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A Microwave Photonic Interference Canceller: Architectures, Systems, and Integration

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A Microwave Photonic Interference
Canceller: Architectures, Systems, and
Integration
Matthew P Chang
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Electrical Engineering
Adviser: Professor Paul R. Prucnal
April 2017
ProQuest Number: 10259699
All rights reserved
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All rights reserved.
Abstract
This thesis is a comprehensive portfolio of work on a Microwave Photonic SelfInterference Canceller, a specialized optical system designed to eliminate interference
from radio-frequency (RF) receivers. The novelty and value of the microwave photonic
system lies in its ability to operate over bandwidths and up to frequencies that are
orders of magnitude larger than what is possible using existing RF technology. The
work begins, in 2012, with a discrete fiber-optic microwave photonic canceller, which
prior work had demonstrated as a proof-of-concept, and culminates, in 2017, with
the first ever monolithically integrated microwave photonic canceller. With an eye
towards practical implementation, the thesis establishes novelty through three major
project thrusts (Fig. 1):
1. Extensive RF and System Analysis to develop a full understanding of
how, and through what mechanisms, MPCs affect an RF receiver. The first
investigations of how a microwave photonic canceller performs in an actual
wireless environment and a digital radio are also presented.
2. New Architectures to improve the performance and functionality of MPCs,
based on the analysis performed in Thrust 1. A novel balanced microwave
photonic canceller architecture is developed and experimentally demonstrated,
and shows significant improvements in link gain, noise figure, and dynamic
range. Its main advantage is its ability to suppress common-mode noise and
reduce noise figure by increasing the optical power.
3. Monolithic Integration of the microwave photonic canceller into a photonic
integrated circuit. This thrust presents the progression of integrating individual
discrete devices into their semiconductor equivalent, as well as a full functional
and RF analysis of the first ever integrated microwave photonic canceller.
iii
Figure 1: The overview and timeline of this thesis.
iv
Acknowledgements
There are countless people that I need to thank for their friendship, love, wisdom,
mentorship, help, encouragement, butt-kicking, perspective, and overall selflessness.
When all is said and done, each of you were the most important elements of my time
at Princeton.
First, to my thesis advisor, Paul Richard Prucnal, I have nothing but the greatest
admiration, respect, and gratitude. You were once asked why, after such a long and
successful career, you continued to push and work so hard. Without hesitation, you
grinned, and responded, “I like winning.” That stuck with me. I could write a
thousand things, but if there are two that I will always be thankful to you for, it is
first, your endless and incredible generosity; and second, that you lit a fire in me. I
learned tremendously about myself, what makes my internal optics run, what I want
to do, and who I want to become under your guidance. I was influenced, both directly
and by example, both in the lab and (more importantly) outside the lab, by you. You
taught me how to lead, how to interact with people in the real world (because face
it, I’ve been in school my whole life), how to focus and get the job done, and how
to treat deserving people with the utmost respect. In your lab, I learned what hard
work and loyalty truly meant, and the enormous trust and faith you placed in me
allowed me to grow both professionally and personally. Your generosity in providing
me unbelievable opportunities after graduate school, all on your own dime and time,
was the most selfless thing I’ve ever experienced outside of family. Speaking of which,
you welcomed me into your family, both lab and personal, in a way that made my
PhD experience truly unlike any other. You are a fantastic cook and a fantastic
father. It was such a privilege being your student. Words simply cannot express the
full extent of my gratitude.
To my lab family, both past and present, you guys defined my PhD, and without
you, my last six years would have been completely different: Dr. Mable Fok, Dr. John
v
Chang, Alex Tait, Mitch Nahmias, Dr. Ben Wu, Dr. Yue Tian, Dr. Zhenxing Wang,
Maddie Lu, Eric Charles Blow, Philip Ma, Thomas Ferriera de Lima, Callie Woods,
Chia-Lo “lil Ben” Lee, Abanti Basak, Shreyas Lakhtakia, Ellen Zhou, Jenny Sun,
Allie Wu, Monica Lu, Ishana Gopaul, and my fearless post-doc, Bhavin J. Shastri.
I learned enormously from each of you, while also enjoying your company, and I
think that’s pretty darn special. I can’t wait to see what great things each of you
accomplish, and I am sure that the carbon pipe will reappear. Thank you for joining
me on the journey.
To my infamous lunch group: Yasmin Afsar, Omer Malik, Mei-Chai Zheng, Coach
Harvey Cheng, Daniel Jiang, Huai-An Chin, Eric J. Zhang, Genny Plant, Jin-Sung
Kim, and Andrew Kim, I would not have made it through my PhD without all of
you. To all those laughs at the lunch table, coffee flips, drink nights, halo nights
(ASSIST?!), basketball games, Golden State victories, jeans, snaps, WeChats and
many other ridiculous shenanigans, it has been so much fun. Growing with you has
been one of the best parts of graduate school. A special shout-out goes to Sunny
Bunny Pandey, my roommate of 4 years and 4 rooms, thank you for your whiskey,
companionship, cooking episodes, and welcoming me into your family in Nepal. To
all of you, here’s to many more, wherever they may be.
I’ve made many friends throughout Princeton that have also shaped my PhD
both socially and professionally: Janam Jhaveri, Ken-Bomb Nagamatsu, Akil WordDaniels (and by extension, Zoey), Sabbir Liakat, Nyan Aung, Evan Peterson, Loan
Le, Arvind Ravikumar, Ryan Jock, Shoham Bhadra, Eric Mills, Bhadri Lalgudi
Visweswaran, Tracy Tsai, Jen-Tang Lu, Carolyn Chou, and Miri Kim. I’d also like to
acknowledge Pat Watson, Bert Harrop, Joe Palmer, Yong Sun, and Claire Gmachl’s
entire research group for their invaluable help in the cleanroom, as well as Larry and
Barry from the SEAS machine shop. Thank you Alex (Calcado) Michaud for being so
on point and pleasant to work with during my tenure with the Graduate Engineering
vi
Council, you were the real MVP. It was also a pleasure working with Cecilia York
and Warren Rieutort-Louis to organize and pull off several successful MelodEEs. A
shout-out goes to Lisa Lewis, Dorothy Coakley, Linda Dreher, and Lidia Himantono
Stokman for being an All-Star administrative team, especially in the 11th hour, and
just good people that are fun to talk to.
To my thesis readers and FPO committee: Profs Mable Fok, Claire Gmachl, Gerard Wysocki, Kaushik Sengupta, and Paul, thank you for your guidance throughout
my PhD and for (hopefully, at the time of this writing) not failing me.
To my industry collaborators and partner funding agencies: Rob Holland and
Elias Kpodzo at L-3 Telemetry East; Andy McCandless, Yanhua Deng, Brian Fix,
and Josh Lala at Bascom-Hunter; Andrew Hofmaier at the Army; Pietro Contu at
PhoeniX; Francisco Soares and Moritz Baier at HHI; and MIRTHE, your assistance
has been invaluable.
Basketball is one of my greatest passions. When experiments and life failed catastrophically, I always took comfort in knowing that there was at least one constant
that had not changed. To anyone who has ever played basketball with me, there are
far too many to name, thank you for letting me feel like I was and still am a baller.
Ball is life and it sure don’t lie.
To Mindy (thought I forgot about you, huh?), Jenny Rose, Katie, and Tobi, thank
you for putting up with me for so long! Despite being terrified of you for about a
year, Mindy, I found friendship in the most unlikely of places. My (and Bearby’s)
last six months in Princeton were somehow fun, quirky, embarrassing, scary, hilarious,
eye-opening, thrilling, and awesome all in one, in large part because of you and Paul.
I promise I won’t disappear, as I’m sure this is just Phase I of your Master Plan.
Thank you for allowing me to be a part of your family. Final prediction: Raven.
My mom, dad, and sister have been my rock my whole life, including the PhD.
Throughout all of the ups and downs, my family has kept me grounded, sane, convii
fident, and loved. My parents worked harder than I ever have and suffered through
hardships that my sister and I never will, to make sure that we would have the opportunities that they never did. My greatest passions were introduced to me by you,
and the values and morals that define who I am are a reflection of you. I truly am
standing on the shoulders of giants. Sarah, I wish you the best of luck in your future
endeavors, but for the next 7 years I will be fully taking advantage of the fact that
I am the only doctor of the sibling-hood. I love you guys with everything I have.
Here’s to continuing our journey together.
viii
To my parents.
ix
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
1 Introduction
1
1.1
The Problem of Wireless Interference . . . . . . . . . . . . . . . . . .
1
1.2
Interference Cancellation
. . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Microwave Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4
Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.5
Major Thrusts of this Thesis
17
. . . . . . . . . . . . . . . . . . . . . .
2 Key Performance Parameters
20
2.1
Cancellation Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2
Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3
Link Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.4
Noise Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.5
Dynamic Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3 Discrete Microwave Photonic Cancellers
3.1
EAM Microwave Photonic Canceller
x
. . . . . . . . . . . . . . . . . .
79
80
3.2
Balanced-Architecture System . . . . . . . . . . . . . . . . . . . . . .
3.3
Wireless Self-Interference Cancellation . . . . . . . . . . . . . . . . . 110
3.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4 An Integrated Optical Delay Line
93
132
4.1
Principles of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.3
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.4
Incorporation in an MPC . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5 An Integrated Microwave Photonic Canceller
155
5.1
Material Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.2
Design and Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.3
Functional Performance . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.4
Radio-Frequency Performance . . . . . . . . . . . . . . . . . . . . . . 178
5.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6 Conclusion and Outlook
201
A Nelder-Mead Simplex Algorithm
204
A.1 Main script: NelderMeadSimplex.m . . . . . . . . . . . . . . . . . . . 205
A.2 subfunction: reflect.m . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
A.3 subfunction: expand.m . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.4 subfunction: contract.m . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.5 subfunction: shrink.m
. . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.6 subfunction: resetsimplex.m . . . . . . . . . . . . . . . . . . . . . . . 212
A.7 subfunction: getSimplexVol.m . . . . . . . . . . . . . . . . . . . . . . 213
Bibliography
214
xi
List of Tables
1.1
Overview of Electronic Interference Cancellers . . . . . . . . . . . . .
15
2.1
Strategies to improve link gain and the effect of modulation technique.
51
2.2
Noise sources and noise figure relationships in an externally modulated
link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.3
Nonlinear Components from a directly modulated laser. . . . . . . . .
68
2.4
Nonlinear Components from an MZM-modulated link . . . . . . . . .
69
2.5
IIP3 and OIP3 of externally modulated links. . . . . . . . . . . . . .
71
3.1
Experimentally measured and extracted EAM Parameters . . . . . .
91
3.2
Key Performance Parameters of the EAM canceller. . . . . . . . . . .
92
3.3
Link Parameters of the balanced MPC. . . . . . . . . . . . . . . . . . 102
3.4
Comparison of Key Performance Parameters . . . . . . . . . . . . . . 109
4.1
Parameters and values used in the simulation of CPO and XGM in an
SOA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1
Comparison of Optical Materials. The more stars the better. . . . . . 157
5.2
Comparison of MPCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.3
Variable definitions and values . . . . . . . . . . . . . . . . . . . . . . 187
5.4
IMPC RF Characterization Summary . . . . . . . . . . . . . . . . . . 196
5.5
Comparison of Key Performance Parameters . . . . . . . . . . . . . . 198
xii
A.1 Coefficients of the Nelder-Mead Simplex Algorithm . . . . . . . . . . 204
xiii
List of Figures
1
The overview and timeline of this thesis. . . . . . . . . . . . . . . . .
1.1
(a) The projected growth of revenue for RF front-end components,
iv
dominated by filters. Taken from [1]. (b) The rise in the number of
distinct-band filters. Source: Triquint. . . . . . . . . . . . . . . . . .
2
1.2
RF Front-End architecture for the 700-900 MHz bands. . . . . . . . .
4
1.3
Examples of self-interference. (a) Transmitter leakage (b) Co-located
antenna systems (c) Adjacent channel interference (d) Full-Duplex
Communications (e) Simultaneous transmit/sense in Cognitive Radio.
7
1.4
A general model of self-interference cancellation. . . . . . . . . . . . .
9
1.5
A general analog optical link, showing RF input and output, and optical signal processing and distribution in the middle. . . . . . . . . .
1.6
A standard RF signal is essentially narrowband when modulated onto
an optical carrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7
14
The first microwave photonic self-interference cancellation system.
Taken from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9
13
A 16-tap active analog interference canceller on an RF circuit board.
Taken from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8
12
16
A multitap microwave photonic self-interference cancellation system.
Taken from [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.10 The overview and timeline of this thesis. . . . . . . . . . . . . . . . .
19
xiv
2.1
Cancellation Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2
Cancellation Measurement Methods . . . . . . . . . . . . . . . . . . .
23
2.3
Linear Interference Canceller . . . . . . . . . . . . . . . . . . . . . . .
25
2.4
Amplitude and Phase Mismatch Curves . . . . . . . . . . . . . . . . .
29
2.5
Cancellation as a function of frequency . . . . . . . . . . . . . . . . .
32
2.6
Operating Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.7
Components of Link Gain . . . . . . . . . . . . . . . . . . . . . . . .
35
2.8
Impedance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.9
Laser Diode Direct Modulation . . . . . . . . . . . . . . . . . . . . .
40
2.10 Mach-Zehnder Modulator . . . . . . . . . . . . . . . . . . . . . . . .
43
2.11 Mach-Zehnder Modulator . . . . . . . . . . . . . . . . . . . . . . . .
45
2.12 photodiode Response . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.13 Example Link Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.14 Noise Figure Concept . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.15 Noise Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2.16 Gain Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.17 Components of Noise Figure . . . . . . . . . . . . . . . . . . . . . . .
63
2.18 Nonlinear Components in an Optical Link . . . . . . . . . . . . . . .
65
2.19 Nonlinear devices in an Optical Link . . . . . . . . . . . . . . . . . .
66
2.20 IIP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.21 Fundamental, IIP2, IIP3 vs. Bias for an MZM-Modulated Link . . . .
72
2.22 Fundamental, OIP2, OIP3 vs. Bias for an MZM-Modulated Link . . .
73
2.23 SFDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
2.24 Link Tradeoff of Bias Angle . . . . . . . . . . . . . . . . . . . . . . .
77
2.25 Link Tradeoff of Vπ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.1
Single-Ended Architecture . . . . . . . . . . . . . . . . . . . . . . . .
80
3.2
EAM Canceller Schematic . . . . . . . . . . . . . . . . . . . . . . . .
82
xv
3.3
EAM Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.4
EAM Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3.5
EAM canceller equalized . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.6
EAM canceller equalized . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.7
RF Equalizer Response . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.8
RF Equalizer Comparison . . . . . . . . . . . . . . . . . . . . . . . .
89
3.9
EAM Transmission Curve . . . . . . . . . . . . . . . . . . . . . . . .
90
3.10 EAM Gain and Noise Figure . . . . . . . . . . . . . . . . . . . . . . .
92
3.11 Balanced Link Concept . . . . . . . . . . . . . . . . . . . . . . . . . .
94
3.12 Balanced Architecture Schematic . . . . . . . . . . . . . . . . . . . .
95
3.13 Experimental cancellation of the balanced interference canceller tuned
to from 2 - 8 GHz in 1 GHz increments. Inset shows cancellation of an
in-band interferer and recovery of an SOI. . . . . . . . . . . . . . . . 100
3.14 Comparison of the RF gain vs. input optical power for single-ended
and balanced MPCs. The data points correspond to experimental measurements.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.15 Comparison of the noise figure vs. RX tap input optical power for
the single-ended and balanced MPC architectures, assuming 30 dB of
RIN suppression. The data points correspond to experimental measurements.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.16 Noise figure slope vs. input optical power. Regions where each type of
noise dominate are explicitly shown.
. . . . . . . . . . . . . . . . . . 108
3.17 Noise figure vs. input optical power for multi-tap MPCs. 30 dB of
RIN suppression is assumed in the balanced case. . . . . . . . . . . . 110
3.18 Power Delay Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.19 Passively Suppressed Channel . . . . . . . . . . . . . . . . . . . . . . 114
xvi
3.20 Block diagram of the WMPC. System inputs and outputs are underlined. αi and τi represent the attenuation and delay of the ith tappeddelay line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.21 Diagram of the outdoor antenna setup, courtesy of Bascom-Hunter
Technologies. Inset: photo of the actual setup. . . . . . . . . . . . . . 117
3.22 Overlay of the WMPC output signal spectra without cancellation
(green dashed curve), with LOS cancellation (blue dotted curve) and
with both LOS and the multipath cancellation (red curve). SOI =
signal of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.23 Self-interference cancellation as measured by a network analyzer for
(A) only LOS cancellation, (B) both LOS and first multipath cancellation and (C) LOS cancellation without the multipath-generating reflector. Line 1 shows the difference between cancellation achieved with
and without the multipath cancellation in the presence of a reflector,
and line 2 demarcates the instantaneous bandwidth where 30 dB of
cancellation is achievable. . . . . . . . . . . . . . . . . . . . . . . . . 120
3.24 (Left) Coherence bandwidth of a self-interference channel and (Right)
number of taps required to cancel self-interference across a 20 MHz
instantaneous bandwidth as a function of the suppression of the LOS
signal. Results are shown for different passive suppression techniques
and RF environments. Taken from [5]. . . . . . . . . . . . . . . . . . 122
3.25 MPC inserted into the analog front-end of the full-duplex mode X310
USRP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.26 The hallway setup used to test the digital radio with MPC. . . . . . . 126
3.27 BER of the SOI as a function of Signal-to-interference ratio at a center
frequency of (a) 2.21 GHz and (b) 2.46 GHz. . . . . . . . . . . . . . . 127
3.28 (Top) Cancellation depth and (bottom) amplitude and phase mismatch.128
xvii
4.1
How CPO results in a dispersion
. . . . . . . . . . . . . . . . . . . . 136
4.2
Simulated gain of the output probe signal as a function of microwave
frequency. Four different pump power and SOA bias current combinations are plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3
Experimental setup used to measure the optical gain and time advance
of the SOA. PC = polarization controller. τ = RF signal’s time of flight
and α = RF gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.4
Relative time advance up to 4 GHz RF frequency as a function of (a)
Ppump (ISOA = 80 mA) and (b) ISOA (Ppump = 5 dBm). Time advance
is measured relative to the signal’s time of flight at Ppump = 0 dBm in
(a) and ISOA = 25 mA in (b). . . . . . . . . . . . . . . . . . . . . . . 142
4.5
Normalized RF gain up to 4 GHz RF frequency as a function of (a)
Ppump (ISOA = 80 mA) and (b) ISOA (Ppump = 5 dBm). Gain is normalized to Ppump = 0 dBm in (a) and ISOA = 25 mA in (b). . . . . . 143
4.6
(a) RF gain and (b) time advance as a function of ISOA and Ppump at
600 MHz RF frequency. RF Gain contours are overlaid on both plots.
Time advance contours are shown in (b) only. . . . . . . . . . . . . . 144
4.7
Output from an oscilloscope while biasing the device along the +3 dB
(top set of curves) and -3 dB (bottom set of curves) gain contours. No
normalization of the amplitude was performed.
4.8
. . . . . . . . . . . . 145
Experimental setup. λ1 = 1550.12 nm, λ2 = 1553.33 nm. PC = Polarization Controller, EDFA = Erbium-Doped Fiber Amplifier, VOA
= Variable Optical Attenuator BPF = Bandpass Filter. . . . . . . . . 148
4.9
(a) Broadband and (b) narrowband cancellation results. The SOI is
recovered after canceling the in-band interference.
xviii
. . . . . . . . . . 149
4.10 (Top) The measured amplitude and phase mismatch between the selfinterference channel and the reference tap from 700 MHz to 1100 MHz.
(Bottom) Measured and simulated cancellation based on the amplitude and phase mismatch. More negative cancellation corresponds to
greater interference cancellation. . . . . . . . . . . . . . . . . . . . . . 151
4.11 (a) Progression of the Nelder-Mead simplex algorithm to adaptively
cancel interference using the SOA bias current and input optical power.
(b) The system is able to locate the interference minimum through 83
ps of delay range. Each curve is averaged over 5 algorithm runs. Insets
show receiver output spectra after 0, 40, and 60 algorithm iterations.
153
5.1
Cross-sectional view of a hybrid evanescent device. Taken from [6].
5.2
Various techniques to monolithically integrate active-passive waveguides. Taken from [7].
5.3
. 158
. . . . . . . . . . . . . . . . . . . . . . . . . . 158
(Top) Block diagram. PD = photodiode. (Middle) Circuit layout on
PhoeniX OptoDesigner. (Bottom) Microscope image of the fabricated
circuit. The white boxes demarcate the actual devices in the block
diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4
Experimental setups used to characterize cancellation using (a) an arbitrary signal of interest and interferer and (b) only a swept-CW interferer.164
5.5
Progression from no cancellation (red, farthest curve) to optimal cancellation (blue, closest curve). . . . . . . . . . . . . . . . . . . . . . . 165
5.6
Measured cancellation across all FDD LTE and WiFi bands. Bands are
plotted in order of ascending center frequency. The gray, thicker bars
indicate the spectral region defined by each band. The red, thinner
bars demarcate the region over which at least −30 dB of cancellation
depth can be achieved at a fixed IMPC configuration. The width of
the bars are magnified by 10x for ease of viewing. . . . . . . . . . . . 167
xix
5.7
Cancellation depth (blue curve) over a sample of LTE and WiFi bands.
The gray areas indicate the spectral region defined by each band. The
red line is a reference line marking −30 dB cancellation depth. . . . . 168
5.8
Relative phase and gain of the reference signal as a function of the SOA
control bias currents. Phase shift at (a) 1.25 GHz and (b) 3 GHz RF
frequency. Gain shift at (c) 1.25 GHz and (d) 3 GHz RF frequency. All
values are measured relative to the phase and amplitude at (20 mA,
20 mA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.9
Independent phase shift range as a function of RF frequency. The
independent gain range is 10 dB, centered upon 13 dB.
. . . . . . . . 172
5.10 Cancellation performance as a function of the SOA control biases for
bandwidths of (a) 65 MHz, (b) 20 MHz, and (c) 1 MHz. The zoomin magnifies the optimal point in the 1 MHz bandwidth case. Center
frequency is 1.963 GHz in all cases. . . . . . . . . . . . . . . . . . . . 174
5.11 Adaptive self-interference cancellation using the Nelder-Mead Simplex
algorithm at a center frequency of 1.963 GHz and 20 MHz bandwidth.
(a) Progression of the simplex as it converges to the optimum point
and (b) output interference power at each iteration of the algorithm.
Insets show the cancellation depths at the specified iteration of the
algorithm. The gray region corresponds to the bandwidth of interest.
175
5.12 Various test setups used to characterize the IMPC. (a) Network analyzer for S-Parameter measurements; (b) Noise figure measurement
using an excess noise ratio (ENR) source, low noise amplifier (LNA),
and a noise figure meter; (c) Linearity measurements using two signal
generators and a spectrum analyzer. . . . . . . . . . . . . . . . . . . . 179
5.13 The circuit (a) |S21| as a function of laser bias and (b) |S11| for a laser
bias of 25 mA. The RX SOA bias is 40 mA. . . . . . . . . . . . . . . . 181
xx
5.14 RF gain as a function of RX laser bias and RX SOA bias at 3 GHz
RF frequency. The dashed box highlights an operating region that
corresponds to increased optical feedback. . . . . . . . . . . . . . . . 182
5.15 Noise figure of the IMPC. Experimental measurements at (a) 1 GHz,
(b) 2 GHz, and (c) 3 GHz. Simulated noise figure at (d) 1 GHz, (e)
2 GHz, and (f) 3 GHz. Bands of high noise figure are highlighted in
dashed regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.16 Measured IIP3 as a function of RF frequency for laser biases of 15 mA,
25 mA, and 35 mA. Each data point is an average of 5 measurements
and the error bars indicate the sample standard deviation. Dashed-line
fits correspond to 40 dB/ decade and 0 dB/ decade. The arrows along
the x-axis indicate the relaxation oscillation resonance frequencies for
each of the bias currents. . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.17 Measured IIP3 as a function of laser bias at 3 GHz RF frequency. Each
data point is an average of 5 measurements and the error bars indicate
the sample standard deviation. The dashed line is a theoretical fit
generated from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.18 Measured P1dB as a function of laser bias (RF frequency = 3 GHz) and
RF frequency (Laser Bias = 35 mA). Each data point is an average
of 5 measurements and the error bars indicate the sample standard
deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.19 SFDR as a function of laser bias (RF frequency = 3 GHz) and RF
frequency (Laser Bias = 25 mA). Above 3 GHz, noise figure simulations
were used together with IIP3 measurements to determine SFDR. Each
data point is an average of 5 measurements and the error bars indicate
the sample standard deviation. . . . . . . . . . . . . . . . . . . . . . . 195
6.1
The overview and timeline of this thesis. . . . . . . . . . . . . . . . . 202
xxi
Chapter 1
Introduction
In this chapter, background information on wireless interference, interference cancellation, and microwave photonics is provided to establish a foundation for the remainder
of the thesis. A brief overview of prior work is also presented to provide readers with
an understanding of where this thesis draws its origins from and where it breaks new
ground.
1.1
The Problem of Wireless Interference
Wireless interference is the single greatest problem in radio-frequency (RF) communications because of the very nature of our shared transmission medium. Our
communications infrastructure is engineered to avoid interference by multiplexing
radio resources such as time (time-division duplexing, TDD), code (code-division duplexing, CDD), and frequency (frequency-division duplexing, FDD1 ) [9]. For decades,
these techniques have successfully permitted multiple users to communicate without
interfering, but they come at a cost: RF spectrum, the universal currency of wireless
communications. As data rates and the number of connected devices continue to grow
1
In the United States, spectrum allocation began with the passing of the Radio Act of 1912,
spurred on in part by the sinking of the Titanic, where emergency radio transmissions were lost due
to interference.
1
(b)
(a)
Figure 1.1: (a) The projected growth of revenue for RF front-end components, dominated by filters. Taken from [1]. (b) The rise in the number of distinct-band filters.
Source: Triquint.
exponentially [10], the limited RF spectrum has become extremely congested, and the
price for bandwidth has soared into the billions USD for 10’s of MHz [11]. Similarly,
the number of communication bands and the market for RF front-end components,
particularly filters, the gateways to different bands, has followed suite (see Fig. 1.1).
Adding more bands and filters may be a temporary solution, but it is not sustainable. Most RF components in our communications infrastructure, like filters,
operate in a single, fixed frequency band (typical bandwidth 20 MHz), meaning that
each band requires a dedicated chain of RF components [1]. However, in the LongTerm Evolution Advanced (LTE) standard alone, there are 44 bands spanning from
450 MHz (Band 31) up to 3600 MHz (Band 22). As a result, for wireless equipment to
operate across multiple LTE bands, their RF front-ends, like those in Fig. 1.2, have
grown extremely complex and expensive. Concurrently, limited circuit board real estate and profit margins in size-sensitive platforms like handsets mean that hardware
designers have to sacrifice functionality for the expanding RF front-end.
2
With the impending arrival of the 5G standard in the next few years, new methods
to operate in the presence of interference, as well as scalable hardware to implement
these new methods, are badly needed. 5G projections indicate at least an order
of magnitude increase in capacity per link. In addition, 5G heralds the arrival of
the Internet of Things (IoT), where an estimated 26 - 50 billion new devices will
be added to a global network of connected cars, smart homes, consumer wearables,
medical equipment, and industrial equipment by 2020 [12]. Any new radio technology
will need to be able to flexibly allocate spectral resources on demand and smoothly
integrate heterogeneous networks, bands, and standards [13]. For hardware, this
translates into wideband and tunable signal processing on a level that today’s RF
technology simply cannot provide.
In this thesis, we present a method of self-interference cancellation instead of interference avoidance, using microwave photonics, instead of traditional RF electronics.
Interference cancellation can increase spectral efficiency, reduce network complexity,
and lower latency in radio access networks (RAN) [14]. It provides an additional
line of defense that can be deployed in tandem with existing interference avoidance
techniques. Of course, it is also of great interest in military and defense applications.
Microwave photonics exhibits the key property that it can operate over extremely wide
bandwidths and operating frequencies. By combining these technologies, a microwave
photonic interference canceller (MPC) can not only increase spectral efficiency over
wide bandwidth channels, but also help enable a true multiband transceiver and
future-proof them to newly defined bands and frequencies.
1.2
Interference Cancellation
Interference cancellation refers to the process of actively removing an interference
signal from a corrupted signal, leaving only the desired signal of interest (SOI). The
3
Low-Noise
Amplifiers
B29 (700 DL)
R1 (700 LD)
R1 (700 UD)
B28 (700 LD)
Antenna
B28 (700 UD)
B12 (700)
GSM PA
Switch
Antenna
Tuner
Switch
B20 (800)
WCDMA PA
B5 (850)
LTE PA
Power Amplifiers
B8 (900)
Filters
Duplexers
Figure 1.2: RF Front-End architecture for the 700-900 MHz bands.
most familiar form of interference cancellation in everyday life is the noise-canceling
headphone, which rejects constant and periodic acoustic noise from the external environment, preserving the fidelity of sound produced internally by the headphones.
The need for interference cancellation arises when interference avoidance techniques
are insufficient and, in some cases, not applicable. For example, in same-channel fullduplex communications, where a communication system simultaneously transmits and
receives on the same frequency, filters are no longer applicable and the strong transmitter signal leaks into the receiver, easily overwhelming the weaker SOI arriving
from distant nodes.
Interference cancellation is often confused for spectral filtering, but the two techniques are different and should be used together, rather than as substitutes, to address
different components of interference. The main advantage of interference cancellation
is its ability to address both in-band (i.e. inside the channel) and out-of-band (i.e.
4
outside the channel) interference; meanwhile, filters only reject out-of-band interference. More specifically, interference cancellation nulls interference without harming
the SOI, regardless of the spectral characteristics of either signal. In contrast, filters
indiscriminately reject (pass) signals outside (inside) of its passband, regardless of
whether it is the SOI or the interference. However, the advantage of interference
cancellation can only be leveraged if the system has prior knowledge of the interferer,
which often translates into increased system complexity. On the other hand, filters
can reject out-of-band interference without any knowledge of the interference.
The performance of an interference canceller depends strongly upon how the interference couples into the receiver and any apriori knowledge about the interferer. A
priori knowledge consists of anything, from the exact waveform of the interference to
its statistical nature, and will in general dictate the signal processing that is required
to cancel the interference. In the most general scenario, the RX signal consists of an
SOI mixed together with an unknown interferer
r(t) = s(t) + n(t)
(1.1)
where s(t) is the SOI and n(t) is the unknown interference signal. The purpose of
the interference canceller is to extract the interference from the combined signal and
subsequently subtract the extracted interference from the total signal. The resulting
output is
y(t) = s(t) + {n(t) − np (t)}
(1.2)
where np (t) is the extracted interference signal, and the term in brackets represents
the residual interference.
At the most basic level, interference can be divided into two categories: known
and unknown. When the interference signal is unknown, statistical methods must
be used to distinguish the interference from the SOI. This is known as blind source
5
separation. Assuming that the SOI and the interference are statistically independent,
techniques such as principal component analysis (PCA) and independent component
analysis (ICA) are often used. These techniques aim to separate the total signal into
two or more components, demarcated by their statistical properties. An interference
canceller would then eliminate the components unrelated to the SOI. There is much
research and interest in the topic of blind source separation, for both commercial and
military interest, and we refer the readers to these sources [15–17]. Herein, we focus
on the situation where the interference signal is either known or has been successfully
extracted.
1.2.1
Self-Interference Cancellation
A specific type of known interference is self-interference. Self-interference is unintentionally generated by one’s own system, and therefore, a reference copy of the interference signal is locally available. The most popular example of self-interference is in
same-channel full-duplex communications, when a transmitter (TX) and a receiver
(RX) simultaneously operate on the same frequency. Full-duplex communications has
been a long sought after goal in wireless communications for its potential to double
spectral efficiency, because the RX and TX no longer have to occupy separate frequency bands for uplink and downlink in FDD systems. In TDD systems, full-duplex
communications could cut latency in half and eliminate the complicated synchronization procedures between communication systems [2, 5, 14, 18–21]. Given the high cost
of spectrum, the potential upside for successfully implementing full-duplex is tremendous, and companies have been started focusing solely on this objective. Despite these
rewards, same-channel full-duplex communications has not been successfully deployed
in commercial networks because, without the protection of filters (remember, samechannel), the self-interference generated by a full-duplex system is enormous. For
example, receiving an SOI at -90 or -100 dBm power when the transmitter is out6
(a)
(c)
(b)
Adjacent Channel
Interference
TX leakage
TX
RX
Co-located Antennas
fc
Frequency
(e)
(d)
Full-Duplex
Communications
Cognitive Radio
Figure 1.3: Examples of self-interference. (a) Transmitter leakage (b) Co-located
antenna systems (c) Adjacent channel interference (d) Full-Duplex Communications
(e) Simultaneous transmit/sense in Cognitive Radio.
putting at +30 or +40 dBm would require > 130 dB of self-interference cancellation!
Other very relevant examples of self-interference are shown in Fig. 1.3.
Because of the sheer power of self-interference, it must typically be canceled in
three stages: passive suppression, active analog cancellation, and digital cancellation [5, 18, 21, 22]. Passive suppression takes advantage of the path loss between the
TX and RX, including antenna directionality and polarization, and is very effective
at suppressing the line-of-sight self-interference, but not so much reflections and multipath [5]. In active analog cancellation, a conditioned canceling signal is generated
and combined with the corrupt RX signal, nulling just the self-interference. This
technique takes advantage of a priori knowledge of the self-interference, but must be
adaptive, since the self-interference incident upon the RX will constantly change due
7
to the dynamic RF environment. In digital cancellation, the self-interference channel
is periodically characterized and used to produce a digital nulling signal. While most
flexible, digital cancellation relies on passive suppression and analog cancellation to
weaken the self-interference so that it does not saturate the analog-to-digital converter
(ADC), which has limited dynamic range. For example, a high-end 16-bit ADC will
have less than 96 dB of dynamic range. If the self-interference is 96 dB stronger than
the SOI, then the SOI will be buried below the quantization noise floor and lost.
As mentioned up previously, in full-duplex communications, the self-interference is
typically over 120 dB stronger than the SOI. In this thesis, we focus specifically
on active analog cancellation.
A model for self-interference cancellation is shown in Fig. 1.4. Here, because
a reference of the transmitted signal is available, the purpose of the interference
canceller is to estimate the self-interference channel, HSI (ω), and then replicate the
self-interference channel using its internal analog circuitry. Then, the reference signal is passed through the replicated channel, HIC (ω), to reproduce the actual selfinterference signal as accurately possible. The replicated self-interference signal is
inverted and added to the actual received signal so that the output of the interference
canceller can be expressed as
Y (ω) = S(ω) + HSI (ω) − HIC (ω) N (ω)
(1.3)
where Y (ω), S(ω), and N (ω) are the Fourier transforms of the output, SOI, and
transmitted signal, respectively. Keeping in mind that the self-interference channel
can include both wireless and wired (internal to the transceiver) components, in most
practical systems, the active analog cancellation system would optimally target the
highest power components of the self-interference, leaving the more numerous, but
weaker components of the self-interference to digital cancellation. To give a sense
8
A
RX Signal
LNA
Interference
Channel
|HSI(ω)|
C
D
+
B Channel
Estimation
|HIC(ω)|
HPA
Transmitter
TX Signal
A
Receiver
C
B
D
A1
A2
Δt
Figure 1.4: A general model of self-interference cancellation.
of what the channel estimation entails, to first order, the self-interference channel
will attenuate and delay the transmitted signal, but more complex channel effects,
such as multipath, reflections within the RF circuitry, amplifier noise, and nonlinear
distortion are also present [2,19]. The ability of the interference canceller to estimate
and mimic these effects determines the maximum amount of interference cancellation
achievable. We quantify how well the interference canceller must estimate the selfinterference channel in section 2.1 and we discuss the wireless self-interference channel
in more detail in section 3.3.
Besides the ability to accurately reproduce an arbitrary wireless channel, there
are a few other properties that an analog canceller should possess. First, it should
possess wideband operation and be frequency tunable. Because an interference
canceller is an extra component in the front-end, it could risk further bloating the
RF front-end. Frequency tunability and wideband operation ensure that an analog
canceller can be tuned to operate across multiple channels, but also that it can meet
the bandwidth demands of 5G. Second, an analog canceller should possess low la9
tency tunability. The latency required to tune the analog canceler adds overhead
to the communication system, reducing its efficiency. The coherence time of a typical
indoor channel, or the time it takes for the channel to change appreciably, is approximately 100 ms [2], so a canceller designed for this environment should ideally have a
tuning latency much less than 1 ms. Third, being an RF front-end component, any
analog canceller should also possess good RF properties, such as noise figure, loss,
and dynamic range. The dynamic range is particularly important for an interference
canceller, because of the need to handle both weak SOIs as well as very strong interferers. We discuss RF key performance parameters in Chapter 2 and analyze the RF
performance of different microwave photonic cancellers throughout this thesis.
1.3
Microwave Photonics
Despite the omnipresence of digital electronics, there has recently been a strong resurgence of interest in RF analog signal processing, primarily because digitizing an analog
signal with high fidelity is extremely demanding. It requires a large number of bits to
achieve high dynamic range and a large sampling rate to represent high bandwidths.
These tall technical requirements have made it cost prohibitive to implement a truly
software-defined radio (SDR) with today’s technology. At the end of the day, an
analog front-end will always be needed to condition the signals for the ADC.
Following the development of the laser and the optical fiber in the late 1960’s,
researchers began investigating the potential of optics to perform high-fidelity transmission, distribution, and processing of analog RF signals [23]. In this process, the RF
signal was first modulated onto an optical carrier (historically 1310 nm and 1550 nm
because of propagation characteristics in optical fiber), which was generated by a
laser. The modulated optical carrier was then transmitted through an extremely lowloss optical fiber over long distances to a receiver, which would convert the optical
10
signal back into an RF signal using a photodetector. The goal was to capture the
natural advantages of optical fiber, namely very low propagation loss and dispersion.
Today, optical fiber exhibits losses as low as 0.5 dB/km and carries aggregate data
rates over 1 Tbps; in contrast, RF cables have losses of 360 dB/km at 2 GHz [24]
and get even lossier at higher frequencies. Military and other weight sensitive platforms also sought lightweight optical fibers to replace heavy RF cables (1.7 kg/km vs.
80-600 kg/km) [25].
Soon after data transmission using fiber-optics had proven successful, researchers
turned towards using optics to perform actual RF signal processing. In 1991, the
term microwave photonics was coined [26] to describe the general field of analog RF
signal processing, distribution, and generation using optics and photonics, and has
since continued to captivate the research community. Microwave photonics seeks
to leverage the physics of optics to achieve functionality and performance that is
impossible to obtain using traditional RF components and circuits. The key advantages of microwave photonics include high dynamic range, low loss, immunity to
electromagnetic interference, reconfigurability, and perhaps most importantly wide
bandwidth [24, 27–29].
The performance of a microwave photonic link, generically shown in Fig. 1.5, is
determined by three main parts: the electrical-optical (EO) conversion, the intermediate optical signal processing, and the optical-electrical (OE) conversion. To speak
a common language with RF engineers, microwave photonic links are often characterized using traditional RF figures of merit: link loss, noise figure, and dynamic
range. Because the EO and OE conversions determine the fidelity of the signal as
it transitions between the electrical and optical domains, the modulators and detectors that perform this conversion have a dominant effect on the RF performance of
the microwave photonic circuit. Over the years, microwave photonic circuits have
shown consistent improvement in all RF figures of merit, and commercial products
11
Transmission Medium
(Fiber or Waveguide)
RF Signal In
RF Signal Out
Electrical-Optical
Conversion
Optical Signal
Processing
Mach-Zehnder Modulator
Electro-absorption Modulator
Direct Modulation
Optical Amplifiers
Attenuators
Resonators
Passive waveguides
Splitters, Couplers, etc.
HEO
Optical-Electrical
Conversion
PIN Photodiode
Avalanche Photodiode
UTC Photodiode
HOE
HSP
Figure 1.5: A general analog optical link, showing RF input and output, and optical
signal processing and distribution in the middle.
are available on the market [30–32], although they are mostly used in high-cost, highperformance applications. The intermediate optical signal processing architecture
determines the circuit functionality. Microwave photonic implementations of FIR
filters [33–35], beamformers [35–38], wideband phase shifters [34, 39, 40], true-time
delays [29, 34, 41], and interference cancellers [42–47] have all been demonstrated.
The wide bandwidth of optics is one of the primary reasons why optics is the
de facto carrier of the Internet, and these benefits apply to microwave photonics as
well. The wideband and tunability of optics can enable enormous cost savings by
consolidating the RF front-end and creating a path towards a true multiband radio.
Optics is able to process extremely wideband signals because of the tiny fractional
bandwidth of an RF signal on an optical carrier. Whereas a single 20 MHz LTE
channel may represent a significant bandwidth when modulated onto a 700 MHz RF
carrier, it is virtually narrowband when modulated onto a 193 THz (1550 nm) optical
carrier (Fig. 1.6). Thus, the same optical component treats multiple RF frequencies
12
TV Bands
LTE
ISM
Radionav. etc.
.4 GHz
Up to 10 GHz
Frequency
Optical
Carrier
Entire Cellular
Spectrum
Frequency
193 THz
Figure 1.6: A standard RF signal is essentially narrowband when modulated onto an
optical carrier.
and bands identically. Modern optical devices can easily process RF signals up to
40 GHz, offering unprecedented flexibility to process channels throughout the entire
cellular spectrum and beyond. Innovations such as traveling wave electrodes [48] and
uni-traveling carrier photodiodes have pushed the upper frequency limits of optical
components well into the millimeter-wave regime [38, 49–51].
1.4
1.4.1
Prior Work
Electronic Interference Cancellers
RF interference cancellers have been previously developed, primarily for military and
defense applications. In the past decade or so, there has been renewed interest because
13
Figure 1.7: A 16-tap active analog interference canceller on an RF circuit board.
Taken from [2].
of the allure of full-duplex communications in telecommunications. While there has
been work focusing on all three stages of interference cancellation (passive, analog,
digital), we focus our review here on active analog approaches. Most active analog
approaches sample the TX signal and apply some form channel-estimation before
combining it with the RX signal, nulling the interference. Various works have differed
in the way they sample the TX signal as well as the channel-estimation process. The
QHx220, offered by Intersil, was one of the first commercial interference cancellers,
which came in a compact RF integrated circuit (RFIC) form. Built for handsets, the
QHx220 could achieve at least −20 dB of cancellation and could operate from 300 MHz
up to 3 GHz. However, it could only cancel narrowband signals, and began to exhibit
nonlinear performance when the input signal power exceeded −40 dBm, limiting the
dynamic range [19, 20, 52]. Duarte et al. [22] demonstrated a full-duplex receiver,
including an analog interference canceller, which used an identical, but separate TX
14
Table 1.1: Overview of Electronic Interference Cancellers
References
Form-Factor
Cancellation (dB)
Bandwidth (MHz)
Frequency Range
[19, 20, 52]
RFIC
−20 dB
Narrowband
0.3 GHz - 3 GHz
[22]
PCB
−30 dB
50 MHz
2.4 GHz Band
[2]
PCB
−47 dB
80 MHz
2.4 GHz Band
radio chain to generate the analog nulling signal on a per-subcarrier basis for OFDM
signals. This prevented siphoning valuable power away from the actual transmitted
signal. It achieved a maximum of −30 dB of cancellation across about 50 MHz in
the 2.4 GHz band. Bharadia et al. [2] built a 16-tap tapped-delay line interference
canceller for full-duplex radios, which used fixed delay lines and variable gain elements
on an RF circuit board, shown in Fig. 1.7, to mimic the self-interference channel.
With 16-taps, the canceller was able to cancel an impressive −47 dB over 80 MHz. A
summary of the most recent RF interference cancellers are shown in Table 1.1. It’s
clear that the limiting factors of electronic interference cancellers are the
bandwidth and the operating frequency range.
1.4.2
Microwave Photonic Interference Cancellers
To address the limitations of RF interference cancellers, a microwave photonic system
for self-interference cancellation was first implemented by Suarez et al. in 2009 [3]
using discrete fiber-optics and optoelectronics. That system, shown in Fig. 1.8 was
composed of a pair of Mach-Zehnder modulators counter-biased in quadrature, with
a variable optical attenuator and mechanical optical delay line in one arm of the
interferometer architecture to perform channel estimation. As a proof-of-concept,
the system demonstrated −30 dB of interference cancellation over 100 MHz, already
surpassing the bandwidth of RF cancellers. In terms of narrowband cancellation, the
system demonstrated an enormous −73 dB of cancellation of a single-tone RF signal.
15
Figure 1.8: The first microwave photonic self-interference cancellation system. Taken
from [3].
In 2011, Cox et al. [47] demonstrated a microwave photonic device, which enabled
isolation between the RX and TX signals. Although this device was not an interference canceller, it worked effectively as an isolator and demonstrated over 40 dB
of isolation up to 6 GHz for a duplexed, single antenna system. It was intended for
simultaneous transmit and receive operation.
In 2013, Chang et al. [4] expanded the original MPC architecture to address
multipath interference by designing amultitap tapped-delay line network to model
the self-interference channel. This multitap MPC, shown in Fig. 1.9 could cancel
the strong line-of-sight signal in addition to 2 multipath components. Tested on a
cabled multipath channel to simulate an actual wireless channel, the same system
was demonstrated at both 900 MHz and 2.4 GHz, showing the frequency tunability
of optics. It achieved −35 dB of cancellation across 200 MHz bandwidth and −45 dB
across 10 MHz in the 2.4 GHz band, successfully canceling the multipath components.
Curious about the RF performance of the MPC, the first measurements on typical
RF figures of merit were made by Lu et al. [53] in 2013. They found that using the
16
Figure 1.9: A multitap microwave photonic self-interference cancellation system.
Taken from [4].
technique presented in [3], the MPC exhibited a link loss of −32 dB, a noise figure of
54 dB, and a spurious-free dynamic range of 102 dB/Hz2/3 .
In just the past 6 years, a number of startup companies, such as Bascom-Hunter
Technologies, have begun developing this technology for commercial and military and
defense markets. These companies have focused on developing the discrete fiber-based
MPC technology primarily for electronic warfare applications, such as the ability to
simultaneous jam and transmit.
1.5
Major Thrusts of this Thesis
This thesis work was motivated and driven by two primary forces, resulting in three
major thrusts to improve the MPC beyond the state-of-the-art. The first force was a
need for practicality. While the discrete MPCs presented in the last section had already demonstrated superior performance compared to their analog RF counterparts,
they were a long way from being feasibly implemented in actual RF systems. In par17
ticular, it was still largely unknown how MPCs would affect the performance of RF
receivers in key performance parameters such as loss, noise, and dynamic range. These
parameters determine receiver sensitivity and selectivity, which have been obsessed
over by RF engineers for decades. Therefore, any front-end component would need to
be fully understood in these terms. Prior studies on this subject were either analytical
or showed that the noise figures and losses of MPCs were far too poor to be used in
an RF receiver [53]. In addition, no MPCs had ever been tested in an actual wireless
testbed or radio, confined solely to the safe calm of cabled self-interference channels, which are not representative of real, chaotic, wireless environments. Therefore,
the first major thrust of this thesis was to establish an analytical and experimental
understanding of all MPCs based on key RF performance parameters (Chapter 2),
and to study them in actual wireless channels and radios (Section 3.3). In parallel,
this led to a second major thrust of this thesis, which was the development of new
MPC architectures designed specifically to improve system performance in actual RF
receivers (Chapter 3).
The second force was the need for scalability combined with the timely rise of
a photonic integrated circuit manufacturing ecosystem. Discrete MPCs, while highperformance, were also extremely expensive. Individual modulators, lasers, and photodetectors could easily reach US$1000 each. Furthermore, scaling discrete MPCs
to include multiple taps to compensate for more complex RF environments quickly
became cost prohibitive. A natural solution was to follow the spirit of electronic
circuits, and integrate MPCs onto a semiconductor platform, allowing them to enjoy
the economies of scale of the billions of dollars already invested into semiconductor
industry. Integration offered obvious advantages in terms of reduced size, weight, and
cost, but also offered unique performance advantages. Fortunately, during this thesis,
a fledgling photonic integrated circuit ecosystem was developing, primarily in Europe
and Asia. Entities from both academia and industry were united by the realization
18
that commercialization of photonic integrated circuits could only be achieved through
standardization. Instead of each party using their own specialized process design kits,
software, and packaging techniques, they combined their efforts, so that their work
and designs could follow suite. This thesis work rides upon the waves of these recent developments, and the third major thrust was photonic integration (Chapters 4
and 5). The final result of this thesis was the world’s first monolithically integrated
microwave photonic canceller.
Figure 1.10: The overview and timeline of this thesis.
19
Chapter 2
Key Performance Parameters
RF receivers must satisfy stringent requirements to detect a weak signal in crowded
and noisy RF environments. Any and all RF front-end components must be chosen carefully so as to either enhance or preserve a receiver’s ability to meet those
requirements. In this chapter, we introduce and define the key RF performance parameters that collectively characterize an MPC’s effect on the RF front-end. These
key performance parameters are
• Cancellation
• Bandwidth
• Link Gain
• Noise Figure
• Dynamic Range
Our goal is to characterize the MPC as if it were an RF component, bridging the
language gap between the optical engineers who develop the circuits and RF engineers
who use them. Moreover, we wish to understand, through experiment and theory,
which optical devices, properties, and parameters drive RF performance, and what
20
trade-offs can be made. To perform this task, this chapter will model a basic analog
optical link, from which the relationships for the key performance parameters will be
extracted. In doing so, we follow in the spirit of Cox. et al. [30,31,54], who performed
extensive RF analysis on optical analog fiber-links. In Chapter 3, these relationships
will be used as building blocks to derive the key performance parameters for specific
MPC architectures.
2.1
Cancellation Depth
The primary key performance parameter of any interference canceller is how much
interference it can eliminate from the RX signal. In this section, our objective is to
quantify interference cancellation, present methods to experimentally measure cancellation, and determine what level of channel-estimation precision is required to
achieve a certain level of cancellation. Along the way, we will introduce the concept
of amplitude and phase mismatch.
2.1.1
Definitions
We define cancellation as
Z
Pc (f )df
C = ZB
(2.1)
Pnc (f )df
B
where C is cancellation in linear units, Pc (f ) is the output interference power of the
MPC with cancellation activated, Pnc (f ) is the output interference power without
cancellation activated, and B is the RF bandwidth. When measuring Pc (f ) and
Pnc (f ), the MPC should be present in the front-end, regardless of whether or not
cancellation is activated. This definition normalizes out any insertion loss of the
MPC, and describes the interference power reduction attributed to activating the
21
-40
Interference Power (dBm)
Without
Cancellation
Cancellation
Depth
-60
-80
With
Cancellation
-100
905
910
915
920
925
RF Frequency (MHz)
Figure 2.1: The definition of cancellation depth.
interference canceller. Note that by definition, cancellation is a function of bandwidth.
Another important quantity is the cancellation depth, defined as
CD(f ) =
Pr (f )
.
Pnr (f )
(2.2)
The cancellation depth reflects the cancellation at a particular frequency (see Fig.
2.1). It is straightforward to see that if the cancellation depth is flat across the entire
bandwidth of interest, then cancellation is identical to cancellation depth.
22
RX
Ref.
Interference
Canceller
Out
Signal of Interest
Signal Generator
RX
+
Interference
Canceller
Ref.
Network Analyzer
Out
Interferer
Signal Generator
Spectrum Analyzer
Port 1
Port 2
(b)
(a)
Figure 2.2: Cancellation measurement techniques using (a) a network analyzer and
(b) a spectrum analyzer.
2.1.2
Measuring Cancellation
There are two methods to measure cancellation that are used extensively in this thesis.
The first method uses a network analyzer’s S-parameter measurements to determine
cancellation depth, as shown in Fig. 2.2a. In this method, the output from Port 1 of
the network analyzer is split by an RF splitter. One splitter output is connected to
the MPC’s received signal (RX) input, while the other splitter output is connected to
the MPC’s reference signal (REF) input. Port 2 of the network analyzer is connected
to the MPC’s output. In this configuration, the network analyzer’s output signal
from Port 1 acts as a test interference signal, and the MPC’s |S21| provides an
effective method to measure the transmission of the interference signal through the
MPC. The difference (in dB) between the MPC’s |S21| with and without any input
to the reference input is a direct measurement of cancellation depth. The main
advantage of the network analyzer measurement method is that cancellation depth
can be determined very quickly over the entire bandwidth of the network analyzer.
The second method to measure cancellation depth uses a spectrum analyzer. In
this method, shown in Fig. 2.2b, two separate signal generators are used. One signal
generator plays the role of the transmitter (i.e. the self-interference source), while the
other signal generator acts the part of the SOI source. The two signals may or may
not be overlapping in frequency. The two signals are coupled together using an RF
23
combiner and used as the RX input of the interference canceller. The interference
signal generator is also tapped by a directional coupler, and the tapped signal is used
as the reference input of the MPC. The output of the MPC is then measured by a
spectrum analyzer.
Cancellation depth is measured by again observing the MPC output signal with
and without any input to the reference input (Pc (f ) and Pnc (f ), respectively). Cancellation depth is calculated by taking the difference between the two spectra (in
dB); note again, that the canceller’s intrinsic insertion loss or gain is normalized out
as a common signal. The advantage of this method is two-fold. First, it permits
an investigation of how the canceller responds to different interference signal characteristics (e.g. modulation formats, peak powers), while also observing how well the
canceller can recover an actual SOI. Note that neither of these are not possible using
the network analyzer method. Fig. 2.1 was generated using the spectrum analyzer
method.
2.1.3
Amplitude and Phase Mismatch
The MPCs demonstrated in this thesis use linear finite-impulse response (FIR) filters,
such as that shown in Fig. 2.3, to perform the self-interference channel estimation.
Channel-estimation using FIR filters enables MPCs to leverage the extensive research
on Wiener filters, error minimization, and search algorithms that have been conceived
for FIR filters [55, 56].
By assuming the FIR architecture, we can quantitatively investigate factors that
will determine the cancellation of an MPC. From Eqn.
(1.3), the general self-
interference cancellation problem can be expressed as
Y (ω) = HSI (ω) − HIC (ω) N (ω)
24
(2.3)
_
RX
Ref.
τ1
α1
τ2
α2
RF_Out
+
…
…
τN
αN
Channel-matching Filter
Figure 2.3: Block diagram of an interference canceller using a channel-matching FIR
filter.
where Y (ω) is the MPC output signal, X(ω) is the TX signal, HSI (ω) is the transfer
function of the self-interference channel, and HI C(ω) is the transfer function of the
MPC’s estimated self-interference channel. As an FIR filter, we can express the
estimated self-interference channel as
HIC (ω) =
N
X
αC,k e−iωτC,k
(2.4)
k=1
where N is the number of FIR taps, and αC , k and τC , k are the weight and delay of
the MPC’s k th FIR filter tap. We can also model the self-interference channel as an
FIR filter
HSI (ω) =
M
X
αI,j e−iωτI,j
(2.5)
j=1
where M is the total number of FIR taps in the self-interference channel, and and αI,j
and τI,j are the weight and delay, respectively, of the self-interference channel’s j th
25
FIR filter tap. Wireless channels are modeled this way to represent the propagation
characteristics of various multipath components; in literature it is called the tappeddelay line model [9]. We will ignore nonlinear distortion and other sources of external
noise for now. Using this approximation, we can rewrite Eqn. (2.3) as
Y (ω) =
M
X
−iωτI,j
αI,j e
X(ω) −
j=1
=
N
X
N
X
αC,k e−iωτC,k X(ω)
k=1
αI,j e
−iωτI,j
− αC,j e
−iωτC,j
k=1
M
X
X(ω) +
αI,k e−iωτI,k X(ω)
(2.6)
k=N +1
where we have assumed that M > N , which is always true in a wireless environment,
since there are infinite multipath components. Without any loss of generality, we
assume that N taps are required to model the self-interference channel, and therefore the N+1 to M terms, given by the second summation term in Eqn. (2.6), are
insignificant. This assumption can be made because in any channel, the higher order
multipath reflections eventually drop below some sensitivity floor.
The residual interference power transfer function of the MPC can now be determined by dividing Y (ω) by X(ω) and calculating its magnitude squared
N
2
Y (ω) 2 X
−iωτI,k
−iωτC,k =
α
e
−
α
e
I,k
C,k
X(ω) k=1
(2.7)
where the goal of the interference canceller is to find the parameters that force this
function to zero.
26
We can now determine the factors that will cancellation. First, we decompose
Eqn. (2.7) into the square-terms and the cross-terms
N
Y (ω) 2 X =
αI,k e−iωτI,k − αC,k e−iωτC,k 2
X(ω) k=1
+
N X
αI,k e
−iωτI,k
− αC,k e
−iωτC,k
iωτI,j
αI,j e
iωτC,j
− αC,j e
(2.8)
k=1
j6=k
where we have assumed that all weights are real-valued. The square-terms physically
represent the residual interference power from a particular multipath component by
itself. The cross-terms represent the residual interference from constructive and destructive interference between multipath components. Looking at the square-terms
first, we can expand the k th square-term as
Y (ω) 2 2
−iωτI,k
− αC,k e−iωτC,k X(ω) = αI,k e
k
2
2
= αI,k
+ αC,k
− 2αI,k αC,k cos(iω(τC,k − τI,k ))
=
2
αI,k
1+
αC,k
αI,k
2
αC,k
−2
cos(ω(τC,k − τI,k ))
αI,k
(2.9)
where it is assumed that all the weights are real valued. We now create two new variables, ∆αk and ∆Φk , which we call the amplitude and phase mismatch, respectively,
and define them as
∆αk =
αC,k
αI,k
∆Φk = ω(τC,k − τI,k )
27
(2.10)
(2.11)
We can now rewrite Eqn. (2.9) as
Y (ω) 2
2
2
X(ω) = αI,k 1 + ∆αk − 2∆αk cos(∆Φk )
k
(2.12)
Equation (2.12) indicates that the propagation of interference through the interference
canceller is closely linked to the amplitude and phase mismatch. When the amplitude
mismatch is 1 (0 dB) and the phase mismatch is 0 radians, then Equation (2.11)
converges to zero i.e., perfect cancellation. Intuitively this makes sense: the better
the match between the FIR filter of the MPC and the self-interference channel, the
more effective the interference cancellation.
To relate Eqn. (2.12) to cancellation depth, we must take the ratio of the transfer
function without cancellation to the transfer function with cancellation. Without
any cancellation, the interference propagates unimpeded through the MPC, and the
transfer function can be given simply as
Y (ω) 2
2
= αI,k
X(ω) k,without
(2.13)
Therefore the cancellation depth simplifies to
CD(ω) =
|Y (ω)/X(ω)|2k,with
|Y (ω)/X(ω)|2k,without
= 1 + ∆αk2 − 2∆αk cos(∆Φk )
(2.14)
Equation (2.14) directly indicates how cancellation relates to amplitude and phase
mismatch. Figure 2.4 plots this relationship as a surface, and indicates what level
of precision is required to achieve a certain cancellation depth. We call these set of
curves, mismatch curves. For example, to achieve a cancellation depth of over 30 dB,
an interference canceller must be able to mimic the self-interference channel within .1
dB in amplitude and 1◦ in phase. Returning to Eqn. (2.8), the cross-terms represent
28
20 dB
25 dB
30 dB
35 dB
40 dB
45 dB
50 dB
55 dB
Figure 2.4: Cancellation depth as a function of amplitude and phase mismatch.
the interaction between the different multipath components, and the FIR filter taps
designed to address them. This encompasses more complex effects such as frequency
fading. It is straightforward to see that these cross-terms will be eliminated if perfect
amplitude and phase matching can be achieved. Although a similar set of mismatch
curves can be constructed for the cross-terms, the overall cancellation and amplitude
depends strongly on the relationship between the two multipath components, which
is a function of the self-interference channel - not the interference canceller.
To summarize, the key takeaways to optimize cancellation are
• Minimize both amplitude and phase mismatch between the interference canceller’s FIR filter and the self-interference channel. It cannot be one or the
other; both must be achieved to obtain high cancellation depth. The mismatch
29
curves shown in Fig. 2.4 provide a guide to how much amplitude and phase
resolution is required.
• If the interference canceller’s FIR filter taps are assigned to cancel interference
multipath components, the strongest multipath components should be targeted
first.
2.2
Bandwidth
The MPC must be able to achieve a high level of cancellation, but it must also be able
to do so over the required bandwidth, which depends upon the receiver’s configuration
(operating channel) and front-end filters. There are two different types of bandwidth
that we are concerned with:
• Instantaneous Bandwidth: The bandwidth over which a system functions
in a static configuration.
• Operating Bandwidth: The full frequency range over which a device can
operate, over all of its configurations.
In this section, we investigate what factors determine each type of bandwidth for an
MPC. We show that there is a cancellation-bandwidth tradeoff that depends not only
on the MPC but also on the receiver and the surrounding wireless environment.
2.2.1
Instantaneous Bandwidth
The instantaneous bandwidth of an MPC is the bandwidth over which the canceller,
in a static configuration, can provide a certain amount of cancellation. In section 2.1,
it was shown that cancellation depth depends on the amplitude and phase mismatch
between the MPC’s channel estimating FIR filter and the self-interference channel.
However, this analysis was performed at one RF frequency. In reality, both amplitude
30
and phase mismatch are functions of frequency because of the frequency variation, or
selectivity, of the self-interference channel. It would be more accurate to write Eqn.
(2.14) as
CD(ω) = 1 + ∆αk (ω)
2
− 2∆αk (ω)cos ∆τk (ω)
(2.15)
Equation (2.15) implies that it may be straightforward to minimize amplitude and
phase mismatch at a single frequency, but doing so over a broad bandwidth is difficult.
It would require the ability to program an FIR filter weight and delay arbitrarily as
a function of frequency. With a large number of filter taps, any channel response
can be constructed to perfectly mimic the self-interference channel; however, this is
intractable and presents a difficult control problem. In practice, the MPC must optimize the cancellation depth at frequencies and bands of high priority. This embodies
the cancellation-bandwidth tradeoff.
Figure 2.5 shows an example of cancellation depth as a function of frequency
(top), together with both the amplitude and phase mismatch of a particular MPC
(bottom). The amplitude and phase are both functions of frequency, dictated by
the self-interference channel. The MPC’s FIR filter was configured to optimize cancellation depth at 2.46 GHz. At that frequency, the phase and amplitude mismatch
both approach the optimal point of (180 + 360n) degrees and 0 dB, respectively, and
therefore cancellation is maximized.
2.2.2
Operating Bandwidth
The operating bandwidth is the entire RF bandwidth over which the MPC has the
ability to operate, given its full range of configurations. The operating bandwidths of
MPCs are typically much wider than their instantaneous bandwidths. This is because,
in contrast to instantaneous bandwidth, which depends to some degree on the wireless
31
Simulation
Experimental
-5
-10
-15
-20
-25
-30
-35
3
Amplitude Mismatch (dB)
-40
2400
2
2420
2440
2460
2480
Frequency (MHz)
560
1
0
580
2500
Zero Mismatch
540
-1
520
Phase Mismatch (deg)
Interference Cancellation (dB)
0
-2
500
-3
2400
2420
2440
2460
2480
2500
Frequency (MHz)
Figure 2.5: (Top) Cancellation depth of an MPC, with the associated (bottom) amplitude and phase mismatch.
32
Figure 2.6: An interference canceller with an operating bandwidth of 2 - 8 GHz.
Frequency call-outs indicate at what center frequency the interference canceller has
been configured to optimize cancellation.
channel, the operating bandwidth is only dependent on the MPC’s internal circuitry.
Operating bandwidth does not have to be paired with a cancellation depth.
Figure 2.6 shows an example of an MPC, which has an operating bandwidth of
2 - 8 GHz. To achieve high cancellation depth at a particular center frequency, the
MPC can be continuously tuned from one frequency to another within the operating bandwidth. This is a unique benefit of an MPC that is not accessible to RF
interference cancellers. The operating bandwidth of the MPC is typically not limited by the optics, which can reach 100’s of GHz. Instead, the operating bandwidth
is typically limited by electronic and RF considerations, such as the optoelectronic
33
device electrodes or packaging. However, with proper engineering, even these have
been increased to well over 40 GHz in commercial devices.
An exception occurs if a physical phenomena that is used in the MPC depends on
a semiconductor carrier lifetime; in this case, the operating bandwidth will typically
be sub-10 GHz, since most semiconductor carrier lifetimes in III-V materials are in
the 10 - 100 ps regime. An example of this is direct modulation of a semiconductor
laser. Additionally, if there are external RF components that must be used in tandem
with the MPC, then these may set the operating bandwidth. In fact, the MPC in
Fig. 2.6 has a much wider operating bandwidth than 2 - 8 GHz; it was artificially
limited by the bandwidth of external RF splitters.
2.3
Link Gain
Link gain describes the change in RF signal power after the signal propagates through
the MPC. Mathematically, it is the ratio of the output signal power to the input signal
power.
G=
Pout
Pin
(2.16)
In this thesis, we use the term “link gain” to refer to either gain or loss (i.e., G > 1 and
G < 1 in Eqn. (2.16)). Link gain is important to an RF receiver because if the signal
is attenuated too much, the overall receiver sensitivity will be reduced. In this section,
the link gain of an optical link will be derived as a function of device parameters and
modulation techniques. Before diving deeper, it should be noted that in this thesis,
link gain refers to the gain of the RF signal and not the optical signal, unless stated
otherwise. This distinction must be made because optical power is converted to and
from electrical current (or voltage), which has a squared relationship with RF power.
Mathematically, this means that 1 dB of optical gain is equivalent to 2 dB of RF gain.
34
Transmission Medium
(Fiber or Waveguide)
Optical Link
Rs
Vs
+
_
Impedance
Matching
Modulator
Optical
Circuit
Photodetector
Impedance
Matching
RL
Figure 2.7: The main stages of an optical link.
We begin by using the block diagram of an optical link, shown in Fig. 2.7, as a
guide to determine link gain. Microwave photonic links consists of five stages: source
impedance matching, the modulator device (EO conversion), optical circuitry, the
photodiode (OE conversion), and finally the load impedance matching. Each of these
stages contributes to the total link gain
H(ω) = |H(ω)| eiφ(ω)
(2.17)
where |H(ω)| is the magnitude response and φ represents the phase response. The
link power gain is given by the norm squared of the transfer function, G = |H(ω)|2 . In
this thesis, we exclusively use intensity modulated direct detection (IMDD) optical
links, where the RF signal is encoded into the intensity of the optical carrier. A
different set of relationships apply for coherent detection. Since the link gain is the
ratio of the power out to the power in, we begin our analysis at the beginning and at
the end of every link: impedance matching.
2.3.1
Impedance Matching
Impedance matching is required to minimize unwanted reflections between components of different impedance, and an optical link is no exception. For an optical link,
both the modulator and the end of the photodiode must be impedance matched to
35
Rs
Vs
RMatch
RM
+
_
...
ZPD
RMatch
RL
Load
Source
Figure 2.8: Resistive source and load impedance matching.
the source and load impedances, respectively. Impedance matching is a fairly underrated part of the optical link design, but should not be ignored because it affects link
gain, bandwidth and frequency response. There are a variety of impedance matching
techniques available to link designers, and a full discussion can be found in [31,54,57].
In this thesis, resistive impedance matching will be used as the impedance matching
technique.
In resistive impedance matching, the modulators input impedance and the photodiode’s output impedance are assumed to be real, and resistors are added either
in series or in parallel to match the source or load. We adopt the most common RF
standard that the source and load impedances are 50 Ω, as is the case for almost all
equipment in our experiments. Figure 2.8 shows circuit schematics of the source and
load impedance matching. At the source, a resistor, Rmatch , is added in series to the
resistance of the modulator (either a laser or an external modulator), RM , so that
the sum resistance is equal to the source resistance, Rs .
Rmatch + RM = Rs
(2.18)
Certain modulator electrodes are designed to be impedance matched to the source,
in which case, they do not require a separate impedance matching circuit. At the
load, because a photodiode under normal operating conditions is treated as a current
36
source with a high output impedance, a matching resistor is inserted in parallel to
the photodiode so that the aggregate output impedance is RL . This is a common
approach used in commercial photodiode packaging.
Rmatch = RL
(2.19)
Resistive impedance matching is lossy because of the energy dissipated in the
matching resistors. However, because of its simplicity and wideband matching characteristics, it is a common impedance matching technique used in commercial optical
devices, including the ones in our experiments. Therefore, to maintain good agreement between experiment and simulation, we adopt resistive matching in our key
performance parameter models as well.
To determine the effect of resistive impedance matching on link gain, note that
on the source side, the impedance matching acts as a voltage divider for the supply
voltage, VS , resulting in a power delivered the modulator
Pin =
(VS /2)2
V2
= S
RM
4RM
(2.20)
This quantity, Pin , represents the power available at the beginning of the link. On
the load side, the impedance matching acts as a current divider, resulting in a power
delivered to the load
Pout = (IP D /2)2 RL =
IP2 D RL
4
(2.21)
where IP D is the RF component of the extracted photocurrent. This quantity, Pout ,
represents the power at the end of the link delivered to the load. Now returning to
37
the overall link gain, it can now be written as
G=
Pout
I 2 RL /4
I 2 R2
= P2 D
= PD2 L
Pin
VS /(4RM )
VS
(2.22)
where we have used the fact that the modulator, source, and load impedances are
matched to 50 Ω. Eqn. (2.22) appears to indicate that resistive impedance matching
induces no losses, but it is concealed within IP D . Let us now look at the inner blocks
of the optical link to find IP D as a function of VS .
2.3.2
Modulator Link Gain
Modulators provide the critical EO conversion in an optical analog link. Together
with the photodiode, they are the most important component of an optical link
because of their often dominating influence on link gain, noise figure, and linearity.
An ideal optical modulator has a perfectly linear response to a stimulating RF signal,
and can achieve lossless conversion of an electrical signal to the optical. In addition,
it should add no noise to the signal and be able to handle arbitrarily large optical
and RF powers. Of course, no such device exists in any of those categories, but
tremendous progress has been made across the board by scientists and engineers
seeking to improve link performance. In this section, the we derive and analyze the
modulator’s contribution to link gain for both direct and external modulation.
Direct modulation refers to the process of using an RF signal to directly modulate
the bias current of a laser. Assuming that the laser is biased above threshold, the
perturbation of the laser bias current results in a similar perturbation of the laser
output power, imprinting the RF signal onto the optical carrier. Direct modulation
is a common modulation technique because of its simplicity; the laser acts as both
the optical source and the modulator device. Typically, only semiconductor lasers
38
are used for direct modulation because of their small cavity lifetimes, allowing them
to respond to high-frequency RF modulation signals [31].
In Fig. 2.9, an example of a laser’s light power vs. forward bias current (LI) is shown. There are three primary parameters of interest for direct modulation:
threshold current (Ith ), DC bias current (IDC ), and slope efficiency (sLD ). Together,
these parameters control the EO conversion efficiency, laser power consumption, and
modulation bandwidth. The laser’s DC bias current sets the operating point of the
laser, upon which the RF signal oscillates
ILAS = IDC + IRF
(2.23)
where ILAS is the total current, and IRF is the RF component of the current. For direct modulation, IDC must be biased above Ith . Furthermore, IDC must be sufficiently
above Ith so that the full swing of the RF signal does not turn the laser off, resulting
in signal clipping. Ideally, a laser is biased at its most linear point in the LI curve;
however, it is well known that a laser’s modulation bandwidth is proportional to the
square root of the output optical power [58]. Therefore, if a higher modulation bandwidth is required, a link designer will have to trade linearity and power consumption
for modulation bandwidth. Direction modulation linearity will be discussed further
in section 2.5.
The slope efficiency (sLD ) of a directly modulated laser is the slope of the LI curve
and has units of Watts per Ampere (W/A). It describes the differential change in
output optical power (P ) that accompanies a differential change in laser bias current
sLD =
dP dILAS I=IDC
39
(2.24)
Output Light Power (mW)
DC Bias
Pt.
PI
Direct Modulation
Ith
IDC
Laser Forward Bias Current (mA)
Figure 2.9: Transfer function of a directly modulated laser.
and is therefore a critical parameter that determines EO conversion efficiency. The
total output optical power of the laser can now be written as
PDM = sLD (ILAS − Ith) = sLD IRF + sLD (IDC − Ith) = Psig + PDC
(2.25)
PDC = sLD (IDC − Ith)
(2.26)
Psig = sLD IRF
(2.27)
where we have used Eqn. (2.23). Assuming that IDC is set at an appropriate point for
the input RF signal to be modulated, the input RF signal current and the intensity
modulated optical signal are linked by the laser’s slope efficiency (Eqn.2.27). Finally,
we can write Eqn. (2.27) in terms of VS , taking into account the source resistance
40
and the resistive impedance matching
Psig,DM = sLD
VS
.
2RS
(2.28)
In external modulation, the RF signal is modulated onto an optical signal using
an auxiliary modulator device, instead of the laser source, which now simply provides
a CW optical carrier to the modulator device. There are two primary types of external intensity modulators used commercially: Mach-Zehnder Modulators (MZM) and
Electro-Absorption Modulators (EAM). Both modulators use the same principle of
biasing at some linear operating point and using the RF signal to modulate the optical carrier’s intensity; however, the physical mechanisms behind how they modulate
the optical intensity are entirely different.
In an MZM, the incident optical carrier is evenly split into two arms of an interferometer. One or both arms of the interferometer contain an optical phase shifter,
in which the application of an electrical field induces a proportional phase shift in
the optical carrier. The physical mechanism behind the phase shifter is most often
the electo-optic effect [59]. After propagating through their respective phase shifters,
the optical signals are recombined and coherently interfere. If the relative phase shift
between the two optical signals is 0, then the optical signal is perfectly transmitted
other than any intrinsic insertion loss, but if the relative phase shift is π, then there is
perfect destructive interference and no output. Thus, the MZM’s optical transmission
is controlled by the electrical signal applied to the optical phase shifter, converting
an electrical signal into an optical signal. MZMs are most commonly built using the
electro-optic material lithium niobate, but recent advances in optical semiconductor
technology have enabled efficient MZMs to be implemented in III-V compounds such
as Indium Phosphide, as well as silicon, through the free-carrier plasma dispersion
effect.
41
The transmission through an MZM can be expressed as [59]
TM ZM
πVRF
πVDC
= 1 − cos
+
Vπ,RF
Vπ,DC
πVRF
= 1 − cos
+ ΦDC
Vπ,RF
(2.29)
where Vπ is known as the half-wave voltage i.e., the voltage required to change the
transmission state of the modulator from 1 to 0 (note that Vπ increases with RF
frequency, hence the distinction between Vπ,DC and Vπ,RF ); VDC is the modulator’s
DC bias voltage; and, ΦDC is the bias angle of the modulator, which is VDC normalized
by Vπ .
ΦDC =
πVDC
Vπ,DC
(2.30)
The MZM’s optical transmission is plotted as a function of bias angle in Fig. 2.10.
Taking into account the input optical power to the MZM as well as its intrinsic
insertion loss, the MZM output optical power can be expressed using Eqn. 2.29 as
PM ZM = PI Tmod TM ZM
PI Tmod
πVRF
=
1 − cos
+ ΦDC
2
Vπ,RF
(2.31)
where PI is the input optical power and Tmod is the modulator’s intrinsic insertion
loss. To determine the relationship for EO conversion at a particular bias angle, we
perform a Taylor series decomposition of Eqn. (2.31)
2
3
PM ZM ≈ PDC,M ZM + Psig,M ZM VRF + P2,M ZM VRF
+ P3,M ZM VRF
+ ...
42
(2.32)
Optical Transmission
1
Bias Pt.
1/2
0
0
π/2
π
Bias Angle (ΦDC )
3π/2
Figure 2.10: Transfer function of a Mach-Zehnder Modulator.
where the coefficients are
PDC,M ZM
PI Tmod
1 − cos(ΦDC )
=
2
(2.33)
Psig,M ZM
PI Tmod
π
=
sin(ΦDC )
2
Vπ,RF
(2.34)
P2,M ZM
2
PI Tmod
π
=
cos(ΦDC )
4
Vπ,RF
(2.35)
P3,M ZM
3
π
PI Tmod
=−
sin(ΦDC )
12
Vπ,RF
(2.36)
The coefficient that describes the EO conversion to the fundamental frequency is
Psig,M ZM . Equation (2.34) indicates that Psig,M ZM is maximized when ΦDC = π/2,
43
or when the DC bias voltage is half of the half-wave voltage (see Fig. 2.10). This
bias point is called the quadrature bias point. Most MZM-modulated links are biased
quadrature because, in addition to exhibiting the largest conversion efficiency, it also
minimizes the second order harmonic (Eqn. 2.35). This does not necessarily mean
that it is always the most advantageous to be biased at quadrature [26, 60]; however,
for the link at hand, we will assume that quadrature biasing is used. For a small
signal, the EO conversion can be expressed as
Psig,M ZM
PI Tmod
π
=
VRF
2
Vπ,RF
(2.37)
and once again taking into the resistive impedance matching between the source and
the modulator, Eqn. (2.37) can be expressed in terms of the source voltage, VS as
Psig,M ZM
PI Tmod
π
=
VS
4
Vπ,RF
(2.38)
An Electro-Absorption Modulator (EAM) is another type of external intensity modulator. Instead of using interferometry to modulate the output optical intensity, it
modulates its optical absorption based on an electric field applied across the propagating waveguide. The application of an electric field across the EAM semiconductor
junction causes the effective optical absorption edges to shift, thereby either increasing or decreasing the optical absorption for wavelengths near the absorption edge.
The physical mechanisms that enable this shift are the Franz-Keldysh effect for bulk
semiconductors and the Quantum-Confined Stark Effect, for quantum well materials [59]. In either case, the wavelength of the propagating optical signal must be
located on or near the optical absorption edge for EAMs to effectively modulate an
optical signal. This is one difference from MZMs, which are largely optical wavelength
independent. EAMs are most often fabricated in III-V compounds, which have direct
bandgaps and clearly defined absorption edges. However, EAMs can be fabricated in
44
Optical Transmission
1
Bias Pt.
T bias
0
V DC
Reverse Bias Voltage (V)
Figure 2.11: Transfer function of an Electro-Absorption Modulator.
any material that exhibits a strong optical absorption effect, which can be modulated
via an electric field. For example, recently, graphene deposited on silicon was used to
create an EAM in silicon-on-insulator [61].
The output optical power of an EAM as a function of the reverse bias voltage
across the EAM is shown in Fig. 2.11. Ideally, when no reverse bias is applied,
the EAM is transparent, except for an intrinsic insertion loss. As the reverse bias is
increased, the EAM becomes more absorptive, increasing the device’s optical insertion
loss. To achieve maximum EO conversion efficiency, the EAM should be biased so
that the optical wavelength is on or near the modulating absorption edge, as shown
in Fig. 2.11.
45
An analytical description of the EAM’s EO conversion depends heavily upon the
device material platform e.g., material composition and quantum well design, which
lies outside the scope of this thesis. Instead, because the EAM EO conversion shares
similar traits to that of an MZM, we model the EAM using the conversion expression
for an MZM (Eqn. 2.38), with an effective Vπ,RF , which is based on the EAM’s
absorption sensitivity at the bias point. To find the effective Vπ,RF for an EAM,
recall that Vπ describes the voltage required for an MZM to switch its transmission
state from 0 to 1. For an MZM, it is related to the slope of the MZM transmission
curve (Fig. 2.10), which, for an MZM biased at quadrature, can be found by taking
the derivative of Eqn. (2.29) at quadrature
π
dTM ZM =
dVRF ΦDC =π/2 2Vπ,RF
(2.39)
from which we can find an expression for Vπ,RF
Vπ,RF =
π
2(dTM ZM /dVRF (2.40)
)
ΦDC =π/2
Thus by measuring the slope of the EAM optical transmission curve at the DC bias
point, an equivalent Vπ,RF can be expressed using Eqn. (2.40) as
Vπ,RF =
π
2(dTEAM /dVRF (2.41)
)
V =Vbias
This effective Vπ,RF term can now be used in Eqn. (2.38) to find the E-O conversion
expression for an EAM
Psig,EAM
PI Tmod
π
VS
=
4
Vπ,EAM
where all the variables have the same definition as in Eqn. (2.38).
46
(2.42)
To summarize the EO conversion relationships for the three modulator devices,
Eqns. (2.28), (2.37), and (2.42) are repeated here
Psig,DM = sLD
VS
2RS
Psig,M ZM
π
PI Tmod
VS
=
4
Vπ,RF
(2.43)
Psig,EAM
PI Tmod
π
=
VS
4
Vπ,EAM
(2.44)
In comparing the different conversion methods in Eqn. (2.44), several noticeable
similarities and differences can be seen:
• Similarity: Dependence on a Sensitivity Parameter. The conversion
efficiency of all three modulator devices depend upon a “sensitivity parameter”,
which is device-specific. For direct modulation, this is the laser slope efficiency,
sLD . For the two external modulators, these are their respective Vπ terms, which
relates the change in optical transmission to the applied voltage. To maximize
gain, sLD should be as large as possible, and Vπ should be as small as possible.
• Difference: Dependence on Optical Power. An extremely important difference between the two modulation techniques is that direct modulation EO conversion is independent of optical power, while external modulation E-O
conversion is directly related to optical power. This difference means that, in
theory, the conversion efficiency and link gain of an external modulator can
be increased arbitrarily by increasing the optical power, eventually achieving
link gain. However, in practice, this is never the case as the maximum optical
power is typically limited by another component in the circuit. Although not
apparent from Eqn. (2.44), the benefit of increasing optical power in a directly
modulated laser is to increase the dynamic range and bandwidth of the link.
47
• Difference: Dependence on Device Insertion Loss. In direct modulation,
optical insertion loss does not exist, since the laser is the optical source. However, in external modulation, any modulator intrinsic insertion loss negatively
affects EO conversion and link gain. Therefore, the optical insertion loss of every component should be minimized. EAMs typically exhibit a larger insertion
loss than MZMs.
• Difference: Bandwidth of Optical Source. A subtle difference between
the modulation techniques is the role that is played by the laser modulation
bandwidth. In direct modulation, the laser modulation bandwidth sets the
bandwidth of the EO conversion, and can be increased by increasing the output
optical power. However, in external modulation, the EO conversion bandwidth
is set by the external modulator, and typically the electrode structure. This
means that any laser, even a very low modulation bandwidth laser, can be used
since it is operated in CW mode. The flexibility to choose different types of
lasers can be very helpful to reduce noise or increase link gain.
2.3.3
Optical Circuitry Link Loss
Following EO conversion, the modulated signal enters the optical circuit, where it
either experiences loss or gain as it propagates through different optical devices. This
is dependent upon the circuit architecture and component devices, and in Chapter 3,
we will introduce the interference canceller architecture. However, for now, we model
the optical circuit’s gain simple as Gopt so that at the optical power at the output of
the circuit (prior to detection) can be written as
Pout = Gopt Pin = Gopt (PDC + Psig )
48
(2.45)
where the optical circuit gain can be assumed to affect both DC and RF components
identically.
2.3.4
Photodiode Link Gain
The last step of the optical link is the OE conversion. The device that performs this,
the photodiode, converts the intensity envelope of an incident optical signal into a
photocurrent; it discards all optical phase information. As a conversion device, the
photodiode, like the modulator, has important impacts upon link performance. Its
conversion efficiency affects link gain; its power handling affects circuit linearity; and
its response bandwidth can put a cap on the maximum link frequency. In addition,
any noise embedded in the incident optical signal is converted into an electrical signal
at the photodiode.
A typical photodiode response is shown in Fig. 2.12. When reverse-biased to
create a sufficient electric field to sweep out photo-generated carriers, the output
electrical current, IP D , is linearly related to the incident optical power.
IP D = rP D Pout
(2.46)
The constant of proportionality, rP D , is known as the photodiode’s responsivity and
has units of A/W. Note also from Fig. 2.12 that at high incident optical powers, the
photodiode’s response saturates. The saturation is caused by excessive photogenerated carriers screening the electric field, which sweeps carriers out of the photodiode’s
active region. In saturation, the photodiode response is much weaker, and becomes
nonlinear. Photodiode saturation is often the upper limit on optical power.
49
0
Photocurrent (mA)
-1
-2
-3
-4
-5
Bias = -3.3V
0
1
2
3
4
5
6
7
Optical Power (mW)
8
9
10
Figure 2.12: Response of a photodiode to an input optical signal.
2.3.5
Total Link Gain
Now that we have determined the workings of the inner blocks of the optical link,
we can compute the total link gain. Starting from Eqn. (2.22), and substituting in
Eqns. (2.44), (2.45), and (2.46) the total link gain can be written as
GDM =
GEM
2
2
2
1
sLD Gopt rP D
4
2 2 2
1 πPI Tmod RL
=
Gopt
rP D
4
2Vπ
(2.47a)
(2.47b)
where GDM and GEM are the link gains due to direct and external modulation respectively, we’ve assumed a matching load and source RL = RS , and we’ve purposefully
50
Method
Circuit/Device
Direct
External
Increase PI
Circuit
N/A
Squared
Improve Sensitivity (sLD ,Vπ )
Device
Squared
Squared
Circuit + Device
Squared
Squared
Increase rP D
Device
Squared
Squared
Impedance Matching
Circuit
Variable
Variable
Reduce Insertion Loss
Table 2.1: Strategies to improve link gain and the effect of modulation technique.
written the expressions with this grouping to emphasize which part of the optical link
contribute to which parts of the link gain. Indeed, Eqns. (2.47) can be rewritten to
emphasize this modular gain contribution
GDM = GIM
GEO Gopt GOE
(2.48)
where GIM , GEO , GOE are the gain contributions from impedance matching, E-O
conversion, and O-E conversion, respectively. Note that the loss from the resistive
impedance matching is now explicitly shown.
A close inspection of Eqns. (2.47) reveals that there are multiple methods to
improve link gain, and they can be split into two general categories: device-level
improvements and circuit-level improvements. Table 2.1 summarizes these strategies
and also indicates how effective they are for the two modulation techniques.
To visualize the differences between external and direct modulation, we plot the
link gain as a function of input optical power for a directly modulated link and an
externally modulated link for a variety of laser slope efficiencies and modulator Vπ
values. The remaining link parameters are Gopt = -2 dB, Tmod = .5, rP D = .72 A/W.
Note the effect that the modulator sensitivity parameters have on link gain, as well
as the difference that optical power has on both links.
51
10
Direct Modulation
External Modulation
Vπ
=
1V
0
Link Gain (dB)
Vπ
=
5V
-10
SLD = .5 W/A
-20
-30
SLD = .1 W/A
-40
-50
0
5
10
15
20
Input Optical Power (dBm)
25
30
Figure 2.13: Link gain for a directly modulated link (sLD = .1 to .5 W/A) and an
externally modulated link (Vπ = 1 to 5 V).
2.4
Noise Figure
A component’s noise figure describes the degradation of signal-to-noise ratio (SNR)
as it passes through [62]. As a result, noise figure has a direct impact on an RF
receiver’s sensitivity. For example, if an RF receiver has a noise figure of 5 dB,
and can detect signals with a minimum power of -100 dBm, then an identical RF
receiver with a noise figure of 10 dB can only detect signals with a minimum power
of -95 dBm. Understanding what factors and contribute to noise figure in an MPC
is therefore essential to ensuring that the benefits of interference cancellation come
with minimum cost to receiver sensitivity.
52
RF
Component
Power
G
SNRout
SNRin
Nout
Nin
Frequency
Frequency
Figure 2.14: A graphical view of the effect of an RF component on signal and noise
power. Noise figure is defined as the ratio between SN Rin and SN Rout .
Noise figure is mathematically defined as the ratio of the input SNR to the output
SNR, and is given on a dB scale. As all optical links have a gain (or loss), we
can express noise figure in terms of link gain and input/output noise levels. In the
following, we write the noise factor, which is the linear version of noise figure, i.e.
F = 10N oiseF igure/10
F =
s(t)/Nin
SN Rin
Nout
=
=
SN Rout
Gs(t)/Nout GNin
(2.49)
where G is the optical link gain, s(t) is the signal at the input, Nin is the input noise,
and Nout is the output noise. All variables are in linear units. One can see that the
noise figure is affected by both the link gain as well as any noise sources within the
link (Fig. 2.14).
When RF components are cascaded, as they are in an RF receiver, the noise figure
of the overall cascade can be expressed in terms of the noise figure and gain of the
53
constituent components, also known as the Friis equation for noise [62]
F = F1 +
(F2 − 1) (F3 − 1)
(FN − 1)
+
+ ... +
G1
G1 G2
G1 G2 ...GN −1
(2.50)
where Fi and Gi are the noise factor and gain of the ith component, respectively, and i
= 1 represents the first element in the RF chain. The presence of the previous stages’
gains in the denominators of Eqn. 2.50 indicates that the first few components have
a weighted importance towards the noise figure of the overall cascade. Designed with
this knowledge, nearly all RF receivers have a high-gain low-noise amplifier (LNA)
with noise figure of 1-2 dB as one of the first two stages. This mitigates the effects
of higher noise figure components later in the chain.
An MPC would likely be inserted very close to the front of an RF receiver chain,
and therefore, its noise figure and gain will have a significant impact on overall receiver
noise figure. Our methodology to determine the noise figure of an interference canceller will be to first derive the noise figure of a optical analog link and subsequently
use this as a building block for more complex architectures.
2.4.1
Noise Sources in an Optical Link
Using Eqn. 2.49 as a guide, we first determine what noise sources are present at
the input and output of an optical link. Optical links generate noise through three
primary mechanisms: thermal noise, shot noise, and relative-intensity noise (RIN).
Thermal noise is generated from the random motion of charge carriers within a dissipative circuit element, and enters optical links through an optoelectronic device’s
electrodes. Thermal noise is a stationary zero-mean Gaussian random process, with
no spectral dependence up to about 1 THz, i.e. it is practically white noise for RF
54
frequencies [63]. The variance of the thermal noise current is given by
2
hIth
i=
4kT B
R
(2.51)
where k is the Boltzmann constant, T is the temperature, B is the bandwidth of
the receiver (also known as the effective noise bandwidth), and R is impedance of
the dissipative circuit element. Thermal noise is also commonly used as the input
noise when calculating noise figure, i.e. Nin = Nth . The output thermal noise has
contributions from both the input thermal noise, which propagates through the optical
link and is subject to link gain, as well as thermal noise from the photodiode at the
end of the optical link. Assuming that the source and load impedance are matched
with value RL , the variance of the output noise variance is
2
hIth,out
i = (1 + G)
4kT B
RL
(2.52)
Shot noise is generated from the random processes of the photon field, and manifests as a noise current when light is incident upon a photodiode [63]. It is modeled
as a stationary Poisson random process with a white spectral density directly proportional to the electron charge, q, and the sum of the average generated photocurrent
(IP D ) and dark current(ID ), IT = IP D + ID . For optical powers typically used in
IMDD links, IT ≈ IP D . The variance of the shot noise current is given by
2
hIsh
i = 2qIP D B
(2.53)
Relative intensity noise manifests as random intensity and phase fluctuations of
a laser output. These fluctuations are caused by spontaneous emission contributing
randomly to the laser’s coherent field [63], as well as noise from the laser driver and
reflections from improper packaging [31]. Unlike the previous two noise sources, RIN
55
has a strong frequency dependence, and mimics the direct modulation response of the
source laser. The most peculiar feature of the RIN frequency response is its peak at
the laser relaxation oscillation frequency and its subsequent f −2 rolloff. Link designers
working with externally modulated links can therefore strategically use source lasers
with extremely low relaxation oscillation frequencies to sidestep the RIN entirely. For
example, solid state lasers have relaxation oscillation frequencies in the kHz [31], but
the optical carrier can be modulated in the MHz or GHz range.
RIN is present in the laser output and is converted into electronic noise along with
the signal at the photodiode. As the name implies, RIN is measured relative to the
laser intensity, and therefore has units of dB/Hz (or 1/Hz in linear units). Modern
semiconductor DFB lasers can have RIN lower than -160 dB/Hz, and techniques,
such as external cavities, can be used to lower RIN even further [64]. Other types
of off-the-shelf lasers, such as solid state lasers and doped fiber lasers, can exhibit
RIN as low as -175 dB/Hz [31]. The variance of the optical power due to RIN can be
calculated as
2
hPRIN
i
=
2
Pout
Z
∞
RIN (f )df
(2.54)
−∞
Equation (2.54) indicates that RIN is a function of both the RF frequency as well
as the average incident optical power, Pout , at the photodiode. To simplify the analysis, we adopt a similar approach as [26] and treat RIN by approximating it as a
frequency-independent and average optical power-independent variable over the frequency range of interest. These effects are aggregated into an effective RIN, which
should be corrected based on the circuit operating conditions and frequency
2
2
hPRIN
i = RIN Pout
B
56
(2.55)
PD
Rmatch
Ith
Ish
IRIN
RL
Isig + Inoise
Figure 2.15: Effective circuit of noise components in an analog photonic link.
and the RIN current variance can then be written as
2
hIRIN
i = RIN (rP D Pout )2 B = RIN (IP2 D )B
(2.56)
where we have substituted IP D = rP D Pout , where rP D is the photodetector responsivity and all variables are given in linear units.
2.4.2
Total Link Noise Figure
At this point, the total output noise power of the optical analog link can be computed
using the effective circuit diagram shown in Fig. 2.15. Since the three noise sources
are statistically independent, the variance of the output noise current is the sum of
the variance of the three output noise currents, and the output noise power can be
expressed as
2
2
2
i + hIsh
i + hIRIN
i kout RL
hNout i = hIth
(2.57)
where RL is the load impedance and kout represents the impedance matching between
the load impedance and the photodetector. As before, we assume resistive matching
in the photodetector package: kout = 1/4.
57
Substituting Eqns. 2.52, 2.53, and 2.56 into Eqn. 2.57 we yield an expression for
the total noise output power
1
hNout i = ((1 + G)4kT B + 2qIP D RL B + RIN (IP2 D )RL B)
4
(2.58)
Finally, substituting Eqn. 2.58 into Eqn. 2.49 yields an expression for noise factor
for an optical link
F =
Nout
1
qIP D RL RIN (IP2 D )RL
=1+ +
+
GNin
G
2GkT
4GkT
(2.59)
where Nin = kT B, because of the resistive impedance matching at the input. This
expression provides valuable insight into what parameters influence noise figure:
• For a resistively impedance matched link with a positive insertion loss (i.e.
G < 1), the noise figure can never be less than 3 dB. This is sometimes called
the passive attenuation limit to the noise figure of an optical link [54] because the
noise figure of a passive RF component is simply equal to its loss. Physically, this
limit stems from the fact that when, G < 1, the photodetector contributes as
much, if not more, thermal noise than the modulator. The passive attenuation
limit can be breached when the link exhibits gain.
• Noise figure is approximately independent of the link bandwidth, at least within
the bandwidth-independent approximation of RIN. This is because the link
bandwidth affects both input and output noise equally. Therefore, an RF component’s noise figure is applicable to any bandwidth signal. However, noise
figure must be cited as a function of frequency, since link and device parameters (e.g. gain) are frequency-dependent.
Figure 2.59 also suggests that increasing link gain and reducing the average detected
photocurrent (IP D ) are keys to reducing noise figure. However, generally speaking,
58
these two variables are not independent; to take the analysis one step further, we
insert the expressions for link gain for both a directly and an externally modulated
link operated in quadrature into Eqn. 2.59
FDM
Nout,DM
4
RIN PI2 RL
2qPI RL
=
=1+
+
(2.60)
+
GDM Nin
(sLD Gopt rP D )2 s2LD Gopt rP D kT
s2LD kT
FEM =
Nout,EM
16V 2
16qVπ2
4Vπ2 RIN
= 1 + 2 2 2 π2
+
+
(2.61)
GEM Nin
π PI Gopt rP D RL2
π 2 PI Gopt rP D RL kT
π 2 RL kT
where FDM and FEM are the noise factors of a directly modulated and externally
modulated link, respectively. These expressions provide additional insight into the
relationships between device parameters and the noise figure of an optical link. In
comparing them, several major differences can be seen, as summarized in the following
list. To assist in analyzing and visualizing these differences, the noise figures of two
optical links, identical save their modulation techniques, are plotted as a function
of input optical power (PI ) and RIN in Fig. 2.16. The common link parameters
were Gopt = −2dB, rP D = .85A/W , T = 290K, and RL = 50Ω. For the externally
modulated link, Vpi = 3V and modulation insertion loss, Tmod = −3dB; for the
directly modulated link, sLD = .3W/A. It is assumed the link behaved linearly
over all operating regimes. For ease of comparison, the small signal link gain is also
reprinted in Fig. 2.16. The main insights can be summarized:
• Optical Power: The most drastic difference between the two modulation techniques is that the noise figure of an externally modulated link decreases with
increasing input optical power, whereas the opposite is true of a directly modulated link. This is a consequence of the fact that the link gain is independent of
optical power for direct modulation, but related to the square of optical power
for external modulation. Because shot noise and RIN both grow with average
59
10
Direct Modulation
External Modulation
Vπ
=
1V
0
Link Gain (dB)
Vπ
=
5V
-10
SLD = .5 W/A
-20
-30
SLD = .1 W/A
-40
-50
0
5
10
15
20
Input Optical Power (dBm)
25
30
50
RIN = -150 dB/Hz
-175 dB/Hz
Noise Figure (dB)
40
RIN = -150 dB/Hz
-155 dB/Hz
30
-160 dB/Hz
-165 dB/Hz
20
-170 dB/H
-175
Direct Modulation
External Modulation
z
dB/H
z
10
0
5
10
15
20
Input Optical Power (dBm)
25
30
Figure 2.16: (Top) Gain and (Bottom) noise figure comparisons for a directly and
externally modulated link. Directly and externally modulated links are given by
dotted and solid lines, respectively. Common link parameters: Gopt = −2dB, rP D =
.85A/W , RL = 50Ω, T = 290K. For the noise figure comparison, sLD = .3W/A and
Vπ = 3 V.
60
optical power, noise figure quickly diverges for a directly modulated link, but
eventually saturates for an externally modulated link as RIN and link gain nullify each other. Link designers should thus be cognizant of the maximum optical
powers that (1) can be handled by the optical link and (2) can be sourced by
lasers. This, together with other link parameters such as RIN, can influence the
choice of modulation technique.
• Relative Intensity Noise: High RIN is detrimental to noise figure for both
directly and externally modulated links, particularly at high optical powers
2
i ∝ PI2 . Fig. 2.16 shows that in externally modulated links, the
because hIRIN
laser RIN determines when the link noise figure saturates with respect to input
optical power. This relationship will be investigated further, below.
• Modulator Sensitivity, Optical Link Loss, and Photodetector Responsivity: Although not explicitly shown in Fig. 2.16, it should be clear from Eqns.
(2.60) and (2.61) that increasing photodiode responsivity, decreasing optical insertion loss, and increasing modulator sensitivity will reduce noise figure for
both modulation techniques. This result is derived from the fact that all three
will improve link gain.
Similar to the situation with link gain, externally modulated links offer more flexibility than directly modulated links because of the circuit-level parameters involved
with external modulation, specifically input optical power and DC bias point. These
parameters can be easily tuned electronically, whereas changing laser slope efficiency
requires an entirely new device. For this reason, and because most of our experimental links are built using external modulators, we will take a deeper dive into the noise
figure relationship for external modulators.
61
Noise Type
Optical Power Dependency
Thermal
Shot
RIN
PI−2
PI−1
const.
Dominant Regime
Low Optical Power
Medium Optical Power
High Optical Power
Table 2.2: Noise sources and noise figure relationships in an externally modulated
link.
2.4.3
Noise Figure of Externally Modulated Links
Equation (2.61) explicitly shows that the noise figure contribution of thermal noise,
shot noise, and RIN have three different relationships versus optical power, as shown
in Table 2.2. Specifically, noise figure contributions from thermal noise and shot noise
decrease with respect to input optical power, while the RIN contribution is constant
with respect to input optical power. Thus, unless the laser has extremely high RIN,
the noise figure vs. input optical power slope will change as the optical power is
scanned from low to high, corresponding to the three sources of noise dominating
the link noise. An example of an optical link with RIN = −170dB/Hz is shown in
Figure 2.17, showing the noise figure passing through these three different regimes.
At high optical powers, the link noise figure vs. input optical power saturates because
of the offsetting effects of increased gain and increased RIN.
We have now established the expressions for noise figure of a directly modulated
(Eqn. (2.61)) and externally modulated (Eqn. (2.61)) link, as well as a basic understanding of how link and device parameters affect noise figure. These will serve as
building blocks when we characterize the noise figures of MPCs.
As a final note, it is worth asking a simple question with regards to Fig. 2.16:
if noise figure is lowest at the extremes of optical power (high power for external
modulation, low power for direct modulation), why not operate the link at these
extremes? The answer: dynamic range.
62
50
Noise Figure (dB)
40
To
tal
No
ise
30
Fig
ur
e
Thermal
Limited ~ P-2
20
Shot Noise Limited
~ P-1
RIN Limited
~ const.
10
0
5
10
15
20
Input Optical Power (dBm)
25
30
Figure 2.17: Noise figure of an optical link. Thermal, shot, and RIN noise figure
asymptotes are superimposed. RIN = −170dB/Hz.
2.5
Dynamic Range
Dynamic range refers to the range of RF input powers that a receiver can handle and
still function up to specification. Interference cancellation naturally requires a large
dynamic range because of the need to handle strong interferers while also resolving
weak signals of interest. Dynamic range is also the Achilles’ Heel of digital cancellation
techniques, because of the limited resolution of analog-to-digital converters. There
are a number of parameters used to characterize dynamic range that we will focus on
in this section:
• IIP3/OIP3: Input or Output Intercept Point
• P1dB: 1-dB Compression Point
• SFDR: Spurious Free Dynamic Range
We have already discussed the lower end of the dynamic range spectrum, namely the
factors that would lead a signal to fall under the noise floor. The upper end of the
63
dynamic range spectrum is limited by nonlinearities, which are generated by high
signal power and adversely affect the RF receiver.
2.5.1
Nonlinear Signal Components
Spurious frequencies are generated when the input signal power is strong enough to
force a device into nonlinear operation. Optical amplifiers and absorbers behave nonlinearly when their excited carrier concentration (i.e., gain medium) is depleted or
saturated, respectively. Even optical fiber can behave nonlinearly if the propagating
optical power is strong enough to induce phenomenon such as Stimulated Brillouin
Scattering. However, in MPCs, modulators and photodetectors, whose transfer functions were shown in Fig. 2.18, are the usual suspects.
There are two categories of spurious frequencies: harmonics and intermodulation
products. Harmonics are integer multiples of the fundamental frequency, while intermodulation (IMD) products are sum and difference frequencies. Figure 2.18 shows
some harmonics and IMD products that are generated when two equipotent signals
with frequencies ω1 and ω2 are sent through a nonlinear component together. Although real signals are not composed of just two tones, by bookending the real signal
bandwidth, this two-tone test offers insight into which nonlinear components are important. For example, if one has a sub-octave link, where the signal bandwidth is
less than twice the lowest frequency in the signal, then all higher order harmonics
will fall outside of the signal band and can be filtered with little difficulty. The most
troublesome nonlinear components are usually the 3rd order IMD products (IMD3),
2ω1 − ω2 and 2ω2 − ω1 , because they fall within the signal band or so close to it that
they become impossible to filter out [58, 62]. However, if a link is multi-octave, then
harmonics must also be monitored.
64
Fundamental
2nd Harmonic
ω1 ω2
2nd IMD
Power
3rd IMD
ω2 + ω1
ω2 – ω1
2ω1
2ω1 – ω2
2ω2
2ω2 - ω1
Frequency
Figure 2.18: Spurious frequencies generated in a nonlinear optical link.
2.5.2
Nonlinearity of Modulated Optical Links
In section 2.3, we saw that modulators and photodetectors are inherently nonlinear
(see Fig. 2.19), but they are (usually) biased to operate in a linear region. We will
now quantify the spurious frequencies generated by a modulator. For this analysis,
we will assume that all other link devices remain linear. Our input signal shall be an
equipotent two-tone signal with frequency components ω1 and ω2
Iin = I0 (cosω1 t + cosω2 t)
(2.62)
Direct Modulation
A directly modulated laser’s nonlinearity and dynamic range is determined by its
bias point as well as the modulating frequency. Specifically, the laser nonlinearity
increases quickly as the modulating frequency approaches the laser’s relaxation os65
1
Optical Transmission
Output Light Power (mW)
(a)
DC Bias
Pt.
PI
Bias Pt.
1/2
Direct Modulation
Ith
(b)
0
0
IDC
π/2
Bias Angle (Φ
Laser Forward Bias Current (mA)
0
(c)
(d)
-1
Bias Pt.
Photocurrent (mA)
Optical Transmission
1
3π/2
π
)
DC
T bias
-2
-3
-4
-5
0
Bias = -3.3V
0
V DC
Reverse Bias Voltage (V)
1
2
3
4
5
6
7
Optical Power (mW)
8
9
10
Figure 2.19: Nonlinear device in an optical link. (a) Laser (b) Mach-Zehnder Modulator (c) Electro-Absorption Modulator and (d) Photodetector.
cillation (RO) frequency, as does RIN. The proper method to determine a directly
modulated laser’s nonlinearity is through a rate equation analysis of the laser’s cavity
photon density and carrier concentration, as was done by Lau and Yariv [58]. They
found that the IMD3 to fundamental power ratio increases with frequency at a rate of
20 dB/decade, temporarily saturating at half of the RO frequency, before continuing
the 20 dB/decade trend after the RO frequency. The frequency dependence comes
from the fact that the fundamental signal is independent of frequency (when significantly below the RO frequency), but the IMD3 is proportional to the square of the
frequency.
66
The IMD3 to fundamental power ratio decreases as a function of the laser average optical power. Qualitatively speaking this occurs because the fundamental signal
has only a weak dependence on the average optical power through the RO resonance
(in fact, at frequencies far below the RO frequency, the fundamental signal in direct
modulation is often stated to be independent of average optical power, as we assumed
in section 2.3). However, the IMD3 is inversely related to the average optical power
squared. At higher average optical powers, this relationship begins to breakdown as
laser heating reduces the fundamental gain and increases nonlinear processes. The
IMD3 also depends on parameters such as the ratio between the photon and carrier lifetimes and the laser linewidth (Q-factor). We will not perform the laser rate
equation analysis in this thesis, as it was also covered in [58]. However, we will use
the rate equation analysis to model and fit nonlinearities and dynamic range to our
experimental data in section 5.4.
External Modulation
In external modulation, the physical mechanism does not depend on a dynamic interaction of photons and electrons. As such, we can model the output signal using a
traditional Taylor Series decomposition of the modulator’s transfer function
PEM (IRF ) = PDC +
∞
X
n
cn Iin
(2.63)
n=1
where the coefficients cn define the strength of each spectral component of the signal,
and PDC is the DC offset. Inserting Eqn. (2.62) into the first three terms (n = 1 to
n = 3) of Eqn. (2.63)
PDM = PDC + sLD I0 (cosω1 t + cosω2 t)
2 c3
3
c2
+
I0 (cosω1 t + cosω2 t) +
I0 (cosω1 t + cosω2 t)
2
6
67
(2.64)
Frequency Component
Term
sLD (IDC − Ith ) + c2 I02
DC
sLD I0 + 49 c3 I03
Fundamental (ω1 , ω2 )
2nd Harmonic (2ω1 , 2ω2 )
1
c I2
2 2 0
3rd Harmonic (3ω1 , 3ω2 )
1
c I3
4 3 0
c2 I02
IMD2 (|ω1 ± ω2 |)
IMD3 (2ω1 ± ω2 , 2ω2 ± ω1 )
3
c I3
4 3 0
Table 2.3: Nonlinear Components from a directly modulated laser.
and grouping together all the common frequency components, we can now analytically
determine the 2nd and 3rd order harmonic and IMD terms. Bypassing the algebra,
these are shown in Table 2.3.
We have already determined the coefficients c1 to c3 when we performed the Taylor
series decomposition of the output optical power from an MZM (Eqns. 2.33 - 2.36).
This is repackaged in Table 2.4, for convenience. Inserting these terms into the
expressions in Table 2.3 directly results in an analytical expression for each of the
spurious frequencies. The interesting feature about the externally modulated link
is that the link’s nonlinear components are strongly affected by the choice of DC
bias. When biased at quadrature, an MZM-modulated link exhibits no 2nd order
nonlinearity, making the second harmonic and IMD2 terms non-factors, and enabling
a multi-octave link. Since the DC bias plays a role in dynamic range, noise figure, and
link gain, it is clearly a lever that controls tradeoffs between performance parameters.
2.5.3
Input and Output Intercept Points
What is more useful than the absolute power of a spurious signal is its power relative to
the fundamental frequency signal, which corresponds to the actual signal of interest.
When plotted against input RF power, the fundamental signal grows linearly i.e.,
68
Coefficient
DC
c1
Power Coefficients
PI Tmod
2
PI Tmod
2
c2
PI Tmod
4
c3
− PI T12mod
Evaluated at ΦDC = π/2
1 − cos(ΦDC )
π
Vπ,RF
π
Vπ,RF
Vπ,RF
PI Tmod π
2Vπ,RF
sin(ΦDC )
2
cos(ΦDC )
3
π
PI Tmod
2
sin(ΦDC )
0
− PI T12mod
π
3
Vπ,RF
Table 2.4: Nonlinear Components from an MZM-modulated link
with a slope of 1 dB/dB. Meanwhile, the nth intermodulation and harmonic terms
grow with a slope of n dB/dB (see Table 2.3). At a particular input RF power, the
spurious signal power curve will intercept the fundamental signal power curve; these
points are known as the input intercept points (IIP). An intercept point of particular
interest to RF engineers and link designers is the third-order intermodulation intercept
point (IIP3), as illustrated in Fig. 2.20, because of the relevance of the IMD3 spectral
location. The IIP3’s output counterpart, the output intercept point (OIP3), is simply
the IIP3 scaled by the link gain. Active RF components often specify the IIP3 in
datasheets as a figure of merit for the power handling capabilities of the component.
For multi-ocative links, the IIP2 is also an important parameter.
To calculate the IIPn of an optical link, we set the expressions for the fundamental
signal power equal to the IMDn power (Table 2.3) and solve for the input signal
current or voltage. For example, to find the IIP3, we calculate
3
|c1 |I0 = |c3 |I03
4
(2.65)
where the absolute value of the coefficients are used because they may be negative
(c3 is often negative due to compressive nonlinearity). We could also replace I0 with
69
20
0
-20
l
ta
n
e
nd
u
F
am
3
-40
IMD
Output Power (dBm)
OIP3
-60
IIP3
-80
-80
-60
-40
-20
0
Input Power (dBm)
20
Figure 2.20: IIP3 and OIP3.
V0 in the case of a input voltage signal. Solving for I0 and subsequently the IIP3
s
I0 =
4|c1 |
3|c3 |
I02 R
2 |c1 |
IIP 3 =
=
R
2
3 |c3 |
=
V02 /R
2 |c1 |
=
2
3 |c3 |R
(2.66)
(2.67)
(2.68)
where R is the source resistance. Using Eqn. (2.68) together with the expressions for
the nonlinear coefficients of an externally modulated link (Table 2.4) and link gain
(Eqns. (2.47a) and (2.47b)), we can find the IIP3 and OIP3 for externally modulated
links. These expressions are shown in Table 2.5.
70
Link Type
External
(MZM @ VDC = ΦDC )
IIP3
2
4Vπ,RF
π2 R
OIP3
1
4
2
PI Tmod rP D Gopt sin(ΦDC ) R
Table 2.5: IIP3 and OIP3 of externally modulated links.
We have previously seen that the bias angle has a strong influence on the system’s spurious frequencies (see Table 2.4); let us investigate closer how it affects the
intercept points. Figure 2.21 shows a plot of the fundamental signal, IIP2, and IIP3
of an MZM-modulated link as a function of the bias angle (the fundamental signal is
plotted relative to the fundamental signal when the MZM is biased at quadrature).
Consistent with our previous findings, we see that the fundamental signal gain is
highest at the quadrature point, falling slowly as the bias point shifts in either direction. Perhaps counter-intuitively, the IIP3 is completely independent of the bias
angle; mathematically this stems from the fact that both the fundamental signal and
the IMD have the same sin(ΦDC ) dependence. However, the IIP2 has a strong bias
angle dependence. As we have seen previously, the second harmonic is completely
eliminated at quadrature, because of the odd symmetry of the MZM transfer function. As such, the IIP2 extends to infinity at quadrature, but plunges as the bias
angle shifts away from quadrature. This means that as the bias shifts significantly
away from quadrature, it is unlikely that the MZM optical link can function as a
multi-octave link.
Figure 2.22 shows the fundamental signal and the OIPn points for the same link
as in Fig. 2.21. Since the OIPn points are the IIPn points scaled by the link gain, the
OIP3 takes on a similar shape as the fundamental signal, i.e. a sin2 (ΦDC ) dependence.
Similarly the OIP2 exhibits a sharp drop-off as the bias moves away from quadrature;
however, the divergence still exists at quadrature.
71
40
IIP2
IIP3
Power (dBm)
20
Fundamental
0
-20
Fundamental
IIP2
IIP3
-40
0
π/4
π/2
3π/4
Bias Angle (rad)
π
Figure 2.21: Fundamental Signal, IIP2, and IIP3 vs. bias angle for an MZMmodulated link. The fundamental signal is given in dB relative to the fundamental
signal biased at quadrature.
2.5.4
1-dB Compression Point
Another measure of the power-handling of an active link or component is the 1-dB
compression point (P1dB). The P1dB corresponds to either the input RF power at
which the fundamental signal’s gain decreases, or is compressed, by 1 dB compared
to a perfectly linear system. The P1dB of a link can be calculated by dividing the
modulator’s transfer function by the perfectly-linear transfer function, and calculating
the input RF power when the ratio is equal to −1 dB
Pf ull (P 1dB) = Psig (P 1dB)
72
(2.69)
40
Fundamental
OIP2
OIP3
OIP2
Power (dBm)
20
Fundamental
0
OIP3
-20
-40
0
π/4
π/2
3π/4
Bias Angle (rad)
π
Figure 2.22: Fundamental Signal, OIP2, and OIP3 vs. bias angle for an MZMmodulated link. The fundamental signal is given in dB relative to the fundamental
signal biased at quadrature.
where Pf ull is the full transfer function of the modulator device. We will forgo the
algebra here, as it provides no additional physical insight than that provided by IIP3.
As a rule of thumb, for RF power amplifiers, the P1dB is typically about 10-15 dB
below the link’s IIP3 point.
2.5.5
Spurious Free Dynamic Range
A very common measure of dynamic range in an optical link is the spurious-free
dynamic range (SFDR). SFDR is defined as the input RF power range over which
the link or system can operate without any significant spurious frequencies. On the
lower-end, it is bounded by the input RF power when the fundamental signal drops
73
-50
-90
-130
Fu
a
nd
m
l
ta
n
e
IMD
3
Output Power (dBm)
-10
1-Hz
Noise Floor
Noise
Figure
SFDR
-170
-170
-130
-90
-50
Input Power (dBm)
-10
Figure 2.23: The 1-Hz SF DR3 of an optical link.
below noise floor. On the upper-end, it is bounded by the input power at which the
largest spurious signal rises above the noise floor. Depending on the link, SFDR may
only be concerned with a subset of spurs; for example, in a suboctave link, it does
not make sense to consider any even intermodulation products. Figure 2.23 depicts
the concept of SFDR.
SFDR can be calculated using knowledge of the link noise floor and the link IIP3.
Mathematically, SFDR is expressed as
SF DRn =
n−1
(IIP n − N F + 174)
n
74
(2.70)
where n is the order of the intermodulation product, NF is the link noise figure, B
is the signal bandwidth in dB (i.e., 1 MHz = 106 ) and -174 is the thermal noise
value in dBm/Hz. The units of SFDR, dB/Hz
n−1
n
, are a little odd because SFDR is
bandwidth dependent. The bandwidth dependence comes from the fact that SFDR
takes into account noise floor, which is a bandwidth dependent quantity. To account
for signal bandwidth, a bandwidth term, B, is added inside the parenthesis of Eqn.
(2.70), which represents bandwidth on a dB scale i.e., 1 MHz = 106 . For interference
cancellation, SFDR is a very useful term, because it provides a good representation
of both the lower end and upper end of link functionality.
2.6
Summary
This concludes our discussion of the RF key performance parameters. As this chapter
has shown, there is far more to an interference canceller than simply the amount of
interference it can eliminate. An MPC must play by the rules of every other RF
component, established to maintain proper receiver selectivity and sensitivity.
We have seen repeatedly that several device and circuit parameters wield a significant amount of influence on the RF system performance. Changing these parameters
may yield improvements in some key performance parameters, but at the expense
of others. With the exception a few parameters, such as reducing insertion loss in
the circuit, nearly every parameter comes with engineering tradeoffs. To connect the
dots in this chapter, we present an engineering tradeoff analysis of two key circuit
parameters relating to the Mach-Zehnder Modulator: DC bias angle and Vπ .
First let us examine the bias angle. Figure 2.24 shows how link gain, noise figure,
IIP3, and IIP2 react to changes in the bias angle of the modulating MZM. This figure
provides a succinct summary why the quadrature point is the most common place to
bias the MZM. It optimizes both link gain and IIP2; in addition, since IIP3 is bias
75
independent, it does not matter where the link is biased from this perspective. For
multi-octave links, or where high link gain is required, the quadrature point is the
optimal bias point. However, it is does not correspond to the lowest noise figure. This
belongs, in this particular case, to a bias angle of about .2π. At this bias angle, the
average DC output power, which is proportional to (1 − cos(ΦDC ))2 is significantly
reduced, thus significantly reducing both shot noise and RIN. Link gain, which is
proportional to sin2 (ΦDC ) is also reduced, but not as much as the link noise; thus,
since noise figure is proportional to the ratio of link noise to link gain, noise figure
is lowest at this bias. This technique is called the low-biasing technique for obvious
reasons, and its sole purpose is to optimize the noise figure of a link. If the RF
receiver on hand is sub-octave, so that IIP2 is irrelevant the low-biasing technique is
an interesting tradeoff to make.
Now let us turn our attention to Vπ . Figure 2.25 shows how link gain, noise figure,
and IIP3 react to changes in the Vπ of an MZM biased at quadrature. IIP2 is ignored
in this case, because it is infinite for a quadrature biased MZM link. Vπ is often a
figure that engineers strive to reduce, and for good reason: a smaller Vπ improves
the modulator sensitivity leading to reduced noise figure and increased link gain, as
indicated by the stars in Fig. 2.25. However, even improving the modulator sensitivity
makes a tradeoff, and in this case, it is linearity (IIP3). Intuitively, reducing the Vπ
means that a smaller signal is able to swing the MZM into the nonlinear portions
of the transfer function and away from the linear quadrature point. This can be
compensated externally to the link by reducing the gain of any pre-amplifier, which
therefore increases the overall receiver IIP3. However, this comes, perhaps with no
surprise, at the expense of overall receiver gain and noise figure - the very parameters
a reduced Vπ sought to improve.
As this summary shows, there are tradeoffs abound when it comes to the parameters of an optical link. In the end, only a thorough cascade analysis or simulation
76
Gain, Noise Figure (dB) or IP2/IP3 (dBm)
40
Noise Figure
IIP3
20
0
IIP2
-20
Link Gain
-40
Vπ = 3 V
0
π/4
π/2
3π/4
Bias Angle (rad)
π
Figure 2.24: Link gain, noise figure, IIP3, and IIP2 vs. bias angle of an MZM. The
arrow represents the quadrature bias point; the stars mark the optimal value of the
labeled curve.
of the entire receiver, as well as an assessment of the application requirements, can
determine what performance tradeoffs should be made. For our purposes, we now
have the tools to analyze the MPCs in the following chapters.
77
Gain, Noise Figure (dB) or IP2/IP3 (dBm)
40
ure
Noise Fig
20
IIP3
0
Link Ga
in
-20
Quadrature
1
2
3
V π (V)
4
5
Figure 2.25: Link gain, noise figure, and IIP3 vs. Vπ of an MZM. The stars mark the
optimal value of the labeled curve.
78
Chapter 3
Discrete Microwave Photonic
Cancellers
The first microwave photonic canceller (MPC) was constructed using discrete fiberoptics [3]. In this chapter, we present the next three architectures of the MPC, which
were also built using discrete fiber-optics. These architectures/systems were each
constructed to advance the system in several new directions:
• Electro-Absorption Modulated System. Begin integrating the system by
converting individual components into their semiconductor equivalent. The end
goal: monolithic integration.
• Balanced-Architecture System. Reduce noise figure.
• Wireless System. Investigate the effects of a real wireless environment.
In Chapter 2, we saw how the key performance parameters of basic optical analog
links related to individual optoelectronic devices and circuit parameters. In this
chapter, we will use these building blocks to investigate how each MPC performs
with respect to these key performance parameters.
79
Optical
Intensity
Signal In
Laser
Source
Modulator
Weighting and
Delaying
Photo
Detector
Electrical
Current
Figure 3.1: Concept of a single-ended link architecture.
3.1
EAM Microwave Photonic Canceller
The first discrete MPC we will present is a single-tap canceller, which uses electroabsorption modulators (EAMs) to perform the electrical-optical (EO) conversion and
a balanced photodetector (BPD) to perform the signal subtraction [42]. The choice
of these particular devices is purposeful, and embodies a shift from previous MPC designs composed of many distinct materials to one composed of a monolithic material,
in this case, InP. The system possesses a single-ended architecture, which is characterized by any intensity-modulated direct detection (IMDD) architecture where optical
signals are detected and converted into an electrical signal relative to zero optical intensity, as shown in Fig. 3.1. It is the optical analog of a single-ended electrical signal,
which is measured relative to ground. The advantage of the single-ended architecture
is its simplicity and low component count. However, the single-ended architecture
usually exhibits fairly high noise figures because of the presence of common-mode
noise. A balanced architecture, which will be presented in section 3.2, is a solution
to this problem.
In this section, we present experimental work on a single-ended architecture and
perform an analysis of the key performance parameters discussed in Chapter 2. We
expand on the experimental work by introducing RF equalization to minimize amplitude mismatch, therefore broadening the cancellation instantaneous bandwidth.
80
3.1.1
Circuit Schematic and Experimental Setup
The circuit schematic of the system is shown in Fig. 3.2. The system accepts two RF
inputs: r(t), the corrupted received signal (RX) and n0 (t), a sampled reference copy
of the transmitted signal (TX)
r(t) = s(t) + n(t)
(3.1)
where s(t) is the signal of interest (SOI) and n(t) is the interference after it has
propagated through self-interference channel. The RF signals are modulated into the
optical domain through their respective EAMs. Each EAM modulates a 1551.72 nm,
10 dBm optical carrier, split from a common DFB laser by a 50/50 optical couipler.
The EAMs are DC biased to maximize the link gain, and the RF signals are coupled
with the DC biases though Bias Ts. The modulated optical signals then propagate
through the RX and reference taps of the interference canceller, which correspond to
the upper and lower circuit paths, respectively, in Fig. 3.2.
The EAM MPC uses a single reference tap to model the TX-RX self-interference
channel. The reference tap models the self-interference channel using a voltagecontrolled MEMS variable optical attenuator (VOA) and a mechanically tuned optical
delay line (TDL) with sub-picosecond resolution. To perform coarse phase matching,
an additional 3.8 meters of optical fiber was spliced into the RX path. After tuning
the VOA and TDL to amplitude and phase match the reference tap with the selfinterference channel, the BPD was used to subtract the processed reference signal
from the RX signal. The output of the BPD is an electrical current
y(t) = G r(t) − αn0 (t − τ )
= G s(t) + [n(t) − n0 (t − τ )]
81
(3.2)
Received Signal
EAM
Laser
Source
EAM
3.8m fiber
TDL
VOA
τ
α
Balanced
Photodetector
+
_
Output
Electrical (RF) Signal
Interference Reference
(Tapped from Transmitter)
Optical Signal
Figure 3.2: Circuit schematic of the EAM MPC. TDL = Tunable Delay Line. VOA
= Variable Optical Attenuator.
where G is the overall link gain, α is the attenuation generated by the VOA and τ
is the time delay generated by the TDL. The term in square brackets in Eqn. (3.2)
represents the residual interference after the MPC. The operating frequency of this
method is set by the BPD bandwidth, which is 10 GHz, covering the vast majority of
used RF spectrum today.
3.1.2
Cancellation Performance
To demonstrate the EAM MPCs ability to reject only the interference while recovering
the SOI, the spectrum analyzer method (see Section 2.1) was used to characterize
cancellation depth. A weak, single-tone SOI 10 dB above the noise floor and an inband interferer were generated by two separate signal generators and coupled together
using an RF combiner. This signal served as the corrupted RX signal. The corrupted
RX signal was modulated onto the RX tap, while a tap of the interference signal was
used as the reference signal and modulated onto the reference tap. Cancellation depth
82
10 kHz
10 kHz
SOI
SOI
(a)
(b)
50 MHz
40 MHz
30 dB
40 dB
SOI
SOI
(c)
(d)
Figure 3.3: RF Spectrum of the MPC output without (dotted line) and with (solid
line) interference cancellation for a (a) 900MHz narrowband interferer, (b) 2.4GHz
narrowband interferer, (c) 900 MHz broadband interferer, and (d) 2.4GHz broadband
interferer. SOI = signal of interest.
was characterized for both a broadband and a narrowband interferer. To demonstrate
the wide operating frequency of the system, the same experiment was performed at
900 MHz and subsequently at 2.4 GHz.
The results of the experiment are shown in Fig. 3.3. For narrowband interference cancellation, a 0 dBm, 10 kHz bandwidth signal with center frequency of either
915.5 MHz or 2.4 GHz was used as the interferer. The VOA and TDL were tuned
until the residual interferer power at the output of the MPC was minimized. The
output of the canceller with and without interference cancellation is shown in Fig.
3.3(a) and 3.3(b) for the 900 MHz and 2.4 GHz bands, respectively. In both cases,
the system reduced the interferer by > 65 dB over the entire 10 kHz bandwidth,
83
Simulation
Experiment
Simulation
Experiment
Figure 3.4: RF Spectrum of the interference canceller output without (dotted line)
and with (solid line) interference cancellation activated for a (a) 900 MHz narrowband
interferer, (b) 2.4 GHz narrowband interferer, (c) 900 MHz broadband interferer, and
(d) 2.4 GHz broadband interferer. SOI = signal of interest.
recovering the SOI. Only weak remnants of the interferer are visible above the noise
floor after cancellation.
To measure broadband cancellation, the same experiment was performed except
the CW interfer was swept over a bandwidth of 50 MHz or 40 MHz, within the
915.5 MHz or 2.4 GHz bands, respectively. After coupling with the CW in-band SOI,
the combined RX signal was input into the MPC. After optimization of the VOA
and TDL, the output of the canceller with and without cancellation are shown in
Fig. 3.3(c) and 3.3(d). Within the 900 MHz band, the system achieved 55 dB cancellation over 10 MHz instantaneous bandwidth. Cancellation degraded as bandwidth
increased, showing the cancellation-bandwidth tradeoff, but the system still achieved
40 dB cancellation over 50 MHz instantaneous bandwidth. For the 2.4 GHz band, the
system achieved 45 dB cancellation depth over 10 MHz and 30 dB over 40 MHz. In
both cases, the SOI could be recovered from the interference.
To confirm the measured cancellation, the network analyzer method was also used
to measure cancellation depth. No SOI was used in these experiments. The experi84
mental results, shown by the thick red curve in Fig. 3.4, demonstrated the extremely
wide instantaneous bandwidth achievable by the system. Cancellation of about 30 dB
and 25 dB was achieved across a 400 MHz bandwidth in the 900 MHz and 2.4 GHz
bands, respectively. The cancellation performance was simulated by measuring the
amplitude and phase mismatches between the MPC’s emulated self-interference channel and the actual self-interference channel, and applying Eqn. (2.14). As shown by
the thin blue curve in Fig. 3.4, the simulation and experimental results match well.
The slight deviation between simulation and experiment was caused by drift that occurred in the time between measuring the experimental cancellation and measuring
the amplitude and phase mismatch.
Cancellation depth varies with frequency because of the inadequacy of a singletap system to emulate the self-interference channel, which in this case was a set of
RF cables, splitters, and combiners used to couple together the interference and the
SOI. Reflections within the channel caused the ripples shown in Fig. 3.4, and can be
considered a form of multipath. Complex channel effects can be compensated with
more filter taps, digital signal processing, and RF equalization.
3.1.3
RF Equalization
RF equalizers are analog circuits inserted into communication links to flatten the
channel’s frequency response, and can be used to address both amplitude and phase.
Equalizers are a form of pre-distortion or post-distortion, whereby the channel’s effects are reversed by an equal and opposite distortion of the signal. In interference
cancellation, an RF equalizer can be used to flatten the self-interference channel so
that it can be matched by a relatively simple interference canceller. This effectively
broadens the instantaneous bandwidth.
An RF equalizer was designed and fabricated to equalize the self-interference channel for the EAM MPC. To design the equalizer, the amplitude and phase mismatch
85
Received Signal
EAM
Laser
Source
EAM
3.8m fiber
TDL
VOA
τ
α
Balanced
Photodetector
+
_
Output
Electrical (RF) Signal
RF
Equalizer
Optical Signal
Interference Reference
(Tapped from Transmitter)
Figure 3.5: Circuit schematic of the EAM canceller, with an RF equalizer to account
for channel effects.
Figure 3.6: An RF equalizer to compensate for amplitude mismatch.
86
between the self-interference channel and the EAM MPC’s single-tap reference network was measured, and the equalizer was designed to have an equal but opposite
frequency response (e.g. same amplitude with 180◦ phase shift). The equalizer was
designed into an RF printed-circuit board (PCB), fabricated, populated, and inserted
prior to the reference-tap input of the EAM MPC, as shown in Fig. 3.5. A photograph
of the machined and populated equalizer is shown in Fig. 3.6. The measured amplitude mismatch between the self-interference channel and the EAM MPC’s reference
tap is plotted in the top curve of Fig. 3.7. Prior to the addition of the RF equalizer,
the amplitude mismatch exhibited a slight low-pass behavior. |S21| measurements
showed that the low-pass response originated in the aggregate RF components used to
construct the self-interference channel as well as the difference between the electrodes
of the EAMs used to modulate the optical carrier. An RF equalizer was designed with
a slight high-pass behavior and the result of adding the equalizer to the canceller is
shown in the bottom curve of Fig. 3.7. There are two prominent results: first, the
original low-pass behavior was eliminated; second, smaller ripples were created. These
smaller ripples were a result of imperfect impedance matching between the equalizer
and the self-interference channel.
The cancellation depth of the EAM MPC with and without the RF equalizer
is shown in Fig. 3.8. The RF equalizer significantly improved the average cancellation depth between 0 - 3 GHz because of the equalization of the amplitude mismatch at low frequency. However, this came at the expense of a deep cancellation
trough at 1.3 GHz, and fluctuating cancellation depth. The fluctuations match up
well with the ripples created by the RF equalizer and indicate the importance of
good impedance matching. Removing the fluctuations would require more precise
RF impedance matching engineering as well as higher component tolerances.
Finally, let us consider practicality of using RF equalizers to improve cancellation
depth. Clearly, the use of RF equalizers are effective in flattening a channel response.
87
Figure 3.7: The amplitude mismatch between the interference coupling channel and
canceller before (green curve, top) and after (blue curve, below) the addition of an
RF equalizer.
However, unless the equalizer is tunable, it is only effective for compensating fixed
channels. For dynamic environments, it is much more practical to use digital cancellation techniques, which can be reconfigured with ease. However, digital cancellers
have poor dynamic range compared to their analog counterparts and can only be used
if the RF signals are within the ADCs dynamic range.
3.1.4
Key Performance Parameters
In addition to the cancellation and bandwidth, the other key performance parameters
of the EAM MPC must also examined. Doing so requires knowledge of the effective
Vπ of the EAMs, and consequently, knowledge of voltage vs. transmission curve. The
88
Cancellation Depth (dB)
-10
-20
-30
-40
Equalized
-50
Unequalized
0
0.5
1
1.5
2
Frequency (GHz)
2.5
3
# 10
9
Figure 3.8: Comparison between the cancellation depth before (black, dashed) and
after (red, solid) the addition of the RF equalizer.
voltage vs. transmission curve of the EAM used in the RX tap was measured by using
a 10 dBm input optical signal at 1551.72 nm and measuring the output optical power
as a function of the reverse voltage bias. This curve enables a calculation of the Vπ
figure, the modulator’s optical insertion loss, and the polynomial coefficients of the
nonlinear transmission curve at each bias point. The transmission curve of the RX
EAM is plotted in Fig. 3.9, and the extracted parameters are shown in Table 3.1.
The other link parameters of interest are the photodiode (Discovery DSC-50S)
responsivity, rP D = .72 A/W; the RX tap gain, Gopt = -2 dB; laser RIN = -145
dB/Hz; and, input optical power to each EAM, PI = 10 dBm. Only the RX tap of
the EAM MPC affects the link gain and linearity, since the RX tap carries the SOI.
This, of course, assumes that optical intensity incident upon the photodetector is
89
1
Transmission
0.8
0.6
0.4
0.2
0
-3
-2.5
-2
-1.5
Bias Voltage (V)
-1
-0.5
0
Figure 3.9: EAM Transmission vs. Reverse Bias Voltage. Data markers show experimentally measured values.
below its saturation power; otherwise, the reference tap would need to be considered
as well. We verified that this was the case. With this information, link gain and IIP3
can be calculated by applying the equations for an optical analog link developed in
Chapter 2. For noise figure, however, the reference path needs to be considered since
it contributes thermal noise, shot noise, and RIN to the total noise. Incorporating
these additional noise sources, the noise factor of the EAM MPC can be expressed as
1
F =M+ +
G
M X
qI
2
RIN (IP,i
)R
+
2GR kT
4GR kT
P,i RL
(3.3)
i=1
where M is the number of taps, including the RX tap, IP,i is the detected photocurrent
contributed by the ith branch, and GR is the link gain of the RX tap. For the EAM
90
Parameter
Value
Receive EAM Vπ
1.93 V
Reference EAM Vπ
1.97 V
Receive EAM Loss
-12 dB
Reference EAM Loss
-16 dB
Receive EAM c2
-.0711
Receive EAM c3
-0.4126
Table 3.1: Experimentally measured and extracted EAM Parameters
MPC, M = 2, and IP,i and GR can be computed from the link parameters. Using
Eqns. (2.47b), (2.68),(2.70), and (3.3), the key performance parameters of the EAM
MPC are calculated and presented in Table 3.2. The canceller exhibits quite a low
link gain and a high noise figure, which are both undesirable for the performance of
the RF receiver. This leads to the natural question: what free system parameters can
be tuned to improve the system performance?
The only truly free parameter (i.e. not dependent on device fabrication or system
architecture) is the detected photocurrent, which is a function of the input optical
power injected into the EAMs. The noise figure and link gain of the canceller are
plotted as a function of the input optical power in Fig. 3.10. The experimentally
measured link gain is demarcated by a circle. Clearly, there was room to improve both
the link loss and the noise figure by injecting a higher optical power into the EAMs.
Unfortunately, EAMs, although they offer the opportunity for monolithic integration,
exhibit fairly high intrinsic insertion losses and limited power handling capabilities;
the maximum power that could be handled by these EAMs was 10 dBm. Therefore,
the power handling limit of the EAMs put a ceiling on the link loss and noise figure
in our experiment. In any case, as Fig. 3.10 shows, the noise figure reduction with
optical power would have saturated very quickly because of the matching growth of
RIN, which is the third term in Eqn. 3.3. This embodies one of the major limitations
91
Architecture
Single-Ended
Link Gain
Noise Figure
IIP3
SFDR
(dB)
(dB)
(dBm)
(dB/Hz 2/3 )
-44.9
49.6
17.2
94.1
-10
60
-20
50
-30
40
-40
30
-50
20
30
0
5
10
15
20
Input Optical Power (dBm)
25
Noise Figure (dB)
Gain (dB)
Table 3.2: Key Performance Parameters of the EAM canceller.
Figure 3.10: Gain and Noise Figure of the EAM canceller. The dotted line represents the input optical power used in the experiment. The circle represents the
experimentally measured link gain.
of a single-ended architecture, which is that it exhibits poor noise figure performance,
because of the inability to reject common-mode noise (e.g. RIN).
To summarize, the EAM MPC represented the first step towards monolithic integration. All the optoelectronic devices and components in the canceller could conceivably be integrated, with the exception of the optical time delay (to be addressed
in Chapter 4). In addition, cancellation depths and bandwidths were competitive
92
with the best published results to date [42]. The EAM MPC’s vulnerability lay in its
inability to reject common-mode noise, which resulted in a high noise figure. In addition, the limited power handling capabilities and high EAM insertion losses limited
the link gain.
3.2
Balanced-Architecture System
Prior to this dissertation, all MPCs employed a single-ended architecture [3, 4, 42,
53, 65–67]. Because of their relatively simplicity, these architectures provided a good
testbed for new optical devices and technologies. However, as seen in section 3.1,
they exhibit high noise figures because of the inability to reject common-mode noise
(RIN). One well-established technique from optical link design to suppress RIN is
through the use of a balanced architecture [60, 68, 69]. A balanced architecture is
composed of an IMDD link where the output photocurrent is a differential signal, i.e.
it is proportional to the difference between two optical intensities, as shown in Fig.
3.11.
In this section, we present the design and demonstration of the first balanced
microwave photonic canceller (B-MPC) [70]. The B-MPC uses a balanced photodetector to perform both RIN suppression as well as RF interference cancellation. It
uses wavelength division multiplexing to create a multi-tap system and also exhibits
higher link gain than its single-ended counterpart. We will first explain and present
the B-MPC concept followed by the B-MPC cancellation performance and key RF
performance parameters. Finally, we will compare its performance to an equivalent
single-ended canceller.
93
Signal In
Laser
Source
Positive-Modulated Output
Optical
Intensity
Weighting and
Delaying
+
Weighting and
Delaying
_
Modulator
Electrical
Current
Balanced
Photo
Detector
Optical
Negative-Modulated Output Intensity
Figure 3.11: Concept of a balanced link architecture.
3.2.1
Circuit Schematic and Experimental Setup
The circuit schematic of the B-MPC is shown in Fig. 3.12, and is considerably more
involved than the single-ended architecture. Not only does the canceller employ a
balanced architecture, but it also supports two reference taps, as opposed to the
single reference tap of the EAM MPC in section 3.1. As usual, the system accepts
two inputs: the corrupted receive signal, RXin and the interference reference signal,
REFin .
Each RF signal is modulated onto a set of optical carriers via two dual-output
MZMs biased at quadrature, which output complementary positive and negative modulated optical signals. The creation of the complementary signals will be central
to canceling RIN, while simultaneously doubling the output power of the RF signals. Now following the RX signal path, the RX signal is modulated onto a 16 dBm,
1548.52 nm optical carrier using the left-most MZM in Fig. 3.12. To be consistent
with the previous architecture, will refer to the output of this MZM as the RX tap.
In the RX tap, the optical power at the output of the dual-output MZMs can be
expressed as
94
αR
AWG
α2-
τ2τΝ-
+
α1+
τ1+
α2+
τ2+
λ4
...
...
AWG
αN-
αN+
τΝ+
Balanced
Photodetector
+
Output
AWG
_
AWG
Dual-Output
MZM
λ3
τ1-
...
+
_
...
λ1
PC
AWG
Laser
REFin
λ2
α1-
...
RXin
τR
_
Figure 3.12: Circuit schematic of the balanced microwave photonic interference canceller. PC = Polarization Controller, AWG = Arrayed Waveguide Grating. α is an
attenuation factor and τ is a time delay.
P±R (t)
=
R
PDC
±
R
PAC
R
PDC
= Po Tmod ·
R
R
PAC
= PDC
s(t) + n(t)
1 + RIN
2
πRs
2Vπ
(3.4)
(3.5)
(3.6)
where P±R (t) is the optical power at each of the MZM’s complementary outputs, s(t)
is the SOI, RIN is the laser relative intensity noise, n(t) is self-interference, and the
remaining link parameters are defined identically to Eqn. (2.47b). The small signal
approximation was assumed in Eqn. 3.4. The term PDC in Eqn. (3.4) represents the
common-mode signal, which consists of a DC offset and RIN. This is the component
that we seek to suppress using the balanced architecture. Meanwhile, the term PAC
consists of the SOI as well as the self-interference. Note that PAC is proportional to
PDC , which contains a RIN term. However, this RIN component is not common-mode
noise and cannot be eliminated by the balanced architecture. The strength of this
RIN component will depend upon the statistics (specifically, the variance) of the RF
signals, s(t) + n(t).
95
Referring again to Fig. 3.11, the positive RX signal is weighted and delayed by
a Santec MEMS-based VOA and a fixed delay line, respectively, before entering the
top arrayed-waveguide grating (AWG); conversely, the negative RX signal enters the
bottom AWG. The output of the top AWG enters the positive port of a balanced
photodiode (BPD) while the output of the bottom AWG enters the negative port.
If precise weighting and delaying of the positive RX signal so that the commonmode components between the complementary RX signals are amplitude and phase
matched, the BPD photocurrent contribution from the aggregate RX tap can be
expressed as
R
IR (t) = rP D P+R (t) − P−R (t) = 2rP D GR
opt PAC s(t) + n(t)) .
(3.7)
where rP D is the balanced photodiode responsivity and GR
opt is the RX tap’s optical
gain (assumed identical for each complementary signal due to matching). Equation
3.7 shows that the effect of the balanced link is to suppress the common-mode signal
while doubling the AC component. This results in a 6 dB gain enhancement of the RF
signals relative to the single-ended architecture. Of course, this gain enhancement
applies to both s(t) and n(t), so this will improve the SNR but not the signal-tointerference ratio (SIR).
Let us now track the reference signal in Fig. 3.12. The reference signal is modulated onto a WDM optical carrier, formed by combining the outputs of a bank of
variable-power lasers, by a second dual-output MZM. The output complementary
96
signals, P∓Ref , can be similarly expressed as
P∓Ref (t)
M X
Ref,k
Ref,k
=
PDC ∓ PAC n0 (t)
(3.8)
k=1
Ref,k
PDC
= Pok Tmod ·
Ref,k
Ref,k
PAC
= PDC
1 + RIN k
2
πRs
2Vπ
(3.9)
(3.10)
where M represents the number of wavelengths, or equivalently, the number of FIR
filter taps being utilized by the system, Pok is the input optical power to the k th
wavelength laser, and n0 (t) is the sampled reference signal. In our experiment, M = 2.
Carefully note that the plus-minus sign is reversed in Eqn. 3.8 compared to Eqn. 3.4.
It reflects the architectural detail that the negative output of the reference dualoutput MZM enters the top AWG and vice versa (see Fig. 3.11). This is a reversal
from the configuration of the RX complementary signals and is a key part of the
self-interference cancellation, as we will discuss later.
The reference signals must now be amplitude and phase matched first to each
other to achieve maximum RIN suppression and second to the RX signal to achieve
maximum self-interference cancellation. To do this, after the dual-output MZM, an
AWG is used to split each of the complementary WDM signals into their k component
wavelengths. Each wavelength component is attenuated and delayed by a separate
MEMS VOA and a motorized delay line, respectively, as shown in Fig. 3.11. The
purpose of the MEMS VOA was solely to amplitude match the complementary reference signals to achieve high RIN suppression. The purpose of the delay lines was
first to phase match the complementary signals for RIN suppression.
After the matching for RIN suppression was achieved, each of the k reference wavelength component had to be matched to an interference component in the RX signal
97
for self-interference cancellation. To do this, the motorized delay lines corresponding
to the k th wavelength complementary signals were locked together, so they moved
in unison to maintain the RIN suppression. Then, the delay line pair was tuned to
phase match to a particular self-interference component in the RX signal. To perform amplitude matching between the k th reference tap and a particular interference
component in the RX signal, the laser power corresponding to the k th wavelength
was tuned. Since optical power is proportional to link gain squared for an externally modulated link, the laser power can control the amplitude of the k th reference
tap. Moreover, changing the optical power of the source laser affects complementary
signals identically, maintaining RIN suppression.
After all matching operations had been performed, the k wavelength components
were recombined by the two AWG’s originally used by the RX signal, as shown in
Fig. 3.11. The negative reference tap signals entered the positive port of the BPD,
and vice versa for the positive reference tap signals. If precise amplitude and phase
matching was achieved between the complementary signals, common-mode RIN will
be suppressed, and the BPD photocurrent output contribution due to the k th reference
tap can be expressed as
Ref,k
IRef,k (t) = rP D P−Ref,k (t) − P+Ref,k (t) = −2rP D GRef,k
opt PAC n(t).
(3.11)
Note again that the common-mode signal has been eliminated while the AC signal is
doubled in amplitude. The total BPD output photocurrent, including contributions
from the RX signal as well as all reference taps, can be written as
I(t) = IR (t) +
M
X
IRef,k (t)
k=1
=
R
2rP D GR
opt PAC s(t)
M
X
Ref,k Ref,k
R
R
+ Gopt PAC n(t) −
Gopt PAC n0 (t − τk ) (3.12)
k=1
98
Eqn. 3.12 shows the self-interference can be canceled if the term in parentheses can
be reduced to zero. This condition is essentially the interference cancellation problem
introduced in previous chapters, where the interference canceller channel estimation
is performed by an FIR filter or a tapped-delay line (see Eqn. (2.3) and (2.4)). The
amplitude tuning of the k th FIR filter tap is achieved by tuning the optical power
of the k th wavelength laser. The delay of each filter tap is tuned by the pair of k th
locked delay lines.
We reiterate the importance of amplitude and phase matching on multiple levels.
First, matching within complementary signal pairs is critical for RIN suppression.
Secondly, matching between taps is critical for cancellation of the RF interference.
This parallel shows that the functions of RIN suppression and RF interference cancellation are very similar in this architecture and one could equivalently view the
balanced architecture as a set of nested balanced links.
3.2.2
Cancellation Performance
The system’s cancellation performance was characterized by creating a cabled selfinterference mixing network with the ability to mimic multipath interference. A
two-multipath network was created to test the B-MPC’s ability to cancel multipath
interference. Fig. 3.13 shows the B-MPC’s interference cancellation performance,
as measured by a network analyzer and a spectrum analyzer (inset). Figure 3.13
contains seven traces, each one representing the cancellation depth achieved by the
system when tuned to optimize cancellation from 2 - 8 GHz in 1 GHz increments.
The B-MPC achieved at least 25 dB cancellation depth over 200 MHz instantaneous
bandwidth and could often exceed this level. Performance was better in the mid
to high frequency ranges of 4-8 GHz, where at least 25 dB of cancellation depth over
1 GHz instantaneous bandwidth was achieved. The performance improved in the mid-
99
Cancellation Overview
−15
−25
−30
2 GHz
3 GHz
8 GHz
−35
5 GHz
7 GHz
-70
−40
6 GHz
4 GHz
−45
Before Cancellation
SI
-80
Power (dBm)
Cancellation (dB)
−20
40 MHz
-90
SOI
-100
40 dB
-110
−50
2
3
4
5
6
5970
5980
Frequency (GHz)
50 MHz
SOI
5990
6000
6010
Frequency (MHz)
7
6020
8
6030
Figure 3.13: Experimental cancellation of the balanced interference canceller tuned to
from 2 - 8 GHz in 1 GHz increments. Inset shows cancellation of an in-band interferer
and recovery of an SOI.
to-high frequency ranges because of easier amplitude and phase matching conditions
at these frequencies.
As usual, over smaller instantaneous bandwidths, cancellation depth improved significantly. The overall operating frequency of the interference canceller was limited to
2 - 8 GHz not by optical devices, but by the components of the self-interference mixing
network. The optical setup itself contains components that are rated to operate from
DC to 10 GHz. For the inset in Fig. 3.13, a 50 MHz bandwidth AWGN interferer
with a 6 GHz center frequency and an aggregate power of 6.8 dBm was used. The SOI
was a single-tone signal at 6 GHz, 10 dB above the noise floor. They were coupled
together using the self-interference network, and a copy of the self-interference signal
was used as the reference input of the B-MPC.
100
The cancellation performance of the B-MPC was lower than the single-ended MPC
presented in section 3.1 for several reasons. First, with two multipath components,
precise amplitude and phase matching was required for both multipath components,
increasing the possibility for amplitude and phase mismatch degrading the cancellation depth. Second, if the complementary taps of the balanced links were not
perfectly matched, this not only reduced RIN suppression, but also presented an additional source of amplitude and phase mismatch that degraded cancellation depth.
Ref,k
terms in Eqn. (3.12) into
This can be modeled by converting the GR
opt and Gopt
complex coefficients to model any amplitude and phase mismatch between complementary signals. Thus, one main finding of this experiment was that it was more
difficult to achieve high cancellation depth because each reference tap now required
two sets of amplitude and phase matching. In this experiment, there were 4x as many
matching conditions as compared to the EAM MPC of section 3.1 (2 per multipath
component, and 2 for balanced links). This can be corrected by a precise control
system and manufacturing.
3.2.3
Key Performance Parameters
We now determine the key performance parameters of the B-MPC. Our focus will
be on link gain, dynamic range, and especially noise figure, given the motivation for
the balanced architecture. We also devise a method to determine when the balanced
architecture actually provides an advantage over the single-ended architecture. To
ensure a fair comparison, we compare the experimental performance of the B-MPC to
a simulated single-ended canceller, which uses the same devices and device parameters
(e.g. Vπ ) as the B-MPC, excepted in a single-ended configuration.
We experimentally measured, or extracted from data sheets, all the device parameters used in the B-MPC that were relevant to link loss, noise figure, and dynamic
range. They are shown in Table 3.3. The link gain vs. input optical power for the
101
Link Parameter
Vpi,RF
Photodetector Responsivity
Modulator Insertion Loss
1
Symbol
Value
Vπ (2GHz)
4.87 V
rP D
.72 A/W
Tmod
-5.5 dB
Receive Optical Circuit Insertion Loss Gopt,RX
RX tap Laser Power
-2.5 dB
PRX
Variable
RX tap Laser RIN
RINRX
-170 dB/Hz
Reference Optical Circuit Insertion Loss
Gopt,REF
-4.5 dB
Reference tap Laser(s) RIN RINREF
Reference tap Laser Power
3
PREF
2
-145 dB/Hz
PRX - 1 dB
Table 3.3: Link Parameters of the balanced MPC.
balanced and single-ended MPCs can be modeled by using Eqn. (2.47b) together
with the parameters from Table 3.3. The simulated link gains are plotted in Fig.
3.14, and experimental measurements of the B-MPC link gain are superimposed for
comparison. The measurements showed excellent agreement with the simulated link
gain. Figure 3.14 also indicates that, all else being equal, the balanced MPC holds a
6 dB gain advantage over the single-ended MPC. The 6 dB (4x) difference originates
from the use of a dual-output MZM and the push-pull effect of the BPD and was
derived in Eqn. (3.7) (the factor of 2 in output current results in a factor of 4 in
output power).
The IIP3 of both single-ended and balanced MPCs are given by the expression
for an externally modulated link in Table 2.5. Since the IIP3 is only a function of
the modulator Vπ , regardless of whether it is dual or single-output, the IIP3 of both
architectures are identically 22.8 dBm. However, the OIP3, which does depend upon
the other link parameters, will tilt 6 dB in favor of the balanced MPC because of its
increased link gain.
1
For a single output in a dual-output MZM
All reference tap lasers have identical RIN
3
PREF and PRX are locked together because the RX and REF branches must always be amplitude
matched in interference cancellation.
2
102
6 dB
Figure 3.14: Comparison of the RF gain vs. input optical power for single-ended and
balanced MPCs. The data points correspond to experimental measurements.
Noise figure was the parameter of highest interest for this comparison. To ascertain
noise figure, we once again begin with the definition of noise figure when the input
noise is thermal in origin
F =
hNout i
.
GkT
(3.13)
Subsequently, we write the expression for the total noise at the output of the MPC
including thermal noise, shot noise, and RIN
2
2
2
i + hIsh,RX
i + hIRIN,RX
i R
hNout i = hIth,RX
+
M
X
2
2
2
hIth,REF,k
i + hIsh,REF,k
i + hIRIN,REF,k
i R
k=1
103
(3.14)
where R = RL = Rs is the source and load impedance, and the subscripts RX
and REF are used to describe the corresponding variable for the RX and reference
taps, respectively. The noise sources can be incoherently summed because they are
independent. The analytical expression for the noise factor, F, can now be written in
a straightforward manner by substituting Eqns. (2.52), (2.53), and (2.56) into Eqn.
(3.14) and accounting for RIN suppression
1
F =1+
+
GR
+
2
)R
q(IRX )R γRX RINRX (IRX+
+
2GR kT
4GR kT
M X
G
REF,k
GR
2
q(IREF,k )R γREF,k RINREF (IREF
+,k )R
+
+
2GR kT
4GR kT
(3.15)
k=1
where GR and GREF,m are the RX and k th reference taps’ link gains, respectively; M is
the total number of reference taps; IRX is the average photocurrent contribution from
the RX tap; IREF,k is the average photocurrent contributions from the k th reference
tap; IRX+ and IREF +,k are the average photocurrent contributions from just the positive RX and k th reference signals; and γRX and γREF,k describe the RIN suppression
of the RX and k th reference tap, respectively. All variables are expressed in linear
units. The RIN suppression can be determined using the expression
γ = 1 + (∆α)2 − 2∆αcos(∆Φ)
(3.16)
where ∆α and ∆Φ represent the amplitude and phase mismatch between complementary signals within a particular reference tap. This expression is identical to Eqn.
(2.14), which describes the cancellation depth achieved by a single-tap interference
canceller. This is no coincidence, as RIN suppression can be viewed as a form of
interference cancellation itself. Thus, Fig. 2.4 can be used to convert amplitude and
phase mismatch to RIN suppression and give a sense of what level of precision is
required to achieve a certain amount of RIN suppression. It is convenient to treat
104
the term γRIN as a new variable called equivalent RIN
RINeq,RX = γRX RINRX
(3.17)
The equivalent RIN can be thought of as the RIN of an equivalent laser that, when
used in a single-ended architecture, would generate the same amount of RIN as the
balanced architecture using the actual laser RIN. Equivalent RIN is useful to enable
a comparison of the effect of RIN between different architectures. Note that the RIN
suppression only applies to the common-mode RIN, but not the RIN associated with
the AC signal. We do not account for this component of RIN, which depends on
the signal statistics; however, as we will see in our experiment it turned out to be
negligible compared to the common-mode RIN.
The noise figures of both balanced MPC and single-ended MPC architectures can
be simulated by using Eqn. (3.15) together with the RF gains measured in Fig.
3.14 and the link parameters in Table 3.3. Fig. 3.15 shows a comparison of MPC
noise figures at 2 GHz RF frequency between the single-ended and balanced MPCs,
assuming 30 dB of RIN suppression, as a function of the RX tap input optical power.
A single reference tap is used. Experimental measurements of the balanced MPC noise
figure are overlaid onto the plot and show excellent agreement with the simulated
noise figure. The advantages of the balanced architecture are now clear: at 16 dBm
input optical power, the balanced MPC not only exhibited a 6 dB gain advantage,
but also a 9 dB noise figure advantage. Moreover, the noise figure advantage grew
with increasing RX tap optical power. We now summarize the key differences in noise
figure between the two architectures:
• The balanced MPC exhibits a lower noise figure than an identical
single-ended MPC at all input optical powers. This stems from the
105
9 dB
Figure 3.15: Comparison of the noise figure vs. RX tap input optical power for the
single-ended and balanced MPC architectures, assuming 30 dB of RIN suppression.
The data points correspond to experimental measurements.
suppression of RIN in the balanced MPC as well as the 6 dB gain advantage
due to the push-pull effect.
• At high optical power, with sufficient RIN suppression, the MPC
noise factor decreases with a
1
x
relationship with input optical power.
When RIN is suppressed, the MPC becomes a shot-noise limited link, which exhibits this type of behavior with optical power (see 2.4). The noise figure can
now be reduced by increasing the optical power until residual RIN is reasserted
as the dominant noise source or thermal noise limits are reached. This is the
ultimate benefit of the balanced architecture. In comparison, the single-ended architecture remains RIN-limited and therefore, noise figure saturates with optical
power.
106
• At medium optical powers, the difference between the balanced and
single-ended MPC noise figures reaches its minimum value because
both architectures are shot noise-limited. Since the detected photocurrent is
twice as high for the balanced architecture than for the single-ended architecture, the extra shot noise begins to offset the 6 dB link gain of the balanced
architecture. In the worst case scenario, the balanced MPC will exhibit only a
3 dB noise figure advantage over the single-ended architecture.
• At low optical powers, both MPC architectures are thermal noiselimited, and RIN cancellation becomes irrelevant. As a result, the noise figure
difference between the balanced and single-ended MPC approaches 6 dB: the
RF gain advantage of the balanced MPC.
Considering the different regimes of noise-limited performance and the additional
complexity of the balanced architecture, we now address the question of when a
balanced architecture is advantageous compared to a single-ended architecture. The
main benefit of the balanced architecture is its ability to suppress RIN and lower
MPC noise figure by increasing optical power. Circuit designers should therefore be
aware of three parameters when determining which noise-limited regime they operate
in, and how much advantage to expect:
• The laser RIN
• The amount of RIN suppression achievable
• The maximum optical power handling and sourcing capability of the system
Since each noise component (thermal, shot, and RIN) exhibits different behavior as
a function of optical power, a plot of the slope of noise figure vs. optical power can
reveal which noise component is dominant. Such a plot for the balanced MPC is
shown in Fig. 3.16 for different levels of RIN suppression. The longer the noise
107
Noise Figure vs. Optical Power Slope
0
RIN
-0.5
Shot
-1
-1.5
Balanced: 30 dB suppression
Balanced: 20 dB suppression
Balanced: 10 dB suppression
Balanced: 0 dB suppression
Single-Ended Architecture
Thermal
-2
0
10
20
30
40
50
Receive Branch Input Optical Power (dBm)
Figure 3.16: Noise figure slope vs. input optical power. Regions where each type of
noise dominate are explicitly shown.
figure slope remains in the shot-noise limited regime, the more noise figure can be
improved by increasing optical power using a balanced architecture. Put another
way, the larger the span of optical power in which the balanced architecture exhibits
a lower magnitude noise figure slope than the single-ended architecture, the more
advantageous it becomes to use the balanced architecture. Thus, for our balanced
MPC, which exceeded 30 dB of RIN suppression, there was an large amount of noise
figure enhancement that we had not yet harvested. Unfortunately, this was limited
by the maximum power output of the RX tap’s laser, which is the reason why having
an understanding of the maximum optical power handling is important. Eventually,
all slopes tend to zero as either residual RIN becomes dominant or the thermal noise
limit is reached.
108
Architecture
Link Gain
(dB)
Noise Figure
(dB)
IIP3
(dBm)
SFDR
(dB/Hz 2/3 )
Single-Ended
Balanced
-44.9
-23.2
49.6
32.2
17.2
22.8
94.1
109.8
Table 3.4: Comparison of Key Performance Parameters
Another question of interest is how the advantage of the balanced architecture
scales with the number of reference taps. A significant advantage of the WDMfriendly architecture of the balanced MPC (see Fig. 3.11) is that the tap count
can be scaled up fairly easily. Each AWG can handle up to 8 distinct wavelengths.
Figure 3.17 shows the simulated noise figure of both a single-ended and a balanced
MPC (with 30 dB of RIN suppression) for up to 4 reference taps. It shows that
the same benefits of the balanced architecture for the single-tap case extend to the
multi-tap cases. In fact, the balanced MPC noise figure scales better with more taps
at medium optical powers. At high optical powers, 30 dB RIN suppression for this
particular interference canceller is unique in that thermal noise and shot noise are
comparable in magnitude so that the balanced MPC scales worse than the singleended architecture for higher tap counts. It can be shown that if RIN suppression
is significantly less than 30 dB, both single-ended and balanced architectures exhibit
the same relationship between noise figure and number of taps; on the other hand, if
RIN suppression is significantly higher than 30 dB, the balanced architecture scales
better than the single-ended architecture.
Finally, the single-ended EAM from section 3.1 and the balanced MPC are compared across the key performance parameters in Table 3.4. The significantly higher
SFDR results from the higher IIP3 and the lower noise figure. In summary, the balanced architecture yields advantages over the single-ended architecture across all key
performance parameters. However, it involves significantly higher complexity and
requires more advanced control to maintain its cancellation performance.
109
50
Noise Figure
40
5.5 dB
30
3 dB
20
10
1 Tap
2 Taps
3 Taps
4 Taps
8 dB
30 dB RIN Suppression
0
0
10
20
30
40
50
Receive Branch Input Optical Power (dBm)
Figure 3.17: Noise figure vs. input optical power for multi-tap MPCs. 30 dB of RIN
suppression is assumed in the balanced case.
3.3
Wireless Self-Interference Cancellation
Up to this point, all MPCs have been tested on stable, cabled self-interference channels. However, a practical RF environment is chaotic and significantly more complex.
In addition, an MPC is only useful if it can function within an actual receiver. In this
chapter, we perform a high-level experimental study of both of these issues. First,
we examine how an MPCs’ performance changes when facing a real RF environment.
Second, we investigate if and how much performance enhancement an MPC actually
enables in a digital receiver.
110
3.3.1
A Wireless Microwave Photonic Canceller
Background
To model the self-interference channel we adopted the tapped-delay line model (see
Fig. 2.3) [9], in which the impulse response of the self-interference channel is approximated by its sampled version
h(t) u
K
X
h(t)δ(t − τk )
(3.18)
k=1
For a cabled self-interference channel, the impulse response exhibits low dispersion
and is not very frequency selective. It is much more challenging to model a real,
wireless self-interference channel because of the additional frequency selectivity. As
shown in Fig. 2.4, even relatively small variations of amplitude and phase, on the level
of 0.1 dB and 1◦ are sufficient to reduce cancellation depth below 30 dB. Variations
of this magnitude are many and close together in wireless channels, affecting both
cancellation depth as well as instantaneous bandwidth.
There are two dominant sources of additional frequency selectivity present in a
wireless channel that are not present in a cabled channel. First, the unguided air
interface is a much more complex environment than a guided channel, because of
the many multipath components, big and small, that arrive over a spread of delays.
In wireless communications, this is known as small-scale fading, where the small
refers to the physical distances over which it acts. When there are many multipath
components, the self-interference channel can be modeled statistically as a Rician
fading channel, because of the presence of a dominant multipath component - in
this case, a very strong line-of-sight (LOS) signal [9]. This is true whether there
are separate transmitter and receiver antennas, or a single antenna duplexed by an
isolator. A plot that reveals information about the self-interference channel is the
power delay profile (PDP). The PDP shows the temporal-dispersion of power from
111
Figure 3.18: Measured power-delay profile of an anechoic chamber vs. a reflective
room. Taken from [5].
the transmitter to the receiver. A large delay spread requires an equally large time
delay range in an MPC to match. Note in Fig. 3.18 that the delay dispersion is much
larger for a reflective room than for an anechoic chamber because of the additional
dispersion added by the multipath reflections.
The second source of frequency selectivity is passive suppression. Passive suppression describes the loss between the transmitter and receiver that provides an
initial level of self-interference cancellation prior to active analog cancellation. For
two separate TX/RX antennas, passive suppression can include path loss by physical
separation, a purposefully placed absorber, polarization, and antenna directivity. For
a duplexed antenna, the passive suppression will only consist of the isolator isolation, which is typically ∼ −20 dB. Passive suppression is typically only effective at
112
eliminating the direct LOS signal, because reflections tend to scramble the signal’s
angle of arrival and polarization. As such, a passively suppressed self-interference
channel does not have a dominant LOS component [5], as shown in Fig. 3.19, and is
dominated by multipath components, resulting in large frequency selectivity. Such a
channel is better modeled by a Rayleigh fading channel.
A common measure of frequency selectivity is the coherence bandwidth of a wireless channel. The coherence bandwidth describes the bandwidth (full-width at half
maximum) over which the relative frequency response of a wireless channel remains
constant. It is related to the delay spread as
Bch u
1
2πSτ
(3.19)
where Sτ is the delay spread. Equation (3.19) is an approximation, because delay
spread often has multiple definitions and is more of a heuristic quantity. An extensive amount of research has been invested into quantifying and modeling wireless
channels. The overview above is meant to provide the basics necessary to interpret
the following results; a full description can be found in one of many wireless communications textbooks such as [9].
Circuit Schematic and Experimental Setup
We now demonstrate over-the-air self-interference cancellation using a wireless microwave photonic canceller (WMPC), which is a discrete MPC that cancels the dominant LOS self-interference as well as up to 6 multipath components [71]. The WMPC
was tested on on a two-antenna transceiver in an outdoor wireless environment. We
investigated the WMPC’s ability to cancel interference, both LOS and multipath,
with the goal of understanding the challenges associated with operating an MPC in
a wireless environment.
113
Figure 3.19: Frequency response of a channel with different levels of passive suppression. The legend refers to different passive suppression techniques. Taken from [5].
Figure 3.20 shows a block diagram of the WMPC and how it interfaces with the
front-end of an RF communication system. Consistent with previous MPCs, the
WMPC takes two RF inputs, a reference signal and a corrupted RX signal. The
RX signal was sourced directly from the RX antenna, while the reference signal is
obtained by tapping the TX output using a balun. A balun is a 3-port RF device,
which converts a balanced signal into an unbalanced signal and vice versa. In this
case, we used the balun to convert an RF signal into a copy of the original signal
and and its inverted version. The non-inverted balun output drives the transmitting
antenna, while the inverted balun output provides the reference signal to the WMPC.
Both RF inputs are subsequently modulated onto optical carriers at 1552.56 nm
and 1550.55 nm, respectively, using two separate electroabsorption modulated lasers
(EMLs). The 1552.56 nm optical carrier, which carries the reference signal, is pro114
Figure 3.20: Block diagram of the WMPC. System inputs and outputs are underlined.
αi and τi represent the attenuation and delay of the ith tapped-delay line.
115
cessed using a FIR filter/tapped-delay line that emulates the self-interference channel.
The FIR filter is implemented by first splitting the signal into N taps using a 1:N
optical splitter. Each tap is independently attenuated by a MEMS VOA and delayed
by a combination of fixed and tunable optical delay lines (TDLs).
Following the FIR filter, each processed reference tap and the RX tap are separate
inputs to a custom-built 8x1 singlemode to multimode (SM-MM) combiner. The
SM-MM combiner maps each singlemode input onto an orthogonal spatial mode of
a multimode fiber [72]. Since each optical signal propagates in an orthogonal spatial
mode, none of the optical carriers interact coherently. This eliminates the need for
a separate optical source (wavelength) for each multipath component as well as the
associated multiplexers/demultiplexers that would be necessary otherwise, greatly
improving the scalability of the system.
Finally, the aggregate optical signal is detected by a multimode photodetector
(MM-PD), which incoherently sums the component optical signals. Since the reference signal was originally inverted by the balun, this subtracts the processed reference
signal from the corrupted RX signal. The system output can be expressed as
y(t) = xRX (t) −
N
X
αi n0 (t − τi )
i=1
= s(t) +
X
N
ni (t) −
i=1
N
X
αi n0 (t − τi )
(3.20)
i=1
where s(t) is the SOI, ni (t) is the ith multipath component, n0 (t) is the copy of
the reference tap, and αi represents the attenuation of the ith reference tap, including
any splitter losses. Optimal cancellation occurs when the term in brackets approaches
zero, such that y(t) → s(t).
In this work, two cancellation taps (N = 2) were demonstrated: one for the LOS
and another for the next dominant multipath component. The VOAs and TDLs
of the FIR filter were tuned by observing the output signal and optimizing for the
116
Figure 3.21: Diagram of the outdoor antenna setup, courtesy of Bascom-Hunter Technologies. Inset: photo of the actual setup.
maximum cancellation in the SOI frequency band. Two optical sources were used
in this experiment only because the optical sources were already integrated with the
EMLs. A single optical source could have been used without coherent interference.
Figure 3.21 show the outdoor antenna setup, which was used to test the WMPC.
A MARS MA-WA25-DP17 directional antenna was used as the RX antenna and an
omnidirectional Ettus VERT2450 antenna was used as the co-located TX antenna.
A reflective block was used to intentionally generate multipath RX-TX components.
A second directional MARS MA-WA25-DP17 antenna was used to generate the SOI.
117
The WMPC was connected to the outdoor antenna setup through a wall-port that
bridged the indoor and outdoor environment.
Experimental Results
In this experiment, the SOI was a weak single-tone signal at 2.39 GHz produced by a
signal generator. The power level of the SOI was chosen such that the received SOI
power was 5 dB above the receiver noise floor. A single-tone SOI was used so that
the SOI and the interference spectra could be visually distinguished; any modulation
format could be used so long as the SOI remained within the bandwidth of the
receiver. The interferer was a swept single-tone signal from 2.35 GHz to 2.43 GHz,
generated by a separate signal generator. It was transmitted such that the interference
power at the receiver was 25 dB above the receiver noise floor when no cancellation
was applied.
The output signal was measured by a spectrum analyzer using a max-hold acquisition to examine the effects of the WMPC upon both the interference signal as well
as the SOI. Fig. 3.22 shows the output spectrum for three separate conditions:
1. No cancellation applied (reference tap EML turned off).
2. Only LOS cancellation (reference tap EML turned on, but first multipath tap
disconnected).
3. Both LOS and first multipath cancellation
As shown in Fig. 3.22, simply implementing the LOS cancellation achieved 17 dB
of cancellation at the SOI frequency. Although significant, canceling the LOS alone
was insufficient because the residual interference was still stronger than the SOI. The
addition of the second reference tap enabled the system to achieve an additional
10 dB of cancellation, reducing the received interference to the spectrum analyzer
118
Figure 3.22: Overlay of the WMPC output signal spectra without cancellation (green
dashed curve), with LOS cancellation (blue dotted curve) and with both LOS and
the multipath cancellation (red curve). SOI = signal of interest.
noise floor at the SOI frequency. As indicated in the call-out of Fig. 3.22, the SOI is
now recoverable from the residual self-interference.
The same experiment was repeated and characterized using a network analyzer,
which provided higher sensitivity than the spectrum analyzer. Port 1 of the network
analyzer was used as the interference source and Port 2 of the network analyzer was
connected to the output of the WMPC. In this configuration, the network analyzer’s
|S21| can be used to study the effect of the WMPC on the self-interference signal; no
SOI was used.
Figure 3.23 shows the results of this measurement with only LOS cancellation and
both LOS and first multipath cancellation. The cancellation profile confirmed that
the inclusion of the multipath cancellation added > 10 dB of cancellation depth to the
total cancellation depth compared to that achieved by only LOS cancellation. With
both LOS and first multipath cancellation, the WMPC achieved 35 dB of cancellation
119
Figure 3.23: Self-interference cancellation as measured by a network analyzer for (A)
only LOS cancellation, (B) both LOS and first multipath cancellation and (C) LOS
cancellation without the multipath-generating reflector. Line 1 shows the difference
between cancellation achieved with and without the multipath cancellation in the
presence of a reflector, and line 2 demarcates the instantaneous bandwidth where
30 dB of cancellation is achievable.
depth over an instantaneous bandwidth of 10 MHz at the SOI frequency, and 30 dB
of cancellation across 20 MHz of instantaneous bandwidth. The cancellation profile
was significantly smoother than the spectrum shown in Fig. 3.22 because of the use
of 10-point averaging and 1% smoothing by the network analyzer. To confirm that
it was the multipath component that was being canceled, the reflector was removed
and the measurement was performed with only LOS cancellation. The cancellation
profile, also plotted in Fig. 3.23, exhibits a similar shape to the cancellation profile
of both LOS and first multipath cancellation with the reflector present.
Let us now examine the performance of the WMPC in the context of a wireless
channel. The results of the WMPC experiment showed that multipath interference
cancellation is an absolute necessity when operating an MPC in a real wireless en-
120
vironment. While the two-reference tap system in the WMPC certainly performed
better than a single-tap system, there was still residual interference, particularly at
frequencies farther from the SOI frequency. This residual interference must be accounted for, since its spectral proximity to the SOI will likely prevent downstream
filters (e.g. IF filters) from removing it. The residual interference has two primary
sources. The first source is uncompensated multipath components. The reflector used
in the experimental setup created a strong multipath component, but many other,
weaker, multipath components were present by nature of the wireless environment.
Once the LOS was canceled, these residual multipath components were revealed in
the form of the frequency selective residual interference spectrum (see Fig. 3.22 green
curve vs. red and blue curves). The second source of interference is the relatively
complex frequency response of the radiating antennas, which exhibit a much larger
frequency selectivity compared to a guided cable. This is a source of interference
because the two-tap FIR filter does not contain enough degrees of freedom to match
the antennas frequency response.
The additional frequency selectivity of the wireless channel can be addressed
through a combination of additional WMPC analog taps as well as digital interference cancellation. The SM-MM combiner has a total of 8 inputs, corresponding
to the corrupted RX signal, the LOS reference tap, and 6 multipath reference taps.
Therefore, there were 5 unused SM-MM inputs for this particular SM-MM combiner.
A logical question to ask is “how many taps are required?”. A study of the coherence
bandwidth and the number of taps required to model a 20 MHz self-interference channel is shown in Fig. 3.24 [5]. The number of taps required were calculated by simply
dividing the channel bandwidth by the channel’s coherence bandwidth. The study
shows that as the suppression of the main LOS is increased, the frequency selectivity
of the remaining multipath components become more dominant. As a result, the coherence bandwidth decreases, increasing the number of taps required. The number of
121
Figure 3.24: (Left) Coherence bandwidth of a self-interference channel and (Right)
number of taps required to cancel self-interference across a 20 MHz instantaneous
bandwidth as a function of the suppression of the LOS signal. Results are shown for
different passive suppression techniques and RF environments. Taken from [5].
taps required is greatly increased for a reflective RF environment (e.g. indoor office),
which has more frequency selectivity (see Fig. 3.19). The number of required taps in
this environment quickly increases to upwards of 20. The situation is not as bad for
a non-reflective environment and is more representative of an outdoor macrocellular
environment due to the lack of scattering elements.
In scenarios where a large number of multipath components are present, postprocessing the signal with a digital interference canceller is an attractive solution.
Digital cancellers are well-suited for dynamic wireless environments, because of the
large number of rapidly reconfigurable digital taps. These are ideal for multipath
components, which may be weak because of a large number of reflections in the
wireless channel, but consequently, highly dynamic. This approach is only feasible
if the residual interference power is weakened enough so that the limited dynamic
range of the ADC can still be used to resolve the SOI. The WMPC and the digital
canceller are thus very complementary. The WMPC acts as a high-dynamic range
front-end that greatly reduces the interference dynamic range prior to hand-off to a
digital interference canceller, which cancels the weak residual multipath that cannot
be addressed by the WMPC’s limited number of taps.
122
3.3.2
Performance in a Digital Radio
The second issue we wish to investigate is the performance of the MPC in the analog
front-end of an actual receiver [73]. In particular, we wish to answer two questions:
1. Does the MPC disrupt normal digital receiver functionality?
2. How much does the MPC improve the receiver’s self-interference rejection?
The Ettus X310 Software Defined Radio
For a digital receiver, we used an Ettus X310 Universal Software Radio Peripherals
(USRP) as a flexible digital radio [74]. The X310 consists of two separate radio units,
each capable of transmitting and receiving digitally encoded signals. Each RX frontend contains a low-noise amplifier (LNA), followed mixers and intermediate frequency
filters required to downconvert the signal from RF into the IQ baseband. We used
Quadrature Phase Shift Keying (QPSK) as the digital modulation format combined
with Orthogonal Frequency Division Multiplexing (OFDM). OFDM is the most commonly used frequency multiplexing technique today and is used in both WiFi and
LTE standards.The X310 transmits in packets, where each packet consists of a certain number of frames. Each frame, in turn, contains of a number of OFDM symbols
in the data payload, which are the IFFT of the QPSK-encoded data modulated onto
all subcarriers. A full description of OFDM can be found in [56].
During normal operation, after the analog signal has been sampled and converted
into the digital domain, the X310 performs several standard digital signal processing
operations to demodulate the OFDM signal. First, it synchronizes the subcarrier frequencies, which can drift due to the Doppler effect, with the local oscillator frequency
and aligns the received packets in time in order to locate the header of each packet,
which contains channel information. Subsequently, the header and payload frames
are separated from each packet, and an FFT is performed to demodulate the data.
123
The X310 then performs its own internal channel estimation using pilot signals built
into the header packets to correct for any channel fading. After correcting for the
channel, the data is sent through a constellation decoder, which makes the hard decision about what QPSK symbol the data represents. The output of the constellation
decoder is the raw bit stream.
To characterize the improvement that the MPC has on the X310’s interference
rejection, we operate the X310 in same-channel full-duplex mode and study how the
introduction of the MPC improves the receiver bit error rate (BER). In full-duplex
mode, the X310 is set to simutaneously transmit and receive on the same band. In
this configuration, there is zero receiver protection from self-interference (the X310
only uses a bandpass filter in the front-end to select the RF channel to receive or
transmit on). Even though the X310 has built-in channel estimation, it does not
protect against self-interference because self-interference is an unknown and, for all
intents and purposes, random data stream, which cannot be modeled as a channel
effect. Therefore, we expect the BER to be very poor without hte MPC because of
the low signal-to-inteference ratio.
Circuit Schematic and Experimental Setup
The experiment used both radio units of the X310; one was operated in full-duplex
mode (herein called the full-duplex radio) while the other transmitted an SOI (herein
called the SOI radio) for the full-duplex radio to receive. The full-duplex radio used
a single directional antenna (Southwest Antennas Broad Panel Antenna with 12 dBi
gain and a 55◦ horizontal beamwidth), which was duplexed via an RF circulator; the
SOI radio transmitted through an omnidirectional antenna (L-Comm HG2403U). A
discrete MPC with a 2-tap FIR filter was inserted into the analog front-end of the
full-duplex X310 radio, after the RX antenna, as shown in Fig. 3.25. Ideally, the
MPC should have been inserted after the LNA, but because the LNA was integrated
124
Signal of Interest
Antenna
Transmit Only
Full-Duplex
Antenna
Transmit & Receive
Ettus Research X310
USRP Software Defined Radio
Figure 3.25: MPC inserted into the analog front-end of the full-duplex mode X310
USRP.
into the X310 receiver board, this was not possible. A 50/50 RF splitter was used to
sample the TX signal prior to entering the RF circulator.
The full duplex radio suffered self-interference from three primary sources: antenna return loss (S11), circulator leakage, and multipath backscatter. Of these
sources, the first two sources were the strongest, as each underwent only ∼ 20 dB
of attenuation prior to entering the receiver. To model and cancel these sources
of self-interference, the MPC used its 2-tap FIR filter, where each tap was used
to compensate a separate source of self-interference. Each reference tap was tuned
to emulate a separate interference source’s channel through a MEMS VOA and a
mechanical TDL. After tuning, the reference taps were combined by a 50/50 optical
splitter and then incident upon the negative port of a balanced photodetector (BPD).
This subtracted the processed reference signal from the corrupted RX signal, which
propagated through the RX tap into the positive port of the BPD. Each reference tap
used a different wavelength optical carrier to avoid coherent beat noise. The output
of the MPC was used as the input into the X310 receiver, as shown in Fig. 3.25.
125
Lab
USRP2
TX
USRP1
TX RX
MPC
RF tap
Circ.
Hallway
Doorway
Signal of Interest
Antenna
(Omni)
Full-Duplex
Antenna
(Directional)
3m
Figure 3.26: The hallway setup used to test the digital radio with MPC.
The full-duplex radio antenna and the SOI radio antenna were separated by 3
meters in a hallway and connected to the X310 RF I/O through long RF cables, as
shown in Fig. 3.26. The full-duplex antenna was pointed directly at the SOI antenna
in a direction parallel to the hallway, to minimize the amount of multipath backscatter
back into the full-duplex antenna. Both radios were programmed to simultaneously
transmit, at the same frequency, different pseudo-random binary sequences (PRBS).
The experiment was performed twice, at center frequencies of 2.21 GHz and 2.46 GHz,
for a signal bandwidth of 1 MHz.
Experimental Results
After optimizing the cancellation of the MPC, the bit-error rate (BER) of the SOI and
the self-interference signal were measured within the full-duplex radio as a function of
the SOI-to-interference ratio (SIR), which was controlled by changing the TX power
amplifier gain of the SOI radio. The results of the BER measurement are shown
126
10
Signal of Interest BER
0
(a)
10-1
(b)
10-1
10-2
10-2
10-3
BER
BER
Signal of Interest BER
100
22 dB
10-4
10
-3
23 dB
10-4
10-5
10-5
10-6
10
With Canceller
Without Canceller
2.21 GHz
10-7
-15
-10
-5
0
5
10
15
-6
With Canceller
Without Canceller
2.46 GHz
20
Signal-to-Interference Ratio (dB)
10-7
-20
-15
-10
-5
0
5
10
15
Signal-to-Interference Ratio (dB)
Figure 3.27: BER of the SOI as a function of Signal-to-interference ratio at a center
frequency of (a) 2.21 GHz and (b) 2.46 GHz.
in Fig. 3.27 both with and without cancellation. These results allow us to make
two key conclusions. First, the digital radio was able to operate using its standard
digital signal processing even in the presence of the MPC in the analog front-end.
Second, we determined that the activation of the MPC improved the self-interference
rejection by over 22 dB at both RF center frequencies. In fact, with the MPC, the
SOI could be received despite having a negative SIR. This was a very positive sign
for the full-duplex radio, which simply could not operate without the MPC, unless
the SOI power was increased enough to brute force overwhelm the self-interference,
which is very unrealistic.
4
Note that the BER could only be measured as low as
10−6 , because the bit streams that were transmitted were only 106 bits long.
To confirm that the MPC was canceling the self-interference, we repeated the
measurement, except this time measuring the BER of the self-interference at the fullduplex receiver. Since the receiver did not have any inherent knowledge of the selfinterference or the SOI, it simply hard-decoded the signal based on the constellation
4
We must note, however, that in a half-duplex system, the MPC actually harms the receiver
sensitivity because of the relatively high noise figure of the MPC. Thus, the MPC should only be
used if the self-interference is strong and is the factor that limits SIR.
127
Simulation
Experimental
-5
-10
-15
-20
-25
-30
-35
3
Amplitude Mismatch (dB)
-40
2400
2
2420
2440
2460
2480
Frequency (MHz)
560
1
0
580
2500
Zero Mismatch
540
-1
520
Phase Mismatch (deg)
Interference Cancellation (dB)
0
-2
500
-3
2400
2420
2440
2460
2480
2500
Frequency (MHz)
Figure 3.28: (Top) Cancellation depth and (bottom) amplitude and phase mismatch.
decoder. Thus, we could measure the BER by comparing the received signal’s bit
stream to the transmitted signal’s bit stream. We found that, indeed, with the MPC
activated, the BER of the self-interference signal was .5 at all SIR ratios, confirming
that the MPC is canceling the self-interference to enable reception of the SOI.
To determine how effective the MPC was at modeling the self-interference channel, we measured the cancellation depth as a function of frequency using a network
128
analyzer and subsequently simulated the cancellation depth by measuring the amplitude and phase mismatch between the MPC’s FIR filter and the self-interference
channel. The results, shown in Fig. 3.28, indicate that the MPC is able to cancel
about 20 dB of self-interference over a 20 MHz bandwidth, with some nulls as deep as
35 dB. This matches well with the additional interference rejection measured in Fig.
3.27. The amplitude and mismatch measurements reveal the frequency selectivity
of the wireless self-interference channel, which cannot be addressed by a 2-tap FIR
filter. Similar to what was observed in section 3.3, the frequency selectivity and the
smaller coherence bandwidth of the wireless self-interference channel are what limit
the cancellation depth and instantaneous bandwidth of the MPC.
3.4
Summary
In this chapter, we performed a wide experimental study of discrete MPCs and architectures. In total, we studied three different types of MPCs: single-ended MPCs,
a novel balanced MPCs, and the first multi-tap (single-ended) MPCs inserted into
wireless channels and a digital radio.
We found that, while single-ended MPCs were the simplest to build and control,
they do not exhibit the best performance in terms of RF key performance parameters. In particular, the inability to remove common-mode noise, such as RIN, results in a high noise figure. Balanced MPCs, which do have the ability to remove
common-mode noise, perform better than single-ended MPCs across all the RF key
performance parameters. Specifically, they exhibit a 6 dB gain advantage over an
otherwise identical single-ended canceller and are able to continuously decrease the
noise figure by increasing the optical power as far as RIN suppression permits. This
directly corresponds to an increase in dynamic range as well. However, the tradeoff is that balanced MPCs are significantly more complicated to build and require at
129
least a two-fold increase in component count. In addition, more sophisticated channel
matching must be performed to maintain the same cancellation performance as an
analogous single-ended architecture.
We also performed the first experimental studies of an MPC in a wireless environment and in a digital radio. We confirmed that the MPC did not fundamentally
disrupt the digital radio’s operation and that the MPC added interference rejection
that matched what we expected from cancellation depth measurements. However, we
found that the additional frequency selectivity and reduced coherence bandwidth of
the wireless channel degraded both the MPC’s cancellation depth and instantaneous
bandwidth. When boiled down to the MPC architecture, the primary challenge that
the wireless channel presents is that it can require a large number of taps to model
(see Fig. 3.24).
This challenge should be addressed with a combination of more analog taps and
digital cancellation. Logically, the analog taps should be used to address the strong
LOS signal and the next strongest multipath components, due to the high dynamic
range achievable by analog MPCs. Meanwhile, the digital canceller should be reserved
for the weak, but numerous and dynamic, residual multipath components, which will
exhibit a larger delay spread and change sporadically due to the high number of
reflections. This highlights the importance of combining multiple types of interference
cancellation techniques throughout a communication system.
Increasing the analog tap count of an MPC is unfortunately a costly proposition
due to the relatively high cost of discrete optoelectronic components such as lasers,
attenuators, delay lines, and AWGs; it is the primary reason why we were limited to
only 2 taps in the experiments in this chapter. However, this problem can be solved if
an MPC can be integrated. If this can be achieved, adding taps becomes an exercise in
circuit layout, design, and ultimately, fabrication yield. The additional size of the new
taps becomes marginal, and the additional cost of the extra chip real estate is much
130
smaller than what would be required for additional discrete components. Essentially,
integrating the MPC enables the potential for enormous scalability, spatially and
financially. Thus, in the next two chapters of this dissertation, we turn our attention
towards integrating the microwave photonic canceller.
131
Chapter 4
An Integrated Optical Delay Line
Photonic integration is one of the holy grails of microwave photonics, because of the
clear benefits that it enables in terms of reduced size, weight, power consumption,
and of course, cost [75–77]. While MPCs have achieved the wideband performance
that their initial impetus promised, all MPCs have been built using discrete optoelectronics, limiting them to high cost, high performance applications, such as military
and defense. Furthermore, active analog cancellation often requires multiple taps to
accurately model the self-interference channel, and scaling up the tap count quickly
renders discrete MPCs size and cost prohibitive. A scalable, integrated solution is
required for MPCs to compete with RF integrated circuits (RFICs) and bring the
benefits of microwave photonics to widespread commercial applications.
To integrate the MPC, we must be able to replace all of the signal processing
building blocks of the discrete system with an integrated counterpart. Instead of
being connected by optical fiber, these integrated building blocks would be connected
by an optical waveguide. In addition, we must be able to sacrifice the flexibility of
different material platforms to monolithically integrate all signal processing blocks
into a single material. We have selected Indium Phosphide (InP) as the material
platform. We will discuss the challenges of monolithic integration and the rationale
132
for our choice for material platform more in Chapter 5. All of the building blocks of
an MPC already have existing integrated counterparts, with the sole exception of the
tunable optical delay line. There are two key challenges to fabricating an integrated
optical delay line:
• Achieving a large tunable delay range with limited chip real-estate. Physical
space is limited on an integrated platform and there are no moving parts.
• Achieving a large delay with low loss. Passive optical waveguides (1 dB/cm [7])
has order of magnitudes higher loss than optical fiber (0.5 dB/km [63]).
In this chapter, we demonstrate an integrated optical delay line using dispersion
in a semiconductor optical amplifier (SOA) [78], also known as slow and fast light.
The general concept behind slow and fast light is that, since there can be no moving
parts, a tunable optical delay is created not by changing the physical length of the
propagating medium, but by changing the speed of information in the propagating
medium. We take advantage of the fact that the phase velocity of light, which can
not be changed much, does not necessarily correspond to the group velocity, or the
velocity of RF envelope, which can be tuned over a much wider range. The chapter
is organized into the following sections:
• Principles of Operation. We introduce the principle of slow and fast light.
• Experimental Setup. We present the experimental setup used to create a
tunable optical delay using an SOA.
• Experimental Results. We analyze the tunable delay range, bandwidth, and
effect that tuning the delay has on the transmission (loss) of the delay line. We
devise a method to use the dispersive SOA as a simultaneous tunable delay line
and a variable optical attenuator.
133
• Incorporation in an MPC. We incorporate the integrated optical delay line
into a discrete MPC as a proof-of-concept. We demonstrate an adaptive system,
which uses the integrated delay line as both a TDL and a VOA.
• Summary
4.1
Principles of Operation
The SOA-based integrated delay line operates under two physical principles: dispersive phase shifting (i.e. slow and fast light) and cross-gain modulation (XGM). Slow
and fast light refers to controlling the group velocity of light in a propagating medium
by generating a material dispersion [79]. The group velocity can be expressed as
vg =
c
dn
n+ω
dω
(4.1)
where c is the speed of light in vacuum, n is the refractive index of the optical medium,
ω is the optical frequency, and dn/dω is the dispersion. Slow and fast light was first
demonstrated in a Bose-Einstein condensate by Hau et al. in 1999 [80]. Since then
the effect has been demonstrated in various media including gases [81, 82], optical
fiber [83–86] and semiconductors [39, 40, 87–91, 91–94].
We are interested in this last group: semiconductors, and in particular SOAs because of their compatibility with InP and their ability to provide gain to compensate
for loss in the delay line. In SOAs, slow and fast light is created via Coherent Population Oscillations (CPO), where the beating between two optical waves leads to
oscillations in the ground state population at the beat frequency [87, 95]. This beating creates a temporal grating within the SOA waveguide, which scatters light around
the optical carrier frequency, locally reducing the semiconductor gain. In microwave
134
photonics, the two optical waves are conveniently played by the optical carrier and
its RF sidebands, so in fact only a single optical carrier is required.
The local reduction in SOA gain from the CPO translates into a dispersion in
refractive index (and therefore, a change in group velocity) through the KramersKronig relations, which links the imaginary and real parts of any analytical function;
in this case, the material gain and refractive index are the imaginary and real parts
of the complex refractive index. This process is illustrated in Fig. 4.1. For an SOA
operated in the gain regime, the CPO results in an increase in group velocity [81, 96]
because it creates a “gain hole” (the reason for this name will be apparent shortly).
Given this, from here on, we use the term advance rather than delay to represent the
time/hase shift achieved by the SOA.
The group velocity can be tuned by modulating the strength of the CPO within the
SOA. This is performed by varying the power of the input optical signals, which we call
the optical pump (Ppump ) or the SOA excited carrier population, which is controlled
by the SOA bias current (ISOA ). Note that CPO is not a result of homogeneous
broadening, as this would exhibit a very large linewidth in an SOA. Instead the CPO
process has a linewidth that is much smaller, approximately equal to 1/2πτ , where τ
refers to the semiconductor carrier lifetime, reflecting the role that the carriers play
in creating the effect. Therefore, when the modulated RF frequency increases beyond
1/2πτ , the effect diminishes significantly. As we will see, this results in a well-known
delay-bandwidth tradeoff [97] when using the dispersive phase-shifting technique.
Note that to group velocity control mechanisms mentioned previously will also
affect the SOA gain, which is undesirable for analog signal processing. Specifically,
to increase the group velocity one must increase ISOA or Ppump ; however, both of
these options increase the overall RF signal gain. To decouple the group velocity
from the gain, we introduce a second physical effect to the SOA, which is cross-gain
modulation (XGM). In XGM, a weak optical probe, detuned from the optical pump
135
Absorption
α
Kramers-Kronig
Relations
Refractive Index
n−n
dn
dω
ω − ωr
Figure 4.1: An illustration of how a local change in gain (top) results in a change in
refractive index (middle) and ultimately dispersion (bottom).
136
by several times the inverse carrier lifetime to avoid inducing unintended CPO with
the pump, is injected into the SOA in the opposite direction of the optical pump.
Due to SOA gain saturation, as Ppump increases, the output probe power decreases
and vice versa [98]. Furthermore, the RF signal modulated on the optical pump is
also modulated onto the probe wave by XGM. Now, the relationship between group
velocity and output power has been reversed for Ppump : a higher Ppump results in a
larger group velocity but a lower RF gain. In contrast, a higher ISOA results in a
larger group velocity and a higher RF gain. Thus, XGM provides a mechanism to
decouple the group velocity and RF gain.
To verify this, we modeled the combined effects of CPO and XGM by using the
rate equation analysis first applied by Mørk et al. [94,98] for the modulation response
of an SOA. The modulated components of the pump and probe waves, ∆P1 and ∆P2
, respectively, can be expressed as
g P¯1 /Ps
d∆P1
· ∆P1
= (g − a) · ∆P1 −
dz
1 + P¯1 /Ps − iωτ
(4.2)
g P¯2 /Ps
d∆P2
= (g − a) · ∆P2 −
dz
1 + P¯1 /Ps − iωτ
× [∆P2 + ∆P1 exp(i∆kz)]
(4.3)
where a is the internal optical loss, Pj is the average power where j = 1, 2 for pump
and probe, respectively, Ps is the SOA saturation power, ω is the microwave frequency,
τ is the semiconductor carrier lifetime, and ∆k = k2 − k1 is the effective propagation
in the reference frame of the probe. g is the standard saturated gain given by
g=
gp
1 + P¯1 /Ps
137
(4.4)
10
P1 = -1 dBm!
-1 ,
XGM Gain (dB)
gp = 150 cm
5
-1 ,
gp = 150 cm
0
−5
gp = 60 cm-1, P1 = 0 dBm!
CW!
Probe!
Modulated!
Pump!
SOA
z=0
−10
P1 = +10 dBm!
-1 ,
cm
g p = 60
L
P 1 = +1
3 dBm !
1000
2000
3000
Microwave Frequency (MHz)
4000
Figure 4.2: Simulated gain of the output probe signal as a function of microwave
frequency. Four different pump power and SOA bias current combinations are plotted.
where gp is the peak gain coefficient. It is assumed that the pump power is significantly
larger than the probe power so that the probe does not contribute to gain saturation
(i.e. does not affect the pump).
To show the independent control of RF gain and group velocity using Ppump and
ISOA , the probe signal (∆P2 ) was solved numerically for four different configurations
of Ppump and ISOA . Changes in Ppump were implemented in the model through the
average input pump power, P1 (L), while changes in ISOA were implemented through
the peak gain coefficient, gp . The initial conditions used in the simulation were P¯1 (L)
= −1 dBm, 10 dBm, and 13 dBm, where L is the SOA length; P¯2 (0) = −5 dBm;
∆P1 (L) = mP1 (L) where m is the modulation index; and ∆P2 (0) = 0. The remaining parameters are listed in Table 4.1. The results of the simulation are plotted in Fig.
138
Parameter
Symbol
Value
SOA Length
L
1 mm
Modulation Index
m
.4
a
15 cm−1
SOA Intrinsic Absorption
Saturation Power Ps
Probe Average Optical Power P¯2
Carrier Lifetime
τ
−1.5 dBm
−5 dBm
800 ps
Table 4.1: Parameters and values used in the simulation of CPO and XGM in an
SOA.
4.2, with the relevant parameters labeled with each curve. The solid curves represent
configurations with similar dispersions and group velocities, but different output powers, showing the ability to control gain independently of group velocity. The dashed
curve represents a configuration with a similar output power as the middle solid curve
(identical gain at 1 GHz), but a significantly weaker dispersion, indicating the ability
to control group velocity independently of gain.
4.2
Experimental Setup
The SOA used in this experiment was a L = 1 mm long device from Amphotonix. It
has a buried heterostructure design with a strained bulk active region to provide low
polarization dependence. The bulk active region is 0.1 µm thick with 0.1 µm thick
separate confinement heterostructures. The SOA has an experimentally measured
transparency current of 15 mA and a saturation power of Ps =−3 dBm. The experimental setup used to characterize the gain and time advance of the SOA is shown in
Fig. 4.3. A 1551.72 nm pump laser is modulated by an electro-absorption modulator
(EAM), which receives an RF input signal from a network analyzer. The network
analyzer will ultimately be used to determine the RF gain and time advance produced by the SOA. The modulated optical pump is boosted to Ppump = 14 dBm by an
139
Probe (P = -5 dBm)
λ2=1553.33 nm
ISOA
SOA
Pump
λ1=1551.72nm
EAM
PC
Isolator
EDFA
x(t)
Bandpass
Filter λ2 Detector
VOA
Ppump
Optical Pump
Optical Probe
Electrical Signal
DUT
Network
Analyzer
Circulator
α x(t − τ ) τ (Ppump , I SOA )
α (Ppump , I SOA )
Figure 4.3: Experimental setup used to measure the optical gain and time advance
of the SOA. PC = polarization controller. τ = RF signal’s time of flight and α = RF
gain.
erbium-doped fiber amplifier (EDFA) before being attenuated to the desired power
by a separate variable optical attenuator (VOA). The resulting pump power, Ppump ,
is our first control parameter. The optical pump is then injected into the SOA via an
isolator and circulator.
Upon entering the SOA, the optical pump beats with its RF sidebands, inducing
CPO. The CPO strength depends on the optical pump power, Ppump , and the SOA
bias current, ISOA . As this occurs, the RF signal in the optical pump is imprinted
onto a separate, strongly detuned CW optical probe via XGM in the SOA as depicted
in Fig. 4.3. The probe is a 1553.33 nm laser with significantly weaker power (P̄2 (0)
= −5 dBm) than Ppump to avoid perturbing the SOA gain. It is injected into the
SOA via a circulator and counter-propagates against the optical pump to improve
isolation between the optical pump and probe at the output. The imprinted optical
probe exits the SOA via a circulator and is passed through an optical filter centered at
140
1553.33 nm to eliminate unwanted wavelengths. Finally, the optical probe is detected,
and the output RF signal is compared to the input RF signal by the network analyzer
to extract the SOA’s induced RF gain and time advance.
4.3
Experimental Results
The effects of Ppump and ISOA on the gain and time advance of the RF signal after
propagating through the SOA are investigated, first as a function of Ppump and ISOA
individually, and then both. These results show how the combined effects of XGM
and CPO can be used to implement an independently tunable gain and advance.
4.3.1
Tunable Time Advance
Figures 4.4a and 4.4b show the experimentally measured time advance up to 4 GHz
RF frequency as a function of Ppump and ISOA , respectively. Both plots exhibit the
delay-bandwidth tradeoff that is characteristic of dispersive phase shifters. In Fig.
4.4a, ISOA is held constant at 80 mA, while Ppump is stepped from 0 dBm to 14 dBm
in increments of 2 dB. Larger pump powers generate a stronger CPO effect, and thus
a larger group velocity in the SOA, so the time advance increases with Ppump , as
observed in Fig. 4.4a. The device achieves a maximum relative time advance of 80 ps
at low RF frequency. From measurements of the half-width half-maximum of the time
advance, τ , the semiconductor carrier lifetime, is 150 ps, within the expected range
of this particular SOA. Figure 4.4b shows similar measurements for time advance,
this time as a function of ISOA , with Ppump held constant at 5 dBm. ISOA is stepped
from 30 mA to 100 mA in increments of 10 mA. The higher the bias current, the
stronger the CPO effect, so time advance increases with ISOA , as observed in Fig.
4.4b. The maximum time advance at low frequency is 150 ps and begins to saturate
around 100 mA due to gain saturation.
141
(a)
(a)
Relative Advance (ps)
60
ISOA = 80 mA
14 dBm
40
20
0
2 dBm
−20
5
1000
(c)
2000
Frequency (MHz)
3000
4000
200
150
2 dBm
(b)
(b)
100
P
pump
= 5 dBm
100 mA
50
0
30 mA
−50
Normalized Gain (dB)
Relative Advance (ps)
Normalized Gain (dB)
80
40
1000
2000
Frequency (MHz)
3000
4000
100 mA
(d)
Figure 4.4:0 Relative time advance up to 4 GHz
RF frequency as a function of (a)
30
−5
Ppump (ISOA
= 80 mA) and (b) ISOA (Ppump = 5 dBm). Time advance is measured
20
−10 the signal’s time of flight at Ppump = 0 dBm in (a) and ISOA = 25 mA in
relative to
10
−15
(b).
I
= 80 mA
P
= 5 dBm
14 dBm
SOA
30 mA
4.3.2
−20
0
1000
2000
Frequency (MHz)
3000
4000
0
0
pump
1000
2000
Frequency (MHz)
3000
4000
Tunable RF Gain
Fig. 4.5a and 4.5b show the effects of Ppump and ISOA on the RF gain, respectively.
For consistency, the same parameters that were used to obtain the data in Fig. 4.5
were also used to obtain the data in Fig. 4.4. As expected, the RF gain increases with
increasing ISOA and decreasing Ppump . In addition, as ISOA and Ppump increase, so
too does the size of the gain hole, consistent with theory. Note that Ppump and ISOA
push time advance in the same direction but push RF gain in opposite directions. In
other words, as Ppump and ISOA increase, so does time advance (see Fig. 4.4a and
4.4b). However, as Ppump increases, RF gain decreases; on the other hand, as ISOA
increases, gain increases (see Fig. 4.5a and 4.5b). As mentioned in section 4.1, the
inverse relationship between RF gain and Ppump originates from XGM in the SOA and
is the mechanism that permits the independent control of RF gain and time advance.
Therefore, by adjusting ISOA and Ppump together in a balanced manner, time advance
and RF gain can be controlled independently.
To illustrate this, we measured RF gain and time advance as functions of both
ISOA and Ppump at 600 MHz RF frequency, and the results are shown as RF gain
and time advance surfaces in Fig. 4.6a and 4.6b, respectively. RF gain is calculated
by subtracting the loss of the modulator-photodetector-RF amplifier link from the
total RF gain to observe only the gain contributions from the SOA in the XGM
142
Relative Advance
14 dBm
40
20
0
2 dBm
−20
5
0
1000
(c)
(a)
2000
Frequency (MHz)
3000
4000
2 dBm
−5
−10
−15
−20
0
14 dBm
1000
2000
Frequency (MHz)
I
SOA
3000
= 80 mA
4000
pump
150
100
100 mA
50
0
30 mA
−50
Normalized Gain (dB)
Relative Advance
Normalized Gain (dB)
60
40
1000
(d)
(b)
2000
Frequency (MHz)
3000
4000
100 mA
30
20
10
0
0
30 mA
1000
2000
Frequency (MHz)
Ppump = 5 dBm
3000
4000
Figure 4.5: Normalized RF gain up to 4 GHz RF frequency as a function of (a) Ppump
(ISOA = 80 mA) and (b) ISOA (Ppump = 5 dBm). Gain is normalized to Ppump = 0
dBm in (a) and ISOA = 25 mA in (b).
configuration. The key observation is that constant RF gain contours exist within
the ISOA - Ppump parameter space. The ±3 dB and ±6 dB RF gain contour lines
are drawn in Fig. 4.6a, but a full range of -15 dB to +15 dB can be achieved.
Furthermore, Fig. 4.6b shows the corresponding time advance surface as a function
of Ppump and ISOA , with time advance contours up to 120 ps. A comparison of the
RF gain and time advance contour lines show that they run nearly orthogonal to each
other in the Ppump − ISOA parameter space, and therefore a mapping exists to achieve
independent tunable RF gain and time advance. The RF gain contour lines from Fig.
4.6a are overlaid onto Fig. 4.6b, and the -6 dB, -3 dB, and +3 dB RF lines each span
across at least 100 ps of time advance.
We now arrive at the main result of this experiment: by adjusting ISOA and
Ppump across different gain and advance contour lines in the ISOA -Ppump parameter
space, the SOA can serve as either an independent RF gain, time advance, or both
simultaneously. To demonstrate this explicitly, in Fig. 4.7, we show the SOA’s ability
to act as a simultaneous RF gain and time advance by measuring the SOA output
using an optical oscilloscope while biasing the SOA along two different gain contour
lines. The RF input is a 0 dBm, 600 MHz CW signal generated by an RF signal
generator. The top and bottom sets of curves, which are offset for clarity, correspond
to biasing the system along the +3 dB and -3 dB gain contour lines, respectively. The
143
(a)!
(b)!
3
3
2
1
2
1
Figure 4.6: (a) RF gain and (b) time advance as a function of ISOA and Ppump at
600 MHz RF frequency. RF Gain contours are overlaid on both plots. Time advance
contours are shown in (b) only.
144
120 ps
150!
100
Power (µW)
SOA!
1
3
2
50
100!
+ 3 dB!
120 ps
50!
0
3
- 3 dB!
0!
−50
−1
−0.5
1
2
0
Time (ns)
0.5
1
Figure 4.7: Output from an oscilloscope while biasing the device along the +3 dB
(top set of curves) and -3 dB (bottom set of curves) gain contours. No normalization
of the amplitude was performed.
biasing conditions used to generate curves 1-3 in both sets of curves are indicated on
the corresponding contour lines in Fig. 4.6b.
In each set of curves, the SOA shows a tunable time advance range of 120 ps,
while exhibiting very little amplitude change along a single gain contour line. On the
other hand, the SOA can act as an optical weight by switching between gain contours,
as shown by the difference between the top and bottom sets of curves. The power
of the top curves is twice the power of the bottom curves as opposed to four times
(i.e. 6 dB) because the oscilloscope used to produce Fig. 4.7 measures optical power,
whereas the RF gain contours in Fig. 4.6 correspond to RF power.
It is worth now discussing a few important properties of the device: latency,
noise, and nonlinearity. The device latency is determined by the time required for
145
adjustments to Ppump and ISOA to alter the SOA’s properties. For Ppump , a change
in pump power takes about 100 ns to propagate to the SOA in our setup. This
corresponds to 20 meters of fiber and can be shortened simply by using shorter fiber.
In an integrated circuit, a change in power would propagate to the SOA much quicker,
easily under to a nanosecond. For ISOA , a change in bias current takes effect at the
inverse of the maximum modulation bandwidth of the SOA, which was 10 MHz or
100 ns in time. Therefore, we estimate that the device latency, in an integrated
system, will be on the order of 100 ns using this particular SOA epitaxial structure.
This can be decreased by designing SOAs with higher bandwidth electrical contacts,
which can readily go above 1 GHz, dropping the latency to under 1 ns. Because of
the speed of the semiconductor system, in practice, the latency is much more likely
to be dictated by the control electronics.
There are three major sources of noise and distortion in the SOA: amplified spontaneous emission (ASE) noise, nonlinearities from power saturation, and group velocity
dispersion. These sources are all functions of Ppump and the ISOA . It is well-known
that SOAs exhibit relatively large ASE backgrounds. The noise figure of this SOA
is 8 dB, which increases with decreasing bias current due to the reduced gain. However, we must also account for the effects of XGM. We experimentally observe lower
SNR at lower bias currents, where excessive gain saturation by the optical pump significantly reduces the XGM gain. The extremes of the two control parameters also
induce nonlinear distortions. For example, at low ISOA and high Ppump , the optical
pump so strongly depletes the gain that it causes signal clipping during XGM. On the
other hand at high ISOA , if Ppump drops significantly below the SOA saturation power,
the pump becomes too weak to drive XGM. Therefore, the extremes of the control
parameters, which are dictated by the SOA saturation power, should be avoided. Finally, group velocity dispersion (GVD) is a consequence of using a dispersive phase
shifter, and can be readily witnessed in Figs. 4.4 and 4.5. The GVD increases at
146
higher Ppump and ISOA , where the induced material dispersion is larger. Therefore,
this technique is not a “true-time delay”, but rather a phase-shifter because of the
frequency-dependent time advance.
4.4
Incorporation in an MPC
In the last section of this chapter, we incorporate the SOA delay line presented in the
previous section into a discrete single-tap MPC as a proof-of-concept [65]. The SOA
performs three critical functions in the MPC:
• A tunable RF gain to perform amplitude matching
• A tunable integrated delay line to perform phase matching
• A inverter so that the reference signal is subtracted from the RX signal upon
detection.
A schematic of the MPC is shown in Fig. 4.8. The output from a λ1 = 1550.12 nm
DFB laser is split to provide the optical carrier for both the RX and reference taps.
The RX signal is modulated onto the optical carrier of the top signal path by a MachZehnder modulator (MZM). The reference signal is modulated onto the optical carrier
of the bottom signal path by a second MZM. The reference signal is then inverted,
and amplitude and phase matched to the RX signal by the SOA. The processed reference signal is optically combined with the RX signal by a 50/50 optical coupler and
then detected by a photodiode to downconvert the signal back into the RF domain.
Note that the SOA also converts the wavelength of the optical carrier from λ1 =
1550.12 nm to λ2 = 1553.33 nm using XGM. This step is required to prevent coherent
beat noise between the RX and reference optical carriers upon optical combination
prior to detection. The self-interference and SOI are generated by two distinct signal
generators. The corrupt RX signal is created by coupling the self-interference and the
147
PC
Mod Output
Signal of Interest
+
Interference
λ1
Interference
Mod = Optical signal
= RF Signal
EDFA
Power
Meter
τ , α
BPF
VOA
Circ.
SOA λ2
Control
Computer
Figure 4.8: Experimental setup. λ1 = 1550.12 nm, λ2 = 1553.33 nm. PC = Polarization Controller, EDFA = Erbium-Doped Fiber Amplifier, VOA = Variable Optical
Attenuator BPF = Bandpass Filter.
SOI using an RF combiner. The reference signal is tapped from the self-interference
signal generator.
The output RF signal is characterized by a network or spectrum analyzer to determine how well the system canceled the self-interference. The output signal was
also used as feedback to adaptively adjust Ppump and ISOA to minimize the interference. Both broadband and narrowband cancellation were measured by varying the
bandwidth of the self-interference signal and examining how well the system could
adaptively cancel the self-interference and recover the SOI.
4.4.1
Experimental Results and Analysis
Broadband and Narrowband Cancellation
Figure 4.9 shows the experimentally measured cancellation after optimizing the SOA’s
RF gain and time advance. The output of the system was measured without (red
148
−20
−30
−40
38 dB!
Signal of Interest!
−50
−60
−70
−80
880
With Cancellation
Without Cancellation
−30
With Cancellation
Without Cancellation
Receiver Amplitude (dBm)
Receiver Amplitude (dBm)
−20
10 kHz!
−40
Signal of !
Interest!
56 dB!
−50
−60
−70
−80
−90
60 MHz!
890
900
910
920
Frequency (MHz)
930
940
−100
950
914.98
914.99
915
915.01
Frequency (MHz)
915.02
Figure 4.9: (a) Broadband and (b) narrowband cancellation results. The SOI is
recovered after canceling the in-band interference.
dashed curve) and then with (blue solid curve) cancellation. The SOI was a weak
915 MHz CW RF signal. Any frequency band can be used, as long as it is within the
range of the CPO bandwidth, as discussed in section 4.1. For the SOA under test,
we measured this frequency to be approximately 1 GHz; this range can be increased
by using an SOA with a smaller semiconductor carrier lifetime.
Broadband cancellation was characterized by sweeping a 0 dBm CW interferer
across a 60 MHz bandwidth centered at 915 MHz and measuring the system output
using an RF spectrum analyzer. The results in Fig. 4.9a show that at least 38 dB of
cancellation was achieved across the 60 MHz bandwidth. More importantly, the SOI,
which had been buried below the interference without cancellation, was recoverable
with cancellation. Narrowband cancellation measurements were performed using a
10 kHz bandwidth interferer and the same SOI. The results are plotted in Fig. 4.9b,
showing the system output with and without cancellation. The maximum amount
of cancellation was 56 dB, and the SOI was recovered. The narrowband cancellation
depth appears larger than the broadband cancellation depth at the same frequency,
but this was an illusion created by the larger spectrum analyzer noise floor due to a
larger IF bandwidth.
An important point to investigate is how the dispersive properties of the SOA
phase shifter affect the amplitude and phase matching over a larger bandwidth. As
149
mentioned in section 4.3, the amplitude and delay of the SOA is not constant with
frequency and this will inevitably have an affect on amplitude and phase matching.
We measured the amplitude and phase mismatch between the self-interference channel and the single reference tap across a 400 MHz bandwidth using a network analyzer
and plotted the mismatch in the top half of Fig. 4.10. Zero mismatch is indicated
by the line representing zero amplitude offset and 180 degrees phase offset. The optimal cancellation is achieved when both amplitude and phase mismatch approach
this line, as seen in the bottom half of Fig. 4.10, which shows experimentally measured cancellation (solid curve) as well as simulated cancellation (dotted curve), using
Eqn. (2.14). The simulation matches the experimental measurements very well, confirming mismatch is the primary limitation to cancellation depth and instantaneous
bandwidth.
There are two qualitatively different contributions to amplitude and phase mismatch in Fig. 4.10: a gradually changing component over the total bandwidth and an
abruptly changing component over smaller frequency ranges. The gradually changing component comes from the SOA dispersion, and because the SOA only exists in
the reference tap it manifests in the amplitude and phase mismatch. The abruptly
changing component comes from the non-flat frequency response of RF components,
which exhibits ripples in amplitude and phase. Despite being only 0.4 dB and 1◦ in
amplitude and phase, respectively, this component of mismatch was actually what
limited the system to 60 MHz bandwidth.
Adaptive Cancellation using the SOA
To test the tunability and control capabilities of the SOA-based MPC, it was made
adaptive by using feedback from the MPC output to iteratively update Ppump and
ISOA . By treating Ppump and ISOA as two independent dimensions, minimizing the
interference power became a two dimensional optimization problem. We used the
150
184
0.2
182
0
No Mismatch Line!
180
−0.2
178
S21 Phase Mismatch (deg)
S21 Magnitude Mismatch (dB)
0.4
−15
Cancellation (dB)
−0.4
−20
700
750
800
850
900
950
1000
1050
176
1100
850
900
950
1000
1050
1100
Frequency (MHz)
−25
−30
−35
−40
−45
−50
−55
700
Simulated Cancellation
Experimental Cancellation
750
800
Frequency (MHz)
Figure 4.10: (Top) The measured amplitude and phase mismatch between the selfinterference channel and the reference tap from 700 MHz to 1100 MHz. (Bottom)
Measured and simulated cancellation based on the amplitude and phase mismatch.
More negative cancellation corresponds to greater interference cancellation.
Nelder-Mead simplex (NMS) algorithm [99] to minimize the interference power because it is a simple and heuristic optimization algorithm that has been shown to work
well for “oddly” shaped error surfaces. More information about the NMS, including
computer code, is available in Appendix A. The NMS algorithm analyzes the system
error surface at each vertex of a simplex, which for a 2-D surface, is a triangle. During
each iteration of the algorithm, the simplex is either reflected, expanded, contracted,
or shrunk, by manipulating the simplex vertex with the worst performance, resulting in a new simplex [99]. Eventually, the simplex converges on the error surface
minimum (although, as with any optimization algorithm, this is not necessarily the
151
global minimum). In this experiment, the error surface was simply defined by the
output power of the MPC. Because the self-interference and the SOI are assumed to
be statistically independent, it can be shown that minimizing the output power of
the MPC is equivalent to minimizing the interference [55].
The algorithm was run on a computer, which controlled Ppump and ISOA through
a GPIB interface and received feedback from an RF power meter. A 10 dBm, 10 MHz
bandwidth AWGN interferer was used along with a single-tone SOI. Figure 4.11a
shows a typical progression as the algorithm adjusts Ppump and ISOA to minimize
interference power. The initial simplex was selected to be far away from the optimal
point, and the algorithm was able to minimize the interference (i.e. achieve amplitude
and phase matching) in nearly all cases. Note that the heat map in Fig. 4.11 was
mapped retroactively; the NMS algorithm has no knowledge of the error surface other
than what it is probing;
To determine the range of time delays accessible by the SOA, the delay of the
reference tap was manually detuned from the optimal value by a variable delay line
before initiating the algorithm. The same initial simplex was used as in Fig. 4.11a.
The results, shown in Fig. 4.11b, show that the SOA is able to cover an 83 ps
continuous delay range, or 26 mm of free space. Cancellation is noise-limited towards
longer delays, where the SOA is operated in a regime that causes it to generate large
amounts of noise and distortion (see discussion in section 4.3).
The SOA’s range, precision, and speed is suited to track delay variations such
as antenna oscillations, which are too fast or small to be compensated by standard
delays. Because of the limited range of the SOA delay line, in a practical system, the
SOA will have to be used in tandem with longer delays, which would perform coarse
adjustments to bring the system close to the optimal delay. The SOA would then be
used to lock onto the optimal delay with high precision and speed.
152
−36
−40
−40
−40
−40
129
SOA Current (mA)
−42
117
−48
105
−45
−45
−45
−45
−65−65
−65
−65
−55 −55
−55
−55
−50 −50
−50
−50
−70−70
−70
−70
94
82
−54
−60−60
−60
−60
−60
70
58
−66
−50
−50
−50 −50
−50
−50
−50
−50
−50 −50
46
1.5
2.7
3.9
5.1
6.3
7.4
SOA Input Optical Power (mW)
−50
−50
−72
8.6
1
−40
2
−50
−60
−40
3
−70
−50
−90
−100
905
910
915
Frequency (MHz)
920
−60
−40
−70
−50
925
−80
−90
−100
905
910
915
Frequency (MHz)
920
Receiver Amplitude (dB)
−80
Receiver Amplitude (dB)
Receiver Output (dBm)!
Receiver Amplitude (dB)
1
−60
−70
925
−80
−90
−100
905
910
915
920
925
Frequency (MHz)
2
3
Itera
t
ion!
ay
Del
)!
(ps
g
n
uni
Det
Figure 4.11: (a) Progression of the Nelder-Mead simplex algorithm to adaptively
cancel interference using the SOA bias current and input optical power. (b) The
system is able to locate the interference minimum through 83 ps of delay range. Each
curve is averaged over 5 algorithm runs. Insets show receiver output spectra after 0,
40, and 60 algorithm iterations.
153
4.5
Summary
In this chapter, we demonstrated an integrated optical delay line based on dispersive
phase-shifting in a semiconductor optical amplifier. The device could independently
control amplitude, phase (delay), or both simultaneously by appropriately controlling
the two control parameters, Ppump and ISOA . The SOA achieved an independent
delay range of 120 ps while simultaneously achieving an independent RF gain range of
±3 dB. We showed that the SOA delay line could be incorporated and used in an MPC
with comparable performance as previous discrete MPCs, and could actually be used
to perform amplitude and phase matching as well as signal inversion. An adaptive
system was also demonstrated by electronically controlling the control parameters of
the delay line. In contrast to fiber-based or mechanical delay lines, several new items
that must be considered are the extra noise, nonlinearities, and dispersion introduced
by the SOA delay line. Having proven the feasibility of an integrated optical delay
line, we now move to integrate an entire microwave photonic canceller.
154
Chapter 5
An Integrated Microwave Photonic
Canceller
In this chapter, we present the first experimental demonstration of a fully integrated
microwave photonic canceller (IMPC) for active analog self-interference cancellation
[100,101]. It is novel not only because it is the first integrated MPC, but also because
it is one of the first integrated microwave photonic devices to possess only RF external
inputs and outputs, confining all optical signal processing to the core of the device. We
call this class of photonic devices a Monolithic Phootnic Integrated Circuit (MPIC).
In contrast to standard photonic integrated circuits, in MPICs, all optical sources and
detectors are monolithically integrated onto a single substrate, obviating the need for
any external fiber coupling. An MPIC has significant performance, manufacturing,
and installation advantages. In terms of performance, monolithic integration reduces
inter-element coupling losses and power consumption by virtue of only requiring a
single thermoelectric element. For manufacturing, the device does not require an
externally coupled fiber, greatly reducing packaging complexity and cost. With its
compact footprint and simple interface, an MPIC could ideally fit onto RF printed
circuit boards, enabling it to merge with mainstream electronic systems. This opens
155
a new window of opportunity for new RF systems, such as 5G small cells and Internet
of Things transceivers, which previously may not have been amenable to the form
factor or cost of fiber-coupled packages, to finally leverage the wideband analog signal
processing of optics.
This chapter is organized into the following sections:
• Material Platform. For monolithic integration, the material platform is critical to determining the performance of the devices. We begin by explaining our
rationale for choosing Indium Phosphide as our material platform.
• Design and Architecture. We present the IMPC design and architecture.
• Functional Performance. We study how the circuit performs in terms of functionality: cancellation, bandwidth, operating frequency, and tunability. Certain
tradeoffs must be made in order to accommodate all devices on a single substrate. How does the functional performance compare to discrete MPCs?
• RF Performance. We investigate standard RF parameters such as gain, noise
figure, linearity, and dynamic range. Again, we ask the question how does
the IMPC compare to discrete MPCs. We also present recommendations for
improving the device in future designs.
• Summary. We summarize the results and present and outlook for the integrated technology moving forward.
5.1
Material Platform
The optical material used to fabricate the integrated circuit is a key design decision
that must be made prior to any circuit design and layout. Especially for a monolithic integrated circuit, where presumably all devices are fabricated into the same
156
Table 5.1: Comparison of Optical Materials. The more stars the better.
Property
InP
SOI
T riP leX T M
Laser and Amplifier (Actives)
***
*
*
Detector
***
**
*
Electro-Optic Effect
***
**
*
* (1 dB/cm)
*** (0.3 dB/cm)
**** (0.045 dB/cm)
Confinement (Bending Radius)
*
***
**
Electronic co-integration
**
***
*
Manufacturing Maturity
***
***
*
Propagation Loss
material, this decision affects everything from device functionality and performance
to, eventually, manufacturing feasibility.
Over time, three materials have risen as the most viable to fabricate wafer-scale
photonic integrated circuits: III-V compounds, such as GaAs and InP, silicon-oninsulator (SOI), and silicon nitride on silicon dioxide (T riP leX T M ). [75,102]. Each of
these material platforms have strengths and weaknesses and the choice of any single
material must take into account the types of devices that can be integrated into
each material, the properties of each material (e.g. electro-optic coefficient, optical
loss, confinement factor), ease of packaging, as well as manufacturing maturity. An
excellent review of the strengths and weaknesses of each material is given in [75]. Here,
we summarize the review using Table 5.1, which compares the material platforms
across several key properties of interest for fabricating MPICs.
For the purposes of monolithic integrated microwave photonic circuits, we selected
InP as our material platform our choice. The decision was made simply because InP
was the only conventional platform that had the ability to integrate both optical
sources, amplifiers, modulators, and detectors onto a single substrate - essentially
all the main ingredients needed for a microwave photonic canceller. The monolithic
integration of active-passive signal processing on one wafer can be achieved through
a variety of techniques, which enable a transition between an active epitaxial struc157
Figure 5.1: Cross-sectional view of a hybrid evanescent device. Taken from [6].
Figure 5.2: Various techniques to monolithically integrate active-passive waveguides.
Taken from [7].
158
ture and a passive epitaxial structure. An overview of these techniques are shown
in 5.2, which are regularly employed in commercial InP PIC fabrication. From a
manufacturing perspectives, InP is by far the oldest of the optical materials, and
has a solid industrial foundation supporting it. Today, 4-inch wafers and even some
6-inch InP wafers are grown and used for optical device fabrication; companies like
Infinera routinely fabricate circuits with hundreds of devices [75, 103, 104]. A wide
variety of multi-project wafer (MPW) services are also available, ideal for university
and academic research.
Another very enticing material is SOI. Silicon photonics, led by the desire to take
advantage of the enormous silicon industry and electronic-photonic co-integration, has
witnessed tremendous growth in the past decade. Besides those advantages mentioned
above, it offers some very attractive benefits such as high device density and low
propagation loss. However, due to our key design goal of monolithic integration, the
inability to fabricate high-quality lasers on-chip excluded SOI from consideration.
That being said, we make note of a fifth material platform, although it is actually
a combination of materials: the hybrid silicon evanescent platform. Hybrid silicon
is composed of SOI for passive signal processing and routing, but with active IIIV devices bonded onto SOI at select areas designed to transition the optical wave
from one material to another [6, 105] (see Fig. 5.1. Lithographic techniques are used
to define III-V active devices on-top of already fabricated silicon photonic passive
devices. Switching between active and passive parts of the device are achieved by
the fabrication of waveguide tapers at the desired locations to push the optical mode
between the two materials. The appeal behind the hybrid evanescent platform is the
ability to combine the benefits of III-V and SOI materials, using each material to
alleviate the weaknesses of the other. This platform could enable an MPIC with the
low loss and compact passive signal processing of SOI juxtaposed with the essential
active signal processing (i.e. lasers) of InP. The hybrid evanescent platform only
159
recently entered commercial feasibility in 2012 through the formation of Aurrion
(now acquired by Juniper Networks), and it still remains to be seen how widespread
it will become. At the time of this work, the hybrid evanescent platform was not
available for MPW services. Nonetheless, it remains a very enticing material for
future integrated microwave photonic devices.
5.2
Design and Architecture
The block diagram of the IMPC is shown in the top part of Fig. 5.3. The architecture
is based off the discrete microwave photonic cancellers that were presented in Chapter
3. It is composed of a single-tap canceller architecture designed to cancel a single
interference signal. The corrupted received signal (RX) follows the bottom signal
path in Fig. 5.3, which we refer to as the RX tap; meanwhile, the reference copy of
the known interference follow the top signal path, which we refer to as the reference
tap. Both the RX signal and the reference signal are intensity-modulated onto optical
carriers by directly modulating the drive current of two separate, C-band distributed
feedback (DFB) lasers on the chip. A pair of monitor photodiodes are used to monitor
the lasers’ average output optical power for diagnostic and feedback purposes.
Following now the path of the RX signal, after it is modulated onto the optical
carrier, it is amplified by a constant-bias RX SOA and is subsequently incident upon
the positive port of a balanced photodiode. The RX tap essentially acts as an analog
photonic link, carrying the signal to the positive port of a balanced photodetector
by way of an SOA that helps compensate for loss associated with the EO and OE
conversions. Meanwhile, on the reference tap, the reference signal must be amplitude
and phase matched to the actual self-interference to optimize the cancellation. To do
this, the signal is passed through a pair of cascaded variable-bias SOAs, which uses
stimulated emission and absorption to tune the amplitude of the signal. To achieve
160
RX In
Monitor
PD
Monitor
PD
Laser
Passive
waveguide
BPD
SOARX
+
_
Laser
SOAref,1
SOAref,2
Reference In
Control
Bias 1
Control
Bias 2
RF
Output
200 μm
Figure 5.3: (Top) Block diagram. PD = photodiode. (Middle) Circuit layout on
PhoeniX OptoDesigner. (Bottom) Microscope image of the fabricated circuit. The
white boxes demarcate the actual devices in the block diagram.
161
a tunable phase shift, it uses a similar slow and fast light effect that was proposed
and demonstrated in Chapter 4. Having two cascaded SOAs enables a decoupling
between the amplitude and phase shift, which would otherwise not be possible with
a single SOA. The amplitude and phase matched reference signal is then incident
upon the negative port of the balanced photodiode. The balanced photodiode, using
direct detection, subtracts the processed reference signal from the RX signal and
the resulting balanced photodiode output current is the IMPC output. All optical
components are connected via passive waveguides, with a documented loss of less
than 1 dB/cm.
The circuits layout was done in PhoeniX OptoDesigner photonic chip design environment, and a portion of the final GDS file is shown in the middle part of Fig.
5.3. The circuit was fabricated on an InP epitaxial substrate with both transmitter
and receiver, active and passive capabilities. The fabrication was performed by the
Fraunhofer Heinrich Hertz Institute through the JePPIX MPW services. A microscope image of the fabricated PIC is shown in the bottom part of Fig. 5.3. The
entire PIC is 4 mm by 6 mm in footprint for three separate IMPC circuits. To test
the device, we mounted it on a temperature-controlled copper stage and probed it
using RF probes for signal input/output and DC probes for power and biasing. The
temperature of the stage was held constant at 20 ◦ C.
5.3
Functional Performance
In this section, we present the IMPC’s performance in terms of its designed functionality. Specifically, our focus is on two key areas. First, we investigate the amount of
cancellation that can be achieved by the IMPC as a function of operating frequency.
We show that, in terms of cancellation, the integrated solution performs as well as a
discrete fiber-optic system. Second, we investigate the control aspects of the IMPC,
162
and determine how much amplitude and phase tunability the IMPC can generate
to perform active analog cancellation. We also study how sensitive the cancellation
performance is to these controls. Using these controls, we then demonstrate an adaptive integrated canceller again using the Nelder-Mead Simplex Algorithm. Finally,
we compare the IMPC to previously published discrete MPCs and then make several recommendation to improve the PIC’s functional performance to conclude this
section.
5.3.1
Cancellation and Operating Frequency
We will characterize the IMPC’s performance through two metrics: cancellation and
operating frequency. While we will discuss instantaneous bandwidth, we do not make
a hard characterization, as this parameter depends strongly on the particular selfinterference channel. Instead, it is better to characterize the range of amplitude and
phase tunability that can be leveraged by the IMPC to compensate for the frequency
variation of self-interference channels; this is done in section 5.3.2.
Interference cancellation was measured using the experimental setup in Fig. 5.4a.
This setup used two vector signal generators to permit an arbitrary selection of the
interference signal and the SOI. The two signals were mixed by an RF combiner, and
a directional coupler was used to extract a portion of the interference for use as the
reference signal. An RF delay line was added to the reference signal input path to
coarse delay match the signal prior to entering the IMPC. Using this setup, additive
white Gaussian noise (AWGN) with a center frequency of 1.963 GHz (LTE Bands 2
and 25), 25 MHz bandwidth, average power of −10 dBm, and peak-to-average power
ratio (PAPR) of 14.75 dB was injected into the IMPC as self-interference. AWGN
was chosen for its uniform frequency coverage as well as its high PAPR to observe
how linearity affected the IMPC’s cancellation performance. The SOI was a −53 dBm
single-tone at the same center frequency.
163
Signal of Interest
RX In
+
AWGN
τ
Integrated
Microwave
Photonic Canceller
RF
Out
Spectrum
Analyzer
REF In
(a)
RX In
τ
REF In
Integrated
Microwave
Photonic Canceller
RF Out
Network
Analyzer
(b)
Figure 5.4: Experimental setups used to characterize cancellation using (a) an arbitrary signal of interest and interferer and (b) only a swept-CW interferer.
The progression in Fig. 5.5 shows the IMPC’s reference tap being tuned from no
cancellation (i.e. reference tap SOAs reverse biased to absorb all light) to optimal
cancellation. With an interference bandwidth of 25 MHz, the IMPC achieved at
least −30 dB of cancellation, given by the fact that the cancellation depth was at
least −30 dB across the band. When the interference fell below −85 dBm, the SOI
could be distinguished from the AWGN interference, confirming that the cancellation
process does not affect the SOI. When the cancellation was nearly optimized, the
frequency selectivity of the channel became visible. At higher interference powers
(not shown), the cancellation began to be limited by the nonlinear response of the
directly modulated lasers. An analysis of linearity will be presented in a separate
publication.
164
AWGN
No Cancellation
Signal of interest
Optimum Cancellation
Figure 5.5: Progression from no cancellation (red, farthest curve) to optimal cancellation (blue, closest curve).
To characterize the IMPC’s operating frequency range, we used the experimental
setup in Fig. 5.4b, which used a Keysight N5222A vector network analyzer (VNA)
as both the source of interference and the signal readout mechanism. For this experiment, the IMPC was tuned to maximize cancellation at various target bands across
the sub-6 GHz spectrum. If the cancellation depth exceeded −30 dB across the band
of interest, thereby guaranteeing that the overall cancellation exceeded −30 dB, then
IMPC was considered suitable to operate in that band. The FDD LTE bands (Bands
1 to 31) and both WiFi bands were chosen as the target bands. This selection was
both realistic and provided a good sample of frequencies and bandwidths.
165
Figure 5.6 catalogs the portion of each target band where at least −30 dB of
cancellation depth could be achieved, given a static IMPC setting. In other words,
the data from each band in Fig. 5.6 reflects a snapshot of the cancellation depth at
a single IMPC setting, selected to optimize cancellation over that band. As given by
the red bars in Fig. 5.6, the IMPC was able to achieve at least −30 dB cancellation
over nearly all FDD LTE and WiFi bands, although performance was not identical
for every band. From this data, we arrive at our first key result, which is that the
IMPC is able to amplitude and phase match our cabled self-interference channel, and
therefore cancel interference, over the entire sub-6 GHz spectrum. Data was not taken
above 6 GHz because the 3-dB cutoff for the direct modulation response was 6 GHz.
To understand why certain bands performed better than others in terms of cancellation bandwidth, the cancellation depth for a sample of the target bands are plotted
in Fig. 5.7. These plots reveal that cancellation depth can vary widely, often by more
than 10 dB, across the target band. This large variation occurs at high cancellation
depths, where the residual interference becomes very sensitive to the slight frequency
variations that can originate from reflections (e.g. multipath) and even component
responses. Ultimately, in this IMPC, the limited amplitude and phase degrees of freedom of a single reference tap prevented the IMPC from achieving high cancellation
across certain bands, consistent with our findings in the discrete MPCs.
To summarize, the IMPC was able to achieve −30 dB cancellation over a majority
of the target operating bands in the sub-6 GHz spectrum. The main limitation on cancellation bandwidth was the use of a single reference tap to match the self-interference
channel.
5.3.2
Amplitude and Phase Control
The importance of accurately matching the self-interference channel leads us to the
second area of investigation: on-chip amplitude and phase control. In the IMPC,
166
WiFi 5 GHz
LTE 22
LTE 7
WiFi 2.4 GHz
LTE 25, 2
LTE 3
LTE 26
LTE 13
0
1
2
3
4
5
6
Frequency (GHz)
Figure 5.6: Measured cancellation across all FDD LTE and WiFi bands. Bands are
plotted in order of ascending center frequency. The gray, thicker bars indicate the
spectral region defined by each band. The red, thinner bars demarcate the region
over which at least −30 dB of cancellation depth can be achieved at a fixed IMPC
configuration. The width of the bars are magnified by 10x for ease of viewing.
amplitude and phase were tuned by using a pair of cascaded SOAs. The SOAs
generated a tunable delay by using the dispersive phase shift properties of an SOA,
similar to the scheme investigated in Chapter 4. Using this method, the RF amplitude
(gain) and phase shift of a single SOA were not independent, but cascading a second
SOA after the first SOA provided an additional degree of freedom that decoupled
the amplitude and phase of the cascaded SOA pair. We refer to section 4.1 for a
discussion of operating principles. Here, we seek to answer three questions. First,
what amplitude and phase range can be achieved using the cascaded SOAs in the
reference tap? Second, since each SOA affects both amplitude and phase, what range
167
LTE 13
LTE 3
LTE 25
WiFi 2.4 GHz
LTE 22
WiFi 5.8 GHz
Figure 5.7: Cancellation depth (blue curve) over a sample of LTE and WiFi bands.
The gray areas indicate the spectral region defined by each band. The red line is a
reference line marking −30 dB cancellation depth.
of independent amplitude and phase control is available? Third, how sensitive is the
cancellation performance to the SOA control biases?
Total Amplitude and Phase Shift
To measure the full range of amplitude and phase achievable by the cascaded SOAs,
we turned off the RX tap and only measured the reference tap’s effect on the RF
signal. The complex S21 of the reference tap was measured by a VNA, which enabled
an observation of both gain and phase. The relative gain and phase of the reference
tap were plotted as a function of both reference SOA bias currents, and results are
shown in Fig. 5.8 for RF frequencies of 1.25 GHz and 3 GHz. This experiment led to
several pertinent results.
168
(a) Phase, 1.25 GHz
(b) Phase, 3 GHz
(c) Gain, 1.25 GHz
(d) Gain, 3 GHz
Figure 5.8: Relative phase and gain of the reference signal as a function of the SOA
control bias currents. Phase shift at (a) 1.25 GHz and (b) 3 GHz RF frequency. Gain
shift at (c) 1.25 GHz and (d) 3 GHz RF frequency. All values are measured relative
to the phase and amplitude at (20 mA, 20 mA).
The first result was that the total phase range at 1.25 GHz was 60◦ , but decreased
to 20◦ at 3 GHz (Figs. 5.8a and 5.8b). This roll-off, which will be quantified further
in the next section, was a side effect of a dispersive material. Since the dispersion was
strongest at the optical carrier frequency, at higher RF frequencies, where the modulation sidebands extended further away from the carrier, the phase shift decreased.
The effect rolls-off at an RF frequency of f = 1/2πτ , where τ is the semiconductor
carrier lifetime and is on the order of 100 ps. There have been several works which
seek to overcome this limitation by using optical filtering and single sideband modulation [40, 89]. The tunable gain range was 17 dB across the SOA control bias space
and was much less sensitive to RF frequency (Figs. 5.8c and 5.8d).
169
The second result was that the RF gain initially increased proportionally to control
bias, but leveled off at a certain band of control bias values, and then began to decrease
with control bias. This effect was a result of the second SOA entering deep power
saturation. To be specific, as the first SOA’s gain increased, its strong optical output
eventually drove the second SOA into deep power saturation, and the second SOA
actually began attenuating the signal rather than amplifying it. The reversal of the
gain behavior with respect to bias current was actually useful for accessing a wider
phase range. Normally, as the SOA gain increases, the phase shift decreases, via the
Kramers-Krönig relations. However, the region where the second SOA was deeply
saturated reversed this relationship. Instead of a single set of phase shift values for
every RF gain value, two sets of phase shift values were accessible.
The third result was that there were certain anomalous bands that were identically located in all the plots in Fig. 5.8. When the control biases entered one of
these bands, the RF gain and phase abruptly changed by 10◦ and 2 dB, respectively.
We traced the origin of these anomalous bands to increased optical feedback from
certain waveguide transition elements back into the DFB lasers, perturbing the laser
dynamics. We also observed that the laser relative intensity noise (RIN) increased
significantly at these bias points, further indicating that optical feedback was the culprit. The anomalous bands became more pronounced at higher RF frequency, closer
to the relaxation oscillation frequency of 5.8 GHz, where RIN was strongest. Optical feedback is unavoidable in any monolithically integrated platform; however, one
group has engineered optical feedback in a monolithic platform to their advantage,
using feedback to sharpen laser linewidth and reduce RIN [64].
Independent Amplitude and Phase Shift
While Fig. 5.8 showed that 17 dB and 60◦ of tunable gain and phase, respectively,
can be generated at 1.25 GHz, the gain and phase were visibly correlated. We now
170
turn our attention towards answering the question of what independent amplitude
and phase range can be achieved using the cascaded SOAs. We also study this range
as a function of RF frequency.
We began by taking the data from Fig. 5.8 and grouping all the control bias pairs
by the RF gain that they produced. Within each “constant gain” group, we mapped
their control bias pairs to their phase shift equivalent and examined the phase range
that was accessible. By definition, the phase was independent of RF gain within
each constant gain group. We subsequently selected a desired range of gain values
and inspected the accessible phase ranges of all the constituent constant gain groups.
The largest minimum phase and the smallest maximum phase over all constituent
constant gain groups bounded the independent phase range that could be accessed
while accessing the selected gain range.
Using this method, we investigated the independent phase range that could be
generated while maintaining an RF gain range of 10 dB, centered upon 13 dB, and
the results are shown in Fig. 5.9. This specific RF gain range was chosen because it
maximized the independent phase range. As expected, the phase range was smaller
than the range represented in Fig. 5.8; at 1.25 GHz, the independent phase range
was 52◦ . Consistent with previous observations, the phase range decreased with RF
frequency to a minimum of 9◦ at 3 GHz. The phase range increased again at 5.5 GHz.
However, this was an artifact produced by the bands of high optical feedback, which
is exacerbated near the relaxation oscillation frequency (5.8 GHz).
5.3.3
Sensitivity of Cancellation to Controls
We now address the question of how sensitive the cancellation is to the control biases.
This question has practical importance, because if the cancellation is too sensitive,
then ultra-precise control electronics will be required to maintain high cancellation
depths, raising the cost and complexity.
171
Phase Range (deg)
50
40
30
20
10
0
1
4
2
3
Frequency (GHz)
5
Figure 5.9: Independent phase shift range as a function of RF frequency. The independent gain range is 10 dB, centered upon 13 dB.
To determine the cancellation sensitivity, we measured the cancellation over the
entire control bias space for bandwidths of 65 MHz, 10 MHz, and 1 MHz. In each
case, the center frequency of the bandwidth of interest was 1.963 GHz (LTE Bands
2 and 25). These cancellation maps are plotted in Fig. 5.10. The maps are similar
for cancellation up to −30 dB, indicating that obtaining −30 dB of cancellation is
relatively independent of bandwidth up to 65 MHz in this self-interference channel.
As the bandwidth shrinks to 1 MHz, pockets of cancellation as high as −50 dB can
be observed (see Fig. 5.10c and zoom-in). This is a direct manifestation of the
frequency-variation of the channel that was also observed in Fig. 5.7. We measured
these pockets to be about 2 mA in either of the control-space dimensions for −40 dB
172
cancellation depth, and 0.5 mA for −50 dB cancellation depth. Note the resemblance
between the cancellation maps of Fig. 5.10 and the phase map in Fig. 5.8, showing
the link between phase matching and cancellation.
To summarize, we studied the control aspects of the IMPC. Our major findings
were that the independent amplitude and phase range of the cascaded SOAs in the
reference tap were 10 dB and 52◦ , respectively, at 1.25 GHz, and decreased as a function of frequency to 10 dB and 9◦ at 3 GHz. The cancellation sensitivity depends
upon bandwidth, operating frequency, and cancellation depth. At 1.963 GHz center
frequency, −30 dB of cancellation could be achieved for all bandwidths up to 65 MHz,
while −40 dB and −50 dB cancellation could only be achieved at lower bandwidths
and had a sensitivity of 2 mA and 0.5 mA, respectively.
5.3.4
Adaptive Interference Cancellation
To demonstrate the IMPC’s ability to cancel interference using the cascaded SOA
controls, we converted the IMPC into an adaptive analog self-interference canceller
by once more using the Nelder-Mead Simplex algorithm (NMS).
The algorithm begins by constructing an N-dimensional initial simplex, where N
is the number of independent system variables (in the Least Mean Squares algorithm,
these would be the filter weights). An N-dimensional simplex has N+1 vertices, so
the algorithm must be provided N+1 initial vertices in the system variable space as
initial conditions. The algorithm progresses by measuring the cost function at each
of its vertices, and based on their values, choose to either reflect, expand, contract, or
shrink the simplex. The parameters that define each of these operations are similar
to the step size parameter in the Least Mean Squares algorithm and can enable the
algorithm to converge quicker or overshoot, depending on the cost function surface.
See appendix A for the code used to run the NMS.
173
(a) 65 MHz
(b) 20 MHz
2 mA
.5 mA
(c) 1 MHz
Figure 5.10: Cancellation performance as a function of the SOA control biases for
bandwidths of (a) 65 MHz, (b) 20 MHz, and (c) 1 MHz. The zoom-in magnifies the
optimal point in the 1 MHz bandwidth case. Center frequency is 1.963 GHz in all
cases.
For our IMPC, there are N = 2 independent variables, and therefore 3 initial
vertices, making the simplex a triangle. The selected cost function was a geometric
mean of the IMPC output RF power across the band of interest
M =10
mB
1 Y
Pr f o +
CF =
M m=1
M
(5.1)
where fo is the lower frequency boundary of the band of interest and B is the bandwidth. M corresponds to the number of samples used for the geometric mean; M = 10
in our cost function. The geometric mean was chosen to compensate for the fact that
cancellation depths can vary by more than 10 dB over the band of interest. Using the
174
0
Iteration
10
(a)
20
30
40
50
Converged
Simplex
60
70
100
Initial Simplex
Co 80
ntr 60
ol B
ias 40 20
2(
mA 0
)
20
40
80
100
A)
ias 1 (m
B
Control
0
60
(b)
Figure 5.11: Adaptive self-interference cancellation using the Nelder-Mead Simplex
algorithm at a center frequency of 1.963 GHz and 20 MHz bandwidth. (a) Progression
of the simplex as it converges to the optimum point and (b) output interference power
at each iteration of the algorithm. Insets show the cancellation depths at the specified
iteration of the algorithm. The gray region corresponds to the bandwidth of interest.
175
geometric mean of the output power as the cost function rewards the algorithm for
driving the interference to extremely low levels in parts of the band of interest, even
if other parts of the band do not exhibit such low interference levels. This might otherwise be ignored by the arithmetic mean, because the portion of the band with poor
cancellation would dominate the cost function. The cost function can be rewritten as
M =10
1 Y
mB
mB
CF =
Pnr fo +
× CD fo +
M m=1
M
M
(5.2)
to explicitly show the presence of the cancellation depth. The cost function was
measured by a network analyzer, which measured the cancellation depth across the
bandwidth of interest at each simplex vertex. A computer running the NMS algorithm updated the reference tap’s SOA control biases based on the cost function.
Figure 5.11a shows the algorithm progressing from the initial simplex to the converged simplex, while Fig. 5.11b shows the average output power at each iteration
of the algorithm. The center frequency and bandwidth of interest for data in Fig.
5.11 were 1.963 GHz and 20 MHz, respectively. The algorithm was able to successfully use the cascaded SOAs to drive the system from no cancellation to a maximum
cancellation of nearly −40 dB.
5.3.5
Comparison with Discrete MPCs
To put the IMPC in context with prior work, we have compared the instantaneous
bandwidth, maximum operating frequency, and cancellation depth of the IMPC with
previously published discrete MPCs within the last 7 years in Table 5.2. While
cancellation depth and instantaneous bandwidth are functions of the self-interference
channel, all of the MPCs in Table 5.2 were tested on cabled self-interference channels,
allowing at least a first-order comparison. When available, 30 dB of cancellation
176
Table 5.2: Comparison of MPCs
Max. Operating
Cancellation
Instantaneous
Frequency (GHz)
Depth (dB)
Bandwidth (MHz)
Discrete
10
30
100
[42]
Discrete
10
30
200
[4]
Discrete
≥ 2.4
40
200
[65]
Discrete
10
30
135
[66]
Discrete
10
30
5500
[106]
Discrete
18
35
25
[67]
Discrete
50
44
50
This Thesis
Integrated
6
30
200
Reference
Form Factor
[3]
depth was used as the common benchmark; when not available, the closest reported
cancellation depth was used.
Table 5.2 shows that the IMPC achieved similar cancellation performance to other
MPCs, despite being fabricated on an integrated platform without the flexibility
allotted by different materials and devices. This is a strong indication that, even in
the near term, an integrated MPC can compete performance-wise with discrete MPCs,
while holding a tremendous advantage in scalability, size, power dissipation, and cost.
In addition, the integrated MPC creates a path to realizing more complex, multi-tap
MPCs, a necessity in real RF environments, without being cost prohibitive. As
integrated photonic manufacturing continues to mature, bolstered by recent demand
and volume generated by datacenter applications, the same benefits will also pass
down to integrated microwave photonic circuits. While not yet price-competitive with
RFICs, integrated microwave photonics possess a performance advantage that may
give it the edge in bandwidth-hungry applications, such as wideband millimeter-wave
communications.
177
5.3.6
Recommendations for Improvement
There are several clear trajectories to IMPC based on these findings. First, the operating frequency of the IMPC can be increased by using an external modulator rather
than direct modulation, which set the operating frequency ceiling of 6 GHz in this
work. Traveling-wave electrode modulators up to 40 GHz bandwidth exist commercially, and the photodetectors in this IMPC already have a cutoff frequency of 35 GHz.
Second, we are investigating methods to improve the IMPC’s independent amplitude
and phase range, which is much less than what can be achieved by discrete attenuator technologies and fiber delay-lines. Dispersion-based approaches, such as the one
studied in this work, inherently link together amplitude and phase via the KramersKrönig relations. In the immediate future, one potential alternative uses switching
delays to completely decouple amplitude and phase. Optical switching delays can
be designed using a crossbar architecture on multiple material platforms, including
InP [107], silicon-on-insulator [108], and silicon-nitride [109]. While some of these
solutions require multiple materials, they can still be monolithically integrated by
using a hybrid silicon technology, which is now commercially active [6, 105]. Another
option involves a switched RF delay line prior to the IMPC. Although this option
will induce losses at higher RF frequencies, it may still be a competitive solution at
lower frequencies while being the most economical. Finally, we look forward to taking
advantage of the scalability of the IMPC platform to introduce additional reference
taps, which will enable the IMPC to compensate for more complex self-interference
channels.
5.4
Radio-Frequency Performance
We now turn our attention to the RF characteristics of the IMPC. The characterization is crucial to understand the device’s effect upon radio-frequency receivers,
178
A Network Analyzer
B
ENR
LNA
RX In
IMPC
C
Sig
Gen 1
Sig
Gen 2
RF Out
B Signal Analyzer
C Spectrum Analyzer
REF In
+
50 Ω
Figure 5.12: Various test setups used to characterize the IMPC. (a) Network analyzer
for S-Parameter measurements; (b) Noise figure measurement using an excess noise
ratio (ENR) source, low noise amplifier (LNA), and a noise figure meter; (c) Linearity
measurements using two signal generators and a spectrum analyzer.
especially given that the IMPC is one of the first photonic integrated circuits to possess only radio-frequency inputs and outputs. We characterize the circuit’s RF gain,
noise figure, input-intercept point, 1-dB compression point, and spurious-free dynamic
range as functions of on-chip device bias points as well as frequency up to 6 GHz, the
3-dB cutoff frequency of the device. We use the analysis to make recommendations
on how to improve the device’s RF characteristics in the next design iteration. At
the end of this section, we present a side-by-side between the RF properties of the
IMPC and the discrete MPCs.
The IMPC was characterized using a variety of test setups, shown in Fig. 5.12,
to be elaborated upon. Note that for the following experiment and analysis, we were
only concerned with the RF characteristics of the signal of interest, which propagated
exclusively through the RX tap. In staying with the style of this thesis, all the
results and analysis presented here are for an intrinsic photonic circuit, i.e. no RF
amplification or external optical components were used. We present all results as RF
parameters and not optical parameters, unless indicated otherwise.
179
5.4.1
RF Gain
The RF gain of the IMPC’s RX tap was measured using a Keysight N5222A vector
network analyzer in configuration A of Fig. 5.12. The |S21| indicated the RF gain
of the RX tap, while the |S11| indicated the RF return loss. We measured the
|S21| as a function of the RX laser bias, with the RX SOA biased at 40 mA, the
balanced photodetector reverse biased at −3.5 V, and the reference branch turned off
and terminated. The results are shown in Fig. 5.13a. For consistency, we refer to the
|S21| as gain even if it corresponded to a loss.
The frequency response of the RX tap resembles a laser modulation response; this
is expected, because the balanced photodetector’s 3-dB cutoff frequency is 35 GHz,
so the frequency response is dominated by the laser. It is well known that at low
frequencies, far below the laser’s relaxation oscillation (RO) resonance frequency, the
linear RF gain of a directly modulated analog photonic link can be expressed as [31]
GRF = (SL Gopt <)2
(5.3)
where SL is the laser slope efficiency in units of W/A, Gopt is the optical gain of the
photonic link in linear units, and < is the photodetector responsivity in units of A/W.
Inserting values of SL = 0.043 W/A and < = 0.8 A/W obtained from test structure
measurements, Gopt = 5 (7 dB) from PICWave simulations of the SOA, and accounting
for a 6 dB loss from passive impedance matching of the photodetector, we calculate
the low frequency RF gain to be −23 dB, which matches very well with the |S21|
measurement in Fig. 5.13a. At this point, we should mention that the nominal laser
slope efficiency was supposed to be 0.35 W/A, but due to a systematic fabrication
error, it was reduced by nearly an order of magnitude. Using SL = 0.35 W/A, and
accounting for the decreased optical gain of the SOA due to saturation, the RF gain
would be −9 dB. There are several additional features of Fig. 5.13 that deserve
180
(a)
(b)
Figure 5.13: The circuit (a) |S21| as a function of laser bias and (b) |S11| for a laser
bias of 25 mA. The RX SOA bias is 40 mA.
attention. First, the overall |S21| decreases with increasing laser bias current because
of SOA saturation. Based on PICWave simulations, as the laser bias increases from
15 mA to 50 mA, SOA saturation causes a 2 dB drop in optical gain, or a 4 dB drop
in RF gain, which matches the measured result. Second, the laser RO resonances are
strong up to 30 mA but becoming increasingly damped at higher bias, consistent with
our fabrication partner’s wafer validation measurements. Third, there is an abrupt
drop in |S21| at 1.5 GHz. This drop appears to be caused by the photodetector and
is likely related to circuit parasitics; its exact source is still under investigation.
The return loss of the DFB lasers are shown in Fig. 5.13b for a bias current of
25 mA. The |S11| increases slightly (i.e. becomes less negative) with increasing bias
current, but not significantly. An |S11| of more than −10 dB indicates that the lasers
are well impedance matched, despite no active attempts at impedance matching.
Although a fortunate coincidence, impedance matching will likely be required if direct
modulation is to be used once the lasers are properly fabricated. To obtain a full
picture of how the RF gain is affected by the laser bias and the RX SOA bias, we
measured the RF gain as a function of both parameters at 3 GHz RF frequency, and
the result is shown in Fig. 5.14. Again, the reference tap was turned off for these
181
Figure 5.14: RF gain as a function of RX laser bias and RX SOA bias at 3 GHz
RF frequency. The dashed box highlights an operating region that corresponds to
increased optical feedback.
measurements. As expected, the major trend is that the RF gain increases with
increasing SOA current: the gain increases by over 10 dB as the SOA bias increases
from 25 mA to 90 mA, with clear signs of power saturation at the higher end of
the range. On the other hand, the laser bias current has two different effects on
gain. At low laser biases, there is a clear decrease in gain as the laser bias increases.
This is because 3 GHz coincides with the RO resonance frequency at 15 mA bias,
but moves off-resonance as the laser bias increases. Above 20 mA laser bias, when
the RO resonance frequency is significantly higher than 3 GHz, a slight decrease in
gain with increasing laser bias occurs due to greater SOA saturation. The second
effect that the laser bias has on RF gain manifests as anomalous, periodic dips in
182
gain; one such region is highlighted by the dashed box in Fig. 5.14. We believe
that these anomalies are caused by increased external optical feedback from the SOA
back into the laser. They are periodic with laser bias because a change in laser bias
induces a linear wavelength shift, which eventually leads to a 2π optical phase shift
between the laser facet and the external reflector. To confirm this hypothesis, we
measured the period of the RF gain dips to be 4.9 mA in laser bias. By measuring
the optical wavelength of a test laser as a function of laser bias, we determined the
wavelength shift to be 0.053 nm/mA, and therefore, the anomalous dips in RF gain
occur every ∆λ = 0.26 nm. Assuming each ∆λ corresponds to a 2π optical phase
shift, the external reflector must be located
L=
2n
2n
−
λo
λo + ∆λ
−1
(5.4)
from the circuit-facing laser facet, where n is the refractive index in the waveguide
and λo is the wavelength at one particular anomaly. Solving Eqn. (5.4) for L using
n = 3, ∆λ = 0.26 nm, and one pair of neighboring anomalies corresponding to λo =
1549.2 nm, we obtain L = 1540 µm. This corresponds closely with the distance between the circuit-facing laser output facet and a waveguide transition element located
directly after the RX SOA. The waveguide transition element transitions the waveguide from a shallow 200 nm waveguide to a 1700 nm deep-etched waveguide, and will
inevitably produce small reflections. The hypothesis that the reflections pass through
the RX SOA (twice) also explains why the anomalies are a function of the SOA bias
as well as why they are more sharply defined at higher SOA biases. A full analysis
of the effects of external feedback are outside the scope of this paper, but we note
that the effects that we observed correspond well with studies on coherent external
feedback in laser diodes [64, 110]. As we will see, the optical feedback into the laser
affects not only the gain, but also the noise.
183
5.4.2
Noise Figure
Noise figure is one of the most important properties in any component intended for
the RF front-end, because of the direct effect it has on the receiver’s sensitivity. Understanding the noise figure of an individual component enables a high-level cascade
analysis of the entire RF system.
Experimental Measurement
To measure noise figure, we used a Keysight 346CK01 excess noise ratio (ENR) source,
an LNA, and a Keysight PXA N9030A Signal Analyzer in configuration B of Fig.
5.12. The noise figure of the IMPC was measured by connecting the ENR-LNA chain
to the RX tap input, while biasing the RX tap under normal operating conditions,
and connecting the signal analyzer to the IMPC output. The LNA was required
because the attenuation through the IMPC caused the output signal to approach
the sensitivity limits of the signal analyzer. We used the Friis noise equation to
back-calculate the noise factor of the IMPC from the noise factor of the LNA-IMPC
chain
FIM P C = GLN A (Fmeas − FLN A ) + 1
(5.5)
where FIM P C , FLN A , and Fmeas are the noise factors of the IMPC, LNA, and the
measured noise factor, respectively, and GLN A is the gain of the LNA. The noise
figure is simply the logarithm of the noise factor. The gain and noise figure of the
LNA used this experiment were 30 dB and 1.4 dB, respectively. Using this technique,
we measured the noise figure of the IMPC as a function of both the RX laser bias and
the RX SOA bias at 1 GHz, 2 GHz, and 3 GHz. For this measurement, the reference
tap was turned off, and the balanced photodetector was reverse-biased at −3.5 V.
The results are shown in Fig. 5.15a-c.
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42 dB
44 dB
(a)
(d)
44 dB
46 dB
(b)
(e)
46 dB
48 dB
(c)
(f)
Figure 5.15: Noise figure of the IMPC. Experimental measurements at (a) 1 GHz, (b)
2 GHz, and (c) 3 GHz. Simulated noise figure at (d) 1 GHz, (e) 2 GHz, and (f) 3 GHz.
Bands of high noise figure are highlighted in dashed regions.
185
To analyze these results, we comment first on the overall trends before focusing on
particular features. The average noise figures of the IMPC (43 dB, 45 dB, and 47 dB
at 1 GHz, 2 GHz, and 3 GHz, respectively) are quite high, especially when compared
to RF standards. These high noise figures are a result of the relatively high RF loss
incurred by the EO and OE conversion observed in the previous section and, as we will
see, high RIN from the lasers. The average noise figure increases with frequency as a
reflection of the increased RIN at higher frequencies; this is particularly noticeable at
3 GHz for low laser biases, when the RO resonance frequency is 3 GHz and RIN is at
its strongest. We will discuss methods to reduce the noise figure in the section 5.4.5.
Focusing now on particular features, notice that the noise figure exhibits anomalies
that share an identical fingerprint as the anomalies seen in the RF gain (Fig. 5.14).
These anomalies, highlighted in the dashed regions of Fig. 5.15, result in abrupt
increases in noise figure by sometimes as much as 10 dB and tend to be localized.
The pattern of periodic anomalies end when the laser bias increases beyond 30 mA,
which coincides with an increased damping of the laser RO resonances. The anomalies
grow in magnitude at higher SOA currents, similar to the gain measurements; in
addition, they become more sharply defined at higher RF frequencies. The qualitative
dependency of the anomalies on bias current and RF frequency suggest that the
source of the additional noise is increased RIN, which we will confirm in the following
subsection. This supports the hypothesis that the anomalies originated from external
optical feedback, which can significantly perturb laser dynamics and increase RIN by
over 10 dB [64].
Before moving on, we note that when operated for interference cancellation, the
noise figure of the IMPC will be larger than that measured in Fig. 5.15. This is due
to the additional noise added by the reference tap, which contributes RIN, shot noise,
and amplified spontaneous emission (ASE) noise. We estimate that the noise figure
will increase by at most 3 dB, because it is unlikely that the reference tap will produce
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Table 5.3: Variable definitions and values
Symbol
q
ID
<
RL
RIN
nsp
Variable
Value
Electron Charge
1.6 × 10−19 C
Average Photocurrent
–
Responsivity
0.8 A/W
Load Resistance
50 Ω
Relative Intensity Noise
–
Spontaneous-Emission Factor
2
h Plank’s Constant
6.63 × 10−34 J s
ν
Optical Frequency
193.5 THz
GSOA
SOA Optical Gain
–
more noise than the RX tap. This is supported by the fact that the reference tap
does not need to produce more optical intensity (and hence RIN and shot noise) than
the RX tap because the actual self-interference is usually weaker than the reference
copy.
Analysis of Noise Sources
Noise figure can be analytically expressed as
SN Rin
N F = 10log
SN Rout
Nout
= 10log
GRF kT
(5.6)
where Nout is the output noise power spectral density, k is the Boltzmann constant,
and T is the temperature. The second equality in Eqn. (5.6) assumes that the input
noise is thermal noise. To determine the main drivers of noise figure in the IMPC,
we first identify the dominant noise sources in any analog photonic link, which are
typically thermal noise, shot noise, laser RIN, and signal-spontaneous beat noise. The
187
power spectral density for each of these noise sources can be expressed as [31]
Nthermal
1
= kT 1 +
GRF
(5.7)
Nshot = 2qID RL
(5.8)
2
NRIN = RIN · ID
RL
(5.9)
Nsig−sp = 4ID <nsp hν(GSOA − 1)RL
(5.10)
where the various variable definitions and values (when fixed) are listed in Table
5.3. Spontaneous-spontaneous beat noise from the SOAs were ignored, because it is
insignificant compared to the signal-spontaneous beat noise. Since the various noise
sources are independent, the output noise power spectral density is simply the sum
of the component power spectral densities
Nout = Nthermal + Nshot + NRIN + Nsig−sp .
(5.11)
To determine the values of each of the noise components, we measured the remaining unknown variables, ID , GSOA , and RIN , by terminating both the RX and
reference inputs, sweeping the RX laser and the RX SOA biases across the values given
in Fig. 5.15, and measuring the average photocurrent of the balanced photodetector
(ID ), the RX laser monitor photocurrent (Imon ), and the RIN using a spectrum analyzer. We calculated GSOA by taking the ratio of Imon /ID , assuming that the monitor
photodiode and the balanced photodiode have the same DC responsivity, and that
there was negligible loss through the passive waveguide. Inserting these measured
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quantities, the values from Table 5.3, and the previously measured GRF into Eqns.
(5.6) - (5.11), we simulated the noise figure of the IMPC, and the results are shown
in Fig. 5.15d-f. The simulated noise figures matched very well with the experimental
measurements and successfully modeled both the average noise figure value as well
as the periodic anomalies. Based on the simulation, the output noise of the IMPC
was dominated by RIN and shot noise, with a total RIN power 3 dB to 10 dB greater
than shot noise; both of these noise sources were at least 10 dB, and often much more,
greater than thermal and signal-spontaneous beat noise. The measured RIN spanned
from −115 dB/Hz at lower bias currents to −138 dB/Hz at higher bias currents and
as expected, the RIN increased with RF frequency. The simulation and RIN measurements confirmed that the increased noise figure in the periodic anomalies were
caused by an increase in RIN.
5.4.3
Linearity and Dynamic Range
The dynamic range of an interference canceller is an important property because a
canceller must simultaneously process high-power interferers and resolve low-power
signals of interest. In this section, we characterize the IMPC’s dynamic range through
the IIP3, P1dB, and SFDR. We quickly discovered that the IMPC was a third-order
limited system, and because this produces nonlinear components that are difficult or
impossible to filter out of the receiver passband, we used the third-order intermodulation product (IMD3) as the limiting nonlinearity for IIP3 and SFDR measurements.
Input Intercept Point (IIP3)
The IIP3 was measured using configuration C of the test setup in Fig. 5.12. In this
configuration, the outputs of two RF signal generators were combined through an RF
combiner and used as the input to the PIC for two-tone measurements (two tones
separated by 1 MHz); the output of the PIC was connected to a Tektronix RSA6120B
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real-time spectrum analyzer to monitor the output frequency components. Because
link gain and noise figure are functions of laser bias, SOA bias, as well as RF frequency,
IIP3 was also studied as a function of these parameters.
IIP3 was first measured as a function of RF frequency up to 6 GHz, which was the
maximum frequency of one of the signal generators. The results are shown in Fig. 5.16
for three different laser biases: 15 mA, 25 mA, and 35 mA. They led to three major
observations. First, the IIP3 generally exhibited different trends corresponding to
different frequency regions relative to the laser RO resonance frequency, fRO . These
trends have been overlaid as dashed lines in Fig. 5.16 to assist in the following
discussion.
• Region 1 (f < fRO /2): In this region, the IIP3 decreased with increasing
frequency at a rate of approximately 40 dB/ decade.
• Region 2 (fRO /2 < f < fRO ): the IIP3 flattened out and approached a
constant level. The duration for which the IIP3 stayed at this level depended
on the laser bias. For example, at 35 mA, the IIP3 remained constant over
nearly the entirety of Region 2; however, at 15 mA, the IIP3 continued its
40 dB/ decade descent without disruption.
• Region 3 (f > fRO ): the IIP3 began to rise at a rate faster than the descent
in region 1.
The trends in frequency regions 1 and 2 can be explained by a laser rate equation
analysis of intermodulation distortion in directly modulated lasers, presented by Lau
and Yariv [8]. In this theory, the effective IIP3 falls at a rate of 40 dB/ decade up
until fRO/2 , consistent with our experimental observations. The behavior in region
2 depends on how damped the RO resonances are; a highly dampened resonance
continues its 40 dB/ decade descent through regions 1 and 2, while an undamped
resonance may actually see a local peak in IIP3 at fRO /2 as the IIP3 approaches a
190
constant level. Here, our experimental observations showed the reverse: at 15 mA,
where the RO resonance was strong, the IIP3 continued its 40 dB/ decade descent
through regions 1 and 2, while at 35 mA, where the RO resonance was weak, the IIP3
approaches a constant level. Frequency region 3 was also completely unexplained
by [8], which predicted that the IIP3 continues to decrease at a rate of 40 dB/ decade.
The second observation was that the IIP3 increased with laser bias. To quantify
this further, we measured the IIP3 as a function of laser bias current at 3 GHz RF
frequency, and the results are shown in Fig. 5.17. Qualitatively speaking, the IIP3
increased with laser bias because the fundamental signal had only a weak dependence
on the average optical power through the RO resonance (at low frequencies, the
fundamental signal in direct modulation is often stated to be independent of average
optical power). However, the IMD3 is inversely related to the average optical power
squared, causing the overall IIP3 to increase with average optical power and therefore
laser bias. To confirm this, we applied the expressions for the fundamental and IMD3
signals from [8] (Eqns. (2) and (6)) combined with measurements of the average
optical power from the monitor photodiodes to generate the dashed-line fit in Fig.
5.17. For this fit, the Q-factor was set to 1, and γ, the ratio between the carrier
and photon lifetimes, was set to 2000. The IIP3 saturates above 30 mA laser bias at
15 dBm.
The third observation is that the IIP3 appears to be completely independent of
the SOA bias current. This is not surprising, as the SOA gain should affect all RF
frequencies evenly and play no part in the RF signal frequency distortion.
5.4.4
1-dB Compression Point (P1dB)
The P1dB represents the power-handling capabilities of the IMPC. The P1dB was
characterized by determining when the gain of the fundamental signal was compressed
by 1 dB. The results are shown in Fig. 5.18 as a function of both RF frequency and
191
40 dB/decade
35 mA
25 mA
15 mA
Figure 5.16: Measured IIP3 as a function of RF frequency for laser biases of 15 mA,
25 mA, and 35 mA. Each data point is an average of 5 measurements and the error bars indicate the sample standard deviation. Dashed-line fits correspond to
40 dB/ decade and 0 dB/ decade. The arrows along the x-axis indicate the relaxation
oscillation resonance frequencies for each of the bias currents.
laser bias. When measuring P1dB as a function of RF frequency, the laser bias was
held constant at 35 mA; conversely, when measuring P1dB as a function of laser bias,
the RF frequency was held constant at 3 GHz. The P1dB showed a similar profile
to the IIP3, but is surprisingly high when compared to the IIP3. This suggests that
there was a natural linearization or pre-distortion occurring at high RF powers, where
the linear extrapolation used to predict IIP3 broke down. Indeed, when measuring
the IMD3 and fundamental signals, the IMD3 saturated earlier than the fundamental,
enabling the fundamental signal to continue to increase linearly so that P1dB was
much closer to the IIP3 than standard power amplifiers. As a function of frequency,
192
Figure 5.17: Measured IIP3 as a function of laser bias at 3 GHz RF frequency. Each
data point is an average of 5 measurements and the error bars indicate the sample
standard deviation. The dashed line is a theoretical fit generated from [8].
the P1dB is relatively flat, exhibiting a slight dip at the RO resonance frequency,
when the IMD3 peaks. These results indicate that, like IIP3, the P1dB saturates
above 30 mA laser bias and biasing the laser above 30 mA is best for power-handling
as well as maximum bandwidth.
Spurious Free Dynamic Range (SFDR)
A common measure of dynamic range in an analog photonic link is SFDR. It measures
the input power range bounded on the lower end by the noise floor and on the upper
end by nonlinearities [54]. For the IMPC, the SFDR is limited on the upper end by
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Figure 5.18: Measured P1dB as a function of laser bias (RF frequency = 3 GHz) and
RF frequency (Laser Bias = 35 mA). Each data point is an average of 5 measurements
and the error bars indicate the sample standard deviation.
the IMD3. Using the expression for IMD3-limited SFDR
SF DR =
2
174 + IIP 3 − N F )
3
(5.12)
where NF is the noise figure and −174 dBm/Hz is the thermal noise floor at room
temperature, we determined SFDR using the experimental data obtained in the previous sections. The results are shown in Fig. 5.19 as a function of both laser bias and
RF frequency, with SOA bias held constant at 65 mA. Because we could not explicitly
measure noise figure above 3 GHz, we used the model presented in section 5.4.2 to
extrapolate the noise figure, and therefore the SFDR, above 3 GHz. The SFDR is
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Figure 5.19: SFDR as a function of laser bias (RF frequency = 3 GHz) and RF
frequency (Laser Bias = 25 mA). Above 3 GHz, noise figure simulations were used
together with IIP3 measurements to determine SFDR. Each data point is an average
of 5 measurements and the error bars indicate the sample standard deviation.
dominated by the IIP3 behavior, because the noise figure is only weakly dependent
upon laser bias and frequency (outside of the periodic anomalies).
5.4.5
Key Observations and Recommendations
With insight from the experiment and analysis in the previous sections, we now
present key observations and recommendations for improving the IMPC across the
different RF properties. A high-level summary is provided in Table 5.4.
Observation: The directly modulated laser dominated all of the RF properties,
by virtue of performing the critical EO conversion. It drove RF gain through the
slope efficiency and frequency response, noise figure through the RIN, and linearity
195
Table 5.4: IMPC RF Characterization Summary
RF Property
Main Driver
Spec. at 35 mA Function of Function of
bias and 3 GHz
Frequency
Laser Bias
Slope Efficiency
−28.55 dB
Decrease
Decrease
Laser RIN
46 dB
Increase
–
IIP3
Laser Dynamics
15 dB ± 0.3 dB
Decrease
Increase
P1dB
Laser Dynamics
15.2 dBm
Decrease
Increase
SFDR
See IIP3 and Noise Fig.
95 dB/Hz2/3
Decrease
Increase
Gain
Noise Fig.
through the strong dependency of the IMD3 on frequency and average optical power.
However, with a higher power laser, both the SOA and the photodetector (10 dBm
saturation power) will play a larger role in the RF characteristics.
Observation: RF gain was typically between −25 dB to −35 dB, however with a
properly fabricated laser, it can be as high as −9 dB. It degraded slightly with higher
laser bias, because of increased SOA saturation.
Recommendation: When the laser is fabricated correctly, optical power will increase by nearly 10 dB. At such high optical powers, the in-line RX SOA is likely
unnecessary because the saturated SOA will only provide a few dB of gain. This will
reduce power consumption and make the RF gain independent of laser bias.
Observation: Noise figure was dominated by laser RIN, which was 3 dB to 10 dB
stronger than the second most powerful noise source, shot noise. Periodic spikes in
noise figure were caused by conditions that promoted external optical feedback. These
largely vanish above 30 mA laser bias.
Recommendation: To reduce noise figure, one must reduce RIN. Besides device
engineering, several architectural approaches can be taken. First, using an external
modulator will help reduce optical feedback from the waveguide transition element,
and hence RIN, because of the insertion loss of the modulator. Second, balanced
architectures specifically designed to suppression RIN should be employed to convert
196
the PIC into a shot-noise limited system. With a shot-noise limited link, increasing
the optical power will result in reduced noise figure. Combined with a laser with 10 dB
more optical power, full suppression of RIN will result in at least a 10 dB reduction
in noise figure.
Observation: The IMPC was IMD3-limited, and the linearity and dynamic range
were lowest at the laser RO resonance frequency. The linearity and dynamic range
saturated at their highest values above 30 mA laser bias.
Recommendation: Higher and less frequency-dependent linearity, can likely be
achieved by using an external modulator. However, one should only be used if sufficient optical power can be generated by the source laser; otherwise, gain and noise
figure will suffer. Another option is RF predistortion of the modulator device.
Recommendation: Operate the IMPC above 30 mA laser bias. In this regime,
linearity and dynamic range were at their highest over the widest frequency range.
In addition, the periodic anomalies caused by the external optical feedback into the
laser were also suppressed. Meanwhile, noise figure was relatively independent of laser
bias because increased optical power was counteracted by decreased RIN. The only
negative tradeoff is that RF gain decreases slightly with increasing laser bias due to
SOA saturation. While operating at a higher biases increases power consumption, the
increase should be marginal compared to the power consumption of the thermoelectric
cooler.
Finally, to put the RF performance of the IMPC in perspective, we compare it
directly with the discrete MPCs previously studied in this thesis work. The comparison in Table 5.5, shows that the IMPC (which uses a single-ended architecture) is
competitive with its discrete counterparts, exhibiting RF performance better than the
single-ended discrete MPC, but worse than the balanced MPC. To project the potential of the IMPC platform, we also show the estimated performance of an IMPC, which
has been improved by implementing the prior recommendations, namely correctly fab197
Architecture
Link Gain
(dB)
Noise Figure
(dB)
IIP3
(dBm)
SFDR
(dB/Hz 2/3 )
Single-Ended
Balanced
IMPC
Improved IMPC
-44.9
-23.2
-28.0
-9.0
49.6
32.2
46.0
36.5
17.2
22.8
25.0
25.0
94.1
109.8
99.0
105.33
Table 5.5: Comparison of Key Performance Parameters
ricating the laser and implementing a balanced architecture. We do not include the
effect of external modulation, because the exact performance depends heavily on the
modulator’s characteristics; however, we note that external modulation is typically
more linear than direct modulation. The projection shows that significant gains can
be obtained with the recommendations made above. In particular, the link gain can
be increased and the noise figure can be decreased significantly, making the SFDR of
the IMPC comparable with that of the discrete balanced MPC. We also note that,
while not shown in Table 5.5, adding more taps would require significant resources in
using discrete fiber-optics, but can be designed into an integrated circuit with only a
marginal increase in chip footprint and cost.
5.5
Summary
We presented a full functonal and RF analysis of the integrated microwave photonic interference canceller, the culmination of this thesis work. The monolithically
integrated IMPC is 4 mm x 6 mm and possesses only RF external inputs and outputs while performing interference cancellation using optical signal processing. The
IMPC is entirely electronically controlled and has the capability to perform on-chip
amplitude and phase tuning. This IMPC used a directly modulated laser and singlecompensating tap architecture.
198
In terms of functional performance, the IMPC was capable of canceling interference from DC to 6 GHz RF frequency. The cancellation was a function of bandwidth
and could generally achieve at least 30 dB cancellation over nearly all LTE/WiFi
bands in the sub-6 GHz spectrum. At 1.25 GHz, the IMPC has a full amplitude and
phase tuning range of 17 dB and 60◦ , respectively, and an independent amplitude
and phase tuning range of 10 dB and 52◦ , respectively. This range decreases with
increasing RF frequency; the minimum phase range for a 10 dB amplitude range is
9◦ . To maintain 50 dB cancellation, the IMPC’s SOA controls must be held within
a 0.5 mA range around the optimum point. This is easily manageable with modern
current supplies; however, we predict that the feedback control system that is used
to track the optimum point will be the limiting factor. In systems that require a
larger phase/delay range, a new delay system should be considered to completely decouple amplitude and phase. Several options considered were photonic and electronic
switching delay structures in a variety of materials.
The IMPC’s RF properties are summarized in Table 5.4. The characterization
encompassed RF gain and S-parameters; noise figure and noise sources; and linearity and dynamic range, including IIP3, P1dB, and SFDR. The characterization was
performed as a function of RF frequency, laser bias, and SOA bias. All RF properties were strongly influenced by the directly modulated laser used to perform E-O
conversion. Noise figure, linearity, and dynamic range improved significantly with
increasing laser bias at the slight expense of RF gain; these improvements saturated
at 30 mA. All properties degraded with increasing RF frequency. External optical
feedback from an on-chip waveguide transition element caused increased noise figure
and decreased RF gain at periodic laser bias intervals, and these operating points
should be avoided. The effect of the external reflections are naturally avoided above
30 mA laser bias. For these reasons, we recommend that the PIC be operated above
30 mA laser bias. Additionally, we recommend operating the RX SOA at as high
199
of a current bias as reasonable, since only the RF gain is strongly affected by the
SOA current. The upper boundary of the RX SOA current should be limited by the
point when the differential RF gain increase no longer warrants the increasing power
consumption and parasitic heating effects.
The greatest challenge that the integrated microwave photonic circuit must overcome to be used practically in an RF front-end are its low RF gain and high noise
figure. The lower than desired RF gain was caused by a systematic fabrication error,
which lowered the laser slope efficiency; with proper fabrication, the RF gain is estimated to increase to −9 dB across the entire sub-6 GHz frequency range, which would
be one of the highest gain analog photonic links that covers such a large frequency
span. To lower the noise figure, we recommend switching to external modulation and,
in parallel, using a balanced architecture to suppress laser RIN. The resulting shotnoise limited link would exhibit reduced noise figure with increasing laser power, and
with full RIN suppression, we predict a noise figure improvement of at least 10 dB.
The circuit linearity can likely also be improved by using an external modulator.
Given these improvements, we are confident that the integrated OICS can perform
equally, if not better, than a discrete OICS. Finally, future PICs should also seek to
test the performance and feasibility of controlling a multiple-tap OICS.
200
Chapter 6
Conclusion and Outlook
This thesis tracked the work on microwave photonic interference cancellers, from
architectures to systems to integration. It built upon prior work and established its
novelty through three major thrusts over five and a half years (Fig. 6.1). During this
time, major new milestones have been reached.
In Chapter 2, a solid foundation for analyzing how microwave photonic cancellers
would affect RF receivers was developed. Key performance parameters, such as link
gain, noise figure, and dynamic range, were experimentally studied in all microwave
photonic cancellers built during this thesis. More importantly, an understanding
of how different optical parameters affect the overall system RF performance was
established.
In Chapter 3, new architectures were developed to improve the RF key performance parameters of microwave photonic cancellers. A novel balanced microwave photonic canceller showed great promise to reduce the noise figure of previous single-ended
architectures. An experimentally constructed balanced canceller showed more than
21 dB improvement in RF gain, 17 dB improvement in noise figure, and 15 dB/Hz2/3
improvement in spurious-free dynamic range compared to the previously built singleended architecture! In this chapter, the first investigations of how microwave photonic
201
Figure 6.1: The overview and timeline of this thesis.
cancellers performed when inserted into a real wireless environment as well as a radio
front-end were also conducted. A microwave photonic canceller was used to demonstrate over a 22 dB improvement in self-interference rejection in a full-duplex radio.
Chapters 4 and 5 were all about monolithic integration, the third and final thrust
of this thesis. In Chapter 4, we demonstrated that a semiconductor optical amplifier
could be used to achieve either an independently controllable optical delay, a variable
optical attenuation, or both, through intelligent operation of the control parameters.
The integrated delay line was incorporated into a microwave photonic canceller to
demonstrate the first all-semiconductor based microwave photonic canceller. This
canceller performed as well as previously demonstrated discrete cancellers and could
be fully electronically controlled.
Finally, in Chapter 5, the capstone of this thesis, we demonstrated the first integrated microwave photonic canceller. This device not only integrated the full function
of a microwave photonic canceller onto a 4 mm x 6 mm InP chip, but was also one
of the first photonic integrated circuits to not require any optical input or output.
202
Through integration, the microwave photonic canceller technology can now leverage
the scalability of the semiconductor industry, greatly reducing the size, weight, power,
and cost of the technology, and opening the door to great performance advantages as
well as new applications and markets.
From a larger perspective, we believe that this thesis work is just the tip of iceberg, and that there is enormous potential still left on the table. Today, the market
for optical technology is as strong as its ever been since 2001, driven largely by the
thriving datacenter market. This has led, in the past few decades, to the development of a fledgling photonic integrated circuit industry and ecosystem, incorporating
design software, design houses, process design kits and libraries, fabrication houses,
and packaging houses, mirroring the early development of the electronic integrated
circuit industry. While there is still a long way to go, and market forces are always
unpredictable, there has already been discussion of a Moore’s Law for Photonics [111].
Whether the technology platform that rises to the challenge is III-V, silicon, or both
still remains to be seen. Nonetheless, the prospects for photonic integration and optical signal processing shine brightly, and new paths and opportunities continue to
surface not just for interference cancellation but for wideband analog signal processing
in general. The question is not if, but rather, when.
203
Appendix A
Nelder-Mead Simplex Algorithm
The following code is a MATLAB script running the core of the Nelder-Mead Simplex
Algorithm. The MATLAB script operates in tandem with a LabVIEW program (not
included), which accepted an input from the VNA measuring the cancellation depth
and outputted commands to the control electronics. The different parameters defining
the actions of the Nelder-Mead Simplex Algorithm were defined in the LabVIEW
program and given in Table
The MATLAB script had several subfunctions, which will be provided following
the main function, NelderMeadSimplex.m:
• reflect.m
• expand.m
Table A.1: Coefficients of the Nelder-Mead Simplex Algorithm
Coefficient
Symbol
Value
Reflection
cref
.7
Expand
cexp
2
Contract
ccon
.6
Shrink
cshr
.6
204
• contract.m
• shrink.m
• resetsimplex.m
• getSimplexVol.m
A.1
Main script: NelderMeadSimplex.m
1
% NelderMeadSimplex .m
2
%
3
% This m− f i l e r u n s t h e main l o o p o f t h e Nelder−Mead Simplex Algorithm .
It
4
% i s t o be used i n c o n j u n c t i o n with a c o n t r o l l i n g LabVIEW program , which
5
% i n t e r f a c e s with t h e c o n t r o l / measurement equipment and r e c o r d s / p l o t s
data .
6
%
7
% This program r e c o r d s t h e s t a t e o f t h e a l g o r i t h m i n s t a t e v a r i a b l e s
8
% d e f i n e d below :
9
%
10
% State 0 = I n i t i a l i z a t i o n
11
% State 1 = Reflection
12
% S t a t e 2 = Expansion
13
% State 3 = Contraction
14
% State 4 = Shrink
15
%
16
% The program i s a l s o d e f i n e d by s e v e r a l key v a r i a b l e s , which p r o v i d e
the
17
% o p e r a t i n g b o u n d a r i e s o f t h e c o n t r o l equipment . These v a r i a b l e s , t o be
18
% d e f i n e d i n t h e e x t e r n a l LabVIEW c o n t r o l l e r a r e
19
% MAX SOA1 [mA]
20
% MIN SOA1 [mA]
205
21
% MAX SOA2 [mA]
22
% MIN SOA2 [mA]
23
% MIN RESET [dBm]
24
% MAX VOL [mAˆ 3 ]
25
% MIN VOL [mAˆ 3 ]
26
%
27
% Sub−f u n c t i o n d e p e n d e n c i e s : r e f l e c t .m, expand .m, c o n t r a c t .m, s h r i n k .m,
g e t S i m p l e x V o l .m , r e s e t .m
28
29
addpath ( ’ / U s e r s /Matt/ Dropbox / Lightwave Lab/LabVIEW Programs / Nelder−Mead
Simplex Algorithm / f u n c t i o n s ’ ) ;
30
iteration = iteration + 1;
% Count number o f a l g o r i t h m i t e r a t i o n s
31
RES ITER = RES ITER + 1 ;
% i t e r a t i o n s s i n c e the l a s t r e s e t
32
33
%
========================================================================
34
% Check f o r t h e r e s e t c o n d i t i o n based on minimum power and max i t e r a t i o n
35
%
========================================================================
36
37
i f ( RES ITER > MAX ITER && min ( power ) > MIN POW)
38
s i m p l e x = r e s e t s i m p l e x ( simplex , o r d e r , RES L ) ;
39
RES COUNT = RES COUNT + 1 ;
40
RES ITER = 0 ;
41
state = 1;
42
end
43
44
% ======================================================
45
% Check f o r t h e r e s e t c o n d i t i o n based on s i m p l e x volume
46
% ======================================================
47
206
48
i f ( g e t S i m p l e x V o l ( s i m p l e x ) > MAX VOL)
49
s i m p l e x = r e s e t s i m p l e x ( simplex , o r d e r , RES L ) ;
50
RES COUNT = RES COUNT + 1 ;
51
RES ITER = 0 ;
52
state = 1;
53
end
54
55
% ======================================================
56
% Check c u r r e n t s t a t e and s e t next s t a t e
57
% ======================================================
58
59
i f ( s t a t e == 0 )
% I n i t i a l i z a t i o n S t a t e . Prepare t o
reflect .
60
state = 1;
61
[ ˜ , o r d e r ] = s o r t ( power ) ;
62
63
e l s e i f ( s t a t e == 1 )
% Reflection State .
i f power ( o r d e r ( end ) ) < power ( o r d e r ( 1 ) )
% If reflection is
b e s t , expand
state = 2;
64
65
e l s e i f power ( o r d e r ( end ) ) < power ( o r d e r ( end −1) ) % I f r e f l e c t i o n i s
b e t t e r but not b e s t , r e f l e c t a g a i n
state = 1;
66
67
else
s i m p l e x ( o r d e r ( end ) , : ) = MEM VERT;
68
% I f r e f l e c t i o n i s bad ,
r e v e r t to the l a s t point
69
power ( o r d e r ( end ) ) = MEMPOW;
70
state = 3;
71
72
73
end
e l s e i f ( s t a t e == 2 )
% Expansion S t a t e
i f power ( o r d e r ( end ) ) < MEMPOW
% I f e x p a n s i o n good , r e f l e c t
state = 1;
74
75
% and p r e p a r e t o c o n t r a c t
else
% I f e x p a n s i o n i s bad ,
r e v e r t to l a s t r e f l e c t e d point
207
s i m p l e x ( o r d e r ( end ) , : ) = MEM VERT;
76
% and p r e p a r e t o r e f l e c t
again
77
power ( o r d e r ( end ) ) = MEMPOW;
78
state = 1;
end
79
80
e l s e i f ( s t a t e == 3 )
% Contraction State
i f power ( o r d e r ( end ) ) < MEMPOW
81
% I f c o n t r a c t i o n good ,
prepare to r e f l e c t
state = 1;
82
else
83
% I f c o n t r a c t i o n i s bad ,
r e v e r t the contracted vertex
s i m p l e x ( o r d e r ( end ) , : ) = MEM VERT;
84
% t o t h e p r e v i o u s p o i n t and
prepare to shrink
85
power ( o r d e r ( end ) ) = MEMPOW;
86
state = 4;
end
87
88
else
% After a shrink , prepare to
r e f l e c t again .
state = 1;
89
90
end
91
92
i f ( g e t S i m p l e x V o l ( s i m p l e x ) < MIN VOL && min ( power ) < MIN POW && RES ITER
> MAX ITER/ 4 )
% I f s i m p l e x s m a l l , but min power met , then
keep s m a l l but p r e v e n t from s h r i n k i n g t o dot [ Converged ]
93
s i m p l e x = r e s e t s i m p l e x ( simplex , o r d e r , MIN VOLˆ ( 1 / 3 ) ) ;
94
RES COUNT = RES COUNT+1;
95
RES ITER = 0 ;
96
state = 1;
97
e l s e i f ( g e t S i m p l e x V o l ( s i m p l e x ) < MIN VOL && min ( power ) > MIN POW)
s i m p l e s m a l l and min power not met , expand s l i g h t l y
98
s i m p l e x = r e s e t s i m p l e x ( simplex , o r d e r , RES L ) ;
99
RES COUNT = RES COUNT+1;
100
RES ITER = 0 ;
208
% If
state = 1;
101
102
end
103
104
% ======================================================
105
% Execute Action
106
% ======================================================
107
108
MEM VERT = s i m p l e x ( o r d e r ( end ) , : ) ;
109
MEMPOW = power ( o r d e r ( end ) ) ;
110
111
switch state
case 1
112
[ ˜ , o r d e r ] = s o r t ( power ) ;
113
MEM VERT = s i m p l e x ( o r d e r ( end ) , : ) ;
114
% Rew rite MEM VERT,
MEMPOW based on new o r d e r
MEMPOW = power ( o r d e r ( end ) ) ;
115
s i m p l e x = r e f l e c t ( simplex , o r d e r , c r e f ) ;
116
ACT VERT = [ o r d e r ( end ) ] ;
117
% The a c t i v e v e r t e x b e i n g
manipulated
case 2
118
s i m p l e x = expand ( simplex , o r d e r , c e x p ) ;
119
ACT VERT = [ o r d e r ( end ) ] ;
120
% The a c t i v e v e r t e x b e i n g
manipulated
case 3
121
s i m p l e x = c o n t r a c t ( simplex , o r d e r , c c o n ) ;
122
ACT VERT = [ o r d e r ( end ) ] ;
123
% The a c t i v e v e r t e x b e i n g
manipulated
otherwise
124
s i m p l e x = s h r i n k ( simplex , o r d e r , c s h r ) ;
125
ACT VERT = o r d e r ( 2 : end ) ;
126
% The a c t i v e v e r t e x b e i n g
manipulated
127
end
128
209
129
130
% ======================================================
131
% Check MIN SOA and MAX SOA Compliance
132
% ======================================================
133
134
check = s i m p l e x ( : , 1 )>MAX SOA1;
135
s i m p l e x ( : , 1 ) = check ∗MAX SOA1 + ˜ check . ∗ s i m p l e x ( : , 1 ) ;
136
check = s i m p l e x ( : , 2 )>MAX SOA2;
137
s i m p l e x ( : , 2 ) = check ∗MAX SOA2 + ˜ check . ∗ s i m p l e x ( : , 2 ) ;
138
139
check = s i m p l e x ( : , 1 )<MIN SOA1 ;
140
s i m p l e x ( : , 1 ) = check ∗MIN SOA1 + ˜ check . ∗ s i m p l e x ( : , 1 ) ;
141
check = s i m p l e x ( : , 2 )<MIN SOA2 ;
142
s i m p l e x ( : , 2 ) = check ∗MIN SOA2 + ˜ check . ∗ s i m p l e x ( : , 2 ) ;
A.2
subfunction: reflect.m
1
% r e f l e c t .m
2
%
3
% This f u n c t i o n i s p a r t o f t h e Nelder−Mead Simplex Algorithm . I t s
purpose
4
% i s t o r e f l e c t t h e w o r s t v e r t e x . I t i s t h e primary s t e p o f t h e Nelder−
Mead
5
% Algorithm .
6
7
f u n c t i o n [ s i m p l e x ] = r e f l e c t ( simplex , o r d e r , c r e f )
8
9
10
c e n t e r = mean ( s i m p l e x ( o r d e r ( 1 : end −1) , : ) ) ;
s i m p l e x ( o r d e r ( end ) , : ) = c e n t e r + c r e f ∗ ( c e n t e r − s i m p l e x ( o r d e r ( end ) , : ) ) ;
11
12
end
210
A.3
subfunction: expand.m
1
% expand .m
2
%
3
% This f u n c t i o n i s p a r t o f t h e Nelder−Mead Simplex Algorithm . I t s
purpose
4
% i s t o expand t h e w o r s t v e r t e x towards a b e t t e r d i r e c t i o n when t h e
5
% r e f l e c t e d p o i n t from t h e p r e v i o u s o p e r a t i o n ended up b e i n g t h e b e s t
6
% vertex .
7
8
f u n c t i o n [ s i m p l e x ] = expand ( simplex , o r d e r , c e x p )
9
10
c e n t e r = mean ( s i m p l e x ( o r d e r ( 1 : end −1) , : ) ) ;
11
s i m p l e x ( o r d e r ( end ) , : ) = c e n t e r + c e x p ∗ ( s i m p l e x ( o r d e r ( end ) , : ) − c e n t e r ) ;
12
13
end
A.4
subfunction: contract.m
1
% c o n t r a c t .m
2
%
3
% This f u n c t i o n i s p a r t o f t h e Nelder−Mead Simplex Algorithm . I t s
purpose
4
% i s t o c o n t r a c t t h e w o r s t v e r t e x o f t h e s i m p l e x when a r e f l e c t e d p o i n t
is
5
% s t i l l the worst point .
6
7
f u n c t i o n [ s i m p l e x ] = c o n t r a c t ( simplex , o r d e r , c c o n )
8
9
10
c e n t e r = mean ( s i m p l e x ( o r d e r ( 1 : end −1) , : ) ) ;
s i m p l e x ( o r d e r ( end ) , : ) = c e n t e r + c c o n ∗ ( s i m p l e x ( o r d e r ( end ) , : ) − c e n t e r ) ;
11
12
end
211
A.5
subfunction: shrink.m
1
% s h r i n k .m
2
%
3
% This f u n c t i o n i s p a r t o f t h e Nelder−Mead Simplex Algorithm . I t s
purpose
4
% i s t o s h r i n k t h e s i m p l e x when t h e c o n t r a c t e d p o i n t i s
5
% point .
s t i l l the worst
6
7
f u n c t i o n [ s i m p l e x ] = s h r i n k ( simplex , o r d e r , c s h r )
8
9
s i m p l e x ( o r d e r ( 2 : end ) , : ) = o n e s ( 2 , 1 ) ∗ s i m p l e x ( o r d e r ( 1 ) , : ) + c s h r ∗ ( s i m p l e x
( o r d e r ( 2 : end ) , : )−o n e s ( 2 , 1 ) ∗ s i m p l e x ( o r d e r ( 1 ) , : ) ) ;
10
11
end
A.6
subfunction: resetsimplex.m
1
% r e s e t .m
2
%
3
% This f u n c t i o n i s used t o r e s e t a s i m p l e x under a c e r t a i n c o n d i t i o n
4
% ( e i t h e r t o o b i g o r has been caught i n a l o c a l minimum ) t o a new
simplex
5
% which c o n s i s t s o f t h e b e s t v e r t e x from t h e o l d s i m p l e x and two
randomly
6
% placed v e r t i c e s .
7
%
8
% L = l e n g t h o f o t h e r v e r t i c e s from t h e r e t a i n e d v e r t e x
9
10
f u n c t i o n [ n e w s i m p l e x ] = r e s e t s i m p l e x ( simplex , o r d e r , L)
11
12
[M,N] = s i z e ( s i m p l e x ) ;
13
n e w s i m p l e x = z e r o s (M,N) ;
212
14
new simplex ( order ( 1 ) , : ) = simplex ( order ( 1 ) , : ) ;
% Re t ai n b e s t
v e r t e x from o l d s i m p l e x
15
16
f o r i = 2 :M
n e w s i m p l e x ( o r d e r ( i ) , : ) = s i m p l e x ( o r d e r ( 1 ) , : ) + L ∗ [ c o s ( p i ∗ rand ( 1 , 1 ) )
17
s i n ( p i ∗ rand ( 1 , 1 ) ) ] ;
18
end
19
20
end
A.7
subfunction: getSimplexVol.m
1
% g e t S i m p l e x A r e a .m
2
%
3
% This f u n c t i o n i s used t o d e t e r m i n e t h e Volume c o v e r e d by a s i m p l e x .
4
5
f u n c t i o n [V] = g e t S i m p l e x V o l ( s i m p l e x )
6
7
[ ˜ , n ] = s i z e ( simplex ) ;
% Get d i m e n s i o n o f s i m p l e x (= # v e r t i c e s −
1)
8
V = abs ( 1 / f a c t o r i a l ( n ) ∗ d e t ( s i m p l e x ( 2 : end , : ) − o n e s ( n , 1 ) ∗ s i m p l e x ( 1 , : ) ) ) ;
9
10
end
213
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