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University of Southern California Dissertations and Theses
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optical communication systems and space-time wave packet generation using frequency combs and spatial modes
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optical communication systems and space-time wave packet generation using frequency combs and spatial modes
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Content
Optical communication systems and space-time wave packet generation using frequency
combs and spatial modes
by
Kaiheng Zou
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
August 2023
Copyright 2023 Kaiheng Zou
This dissertation is dedicated to my parents Yukang Zou and Yunxia Rui.
ii
Acknowledgments
I would like to thank my advisor Prof. Alan E. Willner and my colleagues at OCLab. I could not finish this
PhD dissertation without their support.
First, I would like to deeply thank Prof. Willner for his encouragement, support, and advice during my
PhD study. I feel so lucky to have the chance to work with Prof. Willner and this well-known lab in the
area of optical communications. Besides, I also learned a lot of personal lessons from him, which I believe
will be very helpful in my future life.
Moreover, I would like to thank my colleagues in OCLab, including Dr. Changjing Bao, Dr. Ahmed
Almaiman, Dr. Guodong Xie, Dr. Long Li, Dr. Yinwen Cao, Dr. Peicheng Liao, Dr. Zhe Zhao, Dr. Cong
Liu, Dr. Ahmad Fallhpour, Dr. Fatemeh Alishahi, Dr. Kai Pang, Dr. Runzhou Zhang, Dr. Haoqian Song,
Dr. Hao Song, Huibin Zhou, Xinzhou Su, Amir Minoofar, Narek Karapetyan, Yuxiang Duan, Zile Jiang,
Murale Ramakrishnan, Stanley Ko, Abdulrahman Alhaddad, Yingning Wang, and Ruoyu Zeng.
I would like to thank Prof. Stephan Haas and Prof. Wei Wu for serving on my defense committee. I
would also like to thank Prof. Keith Jenkins and Prof. Mercedeh Khajavikhan for serving on my qualifying
exam committee.
My work in this dissertation was achieved in collaboration with many other institutes. I would like to
thank Prof. Moshe Tur from Tel Aviv University, Dr. Maxim Karpov and Prof. Tobias J. Kippenberg from
´Ecole Polytechnique F´ ed´ erale de Lausanne, Dr. Andrew McClung, Mahsa Torfeh, and Prof. Amir Arbabi
from University of Massachusetts Amherst, and Dr. Murat Yessenov and Prof. Ayman F. Abouraddy from
University of Central Florida.
Last but not least, I would like to thank my girlfriend Zhouting Zhu and my parents, Yukang Zou and
Yunxia Rui, for their support in my life during my PhD study.
iii
Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Optical signal processing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Free space optical communication systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Optical space-time wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Spatial modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Optical frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Digital modulation formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 2: Shared phase recovery for comb-based wavelength division multiplexing optical
communication systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Concept and experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Four 20-Gbaud 16 quadrature-amplitude modulated channels after 50-km fiber transmission . 15
Chapter 3: Optical Volterra filter using nonlinear wave mixing and delays . . . . . . . . . 20
3.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Concept and experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Tunable second-order optical Volterra filter to equalize 10-/20-Gbaud 4-APSK signals . . . . 24
Chapter 4: Free-space optical communication system in the mid infrared (mid-IR) using
multiplexing techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Concept and experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Wavelength division multiplexing (WDM) in the mid-IR . . . . . . . . . . . . . . . . . . . . . 35
4.4 Mode division multiplexing (MDM) and a combination of WDM and MDM in the mid-IR . . 39
Chapter5: Opticalspace-timewavepackets(STWP)generatedusingfrequencycombsand
spatial modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Concept and experimental setup for STWP generation . . . . . . . . . . . . . . . . . . . . . . 45
5.3 STWP carrying orbital angular momentum (OAM) with time-dependent radius . . . . . . . . 49
5.4 STWP carrying dynamically varying OAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Simulation of STWPs with tunable group velocity . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 STWP generation and propagation through multi-mode fiber . . . . . . . . . . . . . . . . . . 61
iv
Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
v
List of Figures
1.1 Potential advantages of optical signal processing systems. . . . . . . . . . . . . . . . . . . . . 1
1.2 Multiplexing in free-space optical communication systems. . . . . . . . . . . . . . . . . . . . . 2
1.3 Optical space-time wave packets generated by correlating the temporal and spatial domains
of the optical wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Beams carrying orbital angular momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 (a) Intensity and phase profiles of Lagurre-Gaussian (LG) modes. (b) Spatial modes decom-
posited as combinations of LG modes. (c) Complex LG spectrum of a beam. . . . . . . . . . 5
1.6 Intensity and phase profiles of Bessel modes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.7 Optical frequency combs and the use of combs in different topics. . . . . . . . . . . . . . . . . 7
1.8 The data signal can be modulated on the amplitude and/or phase of an optical wave [e.g.,
on-off keying (OOK), quadrature phase shift keying (QPSK), and 16-quadrature amplitude
modulation (16-QAM)]. (b) The constellation and the error vector of a QPSK signal. . . . . . 8
1.9 Topics that will be discussed in this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 The concept of using Kerr frequency combs to perform shared carrier phase recovery for KK
detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The experimental setup of the four-channel 20-Gbaud 16-QAM transmission using Kerr fre-
quency combs and KK detection to perform shared carrier phase recovery. . . . . . . . . . . . 14
2.3 Optical spectra (resolution: 0.1 nm) of the generated Kerr combs, (a) I and (b) II. High-
resolution optical spectra (resolution: 100 MHz) of (c) four pairs of comb lines for each
channel, (d) modulated signals, and (e) modulated signals combined with the CW tones of
four channels. High-resolution spectra are measured by a complex optical spectrum analyzer. 15
2.4 Received constellations of the 20-Gbaud 16-QAM channel 4 signal in the BTB scenario (a)
withoutphaserecovery,(b)withindependentphaserecovery,(c)andwiththeestimatedphase
from channel 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 BER of channel 4 as a function of OSNR of the 20-Gbaud 16-QAM signal with independent
phase recovery and shared phase recovery from another channel using (a) coherent comb lines
and (b) two independent lasers as LOs in the BTB scenario. The inset provides a blow-up of
the OSNR detail around the FEC threshold.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vi
2.6 Examples of the individually estimated phase noise of the two channels using (a) coherent
comb lines and (b) two independent lasers as LOs in the BTB scenario. . . . . . . . . . . . . 17
2.7 The individually estimated phase noise after a 50-km transmission for (a) channels 1 and 2
and (b) channels 1 and 4. The inset in (a) shows the standard deviation of the phase noise
difference of channels with different spacings. The insets show the zoomed-in curves. . . . . . 18
2.8 BERs of the four channels after a 50-km transmission at the total launch power of 6 dBm into
the fiber.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 (a) Block diagram of a generic second-order Volterra filter. (b) Concept of an optical second-
order Volterra filter with wave mixing and delays. The delay and tap weights are applied by
an LCoS filter. The multiplexed output is generated through a cascaded sum- and difference-
frequency generation process in a PPLN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 (a)Experimentalsetupoftheopticalsecond-orderVolterrafilterwithwavemixinganddelays.
MLL:mode-lockedlaser. EDFA:erbium-dopedfiberamplifier. LCoS:liquidcrystalonsilicon.
PC: polarization controller. AWG: arbitrary waveform generator. Mod.: modulator. BPF:
band-pass filter. PPLN: periodically poled lithium niobate. Co. Rx: coherent receiver. (b)
Spectrum of the MLL comb source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Received electrical spectrum of the filter output when the input signal is a 10-GHz sinusoidal
wave. (a) The filter has two first-order taps. (b) The filter has two first-order taps and two
second-order taps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Optical spectra at the output of the PPLN waveguide under different tap configurations of
(a) three first-order taps and (b) three second-order taps. . . . . . . . . . . . . . . . . . . . . 25
3.5 Measured first and second-order transfer functions (TFs) with different optical filter configu-
rations. The tap weights are shown above the figures. (a-b) Two cases with first-order taps.
(c-d) Two cases with second-order taps. (e) Combining cases 1 and 3 with both first- and
second-order taps. (f) Combining cases 2 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Equalization of a nonlinearly distorted 20-Gbaud 4-APSK signal with the optical nonlinear
filter. (a) The nonlinear pre-distortion applied to the signal; Eye diagram of (b) the signal
before pre-distortion, (c) the nonlinearly pre-distorted signal, (d) the equalized signal with
three first-order taps, and (e) the equalized signal with three first-order and one second-order
tap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Equalizationofa10-Gbaud4-APSKsignal. Eyediagramof(a)thesignalbeforepre-distortion,
(b) the nonlinearly pre-distorted signal, (c) the equalized signal with two first-order taps, and
(d) the equalized signal with two first-order and one second-order tap. . . . . . . . . . . . . . 28
4.1 Concept for the mid-infrared (IR) wavelength-division-multiplexing (WDM) and orbital an-
gular momentum (OAM)-based mode-division-multiplexing (MDM) free-space optical (FSO)
communication system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Experimental setup of the free-space mid-infrared WDM and MDM communication system.
PC: polarization controller; Col.: collimator; EDFA: erbium-doped fiber amplifier; YDFA:
ytterbium-doped fiber amplifier; PPLN: periodically poled lithium niobate; M: mirror; SPP:
spiral phase plate; HPF: high-pass filter; BPF: tunable band-pass filter; LO: local oscilla-
tor. VOA: variable optical attenuator; OSA: optical spectrum analyzer; DSO: digital storage
oscilloscope; C: fiber-based optical coupler; BS: free-space beam splitter. . . . . . . . . . . . . 34
vii
4.3 (a) Spectrum of the generated mid-IR WDM signals with a resolution of ∼ 1 nm. Arrows
indicatethethreemid-IRWDMchannels. (b)Generatedmid-IRbeampowerasafunctionof
the 1064-nm pump power with different signal power values. The C-band signal wavelength
is set at 1550 nm. (c) Generated mid-IR beam power as a function of PPLN temperature
with different C-band signal wavelengths. (d) Generated mid-IR beam power as a function of
C-band signal wavelength with a PPLN temperature of 49.5
◦ C. . . . . . . . . . . . . . . . . 36
4.4 (a) Spectrum of the WDM signals that are converted back to the C-band. (b) Normalized
optical crosstalk matrix of WDM. (c) Measured bit error rate (BER) as a function of the
received optical signal-to-noise ratio (OSNR) for a C-band generation/detection (gen./det.)
and mid-IR Gaussian beam transmission. In the C-band generation/detection case, C-band
signals are detected by the coherent receiver without wavelength conversion and free-space
propagation. (d) Measured BER as a function of the received OSNR for the three mid-IR
WDM channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 (a) Measured beam profile of the mid-IR OAM beams. Intensity profile and interferogram
with a Gaussian beam of the OAM +1 and OAM +3 beam, respectively. Intensity profile of
the data-carrying multiplexed OAM +1 and +3 beams. Intensity profile of the multiplexed
OAM beam after passing through the second SPP with OAM order − 3. (b) Normalized
crosstalk matrix of MDM. (c) Measured BER of the OAM +3 channel as a function of the
received OSNR for the mid-IR OAM beam transmission when sending both OAM modes and
sending a single OAM mode. (d) Measured BER and OSNR of all the channels, including two
OAM modes with three wavelengths on each mode. . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 Several types of STWPs that will be discussed in this chapter. . . . . . . . . . . . . . . . . . 46
5.2 Concept of generating the space-time wave packets (STWPs) carrying (a-b) OAM with a
time-dependent beam radius and (c-e) dynamically changing OAM values. (b) An example of
the interference leading to the dynamic motion. (c) A single frequency carrying a single OAM
mode. (d)MultiplefrequenciescarryingasingleOAMmode. (e)Multiplefrequenciescarrying
multiple OAMs with different weights on each frequency form an STWP with dynamically
changing OAM values. The constructive combining for generating different OAM modes
appears at different time instants depending on the time-variant relative phase difference
between the comb lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Experimental setup for generating and measuring the dynamic STWP. Col.: collimator; BS:
beam splitter; SLM: spatial light modulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 The optical spectrum at the (a) input and (b) output of the waveshaper. . . . . . . . . . . . . 50
5.5 (a1-c1) Intensity and phase profiles, (a2-c2) the estimated beam radius along with the sim-
ulated radius, and (a3-c3) the OAM purity of the generated OAM-carrying STWPs. In (a)
ℓ = +1 and at 0z
R
, (b) ℓ = +1 and at 0.5z
R
, and (c) ℓ = +3 and at 0z
R
, respectively. All
results are measured in a time duration of 5 ps. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.6 Experimental results of an STWP carrying increasing OAM values from +1 to +6. (a) The
intensity and phase profiles and their corresponding OAM spectra. (b) The OAM purities as
a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.7 ExperimentalresultsofanSTWPcarrying(a)decreasingOAMvaluesfrom+4to+1and(b)
OAMvaluesdecreasingfrom+4to+1andincreasingto+4afterward. (a1, b1)Theintensity
and phase profiles and the corresponding OAM spectra. (a2, b2) The OAM spectrum as a
function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
viii
5.8 The frequency chirp along the azimuthal direction of the experimentally generated STWPs.
(a)ThecasewithamonotonicallydecreasingOAMvalue. (b)Thecasewitha(b1)decreasing
and (b2) increasing OAM value during the period. . . . . . . . . . . . . . . . . . . . . . . . . 54
5.9 SimulationresultsoftheSTWPscarryingdecreasingOAMvaluefrom+6to+1withdifferent
pulse widths in the temporal domain. The STWPs are synthesized with (a) 6 frequency lines,
(b) 12 frequency lines, and (c) 18 frequency lines. . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.10 Simulation results of the STWP carrying a roughly parabolically changing OAM value. (a)
The OAM spectrum as a function of time during a period. (b) Frequency chirps along the
azimuthal direction of the STWP at different time instants. . . . . . . . . . . . . . . . . . . . 55
5.11 (a) For a pulsed Gaussian-like beam without space-time correlation, the group velocities are
different for different frequencies. Due to this, the pulse shape becomes broader after propa-
gation. (b) For an STWP with spatial and temporal spectral correlation, the spatiotemporal
spectrum of the STWP follows the intersection of a light cone with a plane tilted at an angle
of Θ. The group velocity is the same for different frequencies. The pulse shape remains the
same after propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.12 (a)ConceptofgeneratingtheSTWPbycombiningdifferentfrequencies,eachcarryingmultiple
Laguerre-Gaussian modes. (b1-b3) Simulated spatiotemporal profiles of the generated STWP
with different group velocities. The group velocity is estimated by comparing the pulse peak
movement at the two distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.13 The simulated group velocity as a function of propagation distance without the spatial and
temporal spectral correlation, with the correlation between frequency and LG modes, and
with the correlation between frequency and plane waves. As an example, Θ is set to be 30
◦ and the corresponding target value of v
g
is∼ 0.58c. . . . . . . . . . . . . . . . . . . . . . . . . 59
5.14 (a) The simulated group velocity at the propagation distance z = 0 as a function of Θ. (b)
Therangeofpropagationdistancethattheestimatedgroupvelocityremainswithin0.1cfrom
the target value as a function of Θ. (c) The estimated group velocity at the propagation
distance z =0 a different maximum mode number ( |ℓ|+2p+1) of the LG modes used. . . . 60
5.15 Conceptof(a)STWPsgeneratedbysynthesizingmultiplefrequencycomblines,eachcarrying
auniqueBesselmodetohavereduceddiffractionandatunablegroupvelocity;(b)propagating
the STWP through multi-mode fiber (MMF) without mitigating of the mode coupling; and
(c) propagating the STWP through MMF with pre-distortion to mitigate the mode coupling. 62
5.16 Experimental setup for generating an STWP with reduced diffraction and tunable group
velocityaftermulti-modefiberpropagation. AFG:arbitraryfunctiongenerator;ECL:external
cavity laser; LF: lensed fiber; Col.: collimator; BS: beam splitter; SLM: spatial light modulator. 63
5.17 Measured complex transmission matrices of the MMF on one frequency (a) without and (b)
with pre-distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.18 (a) AmplitudesofthemeasuredtransmissionmatricesoftheMMFonthesixfrequencies. (b)
Calculated intensity profiles of the pre-distorted modes on the six frequencies to generate an
STWP with Θ=45 .1
◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
ix
5.19 Measured beam profiles |E(x,y =0,τ )|
2
in free space after MMF propagation for the STWPs
(a) with Θ = 45 .1
◦ without pre-distortion, (b) with Θ = 45 .1
◦ with pre-distortion, and (c)
Θ = 44 .9
◦ with pre-distortion. In each case, two beam profiles are shown with z = 0 (right
at the fiber output) and z =40 mm (with further 40-mm free-space propagation). The insets
are the transverse 2-D complex beam profiles with the corresponding reference delays. . . . . 67
5.20 Measured beam radius after further free-space propagation at the fiber output according to
the time-averaged intensity profiles for (a) an STWP with pre-distortion and a Θ value of
44.9
◦ and (b) a pulse with the fundamental Gaussian mode of the MMF. The insets show the
time-averaged intensity profiles at the corresponding z positions. . . . . . . . . . . . . . . . . 68
x
Abstract
Opticalfrequencycombs,asopticalsourceshavingmultiplediscreteequidistantfrequencylinesintheirspec-
trum, have gained much interest. Frequency combs have many valuable characteristics, including providing
tens or hundreds of narrow-linewidth, mutually coherent, and equidistant optical carriers. Such mutually
coherentcomblinescanbeusedinapplicationslikeopticalcommunicationsystems,opticalsignalprocessing
systems, and optical space-time wave packet (STWP) generation.
Ontheotherhand,opticalbeamscarryingspatialmodeshavebeenunderintenseinvestigationinvarious
areas. One example is the light wave can carry orbital angular momentum (OAM). In general, OAM can be
utilized to characterize the “twisted” helical phase front of a light beam when its wavevector spirals around
the beam axis. Such a helical phase front of an OAM-carrying beam is usually represented by exp(iℓθ ),
where θ is the azimuthal coordinate and ℓ is the number of 2π phase shifts in the phase profile of the beam.
OAM beams with different ℓ values are orthogonal with each other while propagating coaxially. The spatial
modes can be used in both the optical communication systems and the STWP generation.
Thisthesiswilldiscuss(i)optical4× 20-Gbaud16-quadrature-amplitude-modulatedwavelength-division-
multiplexing communication systems using Kerr frequency combs and Kramers-Kronig detection to perform
shared phase recovery, (ii) an optical second-order Volterra filter using wave mixing and delays to equalize
10-20-Gbaud 4-amplitude-phase-shift-keying signals, (iii) 300-Gbit/s free-space optical communications in
themid-infraredusing wavelength- andmode-divisionmultiplexing, and(iv)several typesofopticalSTWPs
generated using frequency comb lines and spatial modes.
xi
Chapter 1
Introduction
1.1 Optical signal processing systems
Optical signal processing (OSP), in which signals are manipulated in the optical domain, has been envi-
sioned as a potential enabling technology to meet the demand of the exponential growth in data generation,
processing, and communications [1, 2]. As shown in Fig. 1.1, several potential advantages of OSP include:
(i) avoiding inefficient optical-to-electrical-to-optical conversion, especially when data is already in the op-
tical domain, (ii) exploiting multiple dimensions of the optical wave (e.g., amplitude, phase, wavelength,
polarization, and space), and (iii) allowing high-speed processing by operating at the line rate [1, 2].
Figure 1.1: Potential advantages of optical signal processing systems.
1.2 Free space optical communication systems
Free-spaceoptical(FSO)communicationshaveattractedmuchinterestduetothepotentiallyhigherdataca-
pacity and lower probability of detection by eavesdroppers when compared to radio-based systems [3, 4]. As
1
shown in Fig. 1.2, to increase the total data capacity of optical communication systems, multiplexing tech-
niques have been utilized in FSO communications. Multiple independent data channels can be multiplexed
using different domains of the optical waves and transmitted simultaneously through the medium. Exam-
ples of the multiplexing techniques include wavelength division multiplexing (WDM), where each channel
occupies a different wavelength, and space division multiplexing (SDM).
Figure 1.2: Multiplexing in free-space optical communication systems.
1.3 Optical space-time wave packets
Optical space-time wave packets (STWPs) are pulsed optical waves that have a correlation between their
spatial and temporal degrees of freedom [5]. Theoretical investigations and experimental demonstrations
of the generation and propagation of STWPs that contain novel properties have been reported, including
dynamicspatialprofiles,near-diffraction-freepropagation,andtunablegroupvelocities[5,6,7,8]. Asshown
in Fig. 1.3, the STWPs can be generated by correlating the temporal domain (e.g. frequency comb lines,
thedetailsofwhichwillbeintroducedinSection1.5)andthespatialdomain(e.g. spatialmodes, the details
of which will be introduced in Section 1.4).
1.4 Spatial modes
Spatial modes have been under intense investigation in various areas [9, 10, 11]. As one example, a light
wave can carry orbital angular momentum (OAM) [11]. In general, OAM can be utilized to characterize the
“twisted” helical phase front of a light beam when its wavevector spirals around the beam axis. Moreover,
2
Figure 1.3: Optical space-time wave packets generated by correlating the temporal and spatial domains of
the optical wave.
such a helical phase front of an OAM-carrying beam is usually represented by exp(iℓθ ), where θ is the
azimuthal coordinate and ℓ is the number of 2π phase shifts in the phase profile of the beam [12]. In
addition, the sign of ℓ (positive or negative) corresponds to the direction of the phase helices (clockwise or
counterclockwise) in the phase profile. Due to the phase singularity in such a helical phase profile, an OAM
beam with a non-zero ℓ (i.e., OAM order) usually has a donut-shaped intensity profile, as shown in Fig. 1.4.
Figure 1.4: Beams carrying orbital angular momentum.
3
OAMcanbetheoreticallyquantifiedasaninfinitenumberofstates, whichmeans ℓcanbeanyinteger. It
should be noted that OAM beams with different ℓ values are orthogonal with each other while propagating
coaxially. For example, considering two OAM beams with ℓ
1
and ℓ
2
, respectively [13],
u
1
(r,θ,z )=A
1
(r,z)exp(iℓ
1
θ ), (1.1)
u
2
(r,θ,z )=A
2
(r,z)exp(iℓ
2
θ ), (1.2)
where (r,θ,z ) is the cylindrical coordinate and refers to the radial, azimuthal position, and propagation
position. Such an orthogonality between different OAM beams can be represented by [13]
Z
2π 0
u
1
(r,θ,z )u
∗ 2
(r,θ,z )dθ =
0 if ℓ
1
̸=ℓ
2
A
1
A
∗ 2
if ℓ
1
=ℓ
2
(1.3)
The orthogonality between different OAM beams is significant when they are utilized in communications,
where they can carry independent data beams, be multiplexed, co-propagate through the medium, and
de-multiplexed with little inherent crosstalk [13].
In general, an OAM beam could refer to any helically phased light beam, irrespective of its radial
distribution. However, a complete two-dimensional (2D) modal basis can generally be characterized by
two modal indices. For example, LG modes have ℓ and p indices, corresponding to azimuthal and radial
distribution, respectively [14]. The electric field of LG modes can be represented by [11]
LG(r,θ,z ;ℓ,p)=
s
2p!
π (p+|ℓ|)!
1
w(z)
r
√
2
w(z)
!
|ℓ|
exp
− r
2
w
2
(z)
L
|ℓ|
p
2r
2
w
2
(z)
exp(iℓθ )exp
ik
0
r
2
z
2(z
2
+z
2
R
)
exp
− i(2p+|ℓ|+1)tan
− 1
z
z
R
,
(1.4)
where the 1/e radius of the Gaussian term is given by w(z) = w(0)[(z
2
+z
2
R
)/z
2
R
]
1/2
[w(0) is the beam
waist], z
R
is the Rayleigh range, and (2p+|ℓ|+1)tan
− 1
(z/z
R
) is the Gouy phase. L
|
ℓ|
p
are the generalized
Laguerre polynomials and (r,θ,z ) is the cylindrical coordinate [13]. The intensity and phase profiles of some
4
LGmodesareshowninFig. 1.5(a). FortheLGbeamwithanon-zero ℓvalue,p+1representsthenumberof
rings in the intensity profile, while ℓ represents the number of 2π phase shifts along the azimuthal direction
inthephaseprofile. Theoretically, LGbeamswithdifferent ℓand/orpvaluesareorthogonalwitheachother
for a given beam waist and propagation distance, which can be represented by [14]
Z
∞
0
Z
2π 0
LG
1
(r,θ,z ;ℓ
1
,p
1
)LG
∗ 2
(r,θ,z ;ℓ
2
,p
2
)rdθdr =
0 if ℓ
1
̸=ℓ
2
or p
1
≠=p
2
1 if ℓ
1
=ℓ
2
and p
1
=p
2
(1.5)
Figure1.5: (a)IntensityandphaseprofilesofLagurre-Gaussian(LG)modes. (b)Spatialmodesdecomposited
as combinations of LG modes. (c) Complex LG spectrum of a beam.
SinceLGmodesareacomplete2Dmodalbasisset,abeamwithagivenspatialdistributioncangenerally
be decomposed into LG modes with different ℓ and p values, as shown in Fig. 1.5(b). In general, the
coefficients of LG components can be represented by a complex modal spectrum, as shown in Fig. 1.5(b).
Moreover, it should be noted that each complex coefficient contains both amplitude and phase information,
which can be tuned such that a specifically structured beam would be generated by a coherent superposition
of these LG components for a desired function [15].
Moreover, therehasbeenmuchinterestinothernoveltypesofspatialmodes. Forexample, Besselbeams
are solutions to the Helmholtz equation in the cylindrical coordinates [16, 17]. They have several unique
proprieties, of which two are interesting: (1) they are theoretically diffraction-free during the propagation
5
and (2) their beam shapes are “self-heal” after being disrupted by a partial obstruction. Theoretically, the
electric field for the n-th-order Bessel beam [in the cylindrical coordinates ( r,θ,z )] is as follows [18]
E(r,θ,z )=a
0
J
|ℓ|
(k
r
r)exp[i(k
z
z± ℓθ )], (1.6)
where J
|ℓ|
is the|ℓ|-th-order Bessel function of the first kind. k
z
and k
r
are the longitudinal and transverse
wave numbers, respectively. High-order (|ℓ| > 0) Bessel beams have helical phase fronts in the azimuthal
direction, which means that they carry OAM of order ℓ. The intensity and phase profiles of several Bessel
modes are shown in Fig. 1.6. Theoretically, the intensity and phase profiles of Bessel beams can be charac-
terized by an azimuthal index ℓ and a continuous radial wave vector k
r
[17].
Figure 1.6: Intensity and phase profiles of Bessel modes.
1.5 Optical frequency combs
Optical frequency combs are optical sources having multiple discrete equidistant frequency lines in their
spectrum 1.7. Frequency combs have many valuable characteristics, including providing tens or hundreds
of narrow-linewidth, mutually coherent, and equidistant optical carriers. There are many ways to generate
frequency combs. The Kerr frequency comb, which is based on integrated microresonators, has gained
much interest and is generated by parametric frequency conversion via coupling a CW laser pump into a
6
microresonator [19]. Depending on the pump power and wavelength, the formation of Kerr combs consists
of several dynamic regimes, including Turing patterns, chaos, low-phase noise combs, and soliton combs [20].
The mutually coherent comb lines provided by the frequency combs can be beneficial to different appli-
cations, as shown in Fig. 1.7. In optical communication systems, the comb lines can be used as the light
sources replacing multiple individual laser sources. Wavelength multiplexing optical communication systems
can be achieved using the frequency combs [21]. Specifically, in optical signal processing systems based on
nonlinear wave mixing, the mutual coherence between the comb lines is of importance [22]. If the pumps in
thenonlinearwavemixingaretakenfromafrequencycomb, thephasenoisefromtheselinescanbecanceled
out [23].
Figure 1.7: Optical frequency combs and the use of combs in different topics.
The frequency comb lines can also be used to generate the STWPs, as shown in Fig. 1.7. For a single-
frequency beam at a given propagation distance: (a) a single-mode beam has a static spatial distribution of
that mode [9], whereas (b) a beam composed of many weighted modes can be tailored to have nearly any
static amplitude and phase spatial distribution [15]. One approach to generate the STWPs is the synthesis
of multiple coherent frequency lines from an optical comb, with each line carrying a different combination
of weighted orthogonal spatial modes [6]. Depending on the elapsed time, the comb frequency spacing (∆ f)
will induce a time-dependent phase shift ∆ ϕ (t)=2π ∆ ft between neighboring lines. A phase shift at a given
7
propagation distance will be frequency dependent, such that the instantaneous constructive or destructive
interference between a spatial mode on one frequency with a spatial mode on a different frequency would
change with time.
1.6 Digital modulation formats
In optical communication systems, digital data signals can be modulated on the amplitude and phase of the
optical wave temporally, as shown in Fig. 1.8(a). In amplitude modulation, the bits in the data streams are
mapped to multiple amplitude levels of the optical wave. For example, an on-off keying (OOK) signal has
two possible amplitude levels representing the bit “0” and “1.” Since only a single photodiode is required at
the receiver to recover the amplitude information, such modulation is usually considered to have a low cost
and simple implementation[24].
0 1
Amplitude and Phase Modulation
Amplitude Modulation
Phase Modulation
00 01
10 11
1111 1110 1101 1100
OOK
0 1 1 0 1 0
01 10 00 11
QPSK
011
0
001
1
16-QAM
A
ref
A
error
In-phase (I)
Quadrature (Q)
φ
E = A cos (2 πft + φ)
(a)
(b)
Constellation and error vector
1011 1010 1001 1000
0111 0110 0101 0100
0011 0010 0001 0000
Figure 1.8: The data signal can be modulated on the amplitude and/or phase of an optical wave [e.g., on-off
keying(OOK),quadraturephaseshiftkeying(QPSK),and16-quadratureamplitudemodulation(16-QAM)].
(b) The constellation and the error vector of a QPSK signal.
8
In contrast, in phase modulation, the bits are mapped to the temporal phase, and the amplitude remains
constant. As an example, the quadrature phase shift keying (QPSK) has four possible phase levels, and
one QPSK symbol can carry two bits. In addition, amplitude and phase modulation can be simultaneously
utilized to increase the number of bits carried by one symbol. As an example, 16-quadrature-amplitude-
modulation (16-QAM) has 16 possibilities mapped on the in-phase and quadrature amplitude axes encoding
four bits of information. Unlike the amplitude-modulated signals, coherent detection is often used at the
receiver to extract the phase information when the temporal phase is also encoded with data [25]. In a
coherent receiver, the received signal is sent to a 90
◦ hybrid along with the continuous-wave (CW) local
oscillator (LO) laser and detected by balanced photodiodes [25].
The quality of the received signal can be characterized by the bit error rate (BER), which is defined as
the ratio between the number of bit errors and the total number of the received bits [26]. Alternatively,
error vector magnitude (EVM) can be used to determine the quality of the received signal [26]. As shown
in Fig. 1.8(b), the received symbols are represented by the points on the constellation according to their
amplitudes and temporal phases. Due to the noise and distortions during transmission, the received symbol
may deviate from the transmitted symbol, resulting in an error vector A
error
. The EVM is calculated by the
ratio between the root mean square of the error vectors A
error
and the reference vector A
ref
[26]:
EVM(%)=
q
1
N
P
N
i=0
|A
error,i
|
2
|A
ref
|
× 100% (1.7)
1.7 Thesis outline
This thesis will discuss several topics about optical communication systems and optical STWP generation
usingfrequencycombsandspatialmodes. Therestofthisthesisisorganizedasfollows. Chapter2willdiscuss
a WDM communication system using Kerr frequency combs to perform shared phase recovery. Chapter 3
will discussan opticalsecond-orderVolterrafilter using wave mixingand delays. Chapter 4will discuss FSO
communications in the mid-infrared (mid-IR) using wavelength- and mode-division multiplexing. Chapter 5
willdiscussseveraltypesofopticalSTWPsgeneratedusingfrequencycomblinesandspatialmodes. Chapter
6 will conclude the thesis.
9
Figure 1.9: Topics that will be discussed in this thesis.
10
Chapter 2
Shared phase recovery for comb-based wavelength division
multiplexing optical communication systems
2.1 Background and motivation
For coherent optical receivers, electronic digital signal processing (DSP) is commonly needed to perform
carrierphaserecovery[27]. Inwavelength-division-multiplexed(WDM)channels,thisphasenoiseestimation
tends to be performed for each data channel individually and independently.
Recent reports have shown that carrier phase recovery in conventional intradyne coherent systems can
be shared by deriving the phase estimation from one WDM channel and using it for other WDM channels
[28,29]. Thisisachievedbyemployingmutuallycoherentlinesoftwoopticalfrequencycombsasbothsignal
sources and local oscillators (LO) [28, 29]. This advance could help to reduce the power consumption for a
WDM system.
Furthermore, there has been interest in using Kramers-Kronig (KK) receivers in optical communication
systems [30, 31, 32, 33, 34]. Such receivers are considered less hardware-complex than conventional coherent
receivers or balanced heterodyne receivers, and yet can provide better performance than direct-detection
receivers [30]. Typically, KK detection is accomplished by adding a single tone adjacent to each WDM
channel in the transmitter or receiver so that a single detector can recover the phase of the signal based on
theKK-relationoftheminimumphasesignal[30,31]. Akeyquestioniswhethertheabove-mentioned“shared
phase estimation” approach used for conventional coherent receivers can also be applied to KK receivers and
11
provide a similar benefit [28, 29], especially given the difference that coherent detection recovers the phase
via a phase-diversity receiver and KK detection recovers the phase from the signal amplitude [30, 25].
In this Chapter, we experimentally demonstrate the KK detection of four 16-quadrature-amplitude-
modulated (16-QAM) 20-Gbaud channels after a 50-km transmission using soliton Kerr combs to perform
shared phase estimation among multiple channels [35]. Using on-chip integrated Kerr frequency combs as
the sources and LOs can reduce the transceiver footprint [36, 37]. The performance of shared phase noise
estimation is investigated for both back-to-back (BTB) and 50-km transmissions. In the former case, the
shared phase noise estimation exhibits an optical signal-to-noise ratio (OSNR) penalty of less than 0.5 dB
at the 7% forward error correction (FEC) threshold. In the latter case, the below-FEC bit error rate (BER)
is achieved for a total capacity of 320 Gbit/s using both independent and shared carrier phase recovery.
2.2 Concept and experimental setup
Figure 2.1 shows the concept of a WDM KK reception communication system that uses two Kerr combs as
light sources and local oscillators (LOs). In the transmitter (Tx), different comb lines are modulated with
differentdatasignalsandcombinedbyamultiplexer. Afterfibertransmission,thereceiveddatachannelsare
detectedbydifferentKKreceivers,inwhichthecorrespondinglinesofKerrcombIIserveasLOs. Ingeneral,
signal processing algorithms compensate for the phase noise arising from the use of different laser sources
[27]. Due to the mutual coherence between Kerr comb lines, the variation in the phase noise extracted from
different KK receivers would be similar. As such, the phase noise estimated in one channel can be directly
applied to other channels [28, 29].
Figure 2.2 illustrates the experimental setup of a four-channel 20-Gbaud 16-QAM transmission system.
Kerr combs are generated by controlling a pump laser, the output of which is amplified before being coupled
to a microresonator. Four comb lines ranging from 1555– 1560 nm are selected from the generated comb
I by a band-pass filter (BPF). A similar configuration is used to generate comb II, except that the pump
wavelength and cavity resonances have a frequency offset of around 10 GHz compared to the first comb.
At the Tx, the selected comb lines are loaded with a 20-Gbaud 16-QAM signal by an in-phase-quadrature
(IQ) modulator. We note that our experiment is different than the general concept of Fig. 2.1, for which
12
Figure 2.1: The concept of using Kerr frequency combs to perform shared carrier phase recovery for KK
detection.
different comb lines would generally be modulated with different data signals. In our experiment, however,
we use one modulator to modulate all the comb lines with the same data. Similar to what was described in
reference [29], we do not de-correlate these different WDM channels mainly due to the following three issues:
(a) The de-correlation would affect the phase noise coherence which is needed for shared phase recovery; (b)
Duetotherelativelylargecomblinespacing(∼ 192GHz)comparedtothesignalbandwidth(∼ 20GHz), the
adjacent channels should have little influence on each other. (c) We do not make use of any inter-channel
data correlation in the signal processing stage, which could unfairly benefit the shared phase recovery and
signaldemodulation[29]. WealsodetectthesignalsusingtwoindependentlasersasLOstoshowthatshared
phase recovery is not enabled by the data correlation. The two independent lasers have the linewidths of
∼ 100 kHz. The signals are transmitted over a 50-km single-mode fiber (SMF), and EDFAs are used for
pre-boosting and loss compensation. Note that in our demonstration, only a single polarization signal is
13
transmitted, and the polarization is aligned by manual adjustment. For a polarization-multiplexed signal,
two polarizations can be detected by two photodiodes, and no polarization tracking is needed [31].
Figure2.2: Theexperimentalsetupofthefour-channel20-Gbaud16-QAMtransmissionusingKerrfrequency
combs and KK detection to perform shared carrier phase recovery.
The received signals are processed offline as follows. First, the signal phase is recovered by the KK
relation. The frequency offset is estimated by a fourth power and fast Fourier transform (FFT) based
algorithm [38]. Subsequently, the signal is equalized by a finite impulse response (FIR) filter-based equalizer
whose taps are pre-converged according to a 1024-symbol training sequence [29]. Next, the phase noise
recovery is performed: (i) with each individual channel phase recovery [27], or (ii) by using the shared phase
estimation from other channels [28, 29]. The individual method [27] consists of: (a) a pilot-aided recovery
with 1 pilot symbol in every 127 data symbols to avoid cycle slip, and (b) the blind phase search (BPS)
recovery with 128-symbol averaging blocks and 64 test angles. To use the shared phase estimation of other
channels, the frequency offset difference is estimated first and removed for the remaining symbols and no
pilot is used.
14
2.3 Four 20-Gbaud 16 quadrature-amplitude modulated channels
after 50-km fiber transmission
The spectra of the generated soliton combs are shown in Fig. 2.3(a) and (b) at a resolution of 0.1 nm.
Comb I is in a single-soliton state, while comb II is in a multi-soliton state, which might be limited by the
pump power. The Kerr combs have a repetition rate of around 192 GHz and a frequency offset of ∼ 10 GHz.
Four comb lines (contained in the black boxes) are selected from each comb. The frequency offset between
the comb lines is illustrated by the high-resolution spectra in Fig. 2.3(c). The comb lines from comb I
are modulated with signals, as shown in Fig. 2.3(d). Figure 2.3(e) shows both the modulated signals and
corresponding LOs.
Figure2.3: Opticalspectra(resolution: 0.1nm)ofthegeneratedKerrcombs,(a)Iand(b)II.High-resolution
optical spectra (resolution: 100 MHz) of (c) four pairs of comb lines for each channel, (d) modulated signals,
and (e) modulated signals combined with the CW tones of four channels. High-resolution spectra are
measured by a complex optical spectrum analyzer.
Figure 2.4 shows the received signal constellations of channel 4 and the corresponding error vector mag-
nitudes (EVMs) in the BTB scenario. Figure 2.4(a) shows the received signal with no phase recovery. The
corresponding EVM is 18.12%, indicating low signal quality without any phase recovery. Figures 2.4(b) and
15
(c) show the received signals with independent phase recovery and use of the estimated phase from channel
1, respectively. The corresponding EVMs are 7.88% and 8.86%, respectively.
Figure 2.4: Received constellations of the 20-Gbaud 16-QAM channel 4 signal in the BTB scenario (a)
withoutphaserecovery, (b)withindependentphaserecovery, (c)andwiththeestimatedphasefromchannel
1.
Figure2.5(a)showsthechannel4BERasafunctionoftheOSNRintheBTBscenariowhenindependent
and shared phase recovery from channel 1 is performed. The OSNR penalty of the BER performance with
shared phase recovery is less than 0.5 dB at the FEC threshold compared to that with independent phase
recovery. This penalty could be the result of the remaining phase drift between the comb lines. It could
also be a result of the path difference between the two receivers. To further confirm that shared phase
recovery is the result of using two combs at both sites rather than the data correlation on adjacent channels,
a comparison using two independent lasers as LOs is performed, as shown in Fig. 2.5(b).
Figure 2.5: BER of channel 4 as a function of OSNR of the 20-Gbaud 16-QAM signal with independent
phase recovery and shared phase recovery from another channel using (a) coherent comb lines and (b) two
independent lasers as LOs in the BTB scenario. The inset provides a blow-up of the OSNR detail around
the FEC threshold.
16
Examples of the individually estimated phase noise of the two channels using coherent comb lines and
two independent lasers are shown in Figs. 2.6(a) and (b), respectively. The phase noise difference in Fig.
2.6(a) remains close to zero, thereby enabling the shared phase recovery of the two channels. In contrast,
the phase noise in Fig. 2.6(b) shows a large difference.
Figure 2.6: Examples of the individually estimated phase noise of the two channels using (a) coherent comb
lines and (b) two independent lasers as LOs in the BTB scenario.
The signals are then transmitted over a 50-km SMF. The pump lasers used for the Tx and Rx combs in
ourexperimenthavethelinewidthsof∼ 20kHzandtheresultingcomblinesalsohavesimilarlinewidths[39].
We note that the transmitted data signals experience chromatic dispersion (CD)-induced walk-off effects,
but the LOs added at the Rx do not experience walk-off. Consequently, the phase noise coherence between
different channels can be degraded after mixing the transmitted signals and LOs at the receiver. However,
thesystemcantolerateacertainamountofwalk-off. Forexample,CDofa50-kmSMFintroducesawalk-off
of∼ 3840 ps for a channel spacing of three comb spacings (3× 192 GHz), which corresponds to∼ 77 symbols
for a 20-Gbaud signal. The averaging block size in our phase recovery is 128 symbol, and it is optimized to
reduce the BER penalty from the phase noise in our experiment. As the optimized averaging block size is
larger than the walk-off length, the CD-induced walk-off by 50-km SMF has a relatively small impact on the
shared phase recovery. If the averaging block size was smaller than the walk-off length, such as when having
a laser with a broader linewidth or having a longer transmission distance, there could be a more significant
impact on the performance difference between shared and independent phase recovery [29].
17
Figure 2.7(a) shows the phase noise of channels 1 and 2 after a 50-km transmission by separate phase
noise recovery, while Fig. 2.7(b) shows the same information for channels 1 and 4. The phase difference
between channels 1 and 4 is larger than that between channels 1 and 2. The inset of Fig. 2.7(a) shows the
standard deviation of the individually estimated phase difference of two channels in the cases of different
channel spacing.
Figure 2.7: The individually estimated phase noise after a 50-km transmission for (a) channels 1 and 2 and
(b) channels 1 and 4. The inset in (a) shows the standard deviation of the phase noise difference of channels
with different spacings. The insets show the zoomed-in curves.
Figure2.8showstheBERresultofthefour-channel20-Gbaud16-QAMsignalsaftera50-kmtransmission.
The total launch power into the fiber of the four channels is 6 dBm. The BERs for channel 2, 3, and 4 with
independent phase recovery and shared phase recovery from channel 1 are shown. In order to fairly compare
theBERsofeachchannelwithtwophaserecoverymethods,wekeeptheOSNRthesameforthatchannel. In
Fig. 2.5(a)theperformancedifferencebetweenindependentandsharedphaserecoveryisshownforchannel4
as a function of OSNR in the BTB case. The performance difference for channel 4 in the 50-km transmission
case in Fig. 2.8 is similar to the difference for channel 4 in the BTB case in Fig. 2.5(a). The reason could be
that the CD-induced walk-off by the 50-km SMF has a relatively small impact on the shared phase recovery.
Although we use channel 1 to provide phase noise estimation, channel 2 or channel 3 could also be used and
might exhibit better performance due to potentially less walk-off relative to other channels [29].
18
Figure 2.8: BERs of the four channels after a 50-km transmission at the total launch power of 6 dBm into
the fiber.
In our experiment, we choose a transmission distance of 50 km to simply demonstrate the concept of
the shared phase recovery for KK receivers using Kerr frequency combs. However, longer distances might
also be possible. In general, the polarization-multiplexed KK receiver could achieve longer transmission
distances [34]. When the transmission distance becomes long, the CD could induce a walk-off between the
phase noise of different channels. Therefore, if no dispersion compensation is performed before detection,
the performance of the shared phase estimation would likely be degraded.
19
Chapter 3
Optical Volterra filter using nonlinear wave mixing and delays
3.1 Background and motivation
Atapped-delay-line(TDL)isafundamentalbuildingblockofelectronicsignalprocessing,anditcanperform
many different functions ( e.g., data equalization). Such a TDL can be achieved in the optical domain to
perform optical signal processing [40, 41] which has the potential advantages of: (i) avoiding inefficient
optical-to-electrical-to-opticalconversionforadatasignalthatisintheopticaldomain,and(ii)simultaneous
high-speed operation on multiple parameters of the optical wave (e.g., wavelength, amplitude, and phase)
[40].
Although signal processing generally relies on linear filtering to achieve various functions such as equal-
ization,correlation,andpulseshaping[40,41,42,43],nonlinearfilterscanenablebettersystemperformance
undermanycircumstances[44,45,46,47,48]. TheVolterrafunction, forexample, canbeusedtoimplement
a nonlinear filter, whose higher-order terms can improve the quality of an output data signal in the case of
degradation by nonlinear distortions [45, 46, 47, 48].
In the past, multiple optical approaches have been adopted to achieve linear filter signal processing
functions [40, 41, 42, 43]. However, there have been few reports of optical nonlinear filtering. For example,
onereportedopticalapproachusedafree-spacesystemtoachieveasecond-orderoperatorthroughautotriple
correlation. Some limitations of that approach include: (i) achieving only the nonlinear function without
the linear term and (ii) it did not show operation on an actual optical data channel [49].
20
In this Chapter, we experimentally demonstrate an optical second-order Volterra nonlinear filter with
wave mixing and delays that: (i) has both linear and nonlinear filtering taps and (ii) can equalize an
opticaldatachannel[50]. Wemeasurethefirst-andsecond-ordertransferfunctionsofthefilterbysendinga
frequency-sweptsinusoidalinput. Subsequently,wedemonstratetheuseofthisfiltertoequalizeanonlinearly
distorted 10-20 Gbaud 4-amplitude and phase shift keying (4-APSK) optical data channel. With three first-
order taps and one second-order tap, the bit error rate (BER) of the 20-Gbaud 4-APSK signal is reduced
from 8.2× 10
− 3
to 3.2× 10
− 3
.
3.2 Concept and experimental setup
Figure 3.1(a) shows a block diagram of a generic second-order Volterra filter [51]. We realize the first-order
filtering function by tapping each delayed copy of the input signal, multiplying it by a tap weight hi, and
addingthetapstogether. Thefunctionofthesecond-orderfilterisrealizedbytappingthedelayedversionsof
the signal, multiplying the two delayed versions with the tap weight h
ij
, and adding the taps together. The
Volterra filtering function is realized when the first-order filter is added to the second-order filter, providing
y(t) as an output [51]:
y(t)=
N
X
i=0
h
i
x(t− iT)+
N
X
j=0
j
X
i=0
h
ij
x(t− iT)x(t− jT) (3.1)
Figure 3.1(b) shows the concept for an optical second-order Volterra filter with nonlinear wave mixing
and delays. An optical frequency comb is used as the source providing equidistant frequency lines that can
efficiently mix around the quasi-phase-matching (QPM) frequency of the periodically poled lithium niobate
(PPLN) waveguide to generate the output. As shown in Fig. 3.1(b), some of the comb lines are sent into an
optical modulator, resulting in signal copies, which are individually delayed, by amounts of Ti=i×T, where
T is the single tap delay. We note that the input of an optical Volterra filter is expected to be an optical
signal. However, theinputinthisworkisabasebandsignalandthesignalcopiesaregeneratedbyanoptical
modulator. If the signal is already in the optical domain, then an optical multicasting stage can be added
to generate the signal copies with a PPLN waveguide [52] or four-wave mixing [53]. These delays, together
21
Figure 3.1: (a) Block diagram of a generic second-order Volterra filter. (b) Concept of an optical second-
order Volterra filter with wave mixing and delays. The delay and tap weights are applied by an LCoS filter.
The multiplexed output is generated through a cascaded sum- and difference-frequency generation process
in a PPLN.
with tap weights, are applied and can be tuned by a liquid-crystal-on-silicon (LCoS) waveshaper (WS). In
general, both the amplitude and phase can be tuned, resulting in complex tap weights. In our experiment,
only real-number tap weights are demonstrated. Other comb lines, which serve as coherent dummy pumps,
are combined with the delayed signal copies and sent to the PPLN waveguide to perform the nonlinear wave
mixing.
First, the three dummy pumps on the left, as shown in Fig. 3.1(b), interact with the three signal copies,
whicharesymmetricallylocatedaroundtheQPMfrequency f
QPM
. Thesegeneratetheirrespectiveproducts
at the frequency of 2f
QPM
, via sum-frequency generation (SFG). Subsequently, the products are converted
to the “Output” as shown in Fig. 3.1(b), via difference-frequency generation (DFG), by another continuous
wave (CW) pump. This “Output” is symmetrically located around the QPM frequency with respect to the
frequency of the CW pump. The first-order part of the filter occurs through this process. Simultaneously,
22
the signal copies, which are closest to QPM on its right, similarly interact with the signal copies with which
they are symmetrical about the QPM frequency. Thereby, they add the second-order contributions of the
filter to the “Output”.
The experimental setup is illustrated in Fig. 3.2(a). A mode-locked laser is used as the optical frequency
comb source. Several comb lines are selected from the comb source with LCoS 1, which has two output
ports. The comb lines of the first output port are used as the dummy pumps, and those of the other output
port are sent to the in-phase-quadrature (IQ) modulator to generate signal copies. In our experiment, we
modulate an APSK signal that has one constellation axis. It might be possible to modulate data on both
axes. The mapping of the 4-APSK signal is shown in the inset of Fig. 3.2(a) [54]. The spectrum of the
MLL comb is shown in Fig. 3.2(b). A delay line interferometer is used to increase the frequency spacing of
the MLL output from 10 GHz to 20 GHz, resulting in the sinusoidal wave-like envelope of some comb lines.
The optical signal-to-noise ratio (OSNR) of the generated signal copies is partially limited by the amplified
spontaneous emission (ASE) noise of the erbium-doped fiber amplifier (EDFA) that is used to increase the
power of the comb lines [53]. Both the dummy pumps and signals have a spacing of 100 GHz. Subsequently,
a second WS, LCoS 2, is used to adjust the delays and weights of the taps. A tap delay of 25 ps is chosen
to match the half-symbol duration of the 20-Gbaud signal to equalize the signal [45]. We note that the tap
delay can be tuned to accommodate other signal baudrates. The filter amplitude weight can be tuned by
the attenuation of the WS with a range of 0 to 35 dB. The delay can be tuned by the WS with a range of
0 ps to 50 ps. The delays are set with a frequency bin (i.e., bandwidth) of ∼ 80 GHz and the phase of the
signals is set to compensate for the phase shift caused by the delays. Moreover, the filter phase weight can
also be tuned by the WS, although we did not demonstrate the phase tuning in our experiment. A laser
source at a wavelength of 1560.42 nm is amplified as the CW pump for the DFG process. All these spectral
lines are combined and sent to the PPLN waveguide, the QPM wavelength of which is at ∼ 1550.5 nm, to
generate a multiplexed output through wave mixing. At the output of the PPLN, the output signal of the
nonlinear filter is selected with a tunable band-pass filter (BPF 3). After amplification, this output signal is
processed by a coherent receiver.
23
Figure 3.2: (a) Experimental setup of the optical second-order Volterra filter with wave mixing and delays.
MLL: mode-locked laser. EDFA: erbium-doped fiber amplifier. LCoS: liquid crystal on silicon. PC: polar-
ization controller. AWG: arbitrary waveform generator. Mod.: modulator. BPF: band-pass filter. PPLN:
periodically poled lithium niobate. Co. Rx: coherent receiver. (b) Spectrum of the MLL comb source.
3.3 Tunable second-order optical Volterra filter to equalize 10-
/20-Gbaud 4-APSK signals
We first characterize the optical second-order Volterra filter by measuring its output when excited by a
sinusoidal (intensity-wise) input light to obtain the transfer functions. As an example, the received electrical
spectra of the filter output when the input signal is a 10-GHz sinusoidal wave are shown in Fig. 3.3. When
the filter only has first-order taps, as shown in Fig. 3.3(a), the output spectrum contains a tone at the
frequency of 10 GHz, which is the same as the input frequency. When the filter has both first- and second-
order taps, as shown in Fig. 3.3(b), the spectrum contains two tones at the frequency of 10 GHz and 20
GHz.
The optical spectrum of the filter output appears in Fig. 3.4 for different tap configurations of the filter.
InFig. 3.4(a),onlythethreedummypumpsandthesignalcopiesthataresymmetricaltothedummypumps
about the QPM wavelength are activated. Therefore, the filter has only first-order taps. In Fig. 3.4(b), six
24
Figure 3.3: Received electrical spectrum of the filter output when the input signal is a 10-GHz sinusoidal
wave. (a) The filter has two first-order taps. (b) The filter has two first-order taps and two second-order
taps.
signal copies, three on each side of the QPM, are activated for wave mixing, resulting in an output of three
second-order taps. The delays of the signal copies are shown as multiples of 25 ps.
Figure 3.4: Optical spectra at the output of the PPLN waveguide under different tap configurations of (a)
three first-order taps and (b) three second-order taps.
Themeasuredtransferfunctionsforthedifferenttapconfigurationsalongwiththetheoreticalsimulation
results are shown in Fig. 3.5. The k-th order sinusoidal input transfer function H
k
(f) is defined as H
k
(f)=
Y(kf)/X(f), where Y(f) and X(f) are the output and input intensity spectra, respectively. Figures 3.5(a)
and(b)showtransferfunctionswithtwoandthreefirst-ordertaps, respectively. Figures3.5(c)and(d)show
transfer functions with two and three second-order taps, respectively. Figures 3.5(e) and (f) show transfer
25
functions with first-order taps and second-order taps together. The difference between the experimental and
simulationresultsmightbeduetoinaccuratetapweightsandthedelaysettingsintheexperimentcompared
to the designed ones. When more taps are present, the difference tends to be larger.
Figure 3.5: Measured first and second-order transfer functions (TFs) with different optical filter configura-
tions. The tap weights are shown above the figures. (a-b) Two cases with first-order taps. (c-d) Two cases
with second-order taps. (e) Combining cases 1 and 3 with both first- and second-order taps. (f) Combining
cases 2 and 4.
AftercharacterizingtheopticalVolterrafilter,wedemonstratetheequalizationofanopticaldatachannel.
As an example, we demonstrate the use of the optical nonlinear filter to equalize a nonlinearly distorted 20-
Gbaud 4-APSK signal. We present the results of the equalization of the 20-Gbaud 4-APSK signal using our
26
nonlinearfilterinFig. 3.6. Figure3.6(b)showsaneyediagramoftheamplitudelevelsofthe4-APSKsignal,
which shows equal amplitude spacing. The eye diagram is obtained by retrieving the electrical signal along
the “I” axis at the coherent receiver. We emulate a nonlinear distortion by applying a nonlinear function, as
shown in Fig. 3.6(a), to the input data at the transmitter. The distortion is applied in the digital domain
before the signal is loaded to the AWG. After pre-distortion, the BER is increased to 0.0082. As shown
in Fig. 3.6(c), the top two levels in the eye diagram are squeezed due to the pre-distortion. We use three
first-ordertapswiththetapweightsof ∼ 0.2,1,0.2anddelaysof0,25,50pstoequalizethesignal. Asshown
in Fig. 3.6(d), the BER is decreased to 0.0060, but the uneven levels cannot be equalized. Subsequently, we
add one second-order tap with a tap weight of 0.12 to equalize the signal further. As shown in Fig. 3.6(e),
with one more second-order tap, the levels become more even, and the BER could be reduced to 0.0032.
The nonlinearly distorted signal may benefit from the use of the optical Volterra filter.
Figure3.6: Equalizationofanonlinearlydistorted20-Gbaud4-APSKsignalwiththeopticalnonlinearfilter.
(a)Thenonlinearpre-distortionappliedtothesignal; Eyediagramof(b)thesignalbeforepre-distortion,(c)
thenonlinearlypre-distortedsignal, (d)theequalizedsignalwiththreefirst-ordertaps, and(e)theequalized
signal with three first-order and one second-order tap.
Subsequently, we tune the tap delay of the filter to equalize a 10-Gbaud 4-APSK signal. The results of
the equalization are shown in Fig. 3.7. The eye diagram of the 10-Gbaud signal without distortion is shown
27
in Fig. 3.7(a) with a BER of 7.88× 10
− 4
. After applying the same pre-distortion in Fig. 3.6(a), the BER
increases to 0.0024. Subsequently, we use two first-order taps with the tap weights of ∼ 1, 0.2 and delays
of 0, 50 ps to equalize the distorted signal. This causes the BER to be reduced to 0.0019 as shown in Fig.
3.7(c). Finally, we use one more second-order tap with a tap weight of 0.12 to further reduce the BER to
0.0012, as shown in Fig. 3.7(d).
Figure 3.7: Equalization of a 10-Gbaud 4-APSK signal. Eye diagram of (a) the signal before pre-distortion,
(b) the nonlinearly pre-distorted signal, (c) the equalized signal with two first-order taps, and (d) the
equalized signal with two first-order and one second-order tap.
In this demonstration, a raised-cosine filter with the 0.35 roll-off factor and the output wavelength of
∼ 1542 nm are chosen. The filter is potentially transparent to signal pulse shape, whereas the lower roll-off
factor may pose a stricter requirement on the tap delays. While the filter can be implemented at other
signal wavelengths within the ∼ 20-nm bandwidth of the PPLN, the corresponding CW pump should be
tuned to be symmetric around the QPM frequency. The tap weights are preset manually to equalize the
distortion in a given situation. For a different situation, the tap weights should be updated according to
the BER performance of the filter output. The optical Volterra filter can be potentially used to equalize
the signals with higher-order modulation formats. In particular, the higher-order modulation formats tend
to be more vulnerable to the nonlinear distortion [46], and the improvement using this equalization might
be more apparent as compared to lower-order modulation formats. To effectively equalize such higher-order
signals, more taps with more precisely tuned weights may be needed to achieve a similar performance [46].
28
Chapter 4
Free-space optical communication system in the mid infrared
(mid-IR) using multiplexing techniques
4.1 Background and motivation
There is growing interest in the mid-IR region for potential applications in communications, sensing, and
imaging in both free space and fiber [55, 56, 57]. For free-space optical communication links, the mid-IR has
several transmission windows that provide a relatively low atmospheric absorption in comparison to the C-
band(1530–1565nm)[41]. Moreover, mid-IRwavelengthstendtohavebetterpenetrationthroughinclement
weather conditions [58]. There have been reports of mid-IR communication systems that have used native
mid-IRtransmitters/receivers(e.g.,quantumcascadelasers)toachieveupto11-Gbit/sdatarateusingdirect
detection [59, 60, 61, 62]. In these systems without wavelength conversion, the transmitter modulates data
directly onto the mid-IR wavelengths and the receiver directly detects the data-carrying mid-IR wavelengths
to recover the data [59, 60, 61, 62]. There have also been reports of mid-IR communication systems that
have used C-band devices and wavelength conversions [63, 64, 65] to achieve 10-Gbit/s quadrature-phase-
shift keying (QPSK) using coherent detection [64, 65]. At the transmitter side, the data channels are
modulated onto the C-band wavelengths, and these data-carrying wavelengths are converted to the mid-
IR wavelengths by wavelength conversion. At the receiver side, the data-carrying mid-IR wavelengths are
wavelength converted to the C-band, and these wavelengths are detected by the C-band receiver to recover
29
thedata[63,64,65]. Importantly,thesetransmissionachievementshaveemployedonlyasingledatachannel
on a single beam.
Similar to communications in optical and radio systems, multiplexing multiple independent data chan-
nels and transmitting them simultaneously has produced dramatic capacity increases. Key examples are
frequency- and wavelength-division multiplexing in RF and optical systems, in which each channel occupies
a different frequency or wavelength [66, 67, 68]. Specifically, wavelength-division-multiplexing (WDM) has
been ubiquitously deployed in the conventional C-band wavelength [67, 68]. However, there were few reports
demonstrating multiple-channel WDM transmission in the mid-IR.
Another capacity-increasing multiplexing technique is space-division-multiplexing (SDM) [69, 70]. One
form of SDM is mode-division-multiplexing (MDM), in which multiple independent data channels are simul-
taneously transmitted on different beams each located on an orthogonal spatial mode [70]. For example, one
modal basis set includes orbital-angular-momentum (OAM) modes, which is a subset of Laguerre-Gaussian
(LG) modes [71]. A beam that carries OAM (i) has a phasefront that “twists” in a helical fashion as it
propagates, (ii) has a central intensity null, and (iii) can be characterized by the OAM order ℓ, which is
the number of 2π phase shifts in the azimuthal direction. Although there have been demonstrations of
generating and detecting an OAM beam [72], there were few reports demonstrating multiple-channel MDM
transmission in the mid-IR.
In this chapter, we experimentally demonstrate a mid-IR FSO communication system using WDM,
MDM, and a combination of WDM and MDM [73]. As the proof-of-principle experiment, we demonstrate
the multiplexing of multiple 50-Gbit/s QPSK channels in only a single domain as follows: (i) WDM only:
three channels with different wavelengths on a single polarization near 3.4 µ m (3.396, 3.397, and 3.398
µ m) each carried by a single Gaussian beam and (ii) MDM only: two channels sent on two different OAM
beams (+1 and +3) at a single mid-IR wavelength and on a single polarization. Additionally, we show the
compatibility of these individual multiplexing approaches and demonstrate WDM+MDM by transmitting
six data channels on a single polarization located at three wavelengths with each carrying two beams on
different modes. The operation is as follows: (i) at the transmitter, a C-band Gaussian beam is QPSK
modulated, wavelength converted to the mid-IR by difference frequency generation (DFG) [63, 64, 65] and
30
then converted into an OAM beam by a spiral phase plate (SPPs); and (ii) at the receiver, converting
the OAM back to a Gaussian beam using an SPP and wavelength converting into the C-band using DFG
[63, 64, 65]. According to the BER results, the OSNR penalties at the forward error correction threshold
are estimated to be: (i) ∼ 2 dB for wavelength conversion; (ii) ∼ 1 dB for wavelength multiplexing; and
(iii) ∼ 0.5-dB for inter-modal crosstalk. Compared with previous demonstrations of data-carrying WDM
systems and OAM-based MDM systems [67, 68, 69, 70, 71], we explore the data transmission/detection in
the mid-IR wavelength region using the scenarios of WDM only, OAM-based MDM only, and a combination
of both multiplexing techniques. By utilizing both multiplexing techniques and the wavelength conversions
betweentheC-bandandthemid-IR,weachievethemultiplexingofsixdatachannelswithatotalcapacityof
300 Gbit/s, corresponding to a 30X increase in comparison to previous single-channel, single-beam mid-IR
demonstrations [59, 60, 61, 62, 63, 64, 65].
4.2 Concept and experimental setup
The conceptual diagram of the mid-IR WDM and MDM FSO communication system is shown in Fig. 4.1.
Multipledata-carryingbeamswithdifferentmid-IRwavelengthsandorthogonalOAMmodesaremultiplexed
andco-propagatethroughfreespace. TousethewidelyavailableC-bandtransceivers,wavelengthconversions
between the C-band and mid-IR are performed in the nonlinear devices. At the transmitter, a Gaussian
beamattheC-bandismodulatedwithQPSKdataandwavelengthconvertedbyaperiodically-poledlithium
niobate(PPLN)waveguide, wherethesecond-ordersusceptibility, χ (2), resultsinathree-wavelengthmixing
process. Specifically, the wave mixing process involves the interaction of three wavelengths, including the
C-band signal wavelength (λ signal
), a pump wavelength (λ pump
), and an idler wavelength (λ idler
) [63, 64, 65].
Theidlerwavelengthcanbegeneratedatthedifferencefrequencyandcanbecalculatedasfollows[63,64,65]:
1/λ
idler
=1/λ
pump
− 1/λ
signal
(4.1)
Thus, mixing a signal at∼ 1550 nm with a pump at 1064 nm results in an idler wavelength of∼ 3400 nm.
Moreover, if the WDM channels are all in the phase-matching bandwidth of the PPLN waveguide, they can
31
be simultaneously converted in the same PPLN waveguide. In addition to the wavelength degree of freedom,
represented by WDM, OAM multiplexing is used to further increase data capacity. Mid-IR OAM beams are
generated by passing the fundamental Gaussian beam through SPPs with different orders. At the receiver,
an SPP with an inverse order is used to convert the corresponding OAM channel back to the Gaussian beam
for signal detection and data recovery. Subsequently, the mid-IR Gaussian beam is mixed with a 1064-nm
pump. AsimilarDFGprocessinanother PPLNwaveguide convertsthemid-IR idlers totheC-bandsignals,
with the wavelengths calculated as follows [63, 64, 65]:
1/λ
signal
=1/λ
pump
− 1/λ
idler
(4.2)
+1
λ
Ch3
+3
Ch4 Ch5 Ch6
λ
mir1
λ
mir2
λ
mir3
C-band WDM signals
OAM +1
OAM +3
Wavelength- and mode-division-
multiplexed FSO channels in the mid IR
OAM -1 OAM -3
Wavelength conversion
Spiral phase plates
Inverse spiral phase plates
C-band coherent detection
Wavelength conversion
λ
C-band Mid IR
λ
λ
1
λ
2
λ
3
λ
C-band Mid IR
λ
λ
1
λ
2
λ
3
Pump
Pump
OAM
Ch1 Ch2
Tx Rx
Figure 4.1: Concept for the mid-infrared (IR) wavelength-division-multiplexing (WDM) and orbital angular
momentum (OAM)-based mode-division-multiplexing (MDM) free-space optical (FSO) communication sys-
tem.
In this experiment, the pump laser is split into two paths and used for the transmitter and receiver.
However, in a typical communication scenario, two different pump lasers are used at the transmitter and
receiver. The different pump lasers could have a phase difference, frequency difference, and power difference,
which can potentially impact the system as follows. (i) The phase difference between the pump lasers adds
phase noise to the recovered C-band signal at the receiver and the variance of the additional phase noise
32
is proportional to the pump linewidth [23]; (ii) The frequency difference causes a frequency shift between
the generated C-band signal at the transmitter and the recovered C-band signal at the receiver, which also
causesdifferentphase-matchingconditionsforthewavelengthconversionsinthetransmitterandreceiver;(iii)
The power difference might cause different conversion efficiencies between the transmitter and the receiver.
Moreover, the phase noise and frequency shift caused by the different pump lasers could be potentially
compensated by the digital signal processing (DSP) at the receiver.
We experimentally demonstrate a mid-IR WDM and OAM-multiplexed FSO communication link using
thesetupillustratedinFig. 4.2. WegenerateC-bandWDMchannelsbymodulatingthreelasersourceswith
two optical in-phase-quadrature modulators, each loaded with a 25-Gbaud QPSK signal. The odd and even
channelsaregeneratedfromdifferentmodulators. Theelectricalsignalisgeneratedbyanarbitrarywaveform
generator with a 92-GSa/s sampling rate. One branch is delayed so that the adjacent wavelength channels
have decorrelated signals. The data channels are amplified with an EDFA (EDFA1) and coupled into a
PPLN waveguide (PPLN1) with a 1064-nm pump laser, which is amplified with a ytterbium-doped fiber
amplifier (YDFA). Polarization controllers are used before the EDFA1 and YDFA to adjust the polarization
of thebeams into the polarization-sensitivePPLN. The PPLNis temperature-controlled to adjustthe quasi-
phase-matching frequency and optimize mixing efficiency. A mid-IR beam is generated through the DFG
process. At the PPLN1 output, two Germanium windows are used as mid-IR band-pass filters to filter
out the high-power pump and input signals. The generated mid-IR fundamental Gaussian beam is split
into two paths and transmitted through two SPPs with OAM orders of +1 and +3. The two OAM beams
are combined, with one path delayed for data decorrelation. Subsequently, the combined beams propagate
co-axially for∼ 0.5 m in free space.
At the receiver, the OAM beams are first de-multiplexed by SPPs of the corresponding inverse order
(− 1 for the OAM +1 beam and − 3 for the other beam). The SPP converts the corresponding OAM input
to a Gaussian beam, while input OAM beams of other orders emerge as ring-shaped, center-null beams to
be blocked by an appropriate spatial filter. The mid-IR Gaussian beam and a 1064 nm pump laser are
coupled into PPLN2, which converts the mid-IR beam back to the C-bandthrough the DFG process. At the
PPLN2 output, a 1500-nm high-pass filter is used to filter out the high-power pump. The received signal is
33
λ1
λ3
λ2
EDFA1
YDFA
1064 nm
Dichroic
mirror
PPLN 1
MIR filter SPP ℓ=+3
SPP ℓ=+1
SPP ℓ=-1
or -3
HPF
25-Gbaud
QPSK
PPLN 2 Dichroic
mirror
1 m
PC3
PC4
PC2
PC1
Col.1
Col.2
Col.3
M1
M2
Free-space
propagation
Mid-IR WDM signal generation (Tx) Mid-IR OAM muxing (Tx)
Mid-IR OAM de-
muxing (Rx)
Mid-IR signal detection (Rx)
Iris
Equalization
Filtering
Carrier phase
recovery
Demodulation
& BER
Rx DSP
BS1
BS2 BS3
C2
C1
EDFA2
Coherent
receiver
LO
BPF
VOA
DSO
OSA
1%
Figure 4.2: Experimental setup of the free-space mid-infrared WDM and MDM communication system. PC:
polarization controller; Col.: collimator; EDFA: erbium-doped fiber amplifier; YDFA: ytterbium-doped fiber
amplifier; PPLN: periodically poled lithium niobate; M: mirror; SPP: spiral phase plate; HPF: high-pass
filter; BPF: tunable band-pass filter; LO: local oscillator. VOA: variable optical attenuator; OSA: optical
spectrum analyzer; DSO: digital storage oscilloscope; C: fiber-based optical coupler; BS: free-space beam
splitter.
then coupled to a single-mode fiber and detected and processed by a C-band coherent receiver. The signal
is amplified by an EDFA and filtered by a band-pass filter to suppress the out-of-band noise before being
detected by the coherent receiver. The signal detected by the coherent receiver is sampled by an oscilloscope
with an 80-GSa/s sampling rate and a 33-GHz bandwidth.
Offline digital signal processing is performed at both the transmitter and receiver. At the transmitter,
the QPSK signal is Nyquist pulse shaped using a raised cosine filter with a roll-off factor of 0.05. The sharp
roll-off factor of 0.05 is chosen for the raised cosine filter to reduce the spectral bandwidth of the signal and
the required guard band between the WDM channels [74]. The pulse-shaped signal is resampled and loaded
to the arbitrary waveform generator. The arbitrary waveform generator operates at a sampling rate of 92
GSa/s. At the receiver, the signal is detected by an optical modulation analyzer (OMA), which consists of
an optical coherent receiver and a digital real-time oscilloscope. The inset of Fig. 4.2 also shows the DSP
sequence at the receiver. The received signal is first digitally filtered with a band-pass filter to suppress the
34
crosstalk from the adjacent wavelengths. Carrier phase recovery (CPR) is performed to compensate for the
frequency offset and phase noise. To compensate for the linear distortions caused by the transmitter and
receiver components (e.g., driver amplifiers, IQ modulators, the optical band-pass filter, coherent receiver,
and the oscilloscope), a finite impulse response feed-forward equalizer with 48 taps is applied to the received
signal. The tap weights of the equalizer are pre-converged in the C-band generation/detection case using
training sequences and kept fixed for the remaining data signals. The BER is measured through error
counting.
4.3 Wavelength division multiplexing (WDM) in the mid-IR
First, we measure the link performance of a WDM system with the mid-IR Gaussian beam. As shown in
the spectrum depicted in Fig. 4.3(a), three WDM channels at the ∼ 3400-nm wavelength (3.396, 3.397, and
3.398 µ m) with a channel spacing of 27.5 GHz (∼ 1 nm @ 3400 nm) are generated through the DFG process.
However, the spectral shapes of the channels are not completely resolved by the optical spectrum analyzer
(OSA) for the mid-IR, as the OSA has a limited resolution bandwidth of 1 nm which is wider than the
data channels themselves. The pump and C-band signal power, as well as the PPLN1 temperature control,
are adjusted to optimize the conversion efficiency of the mid-IR beam generation. Figure 4.3(b) shows the
power of the generated mid-IR beam as a function of the pump power for different signal power values.
The PPLN1 temperature is 49.5
◦ C for Fig. 4.3(b). The mid-IR power generally increases with the pump
power. It also increases with the signal power but tends to saturate at signal power levels higher than 1
W. This might be due to an imbalance between the signal and pump photon numbers, possibly resulting
in pump depletion [75]. Figure 4.3(c) shows the mid-IR power as a function of the PPLN temperature for
three C-band signal wavelengths. The PPLN1 temperature is adjusted to satisfy the quasi-phase-matching
in the DFG process. To determine the temperature that gives the optimal conversion efficiency for 1548.4
nm, the PPLN temperature is tuned by a temperature controller and the conversion efficiency is measured
accordingly. As shown in Fig. 4.3(c), the optimal temperature is 49.5
◦ C for 1548.4 nm. When the C-band
signal wavelength is longer, the optimal temperature tends to increase. When the pump power is 3.86 W
and the PPLN1 temperature is 49.5
◦ C, the conversion efficiency is ∼− 26.5 dB, which is the ratio between
35
the output idler power and the input signal power of the PPLN1. The conversion efficiency could decrease
when the pump power is lower or the PPLN temperature is not optimal. The fact that there are different
optimal temperatures for different C-band signal wavelengths could affect the conversion efficiency of the
WDM channels.
-12
-8
-6
-4
-2
0
Temperature (
°
C)
35 40 45 50 55 60
Signal 1548.4 nm
Signal 1547.6 nm
Signal 1549.2 nm
1064-nm pump power (W)
Mid-IR power (mW)
0 1 2 3 4
0
1
2
3
4
Signal 0.14 W
Signal 0.3 W
Signal 0.52 W
Signal 1 W
Signal 1.54 W
Normalized mid-IR power (dB)
-5
-4
-3
-2
-1
0
C-band signal wavelength (nm)
1546 1548 1550 1552
(b)
(c)
(d)
Power (5 dB/div.)
Wavelength (nm)
3385 3390 3395 3400 3405 3410
Res.: ~1 nm
(a)
Normalized mid-IR power (dB)
-10
Pump power: 0.65 W
PPLN temperature: 49.5
°
C
Pump power: 0.65 W
Figure4.3: (a)Spectrumofthegeneratedmid-IRWDMsignalswitharesolutionof∼ 1nm. Arrowsindicate
the three mid-IR WDM channels. (b) Generated mid-IR beam power as a function of the 1064-nm pump
power with different signal power values. The C-band signal wavelength is set at 1550 nm. (c) Generated
mid-IR beam power as a function of PPLN temperature with different C-band signal wavelengths. (d)
Generated mid-IR beam power as a function of C-band signal wavelength with a PPLN temperature of 49.5
◦ C.
Figure 4.3(d) shows the generated mid-IR power as a function of the C-band signal wavelength when
the PPLN temperature is set at 49.5
◦ C. The pump power at the PPLN1 input is∼ 0.65 W for Figs. 4.3(c-
d). We note that the generated mid-IR power shown in Fig. 4.3(c) is normalized for each C-band signal
wavelength; however, the generated mid-IR power in Fig. 4.3(d) is normalized for a fixed temperature value
36
of 49.5
◦ C. Therefore, the normalized power values in Figs. 4.3(c) and (d) are not directly comparable. At
a ∼ 1.6-nm C-band signal wavelength bandwidth, the generated mid-IR power is > 90% of the maximum
generated mid-IR power. This allows for simultaneous wavelength conversion of a few WDM channels in a
single PPLN waveguide. The wavelength conversion from the mid-IR to the C-band is performed in PPLN2.
Thetotalinputpowerofall3mid-IRchannelsintoPPLN2is0.956mW,andtheoutputpowerofallC-band
channels is 0.0053 mW.
Next, we demonstrate a 150-Gbit/s mid-IR WDM FSO communication system with the Gaussian beam.
Figure 4.4(a) shows the spectrum of the recovered signal at the C-band after PPLN2, in which three WDM
channels and an approximate OSA noise floor level can be seen. The C-band wavelengths shown in Fig.
4.4(a) are chosen such that the central channel wavelength is the optimal C-band signal wavelength in Fig.
4.3(d) with the highest generated mid-IR power. A normalized crosstalk matrix of the WDM channels is
shown in Fig. 4.4(b). The crosstalk matrix is obtained by sending different single wavelength channels
modulated with a 25-Gbuad QPSK signal at the transmitter and measuring the output optical power of a
tunable band-pass filter with different center wavelengths at the receiver. The crosstalk tends to be larger
from the longer- to shorter-wavelength channels than from the shorter- to longer-wavelength channels. This
mightbeduetotheopticalfilterusedinthemeasurement, whichhasahigherextinctionratioattheshorter
wavelengths than at the longer wavelengths. The measured WDM crosstalk matrix shows crosstalk lower
than − 13 dB between adjacent wavelengths. We note that the WDM crosstalk critically depends on the
sharpness of the optical filter, and it can be suppressed by further digital signal processing after coherent
detection.
Subsequently, 25-Gbaud QPSK signals are transmitted on each WDM channel. In the C-band gener-
ation/detection case, the signal generated by the C-band transmitter is directly received by the coherent
receiver without wavelength conversion and free-space propagation. As shown in Fig. 4.4(c), the mid-IR
cases have OSNR penalties compared to C-band generation/detection cases. These OSNR penalties might
be caused by the wavelength conversions where (i) undesired terms might be generated by the PPLN and
overlap with the mid-IR data channels leading to in-band crosstalk and (ii) additional frequency drift and
phasenoisefromthepumplasermightbeaddedtothemid-IRdatachannels[52]. Inboththesingle-channel
37
Tx: Ch. 1 Ch. 2 Ch. 3
Rx:
Ch. 1
Ch. 2
Ch. 3
-0.28 -13.22
-14.89
-0.78
0 -24.04
-31.89 -24.37
-30.4
0 dB
-30 dB
FEC threshold
(b)
(c)
Ch. 1 Ch. 2 Ch. 3
(a)
Power (5 dB/div.)
Wavelength (nm)
1548 1548.2 1548.4 1548.6 1548.8 1549
Res.: 0.01 nm
BER
10
-4
10
-3
10
-2
10
-1
OSNR (dB)
8 10 12 16 14
(d)
FEC threshold
BER
10
-4
10
-3
10
-2
10
-1
OSNR (dB)
10 11 12 16 13 14 15
Channel 1
Channel 2
Channel 3
1-ch. C-band gen./det.
WDM C-band gen./det.
1-ch. Gaussian
WDM Gaussian
Approximate
OSA noise
floor level
Figure 4.4: (a) Spectrum of the WDM signals that are converted back to the C-band. (b) Normalized
optical crosstalk matrix of WDM. (c) Measured bit error rate (BER) as a function of the received optical
signal-to-noise ratio (OSNR) for a C-band generation/detection (gen./det.) and mid-IR Gaussian beam
transmission. In the C-band generation/detection case, C-band signals are detected by the coherent receiver
without wavelength conversion and free-space propagation. (d) Measured BER as a function of the received
OSNR for the three mid-IR WDM channels.
and WDM cases, only mid-IR Gaussian beams are used, and the SPPs are bypassed. At the receiver, we use
an EDFA to amplify the received signal, where the amplified spontaneous emission noise might be added. If
thereceivedsignalpowerchangesduetothefree-spacelinkloss,theOSNRoftheEDFAoutputmightchange
accordingly. We choose the BER-OSNR curve as a figure of merit to investigate the penalties caused by the
wavelength conversion and crosstalk between multiplexed channels [67]. It might also be possible to use the
BER vs. received mid-IR power as a figure of merit [63]. We find that the single-channel mid-IR Gaussian
beam has a∼ 2-dB OSNR penalty at the BER near the FEC threshold of 7% overhead in comparison to the
C-bandgeneration/detectioncase. Tohelpsuppresscrosstalkfromadjacentchannels, weusedigitalfiltering
in the receiver after coherent detection [76]. The mid-IR WDM channel has an additional ∼ 1-dB OSNR
38
penalty compared to the mid-IR single-channel transmission. We note that some of the measured power
penalty might be due to crosstalk induced by the wavelength conversion in which undesired mixing terms
might be generated by the PPLN and interfere with the mid-IR data channels [52]. This could be induced
by distortion during the wavelength conversions. Figure 4.4(d) shows the measured BERs as a function of
the received OSNR for all three WDM channels.
4.4 Mode division multiplexing (MDM) and a combination of
WDM and MDM in the mid-IR
Furthermore,wedemonstrateamid-IRFSOcommunicationsystemusingOAM-multiplexingandacombina-
tionofWDMandOAM-multiplexingwithupto300-Gbit/scapacity. Figure4.5(a)showstheexperimentally
measured beam profiles of the generated 3.4- µ m OAM beams. The intensity profiles of the OAM +1 and
OAM +3 beams are shown in Fig. 4.5(a) on the top row, respectively. The intensity profiles are ring-shaped
due to the phase singularity at the center [11]. To verify the OAM orders of the generated beams, interfero-
grams of the OAM beams with a Gaussian beam are captured and shown in Fig. 4.5(a) on the bottom row.
Theinterferogramshavetwistedarms, andthenumberoftwistscorrespondstotheOAMorderofthebeam.
When two data-carrying OAM +1 and +3 beams are multiplexed, the intensity profile has a ring shape, as
shown in Fig. 4.5(a) on the top right. The intensity profile when the multiplexed OAM beams pass through
the inverse SPP with an OAM order of− 3 is shown in Fig. 4.5(a) on the bottom right. The intensity profile
has a Gaussian-like beam at the center, which corresponds to the OAM +3 beam. The other OAM +1 beam
still has a ring shape after the inversed SPP. This intensity profile indicates that the desired OAM beam can
be converted to the fundamental Gaussian mode and separated from the other beam with a spatial filter.
Figure 4.5(b) shows a normalized crosstalk matrix of the OAM multiplexed channels. The values in the
crosstalk matrix indicate the measured optical power that is converted back to the C-band coherent receiver
when a single mid-IR OAM beam is transmitted, and an inversed SPP is used to receive one OAM mode.
The residual crosstalk between the OAM channels could be caused by the misalignment between the mid-IR
39
OAM +1 OAM +3
OAM
+1
OAM
+3
Tx:
Rx:
0
-0.19
-15.84
-18.81
0 dB
-18 dB
OAM +1 OAM +3
Tx: OAM +1 and +3
Tx: OAM +1 and +3
Rx: SPP -3 to
downcovert OAM +3
Intensity
profile
Interferogram
FEC threshold
w/ WDM and sending both OAMs
w/ WDM and sending single OAM
single wavelength and sending both OAMs
single wavelength and sending single OAM
BER 10
-4
10
-3
10
-2
10
-1
OSNR (dB)
12 14 16 20
FEC threshold
Ch 1:OAM +1 λ1 Ch 4:OAM +3 λ1
Ch 2:OAM +1 λ2 Ch 5:OAM +3 λ2
Ch 3:OAM +1 λ3 Ch 6:OAM +3 λ3
18
(a)
(b)
(c)
(d)
BER (OAM +3)
10
-4
10
-3
10
-2
10
-1
OSNR (dB)
12 14 16 20 18 22
Figure 4.5: (a) Measured beam profile of the mid-IR OAM beams. Intensity profile and interferogram with
a Gaussian beam of the OAM +1 and OAM +3 beam, respectively. Intensity profile of the data-carrying
multiplexed OAM +1 and +3 beams. Intensity profile of the multiplexed OAM beam after passing through
the second SPP with OAM order − 3. (b) Normalized crosstalk matrix of MDM. (c) Measured BER of the
OAM +3 channel as a function of the received OSNR for the mid-IR OAM beam transmission when sending
both OAM modes and sending a single OAM mode. (d) Measured BER and OSNR of all the channels,
including two OAM modes with three wavelengths on each mode.
beam axis and the center of the SPPs, which may degrade the quality of the generated OAM beams and the
back-converted Gaussian beams.
Figure 4.5(c) shows the measured BERs as a function of the received OSNR for the single OAM beam
transmissionandtwomultiplexedOAMbeams. AsshownbytheBERcurves,theOAMmultiplexinginduces
a <1-dB OSNR penalty at the FEC threshold. This could be caused by modal crosstalk between the OAM
channels. In addition, the WDM induces a ∼ 1-dB OSNR penalty compared to the single-wavelength case.
This could be due to the distortion caused by wavelength conversions.
40
In this demonstration, two OAM beams each containing three wavelength channels that are multiplexed,
resulting in a total of six channels. The BER performance of the six channels is shown in Fig. 4.5(d). We
receive one channel at a time in the proof-of-concept experiment. Considering a real communication system
(M modesandN wavelengths),M× N paralleltransmitters/receiverswouldberequiredtotransmit/receive
independent data streams. Each transmitter has a laser source and an IQ modulator, and each coherent
receiver has a local oscillator laser. Moreover, at the transmitter, M PPLNs would be used to convert the
wavelength of the signals, each with a mid-IR filter at the output. At the receiver, another M PPLNs
would be used to convert the mid-IR wavelengths. In our experiment, SPPs and beam splitters are used to
convertandcombinetheOAMbeams. TheMDMcouldalsobepotentiallyrealizedbyothermethods,which
might be more efficient and more compact, such as multi-plane light conversion [77] and Dammann gratings
[78]. However, special designs might be required for them to work in the mid-IR region. The channels have
slightly different performances, which could be due to the different crosstalk values of each channel. For all
the channels, BER below the 7% FEC threshold can be achieved. This indicates that a total gross data
capacity of 300 Gbit/s is transmitted through the mid-IR FSO link.
A few points worth discussing include:
(i) In this approach, wavelength conversion is utilized to convert the signals between the C-band and
mid-IR, such that widely available, high-performance C-band components can be used to enable high-speed
data generation and detection. Although many native mid-IR devices are available (e.g., narrow-linewidth
lasers [79, 80] and optical amplifiers [81]), high-speed mid-IR modulators and photodetectors are still not
easily found but can be used when available.
(ii)Inthisdemonstration, wemultiplexthreewavelengthchannels, whichmightbemainlylimitedbythe
PPLN phase-matching bandwidth. To scale the number of wavelength channels in our scheme, a nonlinear
device with a wider phase-matching bandwidth might be required. Recently, mid-IR generation in nonlinear
devices with wide phase-matching bandwidth have been reported (e.g., 700 nm phase-matching bandwidth
of the thin-film lithium niobate by dispersion engineering [82]).
41
(iii) This work uses a high level of optical power to pump the PPLN waveguides due to the relatively low
conversion efficiency. However, the power requirement can be potentially reduced when more efficient PPLN
waveguides are available [83].
(iv) The data channel mid-IR wavelength depends on the pump wavelength and nonlinear device. Al-
thoughweworkedat∼ 3.4µ m,otherwavelengthscanbedemonstratedbyjudiciouslyusingotherappropriate
pump wavelengths and nonlinear devices [63].
(v)Ingeneral,polarization-division-multiplexing(PDM)ispotentiallycompatiblewithWDMandMDM,
thereby offering another avenue for capacity increase [67]. In the current experimental setup, all channels
are at a single polarization using the single-polarization IQ modulators at the Tx and polarization-sensitive
PPLNs at the Tx and Rx. We believe that PDM can potentially be implemented in our approach, and some
techniques to might help achieve this include: (a) Modulator: a dual-polarization IQ modulator to generate
PDM signals [84], and (b) PPLN: polarization diversity PPLN architecture to perform the wavelength
conversions of the polarization-multiplexed signals [85].
(vi) The results described in this chapter are for free-space communications. However, many of the same
principles for the transmitter and receiver should still be valid in an optical fiber communication system,
and there is interest in low-loss fibers for mid-IR wavelengths [55].
42
Chapter 5
Optical space-time wave packets (STWP) generated using
frequency combs and spatial modes
5.1 Background and motivation
Interestingly, STWPscanbecomposedofstructuredlight[6,7,8], suchthattheycancontainuniquespatial
amplitude and phase profiles [10, 9]. One example of structured light is a beam carrying OAM [12, 11], and
STWPs carrying OAM have been demonstrated [6, 7, 8]. However, using a single frequency carrying either
one OAM mode or multiple LG modes, the resulting monochromatic OAM beam has a static beam radius
or OAM value at each axial distance [10].
Beyond a beam that has a static amplitude and phase profile at some propagation distance, it is possible
to create an STWP that has a “dynamic” spatial profile [10, 86]. To generate a dynamically varying STWP,
multiplespatialmodescanbecoherentlysuperimposedatmultiplefrequencies. Previousreportshaveshown
dynamicbeammotions,suchas(i)aGaussian-likebeamdotmovingalongoneofthetransverseaxes[87],(ii)
alightbeamwithoutcarryingOAMrotatingalongitsazimuthaldirectionorhavingin-and-outmovementin
the radial direction [88], and (iii) a dynamic STWP with two independent and controllable orbital-angular-
momenta [6]. A laudable goal would be to generate STWPs with temporal changes of beam radius in time,
as another type of dynamic motion, by exploiting the coherent combination of multiple spatial modes on
multiple frequencies.
43
Another interesting goal might be to generate an STWP that has a tunable and dynamically changing
OAMvalueℓatagivenpropagationdistance. Previously, anopticalextreme-ultravioletpulsewasgenerated
using high-harmonic generation such that the OAM value dynamically changed during the pulse at a given
propagation distance [89]. This resulted in a dynamic OAM value that: (a) depends on the original OAM
values of two input pulses, and (b) either increases or decreases monotonically [89]. Recently, an STWP
carryingdynamicallyincreasing− 1to+1OAMvaluewasexperimentallygeneratedbysynthesizingmultiple
frequency lines, each having different combinations of spatial modes [90].
OneapproachtogeneratingSTWPsinfreespacewithlowdivergenceandacontrollablegroupvelocityis
to coherently combine multiple different frequencies, each containing a Bessel mode [91, 8]. Typically, these
STWPs are generated at a transmitter and propagate in free space [92, 93]. However, a potentially valuable
issue might be the feasibility of propagating such an STWP in a fiber before it exits into free space, thereby
enabling directional control of the beam for precise applications [94, 95]. Unfortunately, coupling such an
STWP into a multi-mode fiber (MMF) often produces significant modal coupling, which might degrade the
output STWP and distort its propagation characteristics [96].
In this chapter, we experimentally demonstrate and investigate in simulation several different types of
STWPs, as shown in Fig. 5.1 [97, 98, 99]. First, we experimentally demonstrate OAM-carrying STWPs
having a controllable dynamically varying beam radius [97]. In the experiment, (i) six frequency lines
from a Kerr frequency comb are spatially modulated with different spatial patterns to carry the coherent
superpositionofmultipleLGmodes,and(ii)thecomplexbeamprofiles( i.e.,intensityandphase)atdifferent
time instants are captured by using off-axis digital holography and tuning the delay between the STWP and
a reference pulse. In the experiment, the generated STWP (ℓ = +1 or +3) has an OAM purity of (i) >64%
when the propagation distance is 0, and (ii) ∼ 20% when the propagation distance is half the Rayleigh
range. The results indicate that the generated STWP would attain larger beam radii at longer propagation
distances. We also experimentally generate and investigate the tunability of STWPs carrying dynamically
changing OAM values [98]. In the experiment, six Kerr comb lines are combined, each carrying multiple
OAM modes with different ℓ values and complex weights. By changing the modes and complex weights, the
range of OAM values from +1 to +6 or from +1 to +4 is demonstrated. Furthermore, the OAM values
44
can be tuned to increase/decrease monotonically or to decrease first and then increase during the period.
We also investigate the effects of frequency line numbers on the temporal pulse width and the nonlinearly
changing OAM values in the simulation. The simulation results show that (i) the narrower temporal pulse
width of the STWP can be achieved by using more frequency lines and (ii) the nonlinearly varying OAM
values can result in different frequency chirps along the azimuthal direction at different time instants.
We simulate the generation of STWPs by combining multiple LG modes carried by multiple frequencies
[100]. We investigate the group velocity value at different propagation distances and with different numbers
of modes. The simulation results show that (i) the range of the propagation distance that the group velocity
v
g
remains near the target value tends to decrease when v
g
deviates from the vacuum speed of light c and
(ii) the estimated group velocity tends to be closer to the designed value when the maximum mode number
increase from 1 to 15.
In addition, we experimentally demonstrate the generation of an STWP and its propagation over 1-m
graded-index MMF propagation in which fiber modal coupling is mitigated to have reduced divergence and
tunablegroupvelocityatthefiberoutput[99]. TogenerateBesselmodesafterfiberpropagation,eachBessel
mode is decomposed of multiple fiber modes and transmitted through the fiber. Fiber modal coupling is
mitigatedusingtheinversematrixofthemeasuredcomplexcouplingmatrixofthefiber. Thegroupvelocity
is varied from 1.0042c to 0.9967c by tuning the wavenumber of the Bessel mode on each frequency. The
divergence is reduced by generating low-divergence Bessel modes at the fiber output. The measured time-
averaged intensity profiles show that the beam radius of the STWP remains similar after 150 mm further
free-space propagation. The measured results of the STWP after fiber propagation show the group velocity
control and diffraction reduction, which is similar to the STWP without fiber propagation [91, 8].
5.2 Concept and experimental setup for STWP generation
TheconceptofgeneratingtheOAMSTWPswithatime-dependentbeamradiusisshowninFig. 5.2(a). The
STWP is generated by combining coherent comb lines with each frequency line containing a superposition
of LG modes. For different frequency lines, the LG modes with the same ℓ value and different p values
are superposed with a different set of coefficients C
LG
(i,p) to generate the designed structured beams. The
45
Figure 5.1: Several types of STWPs that will be discussed in this chapter.
superposedLGmodesondifferentfrequencieshavearadialintensityprofilefollowingtheHermitepolynomial
distribution with different orders, and a spiral phase profile along the azimuthal direction. They are further
combined with coefficients following a Poisson distribution [87]. As a result, the generated OAM beam has
a time-dependent beam radius. Figure 5.2(b) shows an example of the interference between the different
frequency lines leading to the dynamic motion. For different frequencies, there is a phase difference as a
function of time. Consequently, the constructive or destructive interference between the radial intensity
profiles results in the beam having different beam radii in the temporal domain.
Figures 5.2(c-e) show the concept of generating STWPs carrying dynamically changing OAM values,
which is also achieved by synthesizing combinations of Laguerre-Gaussian (LG) modes on coherent comb
lines. In Fig. 5.2(c), a single frequency line carries a temporally static single OAM ℓ mode (e.g., LG
ℓ,0
).
Figure 5.2(d) represents yet another static case moving to multiple frequency lines carrying a single OAM
mode, resulting in a pulsed OAM in the temporal domain.
AsshowninFig. 5.2(e), puttingcombinationsofOAMmodeswithdifferentcomplexweightsonmultiple
frequencies results in a wave packet having different OAM values at different time instants at a given
46
Figure 5.2: Concept of generating the space-time wave packets (STWPs) carrying (a-b) OAM with a time-
dependent beam radius and (c-e) dynamically changing OAM values. (b) An example of the interference
leadingtothedynamicmotion. (c)AsinglefrequencycarryingasingleOAMmode. (d)Multiplefrequencies
carrying a single OAM mode. (e) Multiple frequencies carrying multiple OAMs with different weights on
each frequency form an STWP with dynamically changing OAM values. The constructive combining for
generating different OAM modes appears at different time instants depending on the time-variant relative
phase difference between the comb lines.
propagation distance. The OAM modes with the same order carried by coherent comb lines could be
coherentlycombinedwithatime-variantrelativephasedifferenceof∆ φ=2π ∆ ft,where∆ f isthefrequency
spacing between the comb lines [5, 6, 7]. When combining the same OAM mode on different coherent comb
lines, they could be constructively combined at some time instant and destructively combined at other time
instantsdependingonthetime-variantrelativephasedifference[6,7]. Asaresult,theconstructivecombining
for generating different OAM modes could appear at different time instants, which results in dynamically
changing OAM values of the generated STWPs. The weight (C
k,j
) of OAM mode (ℓ
k
) on frequency (f
j
)
could be found by applying the temporal Fourier transform to the desired temporal waveform of that mode
[101]. For example, we use six comb lines to generate an STWP carrying changing OAM values from ℓ
1
to
ℓ
6
. The complex weights C
k,j
for OAM mode ℓ
k
can be calculated by the temporal Fourier transform of a
pulse with a different delay ( C
k,j
=exp(− i2πkj/N ), where N =6 is the number of comb lines).
47
As shown in Fig. 5.3, we use a pulse shaping method in the frequency domain to generate the STWPs
carrying. To achieve pulse shaping in the frequency domain, a pulse shaper has been used where two lenses
areplacedinadditiontothegratingsandanSLMisplacedatthefocalpointofthelenses[102]. Inaddition,
such frequency domain pulse shapers in combination with multi-plane light conversion (MPLC) techniques
[103]areusedtoachievearbitraryspatiotemporalfieldgeneration[101,104]. Suchtechniquescanpotentially
be used to generate the STWPs carrying dynamically changing OAM values.
Figure 5.3: Experimental setup for generating and measuring the dynamic STWP. Col.: collimator; BS:
beam splitter; SLM: spatial light modulator.
A Kerr frequency comb at 1550 nm is used as the light source. Six comb lines with a spacing of ∼ 192
GHz are selected by a programmable waveshaper. The comb lines are separated by an optical grating and
directed onto different locations of a spatial light modulator (SLM1). Six phase patterns are loaded on
different locations of SLM1, with each phase pattern controlling the spatial profile of a different comb line
(f
k
). The complex-weighted combinations of LG modes (
P
j
C
k,jLG
ℓ
j,0
, where the OAM order ℓ
j
is chosen
according to the different cases) are generated by spatially modulating the comb lines via appropriate phase
patterns at the corresponding parts of the SLM. Each phase pattern consists of (i) the desired beam phase
profile and (ii) a grating pattern whose diffraction efficiency is designed according to the desired beam
48
amplitude profile [105]. The grating period of each phase pattern on SLM1 is tuned such that the output six
frequenciesaredirectedtothesamepositiononSLM2. Subsequently, thesixfrequenciesarecombinedusing
SLM2. The phase pattern loaded on SLM2 is designed based on a combination of six grating patterns. The
grating periods of these six patterns are tuned such that the output directions of the six frequencies after
SLM2 are the same. The generation of the STWP with our setup has ∼ − 28-dB overall power efficiency
when comparing the optical power before the first optical grating and after the lens 2. This power efficiency
might be due to (i) the insertion loss of the optical elements including the grating, lenses, and mirrors,
(ii) the conversion efficiency of the complex modulation on SLM1, and (iii) the loss of beam combining on
SLM2. WenotethathigherefficiencycouldbepotentiallyachievedbyusingothertechniquessuchasMPLC
[103, 101, 104]. The generated STWP is detected by off-axis digital holography with the help of a reference
beam[106]. Thecomplexbeamprofilesatdifferenttimeinstantsareobtainedbychangingthedelaybetween
the STWP and the reference beam. We also use a 4-f system to image the beam profile generated on SLM1
to the camera. The 4-f system is used to relay the beam on SLM1.
5.3 STWPcarryingorbitalangularmomentum(OAM)withtime-
dependent radius
A Kerr frequency comb is used as the light source to generate the STWP. A waveshaper is used to (i) select
six comb lines, and (ii) adjust the amplitude and phase of the frequency lines to minimize the pulse width
of the output pulse [107]. The input and output optical spectra of the waveshaper are shown in Figs. 5.4(b)
and (c), respectively. The input Kerr frequency comb is in the single soliton state [19].
In Figs. 5.5(a2,b2,c2), the time dependence of the beam radius is investigated both experimentally and
computationally. The beam waist is chosen as 0.3 mm in the experiment as well as the simulation. The
beam radius oscillates from ∼ 0.24 mm to ∼ 0.68 mm. By comparing Figs. 5.5(a2) and (b2), we conclude
that the beam radius increases with distance. The deviation of the measured beam radius with respect to
the simulation values is likely due to the imperfect mode generation by the SLM1. The beam radii of the
generated STWP could have some error due to the power and phase variations of the six frequency comb
49
Figure 5.4: The optical spectrum at the (a) input and (b) output of the waveshaper.
lines. The measurement error could be related to the (i) temperature drifts and mechanical vibrations of
the fiber, which could result in varying phases and polarizations of different frequency lines [108]; (ii) the
relativephasenoisebetweenthegeneratedfrequencylinesofthecombsource,whichcouldresultinarandom
phase shift between the frequency lines [109]; and (iii) the temperature drift of the microresonator and the
frequency drift of the pump laser, which could change the state of the generated comb [110].
OAM purity values are plotted in Figs. 5.5(a3,b3,c3). The variations in measured OAM purity might
be due to the phase variation at given times and this phase distortion reduces OAM purity. OAM purity
values are: (i) ∼ 64% to ∼ 86% for ℓ = +1 and +3 at z ∼ 0, and (ii) ∼ 20% to ∼ 84% for ℓ = +1 at
∼ 0.5z
R
. OAM purity values tend to be lower at larger distances. The relatively low OAM mode purity
might originate from the limited pixel resolution of both SLM1 and SLM2, which affects the quality of the
synthesized spatial modes [111, 112]. The OAM purity can be potentially increased by synthesizing a larger
number of LG modes including higher-order p modes.
In this demonstration, the topological charge of the OAM-carrying STWP is ℓ = +1 or +3. We could
also choose other topological charges (e.g., ℓ = +2 instead of ℓ = +3). However, choosing higher-order
topological charges (e.g., ℓ>+3) for OAM-carrying STWPs may lead to a larger beam size due to increased
divergence of the LG modes [113, 114].
50
Figure 5.5: (a1-c1) Intensity and phase profiles, (a2-c2) the estimated beam radius along with the simulated
radius, and (a3-c3) the OAM purity of the generated OAM-carrying STWPs. In (a) ℓ=+1 and at 0z
R
, (b)
ℓ = +1 and at 0.5z
R
, and (c) ℓ = +3 and at 0z
R
, respectively. All results are measured in a time duration
of 5 ps.
5.4 STWP carrying dynamically varying OAM
We experimentally demonstrate the generation of STWPs carrying six dynamically changing OAM values.
TheintensityandphaseprofilesareshowninFig. 5.6(a). Withthecomplexprofiles,thecorrespondingOAM
spectra are calculated. The OAM spectra show that the OAM value increases from +1 to +6 and the OAM
purityis∼ 70%exceptfortheOAM+6. TheOAMspectraareobtainedby[115](i)decomposingthecomplex
profilesintheLGmodalbasisusing u
ℓ,p
=
RR
E(x,y)· LG
∗ ℓ,p
(x,y)dxdy,whereE(x,y)andLG
ℓ,p
(x,y)denote
the measured complex field and the theoretical complex field of an LG
ℓ,p
mode, respectively; (ii) calculating
the power on OAM order ℓ by summing |u
ℓ,p
|
2
for p from 0 to 9. We note that the mode decomposition
could also be performed based on the Hankel transform, which decomposes the mode as a superposition of
Bessel modes [116]. The OAM spectrum as a function of time is shown in Fig. 5.6(b).
51
Figure 5.6: Experimental results of an STWP carrying increasing OAM values from +1 to +6. (a) The
intensity and phase profiles and their corresponding OAM spectra. (b) The OAM purities as a function of
time.
By changing the order of the spatial modes for each frequency, the OAM values can be tuned. Figure
5.7(a)showsanSTWPcarryingdynamicallychangingOAMvaluesfrom+4to+1duringthe∼ 5.2-psperiod.
The complex profiles and OAM spectra are shown in Fig. 5.7(a1). The OAM spectrum as a function of time
is shown in Fig. 5.7(a2). All the STWPs shown above have monotonically increasing or decreasing OAM
values during their characteristic period. Figure 5.7(b) shows an example in which the OAM value decreases
from +4 to +1 during the first half of the period from 0 ps to 2.6 ps and subsequently increases from +1 to
+4 during the second half of the period from 2.6 ps to 5.2 ps.
We calculate the frequency chirp along the azimuthal direction to investigate the self-torque property of
the generated STWP [89]. As shown in Fig. 5.8, we choose two cases corresponding to the STWPs shown
in Fig. 5.6(b) and Fig. 5.7(b). The frequency chirp is estimated according to the experimentally measured
phase profiles at different time instants. As shown in Fig. 5.8(a), the frequency chirp decreases along the
52
Figure 5.7: Experimental results of an STWP carrying (a) decreasing OAM values from +4 to +1 and (b)
OAM values decreasing from +4 to +1 and increasing to +4 afterward. (a1, b1) The intensity and phase
profiles and the corresponding OAM spectra. (a2, b2) The OAM spectrum as a function of time.
azimuthaldirectionfortheSTWPcarryingadecreasingOAMvalue. AsshowninFig. 5.8(b), thefrequency
chirp is different at different time instants for the STWP carrying an OAM value that is not monotonically
changing.
Additional aspects of the dependence of these STWPs on their parameters are investigated in simulation
only. As shown in Fig. 5.9, we change the temporal pulse width using different numbers of frequency lines.
We note that in the simulation, the comb line frequency spacing is chosen to be 200 GHz, corresponding to
a 5-ps pulse period, which is slightly different than our experimental setup. As shown in Fig. 5.9, using 6
comb lines results in a train of pulses with a pulse duration of the order of 0.83 ps for each pulse, which is
estimatedbydividingthepulseperiodbythenumberoflines. For12and18comblines,thisvaluegoesdown
to 0.42 ps and 0.28 ps, respectively. We assign 6 pulse durations with 6 different OAM values by shaping
the frequency comb lines, where a temporal chirp is induced to the STWP [102]. Therefore, as shown in Fig.
5.9, the total temporal width of the envelope of the train is 2.52 (0.42× 6) ps and 1.68 (0.28× 6) ps when
using 12 and 18 comb lines, respectively.
53
Figure 5.8: The frequency chirp along the azimuthal direction of the experimentally generated STWPs.
(a) The case with a monotonically decreasing OAM value. (b) The case with a (b1) decreasing and (b2)
increasing OAM value during the period.
Figure 5.10 shows the simulated STWP carrying a nonlinearly varying OAM value that increases and
decreases. Such an STWP carrying nonlinearly changing OAM values could be expressed by its azimuthal
phase φ(t,θ ) = ℓ(t)θ [89], where θ is the azimuthal angle. The OAM value ℓ(t) is designed to be a roughly
parabolical function ℓ(t) = (6t/t
s
)
2
when t < t
s
/2 and ℓ(t) = (6(t
s
− t)/t
s
)
2
when t > t
s
/2, where t
s
is the
pulse period. The OAM spectrum as a function of time is shown in Fig. 5.10(a). We estimate the frequency
chirp along the azimuthal direction, as shown in Fig. 5.10(b). For the time instants of 0.92 ps, 1.75 ps, and
2.42ps, althoughthefrequencychirpsallincreasealongtheazimuthaldirectionastheOAMvalueincreases,
the slopes are different for the three curves.
54
Figure 5.9: Simulation results of the STWPs carrying decreasing OAM value from +6 to +1 with different
pulsewidthsinthetemporaldomain. TheSTWPsaresynthesizedwith(a)6frequencylines,(b)12frequency
lines, and (c) 18 frequency lines.
Figure5.10: SimulationresultsoftheSTWPcarryingaroughlyparabolicallychangingOAMvalue. (a)The
OAM spectrum as a function of time during a period. (b) Frequency chirps along the azimuthal direction of
the STWP at different time instants.
5.5 Simulation of STWPs with tunable group velocity
Figure 5.11(a) shows that for a pulsed Gaussian-like beam, each optical frequency carries multiple k
z
com-
ponents satisfying the spatial dispersion relation (k
2
x
+k
2
z
=(ω/c)
2
) [92]. For such a relation, k
x
and k
z
are
the wave vector along the x and z direction, respectively. The speed and the angular frequency of the light
are denoted by c and ω, respectively. As a result, the group velocities v
g
= ∂ω/∂k
z
of different frequencies
(f = ω/2π ) are different. Thus, there is a nonlinear phase shift on different frequencies after propagation
55
andthebeamshapebecomesbroader. AsshowninFig. 5.11(b),thediffraction-freeSTWPcanbegenerated
when the spatiotemporal spectrum of the light beam lies along the intersection of the light cone (the spatial
dispersion relation) and a spectral hyperplane parallel to the k
x
axis with a tilted angle Θ [92]:
(k
z
− k
0
)tanΘ+ k
0
=ω/c (5.1)
wherek
0
isthewavevectorofthecentralfrequencyoftheSTWP.Duetothiscorrelationbetweenspatialand
temporal spectral components, there is a linear relative phase delay ∆ φ = (∆ ωt− ∆ k
z
z) between different
temporal frequencies. Such a ∆ φ induces a time- and propagation-dependent interference between different
frequencies, thusleadingtoapositionchangeofthepulseenvelopeatdifferent z values. Basedonthespace-
timerelation,thegroupvelocityofthepulseisgivenby v
g
=∂ω/∂k
z
=ctanΘ,whichmeansitcanbetuned
bychangingthetiltangleΘ[93]. Inaddition, usingthespatialandtemporalspectralcomponentcorrelation
inEq. (5.1), thewavepacketcanbeexpressedinthereducedform E(x,z,t)=ψ (x,z− cttan(ϕ ))e
i(k0(z− ct))
,
whereψ isaspace-timeenvelope[92]. Thisresultsinanon-diffractingwavepacketwithanenvelopetraveling
at the group velocity v
g
=ctanΘ.
Figure 5.11: (a) For a pulsed Gaussian-like beam without space-time correlation, the group velocities are
different for different frequencies. Due to this, the pulse shape becomes broader after propagation. (b) For
an STWP with spatial and temporal spectral correlation, the spatiotemporal spectrum of the STWP follows
the intersection of a light cone with a plane tilted at an angle of Θ. The group velocity is the same for
different frequencies. The pulse shape remains the same after propagation.
Figure 5.12(a) shows the concept of generating the STWP by combining different frequencies, each
carrying multiple LG modes. It should be noted that we utilize discrete frequencies, which correspond to
56
the separate lines of an optical frequency comb [19]. Compared to the spectrally continuous light source, the
use and manipulation of the amplitude/phase of individual discrete frequency comb lines can enable unique
features, such as tuning the temporal pulse shape [117]. According to the relation between the frequency
and wave vector, each frequency comb line should contain a plane wave with a different spatial frequency.
The real parts of the electrical fields on each frequency along the x direction are sinusoidal waves. Such
plane waves can be represented by synthesizing multiple spatial modes from a complete orthogonal mode set
[118].
Figure 5.12: (a) Concept of generating the STWP by combining different frequencies, each carrying multiple
Laguerre-Gaussian modes. (b1-b3) Simulated spatiotemporal profiles of the generated STWP with different
groupvelocities. Thegroupvelocityisestimatedbycomparingthepulsepeakmovementatthetwodistances.
Three cases of the Θ values are chosen, resulting in the values of v
g
= ctanΘ that are lesser, greater
than the speed of light, or negative, as shown in Fig. 5.12(b). In the simulation, the STWP is generated by
E =
P
i
[exp(iω
i
t)
P
j
C
i,ℓj,pj
LG(z,ℓ
j
,p
j
)], where C
(
i,ℓ
j
,p
j
) is the complex weight of each LG mode, which
are obtained according to the correlation between the spatial and temporal spectral components. The pulse
57
profiles at different propagation distances and time instants can be obtained by changing the values of z and
t.
Foreachcase,thespatiotemporalprofilesalongthe xdirectionandtimetareshown. Thegroupvelocities
at different propagation distances are estimated as follows. We first simulate the pulse profile as a function
of time t at a specific propagation distance z
1
. The pulse peak t
1
is found according to the maximum
intensity point of the pulse envelope. Subsequently, the pulse profile after a small propagation distance
z
2
= z
1
+∆ z is simulated and the pulse peak t
2
is also found. The delay time ∆ t = t
2
− t
1
is estimated
by the movement of the pulse envelope’s maximum intensity. The v
g
of the generated STWP is estimated
by calculating v
g
= (z
2
− z
1
)∆ t [93]. We note that in the case of finite-energy signals, the pulse might be
temporally broadened, and its spatial shape might be distorted after propagation. Therefore, the estimated
group velocity might deviate from v = ctanΘ. The center location of the y axis is chosen (i.e., y = 0) and
the two propagation distances z
1
= 0 and z
2
= 0.1 m are shown. As shown in Fig. 5.12(b1), the beam
profiles indicate that the pulse peak has a time delay with ∆ t at z
1
and z
2
. This indicates that when the
designed v
g
is ∼ 0.58c with the theta of 30
◦ , the pulse is moving forward (i.e., ∆ t > 0) when propagating
from z
1
to z
2
. Similarly, as shown in Fig. 5.12(b2), the beam is also moving forward but with a smaller
time delay. This indicates that the v
g
in this case is larger so that it takes less time delay for the pulse to
propagate with the same distance. Finally, as shown in Fig. 5.12(b3), the pulse profile at z
1
has a time
delay compared to that at z
2
. This indicates that the pulse is moving backward (i.e., ∆ t < 0), resulting in
a negative v
g
.
Figure5.13showsthesimulatedgroupvelocityatdifferentpropagationdistancesofthegeneratedSTWP
bycombiningLGmodes. ThetiltedangleΘis30
◦ inFig. 5.13,resultinginatargetgroupvelocityof∼ 0.58c.
When using LG modes without the spatial and temporal spectral correlation, the generated beam has a v
g
close to the speed of light at different propagation distances [119]. The STWP generated using ideal plane
waves has a v
g
close to the target v
g
. When the STWP is generated by combining LG modes, the generated
beam has the v
g
as a function of the propagation distance. At the propagation distance near zero, the v
g
is
close to the target value. As the beam propagates, the v
g
deviates from the target value and becomes close
58
to c at large distances. This deviation could be due to that the spatial and temporal spectral correlation
could change as the LG modes diverge after propagation.
Figure 5.13: The simulated group velocity as a function of propagation distance without the spatial and
temporalspectralcorrelation,withthecorrelationbetweenfrequencyandLGmodes,andwiththecorrelation
between frequency and plane waves. As an example, Θ is set to be 30
◦ and the corresponding target value
of v
g
is∼ 0.58c.
Subsequently, we investigate the simulated group velocity of the generated STWP as a function of the
value of the parameter Θ. Figure 5.14(a) shows the v
g
at the propagation distance of z = 0 for different Θ values, which fits the target value of the function ctanΘ. As mentioned before, the v
g
is close to the target
valuewithinarangeofthepropagationdistance. Figure5.14(b)showstherangeofthepropagationdistance
that the v
g
deviates from the target values within 0.1c for different Θ values. When Θ is close to 45
◦ (i.e.,
the v
g
is close to c), this range tends to be larger. When Θ deviates from 45
◦ , the range tends to decrease.
This could be because that when the difference between v
g
and c is larger, more spatial modes (i.e., higher
spatial frequency) are required for the spatial and temporal spectral correlation [120]. Therefore, the spatial
and temporal spectral correlation could change at a shorter propagation distance as the LG modes diverge.
We also investigate the simulated v
g
at the propagation distance z = 0 when using a limited number of
LG modes with a maximum mode number (|ℓ|+2p+1), as shown in Fig. 5.14(c). In this figure, Θ is set
to be 40
◦ as an example. When the number of modes is limited, the estimated v
g
is close to the c. This
59
Figure 5.14: (a) The simulated group velocity at the propagation distance z =0 as a function of Θ. (b) The
range of propagation distance that the estimated group velocity remains within 0.1c from the target value
as a function of Θ. (c) The estimated group velocity at the propagation distance z =0 a different maximum
mode number (|ℓ|+2p+1) of the LG modes used.
could be because that when a limited number of modes are used, the generated STWP is close to a pulse
with a single spatial mode without the spatial and temporal spectral correlation. When more LG modes are
used, the v
g
value becomes closer to the target value. The v
g
value remains close to the target value when
an adequate number of modes are used.
60
5.6 STWP generation and propagation through multi-mode fiber
Figure 5.15 shows the concept of the generation of STWPs with reduced diffraction and a tunable group
velocity after MMF propagation. In general, an STWP can be generated by synthesizing optical fields
with different optical temporal and spatial frequency pairs ( f,k
z
) [92, 93]. As shown in Fig. 5.15(a), to
achieveatunablegroupvelocityoftheSTWP,aspecificlinearrelationship( f,k
z
)betweenthetemporaland
longitudinal spatial frequencies should be prescribed. The tilted angle of the linear function between 2πf/c
and k
z
is defined as Θ, where c is the speed of light in vacuum. Due to the relationship k
2
z
+k
2
r
=(2πf/c )
2
in free space, a function between 2πf/c and the radial spatial frequency k
r
can be obtained as shown in
Fig. 5.15(a). In general, a monochromatic Bessel mode in free space corresponds to a specific ( f,k
r
) pair.
Therefore, according to the function between 2πf/c and k
r
, an STWP can be generated by synthesizing
multiple frequencies, each carrying a specific Bessel mode with a unique k
r
value. The generated STWP can
be represented by [92]:
E(r,z,t)=exp[i(k
z1
z− 2πf
1
t)]
X
n
A
n
J
0
(k
rn
r)exp[i(n− 1)(∆ k
z
z− 2π ∆ ft)] (5.2)
where A
n
is the weight of each frequency and J
0
(.) is the 0th-order Bessel function. Due to the reduced
diffraction property of the Bessel modes and the linear relation between f
n
and k
zn
, the STWP expressed
by Eq. (5.2) represents a train of reduced diffraction wave packets that are periodic in both time and
distance. In addition, as the STWP propagates, a phase delay occurs between the neighboring frequencies
∆ φ = ∆ k
z
z− 2π ∆ ft. The STWP’s envelope propagates along the z direction with increasing time due to
the dynamic change of ∆ φ, resulting in a theoretical group velocity of v
g
= ∆ z/∆ t = 2π ∆ f∆ k
z
= ctanΘ.
Therefore, the group velocity of the STWP could be tuned by choosing different Θ values for the linear
function.
If the generated STWP is directly coupled into the MMF, the output beam tends to be distorted by the
randommodecouplingoftheMMF,asshowninFig. 5.15(b). Consequently, thelinearfunctionbetweenthe
f and k
z
components of the output STWP would not be satisfied, and the desired properties of the STWP
would be lost. To generate an STWP with reduced diffraction and a tunable group velocity at the MMF
61
Figure 5.15: Concept of (a) STWPs generated by synthesizing multiple frequency comb lines, each carrying
a unique Bessel mode to have reduced diffraction and a tunable group velocity; (b) propagating the STWP
through multi-mode fiber (MMF) without mitigating of the mode coupling; and (c) propagating the STWP
through MMF with pre-distortion to mitigate the mode coupling.
output,mitigationofthefibermodecouplingisrequired. Onemodecouplingmitigationmethodistodescribe
the fiber mode coupling with the transmission matrix H and apply pre-distortion according to the inverse
matrixH
− 1
[121]. AsshowninFig. 5.15(c),thedesiredBesselmodesondifferentfrequenciesarerepresented
by complex-weighted combinations of LG modes. We note that other modal basis could also potentially be
used. The complex weights on each frequency f
n
can be denoted by a vector [C
n,1
,C
n,2
,...,C
n,m
]
T
. The
pre-distortedinputonthisfrequencycanalsoberepresentedbyadifferentcombinationofLGmodeswiththe
weight vector [C
′
n,1
,C
′
n,2
,...,C
′
n,m
]
T
=H
− 1
· [C
n,1
,C
n,2
,...,C
n,m
]
T
. Therefore, after MMF propagation,
62
due to the mode coupling matrixH, “quasi” Bessel modes can be generated on their original frequencies as
combinations of LG modes.
The experimental setup for generating the STWP with reduced diffraction and a tunable group velocity
is shown in Fig. 5.16. The STWP generation and characterization parts are similar to the setup described
in Section 5.2. The generated STWP is coupled into a 1-m graded-index MMF. The STWP at the output
of the MMF is coupled into free space and detected using off-axis holography [106].
Figure5.16: ExperimentalsetupforgeneratinganSTWPwithreduceddiffractionandtunablegroupvelocity
aftermulti-modefiberpropagation. AFG:arbitraryfunctiongenerator;ECL:externalcavitylaser;LF:lensed
fiber; Col.: collimator; BS: beam splitter; SLM: spatial light modulator.
We first measure the mode coupling transmission matrix of the MMF. An example of the measured
complex transmission matrix of the MMF on one frequency is shown in Fig. 5.17(a). Different modes are
sequentially transmitted at the input. For each input mode, the complex output beam profile is captured
by off-axis holography and decomposed into multiple modes. The matrix entities are obtained according to
the complex weights of the mode decomposition. LG modes are used to describe the transmission matrix
of the fiber. Each input mode and output mode in Fig. 5.17(a) means a single LG mode with different
ℓ and p values. The measured mode coupling tends to be in different mode groups when a graded-index
63
MMF is used. By inverting the measured matrix, a transmission matrix for pre-distortion is generated, as
shown in Fig. 5.17(b). In this matrix, each input mode means a pre-distorted mode as a complex-weighted
combination of LG modes. Each output mode means a single LG mode. As shown in Fig. 5.17(b), the mode
coupling could be mitigated by the pre-distortion. We note that in our experiment only one polarization is
transmittedattheinputandmeasuredattheoutput. Themeasuredmatrixisnotthecompletetransmission
matrix and there tends to be power loss on the modes when the pre-distortion is used [122].
Figure 5.17: Measured complex transmission matrices of the MMF on one frequency (a) without and (b)
with pre-distortion.
Furthermore, we measure the mode coupling transmission matrices of the MMF on the six frequencies
that are used to generate the STWP. The amplitudes of the measured transmission matrices are shown in
64
Fig. 5.18(a). The measured mode coupling shows similar mode groups for different frequencies as shown in
Fig. 5.18(a). According to the measured transmission matrices, pre-distorted input modes are obtained by
inverting the matrices on each frequency to generate “quasi” Bessel modes with different wavenumbers. One
example of the calculated pre-distorted input intensity profiles at the fiber input is shown in Fig. 5.18(b) to
generate an STWP with Θ=45 .1
◦ at the fiber output.
Figure 5.18: (a) Amplitudes of the measured transmission matrices of the MMF on the six frequencies. (b)
Calculated intensity profiles of the pre-distorted modes on the six frequencies to generate an STWP with
Θ=45 .1
◦ .
Subsequently, we characterize the output STWP after the MMF propagation. We first investigate the
group velocity of the STWP by measuring the output beam intensity profiles |E(x,y = 0,τ )|
2
in free space
after the MMF propagation, as shown in Fig. 5.19. When an STWP with a superluminal group velocity is
transmitted through the MMF without pre-distortion, the output STWP is distorted by the mode coupling,
as shown in Fig. 5.19(a). Two profiles are measured with z = 0, right at the fiber output, and with z = 40
mm, after 40-mm further free-space propagation. The pulse peak of the STWP shows little movement after
65
further free-space propagation, which indicates that the pulse envelope of the generated STWP moves at a
speedsimilartothatofthereferencepulse. Asaresult,thedistortedSTWPhasagroupvelocityclosetothe
speed of light and the desired superluminal group velocity property is lost. The insets show the transverse
2D complex profiles at the corresponding reference delay and illustrate the distortion caused by the mode
coupling.
Two cases with mitigation of the fiber mode coupling with pre-distortion are shown in Figs. 5.19(b)
and (c). As shown in Fig. 5.19(b), the pulse peak of the generated STWP moves back by ∼ 0.6 ps after
the 40-mm further free-space propagation. This indicates that the pulse envelope of the generated STWP
moves faster than the reference pulse and corresponds to a superluminal group velocity of ∼ 1.0042c. As
the Θ value is set to be 45 .1
◦ , the designed group velocity is 1.0035c. Another STWP is generated after the
MMF propagation, as shown in Fig. 5.19(c). Its pulse peak moves forward by ∼ 0.4 ps after 40-mm further
free-spacepropagation. ThisindicatesthatthepulseenvelopeofthegeneratedSTWPmovesslowerthanthe
reference pulse and corresponds to a subluminal group velocity of∼ 0.9967c. The Θ value is set to be 44 .9
◦ ,
which results in a designed group velocity of 0.9965c. In both cases, the measured group velocities are close
but slightly deviate from the designed values. The deviation might be due to the imperfect decomposition
of the Bessel modes into the assigned number of LG modes, as well as due to the imperfect inversion of the
mode coupling.
We also investigate the reduced-diffraction property of the generated STWP. The measured beam radius
of an STWP (Θ = 44 .9
◦ ) as a function of further free-space propagation distant z is shown in Fig. 5.20(a).
The beam radius is obtained according to the measured time-averaged intensity profiles of the STWP at the
corresponding z positions. The beam radius is calculated at the point at which the intensity becomes half of
the maximum. These time-averaged intensity profiles are captured by the camera by blocking the reference
pulse. Two intensity profiles at z = 0 and z = 150 mm are shown in the insets as examples. After 150-mm
free-space propagation, the STWP generated at the fiber output remains a similar beam radius. The beam
radiusofanotherpulsewithafundamentalGaussianmodeofthefiberisshowninFig. 5.20(b). Inthiscase,
the input of the MMF is a pulse with a fundamental Gaussian mode. Due to the weak coupling between
the fundamental Gaussian mode and the other mode groups, the output of the MMF is also a pulse with
66
Figure 5.19: Measured beam profiles |E(x,y = 0,τ )|
2
in free space after MMF propagation for the STWPs
(a) with Θ = 45 .1
◦ without pre-distortion, (b) with Θ = 45 .1
◦ with pre-distortion, and (c) Θ = 44 .9
◦ with
pre-distortion. In each case, two beam profiles are shown with z = 0 (right at the fiber output) and z = 40
mm (with further 40-mm free-space propagation). The insets are the transverse 2-D complex beam profiles
with the corresponding reference delays.
the Gaussian mode of the MMF. During further 150-mm free-space propagation, the beam radius tends to
increaseasthepulsepropagates. Wenotethatthemeasuredbeamradiiinbothcasestendtoattaindiscrete
values, which might be due to the limited resolution caused by the pixel size of the camera. Compared to
the pulse in Fig. 5.20(b), the generated STWP at the MMF output has reduced diffraction.
67
Figure 5.20: Measured beam radius after further free-space propagation at the fiber output according to the
time-averaged intensity profiles for (a) an STWP with pre-distortion and a Θ value of 44 .9
◦ and (b) a pulse
with the fundamental Gaussian mode of the MMF. The insets show the time-averaged intensity profiles at
the corresponding z positions.
68
Chapter 6
Conclusion
This dissertation has discussed several topics about optical communications and optical space-time wave
packets using optical frequency combs and spatial modes. Specific examples include the KK-detected WDM
system using Kerr frequency combs, the optical Volterra filter using wave mixing and delays, the free-
space optical communications in the mid-IR using multiplexing techniques, and the STWP generation by
synthesizing frequency comb lines carrying spatial modes.
69
References
[1] P. Minzioni, C. Lacava, T. Tanabe, J. Dong, X. Hu, G. Csaba, W. Porod, G. Singh, A. E. Willner,
A. Almaiman, et al., “Roadmap on all-optical processing,” Journal of Optics, vol. 21, no. 6, p. 063001,
2019.
[2] A.E.Willner,A.Fallahpour,F.Alishahi,Y.Cao,A.Mohajerin-Ariaei,A.Almaiman,P.Liao,K.Zou,
A. N. Willner, and M. Tur, “All-optical signal processing techniques for flexible networks,” Journal of
Lightwave Technology, vol. 37, no. 1, pp. 21–35, 2019.
[3] D. Kedar and S. Arnon, “Urban optical wireless communication networks: the main challenges and
possible solutions,” IEEE Communications Magazine, vol. 42, no. 5, pp. S2–S7, 2004.
[4] H. Kaushal and G. Kaddoum, “Optical communication in space: Challenges and mitigation tech-
niques,” IEEE communications surveys & tutorials, vol. 19, no. 1, pp. 57–96, 2016.
[5] M. Yessenov, L. A. Hall, K. L. Schepler, and A. F. Abouraddy, “Space-time wave packets,” Advances
in Optics and Photonics, vol. 14, no. 3, pp. 455–570, 2022.
[6] Z.Zhao,H.Song,R.Zhang,K.Pang,C.Liu,H.Song,A.Almaiman,K.Manukyan,H.Zhou,B.Lynn,
etal.,“Dynamicspatiotemporalbeamsthatcombinetwoindependentandcontrollableorbital-angular-
momentausingmultipleoptical-frequency-comblines,” Nature communications, vol.11, no.1, p.4099,
2020.
[7] K. Pang, K. Zou, Z. Zhao, H. Song, Y. Zhou, M. Karpov, M. Yessenov, A. Shiri, H. Song, R. Zhang,
et al., “Experimental demonstration of dynamic spatiotemporal structured beams that simultaneously
exhibit two orbital angular momenta by combining multiple frequency lines, each carrying multiple
laguerre–gaussian modes,” Optics Letters, vol. 47, no. 16, pp. 4044–4047, 2022.
[8] M. Yessenov, J. Free, Z. Chen, E. G. Johnson, M. P. Lavery, M. A. Alonso, and A. F. Abouraddy,
“Space-timewavepacketslocalizedinalldimensions,” Nature Communications, vol.13, no.1, p.4573,
2022.
[9] A. Forbes, “Structured light from lasers,” Laser & Photonics Reviews, vol. 13, no. 11, p. 1900140,
2019.
[10] A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nature Photonics, vol. 15, no. 4,
pp. 253–262, 2021.
[11] A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,”
Advances in Optics and Photonics, vol. 3, no. 2, pp. 161–204, 2011.
[12] L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light
and the transformation of laguerre-gaussian laser modes,” Physical review A, vol. 45, no. 11, p. 8185,
1992.
[13] A. E. Willner, K. Pang, H. Song, K. Zou, and H. Zhou, “Orbital angular momentum of light for
communications,” Applied Physics Reviews, vol. 8, no. 4, p. 041312, 2021.
[14] R.L.PhillipsandL.C.Andrews, “Spotsizeanddivergenceforlaguerregaussianbeamsofanyorder,”
Applied optics, vol. 22, no. 5, pp. 643–644, 1983.
70
[15] G. Xie, C. Liu, L. Li, Y. Ren, Z. Zhao, Y. Yan, N. Ahmed, Z. Wang, A. J. Willner, C. Bao, et al.,
“Spatiallightstructuringusingacombinationofmultipleorthogonalorbitalangularmomentumbeams
with complex coefficients,” Optics Letters, vol. 42, no. 5, pp. 991–994, 2017.
[16] J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Physical review letters, vol. 58,
no. 15, p. 1499, 1987.
[17] J. Durnin, “Exact solutions for nondiffracting beams. i. the scalar theory,” JOSA A, vol. 4, no. 4,
pp. 651–654, 1987.
[18] R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher–order bessel
beams,” Optics express, vol. 17, no. 26, pp. 23389–23395, 2009.
[19] T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative kerr solitons in optical
microresonators,” Science, vol. 361, no. 6402, p. eaan8083, 2018.
[20] H.Guo,M.Karpov,E.Lucas,A.Kordts,M.H.Pfeiffer,V.Brasch,G.Lihachev,V.E.Lobanov,M.L.
Gorodetsky,andT.J.Kippenberg,“Universaldynamicsanddeterministicswitchingofdissipativekerr
solitons in optical microresonators,” Nature Physics, vol. 13, no. 1, pp. 94–102, 2017.
[21] J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li,
D. Hillerkuss, et al., “Coherent terabit communications with microresonator kerr frequency combs,”
Nature photonics, vol. 8, no. 5, pp. 375–380, 2014.
[22] A. E. Willner, A. Fallahpour, K. Zou, F. Alishahi, and H. Zhou, “Optical signal processing aided by
optical frequency combs,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 27, no. 2,
pp. 1–16, 2020.
[23] G.-W. Lu, A. Albuquerque, B. J. Puttnam, T. Sakamoto, M. Drummond, R. Nogueira, A. Kanno,
S. Shinada, N. Wada, and T. Kawanishi, “Pump-linewidth-tolerant optical wavelength conversion for
high-order qam signals using coherent pumps,” Optics Express, vol. 22, no. 5, pp. 5067–5075, 2014.
[24] H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-
the-art,” IEEE Communications Magazine, vol. 49, no. 9, pp. 56–62, 2011.
[25] D.-S.Ly-Gagnon, S.Tsukamoto, K.Katoh, andK.Kikuchi, “Coherentdetectionofopticalquadrature
phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology, vol. 24,
no. 1, p. 12, 2006.
[26] W. Freude, R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer,
M. Dreschmann, M. Huebner, et al., “Quality metrics for optical signals: Eye diagram, q-factor, osnr,
evm and ber,” in 2012 14th International Conference on Transparent Optical Networks (ICTON),
pp. 1–4, IEEE, 2012.
[27] T. Pfau, S. Hoffmann, and R. No´ e, “Hardware-efficient coherent digital receiver concept with feed-
forward carrier recovery for m-qam constellations,” Journal of Lightwave Technology, vol. 27, no. 8,
pp. 989–999, 2009.
[28] C.Liu, J.Pan, T.Detwiler, A.Stark, Y.-T.Hsueh, G.-K.Chang, andS.E.Ralph, “Jointdigitalsignal
processing for superchannel coherent optical communication systems,” Optics Express, vol. 21, no. 7,
pp. 8342–8356, 2013.
[29] L. Lundberg, M. Mazur, A. Mirani, B. Foo, J. Schr¨ oder, V. Torres-Company, M. Karlsson, and P. A.
Andrekson, “Phase-coherent lightwave communications with frequency combs,” Nature Communica-
tions, vol. 11, no. 1, p. 201, 2020.
[30] A. Mecozzi, C. Antonelli, and M. Shtaif, “Kramers–kronig coherent receiver,” Optica, vol. 3, no. 11,
pp. 1220–1227, 2016.
71
[31] C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization
multiplexing with the kramers-kronig receiver,” Journal of Lightwave Technology, vol. 35, no. 24,
pp. 5418–5424, 2017.
[32] X.Chen,C.Antonelli,S.Chandrasekhar,G.Raybon,A.Mecozzi,M.Shtaif,andP.Winzer,“Kramers–
kronig receivers for 100-km datacenter interconnects,” Journal of Lightwave Technology, vol. 36, no. 1,
pp. 79–89, 2018.
[33] M.Al-Qadi,G.Vedala,M.O’Sullivan,C.Xie,andR.Hui,“Qd-mll-basedsingle-sidebandsuperchannel
generation scheme with kramers–kronig direct detection receivers,” IEEE Photonics Journal, vol. 11,
no. 4, pp. 1–13, 2019.
[34] R. Rios-Muller, J. Estaran, J. Renaudier, and G. Charlet, “Dual polarization in-phase and quadrature
high speed submarine transmission with only two photodiodes, adcs, mzms and dacs,” in 2017 IEEE
Photonics Conference (IPC) Part II, pp. 1–2, IEEE, 2017.
[35] K. Zou, P. Liao, Y. Cao, A. Kordts, A. Almaiman, M. Karpov, M. H. P. Pfeiffer, F. Alishahi, A. Fal-
lahpour, M.Tur, et al., “Kramers–kronigdetectionoffour20gbaud16-qamchannelsusingkerrcombs
for a shared phase estimation,” Optics Letters, vol. 45, no. 7, pp. 1794–1797, 2020.
[36] P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. Pfeiffer, P. Trocha, S. Wolf,
V. Brasch, M. H. Anderson, et al., “Microresonator-based solitons for massively parallel coherent
optical communications,” Nature, vol. 546, no. 7657, pp. 274–279, 2017.
[37] M. Zhu, Y. Geng, X. Yi, F. Li, H. Ying, J. Zhang, W. Tang, and K. Qiu, “Large scale optical intercon-
nection using kerr frequency comb and direct-detection kramers-kronig receiver,” in 2018 Conference
on Lasers and Electro-Optics (CLEO), pp. 1–2, IEEE, 2018.
[38] A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency estimation in intradyne reception,”
IEEE Photonics Technology Letters, vol. 19, no. 6, pp. 366–368, 2007.
[39] P. Liao, C. Bao, A. Kordts, M. Karpov, M. H. Pfeiffer, L. Zhang, A. Mohajerin-Ariaei, Y. Cao,
A. Almaiman, M. Ziyadi, et al., “Dependence of a microresonator kerr frequency comb on the pump
linewidth,” Optics Letters, vol. 42, no. 4, pp. 779–782, 2017.
[40] S. Khaleghi, O. F. Yilmaz, M. R. Chitgarha, M. Tur, N. Ahmed, S. R. Nuccio, I. M. Fazal, X. Wu,
M. W. Haney, C. Langrock, et al., “High-speed correlation and equalization using a continuously
tunable all-optical tapped delay line,” IEEE Photonics Journal, vol. 4, no. 4, pp. 1220–1235, 2012.
[41] M. S. Rasras, I. Kang, M. Dinu, J. Jaques, N. Dutta, A. Piccirilli, M. A. Cappuzzo, E. Y. Chen, L. T.
Gomez, A. Wong-Foy, et al., “A programmable 8-bit optical correlator filter for optical bit pattern
recognition,” IEEE Photonics Technology Letters, vol. 20, no. 9, pp. 694–696, 2008.
[42] M. Ziyadi, M. R. Chitgarha, A. Mohajerin-Ariaei, S. Khaleghi, A. Almaiman, Y. Cao, M. J. Willner,
M. Tur, L. Paraschis, C. Langrock, et al., “Optical nyquist channel generation using a comb-based
tunable optical tapped-delay-line,” Optics Letters, vol. 39, no. 23, pp. 6585–6588, 2014.
[43] M. Ziyadi, M. R. Chitgarha, S. Khaleghi, A. Mohajerin-Ariaei, A. Almaiman, J. Touch, M. Tur,
C. Langrock, M. M. Fejer, and A. E. Willner, “Tunable optical correlator using an optical frequency
comb and a nonlinear multiplexer,” Optics Express, vol. 22, no. 1, pp. 84–89, 2014.
[44] D.Rafique,M.Mussolin,M.Forzati,J.M˚ artensson,M.N.Chugtai,andA.D.Ellis,“Compensationof
intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Optics
Express, vol. 19, no. 10, pp. 9453–9460, 2011.
[45] F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities
usingafrequency-domainvolterraseriesequalizer,” Optics Express,vol.20,no.2,pp.1360–1369,2012.
[46] G.Stepniak,J.Siuzdak,andP.Zwierko,“Compensationofavlcphosphorescentwhitelednonlinearity
by means of volterra dfe,” IEEE Photonics Technology Letters, vol. 25, no. 16, pp. 1597–1600, 2013.
72
[47] M.Zhu,J.Zhang,S.Hu,X.Yi,B.Xu,M.Xiang,andK.Qiu,“Complexityreductionwithasimplified
mimo volterra filter for pdm-twin-ssb pam-4 transmission,” Journal of Lightwave Technology, vol. 38,
no. 4, pp. 769–776, 2019.
[48] G. sharan Yadav, C.-Y. Chuang, K.-M. Feng, J.-H. Yan, J. Chen, and Y.-K. Chen, “Reducing compu-
tation complexity by using elastic net regularization based pruned volterra equalization in a 80 gbps
pam-4 signal for inter-data center interconnects,” Optics Express, vol. 28, no. 26, pp. 38539–38552,
2020.
[49] Z. Zalevsky, E. Gur, and D. Mendlovic, “Optical implementation of second-order nonlinear volterra
operators with use of triple correlation,” JOSA A, vol. 18, no. 1, pp. 164–169, 2001.
[50] K. Zou, P. Liao, H. Zhou, A. Fallahpour, A. Minoofar, A. Almaiman, F. Alishahi, M. Tur, and
A. E. Willner, “Tunable optical second-order volterra nonlinear filter using wave mixing and delays to
equalize a 10–20 gbaud 4-apsk channel,” Optics Letters, vol. 46, no. 6, pp. 1325–1328, 2021.
[51] M. Schetzen, The Volterra and Wiener theories of nonlinear systems. Krieger Publishing Co., Inc.,
2006.
[52] A. Malacarne, G. Meloni, G. Berrettini, N. Sambo, L. Pot` ı, and A. Bogoni, “Optical multicasting
of 16qam signals in periodically-poled lithium niobate waveguide,” Journal of Lightwave Technology,
vol. 31, no. 11, pp. 1797–1803, 2013.
[53] H.-F. Ting, K.-Y. Wang, J. R. Stroud, K. G. Petrillo, H. Sun, A. C. Foster, and M. A. Foster,
“Wavelength multicasting through four-wave mixing with an optical comb source,” Optics Express,
vol. 25, no. 8, pp. 9276–9284, 2017.
[54] A. Malacarne, F. Fresi, J. Klamkin, and L. Pot` ı, “Versatile offset-free 16-qam single dual-drive iq
modulator driven by binary signals,” Optics Letters, vol. 37, no. 19, pp. 4149–4151, 2012.
[55] N. V. Wheeler, A. M. Heidt, N. K. Baddela, E. N. Fokoua, J. R. Hayes, S. R. Sandoghchi, F. Poletti,
M. N. Petrovich, and D. J. Richardson, “Low-loss and low-bend-sensitivity mid-infrared guidance in a
hollow-core–photonic-bandgap fiber,” Optics Letters, vol. 39, no. 2, pp. 295–298, 2014.
[56] M. Yu, Y. Okawachi, A. G. Griffith, N. Picqu´ e, M. Lipson, and A. L. Gaeta, “Silicon-chip-based
mid-infrared dual-comb spectroscopy,” Nature Communications, vol. 9, no. 1, p. 1869, 2018.
[57] M.J.Walsh,R.K.Reddy,andR.Bhargava,“Label-freebiomedicalimagingwithmid-irspectroscopy,”
IEEE Journal of Selected Topics in Quantum Electronics, vol. 18, no. 4, pp. 1502–1513, 2012.
[58] A. Arnulf, J. Bricard, E. Cur´ e, and C. V´ eret, “Transmission by haze and fog in the spectral region
0.35 to 10 microns,” JOSA, vol. 47, no. 6, pp. 491–498, 1957.
[59] A. Soibel, M. W. Wright, W. H. Farr, S. A. Keo, C. J. Hill, R. Q. Yang, and H. Liu, “Midinfrared
interband cascade laser for free space optical communication,” IEEE Photonics Technology Letters,
vol. 22, no. 2, pp. 121–123, 2009.
[60] X. Pang, R. Schatz, M. Joharifar, A. Udalcovs, V. Bobrovs, L. Zhang, X. Yu, Y.-T. Sun, G. Maisons,
M. Carras, et al., “Direct modulation and free-space transmissions of up to 6 gbps multilevel signals
with a 4.65-mu m quantum cascade laser at room temperature,” Journal of Lightwave Technology,
vol. 40, no. 8, pp. 2370–2377, 2022.
[61] H. Dely, T. Bonazzi, O. Spitz, E. Rodriguez, D. Gacemi, Y. Todorov, K. Pantzas, G. Beaudoin,
I. Sagnes, L. Li, et al., “10 gbit s- 1 free space data transmission at 9 µ m wavelength with unipolar
quantum optoelectronics,” Laser & Photonics Reviews, vol. 16, no. 2, p. 2100414, 2022.
[62] X. Pang, H. Dely, R. Schatz, D. Gacemi, M. Joharifar, T. Salgals, A. Udalcovs, Y.-T. Sun, Y. Fan,
L. Zhang, et al., “11 gb/s lwir fso transmission at 9.6 µ m using a directly-modulated quantum cascade
laser and an uncooled quantum cascade detector,” in 2022 Optical Fiber Communications Conference
and Exhibition (OFC), pp. 1–3, IEEE, 2022.
73
[63] P. S. Cho, G. Harston, K.-D. F. B¨ uchter, D. Soreide, J. M. Saint Clair, W. Sohler, Y. Achiam, and
I. Shpantzer, “Optical homodyne rz-qpsk transmission through wind tunnel at 3.8 and 1.55 micron via
wavelength conversion,” in Atmospheric Propagation VI, vol. 7324, pp. 73–84, SPIE, 2009.
[64] Y.Su,W.Wang,X.Hu,H.Hu,X.Huang,Y.Wang,J.Si,X.Xie,B.Han,H.Feng, etal.,“10gbpsdpsk
transmissionoverfree-spacelinkinthemid-infrared,” Optics Express,vol.26,no.26,pp.34515–34528,
2018.
[65] W. Wang, Y. Zheng, X. Xie, Y. Su, X. Huang, T. Duan, and W. Zhao, “5 gbaud qpsk coherent
transmission in the mid-infrared,” Optics Communications, vol. 466, p. 125681, 2020.
[66] A. F. Molisch, Wireless communications. John Wiley & Sons, 2012.
[67] H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen,
K. M. Birnbaum, et al., “100 tbit/s free-space data link enabled by three-dimensional multiplexing of
orbital angular momentum, polarization, and wavelength,” Optics Letters, vol. 39, no. 2, pp. 197–200,
2014.
[68] E.Ip, N.Bai, Y.K.Huang, E.Mateo, F.Yaman, M.J.Li, S.Bickham, S.Ten, J.Li˜ nares, C.Montero,
et al., “88× 3× 112-gb/s wdm transmission over 50 km of three-mode fiber with inline few mode fiber
amplifier,” in European Conference and Exposition on Optical Communications, pp. Th–13, Optica
Publishing Group, 2011.
[69] C.A.Brackett,“Densewavelengthdivisionmultiplexingnetworks: Principlesandapplications,” IEEE
Journal on Selected areas in Communications, vol. 8, no. 6, pp. 948–964, 1990.
[70] R. G. Van Uden, R. A. Correa, E. A. Lopez, F. Huijskens, C. Xia, G. Li, A. Sch¨ ulzgen, H. De Waardt,
A. Koonen, and C. M. Okonkwo, “Ultra-high-density spatial division multiplexing with a few-mode
multicore fibre,” Nature Photonics, vol. 8, no. 11, pp. 865–870, 2014.
[71] J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur,
et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Na-
ture Photonics, vol. 6, no. 7, pp. 488–496, 2012.
[72] L. Gailele, L. Maweza, A. Dudley, B. Ndagano, C. Rosales-Guzman, and A. Forbes, “Multiplexing of
spatial modes in the mid-ir region,” in Laser Resonators, Microresonators, and Beam Control XIX,
vol. 10090, pp. 86–91, SPIE, 2017.
[73] K. Zou, K. Pang, H. Song, J. Fan, Z. Zhao, H. Song, R. Zhang, H. Zhou, A. Minoofar, C. Liu, et al.,
“High-capacityfree-spaceopticalcommunicationsusingwavelength-andmode-division-multiplexingin
the mid-infrared region,” Nature Communications, vol. 13, no. 1, p. 7662, 2022.
[74] G. Bosco, V. Curri, A. Carena, P. Poggiolini, and F. Forghieri, “On the performance of nyquist-wdm
terabit superchannels based on pm-bpsk, pm-qpsk, pm-8qam or pm-16qam subcarriers,” Journal of
Lightwave Technology, vol. 29, no. 1, pp. 53–61, 2011.
[75] M. Chou, J. Hauden, M. Arbore, and M. Fejer, “1.5-µ m-band wavelength conversion based on
difference-frequency generation in linbo 3 waveguides with integrated coupling structures,” Optics
Letters, vol. 23, no. 13, pp. 1004–1006, 1998.
[76] S. J. Savory, “Digital filters for coherent optical receivers,” Optics Express, vol. 16, no. 2, pp. 804–817,
2008.
[77] N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-gaussian mode
sorter,” Nature Communications, vol. 10, no. 1, p. 1865, 2019.
[78] T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, et al., “Massive
individualorbitalangularmomentumchannelsformultiplexingenabledbydammanngratings,” Light:
Science & Applications, vol. 4, no. 3, pp. e257–e257, 2015.
74
[79] A. Bismuto, Y. Bidaux, S. Blaser, R. Terazzi, T. Gresch, M. Rochat, A. Muller, C. Bonzon, and
J. Faist, “High power and single mode quantum cascade lasers,” Optics Express, vol. 24, no. 10,
pp. 10694–10699, 2016.
[80] H. Yang, R. Q. Yang, J. Gong, and J.-J. He, “Mid-infrared widely tunable single-mode interband
cascade lasers based on v-coupled cavities,” Optics Letters, vol. 45, no. 10, pp. 2700–2703, 2020.
[81] V.Raghunathan,D.Borlaug,R.R.Rice,andB.Jalali,“Demonstrationofamid-infraredsiliconraman
amplifier,” Optics Express, vol. 15, no. 22, pp. 14355–14362, 2007.
[82] J. Mishra, M. Jankowski, A. Y. Hwang, H. S. Stokowski, T. P. McKenna, C. Langrock, E. Ng, D. Hey-
dari, H. Mabuchi, A. H. Safavi-Naeini, et al., “Ultra-broadband mid-infrared generation in dispersion-
engineered thin-film lithium niobate,” Optics Express, vol. 30, no. 18, pp. 32752–32760, 2022.
[83] C.-Y. Cho and Y.-F. Chen, “Compactly efficient cw 3 to 4.5 µ m wavelength tunable mid-infrared laser
in optically pumped semiconductor laser with intracavity opo,” IEEE Journal of Selected Topics in
Quantum Electronics, vol. 28, no. 1: Semiconductor Lasers, pp. 1–6, 2021.
[84] M. Xu, Y. Zhu, F. Pittal` a, J. Tang, M. He, W. C. Ng, J. Wang, Z. Ruan, X. Tang, M. Kuschnerov,
et al., “Dual-polarization thin-film lithium niobate in-phase quadrature modulators for terabit-per-
second transmission,” Optica, vol. 9, no. 1, pp. 61–62, 2022.
[85] H. Hu, E. Palushani, M. Galili, H. C. H. Mulvad, A. Clausen, L. K. Oxenløwe, and P. Jeppesen,
“640 gbit/s and 1.28 tbit/s polarisation insensitive all optical wavelength conversion,” Optics express,
vol. 18, no. 10, pp. 9961–9966, 2010.
[86] B. Sun, P. S. Salter, C. Roider, A. Jesacher, J. Strauss, J. Heberle, M. Schmidt, and M. J. Booth,
“Four-dimensionallightshaping: manipulatingultrafastspatiotemporalfociinspaceandtime,” Light:
Science & Applications, vol. 7, no. 1, pp. 17117–17117, 2018.
[87] D. Auston, “Transverse mode locking,” IEEE Journal of Quantum Electronics, vol. 4, no. 6, pp. 420–
422, 1968.
[88] M. Brambilla, M. Cattaneo, L. Lugiato, R. Pirovano, F. Prati, A. Kent, G.-L. Oppo, A. Coates,
C.Weiss, C.Green, et al., “Dynamicaltransverselaserpatterns.i.theory,” Physical Review A,vol.49,
no. 2, p. 1427, 1994.
[89] L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. San Rom´ an, D. E. Couch, A. Liu,
E. Pisanty, M. Lewenstein, et al., “Generation of extreme-ultraviolet beams with time-varying orbital
angular momentum,” Science, vol. 364, no. 6447, p. eaaw9486, 2019.
[90] L. Chen, W. Zhu, P. Huo, J. Song, H. J. Lezec, T. Xu, and A. Agrawal, “Synthesizing ultrafast optical
pulses with arbitrary spatiotemporal control,” Science Advances, vol. 8, no. 43, p. eabq8314, 2022.
[91] K.Pang,K.Zou,H.Song,M.Karpov,M.Yessenov,Z.Zhao,A.Minoofar,R.Zhang,H.Song,H.Zhou,
et al., “Synthesis of near-diffraction-free orbital-angular-momentum space-time wave packets having a
controllable group velocity using a frequency comb,” Optics Express, vol. 30, no. 10, pp. 16712–16724,
2022.
[92] H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nature Photonics,
vol. 11, no. 11, pp. 733–740, 2017.
[93] H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group ve-
locities in free space,” Nature communications, vol. 10, no. 1, p. 929, 2019.
[94] B.KiblerandP.B´ ejot,“Discretizedconicalwavesinmultimodeopticalfibers,” PhysicalReviewLetters,
vol. 126, no. 2, p. 023902, 2021.
75
[95] C. Ma, J. Di, Y. Zhang, P. Li, F. Xiao, K. Liu, X. Bai, and J. Zhao, “Reconstruction of structured
laser beams through a multimode fiber based on digital optical phase conjugation,” Optics Letters,
vol. 43, no. 14, pp. 3333–3336, 2018.
[96] J. Carpenter, B. J. Eggleton, and J. Schr¨ oder, “Complete spatiotemporal characterization and optical
transfer matrix inversion of a 420 mode fiber,” Optics letters, vol. 41, no. 23, pp. 5580–5583, 2016.
[97] A. Minoofar, K. Zou, K. Pang, H. Song, M. Karpov, M. Yessenov, Z. Zhao, H. Song, H. Zhou, X. Su,
et al., “Generation of oam-carrying space-time wave packets with time-dependent beam radii using a
coherent combination of multiple lg modes on multiple frequencies,” Optics Express, vol. 30, no. 25,
pp. 45267–45278, 2022.
[98] K. Zou, X. Su, M. Yessenov, K. Pang, N. Karapetyan, M. Karpov, H. Song, R. Zhang, H. Zhou, T. J.
Kippenberg, et al., “Tunability of space-time wave packet carrying tunable and dynamically changing
oam value,” Optics Letters, vol. 47, no. 21, pp. 5751–5754, 2022.
[99] K. Zou, K. Pang, H. Song, M. Karpov, X. Su, R. Zhang, H. Song, H. Zhou, T. J. Kippenberg, M. Tur,
et al., “Generating a space-time pulse in free space after multimode fiber propagation in which fiber
modal coupling is mitigated, divergence is reduced, and group velocity is tuned,” in CLEO: Science
and Innovations, pp. STh5E–1, Optica Publishing Group, 2022.
[100] K. Zou, H. Song, Z. Zhao, K. Pang, A. Minoofar, X. Su, H. Zhou, R. Zhang, H. Song, N. Hu, et al.,
“Space–time light sheet with a controllable group velocity and reduced diffraction by combining mul-
tiple frequencies each carrying multiple laguerre–gaussian modes,” Optics Communications, vol. 520,
p. 128477, 2022.
[101] M. Mounaix, N. K. Fontaine, D. T. Neilson, R. Ryf, H. Chen, J. C. Alvarado-Zacarias, and J. Car-
penter, “Time reversed optical waves by arbitrary vector spatiotemporal field generation,” Nature
communications, vol. 11, no. 1, p. 5813, 2020.
[102] A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Optics Communications, vol. 284,
no. 15, pp. 3669–3692, 2011.
[103] G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur, “Efficient and mode
selective spatial mode multiplexer based on multi-plane light conversion,” Optics express, vol. 22,
no. 13, pp. 15599–15607, 2014.
[104] D. Cruz-Delgado, S. Yerolatsitis, N. K. Fontaine, D. N. Christodoulides, R. Amezcua-Correa, and
M. A. Bandres, “Synthesis of ultrafast wavepackets with tailored spatiotemporal properties,” Nature
Photonics, vol. 16, no. 10, pp. 686–691, 2022.
[105] T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of laguerre–gaussian
beamsgeneratedviacomplex-amplitudemodulationusingphase-onlyspatiallightmodulators,” Optics
letters, vol. 34, no. 1, pp. 34–36, 2009.
[106] E.Cuche, P.Marquet, andC.Depeursinge, “Simultaneousamplitude-contrastandquantitativephase-
contrast microscopy by numerical reconstruction of fresnel off-axis holograms,” Applied optics, vol. 38,
no. 34, pp. 6994–7001, 1999.
[107] P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator
detuning measurements in microresonator frequency combs,” Nature communications, vol. 6, no. 1,
p. 5668, 2015.
[108] R. Slav´ ık, F. Parmigiani, J. Kakande, C. Lundstr¨ om, M. Sj¨ odin, P. A. Andrekson, R. Weerasuriya,
S. Sygletos, A. D. Ellis, L. Gr¨ uner-Nielsen, et al., “All-optical phase and amplitude regenerator for
next-generation telecommunications systems,” Nature Photonics, vol. 4, no. 10, pp. 690–695, 2010.
[109] T.Schibli,I.Hartl,D.Yost,M.Martin,A.Marcinkeviˇ cius,M.Fermann,andJ.Ye,“Opticalfrequency
combwithsubmillihertzlinewidthandmorethan10waveragepower,” Nature Photonics,vol.2,no.6,
pp. 355–359, 2008.
76
[110] X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, “Active capture and stabilization of temporal solitons
in microresonators,” Optics letters, vol. 41, no. 9, pp. 2037–2040, 2016.
[111] G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Ro-
manato,“Acompactdiffractivesorterforhigh-resolutiondemultiplexingoforbitalangularmomentum
beams,” Scientific reports , vol. 8, no. 1, p. 10248, 2018.
[112] C. Rosales-Guzm´ an, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a
single hologram,” Journal of Optics, vol. 19, no. 11, p. 113501, 2017.
[113] Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant
laguerre–gaussian beams,” Journal of Optics A: Pure and Applied Optics, vol. 6, no. 11, p. 1005, 2004.
[114] X. Zhong, Y. Zhao, G. Ren, S. He, and Z. Wu, “Influence of finite apertures on orthogonality and
completeness of laguerre-gaussian beams,” IEEE Access, vol. 6, pp. 8742–8754, 2018.
[115] A. D’Errico, R. D’Amelio, B. Piccirillo, F. Cardano, and L. Marrucci, “Measuring the complex orbital
angular momentum spectrum and spatial mode decomposition of structured light beams,” Optica,
vol. 4, no. 11, pp. 1350–1357, 2017.
[116] A. Ruelas, S. Lopez-Aguayo, and J. C. Guti´ errez-Vega, “A hankel transform distribution algorithm
for paraxial wavefields with an application to free-space optical beam propagation,” Journal of Optics,
vol. 18, no. 9, p. 095605, 2016.
[117] F.Ferdous,H.Miao,D.E.Leaird,K.Srinivasan,J.Wang,L.Chen,L.T.Varghese,andA.M.Weiner,
“Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nature Photonics,
vol. 5, no. 12, pp. 770–776, 2011.
[118] C. Schulze, A. Dudley, D. Flamm, M. Duparre, and A. Forbes, “Measurement of the orbital angular
momentumdensityoflightbymodaldecomposition,” New Journal of Physics,vol.15,no.7,p.073025,
2013.
[119] R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” science, vol. 326, no. 5956,
pp. 1074–1077, 2009.
[120] M. Yessenov, L. Mach, B. Bhaduri, D. Mardani, H. E. Kondakci, G. K. Atia, M. A. Alonso, and A. F.
Abouraddy, “What is the maximum differential group delay achievable by a space-time wave packet
in free space?,” Optics express, vol. 27, no. 9, pp. 12443–12457, 2019.
[121] J.Carpenter,B.J.Eggleton,andJ.Schr¨ oder,“110x110opticalmodetransfermatrixinversion,” Optics
express, vol. 22, no. 1, pp. 96–101, 2014.
[122] W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, “Spatiotemporal control
of light transmission through a multimode fiber with strong mode coupling,” Physical review letters,
vol. 117, no. 5, p. 053901, 2016.
77
Abstract (if available)
Abstract
Optical frequency combs, as optical sources having multiple discrete equidistant frequency lines in their spectrum, have gained much interest. Frequency combs have many valuable characteristics, including providing tens or hundreds of narrow-linewidth, mutually coherent, and equidistant optical carriers. Such mutually coherent comb lines can be used in applications like optical communication systems, optical signal processing systems, and optical space-time wave packet (STWP) generation.
On the other hand, optical beams carrying spatial modes have been under intense investigation in various areas. One example is the light wave can carry orbital angular momentum (OAM). In general, OAM can be utilized to characterize the “twisted” helical phase front of a light beam when its wavevector spirals around the beam axis. Such a helical phase front of an OAM-carrying beam is usually represented by $\exp(i\ell\theta)$, where $\theta$ is the azimuthal coordinate and $\ell$ is the number of $2\pi$ phase shifts in the phase profile of the beam. OAM beams with different $\ell$ values are orthogonal with each other while propagating coaxially. The spatial modes can be used in both the optical communication systems and the STWP generation.
This thesis will discuss (i) optical $4\times20$-Gbaud 16-quadrature-amplitude-modulated wavelength-division-multiplexing communication systems using Kerr frequency combs and Kramers-Kronig detection to perform shared phase recovery, (ii) an optical second-order Volterra filter using wave mixing and delays to equalize 10-20-Gbaud 4-amplitude-phase-shift-keying signals, (iii) 300-Gbit/s free-space optical communications in the mid-infrared using wavelength- and mode-division multiplexing, and (iv) several types of optical STWPs generated using frequency comb lines and spatial modes.
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University of Southern California Dissertations and Theses
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Creator
Zou, Kaiheng
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Core Title
optical communication systems and space-time wave packet generation using frequency combs and spatial modes
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Degree Conferral Date
2023-08
Publication Date
06/07/2023
Defense Date
04/24/2023
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free-space optics,mid-infrared,OAI-PMH Harvest,optical communications,optical frequency combs,optical signal processing,optical space-time wave packets.,orbital angular momentum,spatial modes
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Tags
free-space optics
mid-infrared
optical communications
optical frequency combs
optical signal processing
optical space-time wave packets.
orbital angular momentum
spatial modes