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University of Southern California Dissertations and Theses
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Optical and mm-wave orbital-angular-momentum beam generation and its applications in communications
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Optical and mm-wave orbital-angular-momentum beam generation and its applications in communications
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Content
Optical and Mm-wave Orbital-angular-momentum Beam Generation and Its Applications in
Communications
by
Zhe Zhao
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)
December 2020
Copyright 2020 Zhe Zhao
i
Dedication
This dissertation is dedicated to
my loving girlfriend Xin Cheng
for bringing love and sunshine into my life
my parents, Xuewen Zhao and Yanhong Xiang, my sister Hui Zhao
for their love and endless support,
my respectful advisor Prof. Alan E. Willner,
for his great mentor during my Ph.D. career.
ii
Acknowledgements
It takes a village to raise a Ph.D. student. I have been extremely fortunate to have the many loved
ones, mentors, colleagues, and friends, all of whom have supported and encouraged me along the
way to be whom I am today.
I would like to especially thank my respectful advisor, Prof. Alan E. Willner, who is a super
great and excellent mentor. He taught me many priceless lessons about the essence of persistence,
serious attitude, and scientific spirit toward the way to a researcher. His wisdom, not only in
research logics, but also in life philosophy, have always enlightened me to pursue good research
and a better person. The work presented in this dissertation would not have been possible without
so many insights and tremendous support from Prof. Alan E. Willner. I would like to thank Prof.
Wei Wu and Prof. Stephan Haas for serving on my dissertation committee. I would also like to
thank Prof. Moshe Tur, Prof. Jian Wang, Prof. Andreas F. Molisch, and Prof. Robert W. Boyd for
all of the insightful comments and enjoyable discussions.
I sincerely acknowledge my previous and current colleagues, Dr. Hao Huang, Dr. Nisar Ahmed,
Dr. Yan Yan, Dr. Yongxiong Ren, Dr. Morteza Ziyadi, Dr. Guodong Xie, Dr. Changjing Bao, Dr.
Amirhossein M. Ariaei, Prof. Ahmed Almaiman, Dr. Yinwen Cao, Dr. Long Li, Dr. Peicheng Liao,
Mr. Moshe Willner, Ms. Cong Liu, Mr. Ahmad Fallahpour, Ms. Fatemeh Alishahi, Mr. Runzhou
Zhang, Mr. Kai Pang, Mr. Asher Willner, Mr. Haoqian Song, Mr. Kaihang Zou, Mr. Hao Song,
Mr. Huibin Zhou, Mr. Xinzhou Su, Mr. Amir Minoofar, Ms. Nanzhe Hu, and Dr. Jing Du. I also
appreciate the tremendous help and support by the EE staff Ms. Diane Demetras, Ms. Corine
Wong, Ms. Gerrielyn Ramos, Ms. Susan Wiedem, and Mr. Theodore Low.
Last but not least, my grateful thanks are also extended to my collaborators, Professor Miles J.
Padgett, Dr. Giovanni Milione, Dr. Robert Bock, Dr. Solyman Ashra, Dr. Dmitry Starodubov, Dr.
iii
Shilpa Talwar, Dr. Soji Sajuyigbe, and Dr. Brittany Lynn for uncountable fruitful discussions and
valuable advice during my doctoral studies.
iv
Table of Contents
Dedication ........................................................................................................................................ i
Acknowledgements ......................................................................................................................... ii
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Abstract ........................................................................................................................................ xiv
1. Chapter 1 Introduction ........................................................................................................ 1
1.1 Orbital angular momentum beams ........................................................................................ 1
1.2 Generation and detection of OAM beams ............................................................................. 2
1.3 OAM beams multiplexing over communication links .......................................................... 5
1.4 Dissertation outline ............................................................................................................... 6
2. Chapter 2 Generation of optical dynamic spatiotemporal beams containing two forms of
OAM ............................................................................................................................................... 9
2.1 Background and motivation .................................................................................................. 9
2.2 Theoretical principle ........................................................................................................... 12
2.3 Approach for generation of a rotating-revolving LG beam ................................................ 13
2.4 Generalization for the generation of rotating-revolving LG beams .................................... 15
2.5 Simulation results ................................................................................................................ 17
2.6 Summary and discussion ..................................................................................................... 26
3. Chapter 3 Division and multiplication of the state order for optical data-carrying OAM
beams ............................................................................................................................................ 28
3.1 Background and motivation ................................................................................................ 28
3.2 Concept and theoretical principle ........................................................................................ 29
3.3 Experimental results ............................................................................................................ 32
3.4 Summary and discussion ..................................................................................................... 38
4. Chapter 4 Printed-circuit-board-based methods to generate OAM beams for mm-wave
communications ............................................................................................................................ 39
4.1 Background and motivation ................................................................................................ 39
4.2 Mm-wave communications links using thin metamaterial plates to generate data-carrying
OAM beams .............................................................................................................................. 40
4.2.1 Principle ........................................................................................................................ 40
4.2.2 Mm-wave OAM beam generation at 28 GHz .............................................................. 44
4.4.3 Metameterials-based OAM link ................................................................................... 49
v
4.3 Mm-wave communications links using patch antenna arrays to generate data-carrying
OAM beams .............................................................................................................................. 53
4.3.1 Concept and principle ................................................................................................... 53
4.3.2 Patch-antenna-array-based OAM beam generation at 60 GHz .................................... 55
4.3.3 Patch-antenna-array-based OAM communication links ............................................... 63
4.4 Summary and conclusions ................................................................................................... 67
5. Chapter 5 Mm-wave and THz OAM communications link analysis ............................... 69
5.1 Background and motivation ................................................................................................ 69
5.2 Performance of using antenna arrays to generate and receive mm-wave OAM beams ...... 70
5.2.1 Antenna array arrangements ......................................................................................... 71
5.2.2 Comparisons of the beam evolutions between ring antenna arrays and circular antenna
arrays ..................................................................................................................................... 73
5.3 Fundamental system-degrading effects in THz communications using multiple OAM
beams with turbulence ............................................................................................................... 84
5.3.1 Concept and simulation model ..................................................................................... 85
5.3.2 XT performance dependence on frequency .................................................................. 87
5.3.3 XT performance dependence on other parameters ....................................................... 90
5.3.4 XT performance dependence with limited aperture size and misalignment ................ 96
5.4 Summary and conclusions ................................................................................................... 99
6. Chapter 6 Conclusion ..................................................................................................... 100
References ................................................................................................................................... 101
vi
List of Tables
Table 4.1. The measured power matrix of each OAM channel of Rx ℓ = -1 and ℓ = +2 when only
transmitting channel ℓ = -1 or ℓ = +2. ........................................................................................... 65
Table 4.2. The measured power matrix of each OAM channel of Rx ℓ = -1 and ℓ = +2 when only
transmitting channel ℓ = -1 or ℓ = +2. ........................................................................................... 66
Table 5.1 Simulation model parameters definition. ...................................................................... 87
vii
List of Figures
Figure 1.1. Two cases of angular momenta: (a) a spinning object carrying SAM and (b) an
orbiting object carrying OAM. (c) A circularly polarized light beam carrying SAM, and (d) the
phase structure of an OAM-carrying light beam. ........................................................................... 2
Figure 1.2. Three approaches to convert a Gaussian beam into an OAM beam: (a) a spiral phase
plate, (b) a phase hologram with a spiral phase pattern, or (c) a phase hologram with a “fork”
pattern. In this example, the conversion to an OAM beam with ℓ =+3 is depicted. ....................... 3
Figure 1.3. Mid-infrared metasurface phase plate that can generate OAM beams. (a) The
fabricated metasurface phase plate. (b) Magnified structure of the designed metasurface.
(c) Interferogram of the generated OAM beam with ℓ = +1 [12]. .................................................. 4
Figure 1.4. Free-space SDM communication system using OAM beams. (a) Multiple OAM
beams each carrying an independent data stream can be multiplexed at the transmitter, propagate
through free space and be demultiplexed at the receiver. (b) Potential application scenarios might
include data centres and back-haul connections that require high-capacity data transmission [6]. 5
Figure 2.1. Generating a spatiotemporal beam exhibiting both dynamic rotation and revolution.
(a, b) Illustration of a light beam that dynamically rotates around its beam center and revolves
around another central axis; this is analogous to the earth orbiting around the sun, exhibiting both
rotation around its Earth center and revolution around the solar central axis. (c, g) A Gaussian
beam on a single frequency exhibits no dynamic rotation/revolution. (d, h) An LG3,0 beam on a
single frequency exbibits only dynamic rotation. (e, i) Using multiple frequency comb lines, in
which each carries an LGℓ,p mode with a different ℓ value and the same p value, to generate a
Gaussian-like beam dot exhibiting only revolution around a central axis. (f, j) Using multiple
frequency comb lines, in which each carries a superposition of multiple LGℓ,p modes with one
different ℓ value and multiple p values, to generate an LG3,0 beam exhibiting both dynamic
rotation and revolution. ................................................................................................................. 11
Figure 2.2. Coherent interference of multiple frequency comb lines leading to the dynamic
rotation and revolution. (a) LGℓ,0 beam on a single frequency whose center located at the central
axis. (b) The interferogram of multiple LGℓ,p modes with multiple ℓ values and 𝑝 values carried
by a single frequency line. It produces a LGℓ,0 beam whose beam center is offset from the
original central axis. (c) Combining multiple frequency comb lines with each carrying a
superposition of multiple LGℓ,p modes (a different ℓ value and multiple p values). ∆𝜑 : time-
variant relative phase delay between neighbouring LGℓ,p modes, leading the generated light beam
to dynamically revolve around the central axis; ∆𝑓 : frequency spacing; ∆θ: revolving angle; ∆t:
t1 – t2 is the temporal delay. .......................................................................................................... 15
viii
Figure 2.3. Simulation results of a spatiotemporal beam exhibiting both rotation and revolution at
a given distance. For the generated rotating-revolving LG3,0 beam revolving at 0.2 THz, we
simulate its (a) frequency spectrum; (b) spatial LGℓ,p mode distribution, namely the amplitude
and phase of the complex coefficients Cℓ,p of all the LGℓ,p modes used for superposition; (c)
envelope structure (i.e., the iso-surfaces with an amplitude of 1/10 of the peak value), where the
top cap represents the helically twisting phasefront; (d) power distribution on light beams with
different rotating ℓ values; and (e) dynamically rotating and revolving intensity/phase profiles.
Scale bar, 1 mm. The spatiotemporal beam consists of multiple frequency comb lines, in which
each line is a superposition of multiple LGℓ,p modes (same beam waist of 0.3 mm) with one ℓ
value but multiple p values. The dynamic helical phasefront and envelope indicate that the beam
not only rotates around its beam center but also revolves around another central axis 0.75 mm
away from its center. ..................................................................................................................... 19
Figure 2.4. Simulation results of diffraction effects for a spatiotemporal beam exhibiting both
rotation and revolution. Propagation over distance of an offset conventional LG3,0 in (a) near-
field and (b) far-field for t = 0. Propagation along distance of a rotating-revolving LG3,0 in (c) the
near-field and (d) the far-field for t = 0. The first and second rows in (a) to (d) are the intensity
profiles of the propagating beams and the corresponding interferograms with Gaussian beams,
respectively. As the propagation distance increases, the spatiotemporal beam counterclockwise
revolves around the central axis. (e) The mode purity of a generated rotating-revolving LG3,0
beam with different revolving speed (0.02 THz to 2 THz) when the propagation distance is
varied from 0 to 100zR, where zR = 45.6 mm is the Rayleigh range. ............................................ 21
Figure 2.5. Independent control of two momenta of the spatiotemporal beams. (a-c) The 2D
amplitude and phase profiles at time 𝑡 = 0, and the envelope structures in (x,y,t) coordinates of
the generated rotating-revolving LGℓ,0 beams with different rotating ℓ values but the same
revolving speed of 0.2 THz. Scale bar, 1 mm. The phasefronts are exp(i2θ) and exp(i4θ) in one
circle around the beams’ center intensity nulls in (a,b), respectively; (c) is a single beam
combining an array of four rotating-revolving LGℓ,0 beams, where ℓ = 0,1,2,3. (d) The
corresponding profiles/structures of the generated rotating-revolving LGℓ,0 beams with the same ℓ
= 3 value but a different revolving speed. (e) Examples of an LG3,0 beam revolving clockwise at
different speeds from 0.2 THz to 0.6 THz; and (f) an LG3,0 beam revolving counter-clockwise at
a speed of 0.2 THz. Except for the varied parameters and the spatial/frequency spectra, all the
other parameters are the same as those in Fig. 2.3. ....................................................................... 23
Figure 2.6. The relationship between the quality of the rotating-revolving LGℓ,0 beams and the
frequency spectrum. (a) We first calculate the spatiotemporal spectra of the rotating-revolving
beam, and then we select a certain number of frequency lines/modes from the calculated spectra
for power/mode purity calculation in (b) and (c). (b) The power distribution on light beams with
different rotating ℓ values for generating a rotating-revolving LG3,0 beam, when the number of
selected frequency comb lines is varied. (c) The mode purity of a generated rotating-revolving
LGℓ,0 beam (ℓ varies from 0 to +3), when the number of selected frequency comb lines is varied.
(d) We calculate the spatiotemporal spectra of different rotating-revolving beams and count the
number of frequency lines in the 10-dB bandwidth of the frequency spectra; (e) and (f) show the
number of frequency comb lines in the 10-dB bandwidth for generating a rotating-revolving
ix
LGℓ,0 beam with (i) the same beam waist of w0 = 0.2 mm and a revolving radius of R varied from
0.75-1.5 mm; and (ii) the same revolving radius of R = 1.5 mm and a beam waist of w0 varied
from 0.15-0.5 mm, respectively. ................................................................................................... 25
Figure 3.1. The concept of division and multiplication of the OAM state order. PM: phase mask;
1
st
transformation: log-polar to Cartesian coordinate transformation; 2
nd
transformation:
Cartesian to log-polar coordinate transformation. ........................................................................ 30
Figure 3.2. The experimental setup for division and multiplication of the OAM state order. The
red box is our device to achieve division and multiplication. The left side shows generation of
two OAM beams. The right side shows detection of the OAM beams. Col., collimator; M,
mirror; BS, beam splitter; HWP, half-wavelength plate. .............................................................. 32
Figure 3.3. Simulated and experimental intensity profiles of OAM states and the corresponding
interference patterns with a normal Gaussian beam after halving the OAM state order with ℓ
varying from –10 to +10. .............................................................................................................. 33
Figure 3.4. Experimental mode purity after halving the OAM state order of (a) 0, (b) +2, (c) +4,
(d) +6, (e) +8, and (f) +10. ............................................................................................................ 34
Figure 3.5. Simulated and experimental intensity profiles of OAM states and the corresponding
interference patterns with a normal Gaussian beam after doubling the OAM state order with ℓ
varying from –5 to +5. .................................................................................................................. 35
Figure 3.6. Experimental mode purity after doubling the OAM state order of (a) 0, (b) +1, (c) –2,
(d) +3, (e) -4 and (f) +5. The inset in (d) shows mode purity of OAM +3 without doubling the
OAM state order. .......................................................................................................................... 35
Figure 3.7. Simulated power distribution among different OAM states as a function of
displacement (Dt) when (a) halving OAM +8 and (b) doubling OAM +2. The inset shows the
concept of SLMs with/without displacement. .............................................................................. 36
Figure 3.8. The simulated mode purity of the OAM beams (a) and (b) after halving OAM +8 or
(c) and (d) doubling OAM +2 when Dt = 0 µ m and Dt = 40 µ m. The inserts are the
corresponding far-field intensity and phase profiles. .................................................................... 36
Figure 3.9. (a) BERs for multiplex OAM –6 and –2 with/without halving the OAM state order.
(b) BERs for multiplex OAM –3 and –1 with/without doubling the OAM state order. ............... 38
Figure 4.1. (a) Schematic structure and (b) geometric parameters of proposed metamaterials plate
for OAM generation; (c) comparison between the thicknesses of metamaterials plate and spiral
phase plate. .................................................................................................................................... 43
x
Figure 4.2. Illustration of generating OAM beams (OAM ℓ = ± 2). There normally exist three
diffraction order of output beams. Diffraction order m = ± 1 has OAM ℓ = ± 2, respectively. ...... 44
Figure 4.3. The measured and simulated total transmission efficiency dependence on frequency
for each single aperture. ................................................................................................................ 45
Figure 4.4. The phase characteristics of the single rectangular aperture unit for (a) zero-order and
(b) m = ± 1 diffraction order output beams. ................................................................................... 46
Figure 4.5. (a) The measured intensity of the generated OAM ℓ = +1 and ℓ = +3 beams after
Gaussian beam passing through the metamaterials plates and (b) the interference patterns of the
OAM beams with Gaussian beams at 25 GHz, 28 GHz and 31 GHz. .......................................... 47
Figure 4.6. The generated OAM mode power spectra of (a) ℓ = +1 and (b) ℓ = +3 using
metamaterials plates. ..................................................................................................................... 48
Figure 4.7. The dispersion schematic and dependence on frequency for the first-order output
beam. ............................................................................................................................................. 49
Figure 4.8. Experiment setup for (a) multiplexing and (b) demultiplexing two OAM modes,
respectively. .................................................................................................................................. 51
Figure 4.9. The measured crosstalk (CT) dependence on frequency. ........................................... 52
Figure 4.10. BER measurements of 2 Gbaud/s 16-QAM signal for channels ℓ = +1 and ℓ = +3. 52
Figure 4.11. (a) Concept of an mm-wave communication system through stacking multi-layer
patch antenna arrays to multiplex multiple OAM modes; (b) top view of stacked patch antenna
arrays, with each fed with an independent signal from advanced modulation formats (for
example, 16-QAM); (c) comparison of the OAM ℓ = +2 evolution processes generated from
patch antenna arrays with 4 (top) and 8 (bottom) array elements. ................................................ 54
Figure 4.12. Structures of the patch antenna arrays to generate OAM beams of ℓ = ± 1 and ℓ = ± 2.
....................................................................................................................................................... 56
Figure 4.13. Simulated electrical fields of OAM ℓ = +2 at a distance of 5mm to 60mm for (a) 4,
(b) 8 and (c) 12 array elements. ................................................................................................ 57
Figure 4.14. Dependence of the mode purities on the transmission distance when (a) increasing
the number of array elements from 4 to 12, and (b) increasing the array radius from 6mm to
22mm. ........................................................................................................................................... 58
xi
Figure 4.15. The influence of the non-ideal phase delay and power to the different array
elements. ....................................................................................................................................... 59
Figure 4.16. Simulated intensity and phase of OAM beams generated by the manufactured patch
antenna arrays at different distances. ............................................................................................ 60
Figure 4.17. Influence of a co-axial obstruction with size of (a) 0mm, (b) 5mm, (c) 10mm and (d)
15mm on the evolution process. ................................................................................................... 61
Figure 4.18. 8 Measured intensity of generated beams at distances of (a) (d) (h) 5cm, (b) (e) (i)
12.5cm and (c) (f) (j) 25cm. .......................................................................................................... 62
Figure 4.19. The generated OAM mode power spectra of (a) ℓ = -1 and (b) ℓ = +2 using patch
antenna arrays. .............................................................................................................................. 63
Figure 4.20. Diagram of OAM (de)multiplexing using two modes. ............................................ 64
Figure 4.21. Photos of the experiment setup for (a) front view, (b) top view of multiplexing and
(c) demultiplexing two OAM modes, respectively. ...................................................................... 64
Figure 4.22. BER measurement of 2-Gbaud QPSK signal of channels ℓ = -1 and ℓ = +2. .......... 65
Figure 4.23. BER measurement of a 500-Mbaud 16-QAM signal of channels ℓ = -1 and ℓ = +2.
....................................................................................................................................................... 66
Figure 5.1. Structures of (a) ring antenna array and (b) three common antenna arrays with
rectangular or circular boundary shapes. D is the antenna array size and d (dθ , dr ,dx or dy ) is the
lattice period along the corresponding direction. (c) Concept of using antenna arrays as the OAM
source and receiver. All array elements on the same array are fed the same amount of power, and
the total power of each array is the same. ..................................................................................... 72
Figure 5.2. Electrical fields during beam evolutions for ring and circular antenna arrays with D =
10 cm. Parameters of (a) the ring antenna array and (b) the circular antenna array. Electrical
amplitude fields on x-z plane of the OAM beam generated from (c) ring and (d) circular antenna
arrays. (e)-(j) Electrical amplitude fields on x-y plane of two antenna arrays at distance z = 10
cm, 30 cm and 50 cm. ................................................................................................................... 74
Figure 5.3. (a) Intensity and phase profiles, (b) mode purity, and (c) path loss during beam
evolutions for ring and circular antenna arrays with D = 10 cm. P1: the power captured by the
antenna arrays; P2: the total power of the beams within a 10cm diameter circle area. ................. 76
Figure 5.4. (a) Parameters of the antenna arrays. Electrical fields at 0.1 m for (b) circular and (c)
ring antenna arrays when we steer OAM +1 by 0º , 40º , and 70º . ................................................. 76
xii
Figure 5.5. (a) Mode purity and (b) path loss dependence on steering angle at distance of 0.1 m.
....................................................................................................................................................... 77
Figure 5.6. (a) Concept of the lattice shape for antenna arrays. Antenna arrays size is D = 10 cm
and lattice period is d = 2.5 mm. (b) Simulated OAM +1 spectrum with OAM order from -30 to
+30 at 1 m for antenna arrays with different lattice shapes. (c) OAM spectrum for antenna arrays
with triangular, hexagonal and circular lattice shapes. ................................................................. 79
Figure 5.7. (a) Concept of the boundary shape for antenna arrays. Antenna arrays’ size is D = 10
cm, lattice period is d = 2.5 mm and lattice shapes are both circulars. (b) Simulated OAM +1
spectrum with OAM orders from -30 to +30 at a distance of 1 m for antenna arrays with circular
and rectangular boundary shapes. ................................................................................................. 80
Figure 5.8. (a) Simulated OAM +1 spectrum at distances from 1 m to 5 m. Antenna size is D =
10 cm, lattice period is d = 2.5 mm, lattice shape is rectangular and boundary shape is circular.
(b) Simulated OAM +1 spectra at 1 m (left) and 5 m (right). ....................................................... 81
Figure 5.9. (a) Simulated OAM spectrum for OAM 0 to +6 at a distance of 1 m. Antenna size is
D = 10 cm, lattice period is d = 2.5 mm, lattice shape is rectangular and boundary shape is
circular. (b) Simulated OAM +1 and OAM +6 spectra at 1 m. .................................................... 82
Figure 5.10. (a) Structures of three antenna arrays for simulation. (b)-(d) Corresponding
harmonic OAM order power for antenna arrays with lattice periods from 2.5 mm to 7 mm. ...... 83
Figure 5.11. Harmonic OAM order power for antenna arrays of (a) rectangular lattice shape and
circular boundary shape, (b) circular lattice shape and rectangular boundary shape and (c)
circular lattice and boundary shape with size D from 5 cm to 15 cm. .......................................... 84
Figure 5.12. Schematic of a THz communication link using OAM multiplexing. Propagation of
an OAM beam through turbulence leads to the distortion of its phase front. At the receiver side,
such a beam evolves to a beam with both distorted amplitude and phase profiles. A receiver with
limited aperture size and misalignment could further degrade the system performance. Tx:
transmitter; Rx: receiver. .............................................................................................................. 86
Figure 5.13. Schematics of the simulation model for studying the atmospheric turbulence effects
on THz OAM beams. 10 random-generated phase plates with equal spacing of 100 m are used to
emulate the turbulence for a link of a distance up to 1 km. .......................................................... 87
Figure 5.14. Normalized power distribution on different OAM modes when transmitting OAM
+4 with the same beam waist w0 = 1 m through a 1-km link. Parameters are set as (a) Cn
2
= 1 ×
10
-11
m
-2/3
, f = 0.1 THz, (b) Cn
2
= 1 × 10
-11
m
-2/3
, f = 1 THz. ........................................................ 88
xiii
Figure 5.15. Effects of varying the frequency on the normalized power distribution on different
OAM modes. (a) 200-m link with D/r0 = 0.224, w0 = 0.1 m at the transmitter side, (b) 200-m link
with Cn
2
= 1 × 10-11 m-2/3, w0 = 0.1 m. OAM +4 is transmitted for all the cases. The legends in
(a) and (b) are the same. ................................................................................................................ 89
Figure 5.16. Effects of varying the transmitted beam waist on the normalized power distribution
on different OAM modes. (a) a 500-m link with D/r0 = 0.224 at the transmitter side, and (b) a
500-m link with Cn
2
= 1 × 10
-11
m
-2/3
. OAM +4 at f = 0.5 THz is transmitted. The legends in (a)
and (b) are the same. ..................................................................................................................... 91
Figure 5.17. Effects of varying the OAM order on the normalized power distribution on different
OAM modes. (a) The received signal power on the transmitted mode order, and (b) XT1 for 1-
km link with Cn
2
= 1 × 10
-12
m
-2/3
, f = 0.5 THz, and w0 = 0.1 m at the transmitter side without
considering the atmospheric absorption. ....................................................................................... 93
Figure 5.18. Effects of varying Cn
2
on the system performance. Effects of varying Cn
2
on (b) the
signal power, (c) XT1: crosstalk to the neighboring mode. OAM +4 is transmitted for a 1-km
link with w0 = 1 m and f = 0.5 THz. The legends in (a) and (b) are the same. ............................. 94
Figure 5.19. Signal-to-Interference ratio performance for multiple multiplexed OAM beams.
Systems with (a) 3-muxed OAM modes with mode spacing from 1 to 3 (OAM beams of
[+3,+4,+5]; [+2,+4,+6]; and [+1,+4,+7] are transmitted, respectively), and (b) 3-to-7-muxed
OAM modes with a mode spacing of 1 (OAM beams of [+3,+4,+5]; [+2,+3,+4,+5,+6]; and
[+1,+2,+3,+4,+5,+6,+7] are transmitted, respectively). Parameters are set as Cn
2
= 1 × 10
-11
m
-2/3
,
w0 = 0.1 m and f = 0.5 THz. .......................................................................................................... 95
Figure 5.20. Effects of varying the displacement on the normalized power distribution on
different OAM modes. 100-m links with (a) Cn
2
= 0 m
-2/3
, (b) Cn
2
= 1 × 10
-13
m
-2/3
, and (c) Cn
2
= 1
× 10
-11
m
-2/3
. Other parameters are set as w0 = 1 m, f = 5 THz, and OAM +4 is transmitted. ...... 97
Figure 5.21. Effects of varying the displacement on the normalized power distribution on
different OAM modes. 100-m links with (a) Cn
2
= 0 m
-2/3
, (b) Cn
2
= 1 × 10
-13
m
-2/3
, and (c) Cn
2
= 1
× 10
-11
m
-2/3
. Other parameters are set as w0 = 1 m, f = 5 THz, and OAM +4 is transmitted. ...... 98
xiv
Abstract
It was discovered in 1992 that a wave with a helical phasefront possesses orbital angular
momentum (OAM). Any wave (or photon) with azimuthal phase dependence exp(𝑖 ℓ𝜃 ) carries
OAM of ℓℏ per photon (ℓ is the OAM mode order which is an unbounded integer, and ℏ is Plank’s
constant), and its wavefront twists along the propagation axis. This produces a central intensity
null (i.e., phase singularity) and annular ring. OAM beams with different ℓ values are mutually
orthogonal, such that multiple OAM beams can be efficiently (de)multiplexed and co-propagate
with little inherent crosstalk. OAM is also compatible with other degrees of freedom, such as
polarization and wavelength multiplexing.
There have been several techniques for OAM beam generation, including spiral phase plates,
diffractive phase holograms, grating-coupler-based integrated waveguides, metamaterials, and
metasurface. The basic idea of this thesis comes from two main points: (i) only a fixed OAM value
could be added to a “static” beam by utilizing most of these techniques; and (ii) some of these
techniques have been utilized for communications in the optical domain but not yet in the mm-
wave domain. Therefore, we discuss the challenges and potential solutions for OAM beams
generation and its applications in communications in both optical and mm-wave domains,
including: (i) a method to generate an optical dynamic spatiotemporal beam containing two forms
of OAM; (ii) a method to multiplicate and divide the OAM value of optical beams in
communication links; (iii) printed-circuit-board-based methods to generate OAM beams for mm-
wave communications; and (iv) system-degrading effects of mm-wave and THz OAM
communications links.
1
1. Chapter 1 Introduction
This chapter introduces the basic concept of orbital angular momentum (OAM), the OAM beam
generation method, and its application in communications.
1.1 Orbital angular momentum beams
Les Allen, et. al discovered in 1992 that beams of light -- indeed all electromagnetic (EM) waves,
with a helical phase front -- can carry OAM [1]. Since then, there has been growing worldwide
interest in this property, with initial work primarily in the physics community and more recently
being explored by combining science and engineering [2-6].
It is well-known that the concept of linear momentum is usually associated with objects moving
in a straight line. The object could also carry angular momentum if it has a rotational motion, such
as spinning (i.e., spin angular momentum (SAM)), or orbiting around an axis (i.e., OAM), as
shown in Fig. 1.1(a) and (b), respectively. A light beam may also possess these two types of
angular momenta. In paraxial approximation, a light beam carries SAM if the electrical field rotates
along the beam axis (i.e., circularly polarized light), and carries OAM if the wave vector spirals
around the beam axis, leading to a helical phase front, as shown in Fig.1.1(c) and (d) [2]. In its
analytic expression, this helical phase front is usually related to a phase term of exp(iℓθ) in the
transverse plane, where θ refers to the angular coordinate, and ℓ is an integer indicating the number
of intertwined helices (i.e., the number of 2π phase shifts along the circle around the beam axis).
ℓ is an integer and therefore can assume a positive, negative, or even a zero value, corresponding
to clockwise, counterclockwise phase helices or a Gaussian beam (i.e., no helix), respectively [2].
2
Figure 1.1. Two cases of angular momenta: (a) a spinning object carrying SAM and (b) an orbiting object carrying
OAM. (c) A circularly polarized light beam carrying SAM, and (d) the phase structure of an OAM-carrying light beam.
1.2 Generation and detection of OAM beams
Many approaches for generating OAM beams have been proposed and demonstrated. One could
obtain a single or multiple OAM beams directly from the output of a laser cavity [7,8] or by
converting a fundamental Gaussian beam into an OAM beam outside a cavity. The converter could
be a spiral phase plate [2], diffractive phase holograms [9-11], metamaterials [12-15], cylindrical
lens pairs [16], q-plates [17,18], fiber gratings [19], or couplers [20]. There are also different ways
to detect an OAM beam, such as using a converter that creates a conjugate helical phase or using
a plasmonic detector [21].
Among all external-cavity methods, perhaps the most straightforward one is to pass a Gaussian
beam through a coaxially placed spiral phase plate (SPP) [22,23] or to have it reflected by a spiral
phase mirror (SPM) [24]. An SPP is an optical element with a helical surface, as shown in
Fig. 1.2(a). To produce an OAM beam with a state of ℓ, the thickness profile of the plate should
3
be machined as ℓ𝜆𝜃 /2𝜋 (𝑛 −1), with 𝑛 being the refractive index of the medium (note that only half
of the thickness variation is required if using an SPM).
As alternatives, reconfigurable diffractive optical elements, e.g., a pixelated spatial light
modulator (SLM), or a digital micromirror device [11] can be programmed to function as a
refractive element of choice at a given wavelength, as shown in Fig. 1.2(b). Importantly, the
generated OAM beam can be easily changed by simply updating the hologram loaded on the SLM.
To spatially separate the phase-modulated beam from the zeroth-order non-phase-modulated beam
at the SLM output, a linear phase ramp is added to the helical phase mask (i.e., a “fork”-like phase
pattern, as shown in Fig. 1.2(c) to produce a spatially distinct first-order diffracted OAM beam,
carrying the desired charge [10].
Figure 1.2. Three approaches to convert a Gaussian beam into an OAM beam: (a) a spiral phase plate, (b) a phase
hologram with a spiral phase pattern, or (c) a phase hologram with a “fork” pattern. In this example, the conversion
to an OAM beam with ℓ =+3 is depicted.
Some novel material structures, such as metasurfaces, can also be used for OAM generation. A
compact metasurface could be made into a phase plate by manipulating the spatial phase response
4
caused by the structure [12,13]. As shown in Figs. 1.3(a) and 1.3(b) the metasurface is created by
assembling an array of subwavelength antennas, each of which is composed of two arms connected
at one end. Light reflected by this plate experiences a phase change ranging from 0 to 2𝜋 ,
determined by the length of the arms and their relative angle [13]. To generate an OAM beam, the
surface is divided into eight sectors, each of which introduces a phase shift from 0 to 7𝜋 /4 with a
step of 𝜋 /4. The OAM beam with ℓ = +1is obtained after the reflection, as shown in Fig. 1.3(c).
Figure 1.3. Mid-infrared metasurface phase plate that can generate OAM beams. (a) The fabricated metasurface phase
plate. (b) Magnified structure of the designed metasurface. (c) Interferogram of the generated OAM beam with ℓ = +1
[12].
Note that almost all of the mode conversion approaches can also be used to detect an OAM
beam. For example, an OAM beam can be converted back to a Gaussian-like beam if the helical
phase front is removed, e.g., by passing the OAM beam through a conjugate SPP or phase
hologram [2].
5
1.3 OAM beams multiplexing over communication links
Figure 1.4. Free-space SDM communication system using OAM beams. (a) Multiple OAM beams each carrying an
independent data stream can be multiplexed at the transmitter, propagate through free space and be demultiplexed at
the receiver. (b) Potential application scenarios might include data centres and back-haul connections that require
high-capacity data transmission [6].
In general, any orthogonal spatial modal set that can allow for efficient generation,
(de-)multiplexing and detection can be used for space-division-multiplexing (SDM) [4]. For an
SDM system using OAM beams, each channel is identified by an OAM mode with a
6
different ℓ value [4]. As depicted in Fig. 1.4(a), multiple independent data channels, each on a
different OAM beam, are spatially combined and the resulting multiplexed OAM beams are then
transmitted through a single aperture towards the receiver. After coaxially propagating through the
same free-space medium, the arriving beams are collected at the receiver by another aperture, and
subsequently demultiplexed and detected for data recovery. It should be noted that an OAM beam
diverges approximately as the square root of |ℓ| and high-order modes diverge more during
propagation [2]. Since OAM, as a fundamental property, can be carried by any helical-phased EM
waves, including light and radio waves, OAM multiplexing can be thus used in all frequency
ranges [6]. As shown in Fig. 1.4(b), the use of OAM multiplexing might have potential
applications in scenarios such as data centers and back-haul connections.
1.4 Dissertation outline
This thesis focuses on the challenges and potential solutions for OAM beams generation and its
applications in communications in both optical and mm-wave domains, including:
(i) A method to generate an optical dynamic spatiotemporal beam containing two forms
of OAM. Novel forms of beam generation and propagation based on OAM have recently
gained significant interest. In terms of changes in time, OAM can be manifest at a given
distance in different forms, including: (1) a Gaussian-like beam dot that revolves around a
central axis, and (2) a Laguerre-Gaussian (𝐿𝐺
ℓ,𝑝 ) beam with a helical phasefront rotating
around its own beam center. We explore the generation of dynamic spatiotemporal beams
that combine these two forms of orbital-angular-momenta by coherently adding multiple
frequency comb lines. Each line carries a superposition of multiple 𝐿𝐺
ℓ,𝑝 modes such that
each line is composed of a different ℓ value and multiple 𝑝 values. We simulate the generated
beams and find that the following can be achieved: (a) mode purity up to 99%, and (b) control
7
of the helical phasefront from 2π-6π and the revolving speed from 0.2-0.6 THz. This approach
might be useful for generating spatiotemporal beams with even more sophisticated dynamic
properties.
(ii) A method to multiplicate and divide the OAM value of optical beams in communication
links. We experimentally demonstrate all-optical division and multiplication of the state order
ℓ for data-carrying OAM beams. We use linear optical transformations between polar and
Cartesian coordinates to: (i) divide the OAM state order to convert the OAM order from 2ℓ
to ℓ (ℓ = –5, –4, …, +4, +5), and (ii) multiply the OAM state order from ℓ to 2ℓ. We analyze
the OAM mode purity and the bit-error-rate performance of a classical two-mode OAM
multiplexed link for the case of division and multiplication of the OAM state order. The
experimental mode purity for halving and doubling OAM state order can reach around 87%
and 40%, respectively. We further study the dependence of the OAM mode purity on the
displacement of SLMs in simulation. The obtained results show the transformation for
doubling the OAM state order is more sensitive to the increase of the displacement than that
for halving the OAM state order. The link bit error rates are below the forward error correction
threshold of 3.8× 10
–3
for both channels.
(iii) Printed-circuit-board-based methods to generate OAM beams for mm-wave
communications. We experimentally demonstrate short-range OAM-multiplexing mm-wave
communication systems using metamaterials plate and patch antenna arrays, which are fabricated by
using commercial printed-circuit-board (PCB) technology. The metamaterials plate has a thickness of
1.56 mm, and consists of 3.06 × 0.68 mm rectangular apertures with spatial variant orientations. It
generates OAM beams ℓ = +1 and ℓ = +3 with mode purity larger than 77.5% over a bandwidth of
6 GHz (25–31 GHz). We then use these streams to experimentally demonstrate a 16-Gbit/s mm-wave
8
wireless communications link using two multiplexed OAM modes, each carrying a 2-Gbaud 16-QAM
(quadrature amplitude modulation) signal. We also use these patch antenna arrays to demonstrate a
60 GHz wireless communication link using two multiplexed OAM modes, each carrying a 500-Mbaud
16-QAM or 2-Gbaud QPSK (quadrature phase shift keying) signal. A channel crosstalk of less than -
20 dB and bit-error-rates (BER) of less than 3.8 × 10
-3
are achieved.
(iv) System-degrading effects of mm-wave and THz OAM communications links. For mm-
wave OAM links, we evaluate patch antenna arrays with different arrangements as OAM
generators and receivers by simulation. We compare beam evolution processes and steering
performance for circular and ring-antenna-array based OAM links. Mode purity of the
generated OAM +1 beam fluctuates between 10% and 99% for ring antenna arrays with 10
cm diameters while it remains >99% for circular antenna arrays. For Teraherz (THz) OAM
links, we study in simulation the comprehensive diffraction and atmospheric turbulence
effects on THz wireless communication links when using OAM beams. Simulation results
show that: (i) the signal power increases and the crosstalk decreases with frequency and beam
waist, when only related-diffraction effect is considered; and (ii) the signal power tends to
decrease and the crosstalk tends to increase with frequency, beam waist, and OAM order,
when both diffraction and turbulence effects are considered.
9
2. Chapter 2 Generation of optical dynamic spatiotemporal beams containing two forms
of OAM
2.1 Background and motivation
Structured light has recently gained increased interest in that it can accommodate the production
of uniquely propagating beams of light [25-29]. One particularly interesting aspect is the ability of
a structured beam to carry orbital angular momentum (OAM) [30-34]. One form of momentum is
a simple Gaussian beam dot that can rotate in a circular fashion as it propagates, illuminating a
ring shape [35-37]; this OAM is similar to revolution around a central axis. A second form of
momentum is a subset of Laguerre-Gaussian (𝐿𝐺
ℓ,𝑝 ) beams in which the phasefront twists in the
azimuthal direction as it propagates [1,2]. The amount of OAM (ℓ) is the number of 2π azimuthal
phase changes, and 𝑝 + 1 is the number of concentric rings on the intensity cross section for ℓ >
0. The beam rotates around its own beam center with a ring-like vortex intensity profile (Figs.
2.1d,2.1h). This second type of OAM is similar to rotation. Indeed, the earth propagating around
the sun exhibits both rotation around its own Earth center and revolution around a solar central
axis [38].
These two manifestations of momentum can occur in space during propagation, but yet the
beam’s intensity will appear static at any given point of propagation distance [39-42]. This scenario
can be made more complex by enabling the generation and propagation of a dynamic
spatiotemporal beam, such that the beam simultaneously revolves and rotates in the x-y plane in
time at a given propagation distance z. Prior art has produced novel beams by combining different
modes not only on the same frequency [35-37,43-49,51-54] but also on different frequencies [35-
37,43-49]. This ability to produce different modes on different frequencies can be achieved by the
use of optical frequency combs, which have recently undergone much advancement [50].
10
Specifically, it has been previously shown that a light beam can be created to exhibit unique
dynamic features [35-37,43-49], including the following examples: (a) a Gaussian-like beam dot
that exhibits dynamic circular revolution at a given propagation distance by combining multiple
frequency lines in which each line carries a different 𝐿𝐺
ℓ,𝑝 mode (different ℓ but same 𝑝 ) [35-37]
(Fig. 2.1e, 2.1i); (b) a Gaussian-like beam dot formed from multiple Hermite-Gaussian modes,
each at a different frequency, such that the dot can dynamically move up-and-down in a linear
fashion at a given propagation distance [44]; (c) a light beam created by a pair of 𝐿𝐺
ℓ,𝑝 modes with
different ℓ and 𝑝 values at two different frequencies, such that it exhibits dynamic rotation around
its center but no dynamic revolution around another axis at a given propagation distance [47]; (d)
a combination of multiple frequency lines, each of which carries one 𝐿𝐺
ℓ,𝑝 mode with a different
pair of indices (ℓ, 𝑝 ) and can produce 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
[48]; and (e) a light beam, which is created by driving the high-harmonic-generation of two time-
delayed pulses carrying different OAM values, can exhibit dynamic rotation around its center at a
time-dependent speed [49]. A laudable goal would be to produce a more sophisticated beam that
can dynamically rotate and revolve at a tailorable speed and at a given propagation distance.
11
Figure 2.1. Generating a spatiotemporal beam exhibiting both dynamic rotation and revolution. (a, b) Illustration of a
light beam that dynamically rotates around its beam center and revolves around another central axis; this is analogous
to the earth orbiting around the sun, exhibiting both rotation around its Earth center and revolution around the solar
central axis. (c, g) A Gaussian beam on a single frequency exhibits no dynamic rotation/revolution. (d, h) An LG
3,0
beam on a single frequency exbibits only dynamic rotation. (e, i) Using multiple frequency comb lines, in which each
carries an LG
ℓ,p
mode with a different ℓ value and the same p value, to generate a Gaussian-like beam dot exhibiting
only revolution around a central axis. (f, j) Using multiple frequency comb lines, in which each carries a superposition
of multiple LG
ℓ,p
modes with one different ℓ value and multiple p values, to generate an LG
3,0
beam exhibiting both
dynamic rotation and revolution.
12
2.2 Theoretical principle
Our goal below is to generate a self-rotating electric field at 𝑧 = 0 (and more generally at a chosen
distance), that also revolves around a central axis, 𝐎 , distance 𝑅 from its self-rotating axis at a
speed of 𝑓 rev
revolutions per second (or Hz) (i.e., the number of circles per second that the electric
field revolves around 𝐎 ). Revolving the electric field at 𝑧 = 0 of a conventional (i.e., self-rotating
and not revolving) 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam at a speed of 𝑓 rev
with a revolving radius of R, we can obtain a
rotating-revolving electric field:
𝐸 1
(𝑥 , 𝑦 , 0, 𝑡 ) = 𝐿 𝐺 ℓ
̅
,𝑝 ̅
Cartesian
(𝑥 cos𝜑 (𝑡 ) − 𝑦 sin𝜑 (𝑡 ) + 𝑅 , 𝑥 sin𝜑 (𝑡 ) +
𝑦 cos𝜑 (𝑡 ), 0; 𝜔 0
, 𝑤 0
)exp(𝑖 𝜔 0
𝑡 ) (2.1)
We refer to the beam with such a dynamic electric field as a rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam (i.e., a
beam that exhibits dynamic rotation around its beam center as well as revolution around a central
axis). In this equation, 𝜔 0
= 2𝜋𝑓
0
is the angular frequency, 𝑤 0
is the beam waist, and
𝐿𝐺
ℓ
̅
,𝑝 ̅
Cartesian
(𝑥 , 𝑦 , 𝑧 ; 𝜔 0
, 𝑤 0
) is the electric field in Cartesian coordinates of a conventional 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam. 𝜑 (𝑡 ) = 𝜔 rev
𝑡 = 2𝜋𝑓
rev
𝑡 is the revolving angular speed, and(𝑥 cos𝜑 (𝑡 ) − 𝑦 sin𝜑 (𝑡 ) +
𝑅 , 𝑥 sin𝜑 (𝑡 ) + 𝑦 cos 𝜑 (𝑡 ), 0; 𝜔 0
, 𝑤 0
) is the coordinate transformation of (𝑥 , 𝑦 , 0; 𝜔 0
, 𝑤 0
) in a
reference frame rotating in the transverse plane at 𝑧 = 0.
The electric field at 𝑧 = 0 of such a rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam can also be described as a
superposition of multiple frequency comb lines with each line carrying a unique spatial pattern. It
can be written in the form:
𝐸 1
(𝑥 , 𝑦 , 0, 𝑡 ) = ∑ ∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑟 , 𝜃 , 0; 𝜔 0
+ ℓ𝜔 rev
, 𝑤 0
)exp (𝑖 (𝜔 0
+ ℓ𝜔 rev
)𝑡 )
ℓ,𝑝 (2.2)
13
𝐿 𝐺 ℓ,𝑝 (𝑟 , 𝜃 , 0; 𝜔 0
+ ℓ𝜔 rev
, 𝑤 0
) is the electric field of an 𝐿𝐺
ℓ,𝑝 mode in cylindrical coordinates,
where 𝑟 = √𝑥 2
+ 𝑦 2
and 𝜃 = arctan (𝑦 𝑥 ⁄ ) , and (𝑥 , 𝑦 , 𝑧 , 𝑡 ) are the coordinate and time,
respectively. For the ℓ
th
frequency line carrying an 𝐿𝐺
ℓ,𝑝 mode, 𝐶 ℓ,𝑝 is the complex coefficient
and 𝜔 0
+ ℓ𝜔 rev
is the angular frequency. Clearly, the expansion of Eq. (2.2) requires infinite
number of modes to be a perfectly accurate one, but we'll show below that a reasonable number of
a few tens provides an acceptable accuracy, as testified by the purity of the rotating-revolving
𝐿𝐺
ℓ
̅
,𝑝 ̅
beam.
2.3 Approach for generation of a rotating-revolving LG beam
In general, any spatial beam can be generated by a superposition of multiple modes from a
complete spatial modal basis set [55,56], and the dynamic revolution and rotation motions in our
paper can be realized by the judicial selection of spatial modes and frequencies with appropriate
complex coefficients, as described in Eq. (2.2). As an illustrative example, we first consider the
principle of generating a rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam with a zero 𝑝 ̅ value. The generation of an
𝐿𝐺
ℓ
̅
,0
beam that dynamically rotates and revolves in time at a given distance can be explained by
the coherent interference among all the frequency comb lines that each line carries a superposition
of multiple 𝐿𝐺
ℓ,𝑝 modes. The generation can be understood by considering that a simple 𝐿𝐺
ℓ
̅
,0
beam that rotates around its beam center should be firstly offset from the central axis and then be
made to revolve around the original central axis, as the following two steps:
In the first step, we introduce the dynamic rotation with a spatial offset. A single frequency
carrying an 𝐿𝐺
ℓ
̅
,0
mode can generate an 𝐿𝐺
ℓ
̅
,0
beam with a ring-like intensity profile and a twisting
phasefront of exp(𝑖 ℓ
̅
𝜃 ) in a circle around its beam center, which is located at the central axis (Fig.
2.2a). Such a phasefront leads to a Poynting vector with a non-zero azimuthal component. Because
14
the Poynting vector indicates the propagation direction of light beams in free space, the phasefront
of the above-generated beam dynamically rotates around its beam center, which is located at the
central axis, in time at a given propagation distance [57]. Such a structured beam can keep its
intensity and phase profiles, and be made offset by combining several modes from a complete
𝐿𝐺
ℓ,𝑝 modal basis set on a single frequency [57]. The approach is to choose an appropriate complex
coefficient for each mode. As shown in Fig. 2.2b, the constructive interference of multiple 𝐿𝐺
ℓ,𝑝
modes on a single frequency produces a light beam with intensity and phase profiles as the same
as those of an 𝐿𝐺
ℓ
̅
,0
beam, whose beam center is made radially offset from the central axis by a
certain distance. Because the light beam still has a twisting phasefront of exp(𝑖 ℓ
̅
𝜃 ), the phasefront
dynamically rotates around its beam center, which is offset from the central axis, in time at a given
distance.
In the second step, we introduce the dynamic revolution. For the above superposition, the
relative phase delays among all the modes are time-invariant, thus the constructive interference
produces an 𝐿𝐺
ℓ
̅
,0
beam whose intensity profiles appears static at any given point of propagation
distance. An additional dynamic revolution around the central axis could be introduced by
choosing appropriate time-variant relative phase delays among these modes. One possible
approach is to combine different modes located on different frequencies. Here, we introduce a
time-variant relative phase delay of ∆𝜑 = 2𝜋 ∆𝑓𝑡 between the neighboring 𝐿𝐺
ℓ,𝑝 modes (i.e., ∆ℓ =
1) by combining multiple frequency comb lines. Each frequency line carries multiple 𝐿𝐺
ℓ,𝑝 modes
with a different ℓ value and multiple 𝑝 values, where 𝑓 0
is the center frequency, ∆𝑓 is the
frequency spacing between neighbouring frequency comb lines, and 𝜔 ℓ
= 2𝜋 (𝑓 0
+ ℓ∆𝑓 ) is the
angular frequency of each 𝐿𝐺
ℓ,𝑝 mode. In terms of the superposition 𝐿𝐺
ℓ,𝑝 modes on a single
frequency, previous work has found that introducing a relative phase delay of ∆𝜑 between
15
neighboring 𝐿𝐺
ℓ,𝑝 modes will rotate the azimuthal location of the generated light beam by an angle
of ∆θ = ∆𝜑 [57,58]. In our case, the time-variant relative phase delay will lead to dynamic
constructive and destructive interferences, which produce an offset 𝐿𝐺
ℓ
̅
,0
beam exhibiting not only
dynamic rotation around its beam center but also dynamic revolution around a central axis (see
Fig. 2.2c).
Figure 2.2. Coherent interference of multiple frequency comb lines leading to the dynamic rotation and revolution. (a)
LG
ℓ,0
beam on a single frequency whose center located at the central axis. (b) The interferogram of multiple LG
ℓ,p
modes with multiple ℓ values and 𝑝 values carried by a single frequency line. It produces a LG
ℓ,0
beam whose beam
center is offset from the original central axis. (c) Combining multiple frequency comb lines with each carrying a
superposition of multiple LG
ℓ,p
modes (a different ℓ value and multiple p values). ∆𝜑 : time-variant relative phase delay
between neighbouring LG
ℓ,p
modes, leading the generated light beam to dynamically revolve around the central axis;
∆𝑓 : frequency spacing; ∆θ: revolving angle; ∆t: t
1
– t
2
is the temporal delay.
2.4 Generalization for the generation of rotating-revolving LG beams
It is interesting to consider the generalization of our generation method to a broader range so that it can
generate a rotating-revolving 𝐿 𝐺 ℓ
̅
,𝑝 ̅
beams with other (ℓ
̅
, 𝑝 ̅ ) values (e.g., ℓ
̅
values of > 10 or non-zero
16
𝑝 ̅ values). A rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam can be generated by offsetting a conventional 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam
to have an electric field of 𝜓 (𝑥 , 𝑦 , 0)exp(𝑖 𝜔 0
𝑡 ) at 𝑧 = 0 and subsequently dynamically revolving
the beam around a central axis. According to the modal decomposition method [55,56], any offset
conventional 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam with arbitrary (ℓ
̅
, 𝑝 ̅ ) values can be represented by a mode combination,
𝐸 0
(𝑥 , 𝑦 , 0, 𝑡 ) = 𝜓 (𝑥 , 𝑦 , 0)exp(𝑖 𝜔 0
𝑡 ) = ∑ ∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑟 , 𝜃 , 0; 𝜔 0
, 𝑤 0
)exp(𝑖 𝜔 0
𝑡 )
ℓ,𝑝 . When the
beam revolves clockwise around the origin at a speed of 𝑓 𝑟 revolutions per second, the revolution
motion introduces a frequency shift of ℓ𝜔 𝑟 to each 𝐿𝐺
ℓ,𝑝 mode so that 𝜔 0
shifts to 𝜔 0
+ ℓ𝜔 rev
[36].
Such a frequency shift transforms the electric field of a single frequency line carrying multiple modes
into the form
𝐸 1
(𝑥 , 𝑦 , 0, 𝑡 ) = ∑ ∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑟 , 𝜃 , 0; 𝜔 0
+ ℓ𝜔 rev
, 𝑤 0
)exp (𝑖 (𝜔 0
+ ℓ𝜔 rev
)𝑡 )
ℓ,𝑝 (2.3)
which is the electric field at 𝑧 = 0 of a rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam with arbitrary (ℓ
̅
, 𝑝 ̅ ) values.
Equation (2.3) indicates that a rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam can be generated by combining multiple
frequency comb lines with each carrying multiple 𝐿𝐺
ℓ,𝑝 modes. This method could be generalized
to generate rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beams with any (ℓ
̅
, 𝑝 ̅ ) values, by judiciously selecting the
coefficient 𝐶 ℓ,𝑝 to be an integral ∬ 𝜓 (𝑥 , 𝑦 , 0)(𝐿𝐺
ℓ,𝑝 (𝑟 , 𝜃 , 0; 𝜔 0
, 𝑤 0
))
∗
𝑑𝑥𝑑𝑦 [55,56]. However,
when the (ℓ
̅
, 𝑝 ̅ ) values increase, the coefficient 𝐶 ℓ,𝑝 might still have nonnegligible values for 𝐿𝐺
ℓ,𝑝
modes with modal indices out of the ranges shown in the Article. Thus, in this case, we believe that
a higher number of 𝐿𝐺
ℓ,𝑝 modes should be utilized for the combination to generate a rotating-
revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam.
17
2.5 Simulation results
Here, we detail the method for generating a rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam. As an illustrative
example, we simulate the dynamic motion of an 𝐿𝐺
3,0
beam (beam waist 𝑤 0
= 0.3 mm, center
frequency 𝑓 0
= 193.5 THz) revolving around a central axis with a radius of 𝑅 = 0.75 mm at a
speed of 𝑓 rev
= 0.2 THz. We use 61 frequency comb lines with a frequency spacing ∆𝑓 of 0.2 THz.
Each line is a superposition of multiple 𝐿𝐺
ℓ,𝑝 modes containing one unique ℓ value and multiple 𝑝
values, where 𝑝 varies from 0 to 24. The electric field can be represented by
∑ ∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑥 , 𝑦 , 0, 𝜔 ℓ
)
24
𝑝 =0
exp(𝑖 𝜔 ℓ
𝑡 )
30
ℓ=−30
at distance 𝑧 = 0 , where 𝜔 ℓ
= 2𝜋 (𝑓 0
+ ℓ∆𝑓 ) is
linearly dependent on the azimuthal mode index ℓ , and the frequency line at 𝜔 ℓ
carries a
superposition of spatial patterns ∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑥 , 𝑦 , 0, 𝜔 ℓ
)
24
𝑝 =0
.
We characterize the beam’s spatial spectrum (i.e., spatial 𝐿𝐺
ℓ,𝑝 mode distribution) using the
amplitude and phase of its complex coefficients 𝐶 ℓ,𝑝 for each 𝐿𝐺
ℓ,𝑝 mode (Fig. 2.3b). Moreover,
we map the spatial spectrum onto the frequency spectrum based on the linear relationship between
the mode index ℓ and the angular frequency 𝜔 ℓ
(Fig. 2.3a). Specifically, we calculate the total
power on each frequency comb line using the total power of the superposition of
∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑥 , 𝑦 , 0, 𝜔 ℓ
)
24
𝑝 =0
. Additionally, the phasefront and amplitude envelope (equi-amplitude
surface) structures of such a beam are simulated (see Fig. 2.3d), in which the mode purity of the
generated rotating-revolving 𝐿𝐺
3,0
beam is obtained to be ~ 99% (see Fig. 2.3d). As shown in Fig.
2.3e, the dynamic helical phasefront and amplitude envelope indicate that the beam exhibits both
dynamic rotation and revolution in time at a given distance.
18
19
Figure 2.3. Simulation results of a spatiotemporal beam exhibiting both rotation and revolution at a given distance.
For the generated rotating-revolving LG
3,0
beam revolving at 0.2 THz, we simulate its (a) frequency spectrum; (b)
spatial LG
ℓ,p
mode distribution, namely the amplitude and phase of the complex coefficients C
ℓ,p
of all the LG
ℓ,p
modes
used for superposition; (c) envelope structure (i.e., the iso-surfaces with an amplitude of 1/10 of the peak value), where
the top cap represents the helically twisting phasefront; (d) power distribution on light beams with different rotating ℓ
values; and (e) dynamically rotating and revolving intensity/phase profiles. Scale bar, 1 mm. The spatiotemporal beam
consists of multiple frequency comb lines, in which each line is a superposition of multiple LG
ℓ,p
modes (same beam
waist of 0.3 mm) with one ℓ value but multiple p values. The dynamic helical phasefront and envelope indicate that
the beam not only rotates around its beam center but also revolves around another central axis 0.75 mm away from
its center.
To characterize the quality of the rotating-revolving 𝐿𝐺
3,0
beam at various propagation
distances, we analyze the free-space diffraction effects in the near- and far-field. The examples in
Figs. 2.4a,2.4b,2.4c,2.4d show the comparison of the free-space propagation between an offset
conventional 𝐿𝐺
3,0
beam (beam center at (𝑥 , 𝑦 ) = (−0.75 mm, 0)) and the above generated
rotating-revolving 𝐿𝐺
3,0
beam (revolving speed 𝑓 rev
= 0.2 THz, revolving radius 𝑅 = 0.75 mm).
The Rayleigh range is 𝑧 R
(𝜔 0
, 𝑤 0
) = 45.6 mm for the center frequency line. Figures 2.4c,2.4d
show that the spatiotemporal beam counterclockwise revolves around the central axis as a function
of z for 𝑡 = 0. Within the Rayleigh range, the shapes of the intensity profiles of the rotating-
revolving 𝐿𝐺
3,0
beam and its interferograms with Gaussian beams are almost the same as those of
a conventional 𝐿𝐺
3,0
beam. With further propagation, such as at a distance of 70𝑧 R
, the
interferogram of a conventional 𝐿𝐺
3,0
beam remains as a twisting shape, while the interferogram
of the rotating-revolving 𝐿𝐺
3,0
beam is distorted. The mode purity of the rotating-revolving 𝐿𝐺
3,0
beam is > 90% from 0 to 20𝑧 𝑅 ; it decreases to 38% at 70𝑧 𝑅 (See the blue curve in Fig. 2.4e).
Figure 2.4e also shows that the propagation distance of the rotating-revolving 𝐿𝐺
3,0
beam with
20
mode purity of > 90% increases from 2𝑧 𝑅 to more than 100𝑧 𝑅 , when the revolving speed 𝑓 𝑟
decreases from 2 THz to 0.02 THz.
The difference between the diffraction effects of an offset conventional 𝐿𝐺
3,0
beam and a rotating-
revolving 𝐿𝐺
3,0
beam can be understood in the following manner:
(a) The electric field (𝑧 = 0) of an offset conventional 𝐿𝐺
3,0
beam and a rotating-revolving
𝐿𝐺
3,0
beam can be expressed as a superposition of multiple 𝐿𝐺
ℓ,𝑝 modes with the same
mode distribution but different frequency spectra (i.e., one frequency line at 𝜔 0
or multiple
frequency lines at 𝜔 0
+ ℓ𝜔 rev
);
(b) When the frequency difference ℓ𝜔 rev
is ≪ 𝜔 0
and the beam is within the Rayleigh
range, the diffraction effects of the 𝐿𝐺
ℓ,𝑝 mode carried by the frequency line at 𝜔 0
are
almost the same as those of the same mode carried by the frequency line at 𝜔 0
+ 𝑙𝜔
rev
.
Thus, the two superpositions with the same mode distribution but different frequency
spectra are similar to each other in the near-field; and
(c) However, such diffraction effects tend to differ with each other (i) with further
propagation at far-field and (ii) as the frequency difference ℓ𝜔 rev
increases. The diffraction
difference might introduce different spatial amplitude and phase distortions to the same
mode carried by the frequency lines on 𝜔 0
and 𝜔 0
+ ℓ𝜔 rev
. As a result, the superposition
of multiple modes carried by a single frequency line remains as an 𝐿𝐺
3,0
beam, while the
superposition of multiple modes carried by multiple frequency lines is distorted; thus, the
mode purity decreases.
21
Figure 2.4. Simulation results of diffraction effects for a spatiotemporal beam exhibiting both rotation and revolution.
Propagation over distance of an offset conventional LG
3,0
in (a) near-field and (b) far-field for t = 0. Propagation along
distance of a rotating-revolving LG
3,0
in (c) the near-field and (d) the far-field for t = 0. The first and second rows in
(a) to (d) are the intensity profiles of the propagating beams and the corresponding interferograms with Gaussian
22
beams, respectively. As the propagation distance increases, the spatiotemporal beam counterclockwise revolves
around the central axis. (e) The mode purity of a generated rotating-revolving LG
3,0
beam with different revolving
speed (0.02 THz to 2 THz) when the propagation distance is varied from 0 to 100z
R
, where z
R
= 45.6 mm is the
Rayleigh range.
Based on our simulations, the two momenta can be independently and separately controlled by
tuning the rotating ℓ
̅
values and the revolving speed of different rotating-revolving 𝐿𝐺
ℓ
̅
,0
beams.
These two momenta are associated with the dynamic rotation and revolution, respectively (Fig.
2.5). Specifically, we investigate the cases for a rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam (i) revolving
clockwise at a speed of 0.2 THz and carrying a rotating ℓ
̅
value varying from 1 to 3 (Figs.
2.5a,2.5b,2.5c), or (ii) carrying the same rotating ℓ
̅
= 3 value and revolving at a speed varying
from 0.2 THz to 0.6 THz (Figs. 2.5d,2.5e,2.5f). Two phenomena can be discerned from Fig. 2.5.
First, the rotating ℓ
̅
value of the rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam can be controlled by changing the
spatial 𝐿𝐺
ℓ,𝑝 mode distribution on each frequency line. Second, the 𝐿𝐺
ℓ
̅
,0
beam revolves at a speed
equal to the frequency spacing ∆𝑓 . This is because that the dynamic revolution is related to the
time-variant relative phase delay between the neighbouring 𝐿𝐺
ℓ,𝑝 mode for superposition, and its
value is ∆𝜑 = 2𝜋 ∆𝑓𝑡 . Moreover, it is possible to change the direction of revolution of a rotating-
revolving 𝐿𝐺
ℓ
̅
,0
beam by reassigning each modal combination ∑ 𝐶 ℓ,𝑝 𝐿𝐺
ℓ,𝑝 (𝑥 , 𝑦 , 0, 𝜔 ℓ
)
𝑝 , which is
originally carried by a frequency line on 𝜔 ℓ
= 𝜔 0
+ ℓ𝜔 rev
, to be carried by the one on 𝜔 0
−
ℓ𝜔 rev
[36]. Therefore, the total amount of orbital angular momenta associated with these two
motions could be independently controlled by changing the spatial 𝐿𝐺
ℓ,𝑝 mode distribution and
frequency spectrum, respectively.
23
Figure 2.5. Independent control of two momenta of the spatiotemporal beams. (a-c) The 2D amplitude and phase
profiles at time 𝑡 = 0, and the envelope structures in (x,y,t) coordinates of the generated rotating-revolving LG
ℓ,0
beams with different rotating ℓ
̅
values but the same revolving speed of 0.2 THz. Scale bar, 1 mm. The phasefronts are
exp(i2θ) and exp(i4θ) in one circle around the beams’ center intensity nulls in (a,b), respectively; (c) is a single beam
combining an array of four rotating-revolving LG
ℓ,0
beams, where ℓ = 0,1,2,3. (d) The corresponding profiles/structures
of the generated rotating-revolving LG
ℓ,0
beams with the same ℓ = 3 value but a different revolving speed. (e) Examples
of an LG
3,0
beam revolving clockwise at different speeds from 0.2 THz to 0.6 THz; and (f) an LG
3,0
beam revolving
counter-clockwise at a speed of 0.2 THz. Except for the varied parameters and the spatial/frequency spectra, all the
other parameters are the same as those in Fig. 2.3.
Furthermore, we investigate the quality of the dynamic spatiotemporal beam with respect to the
frequency spectrum. Here, all the frequency comb lines carry multiple 𝐿𝐺
ℓ,𝑝 modes with the same
beam waist of 0.3 mm. Figures 2.6a,2.6b,2.6c show the relationship between the power distribution
24
on light beams with different rotating ℓ
̅
values and the number of selected frequency comb lines.
Figure 2.6b shows that when the number of comb lines is selected to be < 10, the power coupling
to the light beams with the undesired rotating ℓ
̅
≠ 3 value is > -5 dB and the mode purity of the
generated rotating-revolving 𝐿𝐺
3,0
beam is < 50%; while when the number of comb lines is > 40,
the power coupling is < -12 dB and the mode purity is > 95%. We can see from Fig. 2.6c that
combining ~ 30 frequency lines could generate a rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam, where ℓ =
0, 1, 2, 3, with mode purity of > 90%. For the cases where a limited number of frequency lines,
such as 20, are used, the mode purity of the generated rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam is higher for
smaller rotating ℓ
̅
values. Figures 2.6d,2.6e,2.6f show the number of frequency comb lines within
the 10-dB bandwidth of the frequency spectra for generating rotating-revolving 𝐿𝐺
ℓ
̅
,0
beams with
different revolving radii or beam waists. The simulation results show that a larger number of
frequency comb lines would generate a rotating-revolving 𝐿𝐺
ℓ
̅
,0
beam with a (i) larger revolving
radius, (ii) smaller beam waist, or (iii) higher rotating ℓ
̅
value. These relationships can be
understood by referring to a Fourier transformation; by looking at the dynamic azimuthal mode
(the generated spatiotemporal beam) at a given time, the beam can be described as a superposition
of multiple 𝐿𝐺
ℓ,𝑝 modes with different azimuthal index ℓ values [55,56,62]. As the light beam’s
(i) revolving radius increases, (ii) beam waist decreases, or (iii) rotating ℓ
̅
value increases, the
azimuthal mode will be spatially distributed within a smaller azimuthal range; thus the number of
comb lines increases after applying a Fourier transformation from the azimuthal spatial domain to
the frequency domain [57].
25
Figure 2.6. The relationship between the quality of the rotating-revolving LG
ℓ,0
beams and the frequency spectrum. (a)
We first calculate the spatiotemporal spectra of the rotating-revolving beam, and then we select a certain number of
frequency lines/modes from the calculated spectra for power/mode purity calculation in (b) and (c). (b) The power
distribution on light beams with different rotating ℓ values for generating a rotating-revolving LG
3,0
beam, when the
number of selected frequency comb lines is varied. (c) The mode purity of a generated rotating-revolving LG
ℓ,0
beam
(ℓ varies from 0 to +3), when the number of selected frequency comb lines is varied. (d) We calculate the
spatiotemporal spectra of different rotating-revolving beams and count the number of frequency lines in the 10-dB
bandwidth of the frequency spectra; (e) and (f) show the number of frequency comb lines in the 10-dB bandwidth for
generating a rotating-revolving LG
ℓ,0
beam with (i) the same beam waist of w
0
= 0.2 mm and a revolving radius of R
varied from 0.75-1.5 mm; and (ii) the same revolving radius of R = 1.5 mm and a beam waist of w
0
varied from 0.15-
0.5 mm, respectively.
26
2.6 Summary and discussion
In this chapter, we explore the generation of spatiotemporal light beams that combine two
independent and controllable orbital-angular-momenta. This scenario is enabled by using multiple
optical frequency comb lines, with each line carrying a superposition of multiple 𝐿𝐺
ℓ,𝑝 modes
containing a different ℓ value but multiple 𝑝 values. As an example, we generate by simulation an
𝐿𝐺
3,0
beam with a beam waist of 𝑤 0
= 0.3 mm, which exhibits dynamic rotation around its beam
center as well as revolution around a central axis with a revolving radius of 𝑅 = 0.75 mm at a speed
of 𝑓 𝑟 = 0.2 THz. We show via simulation that we are able to control not only the spatiotemporal
beam’s helically twisting phasefront but also its dynamic, two-dimensional (2D) motion of rotation
and revolution at a given propagation distance. Specifically, we vary several parameters, including
the rotating ℓ
̅
value, revolving speed, revolving radius, and beam waist of the generated
spatiotemporal light beams.
Although our examples only focus on the generation of rotating-revolving 𝐿 𝐺 ℓ
̅
,0
beams with a
revolving speed of sub-THz, it might be possible to generate spatiotemporal light beams with
different speeds and more sophisticated structures.
The speed of the dynamic motion could be controlled by tuning the frequency spacing between
the frequency lines. It is thus possible to vary the revolving speed from several MHz to sub-THz by
changing the frequency spacing of the frequency comb. Besides, if frequency lines with non-constant
frequency spacing are coherently combined, the generated light beam might exhibit dynamic
motions with time-variant speed.
The structure of the generated spatiotemporal light beam could be tuned by changing the spatial
𝐿𝐺
ℓ,𝑝 mode distribution. For example, it is possible to extend our method to generate rotating-
27
revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beams with non-zero 𝑝 ̅ values. Moreover, if each frequency comb line carries a
superposition of multiple 𝐿𝐺
ℓ,𝑝 modes containing both multiple ℓ values and multiple 𝑝 values, it
might be possible to simultaneously generate multiple rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beams with different
parameters, such as different non-zero 𝑝 ̅ values, or revolving radii. Additionally, a spatiotemporal
light beam would experience spatial beam diffraction when propagating in free space. As a result, it
might not maintain the same dynamic properties at different distances. It might be possible to
generate a non-diffraction rotating-revolving Bessel beam through combining multiple frequency
comb lines with each carrying multiple modes in the Bessel modal basis [59].
We note that our analysis does not include the beam polarization since we are trying to isolate the
effects of orbital angular momenta without considering spin angular momentum. However, we
believe that it might be possible to generate a rotating-revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam that also carries spin
angular momentum. One potential method could be realized in 3 steps: (i) generating two rotating-
revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beams on x- and y-polarizations, (ii) subsequently adding a phase delay of 𝜋 /2 to
one of the beams [2], and (iii) finally coherently combining the two beams.
We also note that our results indicate that we might need to combine a large number of frequency
comb lines with each carrying a large number of 𝐿𝐺
ℓ,𝑝 modes in order to generate a rotating-
revolving 𝐿𝐺
ℓ
̅
,𝑝 ̅
beam with high mode purity. Although these large numbers are difficult to achieve
at present, there have been reports of generating such large numbers of modes and frequency lines
that could potentially be used for spatiotemporal light shaping. For example, reports have shown the
generation and combination of (i) ~210 𝐿𝐺
ℓ,𝑝 modes [60], and (ii) ~90 frequency lines with each
carrying different modes [61]. We believe those techniques indicate the potential to handle the
experimental feasibility of the rotating-revolving 𝐿𝐺
ℓ,𝑝 beams with high mode purity.
28
3. Chapter 3 Division and multiplication of the state order for optical data-carrying OAM
beams
3.1 Background and motivation
OAM-based quantum information processing [63-66] has also been used to increase the usable
alphabet for quantum systems [67]. For certain applications, it might be valuable to manipulate the
OAM states of light, namely, translating one OAM state to another state. This function could be
used for reconfigurable systems, such as switching and routing applications [68,69]. Moreover,
modifying the channel spacing for OAM systems could potentially reduce OAM channel crosstalk
(by increasing the state spacing) or enhance mode and system efficiency (by reducing the state
spacing).
Several elements have been proposed to add or subtract (i.e., shift) a fixed-order number onto
the OAM states of light, including spiral phase plates, spatial light modulator (SLM), integrated
microring resonators, metamaterials, and metasurfaces [9-15]. In addition to shifting the OAM
state order ℓ, it might be valuable to perform the division and multiplication of the OAM state
order; note that frequency dividers and multipliers are fairly useful for signal processing systems
in which the frequency domain is being manipulated.
Previously, OAM state multiplication combined with frequency multiplication has been
achieved using nonlinear harmonic generation [70-73]. Another demonstration of OAM state
multiplication was achieved using linear optical coordinate transformations. However, there have
been few reports of OAM state division, nor has there been data transmission on these beams that
are undergoing for either division or multiplication.
29
3.2 Concept and theoretical principle
Division and multiplication of the OAM state order can be achieved through optical coordinate
transformations in two steps using SLMs, as shown in Fig. 3.1(a). Phase mask 1 and 3 are used to
achieve the coordinate transformations; phase mask 2 and 4 are used to correct the phase of the
transformed beams. For division of the OAM state order by n: (i) unwrap the “donut shape” with
a helical phase to a rectangular shape with a linear phase through an optical log-polar to Cartesian
coordinate transformation
[74] followed by a phase correction pattern; (ii) select 1/𝑛 of the
rectangular shape and wrap it to a donut shape. This second step can be performed through a
Cartesian to log-polar coordinate transformation followed by a phase correction pattern. We block
a part of the incoming beam, resulting in an (𝑛 − 1)/𝑛 power loss in the second step. One possible
choice with less power loss is dividing the rectangular shape into n parts, wrapping each part to a
donut shape, and combining them to one beam. Multiplication of the OAM state order by n can be
also performed through a two-step coordinate transformation: (i) unwrap the donut shape to n
copies of a rectangular shape; (ii) wrap each copy of the rectangular shape to an arc shape with
angles of 2𝜋 /𝑛 and combine all arc shapes to a donut shape. Phase correction patterns are required
in both steps. Figure 3.1(b) shows the intensity and phase profiles in each step for halving and
doubling the OAM state order. The helical phase profiles are converted from exp(iθ) to exp(i2θ),
or inversely, which demonstrates the achievement of a bidirectional transformation between OAM
states |+1⟩ and |+2⟩.
30
Figure 3.1. The concept of division and multiplication of the OAM state order. PM: phase mask; 1
st
transformation:
log-polar to Cartesian coordinate transformation; 2
nd
transformation: Cartesian to log-polar coordinate transformation.
The optical log-polar to Cartesian coordinate transformation, based on two spatially designed
phase masks, can map a log-polar coordinate (𝑟 , 𝜃 ) to a Cartesian coordinate (𝑥 , 𝑦 ) with 𝑥 = 𝑎 1
𝜃
and 𝑦 = −𝑎 1
ln (𝑟 /𝑏 1
), where 𝑎 1
and 𝑏 1
are scaling constants
[75]. These phase masks can be
described as the phase delay
[75].
𝜙 1
(𝑟 , 𝜃 ) = 𝑎 1
2𝜋 𝜆𝑓
(𝑟𝜃 sin 𝜃 − 𝑟 cos𝜃 ln
𝑟 𝑏 1
+ 𝑟 cos𝜃 ) −
𝑟 2
2𝑓 , (3.1)
𝜙 2
(𝑥 , 𝑦 ) = −𝑎 1
𝑏 1
2𝜋 𝜆𝑓
exp(−
𝑥 𝑎 1
) cos (
𝑦 𝑎 1
) +
𝑥 2
+𝑦 2
2𝑓 , (3.2)
where 𝑓 is the lens’ focal length and 𝜆 is the wavelength of light. The last terms of 𝑟 2
/2𝑓 and
(𝑥 2
+ 𝑦 2
)/2𝑓 are the lens functions. The first phase mask unwraps the donut shape to a
rectangular shape, and the second one corrects the phase of the output beams after this
transformation. We can make n copies of the rectangular shape during this transformation by
changing the phase delay of the first phase mask as a sum of multiple 𝜙 1𝑖 (𝑟 , 𝜃 )
[76] with different
parameters 𝑎 1𝑖 and 𝑏 1𝑖 , described by
−
𝑟 2
2𝑓 + ∑ 𝑎 1𝑖 2𝜋 𝜆𝑓
(𝑟𝜃 sin 𝜃 − 𝑟 cos𝜃 log
𝑟 𝑏 1𝑖 + 𝑟 cos𝜃 )
𝑛 𝑖 =1
. (3.3)
31
This phase mask can achieve multiple log-polar to Cartesian transformations simultaneously,
resulting in n copies of rectangular shapes at different positions with 𝑥 = 𝑎 1𝑖 𝜃 and 𝑦 =
−𝑎 1𝑖 ln (𝑟 /𝑏 1𝑖 ) on the Fourier plane (𝑥 , 𝑦 ). Using Eq. (3.2), we can carefully design the second
phase mask to achieve different phase corrections for the corresponding copies of the beam.
In the following transformation, we use the coordinate transformation in reverse to map the
Cartesian coordinate (𝑥 , 𝑦 ) back to the log-polar coordinate (𝑟 ′
, 𝜑 ) with 𝜑 = 𝑥 /𝑎 2𝑖 and 𝑟 ′
=
𝑏 2𝑖 exp (−𝑦 /𝑎 2𝑖 ), where 𝑎 2𝑖 and 𝑏 2𝑖 are scaling constants of the inverse coordinate transformation.
Therefore, we have
𝜑 = 𝑎 1𝑖 𝜃 /𝑎 2𝑖 , (3.4)
𝑟 ′
= 𝑏 2𝑖 (𝑟 /𝑏 1𝑖 )
𝑎 1𝑖 /𝑎 2𝑖 . (3.5)
Let 𝑎 2𝑖 = 𝑛 𝑎 1𝑖 , we can get 𝑟 ′
= 𝑏 2𝑖 (𝑟 /𝑏 1𝑖 )
1/𝑛 and 𝜑 = 𝜃 /𝑛 . Assume the incoming beam has an
OAM state of |ℓ⟩, namely a helical phase front of exp(𝑖 ℓ𝜃 ). After the two-step transformations,
the output beam will have a phase front of exp(𝑖𝑛 ℓ
𝜃 𝑛 ) = exp(𝑖 𝑛 ℓ𝜑 ), indicating that the OAM
state has been transformed from |ℓ⟩ to |𝑛 ℓ⟩.
Similar to the OAM multiplication, we can use the same setup to transform the OAM state |𝑛 ℓ⟩
to |ℓ⟩. We do not copy the rectangular shape in the first step. In the second step, we select 1/𝑛 of
the rectangular shape and wrap it to a donut shape. By setting 𝑎 2𝑖 = 𝑎 1𝑖 /𝑛 , we have
𝑟 ′
= 𝑏 2𝑖 (𝑟 /𝑏 1𝑖 )
𝑛 and 𝜑 = 𝑛𝜃 . Therefore, the phase front of exp(𝑖𝑛 ℓ𝜃 ) will be converted into
exp(𝑖 ℓ𝑛𝜃 ) = exp(𝑖 ℓ𝜑 ), achieving the transformation from |𝑛 ℓ⟩ to |ℓ⟩.
32
3.3 Experimental results
Figure 3.2 shows the experimental setup for division and multiplication of the OAM state order.
A 1550 nm laser is divided into two paths of fibers with different delays and then projected onto
two SLMs (SLM-1 and SLM-2). These SLMs transform Gaussian beams into OAM beams, and a
beam splitter (BS) combines the two OAM beams. The beam is then sent to SLM-3 with phase
patterns described as Eqs. (1a) and (2), which performs the log-polar to Cartesian transformation
on the focal plane at a distance f = 360 mm from SLM-3. SLM-4 is designed to perform the phase
correction and an inverse Cartesian to log-polar transformation simultaneously. Using Eq. (3.2),
we can achieve such phase pattern by simply adding the phase-correction pattern of the first step
and the phase delay of the inverse coordinate transformation’s first phase mask. SLM-5 is used to
correct the phase pattern and demultiplex the OAM beams. Both SLM-4 and SLM-5 are placed at
a distance f = 360 mm from the prior SLM. At the receiver, the beam is split into two paths. We
use a camera to measure the intensity profiles of the output beam, and on the other path, we use a
pair of lens to demagnify the output beam and a filter to block the undesired diffraction orders of
the light. The beam is then coupled to a single mode fiber for detection.
Figure 3.2. The experimental setup for division and multiplication of the OAM state order. The red box is our device
to achieve division and multiplication. The left side shows generation of two OAM beams. The right side shows
detection of the OAM beams. Col., collimator; M, mirror; BS, beam splitter; HWP, half-wavelength plate.
33
We first perform division of the OAM state order. As shown in Fig. 3.3, we measure and
simulate the intensity profiles of the OAM states and the corresponding interference patterns with
a normal Gaussian beam after halving the OAM states from |2ℓ⟩ to |ℓ⟩. The input OAM beams
have an OAM state order varying from –10 to +10. The donut-shape intensity profiles and the
corresponding twisting interference patterns with a normal Gaussian beam demonstrate that the
output beams have an OAM state order varying from –5 to +5, as expected. Figure 3.4 shows the
measured mode purity, characterized by the power distribution among OAM modes, of the output
beams when the input OAM beams have an OAM order of 0, +2, +4, +6, +8, and +10; mode purity
can be defined as the percentage of power that is located in the desired mode. The output beams
have the highest power in the OAM order of 0, +1, +2, +4, and +5. We also find the mode purity
decreases from 87% to 53% when the input OAM state order varies from 0 to +10. A possible
explanation is that when the helical phase front of the input OAM beam twists faster, the phase
front itself introduces higher distortion on the coordinate transformations.
Figure 3.3. Simulated and experimental intensity profiles of OAM states and the corresponding interference patterns
with a normal Gaussian beam after halving the OAM state order with ℓ varying from –10 to +10.
34
Figure 3.4. Experimental mode purity after halving the OAM state order of (a) 0, (b) +2, (c) +4, (d) +6, (e) +8, and (f)
+10.
We now turn to realize multiplication of the OAM state order. In order to double the OAM states
from |ℓ⟩ to |2ℓ⟩, we use the same setup of division and change the phase patterns of SLM-3, SLM-
4, and SLM-5 to the desired phase delay described as Eqs. (1a), (1b) and (2). The input OAM
beams now have an OAM state order varying from –5 to +5. Figure 3.5 shows the measured and
simulated intensity profiles of the OAM beams and the corresponding interference patterns with a
normal Gaussian beam. The output beams still keep the donut shapes, and the interference patterns
indicate the helical phase fronts of the OAM beams with an order from –10 to +10. Figure 3.6(a)–
(f) show the measured mode purity of the output beams when input beams have an OAM order of
0, +1, -2, +3, -4, and +5, respectively. The output beams clearly have the highest power with the
OAM order of 0, +2, -4, +6, -8, and +10. Figure 3.6(d) and the inset show a comparison of the
mode purity with/without doubling the OAM state order of +3, indicating the mode purity
decreases from 85% to 35% after OAM doubling. Meanwhile, the mode purity decreases from 40%
to 28% when the absolute value of input OAM state order increase from 0 to +5. The mode purity
is worse than that of the case for OAM division. The degradation of the mode purity might be
explained by: (i) displacement of the SLMs; (ii) the diffraction effect when copying the beams;
and (iii) an imperfect combination of two semi-annular shapes to a donut shape.
35
Figure 3.5. Simulated and experimental intensity profiles of OAM states and the corresponding interference patterns
with a normal Gaussian beam after doubling the OAM state order with ℓ varying from –5 to +5.
Figure 3.6. Experimental mode purity after doubling the OAM state order of (a) 0, (b) +1, (c) –2, (d) +3, (e) -4 and
(f) +5. The inset in (d) shows mode purity of OAM +3 without doubling the OAM state order.
We further investigate the OAM mode purity through evaluating the displacement of SLMs in
simulation. In an ideal case, the SLMs would be perfectly aligned, namely, all SLMs would be
coaxial. However, in a practical experimental setup, there might be displacements among different
SLMs, as shown in the inset of Fig. 3.7(b). The presence of a displacement causes phase distortions
on the coordinate transformations, resulting in a degradation of the OAM mode purity. Figure
3.7(a) and (b) show the simulated power distribution among different OAM states when there
exists a displacement (Dt) between SLM-3 and SLM-4. We halve OAM +8 or double OAM +2 to
obtain OAM +4 at the receiver. When the displacement increases, the power leaked to other states
increases whereas the power on OAM +4 decreases. For halving the OAM order from +8 to +4,
the crosstalk between OAM +4 and the neighboring state increases from –14 dB to –3 dB when
36
Dt increases from 0 to 80 µ m. For doubling the OAM order from +2 to +4, the crosstalk between
OAM +4 and the neighboring state is –47 dB when Dt = 0, while the power of the neighboring
state becomes greater than that of OAM +4 when Dt = 12 µ m. Figure 3.8(a)–(d) show the mode
purity of the OAM beams after halving OAM +8 and doubling OAM +2 when Dt = 0 µ m and Dt
= 40 µ m, respectively. The mode purity of OAM +4 decreases from 89% to 81% for halving OAM
+8 and from 98% to 18% for doubling OAM +2. The obtained results indicate the mode purity of
the output OAM beams for a doubling OAM system is more sensitive to the increase of Dt than
that for a halving OAM system. These results may explain why the experimental mode purity for
OAM multiplication is much worse than that for OAM division.
Figure 3.7. Simulated power distribution among different OAM states as a function of displacement (Dt) when (a)
halving OAM +8 and (b) doubling OAM +2. The inset shows the concept of SLMs with/without displacement.
Figure 3.8. The simulated mode purity of the OAM beams (a) and (b) after halving OAM +8 or (c) and (d) doubling
OAM +2 when Dt = 0 µ m and Dt = 40 µ m. The inserts are the corresponding far-field intensity and phase profiles.
37
We measure the BER performances of a two-mode OAM multiplexed link for the case of
division and multiplication of the OAM state order. We multiplex two OAM channels, and each
channel carries a 50 Gbit/s Quadrature Phase-Shift Keying (QPSK) signal. First, we turn one
channel on and keep the other off. We transmit a normal Gaussian beam though one channel and
measure the BER of the single channel. We then turn both channels on and transform the OAM
state order to ℓ/2 or 2ℓ and measure the BERs of both channels with halving/doubling the OAM
order. Last, we turn both channels on while erasing the phase patterns for coordinate
transformations on SLM-3, SLM-4, and SLM-5. We can get the BERs of both channels without
halving/doubling the OAM order. Figure 3.9(a) shows the BERs for both channels with OAM
order ℓ = –6 and ℓ = –2 with/without halving the OAM state order. Comparing the BERs of the
case with/without halving the OAM order, we can see there is a BER performance penalty. This
may be because the decreased state spacing introduces more crosstalk between these two channels.
Figure 3.9(b) shows the measured BER performance of two OAM channels with orders ℓ = –3 and
ℓ = –1 with/without doubling the OAM state order. The BERs of both OAM channels can reach
levels lower than the forward error correction (FEC) limit of 3.8× 10–3 [77], demonstrating that
we achieve doubling the OAM order without much penalty on BER performance. The reason is
that doubling the OAM order deteriorates the mode purity but creates wider state spacing, namely,
increasing the OAM channel spacing from 2 to 4. It decreases the interaction between neighboring
states, resulting in less crosstalk.
38
Figure 3.9. (a) BERs for multiplex OAM –6 and –2 with/without halving the OAM state order. (b) BERs for multiplex
OAM –3 and –1 with/without doubling the OAM state order.
3.4 Summary and discussion
Optical devices for division and multiplication of the OAM order can be described as an operator
H, achieving a transformation 𝑯 |ℓ⟩ = |𝑛 ℓ⟩ , where 𝑛 = 1, 2, 3, … for the multiplication or 𝑛 =
1,
1
2
,
1
3
, … for the division. However, our devices act as an operator 𝑯 ′
, transforming the initial state
to 𝑯 ′
|ℓ⟩ = ∑ 𝑏 𝑖 +∞
𝑖 =−∞
|𝑖 ⟩, where 𝑏 𝑖 has the maximum value when 𝑖 = 𝑛 ℓ (i.e., the desired state)
and 𝑏 𝑖 may not be zero when 𝑖 ≠ 𝑛 ℓ (i.e., undesired state introducing crosstalk). This is because
imperfect coordinate mapping results in reduction in mode purity. Minimizing the displacement
of SLMs and enhancing SLMs’ transmission efficiency might be helpful to improve the mode
purity and achieve an operator 𝑯 ′
≈ 𝑯 . OAM state division and multiplication could potentially
bring some benefits to classical information processing. For example, OAM state multiplication
could be used to create wider state spacing, which might help reduce the interactions between
neighboring states, resulting in less crosstalk among OAM multiplexing channels. Moreover,
OAM state division could be used to create dense state spacing, which might enhance mode
efficiency. This might also be useful in OAM-based quantum systems.
39
4. Chapter 4 Printed-circuit-board-based methods to generate OAM beams for mm-wave
communications
4.1 Background and motivation
The typical method for imposing OAM structure on a mm-wave beam is to either pass it through
or reflect it off a spiral plate [6, 20]. The spiral phase plate imposes phase twisting on the wave,
such that the spatial offset thickness of the spiral plate over one rotation around the central axis
introduces an integer multiple of 2π in phase shift. For example, in a 28-GHz system, the thickness
of the spiral plate made of high density polyethylene (HDPE) should be at least 2.07 cm and
6.21 cm for generating OAM values of +1 and +3, respectively [6, 20]. This approach puts a lower
limit on the thickness of the components in the system and does not readily lend themselves to
enabling the co-axial generation of more than one mode. Alternatively, a recent approach has
shown that a thin metamaterial-like plate can impose an OAM value on a fundamental Gaussian
beam (e.g., OAM = 0) both in RF and optical domains [14,79]. Antenna array can be also used to
generate OAM beams in RF [78].
In this chapter, we design and experimentally evaluate thin metamaterial plates and patch
antenna arrays for OAM generation in mm-wave communication systems. The OAM generators
are fabricated by commercial PCB technology. We experimentally employ the metamaterial plates
to demonstrate a 16-Gbit/s mm-wave wireless communication link with data-carrying OAM beams
a carrier-wave frequency of 28 GHz. We also experimentally employ the designed antenna arrays
to demonstrate a 4-Gbit/s (or 8-Gbit/s) mm-wave wireless communication link at a higher carrier-
wave frequency of 60 GHz, using two multiplexed OAM modes, each carrying a 500-Mbaud 16-
QAM or 2-Gbaud quadrature phase-shift keying (QPSK) signal.
40
4.2 Mm-wave communications links using thin metamaterial plates to generate data-
carrying OAM beams
In this section, we experimentally demonstrate OAM multiplexing of a 2-mode, 16-Gbit/s mm-
wave wireless communication link using thin metamaterials-based OAM generators. Each
channel carries a 2-Gbaud 16-QAM signal at a carrier frequency of 28 GHz. The proposed
metamaterials adopt rectangular apertures with spatial variant orientations in the substrate, which
can achieve transformations between Gaussian beams and OAM beams. The metamaterials plates
are manufactured by commercial PCB technology and have a thickness of only 1.56 mm. The
proposed approach can generate OAM beams with a favorable quality at a bandwidth of 6 GHz.
When comparing to spiral phase plates, we note that the metamaterial is best suited for compact
setups and transmissions of short distances: the conversion efficiency is lower, and the different
frequencies show different diffraction spread, which could become problematic at longer distances.
On the other hand, the metamaterial plates are much more compact, which can be of great
importance in practical setups. Summarizing, our experimental results show the possibility of
building a completely metamaterials-based OAM mm-wave communication link with a favorable
BER performance.
4.2.1 Principle
We choose a linearly polarized beam with Jones vector Ein = [1 0]
T
as the input beam to generate
the OAM beam. The well-known Jones matrix of a local linear polarizer with axis of transmission
at angle α(φ) can be written as
(4.1)
( ) ( )
( ) ( )
cos 2 1 sin 2
1
sin 2 1 cos 2 2
+
=
−
Τ
41
If we can design metamaterials-based spatially variant polarizers with orientation of
α(φ) = qφ + α0 (q: polarization order, α0: initial polarization orientation for φ = 0), then the output
beam can be expressed as
0
222
1 1 1 11
0 42
i i q i q
out in
e e e
ii
−
= = + +
−
E T E
(4.2)
where [1 -i]
T
and [1 i]
T
are the two spin eigenstates, corresponding to right and left circular
polarization states (RCP and LCP, respectively). Note that
2 iq
e
and
2 iq
e
-
are the phase changes for
the LCP and RCP components, respectively. Therefore, the spatial variant polarizers convert
linearly polarized light to the LCP (or RCP) component that has an azimuthal phase change of 2πℓ,
where ℓ = 2q (or ℓ = -2q), and thus transform the Gaussian beam to an OAM beam.
To further separate the different components of polarization states, a linear grating phase change
of 2πx/λ can be added on x direction. Therefore, α(φ,x) = πx/λ + qφ + α0, and the output beams can
be rewritten as
0
2 / 2 / 2
1 1 1 11
0 42
i ix i q ix i q
out
e e e e e
ii
− −
= + +
−
E
(4.3)
We can identify by inspection that the output beam has three wave components. The linear
grating phase change results in two diffracted waves at symmetric directions for RCP and LCP
beams with OAM, whereas the linearly polarized Gaussian beam propagates in the normal
direction.
If we choose a circularly polarized beam with Jones vector Ein = [1 ± i]
T
as the input beam, the
output beam will become
42
0
2 /2
11 11
22
i ix i q
out
e e e
ii
=
+ E
(4.4)
which indicates that there are only two beam components in the output. The OAM-carrying LCP
(or RCP) beam is transmited in the diffracted direction, while the normal Gaussian LCP (or RCP)
beam continues in the same direction as the input beam.
Figure 4.1 presents the schematic structure and geometric parameters of the proposed
metamaterial plates. We design a rectangular aperture array with gradually variant orientation on
both sides of a two-layered PCB. Each layer of the PCB is covered by a copper film with thickness
t = 0.03 mm. The laminate substrate has a dielectric constant of n = 2.1 at 28 GHz and a thickness
of h = 1.5 mm. The rectangular aperture has dimensions of 3.06 × 0.68 mm and the lattice constant
d = 3.4 mm. Owing to the excitation of waveguide resonance in the rectangular aperture, the
rectangular aperture array can enhance the transmission efficiency of input beam with linear
polarization, which is perpendicular to the aperture direction. Therefore, each rectangular unit can
be regarded as a local linear polarizer. According to Equations (4.2) and (4.3), we can control the
spatial orientation angle of rectangular apertures to construct the desired spatially variant linear
polarizer. As shown in Figure 4.1(b), the spatially variant polarizer with α(φ,x) = πx/λ + φ/2+ π/2
(λ = 27.2 mm) can generate a ℓ = 1 RCP beam and ℓ = -1 LCP beam. This clearly shows a “fork”
at the center of the metamaterials. To generate a ℓ = 3 RCP beam and ℓ = -3 LCP beam, we need
the set α(φ,x) = πx/λ + 3φ/2 + π/2 (λ = 27.2 mm). The size of the metamaterials-based spatially
variant linear polarizer in our experiment is 150 × 150 mm. They are fabricated by a standard PCB
fabrication process. Figure 4.1(c) shows the comparison between the thickness of the
metamaterials plate with hpcb = 1.56 mm and a normal spiral phase plate with hspp = 20.7 mm for
43
generation of OAM ℓ = +1. We can clearly notice that the metamaterials plate is much thinner than
the spiral phase plate, which can be advantageous in reducing the size of the OAM system.
Figure 4.1. (a) Schematic structure and (b) geometric parameters of proposed metamaterials plate for OAM generation;
(c) comparison between the thicknesses of metamaterials plate and spiral phase plate.
Figure 4.2 depicts the evolution of the helical phase front after passing a regular Gaussian beam
through the metamaterial plate, which indicates generation of two OAM-carrying circularly
polarized beams (OAM order ℓ = ± 2, for example). Note that for a linearly polarized input beam,
there are three diffraction orders: m = 0 and m = ± 1. However, for a RCP (or LCP) input beam,
there exist only two diffraction orders: m = 0 and m = 1 (or m = -1, but not both). The power
transfer to any of these three orders can be easily calculated according to Equations (4.3) and (4.4).
Here, we regard each local liner polarizer as a perfect polarizer without power loss. Owing to the
spatially orientated distribution, 50% of the power of the input beam can be transferred to the
output beam. Therefore, for linearly polarized input, the diffraction efficiency for diffraction orders
at m = ± 1 is 12.5% and at m = 0 is 25%. For circularly polarized input, the diffraction efficiencies
(a) (b) (c)
44
for diffraction orders m = 1 (or m = -1) and m = 0 are both 25%. The OAM order, polarization and
direction of the zero-order output beam remain the same as the input beam. It is worth noting that
the first-order and minus-first-order outputs realize the transformations of three types of
momentum simultaneously, namely OAM, spin orbital momentum (polarization state) and linear
momentum (wave vector).
Figure 4.2. Illustration of generating OAM beams (OAM ℓ = ± 2). There normally exist three diffraction order of output
beams. Diffraction order m = ±1 has OAM ℓ = ± 2, respectively.
4.2.2 Mm-wave OAM beam generation at 28 GHz
We next study the transmission efficiency of the single rectangular aperture unit at the frequency
band from 15 GHz to 40 GHz. Periodic boundary conditions and linearly polarized plane wave
incidences are considered in our full electromagnetic wave simulation. Figure 4.3 shows the
measured and simulated transmission efficiency dependence on frequency. The polarization of the
input beam is perpendicular to the aperture direction. We calculate the total power of the output
beams of all polarization states. We can clearly notice that the measured and simulated results
match each other well. The transmission efficiency Tx is above 50% over a 6 GHz bandwidth (25–
31 GHz) and above 10% over a 8.5 GHz bandwidth (23–31.5 GHz). The transmission efficiency
Gaussian beam
OAM beam
OAM beam
σ = +1
= -1
metamaterials
45
reaches its highest value, 92.7%, at 27 GHz. However, the transmission efficiency Ty for the input
with polarization parallel to the aperture direction is less than 1% over the bandwidth from 15–
40 GHz. The extinction ratio (ER), defined by 10 × log10(Tx/Ty), between transmission
efficiencies of two polarization inputs after the rectangular aperture exceeds 17 dB over 25–31
GHz and 10 dB over 23–31.5 GHz. Therefore, the rectangular aperture can be regarded as a local
linear polarizer with a high performance over a bandwidth around 6~7.5 GHz.
Figure 4.3. The measured and simulated total transmission efficiency dependence on frequency for each single
aperture.
We also investigate the phase characteristics of the single rectangular aperture unit dependence
on frequency and orientation α(φ,x) through simulation. The corresponding phase shift for zero-
order output beams is obtained as a function of frequency, as shown in Figure 4.4(b). The phase
shift decreases from -25.8 degrees at 20 GHz to -180 degrees at 26.3 GHz, and continues to
decrease from +180 degrees at 26.4 GHz down to -52.2 degrees at 35 GHz. As shown in Figure
4.4(c), although the phase shifts for the first-order and minus-first-order output beams are different
at these three frequencies, the relative change in phase shift over a 180-degree range of α(φ,x) is
always +360 degrees (or -360 degrees). Meanwhile, the phase shifts are linearly dependent on the
46
orientation α(φ,x), as shown in Equation (4.2). These results indicate that we can use a single
spatially variant polarizer to generate a same-phase front (i.e., OAM order ℓ helical phase front)
over a 6 GHz frequency bandwidth (25–31 GHz).
Figure 4.4. The phase characteristics of the single rectangular aperture unit for (a) zero-order and (b) m = ±1 diffraction
order output beams.
Figure 4.5 shows the measured intensity of the generated OAM beams after the Gaussian beam
has passed through the metamaterial, and also shows the interference patterns of the OAM beams
with Gaussian beams at three frequencies. A linear-polarization horn lens antenna with a diameter
of 15 cm is used as transmitter. At the receiver, a linear-polarization directional antenna with a
diameter of 1 cm was scanning the xy-plane. We measure the intensity of generated OAM beams
at a scale of 50 × 50 cm with a resolution of 1 cm at frequencies of 25 GHz, 28 GHz and 31 GHz.
(a)
(b)
47
Here, we focus on the first-order output RCP beam. We use a linearly x-polarized beam as input
and detect the electrical component at the y axis of the output RCP beam. Figure 4.5(a) clearly
shows the “donut-shape” intensity of the generated OAM beams, with OAM order ℓ = +1 and
ℓ = +3. The interference patterns of generated OAM beams with normal Gaussian beams clearly
indicate the helical phase front of the output beams with OAM order of ℓ = +1 and ℓ = +3, as
expected.
Figure 4.5. (a) The measured intensity of the generated OAM ℓ = +1 and ℓ = +3 beams after Gaussian beam passing through the
metamaterials plates and (b) the interference patterns of the OAM beams with Gaussian beams at 25 GHz, 28 GHz and 31 GHz.
Figure 4.6 shows the mode power spectra of the generated OAM beams with ℓ = +1 and ℓ = +3,
respectively. To estimate the mode purity of the generated OAM beams, we use a set of so-called
spiral phase plates to transform OAM beams with OAM orders between ℓ = -4 and ℓ = +4 into
Gaussian beams. We then use a linear-polarization horn lens antenna with a diameter of 15 cm as
receiver to measure the power of each OAM mode component. For ℓ = +1, the crosstalk between
neighboring OAM orders can be lower than -12 dB at 25 GHz, 28 GHz and 31 GHz; for ℓ = +3,
25 GHz 28 GHz 31 GHz
(a)
(b)
48
the crosstalk between neighboring OAM orders can be lower than -10 dB at 25 GHz and 31 GHz,
and lower than -19 dB at 28 GHz. These results show that the mode purities are 86.8%, 90.7% and
92.1% for OAM beams with ℓ = +1 at 25 GHz, 28 GHz and 31 GHz, respectively; the mode
purities are 77.5%, 95.7% and 84.5% for OAM beams with ℓ = +3 at 25 GHz, 28 GHz and 31 GHz
respectively. This proves that the proposed metamaterials are capable of generating OAM modes
with high purity over a 6 GHz bandwidth (25–31 GHz).
Figure 4.6. The generated OAM mode power spectra of (a) ℓ = +1 and (b) ℓ = +3 using metamaterials plates.
Figure 4.7 shows the dispersion characteristics and dependence on frequency for the first-order
output beam. In order to separate the different components of polarization states, we add a linear
grating phase change of 2πx/λ on x direction, where λ = 27.2 mm. A dispersion effect will thereby
be introduced to the proposed metamaterials. The inset depicts the schematics of the dispersion.
(b)
(a)
49
The direction angle θ of the first-order output beam varies with frequency. The numerical results
show that the dispersion angle decreases from 27.4 degrees at 24 GHz to 20.2 degrees at 32 GHz.
The measured dispersion angles are 26.6 degrees, 23.6 degrees and 20.6 degrees at 25 GHz,
28 GHz and 31 GHz respectively, which matches well with the numerical results. We have thus
successfully demonstrated a metamaterials-based OAM generation approach.
Figure 4.7. The dispersion schematic and dependence on frequency for the first-order output beam.
4.4.3 Metameterials-based OAM link
We now turn to build a free space communication link using the proposed metamaterials. OAM’s
inherent orthogonality means that different OAM beams can carry different data streams, which
enables a notable increase in capacity for communication. At the receiver, a phase mask with
exp(-iℓφ) can transform OAM beams of order ℓ into a normal Gaussian beam, while other OAM
modes will retain their “donut shape”, allowing each OAM mode to be demultiplexed
independently.
Figure 4.8 provides a diagram of the experimental setup for multiplexing and demultiplexing
two OAM beams. We start from two linear-polarization horn lens antennas, AT1 and AT2, with
50
diameters of 15 cm. Metamaterials PCB1 with α(φ,x) = πx/λ + φ/2 + π/2, λ = 27.2 mm, and PCB2
with α(φ,x) = πx/λ + 3φ/2 + π/2, λ = 27.2 mm, convert x linearly polarized beams into first-order
RCP beams with OAM ℓ = +1 and ℓ = +3, respectively. In order to let OAM ℓ = +1 (or ℓ = +3)
transmit exactly along the z (or x) axis, we tilt the metamaterials PCB1 (or PCB2) to an angle of
23.6 degrees relative to the x (or z) axis. A 50:50 beam splitter (BS) is used to multiplex these two
OAM beams together. The combined OAM ℓ = +1 and ℓ = +3 beams transmit along the same free-
space path for a distance of around 1 m. At this point, we can use another metamaterial PCB3,
with α(φ,x) = πx/λ + φ/2 + π/2 (or α(φ,x) = πx/λ + 3φ/2 + π/2), λ = 27.2 mm, to demultiplex OAM
ℓ = +1 (or ℓ = +3). As shown in Figure 4.8(b), the PCB3 recovers the RCP beam with OAM back
into the minus-first-order LCP Gaussian beam, while another mode remains as an OAM mode. A
lens with focal length f = 1 m and diameter of 45 cm is used to focus the demultiplexed Gaussian
beam. Finally, we use another linear-polarization horn lens antennas with a diameter of 15 cm to
receive the power component on the y axis of the Gaussian beam.
For generation, a 28 GHz continuous-wave signal is used at the transmitter. Limited by the
bandwidth of our electrical amplifier as well as the dispersion effect of the metamaterial plate, we
choose a 2-Gbaud/s signal. We use an arbitrary waveform generator, Tektronix AWG 7102, to
generate one pair of I and Q waveforms, resulting in a 2-Gbaud/s 16-QAM data stream. The 16-
QAM is first amplified by an amplifier and then split into two outputs of equal power, each of
which is connected to one OAM channel. We use cables of different lengths to de-correlate the
signals of the two OAM channels. On the other end, for detection, we use a 80 Gsample/s real-
time scope (Agilent DSA-X 93204A) to capture the mm-wave waveform received by the antenna.
Finally, the recorded signal (2 × 10
6
bits of 2-GBaud 16-QAM signal) is processed offline to
recover the signal and measure the BER.
51
Figure 4.8. Experiment setup for (a) multiplexing and (b) demultiplexing two OAM modes, respectively.
Figure 4.9 shows the measured crosstalk dependence on frequency over 25–31 GHz. In order to
estimate the crosstalk of the OAM ℓ = +1 channel, we turn off the OAM ℓ = +1 channel while
leaving the OAM ℓ = +3 channel on. We use metamaterials with phase mask exp(-iφ) to
demultiplex the OAM mode and measure the received power by an RF spectrum analyzer. For the
OAM ℓ = +3 channel, we use metamaterials with phase mask exp(-i3φ) to demultiplex and repeat
the measurement. As shown in Figure 4.9, crosstalk < -20 dB over a bandwidth of 4 GHz (26–
30 GHz) and < -15 dB over a bandwidth of 6 GHz (25–31 GHz) for both OAM channels are
achieved. The crosstalk dependence on frequency can be explained by the diffraction effect of our
proposed metamaterials. We tilt metamaterials PCB1 and PCB2 at an angle of 23.6 degrees
according to the diffraction effect of the 28 GHz input, and point the combined OAM channels to
the center of the PCB3. The change in frequency will introduce a displacement of the combined
OAM channels and hence increase the crosstalk. Low levels of crosstalk over a bandwidth of 4–6
GHz are achieved for a 1 m link.
52
Figure 4.9. The measured crosstalk (CT) dependence on frequency.
Figure 4.10. BER measurements of 2 Gbaud/s 16-QAM signal for channels ℓ = +1 and ℓ = +3.
We further measure the BER of the two OAM channels. Each channel carries a 2 Gbaud/s 16-
QAM signal. We measure the BER when both OAM channels ℓ = +1 and ℓ = +3 are turned on. In
this case, OAM modes will experience crosstalk from each other. Owing to the low levels of
crosstalk from each channel, the BERs for both channels can reach levels lower than 3.8 × 10
-3
when SNR > 12 dB, as shown in Figure 4.10. The raw BER of 3.8 × 10
-3
is a level that allows the
achievement of a very low package error rate, using appropriate forward error correction. We also
53
show the constellations for OAM channels ℓ = +1 and ℓ = +3 when the SNR is close to 18 dB.
These results collectively demonstrate that the metamaterials-based wireless communication link
has a favorable BER performance.
4.3 Mm-wave communications links using patch antenna arrays to generate data-carrying
OAM beams
In this section, we experimentally evaluate patch antenna arrays for OAM generation at 60 GHz.
We design a 10 × 10 mm
2
(or 22 × 22 mm
2
) patch antenna array for the generation of OAM ℓ = -
1 (or ℓ = +2) at 60 GHz. The generated OAM beam of a patch antenna array, with more array
elements or smaller array radius, evolves to a ring shape more rapidly. We also find phase delay
deviation of 30 degrees leads to mode purity degrading to ~1% and power deviation of 6 dB
reduces the purity by <40%. An obstruction with size of <10mm has not much influence on the
evolution process of OAM beams generated from antenna arrays with r = 8mm, enabling the
stacking of multiple antenna arrays. We then experimentally employ the designed antenna arrays
to demonstrate a 4-Gbit/s (or 8-Gbit/s) mm-wave wireless communication link, using two
multiplexed OAM modes, each carrying a 500-Mbaud 16-QAM or 2-Gbaud quadrature phase-
shift keying (QPSK) signal. We stack two patch antenna arrays to multiplex two OAM beams
(ℓ = -1 and ℓ = +2), each carrying an independent data stream. Both channels can reach BERs of
less than 3.8 × 10
-3
.
4.3.1 Concept and principle
Figure 4.11 shows the concept of an mm-wave communication system through the stacking of
multi-layer patch antenna arrays to multiplexing multiple OAM modes. As shown in Figure 4.11(a)
and 1(b), n layers of patch antenna arrays could be stacked as a “pyramid” shape, which can
54
generate OAM ℓ1, ℓ 2, ℓ 3, …, and ℓ n, simultaneously. There are mi, i = 1, 2, 3, …, array elements
uniformly placed in a circle with a radius of ri (the array radius). Each array element is fed with
the same amount of power, while a 2πℓi/mi phase shift is compared with the neighboring array
element. This introduces a helical phase front of 2πℓi to the generated beams. The size of the upper
layer is smaller than that of the lower layer, in order to ensure that the majority of the power of the
ring-shape OAM beams, generated from the lower layer, is not blocked by the upper layer. Each
OAM channel carries an independent data stream, and transmits coaxially in free space. At the
receiver (Rx) side, we use spatial phase plates (SPP) to convert OAM beams back into Gaussian-
like beams. Horn lens antennas could be used to receive the power of the demultiplexed signal. As
shown in Figure 4.11(a), the intensity profile of OAM beams does not have a perfect ring shape at
the very beginning. When the OAM beams propagate in free space, a ring shape will gradually
form. As shown in Figure 4.11(c), the OAM beams generated from patch antenna arrays with
varying numbers of array elements have differing evolution processes, which will be discussed in
detail in the following sections.
Figure 4.11. (a) Concept of an mm-wave communication system through stacking multi-layer patch antenna arrays to
multiplex multiple OAM modes; (b) top view of stacked patch antenna arrays, with each fed with an independent
55
signal from advanced modulation formats (for example, 16-QAM); (c) comparison of the OAM ℓ = +2 evolution
processes generated from patch antenna arrays with 4 (top) and 8 (bottom) array elements.
4.3.2 Patch-antenna-array-based OAM beam generation at 60 GHz
Figure 4.12 presents the schematic structure and geometric parameters of the proposed patch
antenna arrays for generating OAM ℓ = ± 1 and ℓ = ± 2. We choose RO4003 as the substrate material;
its thickness is 0.203mm. The single antenna has a dimension of 1.689 × 1.265mm
2
and a resonant
frequency of 60 GHz. The size of the patch antenna array is 10 × 10mm
2
for generating OAM ℓ =
± 1 and 22 × 22mm
2
for generating OAM ℓ = ± 2. All patch antenna arrays are printed on PCB
boards with an equal size of 50 × 50mm
2
. For the antenna arrays to generate the OAM beams of ℓ
= ± 1, four identical rectangular patch antennas are placed in a circle with r = 3.5mm, and for the
antenna arrays to generate the OAM beams of ℓ = ± 2, eight identical rectangular patch antennas
are placed in a circle with r = 8mm. A microstrip feeding network is designed to achieve a phase
shift for each array element. The signal is fed into the main feeding line and, finally, equally split
into four or eight paths. For generating OAM ℓ = +1, the phase of the four array elements should
be 0, 90, 180 and 270 degrees in a clockwise direction, and for generating OAM ℓ = +2, the phase
of the 8 array elements should be 0, 90, 180, 270, 360, 450, 540, 630 and 720 degrees in a clockwise
direction. Figure 4.12(b) shows the manufactured antenna arrays to generate the OAM beams of ℓ
= ± 1 and ℓ = ± 2. The antenna arrays generating the OAM beam of ℓ = -1 (or -2) can be obtained
by taking the mirror image from the antenna arrays generating the OAM beam of ℓ = +1 (or +2).
56
Figure 4.12. Structures of the patch antenna arrays to generate OAM beams of ℓ = ± 1 and ℓ = ± 2.
In the ideal case, we assume that there is no coupling effect among different array elements and
each array element is fed with the same amount of power as well as the desired phase shift. We
first use the finite element method (FEM) to simulate the spatial evolution of an OAM ℓ = +2 mode
as we increase the number of array elements from 4 to 12. We focus on one-layer patch antenna
arrays with array radius of 8mm. Figure 4.13 shows the intensity and phase profiles of the
generated OAM beams at a distance from 5 mm to 50 mm. When comparing the intensity and
phase profiles at different distances, namely, the evolution processes, we can clearly see that the
phase and intensity profiles of OAM beams change dramatically at a short distance, while they
become relatively stable at a longer distance. This shows that the phase of OAM beams tends to
evolve to be a helical phase front faster, and the intensity tends to evolve into a ring shape faster,
when increasing the number of array elements. Besides, when increasing the number of array
elements from 4 to 16, we also find that both the continuity of the phase profiles, and the quality
of the ring-shape intensity profiles, become better at the same distance. This shows that the quality
of the generated OAM beams seems to become higher as the number of array elements increases.
57
Figure 4.13. Simulated electrical fields of OAM ℓ = +2 at a distance of 5mm to 60mm for (a) 4, (b) 8 and (c) 12 array
elements.
We then quantitatively explore the influence of the number of array elements and array radius
on the evolution processes of OAM beams, through analysis of the OAM mode purities. Figure
4.14(a) shows the dependence of the mode purities on the transmission distance for different
numbers of array elements, and the same array radius of 8mm. Instead of calculating mode purities
of the entire electrical fields, we focus on the electrical fields in two 2.5mm-wide annuluses with
the highest power. We can clearly see that the mode purities are improved when we increase the
number of the array elements. For example, at a distance of 60mm from the antenna arrays, the
mode purity increases from 48.1% to 96.5% as the number of array elements increases from 4 to
12. We can also see that the mode purity tends to reach the maximum value faster when increasing
the number of the array elements. For example, for 12 array elements, the mode purity changes
58
from 84.5% to 96.5% at a distance of 5mm to 60mm, while for 6 array elements, it changes from
31.3% to 82.2% in the same range of distance. This shows that OAM beams evolve faster as the
number of array elements increases. Figure 4.14(b) shows the relationship between the array radius
and the evolution processes of OAM beams. We focus on an antenna array with 8 elements, to
generate OAM l = +2. The mode purities tend to decrease when we increase the array radius r from
6mm to 22mm. For example, at a distance of 10mm from the antenna arrays, the mode purity
decreases from 84.6% to 18.8% as the array radius increases from 6mm to 22 mm.
Figure 4.14. Dependence of the mode purities on the transmission distance when (a) increasing the number of array
elements from 4 to 12, and (b) increasing the array radius from 6mm to 22mm.
In the non-ideal or practical case, we have to consider non-ideal power and phase delay to the
different array elements and the coupling effect among different array elements. We should also
consider the influence of a co-axial obstruction from another array if we stack multi-layer arrays
together.
We first study the influence of the non-ideal power and phase delay to the different array
elements, neglecting the coupling effect among different array elements. We use an antenna array
with 8 array elements and array radius of 8mm to generate OAM ℓ = +1, +2 and +3, respectively.
59
Figure 4.15(a) shows the mode purity when all array elements has the same amount of power while
the adjacent array elements has a phase shift with deviation (i.e., the phase delay error) of -45 to
+45 degrees to the desire the phase shift. The deviation of the phase shift may reduce the OAM
mode purity. When the deviation reaches +30 or -30 degrees, we can get the desire OAM mode
with purity of ~1%. Figure 4.15(c) shows the mode purity when all array elements has the desire
phase shift while the power deviation between adjacent array elements changes from -10 dB to
+10 dB. When the deviation reaches +6 dB, the purity penalty can be around 20%, 25% and 40%
for OAM l = +1, +2 and +3, respectively.
Figure 4.15. The influence of the non-ideal phase delay and power to the different array elements.
We next study the evolution processes of the OAM beams generated by the designed patch
antenna arrays, in simulation with the feeding network; thus, all the coupling effects, and radiation
from all the elements (antenna, microstrip lines) are considered. Figure 4.16 shows the simulated
results of the evolution processes of the OAM beams. OAM ℓ = 0 is generated by a single patch
antenna, and OAM ℓ = +1 or ℓ = +2 are generated by the designed patch antenna arrays. Simulation
results show the intensity and phase profiles of OAM beams with ℓ = 0, +1 and +2 at a distance of
1mm to 50mm. As shown in Figure 4.16(b) and (c), we can see that the continuity of the phase
60
profiles and the quality of the ring-shape intensity profiles are not as good as those of the electrical
fields as shown in Figure 4.13. It might be caused by the coupling effects among different array
elements, as well as the coupling effects between the microstrip feeding lines and array elements.
Additionally, the radiation pattern from the microstrip feeding lines will also introduce distortions
to OAM beams.
Figure 4.16. Simulated intensity and phase of OAM beams generated by the manufactured patch antenna arrays at
different distances.
We then consider the influence of a co-axial obstruction on the evolution process. We use
another rectangular PCB board, of which both sides are copper layers, as the obstruction. We put
it 5mm away from a patch antenna array with 8 array elements and array radius of 8mm. Figure
4.17 shows the intensity and phase profiles of OAM ℓ = +2 at distance 10mm, 30mm and 50mm,
when we change the size of the PCB board from 0mm to 15mm. When the obstruction size is 5mm,
we cannot observe much influence on the electrical fields. When the obstruction size is 15mm, we
can clear see the decrease of the intensity and deviation of phase profiles.
61
Figure 4.17. Influence of a co-axial obstruction with size of (a) 0mm, (b) 5mm, (c) 10mm and (d) 15mm on the
evolution process.
We use a 2D stage to measure the intensity of OAM beams with ℓ = 0, +1 and +2, at distances
of 5cm, 12.5cm and 25cm from the manufactured antenna arrays. The intensity profiles of the
generated OAM beams are shown in Figure 4.18. The resolution of these profiles is not high
enough because the resolution of our 2D stage is only ~5mm. We stack 2 antenna arrays together.
The fringing intensity distribution of the OAM l = 0 may result from the interference effect of the
reflected beams. We can see the intensity profiles in Figures 4.18(b) and (c), which resemble the
intensity profiles of the simulated OAM beams in Figure 4.16. The evolution processes also show
that OAM ℓ = +1 diffracts faster than OAM ℓ = +2. The beam size of the main lobe for OAM ℓ =
+1 is around 10cm, 30cm and 60cm at distances of 5cm, 12.5cm and 25cm, respectively, while the
size of the main lobe for OAM ℓ = +2 is around 5cm, 10cm and 20cm at distances of 5cm, 12.5cm
and 25cm, respectively.
62
Figure 4.18. 8 Measured intensity of generated beams at distances of (a) (d) (h) 5cm, (b) (e) (i) 12.5cm and (c) (f) (j)
25cm.
We also measure the mode power spectra of the generated OAM beams with ℓ = -1 and ℓ = +2,
as shown in Figure 4.19. To estimate mode purity of the generated OAM beams with ℓ = -1 (or
ℓ = +2), we use a set of spiral phase plates [6] with 2πℓ (ℓ = -7, -6, …, +7 and 8) phase front to
demultiplex the corresponding OAM beams into Gaussian-like beams. We then use a linear-
polarization horn lens antenna with a diameter of 15cm to receive the power of each OAM mode
component. For OAM beams with ℓ = -1 and ℓ = +2, the crosstalk from neighboring OAM orders
can be around 7 dB. This shows that the mode purities are 68.8% and 69.5% for OAM beams with
ℓ = -1 and ℓ = +2, respectively.
63
Figure 4.19. The generated OAM mode power spectra of (a) ℓ = -1 and (b) ℓ = +2 using patch antenna arrays.
4.3.3 Patch-antenna-array-based OAM communication links
We build a free space communication link using the manufactured patch antenna arrays. Figure
4.20 shows a diagram demonstrating the multiplexing and demultiplexing of two OAM beams.
Figure 4.21 shows the photos of our experiment setup. For generation, a 28 GHz continuous-wave
signal is used at the transmitter (Tx). We use an arbitrary waveform generator to generate one pair
of the in-phase component (I) and quadrature component (Q) of waveforms, resulting in a 60 GHz
2-Gbaud QPSK or 500-Mbaud 16-QAM data stream. The data stream is first equally split into two
paths after amplification, with one of them delayed by cable for the signal de-correlation. The two
data streams are then fed into two patch antenna arrays, which are stacked with respect to each
other for multiplexing. We choose OAM beams with ℓ = -1 and ℓ = +2, because the crosstalk
between the two modes can be < -20 dB. The separation between the two layers is ~5mm. The
propagation distance from the antenna arrays to the demultiplexing SPP is 15cm.
At the Rx side, we use an SPP to convert the desired OAM beams into Gaussian-like beams,
while the other OAM modes will retain their ring shape, allowing each OAM mode to be
demultiplexed independently. A HDPE lens with a diameter of 30cm and a focus length of 30cm
is used to focus the demultiplexed beam, before being collected by a linear-polarization horn lens
64
antenna with a diameter of 15cm. After amplification by a low noise amplifier (LNA), the carrier
frequency is down-converted from 60 GHz to 10 GHz. Finally, we use an 80 Gsample/s real-time
scope to record the down-converted signal. The recorded signals are then processed offline for
signal recovery and BER measurement.
Figure 4.20. Diagram of OAM (de)multiplexing using two modes.
Figure 4.21. Photos of the experiment setup for (a) front view, (b) top view of multiplexing and (c) demultiplexing
two OAM modes, respectively.
65
Table 4.1 shows the received power matrix of an 8-Gbit/s link when only turning on Tx ℓ = -1
or ℓ = +2 (2-Gbaud QPSK signal on each channel). Crosstalk levels of less than -20 dB between
channels are achieved. We then measure the BERs of the channels in different scenarios. Owing
to the low levels of crosstalk from each channel, the BERs for both channels, when both Tx l = -1
and l = +2 are on, are close to those when only Tx ℓ = -1 or ℓ = +2 is on, as shown in Figure 4.22.
A BER of <3.8 × 10
-3
can be achieved for both channels when signal-to-noise ratio (SNR) is larger
than 9 dB, which allows for the achievement of very low package error rates, when using
appropriate forward error correction (FEC) [77]. We can see that there is a ~3 dB BER penalty,
compared to a back-to-back (B2B) case, where only a single OAM l = 0 channel is transmitted,
using horn lens antennas at both the Tx and Rx.
Table 4.1. The measured power matrix of each OAM channel of Rx ℓ = -1 and ℓ = +2 when only transmitting channel
ℓ = -1 or ℓ = +2.
Figure 4.22. BER measurement of 2-Gbaud QPSK signal of channels ℓ = -1 and ℓ = +2.
66
We further measure the crosstalk and BERs of the two OAM channels, with each channel carrying
a 500-Mbaud 16-QAM signal to achieve a 4-Gbit/s link. We also measure the received power
matrix of the system when only turning on Tx ℓ = -1 or ℓ = +2. Crosstalk of < -20 dB between
channels can also be achieved. We choose a 500-Mbaud 16-QAM signal because we find that the
quality of the signal becomes worse when using a 16-QAM signal with a higher baud rate, which
might be caused by the bandwidth of the manufactured patch antenna arrays. As shown in Figure
4.23, both channels can reach a BER of lower than 3.8 × 10
-3
when SNR > 20 dB. We can see an
error floor for both OAM channels when SNR > 23 dB. We conjecture that the observed error
floor might be caused by the performance of the patch antenna arrays, such as bandwidth.
Table 4.2. The measured power matrix of each OAM channel of Rx ℓ = -1 and ℓ = +2 when only transmitting channel
ℓ = -1 or ℓ = +2.
Figure 4.23. BER measurement of a 500-Mbaud 16-QAM signal of channels ℓ = -1 and ℓ = +2.
67
4.4 Summary and conclusions
In conclusion, we experimentally demonstrated OAM multiplexing of a 2-mode, 16-Gbit/s mm-
wave wireless communication link using thin metamaterials-based OAM generators. Each
channel carries a 2-Gbaud 16-QAM signal at a carrier frequency of 28 GHz. The proposed
metamaterials adopt rectangular apertures with spatial variant orientations in the substrate, which
can achieve transformations between Gaussian beams and OAM beams. The metamaterials plates
are manufactured by commercial PCB technology and have a thickness of only 1.56 mm. The
proposed approach can generate OAM beams with a favorable quality at a bandwidth of 6 GHz.
When comparing to spiral phase plates, we note that the metamaterial is best suited for compact
setups and transmissions of short distances: the conversion efficiency is lower, and the different
frequencies show different diffraction spread, which could become problematic at longer distances.
On the other hand, the metamaterial plates are much more compact, which can be of great
importance in practical setups. Summarizing, our experimental results show the possibility of
building a completely metamaterials-based OAM mm-wave communication link with a favorable
BER performance.
Moreover, we also experimentally demonstrate OAM multiplexing of a 2-mode, 8-Gbit/s and
4-Gbit/s mm-wave wireless communication link, using patch antenna arrays to generate OAM
beams. Each channel carries a 2-Gbaud QPSK or 500-Mbaud 16-QAM signal at a carrier
frequency of 60 GHz. The patch antenna arrays can be manufactured by commercial PCB
technology, and have a size of 50 × 50mm
2
. The evolution processes of the generated OAM beams
are analyzed. In addition, it could also be interesting to explore the possibility of multiplexing
more OAM beams by stacking more patch antenna arrays to both generate and recover OAM
channels, which might enable a higher data speed, as well as better BER performance. Our work
68
might be beneficial for building an OAM mm-wave communication link with a favorable BER
performance, based on stacked patch antenna arrays.
69
5. Chapter 5 Mm-wave and THz OAM communications link analysis
5.1 Background and motivation
OAM beams have been utilized for mode-division-multiplexing (MDM) in both mm-wave and
optical systems [4-6]. Previous works have shown that there are multiple fundamental issues in
wireless OAM links that could: (a) degrade the OAM beam orthogonality, thus introducing power
coupling from one OAM mode to the other ones, contributing to increased crosstalk (XT) among
the modes; and (b) reduce the signal power of the transmitted OAM beams that could be recovered
at the receiver side [80-87]. The beam diffraction and power attenuation during beam propagation
in the link tend to be key degrading effects for mm-wave OAM links with carrier waves below
100 GHz [6,78]. However, optical OAM links with carrier waves ~ 200 THz tend to be mainly
limited due to the atmosphere-turbulence-induced phase-front distortion [81-83].
For mm-wave applications, there have been several reports of generating and receiving OAM
beams of different values, by using a patch antenna array of multiple elements arranged in a ring
shape [78]. However, it is unclear as to the relative performance of different antenna array
arrangements for use in an OAM-multiplexed mm-wave communication link.
Moreover, as terahertz (THz) wireless communication is attracting increasing attention, it might
be valuable to explore the system performance of THz OAM links [86-91]. Because the carrier
wave frequency of THz links is between the mm-wave and optical ones, diffraction and turbulence
effects might degrade THz OAM links in their own manners. One key issue could be the
investigation of the fundamental system-degrading effects for THz wireless communication links
with atmospheric turbulence when using orbital angular momentum (OAM) beams, especially for
70
those effects which are frequency-dependent, and as such, exhibit different behaviors for mm-
wave or optical systems.
In this chapter, we evaluate antenna arrays with different arrangements as OAM generators and
receivers at 60 GHz through simulation. We compare beam evolution processes and steering
performance for circular and ring-antenna-array based OAM links. Simulation results show that
circular antenna arrays can generate OAM beams with smaller side-lobes and higher mode purity
than those of generated from ring antenna arrays. We also provide a simulation-based assessment
of the fundamental system-degrading effects for THz OAM links with atmospheric turbulence.
Simulation results show that: (i) the signal power increases and the crosstalk decreases with
frequency and beam waist, when only related-diffraction effect is considered; and (ii) the signal
power tends to decrease and the crosstalk tends to increase with frequency, beam waist, and OAM
order, when both diffraction and turbulence effects are considered.
5.2 Performance of using antenna arrays to generate and receive mm-wave OAM beams
In this section, we evaluate and compare the performance antenna arrays with different
arrangements as OAM generators and receivers at 60 GHz through simulation. We find the mode
purity of the OAM +1 beam, generated from a ring antenna array with a 10 cm diameter, fluctuates
between 10% and 99% when the link distance varies from 0 to 0.5 m. As for the OAM +1 generated
from a circular antenna array of the same diameter, the mode purity is almost always greater than
99%. Compared with the ring-antenna-array based OAM links, circular-antenna-array based OAM
links could have a ~10 dB lower power loss at distance of up to 0.5 m. We also find a circular
antenna array with a 5 cm diameter steers an OAM +1 beam up to 80º with a mode purity
degradation of <1%, while a ring antenna array has a mode purity degradation of ~10%. The power
loss could be 40 dB higher when the steering angle increases from 0º to 80º , for both cases. OAM
71
spectrum analysis shows both the lattice shape and the boundary shape of an antenna array could
cause power leakage to harmonic OAM orders from the designed OAM order, leading to a mode
purity degradation and higher inter-mode crosstalk. The effects of the link distance z, designed
OAM order, lattice period d, and array diameter D, on the amount of power leaked to such
harmonic OAM orders, are analyzed through simulations. We find such power leakage increases
as the designed OAM order or the lattice period d increases, and decreases as the array diameter
D increases.
5.2.1 Antenna array arrangements
Figures 5.1(a) and (b) compare the schematic structures and geometric parameters of the different
antenna array arrangements. Here, we investigate three different possible antenna arrangements.
(i) For ring antenna arrays, array elements are homogeneously arranged along a ring of diameter
D. The lattice period dθ, represents the distance between two neighboring elements along the
azimuthal direction. (ii) For rectangular antenna arrays, array elements are homogeneously
arranged in a circle of a diameter D, or a square of a side length D. The lattice periods along x and
y directions are dx and dy, respectively. (ii) For circular antenna arrays, array elements are
homogeneously arranged along several concentric rings in a circular area of a diameter D. The
lattice periods along the azimuthal and radial directions are dθ and dr, respectively. Figure 5.11(c)
shows the system configuration of an OAM-multiplexed mm-wave link using antenna arrays as
OAM generators and receivers. At the transmitter (Tx), neglecting coupling between elements,
each array element is fed the same amount of power, with a ℓθ phase delay, where θ is the
azimuthal angle and ℓ the OAM order. This introduces a helical phase front of 2πℓ, namely, an
OAM value ℓi to the generated beams. At the receiver (Rx), we use the same antenna array to
recover OAM beams. The signal received by each array element at Rx will be delayed by an
72
inverse helical phase of – ℓθ, and then combined into one signal. The OAM beams generated from
antenna arrays with different arrangements have different evolution processes and OAM spectra,
which will be discussed in detail in the following sections.
Figure 5.1. Structures of (a) ring antenna array and (b) three common antenna arrays with rectangular or circular
boundary shapes. D is the antenna array size and d (d
θ
, d
r
,d
x
or d
y
) is the lattice period along the corresponding
direction. (c) Concept of using antenna arrays as the OAM source and receiver. All array elements on the same array
are fed the same amount of power, and the total power of each array is the same.
In the following simulation in this section, we define the dimension of a single array element as
1.2 × 1.2 mm
2
, which could be achieved by etching a copper layer, with a thickness of ~20 μm, on
an RO4003 substrate, with a thickness of ~0.2 mm. We assume the antenna arrays at Tx and Rx
are coaxial. Each array element at Tx generates a Gaussian beam, with a beam waist of 1.2 mm.
These Gaussian beams propagate through free-space, after a distance of z, and coherently interfere
with each other, effectively forming the designed OAM beams. Then, each array element at Rx
efficiently captures (no coupling loss) the power in the corresponding area of dimension, e.g., 1.2
73
× 1.2 mm
2
. The mm-wave has a frequency of 60 GHz and the lattice period d (dθ , dr ,dx or d
y
) is
half-wavelength 2.5 mm, unless marked as variable.
5.2.2 Comparisons of the beam evolutions between ring antenna arrays and circular
antenna arrays
In many applications, it is necessary to design antennas with very high gain to beat the potential
high path loss of mm-wave link. Due to the high divergence of OAM beams, the path loss might
be an important challenge for building a practical OAM link. Enlarging the dimensions of antenna
arrays often leads to higher gain for Gaussian beam sources and it also works for OAM beam
sources. The comparisons of beam evolution processes and beam steering performance between
circular and ring antenna arrays are shown below.
Figure 5. 2 shows the electrical fields during beam evolutions when we use a ring antenna array
and a circular antenna array to generate OAM +1. As shown in Figures 5.2(a) and 5.2(b), the array
size is D = 10 cm and the lattice period is d = 2.5 mm. Figures 5.2(c) and 5.2(d) show the electrical
fields on the x-z plane for these two antenna arrays. Comparing the divergence of the main lobes,
we find the ring antenna array generates a beam with a main lobe that has smaller divergence angle,
as well as greater power loss. There are also several more side-lobes for the ring antenna array.
Figures 5.2(e)-(g) and 5.2(h)-(j) show the electrical fields on the x-y plane for those two antenna
arrays, at distances of 10 cm, 30 cm, and 50 cm, respectively. Both generated beams have dark
spots in the center, and ring shapes. Comparing Figures 5.2(g) and 5.2(j), we find the ring antenna
generates a beam of multiple rings of less power, while the circular antenna generates a beam of
fewer rings with greater power. This indicates that the circular antenna array is more efficient.
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Figure 5.2. Electrical fields during beam evolutions for ring and circular antenna arrays with D = 10 cm. Parameters
of (a) the ring antenna array and (b) the circular antenna array. Electrical amplitude fields on x-z plane of the OAM
beam generated from (c) ring and (d) circular antenna arrays. (e)-(j) Electrical amplitude fields on x-y plane of two
antenna arrays at distance z = 10 cm, 30 cm and 50 cm.
Figure 5.3(a) shows the intensity and phase profiles during beam evolution for the above-
mentioned those two antenna arrays. There are not clear “twisting” phase fronts when the distance
is <5 cm. The phase fronts start to “twist” after 5 cm. The beam generated from the ring antenna
array has a faster phase gradient along the radial direction, which might be explained by the
existence of the multi-ring intensity profiles. Figures 5.3(b) and 5.3(c) show the dependence of
mode purity and power loss as a function of distance. Here we focus on the mode purity of the
electrical fields captured by antenna arrays. The mode purity for the ring antenna array fluctuates
between 10% and 99% during beam evolution. The lowest values are located at distances around
0.11 m, 0.17 m, 0.21 m, 0.28 m, and 0.4 m. However, the mode purity for the circular antenna
arrays is always >99% after 0.05 m. Figure 5.3(c) shows the power loss become higher for longer
distances in both cases, which is due to the beam divergence. We denote P1 as the captured power
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when we use a receiver antenna array that has the same configurations as the transmitter. We find
P1 for the ring antenna array also fluctuates; we believe that this is because such an antenna array
acts as a ring aperture with a fixed radius of 5 cm (see Fig. 5.2(a)), while the intensity has multiple
dark rings of low intensity (see Fig. 5.2(f,g)). When the ring aperture matches with the dark rings,
a valley for P1 is formed. As the distance increases, the ring aperture repeatedly falls on those dark
rings, and thus, less power is captured. The positions for the lowest values of purity and power
loss mirror each other. P1, for the circular antenna array, decreases more smoothly as the distance
increases, and it stays ~15 dB higher than P1 for the ring one. We donate P2 as the total power of
the beams within a 10cm diameter circle area (which is the diameter of the two antenna array).
However, we still find P2 for the circular antenna array is ~10 dB higher than that of ring one,
which indicates the beam generated from the circular antenna array suffers less divergence.
Like the process of introducing a “twisting” phase delay to antenna arrays for generating OAM
beams, we add a tilted phase delay, and a “twisting” phase delay, to antenna arrays for steering
OAM beams. The phase delay for steering OAM ℓ on the x-z plane to an angle of φ, is expressed
as ℓθ + x2πsin(φ)/λ. Figure 5.4 shows the intensity and phase profiles at 0.1 m for circular and ring
antenna arrays when we steer OAM +1 by 0º , 40º , and 70º . The antenna array size is D = 5 cm and
the lattice period is d = 2.5 mm. For both antenna arrays, we observe the intensity profiles are
axially symmetric when the steering angle is 0º . Those intensity profiles tend to lose their axial
symmetry as we increase the steering angle.
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Figure 5.3. (a) Intensity and phase profiles, (b) mode purity, and (c) path loss during beam evolutions for ring and
circular antenna arrays with D = 10 cm. P
1
: the power captured by the antenna arrays; P
2
: the total power of the beams
within a 10cm diameter circle area.
Figure 5.4. (a) Parameters of the antenna arrays. Electrical fields at 0.1 m for (b) circular and (c) ring antenna arrays
when we steer OAM +1 by 0º , 40º , and 70º .
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Figure 5.5 shows the mode purity and path loss dependence on steering angle at distances of 0.1
m. The OAM +1 beam generated by the circular antenna array could be steered up to 80º with a
mode purity degradation of <1%, while for the ring antenna array, the mode purity decreases from
85% to 75% when the steering angle increases from 0º to 80º . The power loss is ~40 dB higher
when the steering angle increases from 0º to 80º for both cases.
Figure 5.5. (a) Mode purity and (b) path loss dependence on steering angle at distance of 0.1 m.
5.2.3 OAM spectrum analysis
In most cases, to generate or recover the designed OAM beams, a spatially dependent phase is
imparted onto the emitted or detected electrical field. Spatial phase distortion during the
impartation processes leads to power leakage from the designed mode to other modes. Antenna
array arrangements always have certain spatial structures, which sample the emitted electrical
fields as generators, or the detected fields as receivers. Such a sampling effect might introduce
spatial phase distortions in non-ideal cases, thus leading to OAM mode purity degradation.
Figure 5.6(a) shows the concept of the lattice shape for antenna arrays. The antenna arrays size
is D = 10 cm, and the lattice period is d = 2.5 mm. Take the triangular lattice shape for example,
all array elements are homogeneously arranged along with several concentric triangles. The lattice
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period, namely, the distance between two neighboring elements, is 2.5 mm, along the directions
with parallel and vertical to the sides of those triangles. Figure 5.6(b) presents the OAM +1 spectra
at 1 m as we change the lattice shape from triangle, rectangle, pentagon, etc., to circle. We observe
a series of periodically spaced harmonic peak orders for antenna arrays with a polygon lattice
shape. The OAM order gap between two neighboring harmonic orders is equal to the number of
the sides of the lattice shape. For example, the OAM beam generated by the triangular antenna
array has peak power on OAM orders of -5, -2, +1, +4, +7, etc. As for the antenna array with a
circular lattice shape, it seems that is no series of such harmonic peak orders. The effect can be
explained by considering the diffraction in cylindrical coordinates. With the n-polygon, n is the
number of the sides, lattice shape can be considered as an azimuthal grating with an angular period
of 2π/n, which introduces a phase front of 2πnm to the m-diffraction order, n and m are integers,
thus resulting in peak power for OAM with order of 1+nm, where n is the period.
The boundary shape of antenna arrays has a similar effect on the OAM spectra. Figure 5.7(a)
shows the concept of the boundary shape of antenna arrays. Both antenna arrays have circular
lattice shapes. One had a rectangular boundary shape, and the other has a circular boundary shape.
We use these antenna arrays to generate OAM +1 and simulate the OAM spectra at a distance of
1 m. For the one with a circular boundary shape, the power leakage to high OAM order, +30, for
example, is relatively high. This might be explained by the sampling effect of the antenna array at
Rx. With a limited number of array elements, the antenna array might introduce an inverse
“twisting” phase delay with a low resolution for a high OAM order. Hence, the detected power on
high OAM orders seems high. For the array with a rectangular boundary shape, we also find a
series of periodically spaced harmonic peak orders with an order gap of 4. The power on the first
harmonic orders, -3 and +5, is even greater than that on the neighboring order, 0 and +2. Such
79
boundary shapes could also be considered as an azimuthal grating, leading to the power leakage
to those harmonic orders.
Figure 5.6. (a) Concept of the lattice shape for antenna arrays. Antenna arrays size is D = 10 cm and lattice period is
d = 2.5 mm. (b) Simulated OAM +1 spectrum with OAM order from -30 to +30 at 1 m for antenna arrays with different
lattice shapes. (c) OAM spectrum for antenna arrays with triangular, hexagonal and circular lattice shapes.
80
Figure 5.7. (a) Concept of the boundary shape for antenna arrays. Antenna arrays’ size is D = 10 cm, lattice period is
d = 2.5 mm and lattice shapes are both circulars. (b) Simulated OAM +1 spectrum with OAM orders from -30 to +30
at a distance of 1 m for antenna arrays with circular and rectangular boundary shapes.
We further study the dependence of the harmonic OAM order power on distance, as shown in
Figure 5.8. The antenna array has a size of D = 10 cm, a lattice period of d = 2.5 mm, a circular
boundary shape, and a rectangular lattice shape. We use this antenna array to generate OAM +1
and simulate the power spectrum at 1 m, 2 m, 3 m, 4 m, and 5 m. We find the harmonic OAM
order gap remains at 4, and the harmonic OAM order power does not change too much, as expected,
when we increased the distance from 1 m to 5 m. For example, the power on the first harmonic
OAM order, -3 and +5, is ~50 dB lower than the main OAM order +1 at both 1 m and 5 m.
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Figure 5.8. (a) Simulated OAM +1 spectrum at distances from 1 m to 5 m. Antenna size is D = 10 cm, lattice period
is d = 2.5 mm, lattice shape is rectangular and boundary shape is circular. (b) Simulated OAM +1 spectra at 1 m (left)
and 5 m (right).
We then study the relationship between the generated OAM order and the harmonic OAM order
power, as shown in Figure 5.9. The antenna array has a size of D = 10 cm, a lattice period of d =
2.5 mm, a circular boundary shape, and a rectangular lattice shape. We use this antenna array to
generate OAM 0 to +6, and simulate the power spectrum at 1 m. We find the harmonic OAM order
gap remains at 4, while the harmonic OAM order power increases when the generated OAM order
increases. When we generate OAM +1, the power on the first harmonic OAM order, -3 and +5, is
~50 dB lower than the main OAM order +1. However, as the generated OAM order increases to
+6, the power on the left-first harmonic OAM order +2 is as high as the main OAM order +6.
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Figure 5.9. (a) Simulated OAM spectrum for OAM 0 to +6 at a distance of 1 m. Antenna size is D = 10 cm, lattice
period is d = 2.5 mm, lattice shape is rectangular and boundary shape is circular. (b) Simulated OAM +1 and OAM
+6 spectra at 1 m.
Figure 5.10 shows the dependence of the harmonic OAM order power on the lattice period d.
We keep the antenna arrays’ size D = 10 cm and increase the lattice period d from 2.5 mm to 7
mm. We select antenna arrays with three different arrangements: (i) rectangular lattice shape,
circular boundary shape; (ii) circular lattice shape and rectangular boundary shape; and (iii)
circular lattice shape and circular boundary shape. Antenna arrays (i) and (ii) are used to analyze
the effects of lattice shape and boundary shape on the harmonic OAM order power, respectively.
OAM spectrum of antenna array (iii) does not have a series of obvious periodically spaced
harmonic orders, and thus, is used for reference. Figures 5.10(b)-(d) show the power on the
designed OAM order +1, left neighboring order 0, and four nearest left harmonic orders, -3, -7, -
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11, and -15. In the simulation model, the mode purity degradation mainly comes from the
neighboring orders and harmonic orders. From Fig. 5.10(d), we observe the harmonic order power
for antenna array (iii) is always lower than the power on the neighboring order 0. The harmonic
order power for antenna array (ii) is always higher than the power on OAM order 0, and keeps
nearly the same level when we increase the lattice period d from 2.5 mm to 7 mm. The reason
might be that the boundary samples the electrical fields using a rectangular boundary shape, thus
the sampling rate remains as 4 per circle, which does not change with the lattice period d. However,
the harmonic power for antenna array (i) increases with the lattice period d. This is because the
lattice shape samples the electrical fields using the array elements, thus the sampling rate decreases
as the lattice period d increases. This means the “azimuthal grating” has a worse phase resolution,
leading to greater power leakage to other modes.
Figure 5.10. (a) Structures of three antenna arrays for simulation. (b)-(d) Corresponding harmonic OAM order power
for antenna arrays with lattice periods from 2.5 mm to 7 mm.
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We then study the influence of antenna array size D on the harmonic OAM order power. We
keep the lattice period d = 2.5 mm and increase the antenna arrays size D from 5 cm to 15 cm. The
arrangements of three antenna arrays are the same as the previous section. The simulated harmonic
OAM order power is shown in Fig. 5.11. The harmonic OAM order power and the neighboring
OAM order power for all three antenna arrays tend to decrease as we increase the antenna array
size D from 5 cm to 15 cm. We find the antenna array (ii) has the highest mode purity degradation.
When the array size is D = 15 cm, the first-left harmonic order power is still as high as -15 dB for
antenna array (ii). However, for antenna array (i) and (iii), all harmonic order power and
neighboring order power are <-40 dB.
Figure 5.11. Harmonic OAM order power for antenna arrays of (a) rectangular lattice shape and circular boundary
shape, (b) circular lattice shape and rectangular boundary shape and (c) circular lattice and boundary shape with size
D from 5 cm to 15 cm.
5.3 Fundamental system-degrading effects in THz communications using multiple OAM
beams with turbulence
In this section, we provide a simulation-based assessment of the fundamental system-degrading
effects for THz OAM links with atmospheric turbulence. Simulation results show that: (i) the
signal power increases and the crosstalk decreases with frequency and beam waist, when only
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related-diffraction effect is considered; and (ii) the signal power tends to decrease and the crosstalk
tends to increase with frequency, beam waist, and OAM order, when both diffraction and
turbulence effects are considered. We find through simulation that the crosstalk to the
neighbouring mode remains < -15 dB when we transmit OAM +4 at 0.5 THz with a beam waist of
1 m through a 1-km link under weak turbulence (Cn
2
< 5 × 10
-13
m
-2/3
). Simulation results also
show that: (i) for 3-OAM-multiplexed THz links the signal-to-interference ratio (SIR) increases
by ~ 5-7 dB if the mode spacing increase by 1; and (ii) the SIR decreases with the multiplexed
mode number.
5.3.1 Concept and simulation model
Figure 5.12 shows a schematic of a THz communication link using OAM multiplexing. The
multiplexed OAM beams diffract and lose power when propagating through free space. The
diffraction effect depends on the beams’ frequency, beam size, and OAM mode order [80]. The
power loss is a function of many factors, including the beam diffraction, as well as atmospheric
absorption [80,82]. Moreover, after propagating through atmospheric turbulence, the phase front
of the transmitted OAM beam might also be distorted, thus leading to power leakage to other OAM
modes. Such power leakage results in crosstalk among different channels when the transmitted
OAM beams are demultiplexed and converted back to Gaussian-like beams at the receiver side.
Besides, a limited receiver aperture size induces additional power loss. Finally, misalignment
between the transmitter and receiver could further increase the inter-channel crosstalk [80].
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Figure 5.12. Schematic of a THz communication link using OAM multiplexing. Propagation of an OAM beam through
turbulence leads to the distortion of its phase front. At the receiver side, such a beam evolves to a beam with both
distorted amplitude and phase profiles. A receiver with limited aperture size and misalignment could further degrade
the system performance. Tx: transmitter; Rx: receiver.
We first build a simulation model to emulate the atmospheric turbulence, and use it to study the
turbulence effects on THz OAM beams. As compared with optical OAM links, the turbulence-
induce distortion might be much weaker in THz OAM links. However, THz OAM beams exhibit
larger diffraction effects, which will also affect system performance. Thus, there might be a
tradeoff between the turbulence effects and diffraction effects. To explore the tradeoff, we emulate
the atmospheric turbulence by using phase plates, which can be written in N-by-N matrix forms of
random phase distributions according to the Kolmogorov turbulence theory [82,83]. Figure 5.13
shows the simulation model for the propagation of THz OAM beams through atmospheric
turbulence. We project an OAM beam onto the 1-st phase plate and then propagate it in free space
for 100 m, where it meets another phase plate emulating the same turbulence. We repeat such
processes until a range of 1-km free-space propagation through turbulence is simulated, which
involves 10 phase plates. The key model parameters are listed in Table 5.1.
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Figure 5.13. Schematics of the simulation model for studying the atmospheric turbulence effects on THz OAM beams.
10 random-generated phase plates with equal spacing of 100 m are used to emulate the turbulence for a link of a
distance up to 1 km.
Parameters Definition
D Transmitted beam diameter
f Transmitted beam frequency
w0 Transmitted beam waist
l Transmitted beam OAM order
Cn
2
Atmospheric structure constant
r0 Fried parameter
Table 5.1 Simulation model parameters definition.
5.3.2 XT performance dependence on frequency
Wave frequency is one of the key parameters that influence both the beam diffraction effects and
turbulence-induced phase distortion effects. One interesting issue could be exploring how the
diffraction and turbulence effects degrade the XT performance of THz links with different
frequencies. We first investigate the distortion of OAM spectrum with two selected frequencies
under atmospheric turbulence. Figure 5.14 shows that when transmitting OAM +4 with a beam
waist of w0 = 1 m through a 1-km link with strong turbulence (𝐶 𝑛 2
= 1 × 10
-11
m
-2/3
). Simulation
results show that the 0.1-THz OAM beam is distorted only a little bit (XT to the neighbouring
mode is ~ -15 dB), while the 1-THz OAM beam experience a large distortion effect (XT to the
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neighbouring mode is ~ -3 dB). The reason that an OAM beam with a higher-frequency THz carrier
wave gets higher XT could be that: (i) the phase distortion induced by the turbulence with the same
𝐶 𝑛 2
gets stronger when we vary the frequency from 0.1 THz to 1 THz [82], and thus (ii) the mode
purity is lower and the XT to other OAM modes is higher in the 0.1-THz link.
Figure 5.14. Normalized power distribution on different OAM modes when transmitting OAM +4 with the same beam
waist w
0
= 1 m through a 1-km link. Parameters are set as (a) C
n
2
= 1 × 10
-11
m
-2/3
, f = 0.1 THz, (b) C
n
2
= 1 × 10
-11
m
-
2/3
, f = 1 THz.
We have shown the distortion of OAM spectrum at two selected frequencies as examples. One
further step could be to study the dependence of the distortion effect on frequency when the
frequency-dependent diffraction effect is considered. We first explore the frequency-dependent
system-degrading effects when isolating the frequency-dependent turbulence effects. Namely, we
propagate THz OAM beams at different frequencies through emulated phase plates with the same
phase patterns. In this case, OAM beams at different frequencies experience similar phase
distortion but different diffraction effects during free-space propagation. We define the crosstalk
from right (higher) nearest mode as XT1 and the crosstalk from right 2
nd
-nearest mode as XT2. As
shown in Fig. 5.15(a), for a THz link where the input beam has the same value of D/r0, when
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frequency gets higher, the signal power on the transmitted OAM mode +4 increases and the XT to
other modes decreases. Specifically, when the frequency increase from 0.03 THz to 10 THz for a
200-m link, the signal power increases by ~ 8 dB and XT1 decreases by ~ 5 dB, where the OAM
beam has the same value of D/r0 = 0.224 at the transmitter side. This might be due to that: (i)
higher-frequency OAM beams diffract slower during free-space propagation, and thus they evolve
to beams with smaller beam size at the same propagation distance; and (ii) the phase distortion
effects get stronger when the beam size of the transmitted OAM beam increases [83].
Figure 5.15. Effects of varying the frequency on the normalized power distribution on different OAM modes. (a) 200-
m link with D/r
0
= 0.224, w
0
= 0.1 m at the transmitter side, (b) 200-m link with C
n
2
= 1 × 10-11 m-2/3, w
0
= 0.1 m.
OAM +4 is transmitted for all the cases. The legends in (a) and (b) are the same.
We further study the dependence of the distortion effect on frequency, regarding both the
frequency-dependent diffraction and turbulence effects. Here, we propagate THz OAM beams at
different frequencies through emulated phase plates with frequency-dependent phase patterns. In
this case, OAM beams at different frequencies experience different phase distortion and diffraction
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effects during free-space propagation. In contrast to the case with the same value of D/r0 as shown
in Fig. 5.15(a), simulation results show that for a THz link where the atmosphere has the same
value of 𝐶 𝑛 2
, the higher frequency, the larger the distortion effect. As shown in Fig. 5.15(b), the
signal power decreases by ~ 11 dB and XT1 increases by ~ 38 dB, where the OAM beam
propagates through turbulence with the same value of Cn
2
= 1 × 10
-11
m
-2/3
. The reason might be
that: (i) the distortion effect gets stronger when the value of D/r0 at each emulated phase plate
increases; and (ii) the Fried parameter r0 decreases with frequency. Although higher frequency
leads to a small diffraction effect, the value of D/r0 still increases with frequency, thus leading to
a larger distortion effect.
5.3.3 XT performance dependence on other parameters
The beam waist is another parameter that influence the beam diffraction effect as well as the
turbulence-induced phase distortion effect. In general, the beam diffraction effect gets weaker
when the beam waist increases. Moreover, increasing the beam waist leads to a larger value of
D/r0 and thus the turbulence-induced phase distortion gets stronger [83]. Therefore, we further
study in simulation the dependence of the distortion effect on the beam waist, when (i) only the
beam-waist-dependent diffraction effect is considered; and (ii) both the beam-waist-dependent
diffraction and phase distortion effects are considered. To simulate these two cases, we transmit
OAM beams through emulated phase plates with phase patterns exhibiting the same value of D/r0,
and the same value of 𝐶 𝑛 2
, respectively. OAM +4 at f = 0.5 THz is transmitted and the link distance
is 500 m. Two phenomena can be discerned from Fig. 5.16. First, when only the beam-waist-
dependent diffraction effect is considered (same D/r0), a larger beam waist leads to a weaker phase
distortion effect. Simulation results show that when the beam waist increases from 0.1 m to 3 m,
the signal power increases by ~ 12 dB and XT1 decreases by ~ 20 dB. This might be due to that:
91
(a) an OAM beam with a larger beam waist diffract slower during propagation, and thus the beam
waist increase slower; and (b) the smaller diffraction effect then leads to a smaller value of D/r0
during propagation. Therefore, the turbulence-induce phase distortion gets weaker [83]. Second,
when both the beam-waist-dependent diffraction and phase distortion effects are considered (same
𝐶 𝑛 2
), the phase distortion effect becomes weaker at first and then becomes larger for beam waist
of > 0.3 m. This might be explained by the comprehensive effects of two factors: (a) beam-waist-
dependent diffraction effect, and (b) the value of D/r0 linearly increases with the beam waist w0.
For the case where w0 is < 0.3 m, increasing the beam waist quickly slows down the diffraction
effect. Even though D/r0 linearly increases with w0 at the transmitter side, the comprehensive
effects still decrease the value of D/r0 as the OAM beam propagates. For the case of w0 > 0.3 m,
the distortion effect is mainly dependent on the linear relation between D/r0 and w0, such that a
larger w0 leads to a larger distortion effect.
Figure 5.16. Effects of varying the transmitted beam waist on the normalized power distribution on different OAM
modes. (a) a 500-m link with D/r
0
= 0.224 at the transmitter side, and (b) a 500-m link with C
n
2
= 1 × 10
-11
m
-2/3
. OAM
+4 at f = 0.5 THz is transmitted. The legends in (a) and (b) are the same.
92
One more key parameter that influences the beam diffraction effect is the OAM order. In
addition, increasing the OAM order also leads to a large value of D/r0, and thus the phase distortion
effect gets stronger. To further explore the OAM-order-dependent system-degrading, we then
simulate the effects of varying the OAM order on the distortion of OAM spectrum. Figures 5.17(a)
and 6(b) show that when transmitting OAM +1, +3, and +6 with a beam waist w0 = 1 m and f =
0.5 THz through a 1-km link with 𝐶 𝑛 2
= 1 × 10
-12
m
-2/3
: (i) at the same propagation distance, the
higher the OAM order, the lower signal power and higher XT1; and (ii) as the propagation distance
increases, the signal power decreases faster while the XT1 increases faster when the OAM order
gets higher. Specifically, when the propagation distance increases from 100 m to 1 km without
considering the atmospheric absorption, the signal power for OAM +1, +3, and +6 decreases by ~
1.5 dB, 4.0 dB, and 7.5 dB, while the corresponding XT1 increases by ~ 10 dB, 11 dB, and 13 dB,
respectively. The reason that the higher-order OAM beam gets a larger distortion effect might be
explained by: (i) for OAM beams with the same beam waist, the beam diameter increases with the
OAM order. Therefore, the higher-order OAM beam has a higher value of D/r0 at the transmitter
side such that it gets a larger distortion; and (ii) moreover, the higher-order OAM beam has a larger
diffraction effect so that the beam diameter increases faster as the propagation distance increases.
Therefore, the value of D/r0 increases faster and the distortion effect gets stronger for higher-order
OAM beams.
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Figure 5.17. Effects of varying the OAM order on the normalized power distribution on different OAM modes. (a)
The received signal power on the transmitted mode order, and (b) XT1 for 1-km link with C
n
2
= 1 × 10
-12
m
-2/3
, f =
0.5 THz, and w
0
= 0.1 m at the transmitter side without considering the atmospheric absorption.
We have shown the dependence of the XT performance for THz OAM links on diffraction-
related parameters with fixed turbulence conditions. To further study the effect of turbulence
conditions, we investigate the XT performance of OAM THz links with different 𝐶 𝑛 2
. Figure 5.18
shows the signal power and crosstalk performance of THz-OAM links with different distances and
carrier frequencies. Figure 5.18(a) shows the case of a 1-km link where an OAM +4 with w0 = 1
m is transmitted. For the case of f = 5 THz, the signal power on the transmitted OAM mode
decreases by ~ 40 dB and XT1 increases by ~ 40 dB, when 𝐶 𝑛 2
increases from 5 × 10
-18
to 5 × 10
-
11
m
-2/3
. This could be because the value of r0 decreases when increasing 𝐶 𝑛 2
, which results in a
larger value of D/r0 such that the distortion effect becomes stronger. Moreover, the results in Figs.
5.18(b) show that for the cases when propagating an OAM +4 with w0 = 1 m through: (i) a 1-km
94
link at f = 5 THz, (ii) a 1-km link at f = 0.5 THz, XT1 can keep < -15 dB when 𝐶 𝑛 2
< 5 × 10
-14
m
-
2/3
, and 5 × 10
-15
m
-2/3
, respectively.
Figure 5.18. Effects of varying C
n
2
on the system performance. Effects of varying C
n
2
on (b) the signal power, (c) XT1:
crosstalk to the neighboring mode. OAM +4 is transmitted for a 1-km link with w
0
= 1 m and f = 0.5 THz. The legends
in (a) and (b) are the same.
We have investigated the XT performance of various THz OAM links when only one OAM
beam is transmitted. Transmitting multiple OAM beams might further degrade the system
performance with turbulence due to the power coupling to other OAM modes. Here, we analyze
the SIR performance of THz links, which multiplex various sets of multiple OAM beams. The SIR
is defined as the ratio of the signal power to the crosstalk from the other channels, where the signal
power is the received power on transmitted mode, and the noise is the total power leakage from
the other channels to the transmitted mode. Figure 5.19 shows the cases when (i) multiplexing 3
OAM beams with mode spacing from 1 to 3 (OAM beams [+3,+4,+5]; [+2,+4,+6]; and [+1,+4,+7]);
(ii) multiplexing 3 to 7 OAM modes with mode spacing of 1 (OAM beams [+3,+4,+5];
95
[+2,+3,+4,+5,+6]; and [+1,+2,+3,+4,+5,+6,+7]). For the cases of 3-muxed OAM beams, the SIR
decreases by ~ 5-7 dB if the mode spacing increase by 1. This might be because that channel
crosstalk between adjacent OAM modes is higher than that with mode spacing of 2 and 3.
Moreover, for the cases of 3-to-7-muxed OAM beams with mode spacing of 1, the SIR also
decreases with the mode number. For example, SIR decreases by ~ 0.5-1 dB if the mode number
increases by 1. This small value change might be explained by that the channel crosstalk from
OAM modes with mode spacing of >1 is small (such as < -15 dB) when the turbulence distortion
effect is not strong.
Figure 5.19. Signal-to-Interference ratio performance for multiple multiplexed OAM beams. Systems with (a) 3-
muxed OAM modes with mode spacing from 1 to 3 (OAM beams of [+3,+4,+5]; [+2,+4,+6]; and [+1,+4,+7] are
transmitted, respectively), and (b) 3-to-7-muxed OAM modes with a mode spacing of 1 (OAM beams of [+3,+4,+5];
[+2,+3,+4,+5,+6]; and [+1,+2,+3,+4,+5,+6,+7] are transmitted, respectively). Parameters are set as C
n
2
= 1 × 10
-11
m
-
2/3
, w
0
= 0.1 m and f = 0.5 THz.
96
5.3.4 XT performance dependence with limited aperture size and misalignment
We explore in the degrading effects due to two main limitations, namely, the beam diffraction and
atmospheric turbulence, for THz OAM links without considering the receiver aperture parameters
or alignment. However, the receiver (Rx) aperture might not (a) recover all the power of the
received beam due to the limited aperture size, or (b) induce crosstalk (XT) to other OAM modes
due to the displacement between the transmitter (Tx) and receiver apertures. One laudable goal
could be to explore the effects of the receiver aperture size and alignment on the system
performance for THz OAM links under atmosphere turbulence.
We first explore the effect of limited receiver aperture size on the crosstalk performance. Figures
5.20(a) to 5.20(c) show that the normalized power distributions on different OAM modes depend
on the receiver aperture size. We consider the cases for 100-m links with 𝐶 𝑛 2
= 0 (no turbulence),
1 × 10
-13
, and 1 × 10
-11
m
-2/3
. Other parameters are set as w0 = 1 m, and f = 5 THz, and the OAM
+4 is transmitted. In Fig. 5.20(a), as the aperture size decreases, the signal power decreases under
no turbulence and the crosstalk to the other OAM modes are all < -100 dB in simulation. Moreover,
as shown in Figs. 5.20(b) and 5.20(c), the aperture size changes the crosstalk performance of those
links under turbulence, which shows dependence on the mode order. Specifically, the XT to the
left neighbouring (lower) OAM modes decreases with the RX size under turbulence, however the
XT to the right neighbouring (higher) OAM modes increases with the RX size under turbulence.
As an example, the XT from OAM +3 increases by ~ 4 dB while the XT from OAM +5 decreases
by ~ 6 dB, when the Rx size decrease from 2 m to 1 m, as shown in Fig. 5.20(c). This might be
because that lower OAM modes diffract slower such that these modes have smaller beam sizes at
the receiver side. As a result, a limited Rx could receive more power on lower OAM modes. By
comparing the cases in Figs. 5.20(b) and 5.20(c), we observe that the signal power under 𝐶 𝑛 2
= 1 ×
97
10
-11
m
-2/3
is smaller than that under 𝐶 𝑛 2
= 1 × 10
-13
m
-2/3
, which might be
due to a larger value of
D/r0 at the transmitter side.
Figure 5.20. Effects of varying the displacement on the normalized power distribution on different OAM modes. 100-
m links with (a) C
n
2
= 0 m
-2/3
, (b) C
n
2
= 1 × 10
-13
m
-2/3
, and (c) C
n
2
= 1 × 10
-11
m
-2/3
. Other parameters are set as w
0
= 1
m, f = 5 THz, and OAM +4 is transmitted.
98
Figure 5.21. Effects of varying the displacement on the normalized power distribution on different OAM modes. 100-
m links with (a) C
n
2
= 0 m
-2/3
, (b) C
n
2
= 1 × 10
-13
m
-2/3
, and (c) C
n
2
= 1 × 10
-11
m
-2/3
. Other parameters are set as w
0
= 1
m, f = 5 THz, and OAM +4 is transmitted.
Besides the limited aperture size, the displacement might also occur at the receiver side. This
could be considered as an additional lateral shift to the phasefront of the received beam, which
might result in channel crosstalk to the system. We then explore the effect of varying the
displacement between Tx and Rx on the crosstalk performance. Figures 5.21(a) to (c) show that
the normalized power distribution on different OAM modes for 100-m links with 𝐶 𝑛 2
= 0, 1 × 10
-
99
13
, and 1 × 10
-11
m
-2/3
, respectively. OAM =4 with w0 = 1 m, and f = 5 THz is transmitted. We
assume the Rx aperture size is larger than the received beam. For the cases with no turbulence and
weak turbulence (𝐶 𝑛 2
= 1 × 10
-13
m
-2/3
), a larger displacement leads to lower signal power and
higher XT. However, for the cases with strong turbulence (𝐶 𝑛 2
= 1 × 10
-11
m
-2/3
), the signal power
and XT fluctuate in a small range of < 5 dB when the displacement increases from 0 to 0.5 m. One
of the reasons for this might be that the power of the received beam is almost averagely leaked to
many neighbouring modes under strong turbulence. Even if there is displacement between Tx and
Rx, the power leakage to the other modes does not change too much.
5.4 Summary and conclusions
In this chapter, we first evaluate patch antenna arrays with different arrangements as OAM
generators and receivers at 60 GHz by simulation. We compare beam evolution and steering
performance of circular- and ring-antenna-array based OAM links. Simulation results show that
circular antenna arrays generate OAM beams with smaller side-lobes, and higher mode purity. We
also find a circular antenna array has a better performance for OAM beam steering than a ring
antenna array. OAM spectrum analysis showed both the lattice shape and the boundary shape of
an antenna array could cause power leakage to harmonic OAM orders from the designed OAM
order, which leads to a mode purity degradation and higher inter-mode crosstalk.
We then study in simulation the comprehensive diffraction and atmospheric turbulence effects
on THz wireless communication links when using OAM beams. Simulation results show that: (i)
the signal power increases and the crosstalk decreases with frequency and beam waist, when only
related-diffraction effect is considered; and (ii) the signal power tends to decrease and the crosstalk
tends to increase with frequency, beam waist, and OAM order, when both diffraction and
turbulence effects are considered.
100
6. Chapter 6 Conclusion
The thesis has made a summary and conclusion in each chapter. As an overall summary, we
investigate approaches to manipulate the OAM state in a dynamic manner as well as with
multiplication/division operators. We also modify and apply optical technologies for OAM beam
generation in the mm-wave domain, which potentially provides a more compact way to generate,
multiplex, demultiplex, and detect mm-wave OAM beams. Moreover, we provide the system
performance of antenna-array-based mm-wave OAM links and system-degrading effects of THz
OAM links with turbulence. There is no doubt that that OAM multiplexing can multiply system
capacity and spectral efficiency, potentially helping address the capacity demands in many
applications. However, it is hard to predict when commercial OAM multiplexing systems will be
available. Moreover, it would not be surprising that the advances in light beams with new OAM
forms find application communication systems.
101
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Abstract (if available)
Abstract
It was discovered in 1992 that a wave with a helical phasefront possesses orbital angular momentum (OAM). Any wave (or photon) with azimuthal phase dependence exp(ilθ) carries OAM of lℏ per photon (l is the OAM mode order which is an unbounded integer, and ℏ is Plank’s constant), and its wavefront twists along the propagation axis. This produces a central intensity null (i.e., phase singularity) and annular ring. OAM beams with different l values are mutually orthogonal, such that multiple OAM beams can be efficiently (de)multiplexed and co-propagate with little inherent crosstalk. OAM is also compatible with other degrees of freedom, such as polarization and wavelength multiplexing. ❧ There have been several techniques for OAM beam generation, including spiral phase plates, diffractive phase holograms, grating-coupler-based integrated waveguides, metamaterials, and metasurface. The basic idea of this thesis comes from two main points: (i) only a fixed OAM value could be added to a “static” beam by utilizing most of these techniques
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Asset Metadata
Creator
Zhao, Zhe
(author)
Core Title
Optical and mm-wave orbital-angular-momentum beam generation and its applications in communications
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
04/11/2021
Defense Date
07/22/2020
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
metasurface,mm-wave,OAI-PMH Harvest,optical communications,orbital angular momentum
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Willner, Alan (
committee chair
), Haas, Stephan (
committee member
), Wu, Wei (
committee member
)
Creator Email
jijyou@gmail.com,zhezhao@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-383827
Unique identifier
UC11666688
Identifier
etd-ZhaoZhe-9038.pdf (filename),usctheses-c89-383827 (legacy record id)
Legacy Identifier
etd-ZhaoZhe-9038.pdf
Dmrecord
383827
Document Type
Dissertation
Rights
Zhao, Zhe
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
metasurface
mm-wave
optical communications
orbital angular momentum