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Integrated silicon waveguides and specialty optical fibers for optical communications system applications
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Integrated silicon waveguides and specialty optical fibers for optical communications system applications
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Content
INTEGRATED SILICON WAVEGUIDES AND SPECIALTY OPTICAL FIBERS
FOR OPTICAL COMMUNICATIONS SYSTEM APPLICATIONS
by
Yang Yue
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
December 2012
Copyright 2012 Yang Yue
ii
Dedication
To my parents, Changhe Yue and Xinhua Fu,
my entire family, research advisor, OCLab members and friends
for their everlasting love, encouragement and support.
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 advisor and colleagues in Optical Communications
Laboratory (OCLab).
First, I would like to express my greatest gratitude to my 'GPS', Dr. Alan E. Willner,
for 'Piling me Higher and Deeper' without 'Permanent Head Damage' through my 'Ph.D.'
study! His knowledge, experience, and his patience and encouragement 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 five years in his group
have been a highlight in my life and opened my eyes to a lot of innovations and
opportunities.
I would sincerely thank Prof. Moshe Tur from Tel Aviv University, Dr. Raymond G.
Beausoleil from HP Laboratories, Prof. Constance Chang-Hasnain from UC Berkeley, Prof.
Ming C. Wu from UC Berkeley, Dr. Samuel Dolinar from JPL, Dr. Kevin M. Birnbaum
from JPL, Dr. Baris I. Erkmen from JPL, and Prof. Siddharth Ramachandran from Boston
University for their guidance and support during my Ph. D study. I also would like to thank
Prof. Solomon W. Golomb, Prof. Peter Beerel, Prof. Andrea Armani, and Prof. Stephan Haas
for guiding me on the dissertation.
I am truly grateful to Prof. Muping Song, Dr. Wei-Ren Peng, Prof. Qiangmin Wang,
Prof. Antonella Bogoni, Prof. Xiaoying Liu, Prof. Guiling Wu, Prof. Junli Wang, Prof.
iv
Changjian Ke, and Prof. Sheng Cui. 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 would also like to express my sincere thanks to my colleagues and comrades in
OCLab. Dr. Louis C. Christen, Dr. Bo Zhang, Dr. Lin Zhang, Dr. Scott Nuccio, Dr. Jian
Wang, Dr. Irfan Fazal, Dr. Jeng-Yuan Yang, Dr. Xiaoxia Wu, Dr. Omer Faruk Yilmaz,
Bishara Shamee, Salman Khaleghi, Mohammad Reza Chitgarha, Hao Huang, Asher
Voskoboinik, Nisar Ahmed, Yan Yan, Yongxiong Ren, Morteza Ziyadi, Guodong Xie, and
Changjing Bao helped me tremendously in my projects. Moreover, I would like to thank my
collaborators, Yunchu Li, Xue Wang, Xiaoxue Zhao, Devang Parekh, Forrest G. Sedgwick,
Christopher Chase, Vadim Karagodsky, Weijian Yang, James Ferrara, Xiaomin Zhang,
Mark Harrison, Martin Lavery and Nenad Bozinovic for all the helpful discussions and
remarkable collaborations.
Last but not least, I express my deepest love to my parents and my family.
v
Table of Contents
Dedication……….. ..................................................................................................... ii
Acknowledgements .................................................................................................... iii
List of Figures ........................................................................................................... vii
Abstract….. ……………………………………………………………………… xiv
Chapter 1 Introduction .......................................................................................... 1
1.1 Optical Communications and Signal Processing .................................... 1
1.2 Integrated Silicon Waveguides ............................................................... 3
1.3 Specialty Optical Fibers ......................................................................... 4
1.4 Thesis Outline ......................................................................................... 6
1.5 References .............................................................................................. 8
Chapter 2 Silicon High-Contrast Grating Hollow-core Waveguide for Linear
System .................................................................................................. 11
2.1 Introduction .......................................................................................... 11
2.2 Principle of Waveguiding ..................................................................... 12
2.3 Basic Effects on Optical Propagation ................................................... 13
2.4 Signal and Subsystem Performance ..................................................... 19
2.5 Summ ary ............................................................................................... 22
2.6 References ............................................................................................ 23
Chapter 3 Three-dimensional Chirped High-contrast Grating Hollow-core
Waveguide ........................................................................................... 25
3.1 Introduction .......................................................................................... 25
3.2 Waveguide Structure and Mode Property ............................................ 26
3.3 Chirped Grating and Its Effect on Waveguide Properties .................... 29
3.4 Summ ary ............................................................................................... 36
3.5 References ............................................................................................ 37
Chapter 4 Silicon Polarization Devices for Polarization Diversity System ..... 40
4.1 Introduction .......................................................................................... 40
4.2 Higher-order-mode Assisted Polarization Rotator ............................... 42
4.3 Slotted-waveguides-based Polarization Splitter ................................... 49
4.4 Summ ary ............................................................................................... 55
4.5 References ............................................................................................ 56
vi
Chapter 5 Highly Efficient Nonlinearity Reduction in Silicon Waveguides
using Vertical Slots ............................................................................. 59
5.1 Introduction .......................................................................................... 59
5.2 Waveguide Structure and Numerical Model ........................................ 60
5.3 Nonlinearity and Dispersion Properties ................................................ 62
5.4 Summ ary ............................................................................................... 68
5.5 References ............................................................................................ 69
Chapter 6 Ultralow Dispersion Silicon-on-nitride Waveguide over an Octave-
Spanning Mid-infrared Wavelength Range ..................................... 71
6.1 Introduction .......................................................................................... 71
6.2 Waveguide Structure and Mode Property ............................................ 73
6.3 Chromatic Dispersion and Nonlinearity ............................................... 75
6.4 Summ ary ............................................................................................... 80
6.5 References ............................................................................................ 81
Chapter 7 UWB Pulse Generation using Two-Photon Absorption .................. 84
7.1 Introduction .......................................................................................... 84
7.2 Principle ................................................................................................ 85
7.3 Results and Discussion ......................................................................... 86
7.4 Summ ary ............................................................................................... 91
7.5 References ............................................................................................ 92
Chapter 8 Orbital Angular Momentum Modes in Specialty Optical Fiber .... 94
8.1 Introduction .......................................................................................... 94
8.2 Properties of Optical Orbital Angular Momentum Modes in a Ring
Fiber ...................................................................................................... 97
8.3 Octave-spanning Supercontinuum Generation of Vortices in a Ring
Photonic Crystal Fiber ........................................................................ 106
8.4 Summ ary ............................................................................................. 111
8.5 References .......................................................................................... 112
Chapter 9 Demonstration of a High Capacity Optical Fiber Link using Two
Orbital-Angular-Momentum Modes with Multiple Wavelength
Channels ............................................................................................ 116
9.1 Introduction ........................................................................................ 116
9.2 Concept and Fiber Properties ............................................................. 117
9.3 Experimental Setup and Results ......................................................... 119
9.4 Summ ary ............................................................................................. 122
9.5 References .......................................................................................... 123
Bibliography………………………………………………………………………125
vii
List of Figures
Figure 1.1 (a) Constellation diagrams of 1024 QAM signal under back-to-back
condition [7]. (b) Cross section of seven-core fiber [9]. (c) Application of
phase plates for multimode conversion [11].......................................................... 2
Figure 1.2 Images of (a) fiber and (b) silicon waveguide. ............................................... 4
Figure 1.3 An assortment of optical and scanning electron micrographs of PCF
structures [32]. ....................................................................................................... 6
Figure 2.1 Schematic of a 2D HCG-HW consisting of two reflecting HCGs. Si
high-index gratings are surrounded by air. Light propagates in the z
direction with TE polarization.. ........................................................................... 13
Figure 2.2 Propagation loss (dB/m) at 1550 nm of a HCG-HW with 15- m core
size as a function of (a) grating period and air gap (b) grating thickness and
air gap. The innermost and outmost loss contour lines in the plots
correspond to 0.2 and 1 dB/m loss, respectively.. ............................................... 14
Figure 2.3 Dispersion profiles for different core sizes.. ................................................ 15
Figure 2.4 Dispersion profiles for different (a) air gaps and (b) grating thicknesses. ... 16
Figure 2.5 FEM analyses of HCG-HW (a) power flow (b) y- polarized electric
field distribution (c) sampled y- polarized electric field at different
waveguide positions (D=15 m, Λ=750 nm, a
g
=430 nm, and t
g
=340 nm). ........ 17
Figure 2.6 Nonlinear coefficient as a function of (a) HCG-HW air gap (a
g
) ( Λ=750
nm, and t
g
=340 nm) (b) HCG-HW grating thickness (t
g
) ( Λ=750 nm, and
a
g
=430 nm) (c) wavelength ( ) ( Λ=750 nm, a
g
=430 nm, and t
g
=340 nm) for
core sizes (D) from 5 to 20 m with a 5- m step. ............................................... 18
Figure 2.7 EOP as a function of bit rate in RZ-OOK data format for different
fabrication deviations in a 10-m-long HCG-HW.. .............................................. 20
Figure 2.8 EOP as a function of HCG-HW length for different fabrication
deviations with a 100 Gb/s RZ-OOK.. ................................................................ 21
Figure 2.9 Carrier-to-nonlinearity ratio as a function of input power for different
HCG-HW length with a 10-GHz sine wave.. ...................................................... 22
Figure 3.1 (a) Schematic of a 3D HCG-HW consisting of two reflecting HCGs in
the y direction. Si high-index gratings are on top of a 2- m SiO
2
layer and
Si substrate. Two different grating designs are chosen as core and cladding
viii
to provide the effective index difference and lateral confinement. Linearly
chirped gratings along the x direction are used to mitigate the effective
refractive index mismatch induced leakage. The light propagates in the z
direction with TE polarization. (b) TE field (power norm) distribution in the
chirped HCG-HW.. .............................................................................................. 27
Figure 3.2 (a) Wavelength dependence of the TE field (power norm) distribution in
the chirped HCG-HW. (b) Effective index difference (between the core and
cladding regions) and effective mode area of the TE mode as a function of
wavelength from 1450 to 1600 nm. ..................................................................... 29
Figure 3.3 (a) Chirping profile for different chirp rate (k). (b) Propagation loss as a
function of chirp rate for HCG-HW with 15 effective gratings in the core
region. .................................................................................................................. 30
Figure 3.4 Propagation loss at 1550 nm as a function of the grating period number
in the chirped region (N
chirp
) for different numbers of periods in the effective
core (N
eff_core
=9, 15, 21). N
chirp
=0 stands for no chirped gratings between the
core and the cladding regions (step-index case). ................................................. 30
Figure 3.5 (a) Propagation loss as a function of wavelength from 1500 to 1600 nm
for different grating period number in the chirped region (N
chirp
=6, 12, 18)
with fixed grating period number in the core region (N
core
=3) and core height
(D=5 m). (b) Propagation loss at 1550 nm as a function of waveguide core
height (D) from 3 to 10 m for different grating period number in the
chirped region (N
chirp
=6, 12, 18) with fixed grating period number in the
core region (N
core
=3). ........................................................................................... 31
Figure 3.6 <1-dB/cm loss bandwidth as a function of grating period number of the
effective core from 7 to 21 for step-index (N
chirp
=0) and chirped (N
core
=3)
HCG-HW with 5- m core height. ....................................................................... 32
Figure 3.7 Contour plot of propagation loss for the HCG-HW (N
core
=3) vs.
wavelength and core height with different number of chirped grating periods
(a) N
chirp
=6 and (b) N
chirp
=18. ............................................................................... 33
Figure 3.8 (a) Chromatic dispersion as a function of wavelength from 1450 to
1600 nm for different grating period number in the chirped region (N
chirp
=6,
12, 18) with fixed grating period number in the core region (N
core
=3) and
core height (D=5 m). (b) Chromatic dispersion as a function of wavelength
from 1450 to 1600 nm for different waveguide core height (D=3, 5, 7 m)
with fixed grating period number in the core region (N
core
=3) and in the
chirped region (N
chirp
=12). ................................................................................... 34
Figure 3.9 (a) Nonlinear coefficient at 1550 nm as a function of the grating period
number in the chirped region (N
chirp
) for different grating period number of
the effective core (N
eff_core
=9, 15, 21). N
chirp
=0 stands for no chirped gratings
ix
between the core and the cladding regions (step-index case). (b) Nonlinear
coefficient as a function of wavelength from 1500 to 1600 nm for different
grating period number in the chirped region (N
chirp
=6, 12, 18) with fixed
grating period number in the core region (N
core
=3). ............................................ 35
Figure 4.1 On-chip polarization diversity scheme. ........................................................ 41
Figure 4.2 Operating principle (a) No coupling case. (b) Weak coupling case. (c)
Proposed 90
o
polarization rotator using wave coupling through an
intermediate multimode waveguide. (d) Schematic of proposed 90
o
polarization rotator. ............................................................................................. 43
Figure 4.3 (a) X-, Y- polarized electric field distributions and electric field lines of
the three supermodes in the proposed waveguide structure. (b) Equivalent E-
field distribution of the supermodes in three waveguides (solid line:
fundamental modes, dot line: TE
01
mode). .......................................................... 44
Figure 4.4 (a) Effective refractive indices of the three supermodes in Fig. 4.3 as a
function of wavelength. (b) Conversion lengths of the fundamental and TE
01
modes................................................................................................................... 45
Figure 4.5 (a) X-polarized normalized power exchange along the propagation
distance in three waveguides. (b) Y-polarized normalized power exchange
along the propagation distance in three waveguides. .......................................... 47
Figure 4.6 PCE as a function of wavelength and >90% PCE Bandwidth. .................... 47
Figure 4.7 (a) PCE@21 m, optimized PCE and its corresponding conversion
length as a function of waveguide distance (D). (b) PCE@21 m, optimized
PCE and its corresponding conversion length as a function of waveguide
width (W). ........................................................................................................... 49
Figure 4.8 (a) PCE@21 m, optimized PCE and its corresponding conversion
length as a function of WG distance variation ( D). (b) PCE@21 m,
optimized PCE and its corresponding conversion length as a function of WG
width variation ( W). .......................................................................................... 49
Figure 4.9 Cross-section view of the horizontally-slotted waveguide polarization
splitter. ................................................................................................................. 51
Figure 4.10 Coupling length of the quasi-TM mode and the coupling length ratio
of the quasi-TE and quasi-TM mode as a function of the slot thickness. ............ 52
Figure 4.11 Exchange of the normalized power along the propagation distance of
the quasi-TE and quasi-TM modes at the input waveguide. ............................... 53
x
Figure 4.12 Extinction ratio of the quasi-TE and quasi-TM mode in a slot
waveguide splitter at one coupling length. .......................................................... 53
Figure 4.13 Coupling length of the quasi-TM mode and the coupling length ratio
of the quasi-TE and quasi-TM mode as a function of the waveguide spacing. ... 54
Figure 4.14 Coupling length of the quasi-TM mode and the coupling length ratio
of the quasi-TE and quasi-TM mode as a function of the slot’s refractive
index. ................................................................................................................... 55
Figure 5.1 Cross sections of strip, single-vertical-slot and double-vertical-slot
waveguides. ......................................................................................................... 61
Figure 5.2 Power distributions of the quasi-TE modes in strip, single-vertical-slot
and double-vertical-slot waveguides. .................................................................. 62
Figure 5.3 (a) Power distributions (b) Effective refractive index and real part of
nonlinear coefficient as a function of slot width in a single-vertical-slot
waveguide. ........................................................................................................... 63
Figure 5.4 Real part of nonlinear coefficient vs. slot width and slot position
variation at 1550 nm (single-vertical-slot waveguide). ....................................... 64
Figure 5.5 (a) Power distributions (b) Effective refractive index and real part of
nonlinear coefficient as a function of the slot spacing in a double-vertical-
slot waveguide. .................................................................................................... 65
Figure 5.6 Real part of nonlinear coefficient as a function of the wavelength in
strip, single-vertical-slot and double-vertical-slot waveguides (s = 150 nm). ..... 65
Figure 5.7 Chromatic dispersion as a function of the wavelength in (a) a single-
vertical-slot waveguide and (b) a double-vertical-slot waveguide (d = 50
nm). ..................................................................................................................... 67
Figure 6.1 The cross-sections of strip, rib, and conformal overlayer SON
waveguides. ......................................................................................................... 73
Figure 6.2 (a) The dispersion property and (b) power distribution (3 m) of the
conformal overlayer SON waveguide (W=2000 nm, H= 800 nm, t=400 nm)
modes................................................................................................................... 74
Figure 6.3 Chromatic dispersion of (a) x- and (b) y-polarized fundamental modes
for strip SON waveguide with different waveguide height. ................................ 76
Figure 6.4 Chromatic dispersion of (a) x- and (b) y-polarized fundamental modes
for strip SON waveguide with different waveguide widths. ............................... 76
xi
Figure 6.5 Chromatic dispersion of x-polarized fundamental modes for (a) rib
SON waveguide with different slab height and (b) conformal overlayer SON
waveguide with different overlayer thickness. .................................................... 77
Figure 6.6 (a) The nonlinear refractive index of silicon and (b) the nonlinear
coefficients in strip, rib and conformal overlayer waveguides. ........................... 80
Figure 7.1 Applications of UWB in a digital home. ...................................................... 85
Figure 7.2 Concept diagram of silicon waveguide-based UWB monocycle pulse
generation using non-degenerated two-photon absorption effect. ....................... 86
Figure 7.3 Experimental setup. Pulsed pump and CW probe are coupled to Si WG.
The inversely modulated probe then combines with the pump to generate
monocycle pulses. ............................................................................................... 87
Figure 7.4 Effects of non-degenerated two-photon absorption. (a) Probe
modulation depth, and (b) modulated probe FWHM as a function of pump
pulse peak power for different pump pulse width. .............................................. 88
Figure 7.5 Ultrawideband signal generation. RF power spectra of generated
monocycle pulse with (a) FWHM
pump
=45 ps, (b) 68 ps, and (c) 93 ps. ............... 89
Figure 7.6 10-dB BW and center frequency of the RF spectrum as a function of (a)
relative delay between the pump and inversely modulated probe
(FWHM
pump
=68 ps); (b) probe wavelength (FWHM
pump
=68 ps,
pump
=1550.4
nm). ..................................................................................................................... 90
Figure 8.1 Comparison of different schemes for multiplexing multiple optical
spatial modes for fiber transmission. ................................................................... 97
Figure 8.2 (a) OAM mode number supported in the single-ring fiber as a function
of the ring outer radius (r
2
) with different ring-cladding index difference
( n). (b) OAM mode number as a function of the wavelength with different
ring-cladding index difference ( n). ................................................................. 100
Figure 8.3 Effective refractive index difference of TE
0,1
and HE
2,1
modes as a
function of (a) the refractive index in the fiber ring region with different ring
outer radius (r
2
), and (b) wavelength with different ring-cladding index
difference ( n). .................................................................................................. 101
Figure 8.4 Intensity and phase distribution of the supported OAM
0,m
modes. ............. 101
Figure 8.5 Intensity and phase variation of OAM
0,3
modes azimuthally along the
center of ring region with different mode walk-off ( ) ( =90
o
provides a
perfect OAM mode). (a) Normalized intensity variation with different mode
walk-off and azimuth. (b) Azimuthally normalized intensity variation for
xii
=90
o
, 120
o
, 150
o
and 180
o
. (c) Phase variation with different mode walk-off
and azimuth. (d) Azimuthal phase variation for =90
o
, 120
o
, 150
o
and 180
o
. .. 102
Figure 8.6 Standard deviation of (a) intensity, and (b) phase azimuthally along the
center of ring region as a function of fiber ellipticity for different OAM
modes................................................................................................................. 103
Figure 8.7 2 and 10-ps walk-off length as a function of fiber ellipticity for
different OAM modes. ...................................................................................... 104
Figure 8.8 Charge weight of (a-h) different well-aligned OAM modes in a =1%
fiber. (i-p) OAM
0,3
modes after 0, 5, 10 and 15-m propagation in a =1%
fiber. .................................................................................................................. 105
Figure 8.9 The demultiplexing efficiency of OAM
0,3
mode as a function of the
propagation length in the ring fiber with different (Periodic vs. 2 walk-off
length). ............................................................................................................... 106
Figure 8.10 (a) Cross section of As
2
S
3
ring PCF. (b) Intensity and phase
distributions of PCF vortex modes (TE
0,1
, TM
0,1
, HE
2,1
even
, HE
2,1
odd
) and
OAM mode (OAM
0,2
=HE
2,1
even
+i×HE
2,1
odd
). ...................................................... 107
Figure 8.11 Effective refractive indices as a function of wavelength for vortex
modes in the designed As
2
S
3
ring PCF. ............................................................. 108
Figure 8.12 Dispersion of vortex mode (TE
0,1
) with different (a) , (b) r
0
, and (c)
r
2
of the As
2
S
3
ring PCF. .................................................................................. 108
Figure 8.13 Nonlinear coefficient and confinement loss of vortex mode (TE
0,1
) in
the designed As
2
S
3
ring PCF. ............................................................................ 109
Figure 8.14 Octave-spanning supercontinuum generation of vortex mode (TE
0,1
) in
the designed As
2
S
3
ring PCF. ............................................................................ 111
Figure 9.1 Conceptual diagram of muxing, transmission and demuxing of 16-
QAM signals over wavelength-division channels carrying 2 OAM beams
through vortex fiber. WDM: wavelength-division multiplexing; :
wavelength; QAM: quadrature amplitude modulation; OAM: orbital angular
momentum; PBS: polarizing beamsplitter; HWP: half-wave plate; QWP:
quarter-wave plate; P: polarizer; SLM: spatial light modulator. ....................... 118
Figure 9.2 Vortex fiber properties. (a) Measured refractive index. Calculated (b)
n
eff
, (c) dispersion and (d) A
eff
for all modes. ..................................................... 118
Figure 9.3 Experimental setup of muxing, transmission and demuxing of 16-QAM
signals over wavelength-division channels carrying 2 OAM beams through
vortex fiber. ....................................................................................................... 120
xiii
Figure 9.4 (a) Spectrum of the modulated signal at the output of WDM 16-QAM
Tx and spectrum of mode A (OAM
+
) after demuxing; (b) Constellations of
16 QAM for demultiplexed mode A (OAM
+
) without and with crosstalk
(XT) at 1550.64 nm; (c) BER as a function of the received power for B2B,
mode A (OAM
+
) and mode B (OAM
-
) without and with crosstalk (XT)
from WDM channels or the other OAM mode at 1550.64 nm. ......................... 121
Figure 9.5 Measured (a) crosstalk and (b) BER as a function of wavelength for
mode A (OAM
+
) and mode B (OAM
-
). ............................................................. 122
xiv
Abstract
In electronics industry, silicon has been served as an enabling material for a few
decades. Recently, by taking advantages of ultrahigh bandwidth in the optical domain,
silicon photonics has attracted a lot of attention. More importantly, advances in silicon
photonics have potentially paved the way for the design and construction of a CMOS-
compatible optoelectronic integration system. In the first part of this dissertation, we design
high-contrast grating hollow-core waveguide for linear systems, polarization rotator and
splitter for integrated polarization diversity system. Dispersion and nonlinearity properties of
silicon waveguide are tailored using different structures and materials for different
communications and signal processing applications. A scheme for generating ultrawideband
monocycle pulse in a silicon waveguide is experimentally demonstrated.
Fueled by the ability to transmit long distance, optical fiber communications will
undoubtedly continue to grow. One of the critical issues in optical communications research
is the challenge of meeting the needs of the inevitable growth in data transmission capacity.
Besides wavelength, polarization, amplitude, phase of the optical field, researchers start to
multiplex data channels carried by different and orthogonal spatial modes. In the second part
of this dissertation, we propose and analyze the use of orbital-angular-momentum modes for
spatial multiplexing in the ring fiber. As
2
S
3
ring PCF is proposed and designed for
supercontinuum generation or vortex modes, making OAM modes very promising in many
optical signal processing applications. Moreover, we demonstrate mode-division and
wavelength-division multiplexing of OAM modes through 1.1-km of vortex fiber.
1
Chapter 1 Introduction
In this Chapter, I will introduce and discuss the motivations and challenges of silicon
photonics and specialty optical fibers as increasingly important building blocks in modern
optical communication and signal processing systems. Material, device, and system level
perspectives will be provided.
1.1 Optical Communications and Signal Processing
Optical communication, as the backbone of today’s telecommunications infrastructure,
supports voice, video and data transmission through global networks [1]. A critical issue in
optical communications research is the challenge of meeting the needs of the inevitable
growth in data transmission capacity. Dense wavelength-division multiplexing (DWDM) has
been proven to be an efficient solution that provides a multiplicative-factor (on the order of
100) increment [2]. Fueled by emerging bandwidth-hungry applications, much work has
focused on increasing the data spectral efficiency by utilizing polarization, amplitude, and
phase manipulations of the optical field [3, 4, 5, 6, 7]. As shown in Fig. 1.1(a), quadrature
amplitude modulation (QAM) multiplicity of 1024 levels in a single-carrier coherent
transmission has been demonstrated in the optical domain [7].
Another potentially complementary approach that has gained much attention recently
is to transmit independent data streams, each in a different core using multi-core fibers
(MCF) or each on a different spatial linearly polarized (LP) mode using few-mode fibers
(FMF) [8, 9, 10, 11, 12, 13, 14]. Figure 1.2(b) and (c) show the cross section of seven-core
fiber and phase plates for multimode conversion used in mode-division-multiplexing
2
(MDM). Increasing the number of spatial modes in the optical fiber can increase the capacity
and the spectral efficiency of the communication link simultaneously. By combined the
above techniques, an aggregated capacity of optical communications link has been
successfully boosted to petabit per second scale [15].
One of the recent promising MDM methods uses optical orbital angular momentum
(OAM) of light. It is known that photons can carry OAM, which is associated with azimuthal
phase dependence of the complex electric field [16, 17]. As OAM has an infinite number of
orthogonal eigenstates, it provides another degree of freedom to manipulate the optical field.
OAM based free-space and fiber optical communication systems have been proposed and
demonstrated for spectrally efficient communication links, which can potentially meet the
latest trend in the field of optical communications [18, 19, 20, 21].
Figure 1.1 (a) Constellation diagrams of 1024 QAM signal under back-to-back condition [7]. (b)
Cross section of seven-core fiber [9]. (c) Application of phase plates for multimode conversion [11].
High speed, low power consumption and enriched function are the laudable goals for
optical signal processing [22]. If successful, one can transmit and manipulate information in
the optical domain, without the need for O-E-O conversion. This can potentially upgrade
system capacity, reduce latency and cost. Optical nonlinearities have been proved to be an
effective enabler to perform nonlinear processing functions, such as pulse manipulating, data
3
routing, and optical logic, especially in silicon devices [23]. All of these are mainly due to
the large nonlinear coefficients and high index contrast provided by silicon [24, 25]. Besides
the third-order nonlinear effects in optical fibers, it is important to notice that silicon has
some unique nonlinear properties, such as two photon absorption (TPA) [26]. These can
potentially enrich the functions of nonlinear signal processing.
1.2 Integrated Silicon Waveguides
Silicon photonics, which uses silicon as a medium for optical and photonic systems, is
quite a fascinating field [27, 28]. One of its advantages is to use the mature semiconductor
fabrication techniques in electronics industry. It is thus quite promising to realize hybrid
integration of electronic and optical devices on a chip, to fully take advantages provided by
both electronics and optics. Another obvious advantage, over free-space or fiber optical
systems, is its compact size and the great potential of energy-efficient operation. From Fig.
1.2 we can see, silicon waveguide is 3 orders of magnitude smaller than single mode fiber in
size. Compared with optical fiber system, silicon-based optical system can potentially reduce
the volume by 9 orders of magnitude.
Although quite promising, silicon photonics still has a few hurdles ahead for
researchers [27]. Compared with optical fiber, its loss, chromatic dispersion and nonlinearity
are remarkably high. The signals of the communications system will be largely distorted by
these effects. It will be quite challenging, especially for the optical systems that require high
linearity. Another feature of silicon waveguide is its high polarization dependence. Moreover,
as the bandgap of silicon crystal is around 1.11 eV, it brings silicon into the 1.1 to 2.2 μm
4
broad TPA wavelength range, which covers the widely used telecom band around 1.55 m
[26].
Figure 1.2 Images of (a) fiber and (b) silicon waveguide.
For a silicon waveguide, silicon typically lies on top of a layer of silica, which is
known as silicon on insulator (SOI) [29]. 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 different
applications typically require unique features from the waveguide, and thus it is quite helpful
to tailor the waveguide properties according to the application. By implementing different
subwavelength structures (such as grating, slot, or thin layer) and materials in the SOI
platform, the waveguide’s properties can be largely tuned. Furthermore, some undesirable
feature, such as TPA, can potentially be used for optical signal processing [26].
1.3 Specialty Optical Fibers
Specialty optical fiber is quite a broad concept, and can include any fiber with special
design or material. It has become an enabling technology for a variety of applications,
ranging from communications to fiber lasers, and environment monitoring. Here, we focus
on the applications on optical communications or signal processing.
5
Commonly used single mode fiber is a step index optical fiber, which guides light in
the high-index central core region by total internal reflection. Rays of light in the core,
striking the interface with the cladding, are completely reflected. This type of fiber forms the
backbone of nowadays optical communications network. Impelled by the ever-growing
demand for data capacity, specialty fibers, such as multi-core and multimode fibers, attract a
great deal of attention in the field of optical communications. Multi-core fiber has a few
high-index cores, each core perform the function as a single mode fiber. Typically,
multimode fibers have step-index profile, while with a larger core size. Thus, it can support
more modes. However, the LP modes transmitted in the multimode fiber have intrinsic walk-
off, and thus needs complex multiple-input-multiple-output (MIMO) system to undo the
inter-mode crosstalk [19].
Vortex fiber, which has a high-index ring region, is designed to suppress intermodal
coupling, and has been proved to maintain OAM modes stably [30, 31]. Encouraging results
of multiplexing multiple OAM modes in a 1.1-km fiber without MIMO have been
demonstrated recently [21]. This specialty fiber potentially provides another scheme for
spatially multiplexing multiple modes for long-haul transmission.
Another famous type of specialty fiber, photonic crystal fiber (PCF) has recently
drawn a great deal of attention from researchers [32]. Figure 1.3 shows a few scanning
electron images of PCF structures. Its micro engineered structure enables a wide range of
unique optical properties that cannot be realized by using conventional single-mode fibers,
such as endless single-mode guiding, high birefringence, large nonlinearity, and tailorable
chromatic dispersion. Moreover, PCF has been proved to be an efficient supercontinuum
generator enabled by its advanced dispersion tailoring [33]. With proper choice of structure
6
and material, it can potentially advance the functions of optical fiber communications and
signal processing to a brand new stage.
Figure 1.3 An assortment of optical and scanning electron micrographs of PCF structures [32].
1.4 Thesis Outline
This dissertation is organized with the following structure: Chapter 2 analyzes
propagation loss, dispersion, nonlinearity of two-dimensional high-contrast grating hollow-
core waveguide. This hollow waveguide has low loss, low dispersion and negligible
nonlinearity and thus can be potentially used to linear systems. Chapter 3 describes a method
of using adiabatically chirped gratings to form a low-loss three-dimensional high-contrast
grating hollow-core waveguide. Chapter 4 presents the designs of silicon polarization
devices (polarization rotator and polarization splitter) for integrated polarization diversity
system. Chapter 5 shows a way of reducing the nonlinearity of a compact silicon waveguides
by using vertical slots. Compared with a strip waveguide, a factor greater than 15 times more
reduction in the nonlinear coefficient is achieved by using double vertical slots. Chapter 6
presents a design of silicon-on-nitride waveguide for octave-spanning ultralow dispersion in
mid-infrared wavelength range, and can be potentially used for efficient supercontinuum and
7
frequency comb generation. Chapter 7 proposes and experimentally demonstrates
ultrawideband monocycle pulse generation using nondegenerate two photon absorption in a
silicon waveguide. Chapter 8 proposes and analyzes the use of orbital-angular-momentum
modes for spatial multiplexing in the ring fiber. Moreover, As
2
S
3
ring PCF is designed for
supercontinuum generation of vortex modes, making it very promising in many optical
signal processing applications. Chapter 9 shows 1.6-Tbit/s muxing, transmission and
demuxing through 1.1-km of vortex fiber carrying 2 OAM beams each with 10 wavelength
channels.
8
1.5 References
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Elsevier, (2008).
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C. M. Ozveren, B. Schofield, R. E. Thomas, R. A. Barry, D. M. Castagnozzi, V. W. S. Chan,
B. R. Hemenway, D. Marquis, S. A. Parikh, M. L. Stevens, E. A. Swanson, S. G. Finn, and
R. G. Gallager, “A wideband all-optical WDM network,” Journal on Selected Areas in
Communications 14, 780-799 (1996).
[3] P. J. Winzer and R.-J. Essiambre, “Advanced modulation formats for high-capacity
optical transport networks,” J. Lightwave Technol. 24, 4711-4720 (2007).
[4] X. Zhou and J. Yu, “Multi-level, multi-dimensional coding for high-speed and high-
spectral-efficiency optical transmission,” J. Lightwave Technol. 27, 3641-3653 (2009).
[5] S. Okamoto, K. Toyoda, T. Omiya, K. Kasai, M. Yoshida, and M. Nakazawa, “512
QAM (54 Gbit/s) coherent optical transmission over 150 km with an optical bandwidth of
4.1 GHz,” in Proc. of ECOC’2010, Paper PD2-3 (2010).
[6] A. H. Gnauck, P. J. Winzer, A. Konczykowska, F. Jorge, J.-Y. Dupuy, M. Riet, G.
Charlet, B. Zhu, and D. W. Peckham, “Generation and transmission of 21.4-Gbaud PDM 64-
QAM using a high-power DAC driving s single I/Q modulator,” in Proc. of OFC’2011,
Paper PDPB2 (2011).
[7] Y. Koizumi, K. Toyoda, M. Yoshida, and M. Nakazawa, "1024 QAM (60 Gbit/s) single-
carrier coherent optical transmission over 150 km," Opt. Express 20, 12508-12514 (2012).
[8] S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new
dimension in fiber optic multiplexing,” Opt. Laser Technol. 40, 1030-1036 (2008).
[9] B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E.
M. Monberg, and F. V. Dimarcello, “112-Tb/s Space-Division Multiplexed DWDM
Transmission With 14-b/s/Hz Aggregate Spectral Efficiency Over a 76.8-km Seven-Core
Fiber,” Opt. Express 19, 16665-16671 (2011).
[10] X. Liu, S. Chandrasekhar, X. Chen, P. J. Winzer, Y. Pan, T. F. Taunay, B. Zhu, M.
Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, "1.12-Tb/s 32-QAM-
OFDM superchannel with 8.6-b/s/Hz intrachannel spectral efficiency and space-division
multiplexed transmission with 60-b/s/Hz aggregate spectral efficiency," Optics Express 19,
958-964 (2011).
[11] C. Koebele, M. Salsi, L. Milord, R. Ryf, C. A. Bolle, P. Sillard, S. Bigo, and G. Charlet,
“40km Transmission of Five Mode Division Multiplexed Data Streams at 100Gb/s with low
MIMO-DSP Complexity,” in Proc. of ECOC’2011, Paper Th.13.C.3 (2011).
9
[12] R. Ryf, A. Sierra, R. Essiambre, A. Gnauck, S. Randel, M. Esmaeelpour, S. Mumtaz, P.
J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki,
“Coherent 1200-km 6 x 6 MIMO Mode-Multiplexed Transmission over 3-Core
Microstructured Fiber,” in Proc. of ECOC’2011, Paper Th.13.C.1 (2011).
[13] S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. Essiambre, D.
W. Peckham, A. McCurdy, and R. Lingle, "6×56-Gb/s mode-division multiplexed
transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization," Opt. Express
19, 16697-16707 (2011).
[14] I. B. Djordjevic, M. Arabaci, L. Xu, and T. Wang, “Spatial-domain-based
multidimensional modulation for multi-Tb/s serial optical transmission, Opt. Express 19,
6845-6857 (2011).
[15] H. Takara, A. Sano, T. Kobayashi, H. Kubota, H. Kawakami, A. Matsuura, Y.
Miyamoto, Y. Abe, H. Ono, K. Shikama, Y. Goto, K. Tsujikawa, Y. Sasaki, I. Ishida, K.
Takenaga, S. Matsuo, K. Saitoh, M. Koshiba, and T. Morioka, "1.01-Pb/s (12 SDM/222
WDM/456 Gb/s) Crosstalk-managed Transmission with 91.4-b/s/Hz Aggregate Spectral
Efficiency," in Proc. of ECOC’2012, Paper Th.3.C.1 (2012).
[16] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular
momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rrv. A
45, 8185 (1992).
[17] L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, Institue of
Physics Publishing, London (2003).
[18] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-
Arnold, “Free-space information transfer using light beams carrying orbital angular
momentum,” Opt. Express 12, 5448-5456 (2004).
[19] Y. Yue, Y. Yan, N. Ahmed, J. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B.
I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, "Mode properties and propagation effects
of optical orbital angular momentum (OAM) modes in a ring fiber," IEEE Photonics Journal
4, 535-543 (2012).
[20] J. Wang, J. Yang, I. M.Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar,
M. Tur, and A. E. Willner, "Terabit free-space data transmission employing orbital angular
momentum multiplexing," Nature Photonics 6, 488-496, (2012).
[21] N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, A. Willner, and S. Ramachandran,
"Orbital Angular Momentum (OAM) Based Mode Division Multiplexing (MDM) over a
Km-length Fiber," in Proc. of ECOC’2012, Paper Th.3.C.6 (2012).
[22] H. Ishikawa, Ultrafast all-optical signal processing devices. John Wiley & Sons (2008).
[23] C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear silicon-on-
insulator waveguides for all-optical signal processing," Opt. Express 15, 5976-5990 (2007).
10
[24] Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon
waveguides: modeling and applications," Opt. Express 15, 16604-16644 (2007).
[25] H. K. Tsang and Y. Liu, "Nonlinear optical properties of silicon waveguides,"
Semiconductor Science and Technology 23, 064007-064015 (2008).
[26] H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake,
and M. Asghari, “Optical dispersion, two-photon absorption, and self-phase modulation in
silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[27] M. Lipson, "Guiding, Modulating, and Emitting Light on Silicon-Challenges and
Opportunities," J. Lightwave Technol. 23, 4222-4238 (2005).
[28] B. Jalali and S. Fathpour, "Silicon Photonics," J. Lightwave Technol. 24, 4600-4615
(2006).
[29] G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction, Wiley, (2004).
[30] S. Ramachandran, P. Kristensen, and M. F. Yan, "Generation and propagation of
radially polarized beams in optical fibers," Optics Letters 34, 2525-2527 (2009).
[31] N. Bozinovic, P. Kristensen, and S. Ramachandran, "Long-range fiber-transmission of
photons with orbital angular momentum," in Proc. of CLEO’2011, Paper CTuB1 (2011).
[32] P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[33] J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal
fiber," Reviews of Modern Physics 78, 1135-1184 (2006).
11
Chapter 2 Silicon High ‐Contrast Grating Hollow ‐core
Waveguide for Linear System
2.1 Introduction
In both electronic and optical systems, time delay is a key basic building block for
many applications, such as radio frequency filtering, data traffic engineering, and signal
processing. Realizing the delay function in the optical domain might take advantage of its
inherent ultrahigh bandwidth and reduce the inefficient optical-electronic conversion. In
fiber systems, large optical delays up to several microseconds have been realized using the
conversion/dispersion scheme [1, 2]. However, the entire system is relatively bulky and
restricts its application. A laudable goal is to make the delay element compact and realize
on-chip integration. Nevertheless, most of the on-chip waveguides tend to experience high
losses, large dispersions and serious nonlinearity [3, 4]. Thus, it is highly desirable to obtain
an on-chip delay element without introducing much signal degradation or distortion.
Hollow-core waveguides (HW) have the potential to achieve all of these features, as light is
primarily confined within the air. HWs using Bragg and antiresonant reflectors have been
intensively studied for decades [5, 6]. However, the loss of on-chip HW remained ~10 dB/m.
High-contrast grating (HCG) demonstrates a very high reflectivity with a broad
bandwidth for surface-normal incident light. It has been incorporated to vertical cavity
surface emitting laser (VCSEL) [7] and high-Q resonator [8]. Recently, an ultra-low loss
(0.006 dB/m) HW was proposed using HCGs as a reflection mirror for small glancing angle
light [9-11]. A 15- m core size HW with semiconductor HCGs demonstrates the possibility
of realizing an ultra-low loss (0.006 dB/m), small dispersion and negligible nonlinearity,
12
simultaneously [9, 12, 13]. It illustrates the significant possibility of forming an ideal on-
chip delay element. Moreover, its unique dispersion property opens up several potential
applications, such as slow light [14].
In this chapter, we analyze the chromatic dispersion (CD) and nonlinearity of two-
dimensional HCG-HW. Its chromatic dispersion is reasonably low for on-chip applications
[13]. Moreover, its nonlinearity is negligibly low, and thus can potentially remain the
systems’ linearity [12]. System-level performance is analyzed for both digital and analog
signals [12, 13]. Due to its low loss, low dispersion, and negligible nonlinearity, high-speed
(>100 Gb/s), long-distance (30 m) and low penalty (<0.2 dB) on-chip digital
communications can be achieved. No observable nonlinearity induced penalty is found for
input power up to 1 W for analog signals.
2.2 Principle of Waveguiding
The schematic of a 2D HCG-HW is depicted in Fig. 2.1. The HCG-HW consists of
two highly reflective gratings that are periodic in the z direction and infinite in the y
direction. The high-index sub-wavelength Si gratings are surrounded by low-index air. The
distance between the two HCGs is the waveguide core size (D). There are three grating
parameters that are highly related with the reflectivity: period ( Λ), air gap (a
g
), and thickness
(t
g
). When a light wave is incident on the periodic grating, it excites the eigenmodes of the
grating bars. With the proper choice of grating parameters, the outside leakage is minimized
with the help of the eigenmodes’ phases, which leads to an extremely high reflectivity [14].
Moreover, as a result of the high index contrast, a broadband high reflectivity can be
achieved [15]. Within this study, the light is launched in the z direction with transverse
13
electric (TE) polarization (i.e., the electric field is parallel to the grating bars, E
y
≠0). With
grating parameter optimization, HCG-HW can achieve similar performance for transverse
magnetic (TM) polarization and even for the case that light propagates parallel to the grating
bars [16].
Figure 2.1 Schematic of a 2D HCG-HW consisting of two reflecting HCGs. Si high-index gratings are
surrounded by air. Light propagates in the z direction with TE polarization.
2.3 Basic Effects on Optical Propagation
The commercial software GSolver 4.2, based on rigorous coupled wave analysis
(RCWA) [17], is used to obtain the HCG reflectivity. The waveguide loss is calculated using
Eq. (1) in Ref. [9]. The refractive index of silicon is obtained by applying the Sellmeier
equation, which was found to be 3.478 at 1550 nm [18], while the refractive index of air is
1.0 in the simulation. At 1550 nm, for an HCG-HW with 15- m core size, the low-loss
region is obtained, as shown in Fig. 2.2. The optimized grating parameters are Λ=750 nm,
a
g
=430 nm, and t
g
=340 nm to achieve low loss and large fabrication tolerance simultaneously.
The contour lines in the loss plots correspond to propagation loss from 0.2 to 1 dB/m with a
14
0.2 dB/m step. The contour plots illustrate that the waveguide loss can be kept below 1 dB/m
as grating parameters vary by tens of nanometers. A 0.005-dB/m propagation loss can be
achieved using a 15- m core size HCG-HW with the optimized grating parameters. As the
loss is inversely proportional to D
2
[9], a larger D not only reduces the loss, it also broadens
the low-loss region.
Figure 2.2 Propagation loss (dB/m) at 1550 nm of a HCG-HW with 15- m core size as a function of
(a) grating period and air gap (b) grating thickness and air gap. The innermost and outmost loss
contour lines in the plots correspond to 0.2 and 1 dB/m loss, respectively.
In general, chromatic dispersion stems from two sources: material dispersion and
waveguide dispersion. Material dispersion represents a wavelength-dependent response of
the material to optical waves, while waveguide dispersion originates from the dependence of
mode confinement on wavelength. As the electric field in the HCG-HW is highly confined
within the air core, the contribution of material dispersion is very limited. Thus, the
chromatic dispersion is mainly from waveguide dispersion. We use rigorous coupled wave
15
analysis (RCWA) to calculate the chromatic dispersion of an HCG-HW. The refractive index
of silicon is obtained using a build-in Sellmeier equation.
Figure 2.3 shows the dispersion with different HCG-HW core sizes from 5 to 20 m.
We note that CD is stable around 1.525 m and the third-order dispersion (TOD) or
dispersion slope decreases with the core size. This is because the loss is inversely
proportional to the square of core size [9]. As the core size decreases, light is confined more
tightly and thus experiences more waveguide dispersion. At 1.55- m wavelength, CD
increases to 216 ps/(nm·km) for a 5- m core size, 50 times larger than the dispersion for a
20- m core size.
Figure 2.3 Dispersion profiles for different core sizes.
Figure 2.4 (a) and (b) show the impact of air gap (or duty cycle) and grating thickness
t
g
on chromatic dispersion. As the air gap and grating thickness drift by 40 nm, away from
the optimized values (i.e., air gap=430 nm and t
g
=340 nm), both CD and TOD are slightly
changed. In contrast, with a parameter drift of 80 nm, the resultant CD and TOD greatly
increase.
16
Figure 2.4 Dispersion profiles for different (a) air gaps and (b) grating thicknesses.
Using a full-vectorial finite element method (FEM), COMSOL Multiphysics 3.4, we
analyze the harmonic propagation property of the HCG-HW. The refractive indices of
silicon and air are the same as used in GSolver. To match the fundamental mode of HCG-
HW, TE polarized light with a cosine mode profile is launched into a 1-mm long waveguide.
Figure 2.5(a) shows the simulated power flow in HCG-HW with the above optimized grating
parameters for D=15 m. In such a case, optical power is highly confined within the hollow-
core air region and the light-material interaction is pronouncedly reduced. The y- polarized
electric field distribution is shown in Fig. 2.5(b). In such a waveguide configuration, the
grating is not uniform along the propagation direction. Moreover, the location of electric
field cycle varies with waveguide parameters and the operating wavelength. Therefore, the
weight of the nonlinear coefficient ( ) at different positions (z) should be considered to
obtain an effective for the entire waveguide. According to Ref. [19], a modified full
vectorial nonlinear coefficient equation is obtained for HCG-HW,
2
4
22
2
0
2
*
0
(, ) ( , )[2 ]
() ( )
ˆ 3( )
nxzn xz e e dx
zk
eh zdx
(1)
17
where k is the wave number,
0
and
0
are the permittivity and permeability in free space, n
is the refractive index, n
2
is the nonlinear refractive index and e and h are the electric and
magnetic fields. Figure 2.5(c) illustrates the sampled E
y
along the waveguide length. As can
be seen, the normalized field varies from -1 to 1; this further demonstrates the necessity of
averaging at different positions.
Figure 2.5 FEM analyses of HCG-HW (a) power flow (b) y- polarized electric field distribution (c)
sampled y- polarized electric field at different waveguide positions (D=15 m, Λ=750 nm, a
g
=430 nm,
and t
g
=340 nm).
In our simulation, we use n
2
=4.5×10
-18
m
2
/W for Si and neglect the nonlinearity of air.
To avoid the impact of boundary conditions at the start and end of the waveguide in the
simulation, we sample the field in the middle section of the simulated waveguide [9]. A
convergence test of the calculated is performed for the sampling range (up to 120 periods),
sampling rate (up to 100 per period), and simulated waveguide length (up to 3 mm). We use
a 1-mm waveguide with 80 sampled periods and 60 samples per period for the following
study to provide a <5% variation within a reasonable computing time. HCG-HW can
18
significantly reduce nonlinear effects as the optical power is highly confined in the air core
region, and only a small fraction of the optical field interacts with the nonlinear Si gratings.
As a result, the nonlinear coefficient is as low as 8.15×10
-13
/W/m for the HCG-HW with
optimized grating parameters at D=15 m.
Figure 2.6 Nonlinear coefficient as a function of (a) HCG-HW air gap (a
g
) ( Λ=750 nm, and t
g
=340 nm)
(b) HCG-HW grating thickness (t
g
) ( Λ=750 nm, and a
g
=430 nm) (c) wavelength ( ) ( Λ=750 nm,
a
g
=430 nm, and t
g
=340 nm) for core sizes (D) from 5 to 20 m with a 5- m step.
For the waveguide with submicron HCGs, the fabrication tolerance is an important
factor that should be taken into account. Figures 2.6(a) and (b) show the impact of grating
parameters on the nonlinearity of HCG-HW with different core sizes. From Fig. 2.2, we can
see that as the air gap and grating thickness increases or decreases from the optimized
19
parameters, the loss always increases. In such cases, the field confinement provided by the
gratings decreases and results in a stronger field-material interaction. Accordingly, the
nonlinear coefficient always increases. We should mention that with the increase in the air
gap or the decrease in the grating thickness, the total area of nonlinear material (Si)
decreases. However, as the significant increase in the interaction between the field and Si
gratings is dominant, it always produces a larger . Here, we note that for a >200-nm grating
parameter variation, is still negligibly low (<10
-10
/W/m). A similar trend is found for the
case of varying the grating period. Moreover, for the HCG-HW with optimized grating
parameters, decreases three orders of magnitude from 6.54×10
-10
to 1.72×10
-13
/W/m as the
core size increases from 5 to 20 m. This is because the loss is inversely proportional to D
2
[9], and the electric field in the grating changes accordingly. Consequently, with the
increasing D, the nonlinear coefficient decreases.
Broadening the operational bandwidth for an on-chip element is of great importance.
HCG-HW exhibits a broad low-loss spectral width [9], as such, a wide ultralow-nonlinearity
wavelength range is also achieved (Fig. 2.6(c)). We can see that less than one order of
magnitude of a nonlinear coefficient increment can be achieved over a 200-nm bandwidth
from 1400 to 1600 nm. Even for a small core size waveguide (D=5 m), is still less than
10
-8
/W/m for wavelength up to 1650 nm. This further shows the broadband operation
capability of the HCG-HW and its potential for different system applications.
2.4 Signal and Subsystem Performance
For high-speed signals, the accumulated chromatic dispersion in on-chip waveguides
can seriously degrade the transmitted signals. In submicron waveguides, the fabrication
20
tolerance is a important factor that should be taken into account. Figure 2.7 shows the eye
opening penalty (EOP) as a function of data rate for different fabrication deviations. We
choose 10 mW input power, -10 dBm receiver sensitivity and a 10-m propagation length.
Signal is in return-to-zero on-off-keying format (RZ-OOK). The original optimized HCG-
HW design has a 5.2×10
-3
dB/m loss and a 8.6 ps/(nm·km) CD. We used Λ=740 nm, air
gap=440 nm, t
g
=330 nm for a 10-nm variation, as it has the highest CD of 21.3 ps/(nm·km)
in ±10 nm fabrication variation, which is 2.5 times larger than that in the original design.
Because of the CD, EOP increases with the bit rate since high data rate signals have a large
spectrum bandwidth. The well fabricated HCG-HW has a penalty below 1 dB for a data rate
< 1 Tb/s. Moreover, for 1 Tb/s signals, error-free transmission with bit error rate below 10
-9
can still be achieved, after 10-m propagation. In contrast, serious signal quality penalty (> 4
dB) is induced for a 640-Gb/s RZ-OOK signal propagating in the 10-m waveguide with 10-
nm structural parameter variations. This signal integrity penalty is attributed mainly to the
increased CD due to the imperfect fabrication, since induced loss change is independent of
signal bit rate.
Figure 2.7 EOP as a function of bit rate in RZ-OOK data format for different fabrication deviations in
a 10-m-long HCG-HW.
21
Long waveguide can provide large delays. For example, a 30-m HCG-HW will give a
100-ns delay. Figure 2.8 shows the EOP as a function of waveguide length for different
fabrication deviations with a 100 Gb/s RZ-OOK signal. The original waveguide supports
high-quality signaling over a distance of > 100 m, with an EOP < 1 dB. Loss could increase
significantly with fabrication deviation. For a 20-nm structure variation, loss can be as large
as 3 dB/m, which causes a large EOP up to 8 dB in a 3-m-long waveguide. The accumulated
CD on the waveguide is too low to give noticeable penalty for a 100 Gb/s signal. Since the
nonlinearity of HCG-HW is negligibly low [12], no nonlinearity-induced penalty is observed.
This allows us to launch high-power signal into the HCG-HW.
Figure 2.8 EOP as a function of HCG-HW length for different fabrication deviations with a 100 Gb/s
RZ-OOK.
By sending a 10-GHz sine wave into the HCG-HW, we further investigate its carrier-
to-nonlinearity ratio (CNR), which is an important parameter to characterize the
performance of the waveguide in an analog link. No observable penalty is found for input
power from 1 mW up to 1 W. Here, the penalty among different waveguide length is mainly
from the chromatic dispersion.
22
Figure 2.9 Carrier-to-nonlinearity ratio as a function of input power for different HCG-HW length
with a 10-GHz sine wave.
2.5 Summary
In this chapter, we analyze the chromatic dispersion and nonlinearity of two-
dimensional HCG-HW. Two-dimensional HCG-HWs have low loss, low dispersion and
negligible low nonlinearity, and thus can potentially provide a linear on-chip medium.
System-level performance is analyzed for both digital and analog signals. In simulation,
high-speed digital signal shows <0.2-dB penalty after 30-m propagation. No observable
nonlinearity induced penalty is found for input power up to 1 W for analog signals.
23
2.6 References
[1] Y. T. Dai, Y. Okawachi, A. C. Turner-Foster, M. Lipson, A. L. Gaeta, and C. Xu,
"Ultralong continuously tunable parametric delays via a cascading discrete stage," Optics
Express 18, 333-339 (2010).
[2] S. R. Nuccio, O. F. Yilmaz, X. Wang, H. Huang, J. Wang, X. Wu, and A. E. Willner,
"Higher-order dispersion compensation to enable a 3.6 mu s wavelength-maintaining delay
of a 100 Gb/s DQPSK signal," Optics Letters 35, 2985-2987 (2010).
[3] P. Dong, W. Qian, S. R. Liao, H. Liang, C. C. Kung, N. N. Feng, R. Shafiiha, J. A. Fong,
D. Z. Feng, A. V. Krishnamoorthy, and M. Asghari, "Low loss shallow-ridge silicon
waveguides," Optics Express 18, 14474-14479 (2010).
[4] Y. Yue, L. Zhang, J. A. Wang, R. G. Beausoleil, and A. E. Willner, "Highly efficient
nonlinearity reduction in silicon-on-insulator waveguides using vertical slots," Optics
Express 18, 22061-22066 (2010).
[5] D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, "Integrated ARROW waveguides
with hollow cores," Optics Express 12, 2710-2715 (2004).
[6] Y. Sakurai, and F. Koyama, "Control of group delay and chromatic dispersion in tunable
hollow waveguide with highly reflective mirrors," Japanese Journal of Applied Physics Part
1-Regular Papers Short Notes & Review Papers 43, 5828-5831 (2004).
[7] M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, "A surface-emitting laser
incorporating a high-index-contrast subwavelength grating," Nature Photon. 1, 119-122
(2007).
[8] Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, "Surface-
normal emission of a high-Q resonator using a subwavelength high-contrast grating," Opt.
Express 16, 17282–17287 (2008).
[9] Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, "A novel
ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings," Opt.
Express 17, 1508-1517 (2009).
[10] B. Pesala, V. Karagodsky, F. Koyama, and C. Chang-Hasnain, "Novel 2-D High-
Contrast Grating Hollow-Core Waveguide," in Proc. of CLEO’2009, Paper CMQ7 (2009).
[11] Y. Zhou, V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, "Ultra-Low Loss
Hollow-Core Waveguides Using High-Contrast Gratings," in Proc. of CLEO’2009, Paper
CFV7 (2009).
[12] Y. Yue, L. Zhang, J. Wang, Y. Xiao-Li, B. Shamee, V. Karagodsky, F. G. Sedgwick, W.
Hofmann, R. G. Beausoleil, C. J. Chang-Hasnain, and A. E. Willner, "A "Linear" High-
24
Contrast Gratings Hollow-Core Waveguide and Its System Level Performance," in Proc. of
OFC’2010, Paper OTuI5 (2009).
[13] Y. Yue, L. Zhang, F. G. Sedgwick, B. Shamee, W. Yang, J. Ferrara, C. Chase, R. G.
Beausoleil, C. J. Chang-Hasnain, and A. E. Willner, "Chromatic Dispersion Variation and Its
Effect on High-Speed Data Signals due to Structural Parameter Changes in a High-Contrast-
Grating Waveguide," in Proc. of IPC’2010, Paper ThB2 (2010).
[14] V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, "Dispersion
properties of high-contrast grating hollow-core waveguides," Opt. Lett. 35, 4099-4101
(2010).
[15] C. F. R. Mateus, M. C. Y. Huang, D. Yunfei, A. R. Neureuther, and C. J. Chang-
Hasnain, "Ultrabroadband mirror using low-index cladded subwavelength grating," IEEE
Photon. Technol. Lett. 16, 518-520 (2004).
[16] M. Kumar, C. Chase, V. Karagodsky, T. Sakaguchi, F. Koyama, and C. J. Chang-
Hasnain, "Low Birefringence and 2-D Optical Confinement of Hollow Waveguide With
Distributed Bragg Reflector and High-Index-Contrast Grating," IEEE Photon. J. 1, 135-143
(2009).
[17] M. G. Moharam, and T. K. Gaylord, "Rigorous Coupled-wave analysis of planar-
grating diffraction," Journal of the Optical Society of America 71, 811-818 (1981).
[18] M. Bass, C. M. DeCusatis, J. M. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V.
N. Mahajan, and E. V. Stryland, Handbook of Optics, 3rd edition, Vol. 4., (McGraw-Hill
2009).
[19] A. V. Shahraam, and T. M. Monro, "A full vectorial model for pulse propagation in
emerging waveguides with subwavelength structures part I: Kerr nonlinearity," Opt. Express
17, 2298-2318 (2009).
25
Chapter 3 Three ‐dimensional Chirped High ‐contrast
Grating Hollow ‐core Waveguide
3.1 Introduction
Hollow-core waveguides (HW), which guide light in a low-index air core, show
promise for achieving low propagation loss, small chromatic dispersion, and negligibly low
nonlinearity [1, 2, 3]. All these features are quite significant for applications that require
small distortion, such as true time delay [4] and high-power ultrashort pulse delivery [5].
Moreover, HWs provide the possibility to fill different gaseous or liquid materials into the
hollow core, and potentially open up a host of applications, such as nonlinear optics, sensing,
quantum optics, and biophotonics [6, 7, 8, 9, 10, 11, 12, 13].
With the advance of integrated optics, it is highly desirable to make the waveguide
element compact and even realize on-chip integration. During the past decade, integrated
HWs have been demonstrated using Bragg and antiresonant Fabry–Pérot reflectors [14, 15,
16, 17, 18]. However, the loss of on-chip hollow-core Bragg and antiresonant reflecting
optical waveguides has remained high. Moreover, multiple-layer deposition tends to
complicate the fabrication process. Recently published works have shown that the ultrahigh
reflectivity of high-contrast-gratings (HCGs) can be used to enable low loss waveguiding,
achieve unique dispersion properties and ultralow nonlinearity in two-dimensional (2D)
waveguides [19, 20]. Moreover, the HCG-HWs tend to require fewer process steps than the
others and, thus, facilitate the fabrication [21].
26
A three-dimensional (3D) HCG-HW can be formed by bonding two HCGs [21, 22].
Due to high reflectivity, the two HCGs can confine the light in the transverse direction. In
order to confine light laterally, “core” and “cladding” regions have to be created by changing
grating parameters, which gives rise to an “effective” index difference. This lateral
confinement might not be perfect and induce unwanted losses. A laudable goal would be to
find a structural design that reduces the losses due to non-perfect lateral confinement.
In this chapter, we propose and simulate the use of chirped gratings between the core
and cladding regions in order to lower the propagation loss in the 3D HCG-HW. Similar to a
graded-index configuration [23], the adiabatically chirped gratings can significantly reduce
the interruptive effective-index mismatch induced leakage through the interface between the
core and cladding without compromising the lateral index-contrast [24]. As a result, a >10×
reduction in loss can be achieved, dropping the loss from 1.62 to 0.13 dB/cm after
introducing the chirped gratings for a waveguide with a ~10- m core width and 5- m core
height. Using a wider core width (~23 m) and optimized core height (~6 m), the
propagation loss can be further reduced to 0.04 dB/cm. The chirped HCG-HW with 6- m
core height also exhibits a 30-nm and 141-nm bandwidth for <0.1-dB/cm and <1-dB/cm
propagation loss, respectively. Moreover, it can potentially achieve a very low nonlinearity
(~10
-5
/W/m).
3.2 Waveguide Structure and Mode Property
The schematic of a 3D HCG-HW is depicted in Fig. 3.1(a), in which light propagates
in the z direction (parallel to the grating bars). It is formed by two parallel silicon-on-
insulator wafers. Each wafer has a 340-nm Si layer, a 2- m SiO
2
space layer on a Si
27
substrate. The waveguide width (W) and core height (D) are 100- m and 5- m, respectively.
There are three grating parameters that are highly related with the grating reflectivity: period
( Λ), air gap (a
g
), and thickness (t
g
). The refractive indices of silicon and silica are obtained
according to the Sellmeier equation in our model [25]. At 1550 nm, they are 3.478 and 1.444,
respectively.
Figure 3.1 (a) Schematic of a 3D HCG-HW consisting of two reflecting HCGs in the y direction. Si
high-index gratings are on top of a 2- m SiO
2
layer and Si substrate. Two different grating designs
are chosen as core and cladding to provide the effective index difference and lateral confinement.
Linearly chirped gratings along the x direction are used to mitigate the effective refractive index
mismatch induced leakage. The light propagates in the z direction with TE polarization. (b) TE field
(power norm) distribution in the chirped HCG-HW.
Commercial software, GSolver 4.2, based on rigorous coupled wave analysis (RCWA)
[26], is used to calculate the HCG reflectivity and design grating parameters. The gratings on
the two wafers can provide light confinement in the y direction. In order to confine light in
the x direction, we choose two different sets of grating parameters for the core and cladding
regions to form a lateral effective index difference. The grating parameters " Λ=1170 nm,
a
g
=450 nm" and " Λ=1080 nm, a
g
=410 nm" for the core and cladding regions are chosen to
achieve high reflectivity and large effective refractive index difference simultaneously. The
grating thickness in all simulations is 340 nm. With the above grating parameters, there is a
7.6×10
-4
effective refractive index difference between the core and cladding region for a
28
HCG-HW with a 5- m core height. The gratings are designed to achieve high reflectivity for
the fundamental transverse electric (TE) mode (i.e., x-polarized). With grating parameter
optimization, HCG-HW can achieve similar performance for the fundamental mode in the
transverse magnetic (TM) polarization and also for the case that light propagates
perpendicularly to the grating bars. By properly controlling the launching condition, the
fundamental mode can be effectively excited [22]. N
core
and N
chirp
are the number of grating
periods in the middle core region and the number of grating periods in the chirped region at
one side. As a result, the number of grating periods in the effective core region N
eff_core
equals
the sum of N
core
and N
chirp
. In the simulation, the nonlinear refractive indices n
2
are 4.5×10
-18
and 2.6×10
-20
m
2
/W, for silicon and silica, respectively [27, 28]. Using a full-vectorial finite
element method (FEM), with perfectly matched layer on the boundaries, we obtain the
electromagnetic field of the TE mode and its propagation loss, while the nonlinear
coefficient is computed using a full-vectorial equation [29]. From Fig. 3.1(b) we can see that
the TE field is well confined within the air core region.
The unique light confinement mechanism of HCG-HW also provides novel
waveguiding characteristics. Figure 3.2(a) shows the TE field (power norm) distributions at
different wavelength in the chirped HCG-HW. Typically, the mode field increases with
wavelength in regular waveguide, while it decreases in the HCG-HW. This unique property
is attributed to the wavelength dependence of the effective refractive index difference
between the core and cladding regions. We investigate the effective refractive indices of the
modes in the slab waveguides (light confinement only in the y direction), which is formed by
uniform gratings as the core region ( Λ=1170 nm, a
g
=450 nm) or cladding region ( Λ=1080
nm, a
g
=410 nm). As shown in Fig. 3.2(b), the effective index difference between core and
29
cladding regions increases from 3.1×10
-4
to 2.2×10
-3
as wavelength increases from 1450 to
1600 nm. Consequently, the mode is confined tighter and, thus, the effective mode area of
the TE mode decreases from 117.8 to 61.9 m
2
.
Figure 3.2 (a) Wavelength dependence of the TE field (power norm) distribution in the chirped HCG-
HW. (b) Effective index difference (between the core and cladding regions) and effective mode area
of the TE mode as a function of wavelength from 1450 to 1600 nm.
3.3 Chirped Grating and Its Effect on Waveguide Properties
The chirping profile of the grating is a key parameter to optimize for achieving
desired features. Here we investigate different parabolic chirping profiles as shown in Fig.
3(a) for lowering the propagation loss according to Eq. (1),
(1)/2
() ( ( 1)/2)
22 (1)/2
k
core clad core clad
nN
nsignnN
N
(1)
where,
core
,
clad
, (n) are the periods of the grating in the core region, the grating in the
cladding region, and the n
th
grating in the chirped region, respectively. N denotes the number
of the grating period in the chirped region, while k is the chirp rate in the parabolic function.
As the green dot line shown in Fig. 3.3(a), a linear profile has a chirp rate k=1.
30
Figure 3.3(b) shows the effect of chirp rate on propagation loss of HCG-HW with
N
eff_core
=15. For small N
chirp
, a smooth chirp profile (k≈0.8) gives the lowest loss. For the
case with enough chirped gratings, a chirp rate k≈1 gives the lowest loss. As k=1 brings
relatively low loss for most cases, we use linearly chirped HCG-HW for the following study.
Figure 3.3 (a) Chirping profile for different chirp rate (k). (b) Propagation loss as a function of chirp
rate for HCG-HW with 15 effective gratings in the core region.
Figure 3.4 Propagation loss at 1550 nm as a function of the grating period number in the chirped
region (N
chirp
) for different numbers of periods in the effective core (N
eff_core
=9, 15, 21). N
chirp
=0 stands
for no chirped gratings between the core and the cladding regions (step-index case).
Figure 3.4 shows the propagation loss at 1550 nm as a function of N
chirp
for different
N
eff_core
. Here N
chirp
=0 stands for no chirped gratings between the core and the cladding (step-
index case). The HCG-HW with a larger core width has smaller loss as the evanescent field
decreases exponentially towards the waveguide edge and gives reduced field intensity at the
31
interface between the core and cladding. FEM simulation shows that the field leakage
through the interface can be significantly reduced by introducing chirped gratings. For a
HCG-HW with N
eff_core
=9, the loss can be reduced by ~12× from 1.62 to 0.13 dB/cm after
introducing the 6 linearly chirped gratings at both sides of the core region.
Figure 3.5 (a) Propagation loss as a function of wavelength from 1500 to 1600 nm for different
grating period number in the chirped region (N
chirp
=6, 12, 18) with fixed grating period number in the
core region (N
core
=3) and core height (D=5 m). (b) Propagation loss at 1550 nm as a function of
waveguide core height (D) from 3 to 10 m for different grating period number in the chirped region
(N
chirp
=6, 12, 18) with fixed grating period number in the core region (N
core
=3).
To take advantage of the inherent broad bandwidth in the optical domain, broadening
the low-loss wavelength range for an on-chip element is of great importance. The
propagation loss as a function of wavelength for the 3D chirped HCG-HW with a 5- μm core
height is shown in Fig. 3.5(a). For this HCG-HW design, the minimal propagation loss is
achieved between 1550 and 1560 nm. The wavelength location of low-loss window can be
well controlled by choosing proper grating parameters [19]. As shown in the figure,
increasing N
chirp
(larger core width) can reduce the propagation loss and broaden the low-loss
spectral width simultaneously. The 3D HCG-HW with 21 effective core grating periods
(N
core
=3, N
chirp
=18) exhibits a 20-nm spectral width from 1545 to 1565 nm for <0.1-dB/cm
propagation loss. As shown in Fig. 3.6, chirping the gratings can pronouncedly increase the
32
low-loss bandwidth, especially for small core width HCG-HW. For a 1-dB/cm loss
requirement, the waveguide can operate over 104 nm.
Figure 3.6 <1-dB/cm loss bandwidth as a function of grating period number of the effective core from
7 to 21 for step-index (N
chirp
=0) and chirped (N
core
=3) HCG-HW with 5- m core height.
If a microelectromechanical system (MEMS) actuator is added to change the HCG-
HW core height, large tunability of waveguide properties can be potentially achieved [14, 15,
16]. The transverse leakage is inversely proportional to D
2
in 2D HCG-HW [19]. Figure
3.5(b) illustrates the impact of core height on the propagation loss of 3D chirped HCG-HW.
With the decrease of core height, the effective index difference between the core and
cladding regions increases and causes a larger leakage through the interface. While for larger
core height, the effective refractive index difference decreases, the lateral confinement is not
strong enough, which results in a larger lateral leakage. Therefore, there is a tradeoff
between the leakage through the interface and lateral leakage. The optimal core height for
the lowest loss is between 5 and 6 μm. For a HCG-HW (N
core
=3, N
chirp
=18) with a 6- μm core
height, a propagation loss as low as 0.04 dB/cm can be achieved at 1550 nm. Simulation
shows that the propagation loss can be further reduced for a wider core or a larger
waveguide width.
33
From Fig. 3.5, we can see that introducing more chirped gratings can efficiently
broaden the bandwidth and increase the core height tuning range for low-loss operation. To
further explore the effect of chirped gratings, the contour plot of propagation loss for the
HCG-HW as a function of wavelength and waveguide core height is shown in Fig. 3.7. In
the figure, we can see that chirping the grating between the core and cladding regions can
efficiently broaden the low-loss window, which means it can simultaneously broaden the
operation range for wavelength and waveguide core height. For a chirped HCG-HW (N
core
=3,
N
chirp
=18) with a 6- μm core height, <0.1-dB/cm loss can be maintained over a 30-nm
spectral width. Moreover, one can achieve <1-dB/cm loss over a 141-nm wavelength range
from 1452 to 1593 nm. Furthermore, as shown in the dark blue region of Fig 3.7(b), <0.1-
dB/cm loss can be achieved with 2- μm core height tuning over 20-nm bandwidth.
Figure 3.7 Contour plot of propagation loss for the HCG-HW (N
core
=3) vs. wavelength and core
height with different number of chirped grating periods (a) N
chirp
=6 and (b) N
chirp
=18.
In general, chromatic dispersion in the waveguide stems from material dispersion and
waveguide dispersion. As light is efficiently confined within the air core region of the 3D
chirped HCG-HW, waveguide dispersion contributes mostly to the total chromatic
dispersion. Figure 3.8(a) shows the chromatic dispersion of the chirped HCG-HW (D=5 m,
34
N
core
=3) with different grating period number in the chirped region. We can see that
increasing the number of the chirped gratings reduce not only the propagation loss, but also
the waveguide and total dispersion. At 1550 nm, the chromatic dispersion decreases from
1.37 to 1.08 ps/(nm·km) after increasing N
chirp
from 6 to 18. Another way of tuning the
chromatic dispersion for HCG-HW is adjusting the waveguide core height, as it can
efficiently control the degree of interaction between the guided mode and high-contrast
gratings on the wafer. Figure 3.8(b) shows chromatic dispersion as a function of wavelength
from 1450 to 1600 nm for different waveguide core height with fixed grating period number
in the core region (N
core
=3) and in the chirped region (N
chirp
=12). As one can see, the
dispersion at 1550 nm increases from 0.5 to 4.5 ps/(nm·km) by decreasing the waveguide
core height from 7 to 3 m. Consequently, we can partially control the total dispersion of
HCG-HW by adjusting the waveguide parameters for different applications. Furthermore,
advanced grating profile might be able to extend the tailorability of the dispersion.
Figure 3.8 (a) Chromatic dispersion as a function of wavelength from 1450 to 1600 nm for different
grating period number in the chirped region (N
chirp
=6, 12, 18) with fixed grating period number in the
core region (N
core
=3) and core height (D=5 m). (b) Chromatic dispersion as a function of wavelength
from 1450 to 1600 nm for different waveguide core height (D=3, 5, 7 m) with fixed grating period
number in the core region (N
core
=3) and in the chirped region (N
chirp
=12).
35
For many on-chip applications, a linear transfer function is highly desired to prevent
the optical wave from being distorted. HCG-HWs can potentially achieve this characteristic.
From Fig. 3.9(a) we can see that the nonlinear coefficients at 1550 nm are on the order of 10
-
5
/W/m and decrease with the effective core width because a larger core width provides
better light confinement in the hollow air region. This results in less interaction between the
field and nonlinear materials. For a HCG-HW with N
eff_core
=21, a nonlinear coefficient as
low as 5.5×10
-5
/W/m is achieved. The wavelength dependence of the nonlinear coefficient
for chirped HCG-HW is shown in Fig. 3.9(b). The nonlinear coefficient of the chirped HCG-
HW increases one order of magnitude as wavelength increases from 1500 to 1600 nm. This
is because with increased wavelength, the mode is confined tighter and, thus, more
interaction with the silicon grating in the transverse direction. Moreover, one can see that
increasing the chirped gratings can reduce the nonlinear coefficient, which is due to the
reduced leakage through the core/cladding interface.
Figure 3.9 (a) Nonlinear coefficient at 1550 nm as a function of the grating period number in the
chirped region (N
chirp
) for different grating period number of the effective core (N
eff_core
=9, 15, 21).
N
chirp
=0 stands for no chirped gratings between the core and the cladding regions (step-index case). (b)
Nonlinear coefficient as a function of wavelength from 1500 to 1600 nm for different grating period
number in the chirped region (N
chirp
=6, 12, 18) with fixed grating period number in the core region
(N
core
=3).
36
3.4 Summary
In this chapter, we propose and simulate 3D chirped HCG-HW. Simulation shows that
HCG-HW possesses novel waveguiding properties due to its unique light confinement
mechanism. The chirping profile of the grating is optimized to achieve low propagation loss.
By adding adiabatically chirped gratings, one can significantly (>10×) reduce the mode
leakage through the interface between the core and cladding without compromising the
lateral index-contrast. A propagation loss as low as 0.04 dB/cm can be achieved using HCG-
HW with a ~23- m core width and ~6-m core height. The 3D chirped HCG-HW also
exhibits a 30-nm and 141-nm bandwidth for <0.1-dB/cm and <1-dB/cm propagation loss,
respectively. For a chirped HCG-HW (N
core
=3, N
chirp
=18), <0.1-dB/cm loss can be achieved
with 2- m core height tuning over 20-nm bandwidth. The chromatic dispersion of HCG-HW
can be partially controlled by adjusting the waveguide parameters. Furthermore, it can
potentially provide an on-chip medium with ultralow nonlinearity (~10
-5
/W/m).
37
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38
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with hollow cores," Optics Express 12, 2710-2715 (2004).
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2-d waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum
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novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,"
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40
Chapter 4 Silicon Polarization Devices for Polarization
Diversity System
4.1 Introduction
Pol-Mux, one of the commonly used advanced modulation formats, provides a
straightforward method to double the spectral efficiency in terms of bits/(s·Hz) [1]. However,
the polarization of an optical wave is at the heart of different data degrading effects, such as
polarization-dependent-loss [2]. Recent reports have shown that differential group delay
(DGD) and polarization mode dispersion (PMD) can have a dramatic impact on the
performance of the Pol-Mux system [3]. Therefore, the control of the signal’s state-of-
polarization is quite desirable.
On the other hand, in on-chip scenario, because of the high index contrast in the
integrated devices, one of the main issues is the serious polarization dependence of their
performance. Theoretically, polarization-independent integrated devices can be achieved by
accurately controlling the geometric parameters [4]. However, it is not trivial in real
fabrication. A 5-nm fabrication variation might introduce a DGD up to 6 ps for a 5-cm long
waveguide [5]. For high-speed communications (>100 Gb/s), 6 ps is close to the time length
of one bit.
To overcome this problem, a polarization diversity scheme as shown in Fig. 11 has
been proposed [5, 6]. The input light is separated according to the polarization state after
passing through the first polarization splitter (green: x-polarized and blue: y-polarized). Then,
the polarization of the x-polarized light is rotated by 90
o
by the polarization rotator so that
original two polarization states of light experience the same responses from the sequent
41
signal-manipulating devices in the two arms as these devices are designed to be identical.
Note that there might also be some shared components between the two arms as shown in
the Fig. 4.1. After the second 90
o
polarization rotator and polarization combiner, the input
light with arbitrary polarization experiences polarization-insensitive operation. The
polarization diversity approach can find an important application in coherent receivers [7].
Compared with conventional detection techniques, coherent detection can provide
compensation for all linear impairments (e.g. dispersion, PMD) and better noise resilience [8]
in high-speed signaling. Note that polarization splitters and rotators are the crucial devices in
the polarization diversity system, and we discuss them in the following sections.
Figure 4.1 On-chip polarization diversity scheme.
In this chapter, we propose and simulate an on-chip 90
o
polarization rotator using
wave coupling through an intermediate, multimode, uniform waveguide (WG) [9]. The
coupling efficiency of the x- and y-polarized fundamental modes between the horizontal and
vertical rectangular waveguides is remarkably enhanced by their proximity to the TE
01
mode
in the multimode waveguide. This newly designed polarization rotator has a very short
polarization conversion length (~21 m). Moreover, we propose and analyze a short
polarization splitter/combiner using two horizontally-slotted waveguides [10]. Introducing
42
horizontal slot remarkably enhances the polarization dependence of the coupling, and the
ratio of the coupling lengths is increased up to 21. This proposed on-chip polarization splitter
can be as short as 46.7 m.
4.2 Higher ‐order ‐mode Assisted Polarization Rotator
The simple act of on-chip rotation of the polarization by 90
o
is not trivial. To realize
polarization rotation, methods such as acousto-optic [11] or electro-optical [12] effects have
been used. Passive polarization rotators have also been studied extensively. Reported
techniques that require precise fabrication of longitudinally varying structures include using
waveguide tapers [13], photonic crystal patterns [14], periodic asymmetric waveguides [15],
and an off-axis double-core structure [16, 17]. Slanted-sidewall waveguides were also used
to realize polarization rotation [18]. A polarization rotator with a uniform and vertical
sidewall structure over propagation direction is highly desired to reduce the fabrication
complexity.
To achieve efficient energy transfer and mode coupling among waveguides, there are
two important rules. One is that the effective refractive indices of the modes should match
each other, and the other is that the directions of the electric fields of the modes in the
coupling area should not be orthogonal to each other. The coupling coefficient is highly
related with these two conditions. Figure 4.2(a) and (b) show the coupling efficiency of
coupled waveguides with the 90
o
rotated mirror structure. The simulation results indicate
that the electric fields of the refractive index matched waveguides, as illustrated in Fig.
4.2(a), cannot couple with each other, which is attributed to the fact that their electric fields
are mostly orthogonal to one another. For the structure in Fig. 4.2(b), the modes of the two
43
waveguides can couple mutually and generate two supermodes, which are analogous to a
symmetric mode and antisymmetric mode in regular directional couplers. Although the
electric fields have parallel components within the coupling region, the coupling effect is
very weak since the coupling length of these two waveguides is relatively long. We find that
the coupling length of scheme (b) is around 1.2 mm for the same structural parameters as in
scheme (c). We introduce WG 2 near WGs 1 and 3 so as to enhance the coupling between
WGs 1 and 3 and make the polarization rotator more compact, as illustrated in Fig. 4.2(c).
The electric field of the TE
01
mode in WG 2 has a parallel electric field component to that of
x-polarized fundamental mode in WG 1. Also, WG 2 has a parallel electric field component
to that of the y-polarized fundamental mode in WG 3. Good electric field alignment
produces a large coupling coefficient, which gives efficient energy transfer. As a result, the
coupling length of scheme (c) is reduced to 21 m, only 1/60 of the coupling length for
scheme (b).
Figure 4.2 Operating principle (a) No coupling case. (b) Weak coupling case. (c) Proposed 90
o
polarization rotator using wave coupling through an intermediate multimode waveguide. (d)
Schematic of proposed 90
o
polarization rotator.
A full-vector finite element method (FEM) is used to analyze the mode properties,
which can give an accurate solution for the field distributions. WGs 1, 2 and 3 are made of
silicon and we choose silica as the cladding. We obtain the refractive indices of silicon and
44
silica according to the Sellmeier equation in our model, taking the material dispersion into
account. The refractive indices of silicon and silica at 1550 nm are 3.4764 and 1.4440,
respectively. To match the effective refractive index of the TE
01
mode in multimode WG 2
with those of the x- and y-polarized fundamental modes in WGs 1 and 3, the parameters of
W=182 nm, D=300 nm and T=500 nm are chosen.
Figure 4.3 (a) X-, Y- polarized electric field distributions and electric field lines of the three
supermodes in the proposed waveguide structure. (b) Equivalent E-field distribution of the
supermodes in three waveguides (solid line: fundamental modes, dot line: TE
01
mode).
There are three supermodes in this three-waveguide structure as it is similar to a three-
core directional coupler [19]. Figure 4.3(a) shows the two orthogonal x- and y- polarized
electric fields of these two symmetric and one antisymmetric modes. The figures in the
bottom row also show the electric field lines within the waveguides for different supermodes.
According to the relationship of electric field lines in adjacent waveguides (WG 1-2, WG 2-
3), we get the equivalent E-field distribution for the three supermodes as shown in Fig.
4.3(b). The solid line represents the fundamental mode, while the dotted line denotes the
TE
01
mode. If the electric field lines of the adjacent waveguides are of the same direction,
then the equivalent E-field distribution will have the same polarity; otherwise, the polarity
45
will change. Figure 4.4(a) shows the effective indices of the three supermodes and Fig. 4.4(b)
illustrates the conversion lengths as a function of wavelength. The modal conversion length
between WGs 1 and 3 is determined by
112
/[2 ( )]
ceffeff
Lnn (1)
while the conversion length for the TE
01
order mode in WG 2 is obtained from
213
/ [2 ( )].
ceffeff
Lnn (2)
Here, n
eff1
, n
eff2
and n
eff3
are the effective refractive indices of the supermode 1, 2 and 3
respectively. The effective refractive indices of the supermodes are equally spaced, i.e.,
13 2 3
2( )
eff eff eff eff
nn n n (3)
to obtain a highly efficient power transfer between WGs 1 and 3. According to these
equations, we can also find that L
c1
=2L
c2
is the optimized case (short conversion length and
high conversion efficiency). From Fig. 4.4(b) we can see that 1543 nm emerges to be the
optimized wavelength for the structure parameters we choose.
Figure 4.4 (a) Effective refractive indices of the three supermodes in Fig. 4.3 as a function of
wavelength. (b) Conversion lengths of the fundamental and TE
01
modes.
46
We obtain the electric field (E) distribution and propagation constant ( ) of the each
supermode through FEM mode solver. In our current configuration, we can calculate E
z
to be
less than 10
-3
of E
x
and E
y
allowing us to consider the x- and y- components of E. By
substituting them into Eq. (4), we further verify the propagation characteristics of the
proposed polarization rotator.
3 12
11 2 2 3 3
(, ) ( , ) (, ) ( , )
jz jz j z
E x y k EA xy e k E x y e k E xy e
(4)
where the vector field in term of the components can be expressed as
ˆˆ ˆ ˆ (, ) (, ) (, ), (, ) (, ) (, ),
xy i ix iy
E x y E xy x E xy y E xy E x y x E x y y
(5)
and k
i
is the normalized coefficient for the linear superposition of the supermodes to generate
an electric field input only in WG 1, where i = 1, 2, 3 for three different supermodes.
Figure 4.5(a) and (b) show the normalized power transfer of the x- and y- polarizations
along the propagation distance. We note that the conversion length of the power from WG 1
to WG 3 is around 21 m, which is twice the conversion length of the TE
01
mode in WG 2.
This result is in good agreement with the conversion length estimated by using Eq. (1) and (2)
since E
z
is relatively small. From 0 m to 10.5 m, the energy is transferred from the x-
polarized fundamental mode in WG 1 to the TE
01
mode in WG 2. Meanwhile, the TE
01
mode
in WG 2 is coupled with the y-polarized fundamental mode in WG 3 and transfers the power
to WG 3. In this process, the power in WG 1 decreases while the power accumulates in WG
2 and WG 3. At the end of this process (10.5 m), the power in WG 1 is equal to the power
of WG 2 along the x-polarization, and the power of WG 2 along the y-polarization is equal to
the power in WG 3. From 10.5 m to 21 m, the powers in WGs 1 and 2 decrease
47
simultaneously and transfer to WG 3, resulting in the power in WG 3 reaching its maximum
at 21 m. The result illustrates that an extinction ratio of 17.22 dB can be achieved, which
shows the best case of polarization conversion from WG 1 to WG 3 using the proposed
structure.
Figure 4.5 (a) X-polarized normalized power exchange along the propagation distance in three
waveguides. (b) Y-polarized normalized power exchange along the propagation distance in three
waveguides.
Figure 4.6 PCE as a function of wavelength and >90% PCE Bandwidth.
PCE is a key parameter in evaluating the property of polarization rotators, which is
defined as the percentage of the power transfer from the input polarized mode to the output
orthogonally polarized mode. We further examine the PCE as a function of wavelength in
order to evaluate the performance of the proposed polarization rotator. Figure 4.6 indicates
48
that, for a PCE above 90%, the polarization rotator with the proposed parameters has a 68-
nm bandwidth around 1543 nm.
For the submicron and high index contrast waveguides, the accuracy of fabrication is
still an important factor which should be taken into account. Figure 4.7 (a) and (b) show the
PCE variation for different waveguide distances (D) and waveguide widths (W), while the
size of multimode waveguide (T) remains the same. The solid lines correspond to the PCE
for a waveguide of 21 m length. We find that the D tolerance is around 40 nm and the W
tolerance is only about 6 nm to keep the PCE above 90%. This is attributed to the fact that
the variation of W significantly affects the effective refractive indices of the three
supermodes compared with the variation of D. The dash-dotted line and dashed line
represent the corresponding conversion lengths for the maximal PCE. In the case of
imprecise D, optimizing the length of WG (L
c max
) accordingly can largely compensate for
the PCE. Figure 4.7(a) shows that for a ±50 nm variation, the maximal PCE can still achieve
95%, while for the case of nonideal W, even with the optimized conversion length L
c max
, the
PCE can only improve a little. Here, the effect of the compensation is related to the slope of
the L
c max
curve. Also, PCE can achieve its maximum with a proper wavelength, where Eq. (3)
holds.
The fabrication tolerance shown in Fig. 4.7(a) and (b) is based on the same D and W
variation of WG 1 and 3. In reality, a different D and W variation is much more common
during fabrication. Figure 4.8(a) and (b) further show the PCE for varying only one the
WG’s D and W values, while keeping other parameters unchanged. For the D and W
variations of one WG, we obtain results similar to the D and W variation of WG 1 and 3,
indicating that the D tolerance is much better than W. From Fig. 4.8(a) we observe that
49
for the optimized conversion length, a 90% PCE can be achieved for a ±40-nm D variation.
For W, a ±5-nm change will decrease the PCE to 70%. There are several steps of
depositing and etching to fabricate the proposed structure. It is important to mention that
there might be some bumps in the layer between WG 1 and 2, 3 during fabrication and thus
bending WG 1.
Figure 4.7 (a) PCE@21 m, optimized PCE and its corresponding conversion length as a function of
waveguide distance (D). (b) PCE@21 m, optimized PCE and its corresponding conversion length as
a function of waveguide width (W).
Figure 4.8 (a) PCE@21 m, optimized PCE and its corresponding conversion length as a function of
WG distance variation ( D). (b) PCE@21 m, optimized PCE and its corresponding conversion
length as a function of WG width variation ( W).
4.3 Slotted ‐waveguides ‐based Polarization Splitter
A Slot waveguide, which exhibits unique mode-propagation properties, was recently
proposed [20]. Slot-waveguide-based modulators [21], sensors [22], polarization-
50
independent couplers [23], dispersion compensators [24] and polarization splitters [25, 26]
have been reported. In these polarization splitters, the polarization dependence of waveguide
coupling is designed to obtain an even multiple (i.e., 2) of the two polarization states’
coupling lengths. This may cause a compromise to the device size or operation bandwidth
(BW) of the polarization splitter.
Strip waveguide coupler exhibits its intrinsic polarization dependence [27], which
means that one polarization state has a stronger coupling than the other. The x-polarization
usually experiences a weaker coupling than the y-polarization. Polarization-independent
coupler was obtained using a pair of vertically-slotted waveguides [23], where the
introduction of the vertical slots shrinks the width of silicon parts and enhances the x-
polarization’s evanescent field and its coupling. As a consequence, the coupling strength is
balanced between the two polarizations. For the horizontal slot waveguide coupler, a fair
amount of the y-polarized mode is confined within the slots and experiences a relatively low
index contrast between the silica slots and air, which increases the coupling of y-polarized
mode and further enhances the polarization dependence. Therefore, the ratio between the x-
and y- polarized coupling length is increased remarkably. In this case, when the power of the
y-polarized (quasi-TM) mode is completely transferred from the input waveguide to the
other waveguide, almost all of the power of the x-polarized (quasi-TE) mode still remains in
the input waveguide. This means that only one coupling length of the strongly coupled
polarization is needed to achieve polarization splitting, which allows for a further reduction
of the splitter’s size and a released requirement on the coupling length ratio.
Figure 4.9 shows the cross-section view of the proposed horizontally-slotted
waveguide polarization splitter. The refractive indices of silicon (n
H
) and silica (n
S
) are
51
obtained according to the Sellmeier equation in our model to take the material dispersion
into account. At 1550 nm, n
H
is equal to 3.4764 and n
S
is equal to 1.4440. d is the spacing
between the two slot waveguides and h
S
is the slot thickness while we keep the total height
(h) and width (w) of slot waveguide at 360 nm and 500 nm, respectively. Using a finite-
element mode solver, we calculate the effective indices of the supermodes for the coupled-
waveguide structure.
Figure 4.9 Cross-section view of the horizontally-slotted waveguide polarization splitter.
Splitting length is an important parameter for a polarization splitter. The coupling
lengths of the two orthogonal polarizations are obtained according to Eq. (6) and (7),
)] ( * 2 /[
sym anti TE sym TE TE
n n L
(6)
)] ( * 2 /[
sym anti TM sym TM TM
n n L
(7)
where L
TE
and L
TM
are the coupling lengths of quasi-TE and quasi-TM modes, where n
TE sym
,
n
TE anti-sym
, n
TM sym
, and n
TM anti-sym
are the effective refractive indices of symmetric and anti-
symmetric supermodes of x- and y- polarizations, respectively. The splitting length equals
the relatively short coupling length L
TM
.
52
The coupling length of L
TM
of the quasi-TM mode and the coupling length ratio L
TE
/L
TM
of the quasi-TE and quasi-TM mode is shown in Fig. 4.10. We observe that the
coupling length of the quasi-TM mode decreases dramatically as the slot thickness h
S
increases from 0 to 60 nm. Further enlarging the slot thickness, L
TM
decreases slowly. This is
because that the y-polarized E-field confined in the low refractive index slot region increases
with the slot thickness for small value h
S
, while it tends to saturate for larger h
S
. However,
for the quasi-TE mode, the E-field in the slot region increases smoothly with the slot
thickness. As a result, L
TE
/L
TM
, which is closely related to ER, has its peak value. In our
simulation, we find that L
TE
/L
TM
reaches its maximum 21 for h
S
=60 nm and =1550 nm
where L
TM
corresponds to 46.7 m.
Figure 4.10 Coupling length of the quasi-TM mode and the coupling length ratio of the quasi-TE and
quasi-TM mode as a function of the slot thickness.
Figure 4.11 shows that the normalized powers of the quasi-TE and quasi-TM modes
in the input slot waveguide of the coupler as a function of the propagation distance. Owing
to the large difference between L
TE
and L
TM
, most of the quasi-TE mode (99.35%) remains in
the input waveguide while the quasi-TM mode is fully coupled to the output waveguide.
Figure 4.12 illustrates the ERs of the quasi-TE mode in input waveguide and the quasi-TM
53
mode in the output waveguide as a function of wavelength at L = 46.7 m. ERs are around
22 dB for both polarization modes at 1550 nm. Furthermore, we examine the wavelength
dependences of the ERs. For the quasi-TM mode in the output waveguide, its ER decreases
monotonously with wavelength. Both L
TE
and L
TM
decrease with wavelength, the ratio L
TE
/
L
TM
decreases. For input waveguide, the power of the quasi-TE mode changes negligibly,
while the power of the quasi-TM mode increases tremendously. Consequently, the ER of the
quasi-TE mode reduces when the wavelength is shifted away from 1550 nm. It also exhibits
an 18-nm BW for ER>20 dB.
Figure 4.11 Exchange of the normalized power along the propagation distance of the quasi-TE and
quasi-TM modes at the input waveguide.
Figure 4.12 Extinction ratio of the quasi-TE and quasi-TM mode in a slot waveguide splitter at one
coupling length.
54
We also investigate the influence of the waveguide spacing d. Figure 4.13 shows that
L
TM
increases from 4.5 to 751.5 m as the value of d increases from 100 to 1000 m. The
reason is that the coupling tends to be weak for large spacing and thus the coupling length
increases. Moreover, the coupling of the quasi-TM modes is less affected when compared
with the coupling of the quasi-TE modes. Accordingly, L
TE
/ L
TM
increases from 5.3 to 108.
It is noted that there is a tradeoff between coupling length and ER of a slot waveguide
splitter.
Figure 4.13 Coupling length of the quasi-TM mode and the coupling length ratio of the quasi-TE and
quasi-TM mode as a function of the waveguide spacing.
Furthermore, we analyze the dependence of splitting length and coupling length ratio
on the refractive index of the slot, as silicon oxynitride can provide a refractive index from
1.47 to 2.3 [28]. As shown in Fig. 4.14, with the increase of the refractive index of the slot,
more modal field is confined within the slot region. As a result, the coupling efficiency
decreases and hence the coupling length increases. Moreover, the coupling length ratio L
TE
/
L
TM
decreases and gives rise to a reduced ER. Consequently, a low refractive index of the
slot or a high index contrast slot is highly desired for achieving a short-length and high-ER
splitter.
55
Figure 4.14 Coupling length of the quasi-TM mode and the coupling length ratio of the quasi-TE and
quasi-TM mode as a function of the slot’s refractive index.
4.4 Summary
In this chapter, we have proposed and analyzed a polarization rotator using wave
coupling through a multimode waveguide and a short polarization splitter using coupled
horizontally-slotted waveguides for on-chip polarization diversity system. A very short 21-
m conversion length is achieved with a 17.22 dB extinction ratio for polarization rotator.
The polarization splitter can be as short as 46.7 m, and exhibits a 22-dB ER for both of the
polarizations at 1550 nm.
56
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57
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/ silica sol-gel
waveguides," Opt. Express 15, 12436-12442 (2007).
[16] H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi,
"Polarization rotator based on silicon wire waveguides," Opt. Express 16, 2628-2635 (2008).
[17] Z. Wang and D. Dai, "Ultrasmall Si-nanowire-based polarization rotator," J. Opt.
Soc. Am. B 25, 747-753 (2008).
[18] H. Deng, D. O. Yevick, C. Brooks, and P. E. Jessop, "Design Rules for Slanted-
Angle Polarization Rotators," J. Lightwave Technol. 23, 432-445 (2005).
[19] J.P. Donnelly, H.A. Haus, and N. Whitaker, “Symmetric three-guide optical coupler
with nonidentical center and outside guides,” IEEE J. Quantum Electron. 23, 401-406 (1987).
[20] V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light
in void nanostructure," Opt. Lett. 29, 1209-1211 (2004).
[21] T. Baehr-Jones, B. Penkov, J. Q. Huang, P. Sullivan, J. Davies, J. Takayesu, J. D.
Luo, T. D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, "Nonlinear polymer-clad
silicon slot waveguide modulator with a half wave voltage of 0.25 V," Appl. Phys. Lett. 92,
163303 (2008).
[22] C. A. Barrios, K. B. Gylfason, B. Sanchez, A. Griol, H. Sohlstrom, M. Holgado, and
R. Casquel, "Slot-waveguide biochemical sensor," Opt. Lett. 32, 3080-3082 (2007).
[23] T. Fujisawa, and M. Koshiba, "Polarization-independent optical directional coupler
based on slot waveguides," Opt. Lett. 31, 56-58 (2006).
[24] L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, "Highly
dispersive slot waveguides," Opt. Express 17, 7095-7101 (2009).
[25] J. Xiao, X. Liu, and X. Sun, "Design of a Compact Polarization Splitter in
Horizontal Multiple-Slotted Waveguide Structures," J. Appl. Phys. 47, 3748-3754 (2008).
[26] Y. Ma, and D. Huang, "A compact slot waveguide directional coupler-based silicon-
on-insulator polarization splitter," 5th IEEE International Conference on Group IV Photonics
(GFP), 297 (2008).
58
[27] H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi,
"Ultrasmall polarization splitter based on silicon wire waveguides," Opt. Express 14, 12401-
12408 (2006).
[28] R. Machorro, E. C. Samano, G. Soto, F. Villa, and L. Cota-Araiza, "Modification of
refractive index in silicon oxynitride films during deposition," Materials Lett. 45, 47-50
(2000).
59
Chapter 5 Highly Efficient Nonlinearity Reduction in Silicon
Waveguides using Vertical Slots
5.1 Introduction
Optical strip waveguides are a key basic building block for integrated optical circuits.
Such waveguides are used to connect different elements and to form integral parts of larger
components, such as on-chip interferometers [1] and arrayed-waveguide gratings [2]. For
many applications, an important characteristic of the waveguide is its ability to maintain a
linear transfer function, preventing the optical wave, which contains an analog or digital
signal, from being distorted [3].
There has been increased interest in using silicon as the base material for photonic
integrated circuits. However, it is fairly difficult to maintain low nonlinearity in silicon
waveguides [4]. A laudable goal would be to modify the strip waveguide in such a way as to
reduce the nonlinear coefficient significantly.
The slot waveguide, which exhibits unique field confinement properties, was
proposed [5, 6] in earlier works. In such a waveguide, the electric field can be highly
confined within the low index slot region [7]. Multi-slot waveguides also have been
proposed and they have been demonstrated experimentally in recent research [8, 9, 10]. As a
result of the introduction of multi-slot waveguides, even more electric field can be
concentrated within the low index slot region. By choosing a highly nonlinear slot material,
it is possible to acquire a waveguide with ultrahigh nonlinearity [11, 12, 13].
60
In this chapter, single and double vertical slots were used to efficiently reduce the
nonlinearity of the on-chip silicon-on-insulator waveguide. By introducing an air slot in the
regular strip waveguide, approximately 50% of the optical power can be confined within the
linear material (air). The real part of nonlinear coefficient ( γ
re
of the waveguide can be
decreased significantly from 120.05 /W/m to 16.11 /W/m. Using a double-air-slot of the
same total slot width as the single slot, γ
re
can be further reduced to 6.77 /W/m. Compared
with strip waveguides of the same sizes, double-vertical-slot waveguides were shown to
reduce γ
re
over a 100-nm wavelength range by a factor greater than 15. The vertical-slot
waveguides also exhibit large negative chromatic dispersions. This can further suppress
nonlinear parametric effects, such as self phase modulation, cross phase modulation, four
wave mixing, etc. These waveguides might be applied in dispersion compensation and
tunable optical delays [14].
5.2 Waveguide Structure and Numerical Model
Figure 5.1 shows the cross sections of a strip, a single- and a double-vertical-slot
silicon waveguide on a 2- m silicon dioxide (SiO
2
) substrate. The low-index slot regions are
filled with air, which has a refractive index of one. The slot mode (x-polarized quasi-TE
mode) has an electric field parallel to the interface of the silicon waveguide and the SiO
2
substrate, and a large fraction of the optical power is confined in the air slot region. As
shown in Fig. 5.1, we consider waveguides with 500-nm height (h) and 500-nm width (w).
The term d is the slot width for both the single- and double-vertical-slot waveguides, while s
is the spacing between the two slots of the double-vertical-slot waveguide. In our model, the
refractive indices of silicon and silica were obtained according to the Sellmeier equation. At
1550 nm, they are 3.4764 and 1.4440, respectively. In the simulation, the nonlinear
61
refractive indices n
2
are 4.5×10
-18
[15, 16] and 2.6×10
-20
m
2
/W, for silicon and silica,
respectively. The two photon absorption (TPA) coefficient ( β
TPA
) of silicon is 7.9×10
-12
m/W
[15, 16], while the TPA coefficient of silica is neglected.
Using a full-vector finite element mode solver (COMSOL Multiphysics 3.4), we
obtain the electromagnetic fields of the quasi-TE modes in different waveguide structures. A
full-vector model [17] that can weigh the contributions of different materials to the nonlinear
coefficient is used to guarantee an accurate result. A nonlinear figure of merit (FOM)
defined as the real part of γ divided by 4 π times the imaginary part of γ, i.e., γ
re
/(4 π γ
im
), is
used to characterize the overall nonlinear performance. A widely used nonlinear FOM has
been defined as n
2
/(λβ
TPA
) [18]. Since γ
re
= 2 πn
2
/( λA
eff
) and γ
im
= β
TPA
/(2A
eff
) in a single,
nonlinear material, the nonlinear FOM we use here is equivalent to the widely used
nonlinear FOM. Here, A
eff
is the effective mode area. By using a perfectly matched layer in
the model, the imaginary part of the effective refractive index is obtained to estimate the
leakage loss.
Figure 5.1 Cross sections of strip, single-vertical-slot and double-vertical-slot waveguides.
62
The power distributions of the quasi-TE modes in the strip, single-vertical-slot and
double-vertical-slot waveguides are shown in Fig. 5.2. Almost all of the power of the strip
waveguide is located in the silicon region, while a large portion of the power of the slot
waveguide is concentrated in the air slot areas. Here, the slot is right in the middle of the
waveguide.
Figure 5.2 Power distributions of the quasi-TE modes in strip, single-vertical-slot and double-vertical-
slot waveguides.
5.3 Nonlinearity and Dispersion Properties
The power distributions of the single-vertical-slot waveguide in different materials are
shown in Fig. 5.3(a). As the slot width increases, the power in Si decreases, while the power
in SiO
2
increases. The air can hold the maximal amount of power when the slot has a width
of 95 nm. Figure 5.3(b) shows n
eff
and
re
as a function of the slot width in a single-vertical-
slot waveguide. For air slot waveguides, since part of the light is confined within the low
refractive index, linear air region, the effective refractive index (n
eff
) and real part of
nonlinear coefficient (
re
of the waveguide decrease as slot width increases. For efficient
mode guiding, the effective refractive index of the mode should be greater than the refractive
index of the substrate material SiO
2
(n
eff
> n
SiO2
) [9]. Also, as n
eff
approaches n
SiO2
, the
leakage loss through the substrate increases. For a 100-nm single-vertical-slot waveguide,
the nonlinearity can be reduced to 16.11 /W/m while maintaining a 1.16 dB/cm leakage loss.
63
Further increasing the slot width gives a larger leakage loss, while the nonlinearity reduction
is not significant. A 110-nm single-vertical-slot waveguide has an 8.53 dB/cm leakage loss
and
re
is 11.11 /W/m. For slot waveguide with low nonlinear coefficient, the effective
refractive index is very close to substrate index. Thus, x-polarization only supports the
fundamental slot mode.
Figure 5.3 (a) Power distributions (b) Effective refractive index and real part of nonlinear coefficient
as a function of slot width in a single-vertical-slot waveguide.
For a nanometer-scale slot, the fabrication tolerance is an important factor that should
be taken into account. Figure 5.4 shows the effect of the deviation of the slot position on
for different slot widths in a single-vertical-slot waveguide at 1550 nm. The slashed region
indicates n
eff
< n
SiO2
. From the contour map, we can see that increases as the slot position
varies. Simulation results indicated that a symmetrical slot structure has the highest amount
of light confined within the air. For asymmetric configuration, as the light-field distributions
in the two silicon regions are unequal, it gives a reduced light confinement in the air slot
region. For larger slot position variation, we can see that
re
first increases and then decreases
with the slot width. In such a case, one silicon region is prominently larger compared with
the other one. Increasing the slot width can first decrease the effective mode area of the
larger silicon region and give an enhanced nonlinearity. Also, from the contour map, we can
64
see that a waveguide with a larger slot width has not only lower nonlinearity, but also
smaller change of nonlinearity with position variation.
Figure 5.4 Real part of nonlinear coefficient vs. slot width and slot position variation at 1550 nm
(single-vertical-slot waveguide).
Since the multi-slot waveguide can provide a higher percentage of optical field
confinement within the slot region [8, 9, 10], we further studied the effect of reducing
nonlinearity by using a double-vertical-slot waveguide. Here, we choose a double-vertical-
slot waveguide with two 50-nm slots to make a fair comparison with the 100-nm single-slot
waveguide. The power distributions of the double-vertical-slot waveguide in different
materials are shown in Fig. 5.5(a). For the waveguide with a slot spacing of 91 nm, the
power in Si reaches its minimal value. Figure 5.5(b) shows the effect of the slot spacing
variation on n
eff
and
re
. When the distribution of optical power in silicon reaches its
minimum, the effective refractive index and real part of nonlinear coefficient of the double-
vertical-slot waveguide also have their minima. A 50-nm double-slot waveguide with a 150-
nm slot spacing can achieve a 6.77 /W/m nonlinearity while maintaining a 3.88 dB/cm loss.
Triple- or multiple-vertical-slot waveguides can further increase the light confinement in the
air region and might further reduce the nonlinearity. Also, larger waveguide size can
65
increase the effective refractive index and thus reduce the leakage loss. A 700×500 nm
2
waveguide with 3 equally distributed 50-nm slots has a 0.19 dB/cm loss and a 4.66 /W/m
re
.
Figure 5.5 (a) Power distributions (b) Effective refractive index and real part of nonlinear coefficient
as a function of the slot spacing in a double-vertical-slot waveguide.
Figure 5.6 Real part of nonlinear coefficient as a function of the wavelength in strip, single-vertical-
slot and double-vertical-slot waveguides (s = 150 nm).
Figure 5.6 illustrates the real part of nonlinear coefficient as a function of the
wavelength. From 1500 to 1600 nm, the nonlinear refractive index of Si increases with
wavelength [15, 16]. For strip waveguide, as field is highly confined within the Si region, the
increase of effective area with wavelength is much less than the change of n
2
(Si). This
induces an increased
re
for strip waveguide. However, for slot waveguides, the increment of
A
eff
is much larger and gives a decreased
re
due to the reduced light confinement at longer
66
wavelengths. For a single-vertical-slot waveguide with a 100-nm slot,
re
is reduced 80%
over a 100-nm wavelength range compared with the strip waveguide. For a double-vertical-
slot waveguide with two 50-nm slots and 150-nm slot spacing, the nonlinearity was reduced
by a factor of more than 15 (>93%) over the 100-nm wavelength range.
Table 5.1 Nonlinearity of the quasi-TE modes in silicon strip and vertical-slot waveguides at 1550 nm
Structure Parameters (w×h_d) (nm) Power in Si (%) A
eff
( m
2
) γ
re
(/W/m)
Strip (300×300) 74.52 0.0984 288.15
Strip (500×500) 96.37 0.1526 120.05
Strip (700×700) 98.43 0.2728 66.78
Single Slot (500×500_100) 26.46 0.2329 16.11
Double Slot (500×500_50_150_50) 19.50 0.1753 6.77
Table 5.1 summarizes the nonlinearity of the quasi-TE modes in the silicon strip and
vertical-slot waveguides at 1550 nm. For the rectangular silicon strip waveguide with a
width from 300 to 700 nm, the portion of the optical power that is confined within the silicon
is always more than 70%. Consequently, the nonlinearity can be reduced by increasing the
effective mode area. One straightforward way to increase the effective mode area is to
increase the size of the waveguide. A
eff
can be increased from 0.0984 to 0.2728 m
2
by
increasing the waveguide width from 300 to 700 nm. Accordingly, the real part of nonlinear
coefficient decreases from 288.15 to 66.78 /W/m. However, to reduce nonlinearity by further
increasing the size of the waveguide is not a good choice, because the effective mode area
does not increase efficiently for larger waveguide. Also, larger waveguides have more
higher-order modes. By introducing a 100-nm vertical air slot in the silicon waveguide, one
can have only 26.46% of the optical power confined in the silicon. Thus, a further reduction
in the
re
to 16.11 /W/m is achieved. In addition, by splitting the 100-nm single air slot into
67
two 50-nm air slots, the waveguide has only 19.50% of the optical power in the silicon and
gives an even larger amount of light to the air slots. This helps to reduce
re
further, to 6.77
/W/m.
Figure 5.7 Chromatic dispersion as a function of the wavelength in (a) a single-vertical-slot
waveguide and (b) a double-vertical-slot waveguide (d = 50 nm).
Compared with the strip waveguide, vertical-slot waveguides exhibit a large negative
chromatic dispersion. From Fig. 5.7(a), it is apparent that the absolute value of the chromatic
dispersion increases with the width of the slot. For a silicon waveguide with a 100-nm slot,
the chromatic dispersion can be as negative as -22.1 ps/nm/m. This large dispersion is
helpful in further reducing nonlinear parametric effects, since it can introduce a phase
mismatch. Figure 5.7(b) shows the chromatic dispersion of the double-vertical-slot silicon
waveguide. By splitting the 100-nm slot into two 50-nm slots, the value of the absolute
dispersion decreases as slot spacing increases. This is due to the reduced waveguide
dispersion for larger Si slot spacing in the middle of the whole waveguide. For a double-
vertical-slot waveguide with a slot spacing of 100-nm, a dispersion variation of less than 1
ps/nm/m over a 232-nm wavelength range (from 1368 to 1600 nm) can be achieved.
68
Together with a wavelength converter, this negative flat dispersion profile could potentially
be used in on-chip tunable optical delay applications [14].
5.4 Summary
In this chapter, vertical slots are used to reduce the nonlinearity of silicon waveguides
efficiently. Up to 50% of the optical power can be confined within the low-refractive, air-
slot region. Compared with a strip waveguide, a nonlinearity reduction of more than 80%
can be achieved over a 100-nm wavelength range using a single-vertical-slot waveguide with
a 100-nm air slot. Also, a reduction in
re
of more than 93% can be achieved for the 100-nm
wavelength range with a double-vertical-slot waveguide. The vertical-slot waveguide also
exhibits a large negative chromatic dispersion, which will further suppress the nonlinear
parametric effects and find applications in dispersion compensation and tunable optical
delays.
69
5.5 References
[1] S. Darmawan, Y. M. Landobasa, P. Dumon, R. Baets, and M. K. Chin, "Resonance
Enhancement in Silicon-on-Insulator-Based Two-Ring Mach-Zehnder Interferometer," IEEE
Photon. Technol. Lett. 20, 1560-1562 (2008).
[2] W. Bogaerts, P. Dumon, D. Van Thourhout, D. Taillaert, P. Jaenen, J. Wouters, S.
Beckx, V. Wiaux, and R. G. Baets, "Compact wavelength-selective functions in silicon-on-
insulator photonic wires," IEEE J. Sel. Top. Quantum Electron. 12, 1394-1401 (2006).
[3] P. P. Mitra, and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre
communications," Nature 411, 1027-1030 (2001).
[4] Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon
waveguides: Modeling and applications," Opt. Express 15, 16604-16644 (2007).
[5] V. R. Almeida, Q. F. Xu, C. A. Barrios, and M. Lipson, "Guiding and confining light in
void nanostructure," Opt. Lett. 29, 1209-1211 (2004).
[6] Q. F. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, "Experimental demonstration
of guiding and confining light in nanometer-size low-refractive-index material," Opt. Lett.
29, 1626-1628 (2004).
[7] A. Armaroli, A. Morand, P. Benech, G. Bellanca, and S. Trillo, "Comparative Analysis
of a Planar Slotted Microdisk Resonator," J. Lightwave Technol. 27, 4009-4016 (2009).
[8] N. N. Feng, J. Michel, and L. C. Kimerling, "Optical field concentration in low-index
waveguides," IEEE J. Quantum Electron. 42, 885-890 (2006).
[9] S. H. Yang, M. L. Cooper, P. R. Bandaru, and S. Mookherjea, "Giant birefringence in
multi-slotted silicon nanophotonic waveguides," Opt. Express 16, 8306-8316 (2008).
[10] H. G. Yoo, Y. J. Fu, D. Riley, J. H. Shin, and P. M. Fauchet, "Birefringence and optical
power confinement in horizontal multi-slot waveguides made of Si and SiO2," Opt. Express
16, 8623-8628 (2008).
[11] C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I.
Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, "All-optical high-speed
signal processing with silicon-organic hybrid slot waveguides," Nat. Photonics 3, 216-219
(2009).
[12] P. Muellner, M. Wellenzohn, and R. Hainberger, "Nonlinearity of optimized silicon
photonic slot waveguides," Opt. Express 17, 9282-9287 (2009).
[13] L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, "Flat and
low dispersion in highly nonlinear slot waveguides," Opt. Express 18, 13187-13193 (2010).
70
[14] J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L.
Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on parametric
wavelength conversion and dispersion," Opt. Express 13, 7872-7877 (2005).
[15] A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr
coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
[16] Q. Lin, J. Zhang, G. Piredda, R.W. Boyd, P. M. Fauchet, and G. P. Agrawal,
"Dispersion of silicon nonlinearities in the near-infrared region," Appl. Phys. Lett. 90,
021111 (2007).
[17] A. V. Shahraam, and T. M. Monro, "A full vectorial model for pulse propagation in
emerging waveguides with subwavelength structures part I: Kerr nonlinearity," Opt. Express
17, 2298-2318 (2009).
[18] V. Mizrahi, K. W. Delong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, "Two-
photon absorption as a limitation to all-optical switching," Opt. Lett. 14, 1140-1142 (1989).
71
Chapter 6 Ultralow Dispersion Silicon ‐on ‐nitride
Waveguide over an Octave ‐Spanning Mid ‐
infrared Wavelength Range
6.1 Introduction
Recently, silicon photonics has attracted a lot of attention due to its great potential for
CMOS compatibility [1]. The band gap energy of silicon crystal is close to 1.11 eV. This
brings the two photon absorption (TPA) wavelength range of silicon devices into ~1.1 to 2.2
m, which covers the widely used telecom band. TPA, which saturates the output power, is
one of the critical barriers of silicon waveguides, especially for nonlinear optical signal
processing. The free carrier generated by TPA can further distort the signal [2, 3]. By
incorporating an integrated p-i-n diode with reverse bias, the carriers can be efficiently
extracted [4, 5, 6].
The mid-infrared (MIR) wavelength region, which covers 3 to 8 m [7], is of great
interest for several applications, such as astronomy, chemical bond spectroscopy,
environmental monitoring, free-space communication, optical sensing, and thermal imaging
[8, 9, 10]. Integrating active and passive components to form an on-chip photonic circuit is a
laudable goal to compact the MIR system, of which the waveguide is the basic building
block [8, 9]. The lack of TPA of silicon in the MIR provides great potential for nonlinear
optics using silicon waveguides [11, 12, 13, 14, 15]. In the past few years, Si waveguides
and devices for MIR have been developed for different applications [16, 17, 18, 19, 20, 21,
22]. Moreover, on-chip silicon waveguides for long wavelength applications are relatively
large in size and thus yield the fabrication accuracy required to achieve desired properties.
72
Broadband low dispersion, high nonlinearity and low loss are desirable goals for many
broadband nonlinear signal processing applications, such as broadband wavelength
conversion, supercontinuum generation and frequency comb generation [14, 23, 24, 25].
Operating at MIR can pronouncedly reduce the nonlinear loss, while the waveguide
materials should be chosen properly to reduce the material absorption induced linear loss.
Silicon dioxide is lossy at wavelengths from 2.6 to 2.9 m and beyond 3.6 m, and thus
becomes improper for MIR applications. Silicon-on-sapphire waveguides can extend the
low-loss wavelength of substrate material to ~4.5 m [9]. It also shows the potential to
achieve small dispersion (±50 ps/(nm•km)) over ~1700-nm bandwidth, which is up to ~4 m
[21]. As a complementary-metal-oxide-semiconductor process compatible platform, silicon
nitride waveguides and devices have shown efficient nonlinear effects over octave
bandwidth range [23, 25]. Promisingly, silicon nitride also has higher nonlinear refractive
index [26, 27] and remains low loss for wavelengths from 1.2 to 6.6 m, which can better
match the broader low-loss wavelength range of Si (up to ~8.5 m) [9]. Thus, the MIR
silicon-on-nitride (SON) waveguide has the great potential to achieve these features [8, 9].
In this chapter, we engineer and flatten the chromatic dispersion of strip, rib and
conformal overlayer SON waveguide in the MIR wavelength region. The rib and conformal
overlayer waveguides provide more structural parameters to tailor the chromatic dispersion
for broadband low-dispersion operation. A low dispersion of ±50-ps/(nm•km) is achieved
over a 4200-nm bandwidth from 2430 to 6630 nm using the SON rib waveguide. With a
proper design, two zero-dispersion wavelengths (ZDWs) can be obtained beyond the TPA
absorption wavelength edge. Moreover, a relatively large nonlinear coefficient can
73
potentially enable nonlinear process, such as supercontinuum and frequency comb
generation, across the over-one-octave bandwidth.
6.2 Waveguide Structure and Mode Property
Figure 6.1 The cross-sections of strip, rib, and conformal overlayer SON waveguides.
The cross-sections of strip, rib and conformal overlayer SON waveguides are shown
in Fig. 6.1, where W, H, h, and t represent the waveguide width, waveguide height, slab
height, and overlayer thickness, respectively. The gray regions are silicon, and the blue
regions are silicon nitride. There is a 2- m silicon nitride space layer on the Si substrate. The
material refractive indices of silicon and silicon nitride are obtained according to the
Sellmeier equations [28, 29] in our model. In the simulation, the dispersion of nonlinear
refractive index (n
2
) for silicon is considered [21]. In our simulation, we develop an 8×8 m
2
model using a full-vector finite-element method (COMSOL). A convergence test shows a 2-
m Si substrate and 1- m perfectly matched layer can provide accurate results. We obtain
the effective refractive indices and mode distributions of the eigenmodes for SON
waveguides with different structural parameters over a broad spectral range. Chromatic
dispersions are obtained by taking the second-order derivatives of the effective refractive
74
indices, and nonlinear coefficients ( ) are calculated from the mode distributions using a full-
vector model [30].
Figure 6.2 (a) The dispersion property and (b) power distribution (3 m) of the conformal overlayer
SON waveguide (W=2000 nm, H=800 nm, t=400 nm) modes.
To achieve good light confinement over a broad bandwidth that expands to MIR, we
use waveguide sizes that are larger than single-mode waveguides for communication band
around 1.55 m. Here, we study the modal characteristics of SON waveguides. Figure 6.2(a)
shows the dispersion property of the conformal overlayer SON waveguide (W=2000 nm,
H=800 nm,
t
=400 nm). TE
11
and TM
11
are the x- and y- polarized fundamental modes, while
TE
21
and TM
21
are the second-order modes with x- and y- polarizations. At 3 m, as shown
in Fig. 6.2(b), the modes are well confined within the silicon core region. At short
wavelengths, the effective refractive indices are close to the material index of Si, and the
index difference between the fundamental and higher order modes is small. The effective
refractive indices decrease with wavelength, while the index difference increases. Moreover,
the effective refractive indices of higher-order modes approach the material index of Si
3
N
4
more quickly. In such case, the higher-order modes leaks out much more quickly than the
fundamental modes at the long wavelength range. At 5.19 m, all higher order modes are cut
75
off, and the x-polarized fundamental mode is still well guided (n
TE11
=2.64). The single-mode
operational wavelength region can be blue shifted by reducing the waveguide size. In
general, the propagation loss increases with wavelength due to the reduced confinement and
more interaction with the rough sidewalls. With a good fabrication process, 1-2 dB/cm
propagation loss can be potentially achieved at long wavelength region [22].
6.3 Chromatic Dispersion and Nonlinearity
In general, chromatic dispersion stems from two sources: material dispersion and
waveguide dispersion. In the long-wavelength region (>3 m), silicon and silicon nitride
have small negative material dispersions. This is because this region is far away from their
band-gap wavelengths. Consequently, waveguide dispersion dominates in the total
dispersion. A large waveguide not only provides good light confinement but also reduces the
waveguide dispersion.
Figure 6.3(a) shows the chromatic dispersions of x-polarized fundamental mode in a
strip SON waveguide with different H. The height of the strip waveguide decreases from
1400 to 600 nm, while its width remains the same (2000 nm). For the low height waveguide,
the mode is confined more tightly and experiences more waveguide dispersion. Thus, the
chromatic dispersion starts to decrease at a relatively shorter wavelength. A strip SON
waveguide with a 2000-nm width and a 600-nm height exhibits a dispersion of ±150
ps/(nm•km) over 3350-nm bandwidth from 1840 to 5190 nm. The effect of strip waveguide
height on the chromatic dispersions of y-polarized fundamental mode is shown in Fig. 6.3(b).
Compared with the dispersion of x-polarized mode, the impact of waveguide height on the
dispersion of y-polarized mode is more pronounced. The chromatic dispersion reaches its
76
maximum 536 ps/(nm•km) at 5630 nm for H=1600 nm. For H=800 nm, the maximum
chromatic dispersion is 766 ps/(nm•km) at 3040 nm.
Figure 6.3 Chromatic dispersion of (a) x- and (b) y-polarized fundamental modes for strip SON
waveguide with different waveguide height.
Figure 6.4 Chromatic dispersion of (a) x- and (b) y-polarized fundamental modes for strip SON
waveguide with different waveguide widths.
Figure 6.4 shows the chromatic dispersions of x- and y-polarized fundamental modes
in strip SON waveguides with different W. The height of the strip waveguide remains as
1200 nm, while its width increases from 1600 to 3200 nm. With the increasing waveguide
width, the x-polarized mode has a released confinement. Both the value and slope of its
chromatic dispersion decrease. A strip SON waveguide with 3200-nm width and 1200-nm
height has a dispersion of 200 ps/(nm•km) over the 5000-nm bandwidth. The impact of
77
waveguide width on the chromatic dispersion for y-polarized fundamental modes is small
due to the fixed structure parameter in y-dimension (H).
The rib waveguide can be fabricated by underetching the high-index guiding layer as
illustrated in Fig. 6.1. The chromatic dispersion of the fundamental mode in x-polarization as
a function of the wavelength in the rib SON waveguide is shown in Fig. 6.5(a). Rib height
(H) is 1200 nm, and rib width (W) is 2000 nm. The slab height (h) of the rib waveguide
increases from 0 to 1000 nm. The increase of slab height gives more space for the mode to
extend and thus yields a lower chromatic dispersion. With the increase of slab height (h)
from 0 to 1000 nm, the chromatic dispersion decreases from 500 to 50 ps/(nm•km) over an
ultrawide bandwidth. The rib waveguide with a 1000-nm slab height exhibits a ±50
ps/(nm•km) dispersion with a 4200-nm bandwidth from 2430 to 6630 nm.
Figure 6.5 Chromatic dispersion of x-polarized fundamental modes for (a) rib SON waveguide with
different slab height and (b) conformal overlayer SON waveguide with different overlayer thickness.
As shown in Fig. 6.1, the conformal overlayer waveguide can be realized by
depositing another low-index layer of silicon nitride film on the strip waveguide [31]. The
chromatic dispersion of the fundamental mode for x-polarization in the conformal overlayer
SON waveguide (W=2000 nm, H=800 nm) is shown in Fig. 6.5(b). The thickness of the
overlayer (t) increases from 0 to 800 with a step of 200 nm. The additional low-index
78
overlayer surrounding the high-index silicon guiding core can be used to control the balance
between material and waveguide dispersion, producing the desired net dispersion
characteristics. Compared between t=0 and 800 nm, not only the value but also the slope
polarity of the chromatic dispersion are tuned. The conformal overlayer waveguide with an
800-nm overlayer has a 3740-nm bandwidth for chromatic dispersion within ±120
ps/(nm•km).
ZDWs are critical to achieve the phase matching condition, especially for nonlinear
four-wave mixing (FWM) process. Table 6.1 shows the ZDWs in the dispersion curves of
the x-polarized fundamental modes for conformal overlayer SON waveguides. There are two
ZDWs for different t from 0 to 800 nm. First, ZDW
1
is around the TPA edge of Si (2200 nm),
and thus the waveguide have pronouncedly reduced nonlinear absorption by operating
around this ZDW. Second, ZDW
2
, which is above 5 m, can efficiently bring the nonlinear
parametric wavelength conversion to a longer wavelength range. With proper design, two
ZDWs can also be achieved in strip and rib SON waveguides between 2 and 6 m.
Table 6.1. Zero-dispersion wavelength in the conformal overlayer waveguides
Structure Parameters (W×H_t) (nm) ZDW
1
(nm) ZDW
2
(nm)
2000×800_0 2090 6097
2000×800_200 2193 6142
2000×800_400 2243 5885
2000×800_600 2250 5380
2000×800_800 2258 5014
For the nonlinear FWM process, the ultrabroadband low dispersion waveguide
facilitates the phase matching for different signals over a broad wavelength range. Another
parameter that affects the efficiency of the parametric wavelength conversion is the
79
nonlinearity of the waveguide. To take the wavelength dependence of material's nonlinear
response into account, the dispersion of n
2
for silicon is considered as shown in Fig. 6.6(a)
[21]. In the simulation, the nonlinear refractive indices n
2
for Si
3
N
4
is 2.4×10
-19
m
2
/W [26].
As the bandgap wavelengths of Si
3
N
4
is below 245 nm, the change of its nonlinear refractive
index at wavelength beyond 1.5 m is small. Moreover, because most of the modal field is
confined within silicon, the impact from the dispersion of nonlinear refractive indices for
Si
3
N
4
to the total waveguide nonlinearity further decreases. Figure 6.6(b) shows the
nonlinear coefficients of strip (W=2000 nm, H=600 nm), rib (W=2000 nm, H=1200 nm,
h=600 nm) and conformal overlayer (W=2000 nm, H=800 nm, t=400 nm) SON waveguides.
For ±150 ps/(nm•km) chromatic dispersion, strip, rib and conformal overlayer waveguide
have 3350-nm, 4800-nm and 3510-nm bandwidths, respectively. From Fig. 6.6(b), we can
see that the strip waveguide has the largest nonlinear coefficient at a short wavelength
( =8.32 /W/m @ 3 m) for up to 6 m and that its nonlinearity decreases fastest. This is
because the strip waveguide has the smallest cross-section among these three waveguides.
Therefore, it provides a tighter light confinement and gives a smaller mode area at a short
wavelength. At longer wavelengths, the strip waveguide is too small to provide good
confinement, while the size of rib and conformal overlayer waveguides are still large enough.
Moreover, the rib and conformal overlayer waveguides provide more waveguide parameters
to engineer chromatic dispersion for broadening low-dispersion bandwidth and enhance the
mode confinement while maintaining comparable nonlinearity in the long wavelength region.
Even at a long wavelength range around 6 m, the nonlinear coefficients of SON
waveguides ( =1.46 /W/m) are still comparable with the ones of integrated Si
3
N
4
waveguides around 1.55 m, which are widely used for octave spanning nonlinear process
80
[23, 25]. Because of the reduced nonlinear coefficient and potentially increased propagation
loss, the efficiency of the nonlinear process is expected to be lower at longer wavelength.
Figure 6.6 (a) The nonlinear refractive index of silicon and (b) the nonlinear coefficients in strip, rib
and conformal overlayer waveguides.
6.4 Summary
In this chapter, we have shown over-one-octave broadband ultralow chromatic
dispersion is achieved using SON waveguide in MIR wavelength region. From 2430 to 6630
nm, a ±50-ps/(nm•km) chromatic dispersion is achieved over a 4200-nm bandwidth using
the rib SON waveguide. Two ZDWs, which are beyond the TPA absorption wavelength
edge, can be obtained with strip, rib and conformal overlayer SON waveguides. Together
with relatively large nonlinear coefficients, they offer great potential to achieve wavelength
conversion, supercontinuum generation, and frequency comb generation over-one-octave
bandwidth.
81
6.5 References
[1] L. Pavesi, and G. Guillot, Optical Interconnects: The Silicon Approach (Heidelberg:
Springer, 2006).
[2] Q. Lin, J. Zhang, G. Piredda, R.W. Boyd, P. M. Fauchet, and G. P. Agrawal, "Dispersion
of silicon nonlinearities in the near-infrared region," Appl. Phys. Lett. 90, 021111 (2007).
[3] Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon
waveguides: Modeling and applications," Opt. Express 15, 16604-16644 (2007).
[4] H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, "High efficiency wavelength
conversion of 10 Gb/s data in silicon waveguides," Opt. Express 14, 1182-1188 (2006).
[5] A. C. Turner-Foster, M. A. Foster, A. L. Gaeta, and M. Lipson, "Large Enhancement of
Wavelength Conversion in Silicon Nanowaveguides via Free-Carrier Removal," in Proc. of
CLEO’2010, Paper CTuEE1 (2010).
[6] J. Cardenas, J. S. Levy, G. Weiderhecker, A. Turner-Foster, A. L. Gaeta, and M. Lipson,
"Efficient Frequency Conversion at Low-Powers in a Silicon Microresonator Using Carrier
Extraction," in Proc. of CLEO’2011, Paper CTuX2 (2011).
[7] J. Byrnes, Unexploded Ordnance Detection and Mitigation (Springer, Netherlands,
2009).
[8] R. A. Soref, S. J. Emelett, and A. R. Buchwald, “Silicon waveguided components for the
long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8, 840–848 (2006).
[9] R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4, 495–497
(2010).
[10] M. Ebrahim-Zadeh, and I. T. Sorokina, Mid-Infrared Coherent Sources and
Applications (Springer, 2007).
[11] V. Raghunathan, D. Borlaug, R. R. Rice, and B. Jalali, "Demonstration of a mid-
infrared silicon Raman amplifier," Optics Express 15, 14355-14362 (2007).
[12] X. P. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, "Mid-infrared optical
parametric amplifier using silicon nanophotonic waveguides," Nature Photonics 4, 557-560
(2010).
[13] X. P. Liu, J. B. Driscoll, J. I. Dadap, R. M. Osgood, S. Assefa, Y. A. Vlasov, and W. M.
J. Green, "Self-phase modulation and nonlinear loss in silicon nanophotonic wires near the
mid-infrared two-photon absorption edge," Optics Express 19, 7778-7789 (2011).
82
[14] R. K. W. Lau, M. Menard, Y. Okawachi, M. A. Foster, A. C. Turner-Foster, R. Salem,
M. Lipson, and A. L. Gaeta, "Continuous-wave mid-infrared frequency conversion in silicon
nanowaveguides," Optics Letters 36, 1263-1265 (2011).
[15] S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S.
Mookherjea, and S. Radic, "Mid-infrared wavelength conversion in silicon waveguides
using ultracompact telecom-band-derived pump source," Nature Photonics 4, 561-564
(2010).
[16] M. M. Milosevic, P. S. Matavulj, P. Y. Y. Yang, A. Bagolini, and G. Z. Mashanovich,
"Rib waveguides for mid-infrared silicon photonics," Journal of the Optical Society of
America B-Optical Physics 26, 1760-1766 (2009).
[17] T. Baehr-Jones, A. Spott, R. Ilic, B. Penkov, W. Asher, and M. Hochberg, "Silicon-on-
sapphire integrated waveguides for the mid-infrared," Optics Express 18, 12127-12135
(2010).
[18] A. Spott, Y. Liu, T. Baehr-Jones, R. Ilic, and M. Hochberg, "Silicon waveguides and
ring resonators at 5.5 m," Applied Physics Letters 97, 213501 (2010).
[19] R. Shankar, R. Leijssen, I. Bulu, and M. Loncar, "Mid-infrared photonic crystal cavities
in silicon," Optics Express 19, 5579-5586 (2011).
[20] G. Z. Mashanovich, M. M. Milosevic, M. Nedeljkovic, N. Owens, B. Q. Xiong, E. J.
Teo, and Y. F. Hu, "Low loss silicon waveguides for the mid-infrared," Optics Express 19,
7112-7119 (2011).
[21] E. K. Tien, Y. W. Huang, S. M. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz,
"Discrete parametric band conversion in silicon for mid-infrared applications," Optics
Express 18, 21981-21989 (2010).
[22] F. Li, S. D. Jackson, C. Grillet, E. Magi, D. Hudson, S. J. Madden, Y. Moghe, C.
O’Brien, A. Read, S. G. Duvall, P. Atanackovic, B. J. Eggleton, and D. J. Moss, "Low
propagation loss silicon-on-sapphire waveguides for the mid-infrared," Opt. Express 19,
15212-15220 (2011).
[23] L. Zhang, Y. Yan, Y. Yue, Q. Lin, O. Painter, R. G. Beausoleil, and A. E. Willner, "On-
chip two-octave supercontinuum generation by enhancing self-steepening of optical pulses,"
Opt. Express 19, 11584-11590 (2011).
[24] J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M.
Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical
interconnects,” Nat. Photonics 4, 37–40 (2010).
[25] Y. Okawachi, K. Saha, J. S. Levy, Y. Henry Wen, M. Lipson, and A. L. Gaeta,
"Octave-spanning frequency comb generation in a silicon nitride chip," Opt. Lett. 36, 3398-
3400 (2011).
83
[26] K. Ikeda, R. E. Saperstein, N. Alic, and Y. Fainman, “Thermal and Kerr nonlinear
properties of plasma-deposited silicon nitride/ silicon dioxide waveguides,” Opt. Express 16,
12987–12994 (2008).
[27] A. Major, F. Yoshino, I. Nikolakakos, J. Stewart Aitchison, and . W. E. Smith,
"Dispersion of the nonlinear refractive index in sapphire," Opt. Lett. 29, 602-604 (2004).
[28] E. D. Palik, Handbook of Optical Constants of Solids. San Diego, CA: Academic, 1998.
[29] T. Baak, "Silicon oxynitride; a material for GRIN optics," Applied Optics 21, 1069-
1072 (1982).
[30] A. V. Shahraam, and T. M. Monro, "A full vectorial model for pulse propagation in
emerging waveguides with subwavelength structures part I: Kerr nonlinearity," Opt. Express
17, 2298-2318 (2009).
[31] X. P. Liu, W. M. J. Green, X. G. Chen, I. W. Hsieh, J. I. Dadap, Y. A. Vlasov, and R.
M. Osgood, "Conformal dielectric overlayers for engineering dispersion and effective
nonlinearity of silicon nanophotonic wires," Optics Letters 33, 2889-2891 (2008).
84
Chapter 7 UWB Pulse Generation using Two ‐Photon
Absorption
7.1 Introduction
In general, ultrawideband (UWB) signals have garnered much interest over the past
several years for their ability to encode high-capacity, broadband data on monocycle pulses
[1]. An application scenario is shown in Fig. 7.1, UWB is used in a digital home to deliver
short-range high speed wireless connection at low power. This has certainly been true in the
electronic domain, and the interest has spilled over into trying to use optics to efficiently
generate UWB pulses for use in microwave and radio frequency (RF) photonics applications.
Optically generating UWB pulses has many advantages, such as high speed, large tunability,
and immunity to electromagnetic interference [2]. Several optical techniques have been
developed for generating UWB pulses, including: (a) phase to intensity modulation
conversion [3], (b) photonic microwave delay-line filter [4], and (c) spectral shaping and
dispersion-induced frequency-to-time mapping [5]. It is highly desirable to make the system
compact and even realize on-chip integration. Some integrated schemes based on cross-
absorption modulation (XAM) in semiconductor optical amplifier and electroabsorption
modulator have been demonstrated for UWB pulse generation [6, 7]. However, these
schemes used complex amplifier or modulator with III-V materials. A laudable goal would
be to generate UWB pulses on the silicon CMOS platform with simplified structure.
Recently, silicon photonics has attracted a lot of attention due to its great potential for
CMOS compatibility [8]. The band gap of silicon crystal is around 1.11 eV. This brings
silicon into the 1.1 to 2.2 m broad two-photon absorption (TPA) wavelength range, which
85
covers the widely-used telecom band [9]. Degenerate or non-degenerate TPA is a process
that simultaneously absorbs two photons of identical or different frequencies in order to
excite a molecule from a lower energy state to a higher energy state. As an ultrafast
absorption effect, TPA has been used for optical switching, logic gates, and pulse
compression in silicon waveguides (Si WGs) [10, 11, 12].
Figure 7.1 Applications of UWB in a digital home.
In this chapter, we propose and experimentally demonstrate UWB monocycle pulse
generation using non-degenerate TPA in a Si WG. A pulsed pump in the Si WG inversely
modulated the continuous-wave (CW) probe during the non-degenerate TPA process. The
monocycle pulse is optically generated after combining the inversely modulated probe and
pulsed pump in the time domain. Using a pulsed pump with a 68-ps full-width at half-
maximum (FWHM), a UWB monocycle pulse with a 143-ps pulse width is generated with a
131.7% fractional bandwidth. The proposed TPA-based UWB monocycle pulse generation
scheme also shows large tunability and the potential for broadband operation.
7.2 Principle
A conceptual diagram of all-optical UWB monocycle pulse generation using non-
degenerate TPA in a Si WG is shown in Fig. 7.2. A CW probe and a pulsed pump are
86
aligned to the Si WG's TE-like mode and coupled together. After passing through the WG,
the CW probe is inversely modulated by the pulsed pump during the non-degenerate TPA
process in the Si WG. By properly controlling the relative delay and power relationship
between the attenuated pump and the inversely modulated probe, positive and negative
UWB monocycle pulses with different shapes can be generated in the optical domain after
their recombination.
Figure 7.2 Concept diagram of silicon waveguide-based UWB monocycle pulse generation using non-
degenerated two-photon absorption effect.
7.3 Results and Discussion
The experimental block diagram of our setup is shown in Fig. 7.3. A CW laser at
pump
=1550.4 nm is first modulated by a Mach-Zehnder modulator (MZM) with 40-Gbit/s
non-return-to-zero (NRZ) amplitude modulation. The pattern generator is programmed to
provide a 200-bit iterative pattern, which corresponds to a 200-MHz repetition rate. The
pattern has only a few consecutive "1"s, while the others are all "0"s. By setting the number
of "1"s from 1 to 5, we can obtain electrical pulses with variable pulse width from 25 to 125
ps. First, the pulsed pump is boosted to 13.8 dBm using an erdium-doped fiber amplifier
(EDFA). Its 90% power port is attenuated by a variable optical attenuator (VOA1), and then
87
combined with a 10-dBm CW probe at
probe
=1569.6 nm and feeds into the Si WG. In this
experiment, a 4.046-cm long waveguide with a 776×300 nm
2
cross section is used. Lensed
fibers with 2.5- m spot diameter (as shown in the inset of Fig. 7.3) are used to couple the
light into and out of the waveguide. The total insertion loss is ~16.2 dB and the waveguide
propagation loss is estimated to be ~2 dB/cm. By aligning both the pump and probe light
polarizations to the TE-like mode of the Si WG, the maximum TPA modulation efficiency
(~80%) can be achieved with a 26-ps pump pulse. In the inset waveform "Modulated Probe",
we can see the long tail (at the rising edge of the inversely modulated probe) generated by
the free-carrier absorption (FCA). From the output of the Si WG, we filtered out the
inversely modulated probe using a 2-nm band-pass filter (BPF). The left 10% pulse pump
after optical coupler (OC1) is then recombined with the modulated probe through OC3.
Using a tunable optical delay line (ODL) and VOA2, the relative time delay and amplitude
relationship between the pump and the inversely modulated probe are well controlled. Then,
the generated UWB monocycle pulse is converted to an electrical signal with a 40-GHz
photodetector for the electrical spectrum analyzer (ESA) and the oscilloscope (Scope). Also,
an optical spectrum analyzer (OSA) is used after OC3 to monitor the optical spectrum.
Figure 7.3 Experimental setup. Pulsed pump and CW probe are coupled to Si WG. The inversely
modulated probe then combines with the pump to generate monocycle pulses.
To obtain a desirable UWB monocycle signal, it is of great importance to increase the
modulation depth of the non-degenerate TPA. Previous research has shown that >90%
88
modulation depth can be achieved using TPA in Si WG with a 1.6-ps pump pulse with very
low energy [11]. Here, we first study the modulation efficiency of the waveguide in our
experiment. By keeping the power of the CW probe at 10 dBm, we increase the pump power
to reach ~15-dBm total input power to the Si WG. From that point, we gradually reduce the
pump power into the Si WG using VOA1 and study its impact on the TPA, as shown by the
B-Spline curves in Fig. 7.4. The probe modulation depth shows a significant reduction. For
the pump pulse with a 26-ps FWHM, the probe modulation depth decreases from 79.4% to
26.1% as the pump pulse peak power decreases from 883.8 to 51.8 mW. From Fig. 7.4(a),
we can see that, at a fixed total input power (rightmost points of the curves), the pump with a
larger pulse width exhibits a smaller probe modulation depth because of the less peak power.
With a 15-dBm total input power into the WG, the probe modulation depths of
FWHM
pump
=26 and 119 ps are 79.4% and 47.9%, respectively. The FWHM of the
modulated probe pulse is relatively stable over pump power as shown in Fig. 7.4(b). The
small increase of the FWHM
probe
is mainly due to the enhanced FCA induced by the
increased pump power.
Figure 7.4 Effects of non-degenerated two-photon absorption. (a) Probe modulation depth, and (b)
modulated probe FWHM as a function of pump pulse peak power for different pump pulse width.
89
Figure 7.5 Ultrawideband signal generation. RF power spectra of generated monocycle pulse with (a)
FWHM
pump
=45 ps, (b) 68 ps, and (c) 93 ps.
To generate UWB monocycle pulses, we choose the FWHM
pump
as 45, 68 and 93 ps,
respectively. The peak powers of the pump and the probe are kept close to each other,
providing a monocycle pulse that has relatively symmetric amplitude. For a positive
monocycle pulse (the pump pulse right ahead of the inversely modulated probe pulse in the
time domain), the FCA generated tail at the rising edge of the inverted probe pulse results in
an asymmetric monocycle pulse and affects the shape of the RF spectrum. By generating a
negative monocycle pulse, the FCA induced pulse tail is largely compensated by the
overlapped pump pulse and produces a symmetric monocycle pulse. From Fig. 7.5 (a)-(c) we
can see that, 45-ps pump has the largest modulation depth, and thus a larger monocycle pulse
amplitude. While the total power of the monocycle pulse is close for input pump with
different FWHM, thus the electrical power spectra are of the similar level. As shown in Fig.
7.5(b), the UWB monocycle pulse generated from a 68-ps pulsed pump has a 143-ps pulse
width. The spectrum of the generated monocycle pulse is measured by an ESA. It can be
seen that the electrical spectrum has a central frequency of 5.62 GHz with a 10-dB
bandwidth of 7.4 GHz, from 1.92 to 9.32 GHz. The fractional bandwidth is 131.7%, which
90
meets the Federal Communications Commission (FCC) requirement of 20%. From Fig. 7.5
we can see that the bandwidth of the electrical spectrum is inversely proportional to the
monocycle pulse width. A pump pulse with 93-ps FWHM provides a monocycle pulse with
6.11-GHz 10-dB RF bandwidth.
Figure 7.6 10-dB BW and center frequency of the RF spectrum as a function of (a) relative delay
between the pump and inversely modulated probe (FWHM
pump
=68 ps); (b) probe wavelength
(FWHM
pump
=68 ps,
pump
=1550.4 nm).
The relative delay of the pump pulse and the inversely modulated probe can be
controlled using ODL in the experimental setup. By tuning the ODL, the width and shape of
the generated monocycle pulse can be controlled properly. As shown in Fig. 7.6(a), this
provides the ability to tune the bandwidth and the center frequency of the RF spectrum for
the generated UWB signal. A maximal 10-dB bandwidth can be achieved with a proper
delay. Also, the center frequency can be tuned from 4 to 6 GHz. In addition, the 10-dB
bandwidth and center frequency of the RF spectrum can be tuned by controlling the FWHM
and the power of the pump pulse.
To take advantage of the ultrahigh bandwidth in the optical domain, broadening the
operational wavelength range of the UWB pulses is of great importance, especially for point-
91
to-multipoint multicasting or multiuser UWB system [13]. Wavelength conversions over 200
nm has been demonstrated using TPA in Si WG [14]. This paves the way for on-chip
broadband applications using non-degenerate TPA. As shown in Fig. 7.6(b), we vary the
probe wavelength from 1570 to 1600 nm while only changing the passing window of the
filters. The 10-dB BW and center frequency of the RF spectrum are relatively stable across a
30-nm optical bandwidth.
The asymmetry of positive and negative monocycle pulses due to the FCA may limit
the application of pulse polarity modulation. There are potentially two alternative ways to
implement data modulation on the proposed monocycle pulse generation scheme. One
straightforward way is to encode data to the time location of the pump pulse for pulse
position modulation (PPM). Second, it is also possible to modulate the UWB monocycle
pulse with on–off keying (OOK) by using an extra optical switch.
7.4 Summary
In this chapter, we propose and experimentally demonstrate ultrawideband monocycle
pulse generation using non-degenerate two-photon absorption in a silicon waveguide. The
free-carrier absorption induced pulse tail at the rising edge of inverted probe pulse is largely
compensated by the overlapped pump pulse and results in a symmetric negative monocycle
pulse. A 143-ps Gaussian monocycle pulse is successfully obtained with a 131.7% fractional
10-dB bandwidth using a 68-ps pulsed pump. The 10-dB bandwidth and center frequency of
the RF spectrum for the generated monocycle pulse can be largely tuned using an optical
delay line. An operational bandwidth of 30 nm is demonstrated experimentally with stable
performance, and larger optical bandwidth is expected.
92
7.5 References
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[9] Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon
waveguides: Modeling and applications," Opt. Express 15, 16604-16644 (2007).
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A. Priem, D. Van Thourhout, P. Dumon, R. Baets, and H. K. Tsang, "Ultrafast all-optical
switching by cross-absorption modulation in silicon wire waveguides," Optics Express 13,
7298-7303 (2005).
[11] T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout,
W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, "High speed logic gate using two-
photon absorption in silicon waveguides," Optics Communications 265, 171-174 (2006).
[12] E.-K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, "Pulse compression and
modelocking by using TPA in silicon waveguides," Optics Express 15, 6500-6506 (2007).
[13] T. Huang, J. Li, J. Sun, and L. R. Chen, "All-optical UWB signal generation and
multicasting using a nonlinear optical loop mirror," Opt. Express 19, 15885-15890 (2011).
93
[14] H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi,
"All-Optical Wavelength Conversion Based on Nonlinear Optical Effects in Si Wire
Waveguides," in Proc. of IPR’2006, Paper ITuH1 (2006).
94
Chapter 8 Orbital Angular Momentum Modes in Specialty
Optical Fiber
8.1 Introduction
Optical fiber communication, as the backbone of today’s telecommunications
infrastructure, supports voice, video and data transmission through global networks. A
critical issue in optical communications research is the challenge of meeting the needs of the
inevitable growth in data transmission capacity. Dense wavelength-division multiplexing
(DWDM) has been proven to be an efficient solution that provides a multiplicative-factor
(on the order of 100) increment. Fueled by emerging bandwidth-hungry applications, much
work has focused on increasing the data spectral efficiency by utilizing polarization,
amplitude, and phase manipulations of the optical field [1, 2]. Another potentially
complementary approach that has gained much attention recently is to transmit independent
data streams, each in a different core using multi-core fibers (MCF) or each on a different
spatial linearly polarized (LP) mode using few-mode fibers (FMF) [3, 4, 5, 6]. Increasing the
number of spatial modes in the optical fiber can increase the capacity and the spectral
efficiency of the communication link simultaneously.
It is known that photons can carry orbital angular momentum (OAM), which is
associated with azimuthal phase dependence of the complex electric field. Light beams
carrying OAM can be described in the spatial phase form of exp(il ) (l=0, ±1, ±2,…). As
OAM has an infinite number of orthogonal eigenstates, it provides another degree of
freedom to manipulate the optical field [7]. OAM based free-space and fiber optical
communication systems have been proposed for spectral and energy efficient
95
communication links, which can meet the latest trend in the field of optical communications
[8, 9, 10, 11, 12]. In free-space optical communication links, multiplexing Laguerre-
Gaussian (LG) modes carrying OAM has been demonstrated to be an efficient way to
increase the spectral efficiency and data capacity [13, 14]. Moreover, the OAM modes based
optical communication system can potentially provide improved security [15]. There are
many other applications of optical OAM modes beyond data communications, including
microscopy, laser cutting of metals, and optical tweezers [7]. Consequently, efficiently
maintaining the OAM modes in an optical fiber, which can potentially facilitate many
applications, is of great importance [16, 17]. Some recent research has shown the capability
of OAM excitation and transmission (1-km) in fiber [18, 19, 20]. Though the reach and
stability of OAM mode propagation in the ring fiber needs further investigation, it can
potentially open the door to a host of different applications.
Generally speaking, OAM mode in fiber is one type of vortices. Vortex beams have
recently attracted great interest because of its unique spatial field distribution, which enables
many applications including high-numerical-aperture focusing, high-resolution imaging,
nanoparticle manipulation, efficient laser machining, electron acceleration and remote
sensing [21]. Free-space optical system can generate vortex beams with a conventional laser
beam [22]. However, the wavelength range of the generated vortex beam is typically limited
by using this technique. A laudable goal would be to have a broadband vortex source to fully
utilize the optical bandwidth. Recently, researchers have shown the capability of generation,
transmission and nonlinear frequency conversion over 250 nm of vortex modes in a ring
fiber [16, 17]. It potentially opens up the door to a host of applications using vortices in an
optical fiber. As traditional methods of generating vortex beams have limited wavelength
96
range, alternative techniques need to be explored, including supercontinuum generation,
which can span hundreds of nanometers wavelength range.
Photonic crystal fiber (PCF) has recently drawn a great deal of attention from
researchers [23]. Its micro engineered structure enables a wide range of unique optical
properties that cannot be realized by using conventional single-mode fibers, such as endless
single-mode guiding, high birefringence, large nonlinearity, and tailorable chromatic
dispersion [23, 24, 25]. Moreover, PCF has been proved to be an efficient supercontinuum
generator enabled by its advanced dispersion tailoring [25]. Supercontinuum of Gaussian
beam has been generated from regular PCF and transferred to vortex beam by spatial phase
change with a spatial light modulator (SLM) [26]. However, the spectrum of the generated
vortex supercontinuum is limited by the bandwidth of the SLM and this scheme is hard to
realize all-fiber system. Chalcogenide materials have shown ultrahigh nonlinear refractive
index, which makes them promising for ultrafast nonlinear optics and ultrahigh-bandwidth
optical systems [27, 28, 29].
In this chapter, we simulate and analyze the mode properties and propagation effects
of OAM modes in the ring fiber. A ring fiber with 0.05 up-doping is designed in simulation
to support up to 10 OAM modes while maintaining single-mode condition radially. Using
optical OAM modes in a multiple-ring fiber for space- and mode-division multiplexing
systems, this scheme can potentially transmit tens of modes in a single fiber, and thus greatly
enhance the system capacity and spectral efficiency. For azimuthal intensity, higher-order
OAM modes are more tolerant to the fiber variations, while odd-order OAM modes have
better azimuthal phase tolerance. OAM
0,4
mode shows <10-ps mode walk-off after 600-km
propagation distance, even in a ring fiber with 1% ellipticity. For the well-aligned OAM
97
modes with different charges in a =1% ring fiber, their crosstalk with other channels is less
than -20dB. Moreover, we propose and design As
2
S
3
ring PCF that can support optical
vortex modes. The design freedom from PCF provides a low dispersion (<60-ps/nm/km
variation in total) over 522-nm bandwidth from 1475 to 1997 nm. The large nonlinear
refractive index of As
2
S
3
enables a 11.7 /W/m nonlinear coefficient of the vortex mode at
1550 nm. By launching a 120-fs pulse with 7.2 pJ pulse energy at 1710 nm into a 1-cm-long
designed As
2
S
3
ring PCF, an octave-spanning supercontinuum spectrum of the vortex mode
is formed from 1196 to 2418 nm at –20 dB.
8.2 Properties of Optical Orbital Angular Momentum Modes in
a Ring Fiber
Figure 8.1 Comparison of different schemes for multiplexing multiple optical spatial modes for fiber
transmission.
98
The block diagram of multiplexing OAM modes in a ring fiber is conceptually shown
in Fig. 8.1. As shown in Fig. 8.1(a), higher-order LP modes are composed of two fiber
eigenmodes with different propagation constants (i.e., LP
2,1
=HE
3,1
+EH
1,1
). While
propagating along the fiber, these two modes walk off, resulting in a highly distorted mode
profile at the detection end. In contrast, fiber OAM modes can be obtained by properly
combining two fiber eigenmodes (i.e.,
odd even
HE i HE OAM
1 , 2 1 , 2 2 , 0
). Since these eigenmodes
have the same propagation constant, they will not undergo any intrinsic mode walk-off.
Therefore, compared with LP modes, OAM modes can better maintain the mode profile after
propagating through a certain length of fiber. However, in multimode step-index fiber,
unwanted radially higher-order modes can easily be excited, thus posing a strict restriction to
the mode coupling. A small change in the launching condition, such as variation of angle,
mode spot size, or deviation from the center, can excite higher-order modes in radial
direction, resulting in serious crosstalk, as shown in Fig. 1(b). On the other hand, a properly
designed high-index single-ring fiber can support only the radially fundamental modes [30]
and thus potentially reduce the crosstalk induced during mode excitement. In order to
multiplex multiple OAM modes simultaneously, the number of the eigenmodes supported by
the fiber should be increased by increasing either the core radius or the refractive index
difference between the core and cladding. In this scheme, a single-ring fiber can provide
one-dimensional mode-division multiplexing of the optical OAM modes. Similar to the
concept of multiple-core fiber [3, 4], by using a multiple-ring fiber with small inter-ring
crosstalk and launching OAM modes with different azimuthal phase into the high-index ring
regions, one can potentially achieve multiplexing of the optical OAM modes in another
spatial dimension. Consequently, this scenario might provide a promising way for spatially
multiplexing more stable modes in a single optical fiber.
99
In this section, we study the OAM mode properties and propagation effects in single-
ring fiber. The ring's inner radius (r
1
), ring's outer radius (r
2
), and the fiber cladding radius
(r
3
) are 4, 5, 62.5 m, respectively. The fiber cladding is made of silica with a refractive
index (n
clad
) of 1.444 at 1550 nm. The refractive index of ring region (n
ring
) can have a 0.05
refractive index increment ( n) by up-doping [31]. Using a commercial full-vector finite-
element mode solver (COMSOL), we can obtain the electromagnetic field distributions and
effective refractive indices of the eigenmodes in the ring fiber. By maintaining the radially
single-mode condition [30], we first investigate the OAM mode number (
odd
m
even
m
HE i HE
1 , 1 ,
and
odd
n
even
n
EH i EH
1 , 1 ,
) that can be supported in the ring fiber. From Fig. 8.2, one can see
that the OAM mode number increases with r
2
and n, while decreases with the wavelength.
A ring fiber (r
1
=4 m, r
2
=5.5 m) with 0.05 up-doping can support 10 OAM modes.
Compared with single mode fiber, this already shows an order of magnitude increment in the
number of multiplexed fiber modes. Moreover, this number can be further increased with
larger ring fiber parameters, such as r
2
and n. As shown in Fig. 8.2(b), a large number of
OAM modes can be supported over a broad bandwidth, which can cross hundreds of
nanometers, in the ring fiber. This further shows the great potential for increasing the link
capacity by utilizing more wavelength-division multiplexing channels.
With the increase of the supported OAM modes number, enlarging the ring-cladding
index difference is a feasible way to reduce the modal coupling in the ring fiber [16]. We
further study the mode index difference between the eigenmodes in the ring fiber with r
1
=4
m at 1550 nm. As shown in Fig. 8.3(a), the effective index difference between TE
0,1
and
HE
2,1
modes increases with n
ring
and r
2
, indicating higher index-contrast and larger ring outer
100
radius are preferred. The effective refractive indices of the eigenmodes in the ring fiber, such
as TE
0,1
and HE
2,1
, always increase with n
ring
and r
2
. However, the mode index difference
does not follow this trend due to the different increment rate of the effective indices for
different modes, especially at high index-contrast case (n
ring
=1.50). Moreover, we investigate
the wavelength dependence of the index difference between TE
0,1
and HE
2,1
modes (r
1
=4 m,
r
2
=5 m), as shown in Fig. 8.3(b). The mode index difference can be maintained above 10
-4
over hundreds of nanometers optical bandwidth.
Figure 8.2 (a) OAM mode number supported in the single-ring fiber as a function of the ring outer
radius (r
2
) with different ring-cladding index difference ( n). (b) OAM mode number as a function of
the wavelength with different ring-cladding index difference ( n).
For the ring fiber with r
1
=4 m, r
2
=5 m, and n=0.05, 8 OAM modes can be
supported by HE
m,1
and EH
n,1
(m=1 ~ 5, n=1 ~ 3). Figure 8.4 shows the intensity and phase
distribution of the OAM modes, which are from the proper combination of the even and odd
HE
m,1
(m=1 ~ 5) modes. For these OAM modes with different charge orders, the field
distributions are well controlled within the high-index ring region. The shape of the ring-like
intensity distribution remains, while its size gradually increases with the OAM mode order.
Moreover, the phase distribution of the OAM
0,m
mode has a 2m change azimuthally and
provides us a chance to efficiently demultiplex these modes with a conjugate phase pattern.
101
Figure 8.3 Effective refractive index difference of TE
0,1
and HE
2,1
modes as a function of (a) the
refractive index in the fiber ring region with different ring outer radius (r
2
), and (b) wavelength with
different ring-cladding index difference ( n).
Figure 8.4 Intensity and phase distribution of the supported OAM
0,m
modes.
Controlling the alignment of two fiber eigenmodes is critical to form an OAM mode
with high purity. Figure 8.5 shows the intensity and phase variation of OAM
0,3
modes
azimuthally along the center of the ring region with different mode walk-off ( ). Note that
=90
o
provides a perfect OAM mode. OAM
0,3
has 3 periods of phase change (- to ) for
each azimuthal circle. A 360° mode walk-off repeats the mode period. From Fig. 8.5(b), one
can see that the intensity distribution is quite flat azimuthally for the well-aligned case
( =90
o
) and the average of maximum intensity variation for the misaligned case ( =180
o
) is
around 5%. As shown in Fig. 8.5(d), for =90
o
, the phase changes linearly along the
102
azimuthal circle. A 30
o
extra mode walk-off ( =120
o
) still gives a relatively smooth phase
change, while it changes with a sharp step when =180
o
.
Figure 8.5 Intensity and phase variation of OAM
0,3
modes azimuthally along the center of ring region
with different mode walk-off ( ) ( =90
o
provides a perfect OAM mode). (a) Normalized intensity
variation with different mode walk-off and azimuth. (b) Azimuthally normalized intensity variation
for =90
o
, 120
o
, 150
o
and 180
o
. (c) Phase variation with different mode walk-off and azimuth. (d)
Azimuthal phase variation for =90
o
, 120
o
, 150
o
and 180
o
.
The non-perfect circularity (ellipticity) of optical fibers can give rise to polarization
mode dispersion (PMD) on the fundamental Gaussian mode. It can also affect the mode
profile or purity of the OAM modes as it induces a difference in the propagation constants of
the decomposed two fiber eigenmodes. In the following, we study the impact of the ring
fiber ellipticity ( ) on the supported OAM modes. In Fig. 8.6(a), the standard deviation (SD)
of the azimuthal intensity increases with fiber ellipticity. The higher order OAM modes
show more tolerance to the ellipticity variations as they have more azimuthal periods, and
103
thus mitigate the effect from fiber ellipticity. From the phase SD as shown in Fig. 8.6(b),
odd-order OAM modes have a stronger tolerance than that of even-order modes. This is
mainly because the ellipticity brings a two-fold symmetry to the ring fiber. Such effect
enhances the phase distribution separation for the OAM modes with even order. For the odd-
order OAM modes, one of its azimuthal 2 phase period can cross the symmetry axis and
thus mitigate this effect.
Figure 8.6 Standard deviation of (a) intensity, and (b) phase azimuthally along the center of ring
region as a function of fiber ellipticity for different OAM modes.
As fiber ellipticity induces an effective refractive index difference to the two
eigenmodes (even and odd) that form the OAM mode, this gives mode walk-off upon
propagation. Here, we use two parameters, 2 walk-off length (L
2
) and 10-ps walk-off
length (L
10ps
), to characterize the intra-mode walk-off of the OAM modes in a non-circular
ring fiber at 1.55 m.
) (
10 55 . 1
1 , 1 , 1 , 1 ,
6
2
m
n n n n
L
even
m
even
m
even
m
even
m
HE HE HE HE
(1)
) (
10 3
1 , 1 , 1 , 1 ,
3
10
m
n n n n
t c
L
even
m
even
m
even
m
even
m
HE HE HE HE
ps
(2)
104
where, , c, t are the wavelength, speed of light in vacuum, and walk-off time, respectively.
L
2
denotes the propagation length when the two eigenmodes walk off to each other with a
2 optical cycle, while L
10ps
means the propagation length after the two eigenmodes has a
10-ps walk-off. Here, we can see that L
10ps
is around 2000 times larger than L
2
. For a 10-
Gbit/s signal, 10-ps intra-mode walk-off is only 10% of one bit length (100 ps). Such a walk-
off will not affect the quality of the detected signal much. One can see from Fig. 8.7, as
higher-order OAM modes have more azimuthal periods, its modal index difference
( even
m
even
m
HE HE
n n
1 , 1 ,
) induced by the ellipticity is smaller. This leads to a longer 2 and 10-ps
walk-off length. OAM
0,4
can propagate more than 600 km with <10-ps mode walk-off, even
in a ring fiber with 1% ellipticity. One can see that the modal index difference increases with
fiber ellipticity and results in a shorter walk-off length.
Figure 8.7 2 and 10-ps walk-off length as a function of fiber ellipticity for different OAM modes.
To effectively demultiplex multiple OAM modes for data communications, we need
to provide channels with small crosstalk. The charge weight of OAM modes is defined as the
electric field overlap integral of the optical field and the eigenmodes in the ring fiber [32].
Fig. 8.8 (a-h) show the crosstalk among well-aligned OAM modes with different charges in
105
a =1% ring fiber. All these OAM modes have <-20dB crosstalk with other channels.
Furthermore, we study the crosstalk of OAM
0,3
modes after 0, 5, 10 and 15-m propagation in
a =1% ring fiber, which is shown in Fig. 8.8 (i-p). With the increase of mode misalignment,
the crosstalk with the other channels also increases. Here, the property of crosstalk is
periodic with the 2 walk-off length of the mode.
Figure 8.8 Charge weight of (a-h) different well-aligned OAM modes in a =1% fiber. (i-p) OAM
0,3
modes after 0, 5, 10 and 15-m propagation in a =1% fiber.
As mentioned above, we can demultiplex these transmitted OAM modes with a
conjugate phase pattern using a phase plate or a spatial light modulator. The demultiplexing
efficiency of OAM
0,3
mode as a function of propagation length is shown in Fig. 8.9. One can
see that it decreases with the ring fiber ellipticity. With the increase of the ellipticity, the
effective index difference between the even and odd eigenmodes increases, and thus the 2
walk-off length decreases. The larger walk-off induces a more severe OAM mode distortion
and thus gives rise to a lower demultiplexing efficiency. The demultiplexing efficiency of
OAM
0,3
mode can be greater than 50% after 300-m long propagation in a ring fiber with
0.5% ellipticity. Consequently, a more circular fiber is preferred to boost the demultiplexing
efficiency and the propagation length of the OAM modes. Moreover, with the MIMO signal
106
processing technique, which is widely employed in currently proposed mode-division
multiplexing systems, OAM modes long-haul transmission is very promising [5, 6]. Here, as
the mode walk-off will recover after 2 period, the demultiplexing efficiency is also periodic
with the 2 walk-off length.
Figure 8.9 The demultiplexing efficiency of OAM
0,3
mode as a function of the propagation length in
the ring fiber with different (Periodic vs. 2 walk-off length).
8.3 Octave ‐spanning Supercontinuum Generation of Vortices
in a Ring Photonic Crystal Fiber
The cross section of the proposed ring PCF is shown in Fig. 8.10(a), where is the air
hole pitch (center-to-center distance between the adjacent holes), r
0
is the radius of the center
air hole, r
2
to r
8
are the radius of the cladding air holes, and is the duty cycle (2r/ ). The
PCF material is As
2
S
3
and its material dispersion is taken into account in our model [33]. In
the simulation, the nonlinear refractive index n
2
for As
2
S
3
is 3×10
-18
m
2
/W [34]. Using a full-
vector finite-element mode solver (COMSOL) with a perfectly matched layer, the modal
distributions of the vortex modes in the designed ring PCF with =0.4 m, r
2
=0.12 m,
=0.9 is obtained as shown in Fig. 8.10(b). From the intensity distribution, we can see that
107
they are well confined within the PCF high-index ring region. Furthermore, optical orbital
angular momentum (OAM) modes can be supported in the ring PCF by properly combining
its two eigenmodes (i.e., OAM
0,2
=HE
2,1
even
+i×HE
2,1
odd
). Since these two eigenmodes are
nearly degenerate, their mode walk-off within 1-cm length is very small. The OAM
0,2
mode
shows smooth phase transition and less than 3% intensity variation.
Figure 8.10 (a) Cross section of As
2
S
3
ring PCF. (b) Intensity and phase distributions of PCF vortex
modes (TE
0,1
, TM
0,1
, HE
2,1
even
, HE
2,1
odd
) and OAM mode (OAM
0,2
=HE
2,1
even
+i×HE
2,1
odd
).
Increasing the modal index difference is a feasible way to reduce the mode coupling
among different vortex modes. The large index contrast between As
2
S
3
and air holes (2.44:1)
of the As
2
S
3
ring PCF proves to be an efficient factor for realizing this goal. Figure 8.11
shows the effective refractive indices of supported vortex modes in the As
2
S
3
ring PCF as a
function of wavelength. The effective index difference between these vortex modes
increases with wavelength due to the increased mode area. At 1.55 m, the index difference,
n of n
TE01
-n
HE21
and n
HE21
-n
TM01
for the designed As
2
S
3
ring PCF are 3.35×10
-2
and 4.75×10
-
2
, which are two orders of magnitude larger than the value in regular ring fiber (~10
-4
) [16].
108
This efficiently breaks the degeneracy and enables good differentiation among different
vortex modes.
Figure 8.11 Effective refractive indices as a function of wavelength for vortex modes in the designed
As
2
S
3
ring PCF.
Figure 8.12 Dispersion of vortex mode (TE
0,1
) with different (a) , (b) r
0
, and (c) r
2
of the As
2
S
3
ring
PCF.
To achieve supercontinuum generation over large bandwidth, a flat and low dispersion
with large nonlinearity are highly desirable. Here, we use =0.4 m, =0.9, r=0.18 m as
109
the key parameters of the ring PCF, and optimize some of them to tailor the dispersion
properties. Figure 8.12(a) shows the dispersion of vortex mode (TE
0,1
) in the designed ring
PCF with different duty cycle. With the decrease of duty cycle, the absolute value of the
dispersion decreases, while the flat-dispersion bandwidth reduces. As shown in Fig. 8.12(b),
the vortex mode experiences a tighter confinement with the increased r
0
, which induces a
reduced and fast-changing dispersion. In Fig. 8.12(c), we reduce the size of innermost
cladding air hole radius r
2
to release the tight confinement to the vortex mode. As a result,
the chromatic dispersion is flattened over a broad bandwidth. The designed ring PCF can
achieve a total dispersion variation of <60 ps/nm/km over a 522-nm bandwidth from 1475 to
1997 nm. This paves the way to the supercontinuum generation of vortex modes.
Figure 8.13 Nonlinear coefficient and confinement loss of vortex mode (TE
0,1
) in the designed As
2
S
3
ring PCF.
The nonlinear coefficient of vortex TE
0,1
mode is calculated from the mode
distributions using a full-vector model [35]. As shown in Fig. 8.13, the nonlinear coefficient
of the designed As
2
S
3
ring PCF decreases with the wavelength mainly due to the increased
effective mode area. At 1.55 m, the mode has a 1.38 m
2
effective mode area with a large
nonlinear coefficient of 11.7 /W/m. The confinement loss is calculated from the imaginary
110
part of the effective refractive index. It increases with the wavelength due to the increased
leakage through the cladding. Even at 2.0 m, the designed ring PCF still has a relatively
low confinement loss of 0.03 dB/m, which is still an order of magnitude smaller than the loss
value achieved in the experiment for As
2
S
3
PCF [36].
Femtosecond laser pulses carried by optical vortices can be generated by mode-locked
laser with some proper spatial hologram [37]. With some free-space optics components, such
as quarter wave plates and lenses, the femtosecond vortex beam can be coupled to the
designed ring PCF [26]. This can potentially provide an optical source for supercontinuum
generation of optical vortices. We use a generalized nonlinear envelope equation to model
the supercontinuum generation process [38]. A 120-fs chirp-free hyperbolic secant pulse is
launched into the designed As
2
S
3
ring PCF ( =0.4 m, r
2
=0.12 m, =0.9) with a peak
power of 60 W, which is equal to pulse energy of 7.2 pJ. The center wavelength of the
launched pulse is at 1710 nm, which is far beyond the two-photon absorption wavelength
range of As
2
S
3
[39]. In the simulation, we choose the loss value as 0.4 dB/m according to the
achievable experiment results of As
2
S
3
PCF [36]. Figure 8.14 shows the evolution of the
frequency spectrum for the vortex mode (TE
0,1
) along the As
2
S
3
ring PCF. The optical
spectrum broadens and flattens with the input pulse propagation. At 10-mm distance, a 1222-
nm supercontinuum spectrum of is formed from 1196 to 2418 nm at –20 dB level, which
covers an octave bandwidth. Although the confinement loss increases to 12.5 dB/m at 2418
nm, this only results in ~0.1 dB loss for a 1-cm fiber. Consequently, the shape of the octave-
spanning supercontinuum spectrum can be maintained. Moreover, the confinement loss can
be further reduced by adding more cladding air holes. With proper As
2
S
3
ring PCF structure
111
design, the broadband supercontinuum can be potentially achieved for other vortices and
OAM modes.
Figure 8.14 Octave-spanning supercontinuum generation of vortex mode (TE
0,1
) in the designed As
2
S
3
ring PCF.
8.4 Summary
In this chapter, we analyze the mode properties and propagation effects of OAM
modes in the ring fiber and propose to use the optical OAM modes in a multiple-ring fiber
for space- and mode-division multiplexing systems. Tens of different modes can be
potentially multiplexed in a single fiber and thus enhance the system capacity and spectral
efficiency. Moreover, we propose As
2
S
3
ring photonic crystal fiber (PCF) for
supercontinuum generation of optical vortex modes. The design freedom of PCFs enables a
low dispersion (<60 ps/nm/km variation in total) over a 522-nm optical bandwidth.
Moreover, the vortex mode has a large nonlinear coefficient of 11.7 /W/m due to the high
nonlinear index As
2
S
3
. An octave-spanning supercontinuum spectrum of the vortex mode is
generated using a 1-cm-long As
2
S
3
ring PCF.
112
8.5 References
[1] X. Zhou, L. E. Nelson, P. Magill, B. Zhu, and D. W. Peckham, "8x450-Gb/s,50-GHz-
Spaced,PDM-32QAM transmission over 400km and one 50GHz-grid ROADM," in Proc.
of OFC’2011, Paper PDPB3 (2011).
[2] L. S. Yan, X. Liu, and W. Shieh, "Toward the Shannon Limit of Spectral Efficiency,"
IEEE Photonics Journal 3, 325-330 (2011).
[3] B. Zhu, T.F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E.
M. Monberg, and F. V. Dimarcello, "112-Tb/s Space-division multiplexed DWDM
transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core
fiber," Opt. Express 19, 16665-16671 (2011).
[4] X. Liu, S. Chandrasekhar, X. Chen, P. J. Winzer, Y. Pan, T. F. Taunay, B. Zhu, M.
Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, "1.12-Tb/s 32-
QAM-OFDM superchannel with 8.6-b/s/Hz intrachannel spectral efficiency and space-
division multiplexed transmission with 60-b/s/Hz aggregate spectral efficiency," Optics
Express 19, 958-964 (2011).
[5] S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R. Essiambre, D.
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transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization," Opt.
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116
Chapter 9 Demonstration of a High Capacity Optical Fiber
Link using Two Orbital ‐Angular ‐Momentum
Modes with Multiple Wavelength Channels
9.1 Introduction
The optical communications community has recently taken more interest in the ability
of mode-division-multiplexing (MDM) to provide yet another degree of freedom to achieve
both higher total capacity and higher spectral efficiency [1, 2, 3, 4]. One of the recent
promising MDM methods uses optical orbital angular momentum (OAM) of light, so beams
with different OAM values l, described in the spatial phase form of exp(il ) (l=0, ±1, ±2,…),
are multiplexed together [5]. Convenient way of creating OAM beams is using for example,
a spiral phase hologram [6]. Encouraging results of MDM using OAM in free-space have
been published [7, 8].
Although results have been achieved in free-space, there is significant interest in the
potential to use OAM for MDM in fiber [9]. A key challenge for fiber-based OAM
multiplexing is that conventional fiber does not readily support an orthogonal set of discrete
OAM modes due to mode coupling [10]. Recent work using new fiber design [10] has
shown the potential for OAM multiplexing and transmission at a single wavelength (1550
nm) 4x100 Gb/s of QPSK data transmission in a MDM scheme [11].
A key potential advantage of using OAM-based MDM for capacity enhancement is
that OAM-MDM should be, in theory, an orthogonal degree of freedom with respect to other
types of multiplexing. Therefore, for a wavelength-division-multiplexed (WDM) optical
system, each independent WDM channel could carry many different orthogonal OAM data-
117
encoded OAM beams, thereby significantly increasing system capacity. A recent publication
has shown the ability to combine OAM with WDM in free space, in which each wavelength
carried 2 OAM values [8]. A laudable goal would be to achieve OAM multiplexing over
multiple wavelengths in a fiber environment.
In this chapter, we demonstrate 1.6-Tbit/s muxing, transmission and demuxing of 20-
Gbaud/s 16-QAM signals through 1.1-km of vortex fiber over 10 wavelength channels each
carrying 2 OAM beams. Enabled by the SLM-based muxing and demuxing techniques, as
well as by vortex fiber, <-17.3-dB crosstalk is achieved over all 20 channels carrying OAM
after km-length fiber propagation. Power penalties of 2.4 dB and 4.55 dB were achieved for
single OAM mode with WDM channels and two OAM modes with WDM channels,
respectively. This represents the first fiber based WDM transmission of OAM over km-
lengths, and also the first MDM transmission without MIMO signal processing. OAM
multiplexing potentially offers a new method for metro-link fiber based systems.
9.2 Concept and Fiber Properties
The conceptual diagram of muxing, transmission and demuxing of 16-QAM signals
over wavelength-division channels carrying 2 OAM beams through vortex fiber is shown in
Fig. 9.1. The WDM laser source with Gaussian intensity profile is firstly modulated by 16-
QAM signal. This source is split into two paths and launched into two spatially-separated
regions of a spatial light modulator (SLM) with 2 clockwise and anti-clockwise helical
phase front, so OAM
+
and OAM
-
beams were formed, each carrying 16-QAM signals over
wavelength-division channels. At the multiplexing stage, half-wave plate (HWP), polarizing
beamsplitter (PBS), and quarter-wave plate (QWP) are used to convert the OAM beams
118
from orthogonally linearly polarized states into left- and right- circularly polarized states
[11]. After propagating through the vortex fiber, the multiplexed OAM modes were
separated using a demuxing stage. The circularly polarized states were first converted back
to linearly polarized states using a QWP and then passing through a polarizer (P) oriented to
the operational polarization of the demuxing SLM. The OAM beams are converted back to
Gaussian beams by reflecting from the SLM with a conjugate phase pattern, and then sent
for coherent detection.
Figure 9.1 Conceptual diagram of muxing, transmission and demuxing of 16-QAM signals over
wavelength-division channels carrying 2 OAM beams through vortex fiber. WDM: wavelength-
division multiplexing; : wavelength; QAM: quadrature amplitude modulation; OAM: orbital angular
momentum; PBS: polarizing beamsplitter; HWP: half-wave plate; QWP: quarter-wave plate; P:
polarizer; SLM: spatial light modulator.
Figure 9.2 Vortex fiber properties. (a) Measured refractive index. Calculated (b) n
eff
, (c) dispersion
and (d) A
eff
for all modes.
Our, so-called, vortex fiber is designed to suppress intermodal couplings by strongly
separating the effective indices (n
eff
) of the antisymmetric vector modes (TE
01
, LP
11
), which
119
avoids the creation of the inherently unstable, conventional LP
11
states. Figure 9.2(a) shows
measured vortex fiber index profile. Using a numerical finite-difference approach, we
calculated several mode properties (Fig. 9.2). We note large n
eff
separation (3x10
-4
) among
higher-order-modes (Fig. 9.2(b)), as designed. Dispersion (Fig. 9.2(c)) and effective area
(A
eff
– Fig. 9.2(d)) of the modes are similar to the conventional values of the LP
01
mode in
SMFs.
9.3 Experimental Setup and Results
Figure 9.3 illustrates the experimental setup of 1.6-Tbit/s data transmission of OAM
beams through vortex fiber. At the WDM 16-QAM transmitter part, 10 DFB lasers with a
100 GHz-spacing from 1546.64nm to 1553.88 nm are combined with an arrayed-waveguide
grating multiplexer (AWG). A narrow-linewidth (~100 kHz) tunable laser is used to replace
the DFB laser at the wavelength of measurement for coherent detection. By using a QPSK
modulator and a QAM emulator, 20-Gbaud/s 16-QAM signal is generated based on the
vector addition of two copies of QPSK signals. By passing the WDM channels through a
100/200 GHz optical interleaver with a relative delay of ~250 symbols, neighboring even
and odd channels are decorrelated. The channels are combined for EDFA amplification, and
then split into two paths, which are decorrelated by adding a 50-m single-mode fiber (SMF)
on one path. The Gaussian beams, each carrying a 16-QAM signal over 10 WDM channels,
from path A and B are collimated and launched into two spatially-separated regions of one
nematic liquid crystal based spatial light modulator (SLM) with 2 clockwise and anti-
clockwise helical phase front. The reflected beams from SLM are converted to orbital
angular momentum beams OAM
+
and OAM
-
with topological charge ±1. The SLM has
800×600 pixels and an effective area of 1.6×1.2 cm
2
. Its diffraction efficiency is greater than
120
90%. After adjusting the polarization and recombining, the OAM modes are converted from
orthogonally linear polarized states into left- and right- circularly polarized states using a
QWP to avoid the excitation of undesired modes. Then the two circularly polarized OAM
modes are coupled into 1.1-km vortex fiber for propagation. The coupling loss is ~1.1 dB
and the vortex fiber has a 1.6-dB/km propagation loss for OAM modes. At the demuxing
stage, a QWP and a polarizer are used to separate the left- and right- circularly polarized
light, while SLM-2 is used to separate the OAM states with topological charge ±1 and
convert one of them to a Gaussian mode. This demuxing scheme is very efficient and <-
20dB inter-mode crosstalk can be achieved. The WDM channels carried on each OAM mode
are selectively filtered out by a 1-nm tunable band-pass filter (BPF) for coherent detection.
Figure 9.3 Experimental setup of muxing, transmission and demuxing of 16-QAM signals over
wavelength-division channels carrying 2 OAM beams through vortex fiber.
Figure 9.4(a) shows the optical spectrum of the modulated signal at the output of
WDM 16-QAM Tx and the optical spectrum of mode A (OAM
+
) after demuxing. We can
see the demuxing WDM channels have a ~25-dB OSNR. In Fig. 9.4(b), one can see that the
constellations of 16 QAM for B2B and demultiplexed mode A (OAM+) without and with
crosstalk (XT) of the channel at 1550.64 nm are gradually distorted. Fig. 9.4(c) depicts the
measured bit-error rate (BER) curves as a function of received power for mode A at 1550.64
121
nm. To better evaluate the source of the penalty, we perform the BER measurement for the
case w/o XT (one wavelength is on, one mode is on), w/ WDM XT (10 wavelengths are on,
one mode is on), w/ all XT (10 wavelengths are on, two mode are on). We can see that BER
for the mode A is slightly better than for the mode B, mainly due to the slightly different
alignment. At a BER of 3.8×10
−3
[12], the average power penalty of mode A and B for the
cases w/o XT, w/ WDM XT and w/ all XT are 1.8 dB, 2.4 dB and 4.55 dB, respectively.
Figure 9.4 (a) Spectrum of the modulated signal at the output of WDM 16-QAM Tx and spectrum of
mode A (OAM
+
) after demuxing; (b) Constellations of 16 QAM for demultiplexed mode A (OAM
+
)
without and with crosstalk (XT) at 1550.64 nm; (c) BER as a function of the received power for B2B,
mode A (OAM
+
) and mode B (OAM
-
) without and with crosstalk (XT) from WDM channels or the
other OAM mode at 1550.64 nm.
Figure 9.5(a) shows the crosstalk from mode B to A and mode A to B as a function of
wavelength from 1540.24 nm to 1559.56 nm with a 100 GHz-spacing. The system is
optimized at the wavelength of 1550.64 nm. Here, the inter-mode crosstalks from B to A and
A to B are -21.41 dB and -20.17 dB, respectively. As we tune away from the optimal
coupling wavelength, we observe that the cross-talk degrades roughly monotonically. This is
caused by the SLM, whose diffraction angle changes as a function of wavelength (which
may be improved with advances in OAM multiplexing technologies), and the fact that the
two OAM states in the vortex fiber exchange energy via a unitary transformation [13], akin
to the polarization state of the fundamental mode of a SMF rotating with wavelength (which
122
may be resolved by the use of polarization-diversity receivers). We choose 10 WDM
channels, from 1546.64nm to 1553.88 nm, to carry 16-QAM signals. The maximal crosstalk
of all the channels is -17.3 dB. As shown in Fig. 9.5(b), the BER varies mainly due to the
crosstalk, and all the 20 channels can achieve a BER of < 3.8×10
−3
. As a result, a total
capacity of 1.6 Tbit/s is achieved with 20-Gbaud/s 16-QAM signal on each channel. Over
20-nm wavelength span from 1540 to 1560 nm, the crosstalk is less than -12.7 dB. Using
alternative modulation format (such as QPSK) with low-density parity-check (LDPC) codes
[14], we hope to potentially expand the operational WDM channels in the future and thus
further increase the system total capacity.
Figure 9.5 Measured (a) crosstalk and (b) BER as a function of wavelength for mode A (OAM
+
) and
mode B (OAM
-
).
9.4 Summary
In this chapter, we demonstrate muxing, transmission and demuxing of 2 OAM beams
over 10 wavelength channels through 1.1-km of vortex fiber. An aggregated total capacity of
1.6-Tbit/s is achieved by using 20-Gbaud/s 16-QAM signal on each channel.
123
9.5 References
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124
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Abstract (if available)
Abstract
In electronics industry, silicon has been served as an enabling material for a few decades. Recently, by taking advantages of ultrahigh bandwidth in the optical domain, silicon photonics has attracted a lot of attention. More importantly, advances in silicon photonics have potentially paved the way for the design and construction of a CMOS-compatible optoelectronic integration system. In the first part of this dissertation, we design high-contrast grating hollow-core waveguide for linear systems, polarization rotator and splitter for integrated polarization diversity system. Dispersion and nonlinearity properties of silicon waveguide are tailored using different structures and materials for different communications and signal processing applications. A scheme for generating ultrawideband monocycle pulse in a silicon waveguide is experimentally demonstrated. ❧ Fueled by the ability to transmit long distance, optical fiber communications will undoubtedly continue to grow. One of the critical issues in optical communications research is the challenge of meeting the needs of the inevitable growth in data transmission capacity. Besides wavelength, polarization, amplitude, phase of the optical field, researchers start to multiplex data channels carried by different and orthogonal spatial modes. In the second part of this dissertation, we propose and analyze the use of orbital-angular-momentum modes for spatial multiplexing in the ring fiber. As2S3 ring PCF is proposed and designed for supercontinuum generation or vortex modes, making OAM modes very promising in many optical signal processing applications. Moreover, we demonstrate mode-division and wavelength-division multiplexing of OAM modes through 1.1-km of vortex fiber.
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Asset Metadata
Creator
Yue, Yang
(author)
Core Title
Integrated silicon waveguides and specialty optical fibers for optical communications system applications
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
10/25/2012
Defense Date
10/19/2012
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OAI-PMH Harvest,optical communications,orbital angular momentum,silicon photonics,specialty fiber
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Willner, Alan E. (
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Tags
optical communications
orbital angular momentum
silicon photonics
specialty fiber