitive to the
beam size. However, when receiving LG01 photons, the lowest crosstalk between LG00 and LG01
photons (< 20 dB) as well as the minimum photon loss for LG01 photons are achieved when R is
0.9 mm, while the photon loss and the crosstalk will become higher for other R values.
Figures 4.5(a1 – a3) show the crosstalk matrices between the LG00 and LG01 modes for R =
0.5, 0.9, and 1.3 mm, respectively. We see that for R = 0.5 or 1.3 mm, the photon loss and crosstalk
between the LG00 and LG01 modes is higher, which might degrade the system performance. Figures
4.5(b) and (c) present the registered photon rates and QSERs for the various R values as a function
of average photon number per pulse (μ) for the LG00 and LG01 modes. Due to the deadtime
limitation of our SPDs, a transmitted data rate of 10 Mbit/s is chosen. Results indicate that

Figure 4.4 Crosstalk analysis between LG00 and LG01 modes as a function of the circle radius (R) of the receiver
pattern using the quantum measurement approach. (a1 - a2) Experimental intensity profiles of the LG00 and LG01
modes in the classical domain. (a3 - a4) Receiver phase patterns for LG00 and LG01 modes. Tx: LG00, Rx: LG00: LG00
is transmitted and LG00 is received. The optimized R value is 0.9 mm.
29

compared with the case of R = 0.9 mm, the registered photon rate is lower and the QSER becomes
higher when R = 0.5 or 1.3 mm. We think this degradation is due to the higher photon loss and
crosstalk caused by the mismatch between R and the beam size of the LG01 modes. We observe
even for R = 0.9 mm, the registered photon rate at a μ of 1 is lower than the transmitted data rate
(10 Mbit/s). This is mainly due to the limited photon detection efficiency of our SPDs (45% at 850
nm).
Besides the case of LG modes with ℓ = 0, we also investigate the orthogonality of LG modes
with higher ℓ (ℓ = 1) and different p (p = 0 or 1) values. Figures 4.6(a1) and (a2) present the
experimental intensity profiles of LG10 and LG11 beams, respectively, in the classical domain.
Figures 4.6(a3) and (a4) show the receiver phase patterns which are utilized to downconvert LG10
and LG11 modes into Gaussian-like (LG00) photons, respectively. Here, for LG10 modes, the
receiver phase pattern is only a special spiral phase pattern. However, for LG11 modes, the receiver
phase pattern also has a circle at the center, which is similar to the one for the LG01 mode. Moreover,
the inside and outside sections of the circle have a π phase difference. Figure 4.6(a) presents the
effect of R values on the normalized registered photon counts, which shows a trend similar to the

Figure 4.5 (a1 – a3) Crosstalk matrices between LG00 and LG01 modes for different R values (0.5, 0.9, and 1.3 mm).
(b) Registered photon rates and (c) QSERs with different R (0.5, 0.9, and 1.3 mm) of the receiver phase pattern as a
function of average photon number per pulse (μ). The transmitted data rate is 10 Mbit/s. The optimized R value is 0.9
mm.
30

one using the LG00 and LG01 modes. We observe that the crosstalk between LG10 and LG11 photons
also achieve its minimum value of ~-14 dB at R = 0.9 mm.

Figure 4.6 Crosstalk analysis between LG10 and LG11 modes as a function of the circle radius (R) of the receiver
pattern using the quantum measurement approach. (a1 - a2) Experimental intensity profiles of the LG10 and LG11
modes in the classical domain. (a3 - a4) Receiver phase patterns for the LG00 and LG01 modes. The optimized R value
is 0.9 mm.

Figure 4.7 (a1 – a3) Crosstalk matrices between LG10 and LG11 modes for different R values (0.5, 0.9 and 1.3 mm).
(b) Registered photon rates and (c) QSERs with different R (0.5, 0.9, and 1.3 mm) of the receiver phase pattern as a
function of average photon number per pulse (μ). The transmitted data rate is 10 Mbit/s. The optimized R value is 0.9
mm.
31

Figures 4.7(a1 – a3) shows the crosstalk matrices between the LG10 to LG11 modes when R
= 0.5, 0.9, and 1.3 mm. The crosstalk and power loss for R = 0.9 mm are lower than the cases of
R = 0.5 or 1.3 mm. The registered photon rates and measured QSERs with different R (0.5, 0.9,
and 1.3 mm) are shown in Figs. 4.7(a) and (b). When R = 0.5 or 1.3 mm, a lower registered photon
rate and a higher QSER than the case of R = 0.9 mm are observed, which is consistent with the
crosstalk behaviors mentioned above.
4.5 Conclusion
In this chapter, experimental demonstration of a 10 Mbit/s free-space quantum
communication link using data encoding on orthogonal LG modes with the same azimuthal index
but different radial indices is presented. Data encoding on two LGℓp modes (i.e., for ℓ = 0, we
encode [“0”, “1”] as [p = 0, p = 1], and for ℓ = 1, we encode [“0”, “1”] as [p = 0, p = 1]) is
demonstrated by employing directly modulated laser diodes and helical phase holograms. The
QSER of < 5%is achieved at an encoding rate of 10 Mbit/s. Moreover, the influence of the R value
of the receiver phase pattern on registered photon rates and QSERs is investigated. Our results
show that a receiver phase pattern whose R does not match the beam size of the LG modes would
induce higher crosstalk between the two encoded quantum branches.
32

Chapter 5 Simulation of Near-diffraction- and Near-
dispersion-free OAM Pulses with a Controllable Group
Velocity by Combining Multiple Frequencies, Each
Carrying a Bessel Mode
5.1 Introduction
Due to their unique amplitude and phase spatial structure, light beams carrying OAM have
recently attracted much interest [4, 54, 62-65]. OAM can be carried by both CW and pulsed light,
where the latter is of special interest in applications that benefit from control in the time domain
[66, 67].
Optical pulse propagation in a medium is governed by several properties, including
diffraction, dispersion, and group velocity. For each, it might be beneficial to achieve the following
goals: (1) minimize its diffraction if power density is desired [68, 69]; (2) minimize its dispersion
to maintain pulse-shape integrity [70, 71]; and (3) manipulate its group velocity to achieve a
controllable temporal delay [72].
Previous reports have achieved: (a) diffraction- and dispersion-free non-OAM light sheets
with one-dimensional (1D) spatial profiles and controllable group velocities in free space [73-75];
(b) diffraction- and dispersion-free 2D non-OAM Bessel-based pulses with superluminal, luminal,
or subluminal group velocities in free space or dispersive media [76-81]; (c) diffraction- and
dispersion-free OAM pulses with superluminal group velocities [82-84]. Therefore, it may be
desirable to achieve all the above goals, producing diffraction- and dispersion-free OAM pulses
with controllable group velocities in free space [85, 86].
To generate various diffraction-free optical pulses, optical nonlinearities or chromatic
dispersion have been exploited previously [76, 84]. In recent reports, a new method was
experimentally demonstrated to generate 1D light sheets with no need for nonlinearity or
dispersion [73-75]. In their approach, a femtosecond pulse was utilized, whose spatial-temporal
spectrum can be built by a diffraction grating and an SLM. Instead of spectrally-broad femtosecond
pulse, we propose to utilize discrete frequency lines with a large frequency spacing (e.g., 200 GHz)
for the generation of 2D OAM pulses. This is because such discrete frequency lines could be
33

efficiently separated due to the large frequency spacing and subsequently spatially modulated in a
real experiment. It should be noted that these discrete frequency lines can be achieved by an optical
frequency comb [87].
In this chapter, we simulate the generation of a near-diffraction-free and near-dispersion-free
OAM pulse with a controllable group velocity by coherently combine multiple optical frequencies.
Each frequency carries a specific Bessel mode with the same ℓ but a different kr (spatial frequency)
values based on space-time correlations. By utilizing a linear relationship between the z-
component of wavevector kz and frequency f based on the form "
!
= "
"
+(2'( ) ⁄ − "
"
)/tan2,
near-diffraction-free and near-dispersion-free properties could be realized for the OAM pulse. In
addition, by varying the tilted angle θ, both positive and negative group velocities could be
achieved and continuously controlled from the subluminal to superluminal values. Moreover, we
also found that as the topological charge (ℓ) of the pulse increases, the group velocity of the
generated OAM pulse is the same while the pulse duration becomes longer. Alternatively, as the
number of frequencies increases, the pulse duration becomes shorter.
5.2 Concept
The In general, Bessel modes are an orthogonal modal basis set that carries OAM, whose
electrical field can be represented by [88]
4(:,j,8) = K
"
exp (O"
%
8)P
&
("
'
:)exp (±Oℓj) (1)
where P
ℓ
is ℓth-order Bessel function, "
%
and "
'
are the longitudinal and radial wavevectors with
" = S"
%
#
+"
'
#
= 2'( ) ⁄ (f is the frequency of the Bessel beam, and c is the speed of light in
vacuum), and :, j, and z are the radial, azimuthal and longitudinal components, respectively.
Therefore, the azimuthal and radial distribution of the Bessel modes can be characterized by the
2D indices (ℓ and kr), respectively [88]. When Bessel beams with the same frequency f and
different spatial frequencies kr are coherently combined, a monochromatic CW OAM beam is
generated [89]. Due to non-diffractive properties of all the Bessel components, the generated CW
OAM beam is also non-diffractive, as shown in Fig. 5.1(a). When Bessel beams with different
frequencies but the same kr value are coherently combined, f is a nonlinear function of kz according
to the equation S"
%
#
+"
'
#
= 2'( ) ⁄ . In this case, group velocities (i.e., vg = ∂2'(/∂kz) are different
for different frequencies due to the nonlinear relation between f and kz. As a result, the temporal
pulse envelop becomes distorted as the pulse propagates, and thus a dispersive OAM pulse is
34

generated, as shown in Fig. 5.1(b).

Figure 5.1 (a) When non-diffractive Bessel beams with the same frequency (f) and different spatial frequencies kr are
coherently combined, a non-diffractive CW OAM beam is generated; (b) When Bessel beams with different f but the
same ℓ and kr value are coherently combined, the pulse envelop becomes distorted as it propagates due to the nonlinear
relation between f and kz. As a result, a dispersive OAM pulse is generated. BM: Bessel mode.
Figure 5.2 illustrates the basic concept of the generation of a near-diffraction-free and near-
dispersion-free OAM pulse with arbitrary group velocities. In our approach, the spatio-temporal
spectrum of such an OAM pulse lies along the intersection of the light-cone with a spectral
hyperplane parallel to the kr-axis and tilted an angle θ with respect to the (kr, kz)-plane "
!
= "
"
+
(2'( ) ⁄ − "
"
)/tan2 (k0 is a constant wavevector). Here, the light cone represents the spatial
dispersion relation of the free-space light beam propagation. In this case, each frequency f is
correlated with a unique kr, which contributes to a classical entanglement between these two
degrees of freedoms. Due to the linear relationship between kz and f, the generated OAM pulse
tends to be near-dispersion-free. In addition, due to non-diffractive properties of all the Bessel
components, the generated OAM pulse is also near-diffraction-free. Moreover, for such an OAM
pulse, the theoretical group velocity vg is given by vg = ∂2'(/∂kz = c×tanθ. Therefore, by tuning the
value of θ, arbitrary group velocities could be potentially achieved [74].
35

Figure 5.2 Each frequency carries a specific Bessel mode with the same ℓ but a unique kr, leading to a linear relation
between f and kz based on "
!
= "
"
+(2'( ) ⁄ − "
"
)/tan2. As a result, a near-diffraction-free and near-dispersion-
free OAM pulse could be generated. In addition, its group velocity could be tuned by changing θ.
5.3 Simulation Results
Figure 5.3 (a) show the simulation results for the dispersive OAM pulse, which correspond
to the cases of Fig. 5.1 (b). The spatio-temporal spectrum is shown in Fig. 3 (a1), in which all the
Bessel beams have different frequencies but the same kr value. Here, 20 discrete frequency lines
are used with a constant frequency spacing of 200 GHz and the central wavelength is around 1550
nm. The temporal pulse shapes in Fig. 5.3 (a3) are calculated at the transverse position where the
intensity is maximum in Fig. 5.3 (a2). We see that there is little intensity at transverse position 0,
which is due to the characteristic ring shape of the OAM beams. We also note that as the
propagation distance (z) increases, the simulated intensity profile (|4(5,7,8,9)|
#
) and the temporal
pulse shape become distorted, as shown in Fig. 5.3 (a2) and (a3). This is due to different group
velocities for different frequencies due to the nonlinear relation between f and kz. However, when
combining Bessel beams with different frequencies and different kr values according to the space-
time correlation in Fig. 5.2, both the simulated intensity profile and the temporal pulse shape
remain invariant with the propagation distance, as shown in Fig. 5.4. We believe that the resulting
near-dispersion-free property is caused by the linear relationship between kz and f. Here, the tilt
angle θ is selected as 3π/10 as an example. Moreover, it is observed that when z is 1 or 10 m, the
36

pulse peak is not at the t = 0. This might be due to different interference conditions between these
Bessel components for different z values.

Figure 5.3 The spatio-temporal spectra for (a1) the dispersive OAM pulse which corresponds to the cases of Fig. 5.1
(b). (a2) The simulated intensity profiles |4(5,7,8,9)|
#
with time at z = 0, 1, 10 m for this pulse; (a3) The simulated
temporal pulse shapes at the transverse position where the intensity is maximum in (a2).
37

Figure 5.4 The spatio-temporal spectra for (a1) the non-dispersive OAM pulse which corresponds to the cases of Fig.
5.2. (a2) The simulated intensity profiles |4(5,7,8,9)|
#
with time at z = 0, 1, 10 m for this pulse; (a3) The simulated
temporal pulse shapes at the transverse position where the intensity is maximum in (a2).
Furthermore, the simulated intensity and phase profiles of such an OAM pulse at different
times (-0.25, 0, 0.25 ps) when z is 0 are shown in Fig. 5.5. In our simulation, when generating an
OAM pulse with a specific ℓ, all the Bessel beam components have the same ℓ value (here, ℓ = 5).
Consequently, we see that the number of 2π azimuthal phase changes is 5 in the phase profiles,
which indicates the ℓ of the resulting pulse is 5. We also find that the intensity and phase profiles
changes with time. This might be due to the specific interference conditions between different
Bessel components. Since their relative phase changes with time, the interference conditions also
change, leading to different intensity and phase profiles at different times.
38

Figure 5.5 Simulated intensity and phase profiles of the non-dispersive OAM pulse at different times (-0.25, 0, 0.25
ps) when z is 0 m.
We also study the diffraction property of such an OAM pulse. Figure 5.6(a) shows the
normalized time averaged intensity profile (=(:,8) = ∫
|4(:,8,9)|
#
>9) at z values. Here, the time
interval of the integration is 5 ps (i.e., the pulse period). We note that the pulse transverse size
tends to remain the same as it propagates along z axis, which indicates it is near-diffraction-free.
This is due to the non-diffractive property of all the Bessel beams [88]. In addition, the group
velocity of such a non-dispersive OAM pulse is studied in Fig. 5.6 (b), which shows the theoretical
and simulated group velocity of the OAM pulse for different values of θ. As mentioned before, the
theoretical group velocity vg is given by vg = ∂2'(/∂kz = c×tanθ. In our simulation, the simulated
group velocity of the OAM pulse is calculated by Dz/Dt for a given smaller time interval Dt.
Consequently, we find that the simulation results are in good agreement with the theoretical values.
Moreover, by changing θ, both positive (in the forward direction) and negative (in the backward
direction) group velocities could be potentially achieved. In addition, it can be controlled from
subluminal to superluminal values. It should be noted that the amplitudes of all Bessel components
are the same and their relative phases are 0 as an example in our simulation. Moreover, we
simulated the generation of such OAM pulses when there is a fixed relative phase difference or
amplitude difference between neighboring frequencies. We find that for both cases, the near-
39

diffraction-free, near-dispersion-free properties, and the controllable group velocity can still be
realized for these OAM pulses.

Figure 5.6 (a) Time averaged intensity profile (I(:,8) = ∫
|4(:,8,9)|
#
dt) at different propagation distances
corresponding to the space-time correlation in Fig. 5.2. (b) Theoretical and simulated group velocity of the OAM pulse
for different values of the tilt angle θ.
Subsequently, the influence of number of frequencies on the pulse duration of the OAM
pulse is explored. Here, the pulse duration is defined as the full width at half-maximum (FWHM)
of the temporal pulse shapes in Fig. 5.3 (a3). As expected, the pulse duration becomes smaller as
the number of frequencies increases. In terms of specific detail of the effect in our simulation, the
pulse duration decreases from 0.6 to 0.158 ps for a ℓ of 5 and when the number of frequencies is
increased from 10 to 40 (see Fig. 5.7(a)). Besides, the effect of the ℓ on the pulse duration is also
explored in Fig. 5.7(b). It is observed that as the ℓ of the pulse increases, the pulse duration
becomes larger, as shown in Fig. 5.7 (b). We note that a similar phenomenon was previously
observed for an OAM-carrying ultrashort pulse that is generated by a combination of multiple LG
beams [90]. In their work, it was also found that the pulse duration is larger for a pulse with a
higher ℓ [90]. However, since the group velocity of the OAM pulse only depends on θ, it remains
the same for all ℓ values.
40

Figure 5.7 (a) Temporal pulse duration of the non-dispersive OAM pulse for different numbers of frequencies. (b) The
pulse duration (the blue curve with square markers) and group velocity (the orange curve with circular markers) of
the non-dispersive OAM pulse in Fig. 5.4 with various ℓ values when the number of frequencies is 20.
5.4 Conclusion
To sum up, we explore via simulation the generation of a near-diffraction-free and near-
dispersion-free OAM pulses with arbitrary group velocities by coherently combine multiple
frequencies. Each frequency carries a specific Bessel mode with the same ℓ but a different kr values
based on space-time correlations. We simulate the propagation of OAM pulses with near-
diffraction-free and near-dispersion-free properties. Moreover, we also find that: (i) both positive
and negative group velocities could be achieved and continuously controlled from the subluminal
to superluminal values. (ii) When the ℓ is varied from 0 to 10, the simulated value of the group
velocity remains the same. However, as the ℓ value increases, the pulse duration becomes longer
for a given number of frequency lines.
The following points are worth mentioning: (1) In this chapter, we call such pulses “near-
diffraction-free and near-dispersion-free OAM pulses”. This is because even though we present
some results in this paper in terms of diffraction and dispersion performance over a certain
propagation distance, we haven’t directly quantified the performance. Moreover, there might exist
some other parameters that might affect the performance. (2) An ideal Bessel beam needs to have
infinite extent in the transverse direction and requires an infinite amount of energy. One typical
approximation of Bessel beams is the apodized Bessel beam [91], such as the Bessel-Gauss beam.
Such a beam can be generated by transmitting a Gaussian beam through an axicon or a diffractive
optical element. Moreover, Bessel-Gauss beams are propagation invariant within the “Bessel
0 2 4 6 8 10
0.2
0.25
0.3
0.35
0.4
0
0.5
1
1.5
2
2.5
10 20 30 40
0.1
0.2
0.3
0.4
0.5
0.6
Topological charge (ℓ)
Pulse duration (ps)
Group velocity (v
g
/c)
z=0m
Number of frequencies
Pulse duration (ps)
z=0m
(b) (a)
41

zone”, which is related to the input Gaussian beam size. Outside the “Bessel zone”, the beam
would experience diffraction [91]. As a result, if OAM pulses discussed in this paper are
experimentally generated by a combination of these apodized Bessel beams, the diffraction- and
dispersion-free length might be limited. (3) Since OAM can also be carried by waves at different
frequencies (e.g., acoustic waves), the OAM pulses discussed in this paper might also apply to
other regimes [92]. (4) Moreover, although we simulate OAM pulses generated by a combination
of multiple Bessel modes, other orthogonal spatial modal sets (e.g., Laguerre Gaussian) might also
be utilized to generate various interesting beams. Furthermore, it might be interesting to explore
other types of OAM pulses which can be potentially generated by combining spatial modes with
different ℓ values.
42

Chapter 6 Experimental Generation of Near-Diffraction-
Free OAM Pulses Having a Controllable Group Velocity
from 1.0069c-0.9933c by Coherently Combining
Different Beams of Multiple Correlated Bessel Modes and
Frequencies
6.1 Introduction
As mentioned in the previous chapter, OAM can be carried by pulses of light, which may
have specific value to applications that make use of high-speed dynamics [4, 93]. Another two
properties that may be of value to short pulses are: (a) the generation of near-diffraction-free pulses,
in which Bessel beams have been shown to exhibit [88], and (b) the control of the pulse’s group
velocity (e.g., time delay) to make it tunable in the superluminal and subluminal domains [72].
There have been previous reports of experimentally generating short pulses that are near-
diffraction-free with some type of controllable group velocity but with no OAM [74, 94-97].
Another report showed OAM pulses with some degree of controllable group velocity but were not
diffraction-free [98]. In the previous chapter, by simulation, it showed that near-diffraction-free
OAM pulses with a tunable, controllable group velocity can be generated by coherently combining
multiple beams of different Bessel modes and comb-line frequencies [85, 86]. In this approach,
each frequency carries a specific Bessel mode with the same ℓ but a different kr value based on a
specific relationship.
In this chapter, we experimentally demonstrate this concept by generating near-diffraction-
free OAM +1 and +3 pulses having a controllable group velocity from 1.0069c to 0.9933c (c is the
speed of light in vacuum). Six frequency lines are utilized, each carrying a specific Bessel mode
with the same ℓ but a different kr value. Consequently, there is a linear relationship between
longitudinal wavevector kz and frequency f based on "
%
= "
"
+(2'( ) ⁄ − "
"
)/9TU2 for the
OAM pulse. By varying the angle 2, group velocities can be controlled from the superluminal to
subluminal values. In addition, modal purities of pulses with the OAM order of +1 and +3 are
43

84.52% and 74.38%, respectively. Also explored are the diffraction of such OAM pulses and the
effect of transmitter aperture size.
6.2 Concept and Experimental Setup
Figure 6.1 illustrates the basic concept for the generation of near-diffraction-free OAM
pulses with a controllable group velocity. Bessel modes are a modal basis set that can carry OAM,
and the azimuthal and radial distribution can be characterized by ℓ (azimuthal index) and kr,
respectively. In general, OAM pulses can be potentially generated by coherently combining
multiple frequency lines, each carrying one or more Bessel modes with the same ℓ value. Due to
the near-diffraction-free property of all Bessel modes, such OAM pulses are also near-diffraction-
free. Moreover, the control of the group velocity of the pulse envelope could be realized by
correlating frequencies and Bessel modes. In our approach, each frequency carries a specific
Bessel mode with the same ℓ but a unique kr value. The spatio-temporal spectrum of such OAM
pulses lies along the intersection of the light-cone with a spectral hyperplane parallel to the kr-axis
and tilted an angle θ with respect to the (kr, kz)-plane: "
!
= "
"
+(2'( ) ⁄ − "
"
)/tan2. Since all
the Bessel modes have the same ℓ value, the OAM order of the generated OAM pulse is also ℓ.
Moreover, due to different frequencies and kz values, there is a relative phase delay Dj between
neighboring Bessel beams. Such a Dj would induce a change in the interference of these Bessel
beams in the propagation direction (z) with time and z position, thus leading to a change of z
position of the pulse envelope with time. Based on the space-time correlation in our approach, the
theoretical group velocity is given by vg = Dz/Dt = 2pDf/Dkz=c•tanθ, where Df and Dkz are
frequency and kz spacing, respectively. Therefore, by tuning the value of θ, the group velocity of
the generated OAM pulse could be potentially controlled.

Figure 6.1 (a1) Concept for the generation of near-diffraction-free OAM pulses with a controllable group velocity.
BM: Bessel mode.
Near-diffraction-free
OAM pulse
qEach frequency carries a specific Bessel
mode with the same ℓ but a unique k
r
value.
z
k
r
k
z
=k
0
+(2!f /c-k
0
)/tan"
k
r
k
z
!
2"f/c
k
z
2"f/c
#
Linear
relation
2"f/c
# BM (ℓ
!
, k
r!
, f
i
)
"
z=z
1
t
1
t
Interference of multiple
freq., multiple modes
z=z
2
t
2
(2#∆"#−∆%
!
&) ∆%
Each Bessel mode is
near-diffraction-free.
(
"
=
∆&
∆*
=
2#∆+
∆%
!
= ,-*./(0)
Controllable group velocity
44

Figure 6.2 shows the experimental setup for the generation and detection of OAM pulses
with a controllable group velocity. A single-soliton Kerr frequency comb with a spacing of ~192
GHz is used. Six frequency lines are first selected by a waveshaper and subsequently amplified by
an EDFA. After a collimator, the output Gaussian pulse is split into two branches. One is the
reference pulse, and the other one is sent to a diffractive grating. Consequently, these six
frequencies are spatially separated, each of which is directed to a different position on SLM 1.
Different phase patterns are loaded for different incoming frequencies to generate the desired
Bessel beams. Subsequently, the resulting beams are coherently combined using SLM 2. Finally,
the generated OAM pulse is combined with the reference pulse and its profile is detected by off-
axis holography. By inducing a time delay between the OAM and reference pulse, the profiles at
different times can be detected.

Figure 6.2 (a) Experimental setup for generation and detection of near-diffraction-free OAM pulses with a controllable
group velocity. EDFA: erbium-doped fiber amplifier; PC: polarization controller; Col.: collimator; SLM: spatial light
modulator; BS: beam splitter. The measured intensity profiles |4(5,7 = 0,8 = 0,9)|
#
with time and the transverse
profiles at the reference time delay of 0 for (b) OAM +1 and (c) OAM +3 pulses.
6.3 Experimental Results
Figure 6.3 (a-b) presents measured intensity profiles |4(5,7 = 0,8 = 0,9)|
#
with time and
the transverse profiles at the reference time delay of 0 for OAM +1 and OAM +3 pulses,
respectively. Based on the phase profiles of the two pulses, the number of the 2p phase shift in the
azimuthal direction is 1 and 3, respectively. This indicates that the OAM orders of the two pulses
are +1 and +3, respectively. The modal purities at the reference time delay of 0 are 84.52% and
45

74.38%, respectively, which can be obtained by decomposing transverse profiles into a set of LG
modes with different ℓ and p values and analyze the coefficient of each component.

Figure 6.3 The measured intensity profiles |4(5,7 = 0,8 = 0,9)|
#
with time and the transverse profiles at the
reference time delay of 0 for (a) OAM +1 and (b) OAM +3 pulses.
Since the reference and the OAM pulse have different group velocities, there would be a time
delay (Dt) for the pulse envelopes when measuring intensity profiles |4(5,7 = 0,9)|
#
at two
different z positions. Consequently, the group velocity of the OAM pulse can be calculated based
on Dz and Dt. Figure 6.4 shows theoretical and measured group velocities of OAM pulses for
different θ values. It is observed that experimental values are close to theoretical ones. We note that
the range of θ values is limited due to the pixel size of our SLM and camera.

Figure 6.4. (a) The theoretical and measured group velocities of OAM pulses for different θ values.
The diffraction property of the OAM pulse is also investigated. Figure 6.5(a) presents time
averaged intensity profiles at different z for OAM pulses based on Laguerre Gaussian (LG) modes
46

and our approach when both have the same initial beam size. Here, all the LG components or the
Bessel components have the same amplitude and phase for the pulse generation. D0.5Imax, defined
as the distance over which the intensity peak at the ring is reduced by a factor of two, is used to
characterize the diffraction properties. Compared with the LG-based OAM pulse, the OAM pulse
based on our approach has a larger D0.5Imax (~37 mm), which indicates it is near-diffraction-free.
Furthermore, the effect of the transmitter aperture size on the diffraction is shown in Fig. 6.5(b).
We see that as the aperture size decreases, D0.5Imax also becomes smaller.

Figure 6.5 (a) The time averaged intensity profile (=(5,8) = ∫
|4(5,8,9)|
#
>9) at different z for OAM pulses based on
(a1) LG modes and (a2) our approach. (b) The measured D0.5Imax values for various transmitter aperture sizes.
6.4 Conclusion
In this chapter, we experimentally demonstrate the generation and detection of near-
diffraction-free OAM +1 and +3 pulses having a controllable group velocity from 1.0069c-0.9933c
in free space by coherently combining multiple frequency comb lines, each carrying a unique
Bessel mode. The diffraction of such OAM pulses and the effects of transmitter aperture size are
also explored. These ST OAM pulses might find applications in imaging, nonlinear optics, optical
communications.
47

Chapter 7 Experimental Demonstration of Dynamic
Spatiotemporal Structured Beams that Exhibit Two
Orbital-angular-momenta Simultaneously Using a Kerr
Frequency Comb
7.1 Introduction
Structured light has gained increasing interest due to the novel amplitude/phase spatial
distributions that can be achieved and used in various applications [73, 99, 100]. For example, a
beam at a given propagating distance from a transmitter can be tailored using interference to
exhibit nearly any fixed spatial distribution by using a complex-weighted combination of multiple
modes from a complete modal basis set located at a single optical frequency [14]. This can be
produced using the LG modal basis set with 2-D indices (ℓ and p) [1].
In terms of changes in time, orbital angular momentum can be manifest at a given distance
in different forms. One form of momentum is a subset of LG beams with a helical phasefront that
rotates around its own beam center [3]. This momentum is similar to rotation. A second form of
momentum is a Gaussian-like beam dot that revolves around a central axis, and this is similar to
revolution [16]. Indeed, the earth propagating around the sun exhibits both rotation around its own
Earth center and revolution around a solar central axis.
It was recently shown by theory and simulation that a dynamic spatiotemporal (ST) light
beam can be generated that simultaneously exhibits these two independent and controllable orbital
angular momenta at a given propagation distance [101]. This scenario is enabled by combining
multiple optical frequency comb lines [102], with each line carrying a superposition of multiple
LGℓ,p modes containing a different ℓ value and multiple p values. The frequency spacing of the
comb lines introduces a time-variant relative phase delay Dj(t) between the LGℓ,p modes on the
neighboring frequencies. Consequently, the Dj(t) will lead to time-dependent constructive and
destructive interference between all the LGℓ,p modes on different frequencies, which produces a
dynamically revolving and rotating LG beam [101].
In this chapter, we experimentally demonstrate the generation of this dynamic spatiotemporal
structured beam that exhibit two types of orbital-angular-momenta simultaneously at a given
48

distance using a Kerr frequency comb [103]. Nine frequency comb lines are coherently combined,
each of which carries a superposition of multiple LG modes containing one unique ℓ value and
multiple p values. At the detection plane, the generated ST beam is reconstructed by off-axis
holography for each frequency and a numerical combination of all the frequencies. Experimental
results show that the mode purity of the reconstructed revolving and rotating LG30 beam is ~89%
when both the beam waist and revolving radius (R) are 0.4 mm. Moreover, we note that a higher
mode purity can be achieved by including more frequency comb lines or reducing the revolving
radius.
7.2 Concept and Experimental Setup
Figure 7.1 illustrates the basic concept for the generation of a dynamically revolving and
rotating LG beam. When a single frequency line carries a single LG mode, a rotating LG beam at
the central axis is generated, as shown in Fig. 7.1(a). When a single frequency line carries multiple
LG modes, a spatially offset rotating LG beam can be generated, as shown in Fig. 7.1(b). The
complex coefficients for each LG mode can be calculated based on the decomposition of such a
light beam into a set of orthogonal LG modes. Moreover, dynamic revolution around the central
axis can be further introduced by adding appropriate time-variant relative phase delay Dj(t) among
the neighboring LG modes with a different ℓ and multiple p values. This could be achieved by
using multiple frequency comb lines with a frequency spacing of Df. The time-variant relative
phase delay Dj(t)=2pDft will lead to dynamic constructive and destructive interferences, which
generates an offset revolving and rotating LG beam, as shown in Fig. 7.1(c).

Figure 7.1 (a) When a single frequency line carries a single LG mode, a rotating LG beam at the central axis is
generated; (b) When a single frequency line carries multiple LG modes, a spatially offset rotating LG beam can be
generated; (c) When combining multiple frequency lines with each carrying a superposition of multiple LG modes
containing one unique ℓ value and multiple p values, a revolving and rotating LG beam is generated.
Figure 7.2(a) shows the experimental setup for generating and detecting the ST beams. A
single-soliton Kerr frequency comb with a spacing of ~192 GHz is used. Nine frequency lines are
(b)
Interference of
multiple modes,
single freq.
!"
!, #
LG
!, #
%
%#&'(
!
)
!, #
Spatially offset
rotating&'
*+, ,
beam
Single freq.,
single mode
(a) LG
-, .
%
%#&'(
!
)
Intensity Phase
Rotating()
*+, ,
beam at the
central axis
f
1
Freq.
f
1
Freq.
Intensity Phase
(c)
!"
/, !
LG
/, !
!
f
1
Freq.
Intensity Phase
Interference of
multiple modes,
multiple freqs.
t
1
t
2 t
1
t
2
Revolving and
rotating&'
*+, ,
beam
∆"
R
∆"
R
!"
&, #
LG
&, #
#
f
2
!"
0, 1
LG
0, 1
1
f
n
∆" ∆# ∆$∝∆"∆&
Phase Angle
+ + +
…
+ + +
…
49

selected by a waveshaper, as shown in Fig. 7.2(b). The output is amplified by an EDFA, and then
sent into a collimator. Subsequently, the generated Gaussian beam is split into two branches by a
beam splitter. One is used as a reference beam. The other one is sent to a diffractive grating.
Consequently, the nine frequency lines are spatially separated and directed to different positions
on the SLM 1, 3. Different phase holograms are loaded on these positions to create specific beams
composed of multiple LG modes. Subsequently, these resulting beams are coherently combined
using SLM 2, 4 and a beam splitter. Finally, the generated ST beam is reconstructed by off-axis
holography for each frequency and a numerical combination of all the frequencies [104].

Figure 7.2 (a) Experimental setup for generation and detection of ST beams. EDFA: erbium-doped fiber amplifier;
PC: polarization controller; Col.: collimator; FROG: frequency-resolved optical gating; SLM: spatial light modulator;
BS: beam splitter. (b) Optical spectrum of the selected nine frequency lines after the waveshaper.
7.3 Experimental Results
Figure 7.3(a) presents the simulated amplitude and phase of the complex coefficients of all
LG modes used for generation. Based on the simulated coefficients of LG modes at each frequency,
specific phase patterns on SLM 1 and 3 are constructed for generation of a beam comprising LG
modes with a specific ℓ but multiple p values. At the receiver, the spatial amplitude and phase
profiles for each frequency are retrieved using off-axis holography [104]. Subsequently, the ST
beam is reconstructed by numerically combining all the frequencies. Figure 7.3(b) present the
intensity and phase profiles of the reconstructed revolving and rotating LG30 beam in time at a
given distance. The dynamic helical phasefront and amplitude profiles indicate both dynamic
rotation and revolution in time. Here, the beam waist and revolving radius are both 0.4 mm. In
addition, the revolving period is ~5.2 ps since the spacing of our frequency comb lines is 192 GHz.
Therefore, we find the beam revolves around a central axis and almost comes back at t = 5 ps.
Moreover, the mode purity is obtained to be ~89%, as shown in Fig. 7.4(a). Furthermore, we also
50

experimentally investigate the mode purity of the ST beam when using different number of
frequency lines for different revolving radii, as shown in Fig. 7.4(b). We see as the number of
frequency lines decreases, the mode purity decreases. In addition, with a given number of
frequency lines, the mode purity is higher for smaller revolving radii.

Figure 7.3 (a) Simulated amplitude and phase of the complex coefficients of all LG modes used for superposition. (b)
The intensity and phase profiles of the reconstructed revolving and rotating LG beam in time.

Figure 7.4 (a) Power distribution on light beams with different ℓ values. (b) The mode purity of the ST beam when
using different number of frequency lines for different revolving radii (R).
7.5 Conclusion
In this chapter, we experimentally demonstrate the generation of dynamic spatiotemporal
structured beams that exhibit two orbital-angular-momenta simultaneously using a Kerr frequency
comb. The mode purity of the revolving and rotating LG30 beam is obtained to be ~89%.
51

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Abstract (if available)

Abstract “Structured light” is a topic of growing interest in the optics community, not only for its inherent properties but also for its possible applications in communications, sensing, imaging, or micromanipulation. In general, structured light refers to an optical beam with a tailored spatial amplitude/phase distribution and corresponding unique properties. One type of structured light involves beams carrying orbital angular momentum (OAM). As a spatially orthogonal modal basis set, OAM modes has been utilized in different applications, such as optical communications, optical sensing, optical imaging.
The first part of this dissertation will study orthogonal spatial modes for optical communications including high-speed classical communications and high-dimensional quantum communications. This part includes: (i) experimental demonstration of free-space mode-division-multiplexing using Laguerre-Gaussian (LG) modes or Hermite-Gaussian (HG) modes; (ii) experimental mitigation of the effects of the limited-size aperture or misalignment by transmitting each of four data channels on multiple LG modes in a free-space optical (FSO) communication link; (iii) experimental demonstration of a quantum link using data encoding on two LG modes.
The second part of the dissertation discusses some interesting spatiotemporal (ST) beams which correlate the spatial modes and frequencies. The potential applications of these dynamic ST beams might include optical communications, optical imaging and sensing. Some simulations are implemented, and then the experimental generation and detection of these dynamic ST beams are also explored. By using Kerr frequency comb, two types of dynamic ST beams are experimentally generated and detected. One is the structured beams that exhibit two orbital-angular-momenta simultaneously. The other one is the near-diffraction-free OAM pulses having a controllable group velocity by coherently combining different beams of multiple correlated Bessel modes and frequencies.

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Asset Metadata

Creator Pang, Kai (author)

Core Title Spatial modes for optical communications and spatiotemporal beams

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2022-05

Publication Date 04/18/2022

Defense Date 03/02/2022

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

Tag free-space optical communications,laser beam shaping,OAI-PMH Harvest,optical vortices.

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Willner, Alan (committee chair), Haas, Stephan (committee member), Molisch, Andreas (committee member)

Creator Email kaipang@usc.edu

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