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Silicon integrated devices for optical system applications

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Silicon integrated devices for optical system applications

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SILICON INTEGRATED DEVICES FOR OPTICAL SYSTEM APPLICATIONS

by
Lin Zhang

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

May 2011

Copyright 2011 Lin Zhang
ii

Dedication

To my parents, Yuantai Zhang and Xiugui Lin,
for the love and support they gave me;
and to Qiuyun Fan
for the love and encouragement she gave me.

iii

Acknowledgements
As a collaborative work, it would not be possible for me to finish this thesis without
the devotion and hard work of my colleagues in Optical Communications Laboratory
(OCLab).
First, I would like to express my greatest gratitude to my advisor, Prof. Alan Willner,
for his support during my Ph. D study. His knowledge, experience, and his patience and
encouragement to me led me through the technical difficulties. Many of the personal lessons
I have learned from him will prove to be as important in my professional career as the
technical ones. The last six years in his group have been a highlight in my life and opened
my eyes to a lot of innovations and opportunities. Thank you, Prof. Willner!
I would thank Dr. Raymond G. Beausoleil at HP Laboratories for guiding and
financially supporting me in my Ph. D study. I also would like to thank Dr. William H.
Steier, Dr. John D. O’Brien, Dr. Michelle Povinelli, and Dr. Andrea Armani for guiding me
on the dissertation.
I am truly grateful to Dr. Changyuan Yu, Dr. Yan Wang, Dr. Ting Luo, Dr. Bo Zhang,
Dr. Muping Song, Dr. Ling Lu and Dr. Qiang Lin. They had been very kind to me in both
research and personal lives and took the efforts to mentor me in many of my research
activities.
I am also thankful to my colleagues in OCLab. Yunchu Li, Jengyuan Yang, Xiaoxia
Wu, Yang Yue, Dr. Jian Wang and Yan Yan helped me in my projects. I thank Scott Nuccio,
Bishara Shamee, Hao Huang, Xue Wang and Xiaomin Zhang for all the helpful discussions.
iv

Last but not least, I express my deepest love to my family. My parents and my
girlfriend gave me love and joy.
v

Table of Contents

Dedication...... ............................................................................................................. ii
Acknowledgements .................................................................................................... iii
List of Figures ........................................................................................................... vii
Abstract............ ........................................................................................................ xiii
Chapter 1 Introduction .......................................................................................... 1
1.1 Perspective on Optical Information Systems ......................................... 1
1.1.1 Optical Communications ........................................................................ 1
1.1.2 Optical Signal Processing ....................................................................... 2
1.2 Material and Device Platforms in Integrated Photonics ......................... 4
1.2.1 Silicon Photonics .................................................................................... 4
1.2.2 Micro-Resonators ................................................................................... 5
1.2.3 Integrated Waveguides ........................................................................... 6
1.3 Thesis Outline ......................................................................................... 7
1.4 Chapter 1 References .............................................................................. 9
Chapter 2 Single Microring Intensity Modulator .............................................. 15
2.1 Modeling of System Performance ........................................................ 16
2.1.1 Signal Generation ................................................................................. 17
2.1.2 Device Characterization ....................................................................... 17
2.1.3 Signal Detection ................................................................................... 23
2.2 Modulator Performance ........................................................................ 24
2.3 Performance in Fiber Transmission ...................................................... 32
2.4 Discussion ............................................................................................. 35
2.5 Summary ............................................................................................... 37
2.6 Chapter 2 References ............................................................................ 38
Chapter 3 Single Microring Phase Modulator ................................................... 44
3.1 Principle ................................................................................................ 45
3.2 Performance Characterization .............................................................. 47
3.2.1 Modulator ............................................................................................. 47
3.2.2 Demodulator ......................................................................................... 54
3.3 Application Scenarios ........................................................................... 59
3.4 Discussion ............................................................................................. 61
3.5 Summary ............................................................................................... 63
3.6 Chapter 3 References ............................................................................ 65

vi

Chapter 4 Embedded Microring Modulator ...................................................... 67
4.1 Introduction .......................................................................................... 67
4.2 Passive Performance ............................................................................. 68
4.3 Comparison of Double-Ring Resonators .............................................. 71
4.4 Embedded Ring Modulator .................................................................. 73
Table 4.1 A comparison of ERRs with other ring EIT structures ................................. 73
4.5 Embedded Ring Resonators with Distributed Coupling ...................... 79
4.6 Summary ............................................................................................... 80
4.7 Chapter 4 References ............................................................................ 81
Chapter 5 Microring-Based Dispersion Compensator ...................................... 83
5.1 Introduction .......................................................................................... 83
5.2 Principle of Microring Dispersion Compensator ................................. 84
5.3 Dispersion-Slope Compensation .......................................................... 88
5.4 Summary ............................................................................................... 90
5.5 Chapter 5 References ............................................................................ 91
Chapter 6 High Dispersion Slot Waveguide ....................................................... 92
6.1 Introduction .......................................................................................... 92
6.2 Narrowband Highly Dispersive Slot Waveguide ................................. 93
6.3 Wideband Highly Dispersive Slot Waveguide ................................... 101
6.4 Summary ............................................................................................. 104
6.5 Chapter 6 References .......................................................................... 105
Chapter 7 Low Dispersion Slot Waveguide ...................................................... 107
7.1 Introduction ........................................................................................ 107
7.2 Slot Waveguide for Flat and Low Dispersion .................................... 108
7.2.1 Chalcogenide Slot Waveguide ........................................................... 110
7.2.2 Silicon Nano-crystal Slot Waveguide ................................................ 114
7.3 Slot/Strip Waveguide for Flat and Low Dispersion ........................... 116
7.4 On-chip Frequency Comb Generation ................................................ 122
7.5 Future Work ........................................................................................ 127
7.6 Chapter 7 References .......................................................................... 128
Bibliography .......................................................................................................... 133
vii

List of Figures
Figure 1.1 Some functions needed in optical communication and communication-
related signal processing........................................................................................ 3
Figure 1.2 Optical communication and signal processing functions enabled by
tailoring dispersion and nonlinearity properties in integrated silicon
waveguides. ........................................................................................................... 7
Figure 2.1 A triple-ring structure with the middle ring only being active, as an
example of building a dynamic model of a microresonator-based device for
system-level characterization. ............................................................................. 21
Figure 2.2 Sing-ring modulators are divided into three types according to their
operation regimes in optical domain: Type I (single-waveguide under-
coupled), Type II (single-waveguide over-coupled), and Type III (dual-
waveguide). A MOS electrode is integrated onto the ring resonator. Phase
transition across resonance is plotted as a function of frequency for all the
three types............................................................................................................ 22
Figure 2.3 Signal quality is evaluated from the eye-diagram in terms of signal Q-
factor and eye-opening penalty. Dash line shows a sampling instant in signal
detection. ............................................................................................................. 24
Figure 2.4 Pulse waveforms and spectra generated by the three types of the
microring modulators. (a) shows the generated pulse trains from Type I, II
and III (top to bottom). (b), (c) and (d) show amplitude (solid blue) and
frequency chirp (dotted red) of a pulse for Type I, II and III. (e), (f) and (g)
show signal spectra for Type I, II and III, with drive voltage of 2, 4 and 6 V,
respectively. All the plots are in the same scale. ................................................. 26
Figure 2.5 Extinction ratio is increased by applying a large drive voltage, which is
saturated for Type I and II, and is improved greatly for Type III. ...................... 27
Figure 2.6 Pulsewidth is examined as a function of drive voltage. Due to the
nonlinearity of the microring-based modulation, Type I and II have a
broader pulse as the voltage increases, while the pulsewidth from Type III
decreases. ............................................................................................................. 28
Figure 2.7 3-dB modulation bandwidth of the microring modulators for all the
three types at drive voltage of 3 V. ...................................................................... 29
Figure 2.8 3-dB modulation bandwidth of the microring modulators for Type II
with a decreased voltage. ..................................................................................... 30
viii

Figure 2.9 Signal quality is examined as the drive voltage increases. Type II
modulators exhibit a better performance at a high drive voltage than the
others. .................................................................................................................. 30
Figure 2.10 Signal quality is examined as laser linewidth increases at drive voltage
of 3 V. All the modulator types have similar signal quality at a large
linewidth. ............................................................................................................. 31
Figure 2.11 Pulse waveforms (dotted blue line) and effective chirps (solid red line)
generated by the Type I and II modulators, with different drive voltages. All
the plots are in the same scale. ............................................................................ 34
Figure 2.12 Power penalty as a function of fiber transmission distance for Type I
and II modulators and a MZM. Type II exhibits negative chirp and thus
greatly extended fiber distance at power penalty of 1 dB. .................................. 35
Figure 3.1 Microring-based NRZ-DPSK modulation and demodulation schemes.
In modulation, CW experiences a phase shift of π across resonance, using a
over-coupled ring structure. A double-waveguide microring filter enables
demodulation of both Duobinary and AMI and thus balanced detection. ........... 46
Figure 3.2 (a) Symbol constellation diagrams for microring-based modulator
(dash), phase modulator (dash dot) and MZM (dot). (b). DPSK signal
bandwidth vs. transit time. (c). DPSK spectrum modulated by MZM. (d).
DPSK spectrum modulated by microring, in the same scale. (10 dB/div. in
power and 20 GHz/div. in frequency). ................................................................ 48
Figure 3.3 Eye-opening penalty of the modulated DPSK signal, compared to a
MZM, versus carrier transit time. Two eye-diagrams, corresponding to
transit times of 16 and 50 ps, are shown in the same scale. ................................ 49
Figure 3.4 Eye-opening penalty of the modulated DPSK signal, compared to a
MZM, versus CW offset. Two eye-diagrams, corresponding CW offsets of 0
and 0.016 nm, are shown in the same scale. ........................................................ 50
Figure 3.5 Signal waveforms from a microring-based DPSK modulator are shown
for different laser linewidths: 10 MHz, 300 kHz and 10 kHz, respectively. ....... 51
Figure 3.6 Eye-opening penalty changes dramatically with laser linewidth for
different cavity Q-factors of the ring resonators, compared to MZM-based
DPSK. When cavity Q-factor is chosen to be 10000 and laser linewidth is
<10 MHz, <1 dB penalty is obtained. Eye-diagrams, corresponding to
linewidths of 30 kHz and 30 MHz with Q=22000, are plotted in the same
scale. .................................................................................................................... 51
Figure 3.7 Signal power linearly increases with phase shift by driving the
modulator harder. ................................................................................................ 52
ix

Figure 3.8 Eye-opening improvement given by driving the ring modulator harder,
as phase shift is more than π. Eye-diagrams are plotted in the same scale,
which are corresponding to the worst and best cases for microring and DLI
demodulators, respectively. ................................................................................. 53
Figure 3.9 A comparison of original logic (a), obtained Duobinary (b) and AMI (c)
data shows correct information is demodulated. ................................................. 55
Figure 3.10 Quality of the demodulated signal is examined with varied cavity Q-
factor. Two eye-diagrams, corresponding to Q=10000 and Q=25000, are
shown in the same scale. ..................................................................................... 56
Figure 3.11 Tolerance of microring-based DPSK demodulator and DLI to varied
signal bit-rate. The proposed approach is more tolerant to a lowered bit rate. .... 57
Figure 3.12 Eye-opening penalty is examined as a function of demodulator
frequency offset. Microring-based demodulator is more tolerant to the offset
than a DLI. Two eye diagrams are corresponding to an offset of 0 and 3
GHz for the microring demodulator. ................................................................... 58
Figure 3.13 Eye-opening improvement is obtained by increasing receiver
bandwidth. Eye-diagrams, corresponding to receiver bandwidth of 0.5 and
1.75 times signal bit rate, are plotted in the same scale. ...................................... 58
Figure 3.14 BER curves given by all microring-based DPSK modulation and
demodulation, with 0.8-dB power penalty, compared to a conventional
DPSK link: MZM + DLI. .................................................................................... 59
Figure 3.15 Power penalty comparison between the all–ring based and
conventional DPSK links, over single mode fiber transmission up to 70 km. .... 60
Figure 3.16 A comparison of eye-diagrams for (a) DPSK, (b) NRZ generated by a
Type I modulator and (c) NRZ generate by a Type II modulator. Cavity Q-
factor is 22000, and drive voltage is 2.2 V. The DPSK signal has 7-dB
higher Q-value. .................................................................................................... 61
Figure 4.1 Structures of previously proposed and embedded ring resonators ............... 68
Figure 4.2 An ERR with point-to-point coupling and its frequency responses at
‘through’ port for (i) m
1
-m
2
=even and (ii) m
1
-m
2
=odd. ...................................... 69
Figure 4.3 Mode distributions with continuous wave inputs at wavelengths λ
1
and
λ
3
for m
1
-m
2
=even, and λ
2
for m
1
-m
2
=odd. ......................................................... 70
Figure 4.4 m
1
-m
2
is an odd number, normalized transmission and delay vs.
coupling between the waveguide and the ring, compared to single- and
double-ring resonators ......................................................................................... 71
x

Figure 4.5 Previously proposed microring-based EIT structures .................................. 72
Figure 4.6 (a) An ERR-based EIT effect with strong coupling is used for high-
speed modulation. (b) Signal eye-diagrams at 20, 25 and 30 Gb/s, as
compared to the signal generated by a single-ring modulator with the same
linewidth and drive voltage. (c) Signal quality are examined when the
coupling coefficient at B area is changed by ±5%. ............................................. 75
Figure 4.7 An embedded-ring-based silicon modulator exhibits an EIT-like
resonance profile. A curved waveguide coupler enhances light coupling. The
inner ring’s resonance is originally blue-shifted relative to the outer ring to
form a stair-like profile, which is then driven towards high frequency by
applying a voltage. .............................................................................................. 76
Figure 4.8 Generated intensity waveform and frequency chirp in a 40 Gb/s NRZ
signal ................................................................................................................... 77
Figure 4.9 Frequency response of the proposed modulator based on embedded ring
resonators. Its 3-dB bandwidth is 28 GHz ........................................................... 78
Figure 4.10 Mode distributions in the ERRs with distributed coupling. m
1
=27. (a)
and (b) m
1
=23, for symmetric and anti-symmetric modes. (c) m
1
=22 and (d)
m
1
=21 are corresponding to independent resonator modes. ................................ 80
Figure 5.1 (a) Structure of the double-ring dispersion compensator with unequal
ring radii. (b) Delay profile of a single over-coupled ring (dash) and the
linearized delay profile in the proposed structure, which is enabled by
employing a combination of over- and under-coupled rings. .............................. 85
Figure 5.2 Intensity (solid) and dispersion (dash) profiles. (a) Dispersion of -530
ps/nm is obtained over 10 GHz bandwidth, and in-band intensity fluctuation
is 0.15 dB. (b) Dispersion can be switched to positive by interchanging the
resonance detunings of the two ring resonators................................................... 86
Figure 5.3 Dispersion is modified by varying resonance detuning. Trade-off
between dispersion and dispersion bandwidth is shown. .................................... 87
Figure 5.4 Dispersion tunability enabled by refractive index variation. Dispersion
fluctuation is also examined as a function of the varied index. ........................... 88
Figure 5.5 Linearly changed dispersion is obtained by cascading a red-shifted
negative dispersion and blue-shifted positive dispersion. Power loss is
decreased to 1.8 dB. ............................................................................................ 89
Figure 5.6 Dispersion slope is flat over 10 GHz bandwidth, which is 3150 ps/nm
2
..... 89

xi

Figure 6.1 Slot and strip modes strongly interact with each other due to index-
matching at the crossing point, producing a sharp index change of
symmetric and anti-symmetric modes. Modal power distributions of the
symmetric mode at different wavelengths. .......................................................... 94
Figure 6.2 Dispersion profiles of the symmetric and anti-symmetric modes, and a
negative dispersion of -181520 ps/nm/km can be obtained from the
symmetric mode. ................................................................................................. 95
Figure 6.3 (a) The dispersion profile red-shifts with a small peak value as the slot
thickness increases. (b) Dispersion value and peak wavelength are examined
as functions of the slot thickness. (c) Dispersion value and peak wavelength
are examined as functions of the silicon-layer thickness. ................................... 96
Figure 6.4 (a) The dispersion profile is fixed at the same wavelength as the
dispersion’s peak value is changed from -181520 to -28473 ps/nm/km by
varying the silica base thickness. (b) A trade-off is found between the
dispersion peak value and dispersion bandwidth. ............................................... 98
Figure 6.5 Dispersion shifts with different slot thicknesses, exhibiting almost
unchanged dispersion value and bandwidth. ....................................................... 99
Figure 6.6 Dispersion properties change with the waveguide width. .......................... 100
Figure 6.7 Dispersion compensation for very high-speed signals transmitted over
11.4-km single mode fiber. Eye-opening penalty increases with bit rate.
Eye-diagrams are in the same scale. .................................................................. 100
Figure 6.8 Waveguides with variable waveguide thickness or width are cascaded
to obtained wideband flattened strong dispersion. ............................................ 101
Figure 6.9 The dispersion shifts over wavelength by changing WT. .......................... 102
Figure 6.10 Flat dispersion of -31300 ps/nm/km over 147 nm. 6.3 ns/m tunable
delay can be obtained by 230-nm wavelength conversion.. .............................. 103
Figure 6.11 Flat dispersion of -46100 ps/nm/km over 91 nm. 6 ns/m tunable delay
can be obtained by 100-nm wavelength conversion. ......................................... 104
Figure 7.1 Slot waveguide with silicon layers surrounding a highly nonlinear slot
layer. .................................................................................................................. 109
Figure 7.2 Dispersion profiles in chalcogenide slot waveguides with different (a)
slot heights, (b) waveguide widths, and (c) upper silicon heights. .................... 111
Figure 7.3 For chalcogenide slot waveguides, nonlinear coefficient γ and FOM are
examined over wavelength with different (a) slot heights and (b) waveguide
widths, respectively. .......................................................................................... 113
xii

Figure 7.4 For Si nano-crystal slot waveguides, (a) dispersion profiles change with
slot height. (b). Dispersion profile red-shifts as lower silicon height
increases. ........................................................................................................... 114
Figure 7.5 For 10-cm Si nano-crystal slot waveguides, dispersion sensitivity
changes with the slot height. ............................................................................. 115
Figure 7.6 (a) Previously reported silicon slot waveguide that exhibits convex
dispersion profile. (b) A strip waveguide is added to produce additional
negative waveguide dispersion. (c) The strip/slot waveguide coupler is
flipped to improve sequent fabrication procedure. (d) A strip/slot hybrid
waveguide with spacing layer removed for achieving flattened dispersion. ..... 116
Figure 7.7 Flattened dispersion profile within 0 ± 16 ps/(nm·km) is obtained from
1562 to 2115 nm. The corresponding dispersion coefficient β2 and modal
field distributions are shown. ............................................................................ 118
Figure 7.8 In silicon strip/slot hybrid waveguides, dispersion profiles are tailored
by changing (a) slot height and (b) waveguide width, respectively. ................. 119
Figure 7.9 In silicon strip/slot hybrid waveguides, dispersion profiles are tailored
by changing (a) lower silicon height and (b) upper silicon height,
respectively. ....................................................................................................... 120
Figure 7.10 (a) In curved waveguides, optical mode feels a bending radius greater
than structural radius R
0
of a ring resonator. Both the "effective" bending
radius R
eff
and effective index strongly depend on frequency. FSR isn't
constant over frequency. A slot waveguide and its mode are shown here. (b)
Chromatic dispersion inside the ring cavity is changed by reducing bending
radius from 16 to 3 microns. Both averaged value and flatness of the
dispersion profiles are varied............................................................................. 124
Figure 7.11 (a) Frequency domain "eye-digram". Spectral response of a double-
waveguide ring resonator is sliced with a freq. step of one FSR. The
obtained spectrum pieces are plotted together to show the uniformity of the
resonance peaks. (b) The silicon strip waveguide produces spectral lines
widely shifted over only 100 nm. (c) Dispersion flattened slot waveguide
enables well-aligned resonance peaks over 563 nm wavelength band. ............. 126

xiii
Abstract
This dissertation presents silicon-based integrated micro-resonators and waveguides
as the key elements of photonic integrated circuits for on-chip optical communication and
signal processing applications.
Electro-optic modulation plays a critical role in implementing space-, power- and
spectrally efficient optical interconnection for high-capacity computing systems. Microring
resonators exhibit a great potential to achieve compact, low power-consumption and high-
speed modulators. In the first part of this dissertation, we briefly review our efforts on
designing and analyzing the microring modulators. Three types of single-ring modulators are
discussed, from device behavior to possible system impact. We then present a novel double-
ring modulator, in which a passive ring resonator is added, enabling higher operation speed
and lower power consumption. We also describe an opportunity of introducing phase
modulation data formats into the on-chip communication environment. Our emphasis is
placed on linking the devices’ physics to their system performance and providing potential
technical solutions to physical-layer challenges of optical interconnection.
Integrated silicon waveguides not only serve as transmission medium of on-chip
signaling but also form various types of devices for communication and signal processing
applications. In the second part of this dissertation, we present some recent progresses in
tailoring the physical properties of the silicon waveguides such as chromatic dispersion and
nonlinearity. Using slot structures, the dispersion could be tailored by more than a few
orders of magnitude, which is highly useful in many signal processing applications.
1

Chapter 1 Introduction
In this Chapter, I would start with a general discussion on the reasons why integrated
photonics plays an increasingly important role in modern optical information systems.
Perspectives at both system and device levels are provided.
1.1 Perspective on Optical Information Systems
Concept of optical information systems could be extremely broad, including
information collection, processing, storage and exchange (or communication). Here, we limit
our discussion to the optical data communication and signal processing systems.
1.1.1 Optical Communications
Optical communication has fueled the global telecom and information technology
revolution in the past two decades [1]. Nowadays, worldwide transport of huge volume of
data relies on optical communication systems. Integrated photonics has attracted a great deal
of attention in recent years not only because it allows for more cost-effective production and
easier packaging but also because smaller chip size assists in realizing faster and less power-
consuming photonic devices to facilitate "green" in-formation technology.
Integrated photonics also plays an increasingly important role in datacom systems.
Optical communication has been investigated in almost every layer of information
technology infrastructure: continent-to-continent, city-to-city, server-to-server, computer-to-
computer, board-to-board, chip-to-chip and finally intra-chip. For example, in contemporary
high-performance computing systems, global interconnection between electronic
2

computation units may cause long latency and high power consumption as clock rates
increase. This has been identified to be one of bottlenecks in the future development of the
integrated circuit industry, according to the International Technology Roadmap for
Semiconductors (ITRS). To alleviate these challenges, many efforts from both academia and
industry have been pursued to explore the possibility of employing optical interconnections.
Advanced integrated photonics is an enabling technology platform to build space-, power-,
and spectrally efficient on-chip photonic interconnection networks that could be seamlessly
integrated with CMOS electronics [2, 3, 4, 5, 6].
1.1.2 Optical Signal Processing
Optical signal processing technology has been aggressively pursued [7], aiming at
higher operation speed and enriched functionality. This holds a great potential to diminish
the need for O-E-O conversion in the data transmission links in order to upgrade system
capacity, reduce latency and cost, and enable a wavelength- and data-format-transparent
optical network.
In general, optical communication and signal processing represent different categories
of devices and subsystems, although it is hard to clearly separate them from each other. In
communication, optical subsystems include, but are not limited to, signal generation,
detection, amplification, filtering and signal degrading effects' monitoring and
compensation. In signal processing, functional units would range even more widely from
pulse manipulating, data routing to optical logic [8, 9, 10, 11, 12], as listed in Figure 1.1.
Pulse manipulation is typically aimed at the time-domain operation and modification of the
pulse waveform and instantaneous phase, while optical logic places an emphasis on bit-level
3

processing of a signal without reshaping the pulses. Data routing is the category that is more
closely related to optical wavelength-division-multiplexing (WDM) communication systems,
in which the carrier wavelength of a signal could serve as a label to indicate what signal-
processing functions need to be carried out. Most of these functions have been demonstrated
using integrated devices.

Figure 1.1 Some functions needed in optical communication and communication-related signal
processing
In practice, an advanced communication or signal processing subsystem may require a
cascade of many optical and electrical devices, which raises at least three system design
concerns: compactness, power consumption and signal quality. As an example, an agile and
ultra-fast arbitrary waveform generator [13] may consist of multi-wavelength optical source,
amplitude and phase modulators, couplers and electrical and optical amplifiers and filters.
Even if each component has very little performance imperfection, it is still possible that the
whole system creates considerable data degradation. Although integrated devices are
potentially able to provide more compact and power-efficient solutions, they do not
4

necessarily produce a satisfactory signal quality. Therefore, from a system point of view, an
emphasis is placed on the signal quality in the integrated devices, which are mostly based on
micro-resonators and novel waveguides in this thesis.
1.2 Material and Device Platforms in Integrated Photonics
In integrated photonics, materials that are available to fabricate various optical devices
include, but are not limited to, semiconductor, dielectric, metal and polymer and so on.
There are miscellaneous optical structures designed to form optical devices, which could be
categorized mainly into a few types: light-guiding, coupling (or interferometric), resonant
and bandgap structures. Each of the materials can be used to build these different types of
the device structures in order to form an integrated photonic platform. Here, we limit our
discussion to the silicon-based resonant and light-guiding structures.
1.2.1 Silicon Photonics
Silicon has long been an enabling material, supporting the revolutionary developments
that have occurred in the integrated circuit industry in the past few decades, and its success
in electronics has shown its undeniable ability to manipulate electrons. Recently, silicon
photonics has attracted a great deal of research interest due to low intrinsic loss in the
infrared wavelength range, high refractive index for light confinement, strong Kerr and
Raman nonlinearity, and cost-effective fabrication [14, 15, 16]. More importantly, advances
in silicon photonics have paved the way for the design and construction of a CMOS-
compatible opto-electronic integration system. Potentially, this can combine the advantages
of electrons in computing with the advantages of photons in communications to facilitate the
development of high-capacity information processing systems.
5

1.2.2 Micro-Resonators
As a key element in integrated photonics, micro-resonators exhibit small chip size,
resonance-enhanced operation efficiency, low power consumption and high wavelength
selectivity. Various types of the micro-resonator-based devices have been proposed and
demonstrated for communication and signal processing applications. Those include lasers
[17, 18, 19, 20], modulators [21, 22, 23, 24, 25], filters [26, 27, 28, 29], amplifiers [30, 31,
32, 33], switches [34, 35, 36], logic gates [37, 38, 39], wavelength converters [40, 41, 42],
delay elements [43, 44], dispersion compensators [45, 46, 47], demodulators [48, 49], format
converters [50] and sensors [51, 52, 53, 54, 55]. With these fundamental building blocks,
enriched functionality can be achieved in on-chip photonic systems. We note that many of
these functions are demonstrated using active micro-resonators, in which thermal tuning,
electrical controlling and optical pumping can be employed, and dynamic performance of
such devices receives a great deal of attention.
It is important to mention that some types of micro-resonators, such as microrings and
microdisks, support traveling waves, which are guided modes in the microring resonators
and are whispering gallery modes (WGMs) formed by periodic internal reflection of light by
sidewalls in the microdisk resonators. These resonators can inherently suppress the ref
lection of light back to the input optical paths, in contrast to Fabry-Perot resonators that form
standing waves. Since on-chip optical isolators have not been widely available so far, the
traveling-wave resonators would be considered advantageous from this point of view.
Therefore, in my work on micro-resonators, I have focused on microring resonators,
although most of the conclusions would hold for other types of the resonators in principle.
6

1.2.3 Integrated Waveguides
Integrated silicon waveguides serve as a transmission medium for on-chip signaling,
and they also are used to form various types of devices for communication and signal
processing applications. We note that many of the communication and signal processing
functions mentioned above are based on nonlinear optical effects.
Optical nonlinearities are significantly enhanced in silicon waveguides, benefiting
from large nonlinear Kerr and Raman coefficients and from the tight light confinement
produced by high index contrast [56, 57, 58, 59]. Various nonlinear effects have been
demonstrated, including self-phase modulation, cross-phase modulation, parametric wave
mixing, two-photon absorption, and Raman scattering. It is well known that chromatic
dispersion plays a critical role in both intra-channel and inter-channel nonlinear interactions,
especially for short optical pulses [60]. Although one may argue that the on-chip silicon
waveguides are typically so short that the chromatic dispersion does not show a significant
effect on picosecond pulse evolution, it does matter in many other important applications,
such as femtosecond pulse propagation [61, 62], ultra wideband wavelength conversion [63,
64], and frequency comb generation [65]. In addition, an optimized dispersion profile could
be beneficial for reducing phase mismatching and releasing the requirement for high pump
power, which has potential to mitigate two-photon absorption and achieve ‘greener’
information processing.
In fact, different combinations of dispersion and nonlinearity in on-chip integrated
waveguides could find a wide variety of communications and signal processing applications,
as shown in Figure 1.2. This has generated a great deal of research interest to effectively
7

tailor the dispersion and nonlinearity properties of the silicon waveguides, and part of these
efforts will be shown in the sequent chapters.

Figure 1.2 Optical communication and signal processing functions enabled by tailoring dispersion and
nonlinearity properties in integrated silicon waveguides.
1.3 Thesis Outline
This dissertation is arranged with the following structure: Chapter 2 presents a silicon
intensity modulator based on a single ring resonator. The modeling of electro-optic
conversion and dynamic multi-resonator coupling is described, and a single-ring modulator
is analyzed in terms of spectral and temporal features. Chapter 3 describes the principle of
using silicon microring modulators to generate and demodulate advanced optical modulation
formats, with an emphasis on differential phase shift keying signals. Chapter 4 presents a
new type of ring-resonator configuration, which is called embedded ring resonators, and
shows a great potential of using such modulator configuration to enhance modulator
performance. Chapter 5 shows effective dispersion compensation and dispersion-slope
8

compensation based on the microring resonators. Chapter 6 presents the development of an
on-chip highly dispersive waveguides using nano-scale slot structures. Both narrowband and
wideband dispersion properties are discussed. Chapter 7 presents a novel technique for
flattening chromatic dispersion in the slot waveguides, and potential applications of this
technique in on-chip frequency comb and supercontinuum generations are presented.

9

1.4 References

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11

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8, pp. 1121-1123, Aug. 1998.

12

[35] V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and
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14

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15

Chapter 2 Single Microring Intensity Modulator
Electro-optic modulation has been viewed as a fundamental function in on-chip
communications. As compared to their peers built in lithium niobate [1, 2], polymer [3, 4]
and compound semiconductors [5, 6], silicon modulators [7, 8, 9, 10] exhibit better CMOS
compatibility but may require relatively high drive voltage and more power consumption. In
addition, though microring modulators exhibit small chip size and capacitance, resonance-
enhanced efficiency and high wavelength selectivity in comparison to Mach-Zehnder
modulators (MZMs) in silicon [11, 12, 13, 14], cavity photon lifetime imposes a limitation
on the ring modulators’ speed [15]. Much research has been done to design electrodes [16,
17], driving schemes [18], and ring structures [19, 20]. Design optimization of the microring
modulators rely on joint consideration of both resonance shift and dynamic loss [19, 20] and
a good understanding of the cavity dynamics [19, 20, 21].
Micro-resonators feature a sharp phase transition near each resonance wavelength.
Although this can be used for phase modulation [22, 23], continuous-wave (CW) input fields
experience a phase variation in intensity modulation as the resonance wavelength shifts,
which adds a frequency chirp to generated signals [19, 24 , 25 ]. In chip-scale
communications [26], one may not see a significant effect of the chirp on signal quality, with
chromatic dispersion limited over short propagation distances, but it plays a critical role in
determining the signal quality in fiber transmission [19, 24]. The single-ring modulator’s
performance in fiber telecom systems has been discussed in detail [27].
In this Chapter, we numerically analyze single-ring silicon modulators, in terms of
extinction ratio, chirp, signal spectrum, pulsewidth and signal Q factor, focusing on a bit rate
16

of 10 Gb/s. Three types of single-ring modulators are considered, based on their
configurations in the optical domain. The dynamic responses of these modulators greatly
affect their performance when used for intensity modulation, particularly in characteristics
such as modulation speed, power consumption and signal quality. We show that resonance-
induced negative chirp assists signal transmission over single mode fibers. The power
penalty in 80-km fiber transmission, with a drive voltage of 1 V, is only 0.8 dB, even
without dispersion compensation.
2.1 Modeling of System Performance
A comprehensive system model of the micro-resonator devices includes signal
generation in pseudo-random bit sequence (PRBS), simulation of device characteristics, and
signal detection and evaluation. For a passive device, typically only a static model is needed.
For an active device, there could be different driving and controlling techniques, using all-
optical, electro-optic or thermo-optic effects. Here, we describe a model for an electro-optic
semiconductor microring resonator.
Generally speaking, the resonance is tuned when a drive voltage or current is applied
onto an electrode integrated with the resonator, which changes carrier density inside the
cavity and thus the effective index of refraction of the guided mode. The index change is
translated into a resonance shift in frequency domain. Note that the carrier density variation
may also cause a change in propagation loss of the guided mode, which results in the
reshaping of the resonance profile.
In the following, I describe the signal generation, device characterization and signal
detection, respectively.
17

2.1.1 Signal Generation
At the beginning of the modeling, logic data in PRBS at a bit rate of interest is
encoded according to the specific modulation format (this is needed for a format other than
binary on-off-keying (OOK) signals). A long PRBS word is desirable for system-level
evaluation but also requires strong computation capability. The PRBS length is 2
13
-1=8191,
which is usually long enough to examine data pattern dependencies induced by the resonator
devices. Then, the data stream is sent to a driver, and a square-wave voltage signal in a non-
return-to-zero (NRZ) format is generated, which may need to be biased according to the type
of electrode used. The signal then passes through an electrical filter with a relatively large
bandwidth. The filter bandwidth is chosen depending on whether or not the driving circuit's
speed limitation needs to be considered. It is normally set to be 1.5~3 times the signal bit
rate so that the circuit limitation becomes negligible. In this way, we obtain a driving signal.
2.1.2 Device Characterization
When a drive voltage signal is applied to the electrode of a semiconductor modulator,
there have been many ways to convert it to a carrier density variation in the cavity, based on
different materials (e.g., III-V compound semiconductor or silicon) and electrode designs.
For silicon-based devices, typical electrode designs include metal-oxide-semiconductor
(MOS) capacitors [7, 11, 28, 29, 30], p-n junctions [17, 31, 32] and p-i-n diodes [8, 10, 13,
14, 18, 33 , 34 ]. These have different carrier transit times and RC time constants. In
particular, parasitic capacitances may change with drive voltage. Therefore it is difficult to
build a uniform electrical model to describe the behavior of these devices.
18

Here, we focus on the MOS capacitors operated at hole accumulation regime, since
there is low dc power consumption and no carrier confinement required. A typical MOS
capacitor electrode structure can be found in [30]. In the time domain, time-varying
capacitances of the accumulation layer in the MOS capacitor are negligible [30], with only
the fixed oxide gate capacitance left. The carrier density in the electrode thus follows an
exponential transition over time. The carrier transit time, from 10% to 90% of the maximum
carrier density, is ranged from 12 to 100 ps [7, 11, 28, 29, 30]. The dependence of the transit
time on drive voltage is ignored. We also assume that the transit time at the rising and falling
edges is the same [7, 29]. Silicon-based microresonators could have a radius below 10 μm,
and the RC time constant is estimated to be <3 ps for the proposed structure in [30] as an
example, which is negligible as compared to the carrier transit time.
Carrier-induced index change in silicon is caused mainly by the plasma dispersion
effect [35], because unstrained pure crystalline silicon has no linear electro-optical (Pockels)
effect due to crystal symmetry, and has a weak Franz-Keldysh effect [36]. Spatially, the
carrier density is a 2D distribution over the cross section of the electrode. It can be simulated
based on specific MOS structures to obtain the effective index change of an optical guided
mode using commercially available mode solvers [7, 9, 28, 29, 30]. In a simplified model, a
spatially averaged carrier density with a steady-state value, linearly proportional to the peak-
to-peak value V
pp
of the drive voltage is used, so that the effective index change is calibrated
to be 2.4×10
-4
at V
pp
=5V, which can be obtained using several proposed electrode designs
[11, 28, 29, 30]. The imaginary part of the index change causes dynamic loss, which is also
calculated from [35] at the wavelength of interest. The loss variation results in a dynamically
changed cavity Q-factor and thus a reshaped resonance profile when it is shifted [8, 9, 17].
19

Note that, for other material systems such as polymer, there may be no dynamic process of
carriers. In that case, the index change can be obtained directly from the electro-optic effect
[3, 4].
The effective index change over time causes a shift of the resonance (angular)
frequency, which is reflected in the coefficients in a set of dynamic coupled mode equations
describing the micro-resonator system in the optical domain. The dynamic model can
accurately predict a transient evolution of the optical field in the cavities, to account for the
effect of the cavity photon lifetime on output waveform and frequency chirp. The model
given in [37,38] abstractly treats an optical wave traveling around the cavity as a whole, and
it is applicable to both traveling-wave [37] as well as standing-wave cavities [39]. One may
solve the dynamic equations for the energy amplitude that represents the total energy stored
in the ring [37, 38], with a time step not limited by cavity round-trip time. The model works
only when one has no need to examine the walk-off of optical waves traveling along
different paths in ring resonators [8, 19]. Another model adopted in [21], in a more intuitive
way, tracks the optical wave along its path in a cavity and is inherently suitable for cases,
where the optical fields at different parts of the rings have to be resolved to examine the
interference and evolution of the optical waves [10, 20, 40]. However, this means that a ring
resonator is divided into different parts separated by ring-to-ring and ring-to-waveguide
couplers, requiring a time step less than the cavity round-trip time in simulations, which
means more memory and computation time. For the devices discussed in this chapter, the
two models would generate simulation results in good agreement with each other [41]; we
use the first model here.
20

For a specific resonator system, one can write corresponding dynamic equations based
on the first modeling technique [37, 38]. Figure 2.1 shows an example, which is a triple-ring
structure with only the middle ring being active. The dynamic equations are:

1
r1 1 1 in 2 2
l1 c1
a (t) 1 1
j a (t) j E (t) j a (t)
t
  ∂
= - ω - - + μ + μ
 
∂ τ τ
 
(2.1)

2
r2 2 2 1 3 3
l2
a (t) 1
j (t) a (t) j a (t) j a (t)
t (t)
  ∂
= - ω - + μ + μ
 
∂ τ
 
(2.2)

3
r3 3 3 2
l3 c4
a (t) 1 1
j a (t) j a (t)
t
  ∂
= - ω - - + μ
 
∂ τ τ
 
(2.3)

) ( ) ( ) (
1 1 1
t a j t E t E
in out
μ + =
(2.4)

) ( ) (
3 4 2
t a j t E
out
μ =
(2.5)
The energy amplitude is defined as a
i
(t) =A
i
(t)e
-jωt
, and ω and ω
ri
are the carrier and
the i
th
ring's angular resonance frequencies, respectively. E
in
, E
out1
and E
out2
represent the
input and output signals, τ
li
is the i
th
ring's amplitude decay rate due to cavity loss, and τ
c1

and τ
c4
are the amplitude decay rates due to ring-to-waveguide coupling, that represents
equivalently a loss to the ring [37]. In the j
th
coupling area, coupling factor μ
j
satisfies μ
j
2
=
κ
j
2
v
g
/2πR = 2/τ
cj
, where κ
j
2
is the power coupling coefficient, and v
g
is the group velocity in a
ring with a radius R [37]. Since only the middle ring is driven, only its resonance angular
frequency and loss are temporal functions. The differential equations can be solved using the
implicit Euler method with a time step of ~1ps, which is much longer than a round-trip time
in a ring resonator with a radius of a few μm.
21

Figure 2.1 A triple-ring structure with the middle ring only being active, as an example of building a
dynamic model of a microresonator-based device for system-level characterization.
Note that, for a single resonator, there are three different operating regimes: over-
coupled, critically coupled and under-coupled, depending on whether the coupling to the
resonator is stronger, equal or weaker as compared to its round-trip loss [42, 43]. Each
regime features a unique phase transition in the vicinity of the resonance frequency and may
have dramatically different dynamic properties [27]. Although one can use a uniform
dynamic model as described above to simulate all these types of devices, a static model is
still valuable for an active device, since it indicates the operating regime of a resonator by
showing its phase profile. Moreover, a static model provides information of steady-state
operations of the device (e.g., extinction ratio) and allows one to estimate, say, pump power,
drive voltage or heater requirement when the device employs all-optical, electro-optic or
thermo-optic effects.
In the described model, nonlinear Kerr and Raman effects and two-photon absorption
(TPA) of the traveling waves are ignored. Therefore, the model works for an all-optical
operation only when a pump is launched in the direction normal to the resonator (or wafer)
plane [40, 44, 45], because in this case, although pump-induced carrier density variation by
TPA is considered, the pump wave is not traveling around the cavity, and there is no need to
consider its waveform change by Kerr and TPA effects.
22

This chapter is focused on only a single ring resonator, and it typically has a
Lorentzian spectral response when coupled to one or two waveguides. As shown in Figure
2.2, in the single-waveguide case, the ring can be under-coupled (Type I) or over-coupled
(Type II), depending on whether the coupling between the ring and the waveguide is weaker
or stronger than the ring’s round-trip loss. In these cases, a notch resonance profile is seen at
the output port. In the dual-waveguide case (named Type III), the output port has a band-
pass profile. Each of the operating regimes of the ring modulators features a unique phase
transition over frequency, accompanying the sharp amplitude response, as shown in Figure
2.2.

Figure 2.2 Sing-ring modulators are divided into three types according to their operation regimes in
optical domain: Type I (single-waveguide under-coupled), Type II (single-waveguide over-
coupled), and Type III (dual-waveguide). A MOS electrode is integrated onto the ring
resonator. Phase transition across resonance is plotted as a function of frequency for all the
three types.
We use the model above. The Type I and II modulators are simulated using the
following equations, and the next two equations are for Type III.
23

) ( ) ( )
1
) (
1
) ( (
) (
t E j t a
t
t j
t
t a
in
c l
r
μ
τ τ
ω + - - - =
∂
∂
(2.6)
) ( ) ( ) ( t a j t E t E
in out
μ + = (2.7)

) ( ) ( )
1 1
) (
1
) ( (
) (
2 1
t E j t a
t
t j
t
t a
in
c c l
r
μ
τ τ τ
ω + - - - - =
∂
∂
(2.8)
) ( ) ( t a j t E
out
μ = (2.9)
The energy amplitude is defined as a(t) = A(t) e
-jωt
, and ω and ω
r
are the carrier and
resonance angular frequencies, respectively. E
in
and E
out
represent the input CW and
modulated signal. τ
l
is the amplitude decay rate due to cavity loss, and τ
c
, τ
c1
and τ
c2
are the
amplitude decay rates due to ring-waveguide coupling. In the symmetric configuration of
Type III, τ
c1
= τ
c2
, which means that it is also an under-coupled ring since the output
waveguide equivalently adds loss to the cavity. Coupling factor μ satisfies μ
2
= κ
2
v
g
/2πR =
2/τ
c
, where κ
2
is the power coupling coefficient, and v
g
is the group velocity in a ring with a
radius of R. The differential equations are solved using the implicit Euler method with a time
step of 0.8 ps.
2.1.3 Signal Detection
At the receiver end, we typically set the receiver sensitivity to be - 25 dBm for a BER
at 10
-9
in the back-to-back case. There is no optical amplifier (and thus no optical filter)
used. The received power at photodiodes is not held fixed, which is typical for on-chip
communications. For performance evaluation in fiber transmission, we set the receiver
sensitivity to be - 20 dBm. Pre-amplification is used to compensate for the fiber loss and to
24

keep the same received power. An optical filter with 4 times bit-rate bandwidth is built into
the receivers to remove amplifier noise. In all cases, an electrical filter with a bandwidth of
0.7 times the bit-rate is used in the receivers.
System performance of the micro-resonator devices can be evaluated using a signal Q-
factor, eye-opening penalty (EOP) and power penalty [46]. We usually use signal Q-factor
and eye-opening penalty, defined in Figure 2.3, in the back-to-back case, and power penalty
in fiber transmission.

Figure 2.3 Signal quality is evaluated from the eye-diagram in terms of signal Q-factor and eye-
opening penalty. Dash line shows a sampling instant in signal detection.

2.2 Modulator Performance
In our analysis, the ring has a diameter of 5.4 μm, with a mode number m = 28 at f =
193 THz. The cavity Q-factor is 19000, corresponding to a resonance linewidth of 10 GHz
and a photon lifetime τ = Q/2πf = 15.7 ps. This requires that Type I, II and III have an
intrinsic loss of 13.9, 9.3 and 1 dB/cm, respectively. In reality, a Type III modulator cannot
25

have a loss as low as 1 dB/cm. However, for purpose of comparison, we set the same Q-
factor for all three types. In Figure 2.2, when the voltage is logic zero, Type I and II are on
resonance, but Type III is off resonance. In this way, the generated optical signals have a
non-inverted logic pattern as compared to the driving signal.
A 10 Gb/s NRZ on-off-keying signal is generated by each type of modulator. Figure
2.4 (a) shows output pulse trains, while Figure 2.4 (b)-(d) show an optical pulse’s
instantaneous amplitude for logic data ‘00111000’ generated from Type I, II and III
configurations, with a drive voltage of 5 V. The time window is 800 ps, and the vertical axis
is 0~1.4 (mW)
½
. There is an overshoot at rising edges of the pulses for Type I and II, which
results from interference between the CW input field and the optical wave coupled out of the
ring. The voltage-induced index change in the cavity adds an instantaneous phase shift to the
optical wave traveling around, which is an equivalent frequency change to the carrier [34,
47,48]. When coupled out of the ring, the optical wave interferes with the CW, resulting in
the overshoot (similarly, an undershoot at falling edge in Type III).
The interference also induces phase oscillation, reflected as a strong frequency chirp
in Figure 2.4. This is why a static model would not correctly predict signal chirp [27]. All
the chirps are plotted in the vertical axis from -120 to 80 GHz. Note that peak value of the
chirp may occur at low instantaneous optical power, making it less effective. In contrast, the
chirp accompanied by the overshoots causes an asymmetric spectral shape of the signals.
Therefore, Type I and II have the opposite chirp peak values, but both of them have spectral
red-shift broadening as shown in Figure 2.4, while Type III has a spectrum with strong blue-
shifted components. In Figure 2.4 (e)~(g), vertical and horizontal axes are 10 dB/div. and 20
26

GHz/div. We plot the generated 10 Gb/s NRZ spectra at drive voltage of 2, 4 and 6 V, which
become more asymmetric as the voltage increases.

Figure 2.4 Pulse waveforms and spectra generated by the three types of the microring modulators. (a)
shows the generated pulse trains from Type I, II and III (top to bottom). (b), (c) and (d) show
amplitude (solid blue) and frequency chirp (dotted red) of a pulse for Type I, II and III. (e),
(f) and (g) show signal spectra for Type I, II and III, with drive voltage of 2, 4 and 6 V,
respectively. All the plots are in the same scale.
Note that in Figure 2.4 (b) and (c) the pulse from Type II has a stronger overshoot and
an intensity dip that follows the falling edge. This is because an over-coupled ring cavity
stores more energy than an under-coupled one when they are on-resonance, although they
have the same steady-state transfer function. At the falling time, photons are accumulated in
27

the cavity, and some of them are coupled out and destructively interfere with the CW field.
The Type II cavity has an energy that increases to a high enough value to deplete the CW
and then produces output light after the intensity dip. In contrast, Type I never has enough
cavity energy to deplete the CW. The difference between the Type I and II dynamic
waveforms becomes more significant when the resonators are operated further away from
critical coupling [42, 43].

Figure 2.5 Extinction ratio is increased by applying a large drive voltage, which is saturated for Type I
and II, and is improved greatly for Type III.
Figure 2.5 shows the extinction ratio of the modulated signals as the peak-to-peak
drive voltage increases from 1.2 to 8 V. A 2.2 V drive voltage blue-shifts the resonance peak
by 10 GHz. The carrier-induced dynamic loss depends on doping density and is 2.5, 5 and
7.5 dB/cm for a drive voltage of 2, 4 and 6 V, respectively. Modulator Types I and II have
the same extinction ratio since they share a common amplitude response. The extinction
ratio saturates quickly, and the 4-V drive voltage produces an extinction ratio of 13.8 dB.
When the ring resonator is operated near critical coupling, the saturated extinction ratio can
be higher. In contrast, for Type III, the high voltage and thus dynamic loss apply to ‘1’-bits,
28

producing relatively low signal peak power and small extinction ratio under the optical bias
condition. However, an increased voltage reduces ‘0’-level in the signal and improves the
extinction ratio, as shown in Figure 2.5.

Figure 2.6 Pulsewidth is examined as a function of drive voltage. Due to the nonlinearity of the
microring-based modulation, Type I and II have a broader pulse as the voltage increases,
while the pulsewidth from Type III decreases.
Pulsewidth at full width at half maximum (FWHM) of a single ‘1’ pulse is plotted in
Figure 2.6, with the same parameters as in Figure 2.5. The pulses become broader for Type I
& II and narrower for Type III as the voltage increases, because of the modulation
nonlinearity caused by the Lorentzian resonance profile and cavity dynamics [41]. The
pulsewidth, together with the frequency chirp, affects the signal spectra. Unlike the
extinction ratio that is mostly related to the steady states of the modulators, the pulsewidth
reflects modulation dynamics. The Type II modulator produces a narrower pulsewidth than
Type I, since an over-coupled ring modulator is faster than an under-coupled one with the
same Q-factor [19, 20, 21]. At the rising time of the generated pulses from Type I and II, the
resonance shifts away from the CW, and the photons stored in the ring escape outwards by
both loss and coupling. With the same Q-factor and the slightly different photon storage
29

mentioned above, Types I and II have a similar rising edge in Figure 2.4, associated with
photon lifetime. However, at the falling time when the ring becomes on-resonance again, the
loss discounts the efforts of incoming photons to re-build the optical field in the cavity. The
falling time thus must be longer than the photon lifetime, which is why both Type I and II
have a falling time longer than the rising time. Also, a stronger coupling in Type II allows
for a faster build-up of optical field, and the falling edge for Type II is steeper. Type II thus
has narrower pulses than Type I as shown in Figure 2.6.

Figure 2.7 3-dB modulation bandwidth of the microring modulators for all the three types at drive
voltage of 3 V.
Figure 2.7 shows 3-dB modulation bandwidth for the three types of modulators with
drive voltage at 3 V. Type I, II and III have a bandwidth of 15.5, 19.5 and 9.6 GHz. There is
a peaking effect for Type II, which is associated with over-coupling (small-damping). The
overshoot and intensity dip cause steep pulse edges and a fast response of the modulator.
The peaking effect in Type II decreases with drive voltage from 3 to 1 V, as shown in Figure
2.8. Accordingly, 3-dB bandwidth is reduced from 19.5 to 10.5 GHz. This is because, with a
small voltage, the photons in the cavity are not fully removed at logic ‘1’, producing a small
30

overshoot. With driving circuit’s effect on modulation speed ignored here, the bandwidth is
limited by carrier transit time and photon lifetime. A fast optical response in Type II makes it
more tolerant to a long transit time.

Figure 2.8 3-dB modulation bandwidth of the microring modulators for Type II with a decreased
voltage.

Figure 2.9 Signal quality is examined as the drive voltage increases. Type II modulators exhibit a
better performance at a high drive voltage than the others.
We examine how the drive voltage affects signal quality, keeping other modulator
parameters the same. As shown in Figure 2.9, signal Q factor is improved by increasing the
31

voltage, which is mainly due to a better extinction ratio. When voltage is larger, Type III
becomes worse with reduced pulse peak power and a small eye-opening, due to the dynamic
loss, although a high voltage produces a low ‘0’-level and a good extinction ratio, as shown
in Figure 2.5. Type I and II have much better signal quality but it saturates at a high voltage.
Type II keeps improving, with steep pulse edges. In contrast, Type I suffers from a long
pulse tail at the falling time (see Figure 2.4) and an ever-increasing pulsewidth, which causes
the signal Q to slightly drop at a high voltage. We believe that, when many data channels are
densely multiplexed and a relatively narrow filter is added to each channel, the signal quality
would decrease with the drive voltage after an optimal value.

Figure 2.10 Signal quality is examined as laser linewidth increases at drive voltage of 3 V. All the
modulator types have similar signal quality at a large linewidth.
Signal quality also strongly depends on laser phase noise and linewidth. Figure 2.10
shows that signal Q decreases quickly with the laser linewidth more than 100 MHz, at drive
voltage of 3 V. All the three types perform quite similar at a large linewidth. Signals
degrade because the laser phase noise is converted to amplitude noise by the resonator-based
32

active devices [49, 50], which may become a critical issue when an on-chip laser with a
large linewidth [51] is integrated together with the modulators.
Based on the simulation results given above, we note that the Type III modulator
requires low cavity loss to obtain the same cavity Q as Type I and II, and that it does not
perform well due to a small extinction ratio at low drive voltage. For a feasible cavity loss,
Type III has a lower cavity Q, and drive voltage has to be much higher to produce an
acceptable extinction ratio, which is power-consuming. Hence, in the following section, we
limit our discussion only to Type I and II modulators.
2.3 Performance in Fiber Transmission
Frequency chirp in the modulated signals has not been fully discussed so far. It is
negligible for 10 Gb/s signals in short-reach optical communications where signals travel
over a few meters at most, even if the waveguide dispersion can be as high as several
thousand ps/(nm·km) [52, 53]. However, the frequency chirp has a significant effect on data
transmission over optical fibers. Besides the unique steady-state phase properties shown in
Figure 2.2, cavity dynamics also strongly affects the signal chirp [25]. As a result, the three
types of modulators have a quite different chirp waveform in Figure 2.4.
It is important to mention that the chirp is difficult to be evaluated using conventional
α-parameter [54], because, based on its definition
) / ( E E ∂ ∂ ⋅ = ϕ α (2.10)
where E and φ are the amplitude and phase of an optical field, the calculated α-parameter
becomes a temporal function oscillating quickly from negative to positive at bit transitions.
33

This is partially originated from the interference effect explained above. Instead, we define
“effective chirp” as δf(t)·p(t) to evaluate chirp effect, where δf(t) is the instantaneous carrier
frequency variation in GHz, and p(t) is the instantaneous optical power in mW [19]. This
generates a chirp waveform weighted by power so that the absolute chirp peak occurring at
low power duration would not be concerned much.
In Figure 2.11, we plot the amplitude and effective chirp for the pulse ‘00111000’
generated from the Type I and II modulators. Since a high drive voltage produces a very
asymmetric signal spectrum, we consider a voltage <3 V for data transmission over fibers.
Figure 2.11 (a) and (b) compare Type I and II with a drive voltage at 1.25 V, and Figure 2.11
(c) and (d) show Type II with a voltage reduced to 1 and 0.75 V. In the same scale,
horizontal axis show an 800-ps time window, and vertical axes are 0.1~0.9 (mW)
½
for the
amplitude and -1~0.4 GHz·mW for the effective chirp. We note that Type I essentially has
red-shifted frequency components at rising time when the chirp is weighted by optical
power, which is different from the absolute chirp shown in Figure 2.4. Importantly, the
frequency components at falling time are also red-shifted, which means that the high
frequency components at both pulse edges will walk off with carrier frequency due to
chromatic dispersion in fibers, producing data distortion. In contrast, Type II has red-shift at
rising time and blue-shift at falling time, so-called negative chirp [54], which is useful to
compensate for the positive dispersion in standard single mode fibers. As voltage increases
from 0.75 to 1.25 V, the effective chirps at the pulse edges become more unbalanced, with
the red shift enhanced partially by the increased overshoot and the blue shift remaining
almost the same.

34

Figure 2.11 Pulse waveforms (dotted blue line) and effective chirps (solid red line) generated by the
Type I and II modulators, with different drive voltages. All the plots are in the same scale.
We simulate signal transmission in a single-channel fiber link, in which chromatic
dispersion is 16.3 ps/nm/km at 193 THz. Fiber nonlinearity is negligible since launched
signal power of -10 dBm is relatively low. Polarization mode dispersion in the fibers is
ignored. Type I and II modulators with 1-V drive voltage are compared to a 10-GHz MZM
that has V
π
=4 V, 30-dB extinction ratio, and no chirp. Power penalty is measured as the
difference of required received power for signal detection at bit error rate of 10
-9
with and
without fibers. Figure 2.12 shows power penalty over fiber length for the Type I and II
modulators and the MZM, without dispersion compensation. One can see a great reach
extension for Type II due to its negative effective chirp. In contrast, Type I suffers from the
chirp-induced data distortion and performs worse than the MZM. At 1-dB power penalty,
fiber transmission distance is 16 km, 83 km, and 30 km for Type I and II, and MZM,
respectively.
35

Figure 2.12 Power penalty as a function of fiber transmission distance for Type I and II modulators
and a MZM. Type II exhibits negative chirp and thus greatly extended fiber distance at
power penalty of 1 dB.
2.4 Discussion
The under- and over-coupled ring resonators have been differentiated in terms of
group delay [42, 43], but it is not necessarily related to the photon lifetime that is a key
parameter when the resonators are used as modulators. The under- and over-coupled rings
can have dramatically different group delay but the same photon lifetime, as long as they
have the same resonance wavelength and cavity Q. Although an over-coupled ring
modulator is potentially better than an under-coupled one in terms of modulation speed,
based on the analyses above, it is important to mention that the over-coupled ring (Type II)
needs lower loss for achieving the same cavity Q, which may require a reduced doping
density in MOS electrodes, assuming the same waveguide scattering loss and bending loss
for both Type I and II modulators. Less doping may in turn cause a longer carrier transit
time, tightening the electrical limitation on modulation speed. Therefore, the claim that a
Type II modulator is faster and more tolerant to a long transit time is true only when photon
lifetime is the limiting factor of the modulation speed.
36

In this paper, the photon lifetime is 15.7 ps for a cavity Q = 19000, and the transit
time is set to be 23 ps. However, this does not mean that the photon lifetime has less
limitation on the modulator bandwidth. This can be understood by recalling our explanation
on the pulsewidth trend in Figure 2.6. At pulse falling edge for Type I and II modulators, the
cavity loss counteracts the waveguide-to-ring coupling, which extends the falling time
beyond the photon lifetime. On the other hand, we also note that the pulse transition time
may be less than a summation of the carrier transit time and the photon lifetime (e.g., at
pulse rising time), because the two transient processes go in parallel in electrical and optical
domains, respectively.
Due to the dynamic loss caused by the transient carrier density, the resonance profile
is reshaped when shifted. One can tell that it is an under-coupled ring modulator if the
resonance notch becomes shallower when a voltage is applied [8, 9]. This because, for an
under-coupled ring, the cavity loss is originally greater than coupling, and adding the
dynamic loss makes the ring further away critical coupling and shows the shallower notch.
Otherwise, it is an over-coupled modulator if the notch is deepened first [17]. One thing that
might be interesting is that applying a high voltage can induce a dynamic loss strong enough
to switch an originally over-coupled ring to under-coupled, which is similar to the process of
switching slot light to fast light [55]. As a modulator, this causes a ‘hybrid’ chirp property
that is under current investigation.
The single-ring silicon modulators have some performance trade-offs. For example, a
high-Q resonator has a narrow linewidth, allowing for low drive voltage and power
consumption, but the photon lifetime limits modulation speed. In other words, increasing the
ring-waveguide coupling makes the modulator faster but leads to an increased resonance
37

linewidth and stronger overshoot. One possible way to provide improved performance is to
use multiple-ring structures [19, 20]. Recently, the coupling modulation scheme is proposed
to enhance modulator speed and reduce frequency chirp [40, 56, 57], which would exhibit a
great potential in the future if electrodes for controlling the coupler could be made short and
low loss so that resonance Q-factor can be high enough and the speed limitation imposed by
carrier transit time can be mitigated.
Temperature insensitive microring modulators are highly desirable. There have been
some reports on eliminating the temperature drift of resonance wavelength of ring resonators
[58, 59], in which polymer with the opposite thermo-optic coefficient is integrated into
silicon slot waveguides. This increases the feasibility of using microring modulators in on-
chip WDM systems.
2.5 Summary
In this chapter, we have numerically analyzed silicon microring modulators for
intensity modulation in both the back-to-back case and for fiber telecom systems. Three
types of single-ring modulators have shown dramatically different performance. An over-
coupled ring modulator has a potential to be useful in fiber systems. A 0.8 dB power penalty
could be achieved in 80-km data transmission over single mode fibers, without dispersion
compensation.

38

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44

Chapter 3 Single Microring Phase Modulator
Advanced data modulation formats have become quite important within the optical
communications community, and of particular interest is differential-phase-shift-keying
(DPSK) [1]. When compared to amplitude-shift-keying, DPSK exhibits a 3-dB increase in
receiver sensitivity and is more tolerant to fiber nonlinearities. DPSK signal, which carries
information via the phase difference of adjacent symbols, is typically modulated using either
a phase modulator or Mach-Zehnder modulator (MZM), while a delay-line interferometer
(DLI) is used as a demodulator for balanced detection [2]. More recently, several novel
demodulation schemes have been reported [3, 4, 5]. But these modulators and demodulators
are relatively complicated, which causes higher cost of DPSK transmitter and receiver.
Moreover these devices are fairly large, on the order of a centimeter, which may mean large
power consumption.
Micro-ring based devices have attracted a great deal of attention in recent years.
Silicon-based microring modulators, switches and filters, fabricated in silicon-on-insulator
(SOI) platform, have been demonstrated [6, 7, 8]. In this chapter, we discuss how to use
integrated ring resonator structures to (de)modulate DPSK signals, which replace the
interferometers for DPSK-based communication and interconnect systems. As compared to
conventional MZMs and DLIs, microring structures require a relatively small chip area and
low driving power, and are easy to fabricate into arrays. We numerically outline and analyze
the use of the π phase shift from a ring resonator to generate DPSK as well as using a ring
resonator surrounded by two waveguides to achieve a DLI-type filtering function for
demodulation. A trade-off between data pattern dependence of the Duobinary signal and
45

power loss of the alternate-mark inversion (AMI) signal is identified, as cavity Q-factor of
the demodulator varies. Power penalty is diminished to 0.8 dB with an optimal cavity Q-
factor of 22000 for 10 Gb/s non-return-to-zero (NRZ) DPSK, which is relatively low
compared to some published demodulation schemes.
3.1 Principle
Typically, DPSK is generated by using a MZM to introduce a π phase shift between
bits if the sign of the second bit is meant to flip from a “1” to “0” from the first bit. Phase
shift of π is realized across the minimum point of transfer function of a MZM, which causes
intensity dips in modulated signals. An alternative way to modulate DPSK can be direct
phase modulation, which induces undesired frequency chirp.
Microring modulators were demonstrated only for intensity modulation by switching
output optical power on and off through shifting of the ring resonance peak. This shifting
can be realized by varying carrier density and thus refractive index in the ring cavity, when a
voltage is applied or carriers are injected. in the electrical domain, the microring modulators
are typically based on MOS capacitors [6, 7] or p-i-n diodes [8]. Many novel structures and
driving schemes have already been proposed to increase modulation speed and efficiency
[9]. Here a MOS-capacitor-based modulator is considered since it is more suitable for high-
speed operations. In the optical domain, a microring in a single-waveguide configuration can
be over-coupled (i.e., waveguide-ring coupling > ring loss), causing phase shift of 2π across
resonance [10, 11, 12]. In this case, the transfer function of the over-coupled ring is obtained
from
46

) exp( 1
) exp(
ϕ
ϕ
i ta
i a t
E
E
in
out
- - = (3.1)
where t and a stand for the transmission coefficient and loss, and ϕ is phase change in one
round trip [10]. When the resonance peak is shifted, the continuous wave (CW) laser source
can experience a phase shift of π across the center of the phase profile, as shown in Figure
3.1, and have the same power in each bit duration. NRZ DPSK data format is generated this
way. It is important to note that there are intensity dips at bit transitions in the modulated
signal, which are caused by the notched profile of the ring resonator. On the other hand,
when the resonance profile is moved, the optical signal also experiences a fast phase shift,
which will give rise to frequency chirp at the bit transitions. A major difference between our
microring-based DPSK modulator and a conventional phase modulator or MZM is that there
are both intensity dips and frequency chirp in the modulated signal.

Figure 3.1 Microring-based NRZ-DPSK modulation and demodulation schemes. In modulation, CW
experiences a phase shift of π across resonance, using a over-coupled ring structure. A
double-waveguide microring filter enables demodulation of both Duobinary and AMI and
thus balanced detection.
Demodulation of DPSK signals can be achieved using a single-ring double-waveguide
filter, in which one output port works as a band-pass filter to obtain the Duobinary signal,
47

and the other as a notch filter to produce the AMI signal. The transfer function of the
microring filter can be derived [10], as shown below,

) exp( 1
) exp( ) 1 (
2
2
ϕ
ϕ
i a t
i a t
E
E
in
out
- - = (3.2)
where t, a and ϕ are defined as before. This microring filter actually functions as a
discriminator to convert the phase-modulated signal to amplitude-modulation for direct
detection [4, 5]. Duobinary and AMI are detected individually and then combined
electrically to obtain balanced detection of DPSK signal.
3.2 Performance Characterization
3.2.1 Modulator
We first plot the symbol constellation diagram for the proposed DPSK modulation
scheme, in which the electric field of the modulated optical wave is mapped into the
complex plane, as shown in Figure 3.2 (a). The microring-based DPSK modulator (dash) is
compared to those obtained from phase modulator (dash dot) and MZM (dot). Frequency
chirp will occur unless the modulated optical field transits between two symbols (black dots)
exactly along the real axis, so only a MZM can produce a chirp-free signal. On the other
hand, the intensity dip at a bit transition can be removed only when the transition between
the two symbols is along an arc, like those created by a phase modulator [1]. From Figure
3.2 (a), we note that the microring-based DPSK modulator induces both frequency chirp and
intensity dips. But fortunately, the frequency chirp and intensity dips always occur together
meaning that the chirp is at low power. The DPSK signal modulated with microring has a
48

transition close to the real axis making it very similar to that of a MZM. This is verified in
Figure 3.2 (b), in which -3 and -10 dB signal bandwidths are examined as a function of
transit time. For a long transit time, signal bandwidth can be narrower, compared to a MZM.
Spectra of a MZM and a microring modulator are plotted with the same scale (10 dB/div. in
power and 20 GHz/div. in frequency) in Figure 3.2 (c) and (d), when the carrier transit time
is 16 ps. We note that the microring-based DPSK modulation produces a spectrum very
similar to that given by the MZM, although it is a little broader in the low-power regions far
away from signal carrier due to the chirp. Since the intensity dips keep the frequency chirp at
low power levels, the chirp-induced spectrum broadening is small.

Figure 3.2 (a) Symbol constellation diagrams for microring-based modulator (dash), phase modulator
(dash dot) and MZM (dot). (b). DPSK signal bandwidth vs. transit time. (c). DPSK spectrum
modulated by MZM. (d). DPSK spectrum modulated by microring, in the same scale. (10
dB/div. in power and 20 GHz/div. in frequency).
To evaluate the performance of the microring-based DPSK modulator, a DLI-based
demodulator is used to isolate the effects of the modulator on signal quality. Figure 3.3
shows eye-opening penalty, compared to a MZM, as a function of transit time. We consider
a well-biased case, in which the resonance peak is shifted by 0.08 nm, and CW carrier is
originally located 0.04-nm away from the resonance peak. Modulated signal has the same
maximum transmission. The eye-opening penalty increases with transit time. In Figure 3.3
49

eye-diagrams are shown in the same scale and correspond to transit times of 16 and 50 ps,
respectively. Tracks in the eye-diagrams split at long transit time because the refractive
index at for a single ‘1’-bit has a smaller modulation depth, compared to consecutive ‘1’-
bits, and thus the maximum transmission is not reached for a single ‘1’-bit. Timing jitter in
the modulated signal also becomes worse as seen in the eye-diagrams. Nevertheless, the
microring-based modulator can produce good DPSK signals with no visible imperfections,
as long as the transit time is shorter than 30 ps, which can be easily achieved [6, 9].

Figure 3.3 Eye-opening penalty of the modulated DPSK signal, compared to a MZM, versus carrier
transit time. Two eye-diagrams, corresponding to transit times of 16 and 50 ps, are shown in
the same scale.
Modulation performance is examined for different bias conditions. CW offset, defined
as (λ
resonance
-λ
CW
-0.04nm), is set from 0 to 0.016 nm. As shown in Figure 3.4, the eye-
opening penalty almost linearly increases with CW offset. Data quality is degraded by ‘1’-
level splitting in the modulated signal, with an almost unchanged ‘0’-level, as shown in eye-
diagrams in Figure 3.4. Time jitter in the modulated signal is more severe. Compared to
50

Figure 3.3, it is noted that the bias condition is more critical to keep good modulation
performance.

Figure 3.4 Eye-opening penalty of the modulated DPSK signal, compared to a MZM, versus CW
offset. Two eye-diagrams, corresponding CW offsets of 0 and 0.016 nm, are shown in the
same scale.
We examine the dependence of signal quality on laser linewidth with a Lorentzian
shaped spectrum. A drive voltage of 1 volt causes a phase shift of π, corresponding to a
cavity Q-factor of 16000. It is noted that large linewidth may significantly degrade the
DPSK signal waveform, as illustrated in Figure 3.5. Signal distortion is quite severe for a
laser with a 3-dB linewidth of 10 MHz. In contrast, signal quality is similar when the laser
linewidth is 300 kHz and 10 kHz. This waveform distortion is attributed to the fact that the
phase noise contained in the laser linewidth is greatly enhanced in the microring-based
resonant structure [13]. When light stored in the resonator is coupled out, it would beat with
light traveling along the waveguide, which converts the phase noise to amplitude noise [13].
51

Figure 3.5 Signal waveforms from a microring-based DPSK modulator are shown for different laser
linewidths: 10 MHz, 300 kHz and 10 kHz, respectively.

Figure 3.6 Eye-opening penalty changes dramatically with laser linewidth for different cavity Q-
factors of the ring resonators, compared to MZM-based DPSK. When cavity Q-factor is
chosen to be 10000 and laser linewidth is <10 MHz, <1 dB penalty is obtained. Eye-
diagrams, corresponding to linewidths of 30 kHz and 30 MHz with Q=22000, are plotted in
the same scale.
Figure 3.6 shows eye-opening penalty (EOP) of the microring-modulated DPSK as a
function of the laser linewidth from 30 kHz to 30 MHz, for different cavity Q-factors
ranging from 10000 to 22000. Drive voltage is modified according to a different cavity Q-
factor to keep the phase shift fixed at π. Transit time is set to be 23 ps, and a DLI is used as a
52

demodulator. These results are also compared to those obtained using a MZM-based
modulator. We note that a high-Q ring modulator suffers from the linewidth problem much
more than a low-Q resonator or a MZM does, and EOP caused by the laser phase noise
increases approximately exponentially with laser linewidth and can be up to 6 dB. It is very
important to carefully deal with this issue because, once the signal is degraded by the laser
phase noise, one could not improve the detection bit-error-rate (BER) by simply adding an
optical filter or increasing the received power in the receiver. Thus design guidance is highly
needed. In particular, it is found out that a microring-based DPSK modulator with Q of
10000 works well with a laser of <10 MHz linewidth, in which EOP is <1 dB, and the
required drive voltage is 1.88 volt.

Figure 3.7 Signal power linearly increases with phase shift by driving the modulator harder.
We also examine the effect of drive voltage on the DPSK signal quality, which can
change output power and phase shift in the modulated signal. The laser linewidth problem
can be partially solved when the microring modulator is driven harder. In this way, the
required drive voltage is higher, and the resonance shift becomes larger, which causes a
53

higher transmittivity through the ring resonator and more phase shift. We choose cavity Q-
factor to be 22000, since a high-Q resonator suffers more from laser phase noise. Drive
voltage is increased from 0.77 to 2.2 volt, and correspondingly the induced phase shift is
changed from exactly π to 1.41π. As shown in Figure 3.7, signal power is increased almost
linearly with induced phase shift.

Figure 3.8 Eye-opening improvement given by driving the ring modulator harder, as phase shift is
more than π. Eye-diagrams are plotted in the same scale, which are corresponding to the
worst and best cases for microring and DLI demodulators, respectively.
We note from Figure 3.8 that over-driving of the microring modulator improves eye-
opening dramatically by up to 7 dB for microring-based demodulation as well as by 5 dB for
DLI-based one, with 3-MHz laser linewidth. It is important to mention that the DLI
demodulator is based on interference of light waves and requires an accurate π phase shift.
When the phase shift is changed, the DLI demodulator would have a relatively low
extinction ratio that discounts the benefit from the signal power increment. But, a varied
phase shift does not change the spectral distribution of duobinary and AMI signals in DPSK
signal spectrum. Thus the microring demodulator operating in the frequency domain relaxes
the requirement for an exact π phase shift. The over-driving of the modulator results in a 7-
54

dB eye-opening improvement, which could compensate for the EOP caused by the laser
linewidth problem. It should be noted that, for telecom applications, the phase shift in DPSK
signals is desired to be exactly (or close to) π, which makes the signal more tolerant to phase
noise induced by in-line amplifiers [1]. In this case, one may not want to over-drive the
modulator too much. But for short-reach or on-chip interconnects [14], system performance
would benefit from over-driving the microring modulator.
3.2.2 Demodulator
To demodulate a DPSK signal, the microring-based double-waveguide filter is used to
extract Duobinary and AMI signals from the DPSK by spectrum slicing. The micro-ring
demodulator has a cavity Q-factor of up to 25000 and a radius of 5 μm. A MZM is used as a
DPSK modulator to isolate effects of the demodulator. First, we examine whether or not
correct data pattern is demodulated by the spectrum-slicing scheme, because it may cause a
clean eye-diagram but incorrect information. For example, if a two-bit-time DLI is used,
which functions as a periodic spectrum slicer with a FSR of the half bit-rate, then good eye-
diagrams can be obtained, but the demodulated information is not the same as what is
modulated and must be post-coded to retrieve the correct data.
Figure 3.9 shows original logic data bits that are then pre-coded and modulated onto a
CW carrier. We note that the demodulated Duobinary and AMI signals contain the correct
data pattern. It can be seen that, in the Duobinary signal, optical power rises, but this
happens only between adjacent ‘0’-bits, which will not be sampled in signal detection. This
means that a microring-based filter can be used as a DPSK demodulator in principle.
55

Figure 3.9 A comparison of original logic (a), obtained Duobinary (b) and AMI (c) data shows correct
information is demodulated.
As a key parameter, cavity Q-factor of the microring-based demodulator is optimized.
As shown in Figure 3.10, using MZM as DPSK modulator, eye-opening of the demodulated
signals increases with the Q-factor and reaches the maximum when Q=22000. This is
attributed to the fact that, when Q-factor is higher, the notch filter profile of the demodulator
becomes narrower and leaves more power in the AMI port. Since optical power of the AMI
signal is increased, the eye-diagram is more open in balanced detection, as seen from the
eye-diagrams for Q-factors of 10000 and 25000 in Figure 3.10. However, correspondingly,
the Duobinary signal is cut by the narrowed bandpass filtering, which causes pattern
dependence in the Duobinary data. A single ‘1’-bit has a lower instantaneous power than
consecutive ‘1’-bits at the sampling instant, as shown in the eye-diagrams, and this may
decrease eye-opening with Q-factor higher than 22000. The trade-off between AMI signal
power and Duobinary data pattern dependence originates from the Lorentzian shape of
micro-ring resonance. Comparatively, a traditional DLI has a cosine-squared filter shape for
its transfer function. The micro-ring DPSK demodulator has to have a balance between
keeping AMI power and maintaining the Duobinary spectrum. An optimal cavity Q-factor is
56

chosen to be 22000, and beyond this, eye-opening drops due to the pattern dependence in the
Duobinary data.

Figure 3.10 Quality of the demodulated signal is examined with varied cavity Q-factor. Two eye-
diagrams, corresponding to Q=10000 and Q=25000, are shown in the same scale.
Figure 3.11 shows the tolerance of demodulation performance to varied signal bit-rate.
DPSK signal is generated by a MZM, and a microring-based DPSK demodulator with Q-
factor of 22000 is compared to a DLI that is designed for 10 Gb/s DPSK signal. Eye-opening
penalty is caused when the bit-rate is changed away from 10 Gb/s. We note that the
microring demodulation scheme exhibits better tolerance to a lowered bit-rate, while it
induces a larger penalty than a DLI, for an increased bit-rate, which is because the
Duobinary signal suffers very much from the pattern dependence. In fact, optimal cavity Q-
factor in demodulation for 10 Gb/s DPSK is between 19000 and 22000 but very near 22000.
Thus, when the cavity Q-factor is fixed at 22000, 9 Gb/s signal shows a better demodulation
performance.
57

Figure 3.11 Tolerance of microring-based DPSK demodulator and DLI to varied signal bit-rate. The
proposed approach is more tolerant to a lowered bit rate.
The quality of the demodulated DPSK signal at 10 Gb/s is examined when a
frequency offset between the demodulator resonance frequency and the signal carrier is
increased from 0 to 3 GHz. Here, we use a MZM as a DPSK modulator and choose the
demodulator bandwidth of 8.7 GHz, corresponding to cavity Q-factor of 22000, which has
been optimized. The microring-based DPSK demodulator is much more tolerant to
demodulator frequency offset than a DLI, because DLI demodulation scheme is based on
interference of optical waves and relies on accurate phase control that is reflected as
frequency shift of DLI transfer function. According to the periodic property of DLI transfer
function, the demodulated DPSK signal from a DLI would be completely distorted with ¼
bit-rate offset (that is, 2.5 GHz) and become inverted with the ½ bit-rate offset. In contrast,
microring demodulator works in frequency domain and has a huge free spectrum range (e.g.,
more than ten nanometers for a ring radius of 1.5 μm) and a relatively stable performance
with much smaller EOP, compared to a DLI.
58

Figure 3.12 Eye-opening penalty is examined as a function of demodulator frequency offset.
Microring-based demodulator is more tolerant to the offset than a DLI. Two eye diagrams
are corresponding to an offset of 0 and 3 GHz for the microring demodulator.

Figure 3.13 Eye-opening improvement is obtained by increasing receiver bandwidth. Eye-diagrams,
corresponding to receiver bandwidth of 0.5 and 1.75 times signal bit rate, are plotted in the
same scale.
We also note from Figure 3.13 that eye-opening can be improved by optimizing the
receiver bandwidth, in which the PIN photo-detector bandwidth varies from 0.5 to 2.3 times
the bit-rate, and the demodulator bandwidth is 8.7 GHz. Eye-opening improvement is
increased by >1 dB as the normalized receiver bandwidth by bit-rate is changed from 0.7 to
59

1.5. As shown in the eye-diagrams in Figure 3.13, the eye-opening improvement comes
mainly from better detection of demodulated AMI signal that contains more high frequency
components.
3.3 Application Scenarios
Employing microring structures as both the modulator and demodulator, overall
performance of a DPSK link is examined for the back-to-back case and compared to using
regular MZM and DLI. Transit time in the modulator is set to be 16 ps, and Q-factor in the
demodulator is chosen to be 22000, in a well-biased case. With receiver sensitivity of –25
dBm, BER curves are plotted in Figure 3.14. Error-free detection can be easily achieved by
increasing received power, with a power penalty of about 0.8 dB between the microring-
based and traditional scheme. As described above, this penalty comes mainly from non-
ideality of the microring-based demodulator and is expected to be decreased further by
improving the ring structure in the future.

Figure 3.14 BER curves given by all microring-based DPSK modulation and demodulation, with 0.8-
dB power penalty, compared to a conventional DPSK link: MZM + DLI.
60

Figure 3.15 Power penalty comparison between the all–ring based and conventional DPSK links, over
single mode fiber transmission up to 70 km.
Overall performance of the proposed microring-based DPSK modulator and
demodulator in fiber transmission is compared to conventional scheme (MZM+DLI) in
terms of power penalty, as shown in Figure 3.15. Laser linewidth is 100 kHz, and transit
time in the modulator is set to be 16 ps. Q-factors in the modulator and demodulator are
chosen to be 10000 and 22000, respectively. The bias condition in the microring modulator
is controlled to generate a phase shift of π. Receiver bandwidth is 17 GHz for the all-ring-
based DPSK and 7 GHz for MZM+DLI. The power penalty is measured at 10
-9
BER, in a
pre-amplified receiver configuration. Over a 70-km-long single-mode fiber without
dispersion compensation, the all-ring-based technique exhibits negative power penalty
within 30-km transmission possibly due to frequency chirp and becomes worse than the
conventional scheme after 50 km.
For on-chip applications, we note that an absence of ASE noise induced by optical
amplifiers makes the performance of two types of DPSK links closer to each other, since
DLI-based demodulation plus a balanced detection would reduce ASE-related signal
61

distortion, which means, for telecom applications, the microring demodulator is a little
disadvantageous in this sense.
To save electrical driving power at transmitter side, using DPSK data format is
advantageous over the amplitude modulated NRZ signals generated from Type I and II in
terms of signal quality. In Figure 3.16, we compare these three situations with the same
cavity Q-factor of 22000 and 2.2-V drive voltage. Without suffering from overshoots in the
pulses, DPSK has a signal Q of 28 dB, which is 7-dB higher than the two NRZ signals. From
Figure 3.7, if the DPSK modulator sacrifices this 7-dB benefit, it needs only 1.3 V drive
voltage.

Figure 3.16 A comparison of eye-diagrams for (a) DPSK, (b) NRZ generated by a Type I modulator
and (c) NRZ generate by a Type II modulator. Cavity Q-factor is 22000, and drive voltage is
2.2 V. The DPSK signal has 7-dB higher Q-value.
3.4 Discussion
It would be desirable to design compact microring-based DPSK modulator and
demodulator compatible to the SOI platforms, with two potential advantages: (i) increased
laser power utilization and (ii) reduced drive power consumption. First, since logic data is
carried by the phase of an optical wave, laser output power could be maintained at every bit
instead of being removed as in on-off keying. Second, a sharp phase transition occurs across
resonance, and π phase shift can be obtained by shifting the spectral response of a resonator
62

very little using a low voltage. We believe that the second issue is more important since
electrical power consumption is dominant for a microring modulator [14]. When the phase-
modulated data format is introduced into chip-scale optical interconnects, the standard
scheme (MZM plus DLI) in telecom systems [1] requires too much “real estate” and would
not be affordable.
In high-capacity computing systems, the chip itself will be heated up when the system
starts fully running. An on-chip modulator as an active device may have a local thermal
problem and thus a frequency drift, while its demodulator partner that serves the same
wavelength channel with it might be several millimetres away from it and does not have the
same frequency drift. In this case, the demodulator has an equivalent frequency offset
relative to the modulated data stream. As shown before, the microring-based DPSK
demodulator is much more tolerant to demodulator frequency offset than a DLI.
We note that a microring-based DPSK demodulator has been successfully
demonstrated very recently [15], exhibiting a good flexibility when used for demodulating
DPSK signals at variable data bit rates.
In addition to the DPSK format, other phase-modulated data formats include DQPSK,
duobinary and so on. As a multi-level phase shift keying signal, DQPSK allows to double
the volume of information that is being transmitted without extending signal spectral width.
Microring-based DQPSK generator and demodulator have been proposed [16], but high
drive voltage has to be applied to achieve low bit-error-rate detection. Another way to
increase the interconnection capacity is to employ duobinary modulation format, which
occupies only half-rate bandwidth and can be detected with no need for a demodulator [17].
63

Although the phase-modulated signaling provides a potential opportunity to improve
system performance as described above, this sometimes (except duobinary format [17])
requires a modification of the receiver part, for example, adding a demodulator for phase
shift keying signals and requiring two photodiodes for balanced detection of a single data
channel. It is still pending whether the whole system would benefit from employing phase
modulation formats.
3.5 Summary
We have discussed silicon single-ring-based modulation and demodulation of DPSK
signals, which enables integrated ultra-small DPSK transmitters and receivers. Single-
waveguide microring modulator with over-coupling is used to achieve a phase shift of π, and
eye-opening penalty can be as low as 0.35 dB, compared to a MZM. Demodulation of a
DPSK signal has been realized using a double-waveguide microring filter, which extracts the
Duobinary and AMI formats from the DPSK spectrum simultaneously for balanced
detection.
We have analyzed signal quality dependencies of integrated ultra-small silicon
microring-based DPSK modulation and demodulation on laser linewidth, drive voltage,
demodulator offset and receiver bandwidth. It is shown that the microring-based DPSK
modulator is more sensitive to laser phase noise, while the microring-based DPSK
demodulator can be more tolerant to a change in phase shift away from π and frequency
offset between the demodulator center wavelength and the signal carrier. Data quality of the
microring-based DPSK can be optimized with an eye-opening improvement of up to 7 dB.
We also compared the data transmission performance of the microring-based DPSK over 70-
64

km single mode fiber to that in a conventional DPSK link consisting of MZM and DLI,
which shows that lower power penalty is obtained in a distance of <50 km using the
microring-based scheme. Advantages of using phase-modulated signals in on-chip
communications have also been discussed.

65

3.6 References

[1] A. H. Gnauck, and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave
Technol. 23, 115-130 (2005).
[2] E. A. Swanson, J. C. Livas, and R. S. Bondurant, “High sensitivity optically preamplified
direct detection DPSK receiver with active delay-line stabilization,” IEEE Photon. Technol.
Lett. 6, 263-265 (1994).
[3] E. Ciaramella, G. Contestabile, and A. D'Errico, “A novel scheme to detect optical DPSK
signals,” IEEE Photon. Technol. Lett. 16, 2138-2140 (2004).
[4] I. Lyubomirsky and C. Chien, “DPSK demodulator based on optical discriminator filter,”
IEEE Photon. Technol. Lett. 17, 492-494 (2005).
[5] L. Christen, Y. K. Lize, S. Nuccio, J.-Y Yang, S. Poorya, A. E. Willner, L. Paraschis,
“Fiber Bragg grating balanced DPSK demodulation,” in Proceedings of IEEE LEOS Annual
Meeting 2006 (Institute of Electrical and Electronics Engineers, Montreal, Canada, 2006),
pp. 563-564.
[6] C. A. Barrios and M. Lipson, “Modeling and analysis of high-speed electro-optic
modulation in high confinement silicon waveguides using metal-oxide-semiconductor
configuration,” J. Appl. Phys. 96, 6008-6015 (2004).
[7] R. D. Kekatpure and M. L. Brongersma, “CMOS compatible high-speed electro-optical
modulator,” Proc. SPIE 5926, paper G1 (2005).
[8] Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic
modulator,” Nature 435, 325-327 (2005).
[9] C. A. Barrios, “Electrooptic modulation of multisilicon-on-insulator photonic wires,” J.
Lightwave Technol. 24, 2146-2155, (2006).
[10] Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt.
Soc. Am. B, 20, 2125-2132 (2003).
[11] J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical
transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-
730 (2004).

66

[12] A. Stapleton, S. Farrell, H. Akhavan, R. Shafiiha, Z. Peng, S.-J. Choi, J. O’Brien, P. D.
Dapkus, and W. Marshall, “Optical phase characterization of active semiconductor
microdisk resonators in transmission,” Appl. Phys. Lett. 88, 031106 (2006).
[13] J. Caprnany, “Investigation of phase-induced intensity noise in amplified fibre-optic
recirculating delay line,” Electron. Lett. 29, 346-348, (1993).
[ 14 ] R. G. Beausoleil, “Nanophotonic Interconnect for High-Performance Many-Core
Computation”, in Proceedings of IEEE LEOS Annual Meeting 2007 (Institute of Electrical
and Electronics Engineers, Orlando, USA, 2007), pp. 523-524.
[15] L. Xu, C. Li, C. Y. Wong and H. K. Tsang, "Optical differential phase shift keying
demodulation using a silicon microring resonator," IEEE Photonics Technology Letters, vol.
21, no. 5, pp. 295-297, Mar. 2009.
[16] L. Zhang, J.-Y. Yang, Y. Li, R. G. Beausoleil and A. E. Willner, "Monolithic modulator
and demodulator of DQPSK signals based on silicon microrings," Optics Letters, vol. 33, no.
15, pp. 1428-1430, July 2008.
[17] L. Zhang, M. Song, J.-Y. Yang, W.-R. Peng, S. Nuccio, R. G. Beausoleil and Alan E.
Willner, "Generating spectral-efficient duobinary data format from silicon ring resonator
modulators," in European Conference on Optical Communication (ECOC) 2008, (IEE,
September 21-25, 2008, Brussels, Belgium), paper Tu.3.C.4.
67

Chapter 4 Embedded Microring Modulator
4.1 Introduction
Micro-resonators have exhibited great design flexibility and unique advantages for
achieving various compact devices, as we have discussed, including lasers, modulators,
switches, filters, delay elements, signal processing units and sensors. Much progress has
been made in designing and fabricating these sophisticated devices by employing multiple
rings that are cascaded in a parallel [1], serial [2], 2D-arrayed [3] or vertically coiled [4]
configuration, as shown in Figure 4.1. However, a cascade of many ring resonators may
require an increased chip-size.
In this chapter, we discuss an embedded configuration as another way to cascade the
ring resonators. Typically, the rings are embedded with coupling in either a point-to-point or
distributed manner, as shown in Figure 4.1. The embedded ring resonator (ERR) may enable
a smaller footprint and unique amplitude and phase characteristics. For example, the ERR
structure can produce an electromagnetically-induced-transparency (EIT) -like effect that
could be used for high-speed modulation up to 30 and even 40 Gb/s, which is hardly
achieved using previously reported EIT-like microring structures [5, 6, 7]. In the following,
we mostly focus on the point-to-point coupling configuration, and the distributed coupling
configuration is discussed very briefly at the end of this chapter. Since the embedded ring
resonators are a type of resonator structures, it is necessary to earn a basic understanding on
their passive performance before one can design an active device using them.
68

Figure 4.1 Structures of previously proposed and embedded ring resonators
4.2 Passive Performance
Let us consider a simplified ERR with point-to-point coupling, in which only two ring
resonators are embedded and placed between waveguides as shown in Figure 4.2. One can
derive its transfer function using coupled mode theory [8]. For simplicity, the coupling
coefficients at A and B are set to be the same, while the coupling coefficients at C and D are
the same as well. We obtain transfer functions at ‘through’ and ‘drop’ ports
A e e r t TF
drop
/ ] 1 [
2
2
2 1 2
2
1
2
1
- = τ τ (4.1)
A e r r e e r t r e e r r TF
through
/ } ) 1 ( ] [ {
2
2
2 1
2
2 2 1
2
1
2
1 2
2
1
2
2 2
2
2 1 1
4
1 1
τ τ τ τ τ - + + - + = (4.2)
where 1 ] [ 2
2
2
2 1
2
1
4
1
2
1 2 1
2
1 1 2
2
1 2 1
2
1
2
2
4
1
+ + - + = e e r r e e t r e e r A τ τ τ τ τ τ . Coupling is assumed to
be lossless. (r
1
, t
1
) and (r
2
, t
2
) represent the amplitude coupling coefficients between the outer
ring and the waveguides and between the two rings, respectively, satisfying r
1
2
+t
1
2
=1 and
69

r
2
2
+t
2
2
=1. φ
1
and φ
2
are round-trip phases in the outer and inner rings with e
1
=exp(jφ
1
) and
e
2
=exp(jφ
2
), while τ
1
and τ
2
are amplitude transmission coefficients within the quarter round-
trip in the outer ring and half round-trip in the inner ring, respectively. The two rings have
their own resonance wavelengths λ
R1
and λ
R2
, satisfying n⋅L
1
=m
1
⋅λ
R1
and n⋅L
2
=m
2
⋅λ
R2
,
where n is the effective refractive index; L
1
and L
2
are perimeters of the outer and inner
rings; and m
1
and m
2
are integer numbers.

Figure 4.2 An ERR with point-to-point coupling and its frequency responses at ‘through’ port for (i)
m
1
-m
2
=even and (ii) m
1
-m
2
=odd.
When λ
R1
and λ
R2
are set to be the same, the ERR has two typical working regimes:
(i) m
1
-m
2
is an even number (here, m
1
=46, m
2
=32), in which case the transfer function
at ‘through’ port features a doublet in amplitude response. As shown in Figure 4.2, two
notches occur at wavelengths λ
1
and λ
3
that are equally shifted from the common resonance
wavelength λ
2
of the two rings. FDTD simulations for TE mode show the symmetric and
anti-symmetric field distributions at coupling areas A and B, excited at wavelength λ
1
and
λ
3
, respectively. In this case, waveguide width is 300 nm and waveguide spacing in four
70

coupling areas is 160 nm. Size of the ring resonators are set to match integers m
1
=46 and
m
2
=32.
(ii) m
1
-m
2
is an odd number (here, m
1
=47, m
2
=32), in which case the through-port
transfer function shown in Figure 4.2 features an EIT-like profile centered at wavelength λ
2
.
With input continuous wave at λ
2
, FDTD simulation shows that half of the inner ring is
brighter than the other half, as illustrated in Figure 4.3. This is because the phase difference
between the two optical waves traveling over half round-trip of the two rings is (m
1
-m
2
)π
and m
1
-m
2
is odd.

Figure 4.3 Mode distributions with continuous wave inputs at wavelengths λ
1
and λ
3
for m
1
-m
2
=even,
and λ
2
for m
1
-m
2
=odd.
The ERR is characterized with varied coupling coefficient between the waveguide and
the outer ring and is compared to single-ring and double-ring resonators [9]. When m
1
-m
2
is
an odd number (m
1
=47, m
2
=32), the normalized output power at resonance wavelength at
‘through’ port increases with the coupling in Figure 4.4. The power coupling coefficient
between the two rings is set to be 0.13, and the loss is 2.23 dB/cm. The group delay can be
71

enhanced in an EIT-like profile. Compared to a single- or double-ring resonator with the
same structural parameters, the group delay is increased by ten times using ERRs. However,
we note that the resonance linewidth also becomes ten times narrower, so the delay-
bandwidth trade-off still holds. ERR structures have Vernier effect, and the overall free
spectral range (FSR) can be designed by changing individual FSRs of the two rings, FSR
1

and FSR
2
, and also their ratio FSR
2
/FSR
1
. An effective FSR extension has been reported by
embedding a small ring into a bigger one with an output waveguide coupled to the inner ring
[10].

Figure 4.4 m
1
-m
2
is an odd number, normalized transmission and delay vs. coupling between the
waveguide and the ring, compared to single- and double-ring resonators
4.3 Comparison of Double-Ring Resonators
We note that some previously reported double-ring structures have also exhibited EIT-
like profiles [5, 6, 7], as shown in Figure 4.5. Before we show more details on the
modulation performance of ERR-based E/O modulators, it is necessary to estimate the
difference of the proposed resonators.
72

Figure 4.5 Previously proposed microring-based EIT structures
Let us consider the high-speed modulation. The single-ring modulators have
encountered some performance trade-offs. The photon lifetime of a resonator, which
essentially affects how fast photons can be coupled into and out of the resonator, plays an
important role in determining the microring modulator’s speed. A high-Q resonator has a
narrow linewidth and requires low drive voltage and power consumption, although it limits
modulation bandwidth. Such a trade-off originates from the nature of optical resonators and
holds for both the silicon-based ones and those with various materials and electrodes [11,
12]. Therefore, there are three major requirements for the modulator: (i) The structure has a
narrow resonance peak, even if the coupling is designed to very strong to achieve high-speed
modulation. (ii) When the resonance peak is shifted, a good extinction ratio can be obtained.
(iii) The position of the resonance peak is sensitive to an applied voltage so that drive
voltage does not have to be too high.
We have modeled the three types of previously proposed microring-based EIT
structures (A, B and C) shown in Figure 4.5. The ERR is compared to them in terms of how
well they can meet the three requirements above, as listed in Table 4.1.
73

Table 4.1 A comparison of ERRs with other ring EIT structures

Modulator Requirements Reference
(i) (ii) (iii)
Structures
A
× × √ [5]
B √ √ × [6]
C × √ √ [7]
ERR √ √ √

Although the three previous structures may not be fully optimized as compared to the
ERRs, the three structures show a very similar EIT-like profile to that given by ERRs, which
is believed to be a relatively fair comparison. For modulation, these EIT structures exhibit
quite different performances. We list the advantages and disadvantages for each of them in
Table 4.1, which shows the ERRs can be better than others.
4.4 Embedded Ring Modulator
ERRs can be used for high-speed digital E/O modulation. When the electrical design
of a microring modulator is improved, the modulation speed is limited in optical domain by
the photon lifetime of the resonator that determines how fast light can be coupled into and
out of the resonator. For a single-ring modulator, weak coupling allows increasing cavity Q
and generating good extinction ratio by applying relatively low voltage, but this limits
modulation speed. For example, in 10 Gb/s modulation, the power coupling coefficient
between the ring and the waveguide has to be ~0.02 (for 5-μm radius) to obtain 10-GHz
linewidth (i.e., cavity Q = 19000).
74

In contrast, an ERR can have a 10-GHz resonance in the EIT-like profile, even if all
power coupling coefficients are up to 0.13 (i.e., cavity Q = ~1500). The narrow profile
results from the interaction of two low-Q resonators, which greatly relaxes the limitation of
modulation speed imposed by the photon lifetime. As shown in Figure 4.6 (a), when a MOS
capacitor is integrated onto the inner ring with carrier transit time of 16 ps, the resonance
peak can be shifted for intensity modulation by applying a voltage of 4.5 V and thus varying
the refractive index of the inner ring [13], which is corresponding to a frequency shift of ~10
GHz of the inner ring (Δm
2
≈ 2×10
-3
). One may not want to drive the two rings at the same
time because this costs more driving power, needs a different drive voltage for each ring and
requires very accurate fabrication of two electrodes. A dynamic model is developed to
simulate the performance of this modulator. Figure 4.6 (b) shows eye-diagrams for 20, 25
and 30 Gb/s intensity modulations respectively, in comparison with a 30 Gb/s signal
generated by a single-ring modulator with the same linewidth and drive voltage. The ERR-
generated 30 Gb/s signal exhibits much larger eye-opening with an extinction ratio of 11.5
dB. Corresponding signal Q-factor is 16.7 dB, which indicates that an error-free detection
(BER<10
-9
) can be obtained. Silicon ERR-based EIT enables digital modulation at even 30
Gb/s with extinction ratio of > 11 dB by applying 4.5-V voltage, which is hardly achieved
using previously reported EIT-like microrings. This ERR-based modulator could be tolerant
to a variation of coupling coefficients. We examine the generated signal quality at 30 Gb/s
when the power coupling coefficient at B area is changed by ±5%, which causes asymmetric
coupling between the two rings. As shown in Figure 4.6 (c), the signal eye-diagram remains
almost unchanged for the variation of the coupling coefficient by 10% in total, and the signal
Q-factor is 16.7, 16.6 and 16.3 dB, respectively. This good stability can be attributed to the
fact that this ERR used here for signal modulation is made highly over-coupled by
75

increasing coupling coefficients, and a relatively small perturbation to coupling coefficients
can hardly change the resonator to under-coupling.

(a)

(b)

(c)
Figure 4.6 (a) An ERR-based EIT effect with strong coupling is used for high-speed modulation. (b)
Signal eye-diagrams at 20, 25 and 30 Gb/s, as compared to the signal generated by a single-
ring modulator with the same linewidth and drive voltage. (c) Signal quality are examined
when the coupling coefficient at B area is changed by ±5%.
To further increase the modulation speed, one can use an asymmetric ERR, as shown
in Figure 4.7. Here, we set κ
11
=0.3, κ
12
=0.21, κ
21
=0.23, and κ
22
=0.13 to make the ring
resonators more over-coupled. Very strong coupling releases the requirement for a low
76

cavity loss. Although this may dramatically widen resonance peak and requires a high drive
voltage as for a single-ring modulator, the spectral profile can still have a sharp edge when
the inner ring’s resonance wavelength is originally blue-shifted relative to the outer ring’s
resonance, as illustrated by a solid line in Figure 4.7.

Figure 4.7 An embedded-ring-based silicon modulator exhibits an EIT-like resonance profile. A
curved waveguide coupler enhances light coupling. The inner ring’s resonance is originally
blue-shifted relative to the outer ring to form a stair-like profile, which is then driven towards
high frequency by applying a voltage.
Applying a drive voltage will shifts the resonance profile again towards high
frequency (dash line), which results in an intensity modulation of the CW light. In our case,
the inner ring radius is 3.1 μm, and this may not allow for a strong coupling if the waveguide
is straight [14]. In Figure 4.7, finite-element simulation shows that the coupling can be
effectively enhanced by curving the waveguides in each coupler. One has to narrow down
waveguide width of the outer waveguide with a smaller curvature for index-matching [14].
A power coupling coefficient is shown to be 0.2, which means that specified coupling
coefficients are achievable in reality. Little of higher order mode is excited in the coupler,
which may require more special waveguide design. With the strong coupling, the cavity loss,
including scattering loss, bending loss and carrier-induced loss, can be as high as 28 dB/cm.
This means that more carrier-induced loss can be affordable, which allows for a higher
77

doping density of carriers in silicon and thus a shorter transit time. In this case, transit time is
16 ps for a MOS capacitor electrode. A dynamic model has been developed to evaluate the
effect of photon lifetime on modulation performance. This is done by re-writing the steady-
state coupled mode equations into a time-dependent form. Optical wave traveling over the
inner and outer rings is discretized in time domain with an individual time step that is equal
to the time that the optical wave takes to go through a half of the inner ring and a quarter of
the outer ring, respectively.
Figure 4.8 shows the generated 40 Gb/s NRZ signal, when the drive voltage is 5.5 V.
We note that extinction ratio is 4.2 dB, limited by the applied voltage and bias condition.
There is a reduced overshoot in the intensity waveform, which is attributed to a stair-like
profile on the right side of the sharp transfer function as show in Figure 4.7. We also
examine the instantaneous frequency chirp, and it is negative with a peak value within ±3
GHz at pulse edges.

Figure 4.8 Generated intensity waveform and frequency chirp in a 40 Gb/s NRZ signal
78

Frequency response of the ERR-based modulator is shown in Figure 4.9, and 3-dB
modulation bandwidth is found to be 28 GHz, which is 0.7 times signal bit rate for 40 Gb/s
NRZ signal generation. This operation speed is partially limited by the carrier transit time in
the MOS electrode.

Figure 4.9 Frequency response of the proposed modulator based on embedded ring resonators. Its 3-
dB bandwidth is 28 GHz
The extinction ratio is examined as a function of the drive voltage, with a 16-ps transit
time. We note that the extinction ratio almost linearly increases from 2.3 to 8.4 dB with the
drive voltage. In this case, the CW wavelength is fixed, which is 1.03 nm away from the
resonance wavelength of the outer ring. The fixed optical bias ensures the signal quality even
if the extinction ratio is lowered to 2.3 dB, otherwise a smaller offset between the CW and
the outer-ring resonance would induce a strong pattern dependence. We believe that such an
extinction ratio would be acceptable to some of communication scenarios where the
microring modulators are expected to find their applications. Those include chip-scale
interconnection, data center communication and local access networks.
79

Laser linewidth may also play an important role in determining the signal quality of
the ring modulators. The signal Q-penalty increases rapidly with the laser linewidth. A
threshold of a 200-MHz linewidth is found for error-free (bit error rate is below 10
-9
) signal
generation. This might serve as a design goal of integrated lasers that are possibly
monolithically fabricated together with the modulators, especially when the size of the lasers
greatly shrinks in the future.
4.5 Embedded Ring Resonators with Distributed Coupling
ERRs can interact with each other by distributed coupling. They exhibit EIT-like
profiles if m
1
-m
2
is odd and are expected to be useful as modulators as well. Concentric
structures have been proposed [13-15] in order to form a single resonator with desired
properties, in which mode distributions in all the rings contain the same number of optical
cycles. In our case, ERRs can have different working regimes. We choose a radius of 2.4 μm
(m
1
=27) for the outer ring and shrink the inner ring. When inner-ring radius is 2.06 μm,
symmetric and anti-symmetric modes are formed as shown in Figure 4.10 (a) and (b). In this
case, the two rings form a single resonator. Field distributions are captured at 5 ps. The
symmetric mode mainly stays in the outer ring and is built quickly (i.e., a low cavity Q),
while the anti-symmetric mode is mostly concentrated in the inner ring with a higher Q. In
contrast, as the inner ring is shrunk, due to the phase difference between the two traveling
modes, each ring becomes an independent resonator that has its own mode number m
2
. There
are (m
1
-m
2
) power-fluctuated areas, separated by solid lines in Figure 4.10 (c) and (d) where
the inner-ring radius is 1.96 and 1.86 μm with m
2
=22 and 21, respectively. Different from a
well-separated mode pattern of the inner ring in Figure 4.10 (c), the inner ring has an almost
uniform field distribution in Figure 4.10 (d).
80

Figure 4.10 Mode distributions in the ERRs with distributed coupling. m
1
=27. (a) and (b) m
1
=23, for
symmetric and anti-symmetric modes. (c) m
1
=22 and (d) m
1
=21 are corresponding to
independent resonator modes.
4.6 Summary
In this chapter, we discuss a new type of optical resonator that consists of embedded
ring resonators. The resonators exhibit unique amplitude and phase characteristics and allow
designing compact filters, modulators and delay elements. A basic configuration of the
ERRs with two rings coupled in a point-to-point manner is discussed under two operating
conditions. An ERR-based microring modulator shows a high operation speed up to 40 GHz.
ERRs with distributed coupling are briefly described as well..

81

4.7 References

[1] A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a
proposal and analysis,” Opt. Lett. 24, 711 (1999).
[2] J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel
propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722
(2002).
[ 3 ] Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-D
microresonator lattice optical filters,” IEEE J. of Quantum Electronics 41, 1410 (2005).
[4] M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12, 2303 (2004).
[5] S. T. Chu, B. E. Little, W. Pan, T. Kaneko and Y. Kokubun, “Second-order filter
response from parallel coupled glass microring resonators,” IEEE Photonics Technol. Lett.
11, 1426 (1999).
[6] D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger and R. W. Boyd, “Coupled-
resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[7] S. J. Emelett and R. A. Soref, “Analysis of dual-microring-resonator cross-connect
switches and modulators,” Opt. Express 13, 7840 (2005).
[8] A. Yariv, “Universal relations for coupling of optical power between microresonators
and dielectric waveguides,” Electron. Lett. 36, 321 (2000).
[9] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator
channel dropping filters,” J. Lightwave Technol. 15, 998 (1997).
[10] I. S. Hidayat, Y. Toyota, O. Torigoe, O. Wada, R. Koga, “Application of transfer matrix
method with signal flow-chart to analyze optical multi-path ring-resonator,” Mem. Fac. Eng.
Okayama Univ., 36, 73 (2002).
[11] T. Sadagopan, S. J. Choi, K. Djordjev, and P. D. Dapkus, “Carrier-induced refractive
index changes in InP-based circular microresonators for low-voltage high-speed
modulation,” IEEE Photon. Technol. Lett. 17, 414-416, (2005).
[12 ] T. Sadagopan, S. J. Choi, S. J. Choi, P. D. Dapkus, and A. E. Bond, "Optical
modulators based on depletion width translation in semiconductor microdisk resonators,"
IEEE Photon. Technol. Lett. , vol. 17, no. 3, pp. 567-569, Mar. 2005.

82

[13] C. A. Barrios and M. Lipson, “Modeling and analysis of high-speed electro-optic
modulation in high confinement silicon waveguides using metal-oxide-semiconductor
configuration,” J. Appl. Phys. 96, 6008 (2004).
[14] Q. F. Xu, D. Fattal, and R. G. Beausoleil, Optics Express, vol. 16, pp. 4309-4315, 2008.
83

Chapter 5 Microring-Based Dispersion Compensator
5.1 Introduction
It has been clear since the late 1990’s that 10-Gbit/s systems covering >100 km will
likely require some form of chromatic dispersion compensation. Of course, higher-bit-rate
systems will necessitate dispersion compensation at much shorter distances. Moreover,
conventionally deployed dispersion compensating fiber tends to provide coarse
compensation, often necessitating some form of receiver-based compensation for the
residual dispersion that might be performed on a per-channel basis. In this scenario in which
the final compensator is in each receiver, issues of size and cost could become quite
important, such that smaller might be less expensive, less power-consuming to tune, and
easier to fabricate and package.
Solutions for chromatic dispersion compensators in the receiver include: (i) chirped
fiber Bragg gratings (FBG), which tend to be several centimeters in length [1], and (ii) all-
pass ring-resonator phase filters, which tend to be a few millimeters in diameter [2, 3]. Such
ring resonators can be single- or multiple-ring configurations, and they tend to assume
negligible loss. Recently, silicon microring resonators with high index contrast have
attracted a great deal of attention and are expected to provide a platform for designing
various functional devices [4, 5, 6, 7]. Many of these devices are very compact, typically in
chip size of tens of μm
2
. To date, there has been little published using very small, 10-micron-
sized ring resonators to achieve high dispersion values, which is potentially for use in
receivers.
84

In this chapter, we show a new microring-based dispersion compensator consisting of
two mutually-coupled microring resonators with unequal radii, in which one is over-coupled
and the other is under-coupled [8]. The combination of the under- and over-coupled rings
greatly improves the linearity of the delay curve in the proposed structure, enabling flat
dispersion and intensity profiles. A -530 ps/nm chromatic dispersion with 10-GHz
bandwidth is achieved over 8.5 μm, resulting in an equivalent dispersion of up to 6.23×10
10

ps/nm/km. We show that both positive and negative dispersions can be designed.
Dispersion-slope compensation can also be achieved by cascading two such structures that
exhibit the opposite dispersions. Dispersion slope of 3150 ps/nm
2
is obtained.
5.2 Principle of Microring Dispersion Compensator
Figure 5.1 (a) shows the structure of the proposed microring-based dispersion
compensator, which consists of an over-coupled ring resonator I and an under-coupled ring
resonator II. The two rings are placed within a double-waveguide configuration, and the
coupling coefficients κ
1
and κ
3
between the rings and waveguides are kept the same, while
the coupling coefficient κ
2
between the two rings is relatively small. Note that the two rings
have a detuned resonance ±Δλ relative to the center wavelength of the compensator. As
shown in Figure 5.1 (b), a single over-coupled ring I alone has a Lorentzian-like delay
profile [8], indicating that the delay curve is far away from a linear function of wavelength.
When adding the under-coupled ring II that has a frequency offset 2Δλ relative to the ring I,
one may note a notch opened in the delay profile, which improves the linearity of the delay
curve between the resonance wavelengths of the two ring resonators.
85

On the other hand, a flat amplitude response is desired. Although scattering loss and
bending loss of such small ring resonators do not allow us to design an all-pass filter in
reality, the spectral loss needs to be flat to minimize the signal distortion that might be
caused by a tilted amplitude profile. In our approach, the radii of the two rings have to be
carefully chosen in order to obtain constant attenuation over the high-dispersion wavelength
band and simultaneously maintain the linearity of the delay curve.
The passive microring dispersion compensator is modeled using 2×2 scattering matrix
method. Coupling loss is ignored. Intrinsic loss is set to be 2.75 dB/cm, which is feasible
with current fabrication technology.

Figure 5.1 (a) Structure of the double-ring dispersion compensator with unequal ring radii. (b) Delay
profile of a single over-coupled ring (dash) and the linearized delay profile in the proposed
structure, which is enabled by employing a combination of over- and under-coupled rings.
We set the radii of the two rings R
1
≈2.5 μm and R
2
≈4.23 μm, which satisfy the
resonance detuning condition: m
1
=26-Δm, m
2
=44+Δm and Δm=0.0008. For each ring,
2πR
i
n=m
i
λ
0
, i=1,2, λ
0
=1550 nm, and n is the effective refractive index in silicon waveguide.
Correspondingly, the power coupling coefficients between the ring and the waveguide and
between the two rings are 0.015 and 0.00004, respectively.
86

(a)

(b)
Figure 5.2 Intensity (solid) and dispersion (dash) profiles. (a) Dispersion of -530 ps/nm is obtained
over 10 GHz bandwidth, and in-band intensity fluctuation is 0.15 dB. (b) Dispersion can be
switched to positive by interchanging the resonance detunings of the two ring resonators.
Similar to a multiple-ring filter [5], the coupling between rings have to be weak in
order to keep the resonance peaks of the two rings close to each other. As shown in Figure
5.2 (a), a large and flat dispersion at -530 ps/nm can be obtained over 10-GHz bandwidth
centered at 1550 nm, and the dispersion fluctuation is 8 ps/nm. We note that the amplitude
response is also flat over this wavelength region, with a loss of 5.2 dB and 0.15-dB intensity
87

flatness. Moreover, one can switch the sign of the dispersion by simply interchanging the
resonance detuning, that is, the resonance wavelength of each ring resonator is detuned in
the opposite directions but by the same amount. As a result, a positive dispersion of 530
ps/nm is achieved while keeping the intensity response almost unchanged, as shown in
Figure 5.2 (b).

Figure 5.3 Dispersion is modified by varying resonance detuning. Trade-off between dispersion and
dispersion bandwidth is shown.
We note that the dispersion can be reduced or enlarged by increasing or decreasing the
resonance detuning. A corresponding modification of the ring radii and the coupling
coefficients is needed. An important issue in designing this type of microring-based
dispersion compensators is the trade-off between dispersion and dispersion bandwidth,
which stems from the well-known trade-off between delay and bandwidth. Since adding the
second ring II mainly improves the linearity of the original single-ring delay profile, the
delay slope (i.e., dispersion) is closely related to the cavity Q-factor of the ring I. As can be
seen in Figure 5.3, the dispersion bandwidth decreases as the dispersion becomes more
negative. We expect that the dispersion-bandwidth-product could be increased by adding
88

more ring resonators, which certainly provides more degrees of freedom in designing.
Moreover, it should be mentioned that we keep the loss coefficient fixed in this case. The
trade-off might be relaxed to a certain extent by optimizing the loss coefficient.
We also show the tunability of the induced dispersion by changing the refractive index
in the ring II. With the same parameters as in Figure 5.2 (a), we only change Δm for the
second ring, which corresponds to a small amount of the refractive index variation, around
2×10
-5
. Accordingly, the dispersion is tuned from -500 to -560 ps/nm, as shown in Figure
5.4. It is noted that the dispersion fluctuation could be larger, which may cause more signal
distortion.

Figure 5.4 Dispersion tunability enabled by refractive index variation. Dispersion fluctuation is also
examined as a function of the varied index.
5.3 Dispersion-Slope Compensation
When looking at Figure 5.2 (a) and (b), one may note that a linear dispersion can be
obtained by cascading a red-shifted negative dispersion and blue-shifted positive dispersion,
which becomes a four-ring configuration. In this case, m
1
=26-0.001, m
2
=44+0.001, and the
89

power coupling coefficients between the ring and the waveguide and between the two rings
are 0.016 and 0.00003, respectively. Each dispersion compensator is then red- or blue-
shifted by 0.2 nm, as shown in Figure 5.5. A linear dispersion profile occurs around 1550
nm, with an increased transmittivity. Power loss is lowered to 1.8 dB.

Figure 5.5 Linearly changed dispersion is obtained by cascading a red-shifted negative dispersion and
blue-shifted positive dispersion. Power loss is decreased to 1.8 dB.

Figure 5.6 Dispersion slope is flat over 10 GHz bandwidth, which is 3150 ps/nm
2

90

We calculate the dispersion slope, as shown in Figure 5.6. Up to 3150 ps/nm
2

dispersion slope is observed over a bandwidth of 10 GHz.
5.4 Summary
In this chapter, we propose ultra-small silicon microring-based dispersion
compensators, with -530 ps/nm dispersion over 8.5 μm and power fluctuation of 0.15 dB.
Equivalent dispersion is up-to 6.23×10
10
ps/nm/km. Dispersion slope compensation is also
achieved.

91

5.5 References

[1] Z. Pan, Y. W. Song, C. Yu, Y. Wang, Q. Yu, J. Popelek, H. Li, and A. E. Willner,
“Tunable chromatic dispersion compensation in 40-Gb/s systems using nonlinearly chirped
fiber Bragg gratings,” J. Lightwave Technol., 20, 2239-2246 (2002).
[2] C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Capuzzo, L. T. Gomez, T. N. Nielsen, and I.
Brener, “Multistage dispersion compensator using ring resonators,” Opt. Lett., 24, 1555-
1557 (1999).
[3] C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti,
“Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,”
Photon. Technol. Lett., 11, 1623-1625 (1999).
[4] C. A. Barrios and M. Lipson, “Modeling and analysis of high-speed electro-optic
modulation in high confinement silicon waveguides using metal-oxide-semiconductor
configuration,” J. Appl. Phys., 96, 6008-6015 (2004).
[5] S. Xiao, M. H. Khan, H. Shen, and M. Qi, “A highly compact third-order silicon
microring add-drop filter with a very large free spectral range, a flat passband and a low
delay dispersion,” Opt. Express, 15, 14765-14771 (2007).
[ 6 ] C. Li, L. Zhou, and A. W. Poon, “Silicon microring carrier-injection-based
modulators/switches with tunable extinction ratios and OR-logic switching by using
waveguide cross-coupling,” Opt. Express, 15, 5069-5076 (2007).
[7] L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner,
“Microring-based modulation and demodulation of DPSK signal,” Opt. Express, 15, 11564-
11569 (2007).
[8] Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt.
Soc. Am. B, 20, 2125-2132 (2003).
92

Chapter 6 High Dispersion Slot Waveguide
6.1 Introduction
In preceding chapters we have been focused on micro-resonators and their
applications in chip-scale optical communications. Starting from this chapter, we direct our
discussions to integrated waveguides, especially to slot waveguides. In fact, the microring
resonators are still formed by waveguides, and thus some of results presented in this and the
following chapters could be applied to microring resonator devices. Another direction
modification is that the application scenario is more about optical signal processing, even
though it is believed to be communication-oriented signal processing.
The dispersive elements could also be essential in various signal-processing functions,
such as analog-to-digital conversion [1], tunable delays [2, 3] and optical correlation [4].
Moreover, there is the need for integrated receiver-based compensation for the chromatic
dispersion that may be performed on a per-wavelength-channel basis. Reports of integrated
dispersion elements include: (i) ring resonators that have bandwidths on the order of tens of
GHz [5, 6], (ii) wideband silicon waveguides that have dispersions of ~4000 ps/nm/km [7,
8], and (iii) sidewall-modulated Bragg gratings that exhibit a large dispersion and good
bandwidth in reflection port [9]. A laudable goal would be to design on-chip dispersion
compensators that are capable of producing a large and tailorable dispersion [10].
Recently, slot waveguides have been proposed [ 11 ] for on-chip applications,
exhibiting enhanced birefringence [12], nonlinearity [13] as well as improved modulation
efficiency [ 14 ]. A slotted structure could provide some design freedom to tailor the
93

waveguide dispersion [ 15 ]. A potential application that would make use of the slot
waveguides would be to form on-chip highly dispersive elements with a large dispersion-
bandwidth product.
In this chapter, we propose a dispersive slot waveguide based on a strongly
wavelength-dependent coupling between a slot waveguide and a strip waveguide. On a SOI
platform, the two waveguides are vertically coupled, producing a strong negative dispersion
due to an anti-crossing of the two modes [16, 17]. Simulations show that a negative
dispersion of up to -181520 ps/nm/km is obtained. Dispersion compensation for a 320 Gb/s
RZ signal has an eye-opening penalty of ~4 dB after 10.87-km single-mode fiber
transmission. The dispersion bandwidth is greatly increased to147 nm for a dispersion of -
31300 ps/nm/km, with a <1% variance, which corresponds to a 6.3-ns tunable delay given by
a 1-m-long slot waveguide.
6.2 Narrowband Highly Dispersive Slot Waveguide
The proposed structure consists of a strip waveguide and a slot waveguide that are
vertically coupled to each other. The effective index of quasi-TM mode (vertically polarized)
in the strip waveguide decreases with wavelength faster than that in the slot waveguide, and
a strong mode coupling occurs around a certain wavelength where the effective indices are
close to each other, as conceptually shown in Figure 6.1. At the crossing-point, formed
composite modes, including symmetric and anti-symmetric modes, experience a sharp
transition of mode shape from short to long wavelength in Figure 6.1, which induces a high
dispersion [16, 17]. The strip waveguide with a thickness of 255 nm and width of 500 nm is
placed on the top of the slot waveguide, separated by a 500-nm-thick silica base layer. The
94

low-index slot is a 40-nm silica layer, surrounded by two 160-nm silicon layers. Such a
vertical placement allows for more accurate control of the structural parameters during
fabrication and reduced propagation loss to several dB/cm [18]. Since the symmetric mode is
used, we have to design a mode converter to excite it [16]. At crossing wavelength 1.489-
μm, the index difference of the symmetric and anti-symmetric modes equals 0.019, which
produces the coupling length from a slot mode to a strip mode for 100% coupling of 39.2
μm. To excite the symmetric mode, ~50% coupling is needed, and mode converter length is
~20 microns. We use a mode solver based on finite-element algorithm to obtain effective
index with an element size equal to 3, 5, and 15 nm for slot, silicon and other parts. Material
dispersion is considered using Sellmeier equations for both silicon and silica, and dispersion
profiles are calculated from the effective index [15].

Figure 6.1 Slot and strip modes strongly interact with each other due to index-matching at the crossing
point, producing a sharp index change of symmetric and anti-symmetric modes. Modal
power distributions of the symmetric mode at different wavelengths.
At the crossing wavelength 1.489-μm shown in Figure 6.2, the symmetric mode has -
181520 ps/nm/km dispersion. It is obtained over a relatively small wavelength range of 3.5
nm for a 1% dispersion variation. The anti-symmetric mode has almost the same amount of
positive dispersion.
95

Figure 6.2 Dispersion profiles of the symmetric and anti-symmetric modes, and a negative dispersion
of -181520 ps/nm/km can be obtained from the symmetric mode.
Keeping all other parameters the same, we show in Figure 6.3 (a) that an increment in
slot thickness (ST) shifts the dispersion profile towards a longer wavelength, and the
symmetric mode becomes less dispersive. Figure 6.3 (b) shows that dispersion peak
wavelength almost linearly shifts by 77 nm, as the slot increases from 40 to 60 nm.
Accordingly, the dispersion value decreases by 38%. This can be attributed to the fact that
the increased ST lowers the effective index of the slot mode and red-shifts the crossing
point, where the indices of the two guided modes have closer slopes over wavelength. This
causes a less dispersive symmetric mode. The silicon layers surrounding the slot also modify
the dispersion properties of the proposed structure. In Figure 6.3 (c), an increment (from 150
to 170 nm, with a 56-nm slot) in the thickness of the silicon layers blue-shifts the dispersion
by 103 nm and makes dispersion more negative, from –79683 to –182800 ps/nm/km. It is
important to mention that the dispersion profile becomes wider, while the symmetric mode is
less dispersive.
96

(a)

(b)

(c)
Figure 6.3 (a) The dispersion profile red-shifts with a small peak value as the slot thickness increases.
(b) Dispersion value and peak wavelength are examined as functions of the slot thickness. (c)
Dispersion and peak wavelength are examined as functions of the silicon-layer thickness.
97

Although changing the ST or silicon-layer thickness can slightly modify the peak
value of dispersion, it might be desirable to dramatically change dispersion for various
applications. The thickness of the silica base between the strip and slot waveguides plays a
critical role in tuning the dispersion while almost keeping the dispersion peak wavelength. In
Figure 6.4 (a), the dispersion decreases greatly from -181520 to -28473 ps/nm/km, as the
base thickness varies from 500 to 200 nm. To keep the same peak wavelength, one has to
change the thickness of the strip waveguide to 255, 255.5, 257 and 259 nm for the silica base
of 500, 400, 300 and 200 nm, respectively. As the base decreases, effective indices of the
slot and strip waveguides increase, causing a small shift of the crossing wavelength. The
strip thickness is thus changed to balance this effect. We note a trade-off [16] between peak
dispersion and dispersion’s full width at half maximum (FWHM), from Figure 6.4 (b). The
dispersion’s FWHM drops from 177 to 26 nm as the waveguide is more dispersive. A
thicker base helps separate two modes well, and thus the strong interaction between them
occurs at a smaller wavelength band where their effective indices are very close to each
other. The symmetric mode thus experiences a sharper transition from the strip mode to slot
mode and higher dispersion.
98

(a)

(b)
Figure 6.4 (a) The dispersion profile is fixed at the same wavelength as the dispersion’s peak value is
changed from -181520 to -28473 ps/nm/km by varying the silica base thickness. (b) A trade-
off is found between the dispersion peak value and dispersion bandwidth.
It may be desirable to design a dispersion-shifted profile without changing its peak
value and bandwidth. One can achieve this by thickening both the slot and the silica base. A
thickened slot makes the dispersion profile red-shifted but reduces its peak value as well,
while a larger silica base makes the dispersion deeper and mostly keeps its peak wavelength.
In Figure 6.5, the dispersion profile is shifted relative to the original 40-nm slot by 37 and 77
99

nm, with ST changed to 48 and 60 nm, respectively. The base thicknesses are 500, 555 and
606 nm, and the maximum dispersion values are -181520, -182585, and -182028 ps/nm/km,
respectively.

Figure 6.5 Dispersion shifts with different slot thicknesses, exhibiting almost unchanged dispersion
value and bandwidth.
Figure 6.6 shows that shrinking the waveguide width from 500 to 200 nm can also
vary the dispersion, when other parameters are kept the same as in Figure 6.2. However, we
note that the dispersion is less sensitive to the width change as compared to changing the
vertical dimensions. This might be because the fact that horizontal light confinement for the
y-polarization state is not as tight as in the vertical direction for the 500-nm width. Later, I
will show how to use the dispersion change induced by varying the waveguide width to
obtain ultra-wideband and flattened dispersion profile.
100

Figure 6.6 Dispersion properties change with the waveguide width.

Figure 6.7 Dispersion compensation for very high-speed signals transmitted over 11.4-km single
mode fiber. Eye-opening penalty increases with bit rate. Eye-diagrams are in the same scale.
We simulate the dispersion compensation of high-speed RZ on-off-keying signals
after 10.87-km single-mode fiber transmission, using a 1-m-long waveguide. Data rate
ranges from 160 to 400 Gb/s. The waveguide width is 500 nm, and the strip waveguide
thickness is 255 nm. The slot is 48-nm thick, surrounded by two 160-nm silicon layers. The
silica base layer is 555 nm. Signal carrier is aligned to the peak dispersion wavelength,
where the single mode fiber has a dispersion of 14.4 ps/nm/km. In Figure 6.7, the dispersion
101

compensator works for the 160 Gb/s signal with almost no penalty. However, with the
increased bit rate, the signals suffer from the third-order dispersion in the fibers, which
would not be compensated by the waveguide in this case. An eye-opening penalty of 4 dB is
induced to a 320 Gb/s RZ signal, relative to the case at 160 Gb/s.
6.3 Wideband Highly Dispersive Slot Waveguide
Broadband and strong dispersion could be useful in both telecom systems and optical
signal processing, e.g., for achieving multi-channel dispersion compensation or a tunable
optical delay [2, 3]. Dispersion can be flattened by a cascade of the waveguide sections with
the modified structural parameter, each with a shifted dispersion profile. As an example,
Figure 6.8 shows that the strip waveguide has a slightly tailored waveguide thickness (WT)
by depositing silicon, and the dispersion curve shifts over wavelength, as shown in Figure
6.9. Similar trend can be seen in Figure 6.3 (a) and Figure 6.6 as other structural parameters
are changed.

Figure 6.8 Waveguides with variable waveguide thickness or width are cascaded to obtained
wideband flattened strong dispersion.
102

Figure 6.9 The dispersion shifts over wavelength by changing WT.
The length of each modified waveguide section is calculated by solving the following
linear equations:












=












⋅












0
0
0
2
1
2 1
2 2 2 2 1
1 1 2 1 1
) ( ) ( ) (
) ( ) ( ) (
) ( ) ( ) (
D
D
D
c
c
c
D D D
D D D
D D D
n n n n n
n
n
⋮ ⋮
…
⋮ ⋱ ⋮ ⋮
…
…
λ λ λ
λ λ λ
λ λ λ

where D
i
(λ) (i=1, 2, …, n) is the dispersion profile of the i
th
waveguide section; D
0
is the
desired dispersion value after flattening. Length coefficients c
1
, c
2
, … , and c
n
are solved to
determine the length ratio of each modified section to the total waveguide. This forms a
dispersion profile with n dispersion values clamped to D
0
at wavelengths λ
1
, λ
2
, … , and λ
n
.
The waveguide width is 500 nm. The slot is 40-nm thick and surrounded by two 150-
nm-thick silicon layers. The silica base is 500 nm. There are six waveguide sections. The
shifted dispersion profiles are shown, when WT are 246, 249, 252, 255, 259 and 265 nm,
103

respectively. It is important to choose the WT values according to the clamping wavelengths
to make the overall dispersion as flat as possible. The length ratios are 15%, 11%, 12%,
12%, 17% and 33% accordingly. In Figure 6.10, a dispersion of -31300 ps/nm/km is
obtained over 147 nm bandwidth, with a variance of 305.5 ps/nm/km, <1% of the mean
dispersion. Such dispersive media could be used to introduce a tunable optical delay of 6.3
ns/m for on-chip signal processing by converting wavelength from 1450 to 1680 nm, as
shown in Figure 6.10.

Figure 6.10 Flat dispersion of -31300 ps/nm/km over 147 nm. 6.3 ns/m tunable delay can be obtained
by 230-nm wavelength conversion..
Flat dispersion can also be obtained by varying the waveguide width. With the same
vertical dimensions of the slot-waveguide and silica base, we keep WT=255 nm and vary the
width from 565 to 500, 445, 390 and 340 nm, and flat dispersion of -46100 ps/nm/km is
from 1473 to 1564 nm as shown in Figure 6.11. The length ratios are 26%, 17%, 17%, 7%
and 33%, respectively, with a variance is 623 ps/nm/km, <1.4% of the mean dispersion.
Varying the waveguide width is more fabrication friendly and can be easily realized by
lithography.
104

Figure 6.11 Flat dispersion of -46100 ps/nm/km over 91 nm. 6 ns/m tunable delay can be obtained by
100-nm wavelength conversion.
The dispersion bandwidth is extendable if adding more sections. The overall
dispersion can be tailored in other ways, e.g., changing with wavelength linearly when the
structural parameters are chosen appropriately. Nevertheless, it is still challenging in current
fabrication technology to make signal loss in a long waveguide (a few meters) acceptably
low. It might be inevitable to adopt optical amplification in the waveguides to mitigate this
problem.
6.4 Summary
In this chapter, we analyze a slot-waveguide with high dispersion, in which a slot
waveguide is coupled to a strip waveguide. A negative dispersion of up to -181520
ps/nm/km is obtained due to a strong interaction of the slot and strip modes. A flat and large
dispersion is achievable by cascading the dispersive slot-waveguides with varied waveguide
thickness or width for dispersion compensation and signal processing applications. We show
-31300 ps/nm/km dispersion over 147-nm bandwidth with <1% variance.
105

6.5 References

[ 1 ] A. S. Bhushan, F. Coppinger, and B. Jalali, "Time-stretched analogue-to-digital
conversion," Electron. Lett. 34, 1081-1083 (1998).
[2] Y. Wang, C. Yu, L. -S. Yan, A. E. Willner, R. Roussev, C. Langrock, and M. M. Fejer,
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107

Chapter 7 Low Dispersion Slot Waveguide
7.1 Introduction
Integrated highly nonlinear waveguides and photonic nano-wires can form the
backbone of on-chip optical signal processing [1, 2, 3], especially for high bit-rate signals
[4]. Such waveguides can be composed of silicon [1, 2, 3, 5, 6, 7, 8], silicon nitride [9, 10],
Si nano-crystals (Si-nc or Si-rich Si dioxide) [11, 12], III-V compound semiconductors [13,
14], and chalcogenide glasses [4, 15, 16]. Flat and low chromatic dispersion over a wide
wavelength range is crucial for enhancing nonlinear interactions of optical waves.
Recently, there has been much interest in tailoring dispersion in the waveguides [17,
18, 19, 20, 21]. Due to high index contrast and tight light confinement, waveguide dispersion
is strong compared to material dispersion in the silicon waveguides, which means that one
can modify some geometry parameters to tailor the overall dispersion. In fact, the strong
waveguide dispersion may change rapidly over wavelength, which makes it difficult to
obtain a wideband and flat dispersion profile. A slotted waveguide structure [22] could
provide extra design freedom to tailor the waveguide dispersion [23, 24, 25, 26, 27], while
keeping a large fraction of the guided mode in a thin slot layer. However, there has been
little reported on the wideband low dispersion in these waveguides.
In this chapter, we first present highly nonlinear slot waveguides for achieving a flat
and near-zero dispersion profile. Two types of nonlinear materials are considered: (i)
chalcogenide glass (As
2
S
3
), which reduces the overall two-photon-absorption (TPA)
coefficient and (ii) Si-nc, which greatly increases the nonlinear Kerr coefficient compared to
108

silicon strip waveguides. The As
2
S
3
slot waveguide exhibits a flat dispersion of 0±0.17
ps/(nm·m) over a bandwidth of 249 nm, with a nonlinear coefficient of 160 /(W·m) and a
significantly improved nonlinear figure of merit (FOM). The silicon-nc slot waveguide
features a flat dispersion of 0±0.16 ps/(nm·m) over a bandwidth of 244 nm, with a high
nonlinear coefficient of 2874 /(W·m) at 1550 nm.
We then propose and analyze a silicon strip/slot hybrid waveguide that generates low
dispersion of 0 ± 16 ps/(nm·km) over a 553-nm wavelength range, which is more than 20
times better than previous results in terms of dispersion flatness. Different from previously
reported slot waveguides, the proposed waveguide has a critically important feature, that is,
mode transition from a strip mode to a slot mode, which achieves flattened dispersion by
introducing additional slightly negative and broadband waveguide dispersion. A dispersion
profile with three ZDWs is, for the first time, obtained in silicon waveguides. We also show
that dispersion slope and the average value of dispersion can be tailored almost
independently by modifying different geometry parameters. The waveguide exhibits
flattened dispersion from 1562-nm to 2115-nm wavelength, which is potentially useful for
both telecom and mid-infrared applications.
7.2 Slot Waveguide for Flat and Low Dispersion
As shown in Figure 7.1, a horizontal slot is surrounded by two silicon layers with air
cladding, and waveguide substrate is 2-μm buried oxide. For the quasi-TM mode (vertically
polarized), due to the discontinuity of its electric field at the interfaces of the slot and the
silicon layers, a large fraction of the guided mode is confined in the slot layer. The effective
index of quasi-TM mode as a function of wavelength is obtained by using a finite-element
109

mode solver (COMSOL Multiphysics 3.4), with an element size of 5, 40, and 100 nm for
slot, silicon and other regions, respectively. Material dispersion is taken into account for the
slot (As
2
S
3
[28] and Si nano-crystal [25]), silicon [29] and silica substrate. Group velocity
dispersion, D = -(c/λ)·(d
2
n/dλ
2
), is calculated, where n is the effective index of refraction,
and c and λ are the speed of light and wavelength in vacuum, respectively.

Figure 7.1 Slot waveguide with silicon layers surrounding a highly nonlinear slot layer.
We note that the Kerr nonlinear index of refraction n
2
and TPA coefficient β
TPA
,
corresponding to the real and imaginary parts of nonlinear coefficient γ respectively, vary
with wavelength [30, 31]. For silicon, the measurement results given in Refs. 30 and 31 are
fitted using six-order polynomials and averaged to take the dispersion of nonlinearity into
account, when we calculate γ as a function of wavelength. For As
2
S
3
, different n
2
and β
TPA

values have been reported [32, 33, 34, 35, 36] at individual wavelengths, and the measured
n
2
is 2.5×10
-18
[32], (averaged) 3×10
-18
[33, 34], 2.93×10
-18
[35], and 3.05×10
-18
m
2
/W [36]
at 1.064, 1.31, 1.54 and 1.57 μm, respectively, which is roughly unchanged. We thus choose
constant n
2
= 3×10
-18
m
2
/W and β
TPA
= 6.2×10
-15
m/W for our simulations at all wavelengths.
For Si nano-crystal with 8% silicon excess, annealed at 800
o
C, we choose n
2
= 4.8×10
-17

110

m
2
/W and β
TPA
= 7×10
-11
m/W at 1550 nm [11]. For silica, n
2
= 2.6×10
-20
m
2
/W is used, and
β
TPA
is neglected. Nonlinear coefficient γ is computed with a space step of 1 nm using a full-
vector model [37], in which the contributions of different materials to nonlinearity are
weighted by optical mode distribution, and the longitudinal electric field of the mode is also
considered. We use a FOM defined as γ's real part divided by γ's imaginary part times 4π,
i.e., γ
re
/4πγ
im
. In a scalar model with a single nonlinear material, γ
re
= 2πn
2
/λA
eff
and γ
im
=
β
TPA
/2A
eff
, where A
eff
is effective mode area. The FOM that we use here is equivalent to the
widely used FOM n
2
/λβ
TPA
[38].
7.2.1 Chalcogenide Slot Waveguide
For As
2
S
3
slot waveguides, we set W = 280 nm, H
u
= H
l
= 180 nm, and H
s
= 115 nm.
Figure 7.2 (a) shows a flat dispersion profile within 0±0.017 ps/nm obtained from 1460 to
1709 nm wavelength (bandwidth of 249 nm) for a 10-cm-long waveguide on chip. There are
two zero dispersion wavelengths (ZDWs), located at 1500 and 1677 nm, respectively. The
maximum dispersion of 0.1695 ps/(nm·m) occurs at 1595-nm wavelength. The flat and low
dispersion is obtained by employing slot structures. As shown in Ref. 27, the effective index
of a slot waveguide can change rapidly with wavelength, and this makes the effective index
of the slot mode closer to that of a substrate mode at long wavelengths (e.g., around 2100 nm
in this case), causing mode coupling and negative dispersion [27]. Thus, the total dispersion
curve is bent and becomes small and flat at the wavelength of interest near 1550 nm.
Varying slot height can greatly increase the dispersion. Figure 7.2 (a) shows an increase in
the dispersion peak value from 0.0971 to 0.3301 ps/(nm·m) as the slot height decreases from
120 to 105 nm. Accordingly, dispersion peak blue-shifts from 1600 to 1580 nm.
111

(a)

(b)

(c)
Figure 7.2 Dispersion profiles in chalcogenide slot waveguides with different (a) slot heights, (b)
waveguide widths, and (c) upper silicon heights.
112

Dispersion can be tailored by changing the waveguide width, as shown in Figure 7.2
(b). Widening the waveguide shifts a dispersion curve to longer wavelength. The right ZDW
red-shifts by 111 nm, from 1677 to 1788 nm as the waveguide width is changed from 280 to
310 nm, while the left ZDW blue-shifts by only 32 nm. Changing H
s
and W together enables
tailoring the center wavelength and ZDWs while keeping flat and low dispersion. For
example, one can start with the structure parameters given above and then increase both H
s

and W. These produce a red-shift of dispersion curves while maintaining the low peak value
of dispersion. Figure 7.2 (c) shows the tailored dispersion as the upper Si height H
u
is
reduced from 190 to 160 nm. It is noted that the right ZDW has a shift towards long
wavelength by 65 nm, larger than the left ZDW. The peak dispersion value is relatively
tolerant to the change of H
u
, and the dispersion flatness is improved by increasing H
u
.
We examine the nonlinear coefficient γ and FOM as a function of wavelength with
varied slot height in Figure 7.3 (a). For H
s
= 115 nm, γ decreases from 178.3 to 135.6 /(W·m)
as wavelength increases from 1400 to 1800 nm. A similar trend is found for other slot
heights. In contrast, the FOM increases with wavelength from 0.87 to 1.52 for H
s
= 115 nm.
This is explained as follows. First, the dispersion of nonlinearity in silicon is considered.
From 1400 to 1800 nm, Si material FOM defined as n
2
/λβ
TPA
increases from 0.252 to 0.739.
Second, we assume that n
2
and β
TPA
in As
2
S
3
do not change with wavelength, so As
2
S
3

material FOM decreases with wavelength. Third, the material index and mode distribution
change with wavelength, and contributions of different materials to FOM depend on
wavelength. With H
s
of around 120 nm and a smaller index contrast between Si and As
2
S
3

(compared to Si and Si-nc), the field enhancement in the As
2
S
3
slot is less than that shown
later for an 120-nm Si-nc slot, and contribution of silicon layers to FOM is more than that of
113

the As
2
S
3
slot. This is why the overall FOM increases with wavelength. The third factor has
limited effect. For a larger H
s
, more power is confined in the As
2
S
3
slot, and the FOM value
becomes higher. There are similar changes in γ and FOM versus wavelength in Figure 7.3
(b), with the waveguide width changed, but the FOM is insensitive to the width change.

(a)

(b)
Figure 7.3 For chalcogenide slot waveguides, nonlinear coefficient γ and FOM are examined over
wavelength with different (a) slot heights and (b) waveguide widths, respectively.

114

7.2.2 Silicon Nano-crystal Slot Waveguide

(a)

(b)
Figure 7.4 For Si nano-crystal slot waveguides, (a) dispersion profiles change with slot height. (b).
Dispersion profile red-shifts as lower silicon height increases.
For silicon nano-crystal slot waveguides, we choose W = 500 nm, H
u
= H
l
=180 nm,
and H
s
= 47 nm. Figure 7.4 (a) shows a flat dispersion profile within 0±0.16 ps/(nm·m)
obtained over a 244-nm wavelength range, from 1539 to 1783 nm. There are two ZDWs at
1580 and 1751 nm, respectively. The peak dispersion of 0.156 ps/(nm·m) is found at 1670
nm. Figure 7.4 (a) shows that the dispersion peak value is decreased from 0.2101 to 0.0508
ps/(nm·m) as slot height Hs varies from 46 to 49 nm, at a rate of 0.053 ps/(nm·m) per nm.
115

We note that, as compared to As
2
S
3
slot waveguides, the Si nano-crystal slot waveguides
have a larger index contrast between the slot and silicon layers and a smaller slot height, and
this causes stronger field enhancement in the slot. The overall dispersion is dominated by
waveguide dispersion, which results in the high sensitivity of dispersion to the slot height.
We examine the dispersion change caused by increasing the lower silicon height H
l
, as
shown in Figure 7.4 (b). The thicker the lower silicon layer, the flatter the dispersion profile
near its peak value. The dispersion curve is red-shifted for a larger H
l
. The right ZDW shifts
by 106 nm, from 1696 to 1802 nm as H
l
is changed from 170 to 190 nm. Generally, similar
trends of dispersion tailoring are found for the As
2
S
3
and Si nano-crystal slot waveguides as
a structural parameter is changed, so we do not show dispersion profiles with varied W and
H
u
repeatedly.

Figure 7.5 For 10-cm Si nano-crystal slot waveguides, dispersion sensitivity changes with the slot
height.
However, for a smaller slot height, dispersion is more sensitive to a change in the slot
height. As an example, Figure 7.5 shows dispersion profiles modified by a 10-nm change of
H
s
in a 10-cm-long Si nano-crystal slot waveguide when H
s
= 40, 80 and 120 nm, (W = 500
nm, H
u
= H
l
=180 nm), and accordingly dispersion value is changed by 0.698, 0.339, and
116

0.195 ps/(nm·m) at 1650-nm wavelength. A small H
s
induces strong field enhancement
shown in Figure 7.5 insets, and light is tightly confined in a small area, which causes an
increased dispersion sensitivity to the slot height.
Calculated nonlinear coefficient γ and FOM with the slot height of 47 nm are 2874
/(W·m) and 0.447, respectively, at 1550-nm wavelength. A small change in the slot height,
from 46 to 49 nm, does not change γ and FOM much. Due to the strong field enhancement in
the slot, FOM is dominated by the material properties of the Si-nc slot. This is confirmed by
noting that silicon's material FOM is 0.352 at 1550-nm wavelength, but the Si-nc’s material
FOM n
2
/λβ
TPA
is 0.4424, which is close to the computed FOM. Due to the lack of
measurements on the dispersion of nonlinearity in Si-nc, γ is not calculated as a function of
wavelength.
7.3 Slot/Strip Waveguide for Flat and Low Dispersion

Figure 7.6 (a) Previously reported silicon slot waveguide that exhibits convex dispersion profile. (b) A
strip waveguide is added to produce additional negative waveguide dispersion. (c) The
strip/slot waveguide coupler is flipped to improve sequent fabrication procedure. (d) A
strip/slot hybrid waveguide with spacing layer removed for achieving flattened dispersion.
Although previous work has shown that relatively low dispersion can be obtained with
a chalcogenide or Si-nc slot as shown in Figure 7.6 (a), there is always a convex dispersion
profile with a dispersion variation of >300 ps/(nm·km) over about 250-nm bandwidth.
Further flattening the dispersion requires introducing additional waveguide dispersion,
117

which is broad and slightly negative in order to compensate for the existent convex profile.
As discussed in the preceding chapter, strong negative dispersion can be produced by an
anti-crossing effect due to mode coupling from a strip mode to a slot mode in a strip/slot
waveguide coupler. Figure 7.6 (b) shows such a coupler, in which a strip waveguide is
integrated on the top of a slot waveguide. The peak value and bandwidth of the negative
dispersion are controllable by modifying the spacing between the two waveguides. The SiO
2

spacing layer here has to be reduced or even removed to broaden the newly induced
waveguide dispersion and make it slightly negative. However, a problem in doing this is that
one end up having a thick top silicon layer, which is likely to be lossy amorphous silicon
deposited in fabrication process. We thus flip the strip and slot waveguides and then remove
the SiO
2
spacing layer as shown in Figure 7.6 (c) and (d).
Finally, in the proposed structure, the lower silicon layer of the slot waveguide and the
silicon strip merge. Although it looks like a slot waveguide, we would emphasize that this is
essentially a strip/slot hybrid waveguide according to the light-guiding physics. At short
wavelengths, the low-index slot layer serves as a barrier to confine most of light within the
lower silicon part, which forms a strip mode. As the wavelength becomes longer, light
extends more to the upper silicon part, and the electric field in the slot is enhanced to form a
mode that is more like a slot mode. Such mode transition, although subtle, is in principle
similar to the mode coupling described above and produces additional shallow and concave
waveguide dispersion to compensate for the existent convex dispersion. This light-guiding
mechanism differentiates the proposed waveguide from previously reported slot waveguides.
We choose the following structural parameters: upper silicon height H
u
= 265 nm, slot
height H
s
= 50 nm, lower silicon height H
l
= 510 nm, and waveguide width W = 500 nm. The
118

silica substrate is 2-μm-thick. A quasi-TM mode (vertically polarized) forms a slot mode.
The slot material is silicon nano-crystal with silicon excess of 8%, annealed at 800
o
C. It
exhibits relatively high nonlinearity compared to samples with higher silicon excess.

Figure 7.7 Flattened dispersion profile within 0 ± 16 ps/(nm·km) is obtained from 1562 to 2115 nm.
The corresponding dispersion coefficient β2 and modal field distributions are shown.
Figure 7.7 shows the calculated dispersion for the proposed slot waveguide with the
parameters mentioned above, and the corresponding dispersion coefficient β
2
is also
presented. Flat dispersion of 0 ± 16 ps/(nm·km) is obtained over a 553-nm wavelength range,
from 1562 to 2115 nm, and β
2
is within 0 ± 0.04 ps
2
/m. Three ZDWs are located at 1595,
1828, and 2062 nm. We show the modal distributions at 1550, 2150, and 2750 nm in Figure
7.7. At a short wavelength, the mode looks like a quasi-strip mode and remains mainly in the
lower silicon part, while it starts to transition to a quasi-slot mode at long wavelengths. This
confirms our explanation on the principle of dispersion flattening. The contributions of
silicon and Si-nc to the overall dispersion vary with wavelength due to the mode transition.
The optical power within the slot increases from 2.2 to only 10.3 percent of the total power
as the wavelength is increased from 1400 to 2200 nm. This means that one could replace the
Si-nc slot with other materials with similar refractive index such as SiO
2
, keeping the low
and flat dispersion.
119

(a)

(b)
Figure 7.8 In silicon strip/slot hybrid waveguides, dispersion profiles are tailored by changing (a) slot
height and (b) waveguide width, respectively.
We now examine the dependence of the dispersion properties on the slot height, H
s
,
with H
u
= 265 nm, H
l
= 510 nm, and W = 500 nm. Figure 7.8 (a) shows that, as H
s
increases
from 44 to 56 nm, the dispersion curve rotates, with a nearly zero dispersion value at around
1866 nm. In this way, one can modify the third order dispersion (i.e., dispersion slope),
without changing the average dispersion much. Another important parameter is the
waveguide width. With H
u
= 265 nm, H
s
= 50 nm, and H
l
= 510 nm, we calculate the
dispersion curves for different widths ranging from 420 to 540 nm. As seen in Figure 7.8 (b),
120

the dispersion at a long wavelength is more sensitive to the width and becomes more
negative, with the local minimum value shifting from a wavelength of 1902 nm to 2025 nm
as the width decreases. Dispersion flatness is thus destroyed. This is because the decreased
width causes a reduced effective index of the guided mode and makes it more likely to be
coupled with the substrate mode, and such a mode transition also results in negative
dispersion at longer wavelengths.

(a)

(b)
Figure 7.9 In silicon strip/slot hybrid waveguides, dispersion profiles are tailored by changing (a)
lower silicon height and (b) upper silicon height, respectively.
121

The influence of the lower silicon height, H
l
, on the dispersion is also examined, with
H
s
= 50 nm, H
u
= 265 nm, and W = 500 nm, as shown in Figure 7.9 (a). The dispersion
profile is almost entirely moved up without a dramatic change in its shape and slope, as the
lower silicon height H
l
is increased from 490 to 530 nm. At 1850-nm wavelength, the
dispersion value increases from –104 to 80 ps/(nm·km), with a rate of 45 ps/(nm·km) per 10
nm. As mentioned above, the dispersion is flattened by the mode transition. When the lower
silicon height is decreased, the effective index of the strip mode drops with wavelength more
rapidly, which causes a sharper mode transition from the strip mode to the slot mode and
over-balanced dispersion, and moves the whole dispersion curve down. The averaged
dispersion value can be changed more effectively by modifying the upper silicon height H
u
.
When keeping H
s
= 50 nm, H
l
= 510 nm, and W = 500 nm, we change H
u
from 235 to 295
nm. As shown in Figure 7.9 (b), the dispersion curve is moved down, and the dispersion
value at 1850-nm wavelength is reduced from 162 to -153 ps/(nm·km), with a rate of 52
ps/(nm·km) per 10 nm. In this case, the dispersion slope is slightly changed.
Although the goal of this paper is to show a flattened near-zero dispersion profile, we
would emphasize that this just serves as an example of dispersion tailorability enabled by the
mode transition in the proposed strip/slot hybrid waveguide. For a specific application, a
perfectly flat dispersion may not be most desirable, since many other important parameters
such as the nonlinear coefficient and loss are not as flat as the dispersion over wavelength.
We find that an appropriate combination of the slot height and lower silicon height provides
a very effective way to tailor the dispersion profile, which is critically important for
manipulating ultrafast pulse dynamics and nonlinear signal processing in silicon photonics.
More sophisticated dispersion tailoring (e.g., designing 4th- and 5th-order dispersion terms)
122

may be needed for a specific application, in which case advanced optimization algorithm
such as generic algorithm could be used.
To calculate the nonlinear coefficient, we choose a nonlinear Kerr index n
2
= 4.8×10
-
17
m
2
/W and TPA coefficient β
TPA
= 7×10
-11
m/W for Si-nc with 8% silicon excess, annealed
at 800
o
C [11], along with n
2
= 4.06×10
-18
m
2
/W and β
TPA
= 7.45×10
-12
m/W for silicon at
1550 nm [30, 31]. To obtain the parameters in silicon, we fit the measurement results in
Refs. 30 and 31 using six-order polynomials and average them. Using a full-vector model
[37], the nonlinear coefficient γ in high-index-contrast waveguides is computed with
contribution of both transverse and longitudinal field components. We obtain γ = 134+j28
/(W·m) at 1550-nm wavelength, and the imaginary part of γ characterizes the TPA property
of the waveguide.
The mode transition to a quasi-slot mode at long wavelengths helps lift the guided
mode up, which is beneficial to reduce the loss by substrate leakage. The proposed
waveguide is multi-mode, and the higher-order mode has greatly different dispersion
profiles, which requires a careful mode excitement. A single-mode nano-taper [39] may be
helpful in doing beam coupling and mode screening simultaneously. The dispersion is highly
polarization dependent, and the quasi-TE mode has a strong anomalous dispersion >1000
ps/(nm·km) when the wavelength is beyond 1500 nm.
7.4 On-chip Frequency Comb Generation
Optical frequency combs have attracted a great deal of research interest in recent years
[40] and become a powerful tool for a variety of important applications, such as optical
123

frequency metrology [41], molecular detection [42], optical atomic clocks [43], and optical
arbitrary waveform generation [44].
On-chip frequency comb generation in integrated photonics platforms would be
highly valuable. High-Q micro-resonators could achieve strong nonlinear wave-mixing and
generate wideband frequency combs with low pump power [ 45 , 46 ]. Equally spaced
resonance peaks are desired not only for improving measurement precision but also for
further reducing pump power requirement. However, highly nonlinear waveguides (or
photonic nano-wires) typically have strong index contrast and waveguide dispersion, making
it difficult to obtain equally spaced (i.e., uniform) spectral lines in the generated frequency
combs. Here, we present how the low chromatic dispersion of ±15 ps/nm/km over a 563-nm
wavelength range in a silicon waveguide can be used to generate a frequency comb with
uniform spectral lines.
In principle, guided optical modes in waveguides have a frequency-dependent
effective index. When the waveguide is bended to form a ring resonator, the optical mode is
extended outwards and feels a bending radius greater than designed structural radius R
0
of a
ring resonator. As shown in Figure 7.10 (a), the guided mode has an effective bending radius
R
eff
, which is also frequency-dependent. Therefore, under resonance condition
2πR
eff
(f)n
eff
(f)f=mc, where m is an integer number and c is the speed of light in vacuum, a
ring resonator's FSR typically is not constant over frequency. However, considering an ideal
waveguide with zero-dispersion over a wide frequency band, in which dispersion
relationship satisfies n
eff
(f)=n
0
-kλ=n
0
-kc/f (n
0
and k are constants), one can find the FSR
equal to a constant c/(2πRn
0
), assuming the structural bending radius R
0
is so large that
essentially R
eff
≈ R
0
. This means that, starting with a design of a straight waveguide that
124

achieves flat and near-zero dispersion over a broad bandwidth, we would have a chance to
greatly improve the uniformity of spectral lines in the ring-resonator transfer functions.

(a)

(b)
Figure 7.10 (a) In curved waveguides, optical mode feels a bending radius greater than structural
radius R
0
of a ring resonator. Both the "effective" bending radius R
eff
and effective index
strongly depend on frequency. FSR isn't constant over frequency. A slot waveguide and its
mode are shown here. (b) Chromatic dispersion inside the ring cavity is changed by reducing
bending radius from 16 to 3 microns. Both averaged value and flatness of the dispersion
profiles are varied.

125

Here we consider a horizontally slotted waveguide that consists of upper and lower
silicon layers (high refractive index) and a low-index silica slot. With structural parameters:
upper silicon height H
u
=286 nm, slot height H
s
=32 nm, lower silicon height H
l
=510 nm, and
waveguide width W=500 nm, quasi-TM mode (y-polarized state) exhibits a flat dispersion
within ±15 ps/nm/km over a 563-nm-wide wavelength range, from 1566 to 2129 nm, as
shown in Figure 7.10 (b). Simulations are carried out, using a finite-element-method
software (COMSOL) with minimal element size of 5 nm for the thin slot layer. For bended
waveguides, eigen-frequency of the guided mode is obtained [ 47 ] iteratively at each
wavelength and is used, together with corresponding R
eff
measured from modal distribution,
to determine the effective index.
Free spectral range (FSR) as a function of wavelength has a standard deviation down
to 0.023 GHz, 250 times smaller than that produced by a strip waveguide in the same size
without a slot structure. Nonlinear coefficient is ~70.8 /W/m at 1800-nm wavelength.
Simulations show that waveguide bending in microring resonators can significantly affect
uniformity of the spectral lines in the generated frequency combs.
We note that, although a high index contrast in integrated waveguides allows for
tightly curving of the waveguides, it may significantly change not only waveguide loss but
also dispersion properties. Figure 7.10 (b) shows chromatic dispersion profiles of the bended
waveguides as R
0
is reduced from infinity down to 3 μm. Little change in dispersion has
been seen for a bending radius R
0
>11 μm. In contrast, the averaged dispersion shifts by up
to >100 ps/nm/km rapidly when R
0
is reduced from 6 to 3 μm, because much more optical
field is pushed outwards, which causes strong wavelength dependence of the effective index.
126

Although the wavelength dependence of both n
eff
and R
eff
contributes to a shifted dispersion
profile, simulations show that the effect of the R
eff
change is dominant.

Figure 7.11 (a) Frequency domain "eye-digram". Spectral response of a double-waveguide ring
resonator is sliced with a freq. step of one FSR. The obtained spectrum pieces are plotted
together to show the uniformity of the resonance peaks. (b) The silicon strip waveguide
produces spectral lines widely shifted over only 100 nm. (c) Dispersion flattened slot
waveguide enables well-aligned resonance peaks over 563 nm wavelength band.
Since the straight waveguide has flat and low dispersion, the FSR is nearly constant
with a standard deviation as low as 0.022 GHz for R
0
=32 μm from 1566 to 2129 nm
wavelength. The averaged FSR is 376.3 GHz, and the averaged resonance linewidth is 2.97
GHz. In contrast, a silicon strip waveguide with a height of 510 nm and a width of 500 nm
produces a fast-changing FSR over the same wavelength band. The FSR standard deviation
is 10.53 GHz, and the linewidth is 2.79 GHz. In both cases, we assume 2 dB/cm waveguide
loss, and 2% power coupling ratio. To illustrate the uniformity of the spectral lines, we plot a
frequency-domain "eye-diagram". As shown in Figure 7.11 (a), the frequency axis is
discretized with a frequency spacing of one FSR, and each of these spectrum pieces are
127

plotted onto the same figure. If the FSR is not constant vs. frequency, the resonance peaks
will be shifted relatively to each other in the "eye-diagram". Figure 7.11 (b) and (c) compare
the strip waveguide over 100-nm bandwidth and the slot waveguide over 563-nm bandwidth
in the frequency "eye-diagram", which shows the uniformity improvement by using the low-
dispersion slot waveguide.
7.5 Future Work
Achieving the flattened dispersion in integrated waveguides provides a great
opportunity of generating supercontinuum for on-chip applications. Here, we try to outline
three major mechanisms for efficiently generating supercontinua.
• In the normal dispersion regime, self-phase modulation plays a major role in
broadening the spectrum, producing high spectrum coherence. The generated
supercontinuum is mostly symmetric, centered at the input wavelength.
• In the anomalous dispersion regime, higher-order soliton fission is identified as a
key effect responsible for spectrum broadening. Another key effect is dispersive wave
generation. This is useful for on-chip frequency metrology.
• Enabled by sophisticated dispersion tailoring, ultraflat & low dispersion can be
achievable. This unique dispersion profile greatly enhances self-steepening of the pulse. The
steepened falling edge could be as short as 5 fs, which effectively produces blue-shifted
frequency components and generates highly asymmetric supercontinuum.

128

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Abstract (if available)

Abstract This dissertation presents silicon-based integrated micro-resonators and waveguides as the key elements of photonic integrated circuits for on-chip optical communication and signal processing applications.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Zhang, Lin (author)

Core Title Silicon integrated devices for optical system applications

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 05/04/2013

Defense Date 10/29/2010

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

Tag dispersion,integrated photonics,micro-resonator,modulator,nonlinearity,OAI-PMH Harvest,phase shift keying,silicon,slot waveguide

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Willner, Alan E. (committee chair), Armani, Andrea M. (committee member), Povinelli, Michelle L. (committee member)

Creator Email linzhang.lzh@gmail.com,linzhang@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3901

Unique identifier UC1231731

Identifier etd-Zhang-4221 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-472729 (legacy record id),usctheses-m3901 (legacy record id)

Legacy Identifier etd-Zhang-4221.pdf

Dmrecord 472729

Document Type Dissertation

Rights Zhang, Lin

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu