A NOVEL TUNABLE FILTER FOR WAVELENGTH DIVISION MULTIPLEXED
COMMUNICATION SYSTEMS
By
CHRISTOPHER PARKE HUSSELL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995
Copyright 1995
by
Christopher Parke Hussell
To my loving wife, Mary.
ACKNOWLEDGEMENTS
Many people have assisted me with my work at the University of Florida and it
would be impossible to acknowledge all of them. However, I would like to acknowledge
several individuals in particular.
First, I want to thank Dr. Ramu V. Ramaswamy for his hard work and devotion
to his students in providing funds, wellequipped laboratory facilities, and a stable
working environment conducive for research. His devotion towards generating funds
for the Photonics Research Laboratory and the University is truly commendable.
His success in doing so allowed me to take a novel concept and buy the necessary
equipment for its demonstration in a very reasonable time frame. His willingness to
share his broad knowledge of this and related areas proved to be essential for the
completion of this work.
I would also like to thank Dr. Robert Tavlykaev for constantly providing helpful
discussions. I found him to be very knowledgeable of the current as well as past
literature. This has made him an invaluable resource for me.
Very conducive discussions with Dr. K. Thyagarajan helped me to have a deeper
understanding of the problem at hand. I found him to have a very broad theoretical
knowledge. He was always willing to assist.
I am grateful to Dr. Sanjai Sinha for assisting me with MOCVD growth. I also
found him to be very helpful in providing me with ideas. In fact, when looking for
the best materials to use for a dielectric mirror, he suggested using silicon. Not long
after that, I formulated the idea of using silicon as a buffer layer for DBR grating
waveguides.
I am indebted to Hsing Chien Cheng. He was very eager on countless occasions to
help me with theoretical modeling, fabrication, and measurement. I greatly respect
his hard work and pleasant attitude.
I want to thank the professors in the area: Dr. Peter Zory, and Dr. Ramakant
Srivastava. Their work at the University is commendable.
I want to thank Dr. Scott Samson for reviewing the rough draft of this dissertation.
His comments helped me to convey the information in this work more effectively.
Also, I appreciate my other colleagues, particularly, Wuhong Li, Suning Xie, Dave
Marring, Kirk Lewis, Ron Slocumb, Dr. Amalia Miliou, Jamal Natour, Dr. Sang K.
Han, Dr. Hyoun S. Kim, Dr. Song Jae Lee, Dr. Young Soon Kim, Dr. Sang Sun
Lee. Through personal discussions and group meetings, these people have been very
helpful.
Several members of the department have also provided me with necessary as
sistance in their areas of expertise. The machinist, Allan Herlinger, professionally
modeled several optical measurement devices for me. Keith Rambo helped me to
plan the installation of the RIE, provided me with spare parts, and assisted me with
finding vendors of vacuum equipment. The Microelectronics Laboratory managers,
James Chamblee, Tim Vaught, Steve Schein, and Jim Hales, provided me with two
wellmaintained laboratories.
The brotherhood in Gainesville at the Campus Church of Christ have warmly
opened their homes to me and my family constantly from the time we first moved
into the area. These people, far too many to list here, have kept me in their prayers
and offered much moral support.
My father, Lewis, and mother, Rosalee, have been supportive to me towards
fulfilling my dreams since childhood. Their love and support continue to encourage
me to attain higher goals. I am truly grateful for their financial support. I love them
very much.
My greatest appreciation goes to my wife, Mary. She has unceasingly supported
me in keeping the home and family in order. She is very devoted and giving to our
lovely children, Ashley and Kristin, who take after their mother in beauty. She has
endured much and keeps on smiling. Truly, she has made it all worth while. My love
for her is unending. My children truly bring joy to my life.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ............................ iv
LIST OF FIGURES .................... .......... x
ABSTRACT .................................... xvi
CHAPTERS
1 INTRODUCTION ............................... 1
1.1 M otivation . . . . . . . . . . . . . . . . 1
1.1.1 Bandwidth of Optical Fiber .................. 2
1.1.2 Practical WDM Communication Systems ........... 4
1.1.3 Wavelength Selective lOCs ................... 6
1.2 A New Wavelength Selective Tunable Filter ............. 11
1.3 A Novel Waveguide Technology .................... 14
1.4 Chapter Organization ........................... 17
2 THEORY OF OPTICAL FIELDS IN MULTILAYER STRUCTURES ... 20
2.1 Basic Equations .............................. 21
2.2 Reflection and Transmission From a Single Boundary ........ 27
2.3 Characteristic Matrix of a Thin Film ................. 31
2.4 Characteristics of a Multilayer Structure ............... 33
2.5 Alternative Matrix Formulation .................... 36
2.6 Connection Between the Two Formalisms ............... 37
2.7 Applications of the Formalisms to Real Structures . . . . ... 39
2.7.1 Reflectance from Periodic QuarterWave Structures . . 39
2.7.2 Basic Laser Relationships . . . . . . . . . . 40
2.7.3 Basic Waveguide Relationships . . . . . . . . 43
2.7.4 Application to Corrugated Waveguides . . . . . ... 44
2.8 Spectral Index Method .......................... 47
2.9 Summary of the Numerical Methods . . . . . . . . . 51
3 A NEW WAVELENGTH SELECTIVE FABRYPEROT FILTER .... 52
3.1 Thin Film Dielectric FabryPerot Filters . . . . . . . . 52
3.1.1 Characteristics of a Thin Film Dielectric Mirror . . ... 52
3.1.2 Characteristics of a Thin Film Dielectric FabryPerot Filter 53
3.2 DBR Corrugated Waveguide FabryPerot Filters . . . . . ... 55
3.2.1 Characteristics of a DBR Corrugated Waveguide . . ... 55
3.2.2 Characteristics of a DBR Corrugated Waveguide FabryPerot
F ilter . . . . . . . . . . . . . . . . 58
3.3 Proposal: Asymmetric FabryPerot Filter . . . . . . . . 59
3.3.1 Passive ASFP Filter .... ................... 60
3.3.2 Tuning an ASFP filter ..................... 62
3.3.3 Tunable DFB laser ....................... 65
3.3.4 Sum m ary ............................ 66
4 A NOVEL WAVEGUIDE GEOMETRY . . . . . . . . . 68
4.1 The Need for Better DBR Corrugated Waveguides . . . . ... 68
4.2 Corrugated Waveguide Technologies (Previous Work) . . . ... 69
4.2.1 Investigations of DBR Corrugated Waveguides . . . .... 70
4.2.2 Passive DBR Corrugated Waveguide Devices . . . .... 71
4.3 Difficulties in Achieving High Reflectance . . . . . . . . 72
4.4 Proposal: Channel Waveguides With HighIndex Overlays . . 74
4.5 Benefits of Silicon as a HighIndex Overlay On LiNbO3 . . ... 78
4.5.1 Previous Reports of Waveguides with Silicon Overlays . . 78
4.5.2 Additional Benefits of Silicon on LiNbO3 . . . . . ... 80
4.6 Misconceptions of Guiding in ThickFilm Structures . . . .... 80
4.7 Properties of LiNbO3 Waveguides With Varying Silicon Overlay Thick
ness . . . . . . . . . . . . . . . . . . 85
4.8 Sum m ary . ... .... .... ... .... .... .. .. .. .. 88
5 DEVICE FABRICATION TECHNIQUES . . . . . . . . . 89
5.1 Reactive Ion Etching .......................... 89
5.1.1 Cl2 Reactive Ion Etching . . . . . . . . . . 93
5.1.2 02 Plasma Ashing ....................... 96
5.2 Waveguide Fabrication ......................... 100
5.3 Grating Lithography and Transfer . . . . . . . . . . 103
5.4 Polishing . .. . . . . . . . . . . . . . . 107
5.5 Endface DBR Fabrication ....................... 108
5.6 Sum m ary ... .. .. ... .... .... .... .. .. ... . 110
6 EXPERIMENTAL RESULTS OF THE NEW WAVEGUIDE GEOMETRY 113
6.1 Near Field Profile Measurement .
6.1.1 Measurement Technique .
6.1.2 Field Profile Results ...
6.2 Loss Measurement ........
6.2.1 Waveguide Device Reflection
6.2.2 Loss Results .........
6.3 Mode Index Measurement ....
6.3.1 Prism Coupler Technique. .
6.3.2 Prism Coupler Results . .
6.4 Summary .............
. . . . . .
. . . . . .
. . . . . .
. . . . . .
and Transmission
o.........
o.o.......
. . . . . .
. . . . . .
. . . . . .
..oo.o.
.....o..
Measurement
. . . . o .
........
7 EXPERIMENTAL RESULTS OF GRATING FILTERS IN LiNbO3 .... 129
7.1 Waveguide Grating Results ........................ 129
7.1.1 The First Sample: APE:LiTaOs . . . . . . . . 131
7.1.2 The First APE:LiNbO3 Sample ................ 132
7.1.3 The First Ti:LiNbO3 Sample .................. 136
7.1.4 Short Period DBR Corrugated Waveguide . . . . ... 137
7.1.5 The Best Reflection Measurements . . . . . . . 137
7.2 ASFP Filter Results .......................... 140
7.2.1 Results of ASFP #1 ...................... 140
7.2.2 ASFP Filter Results with EndfaceDeposited DBR Mirror on
ASFP # 2 .... ... .... ... .... .. .. ... .. 144
7.3 Prospects for Improvement ........................ 146
7.3.1 Improvement of Grating Fabrication . . . . . . . 147
7.3.2 Optimization of APE:LiNbO3 Waveguides . . . . ... 151
7.3.3 Optimization of the Silicon Thickness . . . . . . . 151
7.3.4 Optimize Structure With Buffer Layer . . . . . . 152
7.4 Sum m ary ................................ 152
8 CONCLUSIONS AND FUTURE WORK . . . . . . . . . 156
8.1 Conclusions . . . . .. . . . . . . . . . . 156
8.2 Future W ork ................................ 158
8.2.1 Detailed Study of Waveguides With HighIndex Overlays . 159
8.2.2 Optimization of Grating Performance . . . . . . . 160
8.2.3 High Performance lOCs .................... 161
APPENDICES
A DISPERSION OF SILICON ................ .... ... 166
B DISPERSION OF LITHIUM NIOBATE . . . . . . . . . 171
REFERENCES ................................... 173
BIOGRAPHICAL SKETCH ............................ 180
LIST OF FIGURES
1.1 Attenuation of a lowloss optical fiber (reference [45]). . . . . 3
1.2 Simplified schematic of a WDM distribution system . . . . . 4
1.3 Simplified schematic of a WDM ring network. . . . . . . 5
1.4 Polarization insensitive acoustically tunable optical filter. . . . 7
1.5 Cascaded MachZehnder interferometer. . . . . . . . . 9
1.6 Resonant grating assisted directional coupler filter. . . . . . 9
1.7 Proposed Tunable Wavelength Selective Filter. . . . . . .. 12
1.8 Comparison of the reflection spectra of a dielectric endface mirror and
a corrugated waveguide. a) shows that the reflection band of a corru
gated waveguide is very narrow compared to the spectra of the dielec
tric mirror. b) shows an expanded view of the corrugated waveguide
spectra. . . . . . . . . . . . . . . . .. . 13
1.9 Transverse cross section of a gradedindex channel waveguide with an
overlaying film of refractive index near or higher than the core of the
w aveguide .. . . . . . . . . . . . . . . . . 15
1.10 Longitudinal cross section of a DBR corrugated waveguide made from
a highindex overlay. ......................... 15
2.1 k vector relationships with light incident on a single surface. ... . 27
2.2 Conventions for the positive directions of the E and H field vectors for
the (a) TE or spolarized waves and (b) TM or ppolarized waves. . 28
2.3 A single dielectric planar layer on a substrate. . . . . . ... 31
2.4 A general multilayer thinfilm structure. . . . . . . .. 34
2.5 Dielectric mirror made from a periodic quarterwave stack. .. . . 40
2.6 Singlefilm laser.. ............................. 41
2.7 Singlefilm waveguide. ........................ 43
2.8 Single mode waveguide with an abrupt step. . . . . . ... 45
2.9 Dual rib waveguide cross section. The dashed line represents the ef
fective movement of the boundary by the GoosHiinchen shift. . . 48
3.1 Simple DBR consisting of a multilayer dielectric stack (a), and its
reflectance and transmission spectra (b). The parameters are no = 1,
ns = 2.14, nL = 1.2, nH = 2.2, and 4 periods . . . . . .... 53
3.2 FabryPerot filter consisting of two multilayer dielectric stack DBRs
separated by a short waveguide cavity (a), reflectance spectra of the
structure with a cavity length of 10pm (b). The parameters for each
mirror are no = 1, n, = 2.14, nL = 1.2, nH = 2.2, and 4 periods. . 54
3.3 DBR corrugated waveguide (a), reflectance spectra (b). The parame
ters used in the calculation are NH = 2.1406 and NL = No = N, = 2.14
and 10,000 periods. Loss of 0.35dB/cm is also included in the calculation. 56
3.4 FabryPerot filter consisting of two DBR corrugated waveguides sepa
rated by a short waveguide cavity (a), reflectance spectra of the struc
ture (b) . . . . . . . . . . . . . . . . . 58
3.5 Asymmetric FabryPerot filter consisting of a DBR corrugated waveg
uide with a dielectric mirror deposited on the endface . . . .... 60
3.6 Transmission spectra of an asymmetric FabryPerot filter (a); expanded
view of the transmission peak (b). The parameters used for the dielec
tric mirror were the same as those used for figure 3.1(b) and those used
for the corrugated waveguide were the same as those use for figure 3.3(b). 61
3.7 Transmission spectra of an ASFP filter for a corrugated waveguide loss
of .35dB/cm (solid) and 3.5dB/cm (dotted). . . . . . .. 62
3.8 Tunable ASFP filter. .......................... 63
3.9 Transmission spectra of a Tunable ASFP filter for several values of the
mode index change ............................. 64
4.1 Longitudinal cross section of a gradedindex waveguide with an over
laying film grating (a); local normal mode index profile along the prop
agation direction (b) ............................ 75
4.2 Corrugated waveguide structure used by Adar et al. [2] (a); Local nor
mal mode index profile of the structure along the propagation direction
(b); Local normal mode index profile of the structure with Si3N4 only
in the valleys of the grating (solid) and with no Si3N4 at all (dotted) (c). 77
4.3 Simple ridge waveguide structure used to illustrate a case were the
effective index method gives the wrong number of modes. This waveg
uide is single mode, but the effective index method predicts five modes. 82
4.4 The effective index method is applied to the waveguide structure in
figure 4.3 by finding the mode indices of asymmetric planar waveguides
with film thicknesses of D (a) and H (b). These mode indices are
applied to a symmetric laterally confined planar waveguide (c). The
mode indices of the waveguide depicted in (c) are generally accepted
to be equivalent to the mode indices of the structure in figure 4.3. . 83
4.5 Viewing the proposed structure (figure 1.9) upside down, we see some
similarity between it and the structure in figure 4.3. In this case, the
rectangular ridge is replaced by a lower refractive index strip load with
larger cross section ............................. 84
4.6 Highindex striploaded waveguide structure used for spectral index
calculations . . . . . . . . . . . . . . . . 85
4.7 Dependence of the mode index of the device in figure 4.6 on silicon
strip thickness for several strip widths. . . . . . . . .. 86
5.1 Schematic illustration of the voltage drop across a plasma discharge. 90
5.2 RIE machine used for etching the devices in this work. . . . . 91
5.3 Diagram of the RIE system pictured in figure 5.2. . . . . .... 92
5.4 A trench 3pm wide by 9pm deep, etched in GaAs. . . . . .... 94
5.5 A 1/m period grating etched into GaAs by Cl2 RIE. . . . .... 94
5.6 Schematic of a multilayer mask that can be used for very deep RIE or
thick liftoff, a) shows the structure just after developing the imaging
layer and b) shows the completed structure. . . . . . .. 97
5.7 Picture of the multilayer photoresist mask from an optical microscope. 98
5.8 SEM picture the result of etching GaAs using a multilayer structure
without the SiO2 layer and nonoptimized 02 etching parameters. . 100
5.9 Standard fabrication process for annealed proton exchanged LiNbO3
waveguides. . . . . .. . . . . . . . . . .. . 102
5.10 Diagram of the grating lithography setup. . . . . . . ... 104
5.11 Plot of equation 5.1. ......................... 105
5.12 Setup used to verify the period and measure the chirp of the gratings. 105
5.13 SEM picture of an etched grating on a silicon substrate . . . .
5.14 Picture of two stages of an LiNbO3 endface polish. After polishing
with 0.25/m diamond paste (a), and after using a colloidal alumina
polishing compound (b,c). The LiNbO3 substrate (bottom) with a
thin silicon layer on top of it, followed by a 2/pm thick layer of SiO2.
The silicon layer is barely visible in the picture. Above the SiO2 layer
is a thin layer of wax, which is thick enough to be visible only in (c),
that does not polish well. A silicon polishing block is on the top. The
picture was taken by an optical microscope with the contrast enhanced
to show any scratches. ........................ 109
5.15 Diagram of the sample holder/shutter assembly (a), and picture (b) of
the ebeam deposition system ....................... 110
5.16 Calculated and measured transmission of two dielectric mirrors de
posited by the system shown in figure 5.15 (a) onto glass microscope
slides, and calculated reflection of the same structure (b). . . .... 111
6.1 Near field measurement setup. . . . . . . . . . .. 114
6.2 Contour plots of the optical intensity profiles for the fiber (upper left)
and waveguides of several mask widths with a 1775A silicon overlay. 120
6.3 Experimental setup used to measure reflection, transmission, and loss
of waveguide device samples. . . . . . . . . . . ... 121
6.4 Loss measurements of zcut APE:LiNbO3 waveguides proton exchanged
in pyrophosphoric acid at 200C for 71 minutes, and annealed for 7
hours at 3500C for two sample lengths. . . . . . . . ... 123
6.5 Loss measurements of zcut APE:LiNbO3 waveguides proton exchanged
in pyrophosphoric acid at 200C for 71 minutes, and annealed for 7
hours at 350C for three thicknesses of silicon: none, 4120A, and 8200A. 124
6.6 Schematic diagram of the prism coupler. . . . . . . ... 125
6.7 Measurements of the mode index of APE:LiNbO3 waveguides with
silicon overlays of different thicknesses. . . . . . . . .. 127
7.1 Transmission spectra of a 5pm wide APE:LiTaO3 waveguide on the
first DBR corrugated waveguide sample (DBR #1), fabricated with
the highindex overlay technology. . . . . . . . . . . 132
7.2 Transmission spectra of an 8.5/tm waveguide on the first APE:LiNbO3
DBR corrugated waveguide sample (DBR #2). . . . . . .. 133
106
7.3 Exact solution of the mode index of the first three modes of a planar
waveguide with a cladding index of 1.0, a film index of 3.5 (silicon), and
a substrate index of 2.14 for varying film thicknesses at a wavelength
of 1534nm .. . . . . . . . . . . . . . . . . 134
7.4 The grating not only causes coupling between the forward and re
verse guided modes, but also between the forward guided mode and
reverse radiation modes. The planar highindex overlay allows for one
dimensional radiation with mode indices above those of the channel
w aveguide .. . . . . . . . . . . . . . . . . 135
7.5 Transmission spectra of the TM mode of a 5.5/pm waveguide on the
first Ti:LiNbO3 DBR corrugated waveguide sample (DBR #3). . . 136
7.6 Transmission spectra of a 10pm waveguide on a DBR corrugated waveg
uide sample with a shorter period. After these measurements, the
silicon grating was removed and a new one fabricated on the sample
(D BR # 4). . . . . . . . . . . . . . . .. . 137
7.7 Reflection spectra of a 10ym waveguide on an APE:LiNbO3 DBR cor
rugated waveguide sample (DBR #4). . . . . . . . ... 138
7.8 Calculated and measured reflection and transmission of a DBR cor
rugated waveguide with ~ 10,300 periods, AN = 2.8 x 104, A =
358.55nm, and a 50:50 duty cycle (DBR #5). . . . . . .. 139
7.9 SEM picture of a grating over an APE:LiNbO3 waveguide. .. . . 139
7.10 Structure of sample ASFP #1. The cavity is of sufficient length to
add electrodes. .. ....... .... ... .... .. .. ... 141
7.11 Reflection spectra of the sample configuration depicted in figure 7.10. 142
7.12 An external dielectric mirror was index matched and buttcoupled to
the device of figure 7.10. ....................... 142
7.13 Reflection spectra of the sample with an external buttcoupled mirror
as depicted in figure 7.12. ...................... 143
7.14 Measured and calculated transmission of the device in figure 7.12. . 143
7.15 ASFP #1 after fabricating the electrodes. . . . . . . ... 144
7.16 Two sets of measured results for OV and 30V for the device in figure 7.15.145
7.17 A dielectric mirror was deposited on the end of a device as shown. . 145
7.18 Calculated and measured reflection spectra for the device in figure 7.17.146
7.19 SEM picture of the grating profile produced on a silicon substrate by
R IE . . . . . . . . . . . . . . . . .. . 147
xiv
7.20 Calculated reflection spectra of a DBR corrugated waveguide with a
30:70 duty cycle. ........................... 148
7.21 A grating with a curved longitudinal mode index profile (a) was used
to calculate the effect of a nonsquare grating on the reflectance spectra
(b ) . . . . . . . . . . . . . . . . . . . 149
7.22 SEM picture of an etched grating on a silicon substrate produced by
under exposure ............................... 149
7.23 Reflection spectra of a chirped DBR corrugated waveguide. .. . . 150
8.1 Conceptual drawing of an ASFP directional coupler. . . . .... 162
8.2 A tunable Er:LiNbO3 single longitudinal mode laser. . . . .... 162
8.3 Application of high index overlay strips to reduce radiation loss at
waveguide bends. ........................... 163
8.4 Conceptual drawing of a WDM Mux/Demux Chip. . . . .... 164
A.1 Match between the measured and theoretical transmission through an
8050A thick silicon film on a glass microscope slide after iteration of
the dispersion curves ............................ 167
A.2 Comparison between the known dispersion curves of single crystal sil
icon and the ebeam deposited 8050A film . . . . . . ... 168
A.3 Match between the measured and theoretical transmission through a
4210A thick silicon film on a glass microscope slide using the ebeam
Si dispersion curves in figure A.2. . . . . . . . . .. 168
A.4 Match between the measured and theoretical transmission through a
4180A thick silicon film on a glass microscope slide using the ebeam
Si dispersion curves in figure A.2. The poor result is attributed to
oxidation of the Si film caused by evaporating SiO2 immediately prior
to the Si evaporation ............................ 169
A.5 Comparison of the dispersion curves of single crystal silicon, uncontam
inated ebeam deposited silicon, and contaminated ebeam deposited
silicon . . . . . . . . . . . . . . . . . . 170
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A NOVEL TUNABLE FILTER FOR WAVELENGTH DIVISION MULTIPLEXED
COMMUNICATION SYSTEMS
By
Christopher Parke Hussell
August 1995
Chairman: Dr. Ramu V. Ramaswamy
Major Department: Electrical and Computer Engineering
Two essential integrated optical components of wavelength division multiplexed
(WDM) systems are tunable lasers and tunable wavelength selective filters, which we
address in this dissertation within a common framework by proposing a new concept
that has commercial potential.
Conventional FabryPerot filters lack the free spectral range (FSR) required in
WDM systems. A new approach to achieve transmission filters with large FSR,
tunable over many channels using the linear electrooptic effect, is proposed. By
simply adding gain to the waveguide, this configuration is extendible as a tunable,
single longitudinal mode laser The most significant obstacle in realizing these filters
is achieving highreflectance corrugated waveguidesan essential consideration for
achieving a useful filter.
The primary contribution of this work is the novel approach used to achieve
highreflectance corrugated waveguides. The concept involves channel waveguides
with planar films of higher refractive index on top. This influences the propagation
constant of the channel waveguide mode. However, the film does not become the
primary guiding layer (the majority of the guided power remains in the lowerindex
material) even for a considerably thick film. A somewhat startling finding is that
planar films with thicknesses capable of supporting multiple planar guided modes
xvi
do not necessarily have a measurable affect on the loss and mode field profile of the
channel waveguide. The effect on the propagation constant is small, but measurable.
This new technology forms the basis of an ideal geometry for creating corrugated
waveguide devices.
These new concepts are applied and demonstrated in a material system in which
it is difficult to achieve high reflectance from a corrugated waveguideLiNbO3. We
choose this material not only to prove the robustness of the new technology, but
because it is one of the few material systems in which integrated optical circuit prod
ucts are commercially available. Preliminary structures exhibit the highest reflection
reported to date in LiNbO3 corrugated waveguides (80%) with an order of magni
tude shorter length (3.7mm) than that has recently been reported for devices with
comparable reflection. The results of the proposed wavelength selective filter are in
excellent agreement with the theory.
xvii
CHAPTER 1
INTRODUCTION
Our motivation for this dissertation begins with a discussion of how wavelength
division multiplexing in optical fiber communication systems can be used to meet the
growing need for highbandwidth services. Most of the components in these commu
nication systems such as tunable laser sources, external modulators, star couplers,
fiber amplifiers, and detectors are already commercially available. Several concepts
for the remaining crucial component, the tunable wavelength selective filter, have also
been reported; however, they are either unrealizable for commercial applications or
inadequate in meeting the needs of practical WDM systems. We propose a new filter
device that can meet the needs of even the most complex WDM system. However, the
primary difficulty of this filter is that it requires a totally new technology: the fabri
cation of a high reflectance corrugated waveguide in an electrooptic material. This
problem is addressed and a novel waveguide geometry that overcomes the difficulties
inherent to most other waveguide geometries associated with high reflectance corru
gated waveguides is proposed. Section 1.4 presents the organization of the ensuing
chapters of this dissertation.
1.1 Motivation
This information age that we are living in now continues to demand more and more
communications bandwidth. However, the bandwidth of typical residential telephone
services has remained virtually the same for decades. Integrated Services Digital
Network (ISDN) offers a modest improvement in digital transmission rates, but much
higher bandwidth is required in order to move into a new era of communications along
with the continuing advancements of computer hardware that is propelling us into
this explosive information age.
There is great demand for broadband distribution networks such as CATV, but
consumers also require services that can only be provided by twoway, routable net
works. These broadband communication networks are the key to the future of many
recently emerging technologies such as video on demand, video teleconferenceing, re
mote medical imaging, home shopping, and virtual reality networking, just to name
a few. These technologies will further enable us to meet in a virtual world to discuss
and share ideas, to conduct business transactions, and to embark towards new lev
els of entertainment, all in the comfort of our homes. Recently, all the processing,
storage, and imaging technology required to do these things has become available at
affordable prices. However, these services will be severely limited until the realization
of Broadband ISDN.
It is inevitable that a broadband communications era is on the horizon. The
question is, "What is the most efficient way to realize broadband communication
systems?"
1.1.1 Bandwidth of Optical Fiber
The question just posed is easy to answer because conventional optical fiber offers
the bandwidth we needmuch more than what is presently being utilized. To deter
mine the bandwidth of an optical fiber, we recall that the bandwidth B is related to
the spectral width AA by
B=AAf= (1.1)
10

1000 1200 1400 1600 1800 2000
Wavelength (nm)
Figure 1.1. Attenuation of a lowloss optical fiber (reference [45]).
where c and A are the speed of light and its wavelength in vacuum, respectively. The
theoretical maximum binary signaling efficiency is
R 2 bits
 (1.2)
B Hz sec
Nationwide fiberoptic networks are based upon single wavelength transmission
at 1.3 or 1.55[m with bit rates of about 565 Mb/s. This data rate is high enough for
more than 7600 multiplexed voice or about 8 video channels, and equations 1.1 and 1.2
suggest that this transmission rate can be contained within a spectral width less than
0.003nm when A = 1.55[/m. Typical systems, however, require more bandwidth due
to laser chirp, but this can be avoided by using externalcavity modulators.
Figure 1.1 shows that the fiber provides low loss regions around the wavelengths
of 1.3/tm and 1.5/im that are about 100nm wide. In principle, the fiber has more
than 10 THz of bandwidth to offer in the 1.51tm region. That is theoretically enough
bandwidth for about 300,000 high quality uncompressed video transmissions (70Mb/s
each). Since the operating bandwidth of lowcost electronics is presently far below
100 GHz, it appears that the only available means of utilizing more of the fiber
bandwidth is to move towards wavelength division multiplexing (WDM).
Ch 1 L' Narrow Bandpass Z ,
Ch 2 L 2 Wavelength Filter
Ch 2 LD 2 7Star
Coupler
ChM m LD I Controller Chx
 j
Figure 1.2. Simplified schematic of a WDM distribution system.
1.1.2 Practical WDM Communication Systems
Figure 1.2 shows a simple WDM distribution system. First, many electrical chan
nels are multiplexed together using a scheme such as time division multiplexing. Each
of those are used to externally or internally modulate a laser for one of the wavelength
channels. Each laser output becomes an individual optical channel. The optical chan
nels are multiplexed together by a star coupler and then distributed. Each receiver
has the ability to select the desired optical channel using a narrow bandpass tunable
optical filter with high free spectral range. After detection, the electrical signal is
then further demultiplexed for the user.
Recently, a couple of groups have successfully demonstrated the feasibility of such
systems. In 1990, H. Toba et al. [79] from NTT Laboratories in Japan demonstrated a
100 channel WDM transmission system at 622 Mb/s over 50 km. They have since ex
tended their system to 128 channels over 480 km using erbium doped fiber amplifiers
(EDFAs) [50]. A. Chraplyvy et al. [14] from AT&T Bell Laboratories demonstrated
a WDM system transmitting over 0.3 Tb/s through 150 km of fiber. That is enough
bandwidth for almost 5000 uncompressed high quality video transmissions, which is
5
WDM Mux/Demux Chip
 y
Tunable LD O utput'
Input t Output
 I 11   **Otu
In
ANe )k Router
Figure 1.3. Simplified schematic of a WDM ring network.
almost as many video channels as today's systems offer for voice conversations. By
using newly developed video compression schemes, this figure can easily be increased
by one or two orders of magnitude. Although these experiments were conducted at
a wavelength of 1.55/m, they utilized conventional 1.3pm zero dispersion fiber. This
means that the existing fiber network can be dramatically enhanced by simply re
placing transmitter, receiver, and repeater hardware, instead of laying thousands of
miles of new fiber.
These concepts are expected to be applied to ring networks where each subscriber
can transmit and receive data on a network. Figure 1.3 shows a simplified diagram of
an optical WDM ring network which is then routed to a wide area network (WAN).
The expanded view in the inset of figure 1.3 shows a conceptual view of the optical
hardware of one of the subscribers to the ring. In this case, each subscriber has a
tunable laser diode and a WDM multiplexing/demultiplexing chip. The tunable laser
allows the subscriber to transmit information on any of the wavelength channels. The
purpose of the WDM multiplexing/demultiplexing chip is threefold: extract all the
optical power of only the desired channel from the fiber optic bus, route the rest of the
channels back onto the bus for other subscribers, and retransmit a new signal from
the tunable laser tuned to the same wavelength channel as the filter. In this simplified
conceptual system, to prevent interference of signals from different sources, it would
be necessary for the laser and filter to be tuned to the same frequency channel and
for the filter to extract most of the input light in that channel (perhaps with 15dB
or less transmitted).
Many essential components of WDM systems such as tunable semiconductor laser
diodes, external modulators, waveguide couplers, detectors, and EDFAs are com
mercially available. However, commercially viable technologies for filters of densely
spaced WDM systems are yet to be reported. As will be discussed below, the filters
used in previous systems [79, 50, 14] do not meet all of the requirements of a practical
system.
1.1.3 Wavelength Selective IOCs
So far, we have presented a motivation for WDM as a realistic means of obtaining
very high bandwidth communication systems. Crucial to the realization of WDM in
commercial communication systems is the advent of practical wavelength selective
integrated optical circuit (IOC) filters. Many wavelength selective IOC filter designs
have been reported for WDM applications, but few are adequate for densely spaced
WDM. Typically, the devices are very long, have rather broad passband spectra,
require high power, and often involve complex fabrication processes. The remainder of
Acoustic Transducer TETM Splitter
TTM Splitter Acoustic Bea
Input
Unfiltered
Filtered
Figure 1.4. Polarization insensitive acoustically tunable optical filter.
our motivation will discuss some of the most common designs of wavelength selective
IOC filters reported in the literature. The pitfalls of these devices as they apply
to densely spaced WDM systems will be discussed. Then the goals of this research
work will be presented in section 1.2 as it presents a proposal for a new wavelength
selective IOC filter more suitable for densely spaced WDM systems.
Acoustically Tunable Optical Filter
Figure 1.4 shows a diagram of a polarization insensitive acoustically tunable op
tical filter (ATOF) [71]. The two waveguide polarizations are first separated by a
polarization splitter and fed through two identical waveguides that are under the
influence of a surface acoustic wave (SAW). Within a narrow band of wavelengths,
the difference between the propagation constants of the TE and TM modes equals
the propagation constant of the SAW and the two modes can then couple.
The ATOF is often regarded as one of the most promising filters for WDM sys
tems because it can easily be made polarization insensitive, it is tunable over more
than 100nm, and it can route many channels at one time by creating a SAW with
multiple frequency components. However, the ATOF is not suitable for systems such
as those in the most notable demonstrations of WDM systems [79, 50, 14]. This is
primarily because the passband is too broad (~'nm). Equation 1.1 shows that Inm
bandwidth corresponds to a 125 GHz channel at A = 1.55Lm. This is theoretically
enough bandwidth for a channel capacity of over 200 Gb/s. Since even very expen
sive electronic and electrooptic hardware presently cannot achieve these rates, the
ATOF would result in wasted bandwidth. The ATOF makes up for this deficiency
by having a very broad tuning range, but currently available tunable semiconductor
lasers and laser arrays (excluding external cavity lasers) can only operate within a
wavelength range of only a few nanometers. Therefore, a filter with a very narrow
passband is needed to more fully utilize the band of wavelengths within the tuning
range of commercial lasers.
Other disadvantages of the ATOF are that it is several centimeters long and
requires high power to generate the surface acoustic wave (SAW), typically more
than 100mW. In turn, the requirement for heat dissipation can make it difficult to
maintain stable operation. Recently, significant advances have been made to maintain
temperature stability in a commercial device [80].
Cascaded Filters
The systems demonstrated in references [79, 50, 14] used cascaded tunable band
pass filters. A cascade of three filters was used in reference [14] to achieve a 0.65nm
full width half maximum (FWHM) passband. This provided enough bandwidth for
a bit rate of over 100 Gb/s, although only 20 Gb/s was achieved.
The filter cascade used in references [79, 50] had a much narrower bandwidth
of 0.04nm (5 GHz), which corresponded to the bandwidth of the frequency shift
keyed (FSK) optical signal they used. This filter consisted of a cascade of seven
MachZehnder filters. Figure 1.5 shows a diagram of a similar device with only three
Thin Film Heater
Figure 1.5. Cascaded MachZehnder interferometer.
Figure 1.6. Resonant grating assisted directional coupler filter.
MachZehnder filters. The total device length was about 21cm which required a chip
size of 5 x 6cm2. Although it was not mentioned, it is reasonable to assume that the
filter was fabricated around the perimeter of the chip. The polarization dependence
of each MachZehnder filter was compensated using a laser trimming technique. Not
only is this a very long device which would be very difficult and costly to mass
produce, but each of the 7 MachZehnder stages must be individually controlled to
achieve wavelength selection.
Resonant Optical Directional Coupler
The resonant optical directional coupler (RODC) proposed by Kazarinov et al. [33]
is shown in figure 1.6 is a modification of the original proposal by Haus et al. [26].
The structure consists of two waveguide FabryPerot interferometers with zerolength
cavities (quarterwave shifted gratings) coupled to a center waveguide that acts as
the signal transmission bus. Light couples out of the upper grating waveguide only
at wavelengths for which the FabryPerot waveguide is resonant. The second Fabry
Perot waveguide below the center waveguide permits a complete power transfer to
the filtered output of the top waveguide at the resonant frequency if both Fabry
Perot waveguides are resonant at the same wavelength. Without the second resonant
waveguide, only half of the power at the resonant wavelength would be coupled to
the filtered output channel with a quarter of it being transmitted to the unfiltered
output and the other quarter being reflected to the source.
The theoretical device transmission characteristic is very narrow (~ .01nm), but
has yet to be adequately demonstrated. The first attempt at demonstrating a sim
ilar device without the lower resonant waveguide was reported in reference [39] in
GaAs/AlGaAs. The theoretical response of the RODC indicates that it is well suited
for WDM applications, but there are two fundamental limitations for which no solu
tions have been reported: it requires excellent DBR corrugated waveguides and has
a narrow stopband (typically ~ 2nm).
The new innovations in waveguide grating fabrication and the concepts that are
presented in this dissertation may prove to be invaluable towards circumventing the
limitations of the RODC and ultimately realizing an electrooptically tunable RODC
as well as other FabryPerot filters that are suitable for WDM. The filter we propose
is a FabryPerot waveguide structure that can be used in the RODC to extend its
stopband (see section 8.2.3). Section 1.3 will introduce a new waveguide geometry
well suited to make high reflection corrugated waveguides in commercially viable
electrooptic materials such as LiNbO3.
1.2 A New Wavelength Selective Tunable Filter
A wavelength selective filter most suitable for densely spaced WDM systems
should have a very narrow passband (~ 0.01nm), a very broad stopband (> 100nm),
and a high extinction ratio (> 14dB). The narrow passband is adequate for moderate
bit rates (< 2 Gb/s) and enables the fiber bandwidth to be utilized more efficiently
as the wavelength channels are densely spaced. The broad stopband and high extinc
tion ratio allows a receiver with a discriminant signal detection scheme to isolate one
of the wavelength channels without crosstalk. Other desirable aspects of a tunable
wavelength selective filter are the ability to operate at high speed with low electrical
power, a simple biasing scheme, and few controllers. In order for the device to be
commercially practical, it should be as compact as possible and be manufactured with
standard microelectronics technology. Most wavelength selective filters reported to
date, including all of those described in section 1.1.3, lack one or more of the above
requirements. We propose a new filter scheme that has the potential to satisfy all
these requirements.
Figure 1.7 shows the proposed filter structure. It consists of a corrugated waveg
uide that exhibits high reflectance in a narrow wavelength spectrum in cascade with
a broadband high reflectance mirror deposited on the endface. The reflection spectra
of these two mirrors is shown in figure 1.8. The reflection band of the corrugated
waveguide is shown as a sharp spike in figure 1.8 (a). Comparing the width of the
reflection bands of the dielectric mirror from the plot in figure 1.8 (a) with that of the
corrugated waveguide in the expanded view in figure 1.8 (b), we see that the reflec
tion band of the dielectric mirror is over three orders of magnitude wider than that
of the corrugated waveguide. These two mirrors, fabricated in cascade constitute an
asymmetric FabryPerot (ASFP) interferometer. We use the term "Asymmetric" to
Corrugated Waveguide
Reflector
AR Coating
Figure 1.7. Proposed Tunable Wavelength Selective Filter.
refer to the use of two mirrors with vastly different reflection spectra as opposed to
the conventional case where two similar mirrors are used.
The two conventional cases of waveguide FabryPerot filters, where similar mirrors
are used on both ends, are not adequate for WDM. For the case where two dielectric
mirrors are separated by a cavity, there exist many transmission peaks. The use of
two corrugated waveguides separated by a cavity may exhibit a single transmission
peak, but the stopband is very narrow. It will be shown that the proposed struc
ture in figure 1.7 utilizes the desirable characteristics of both mirrors: a very broad
stopband contributed by the dielectric endface mirror, and the selection of only one
transmission peak by the narrow reflection band of the corrugated waveguide mirror.
The resulting transmission spectra consists of a single transmission peak with an
extremely high free spectral range (FSR). It will be shown that by using the electro
optic effect via an appropriate electrode scheme, the transmission peak can be shifted
to accommodate many wavelength channels in a WDM system. The reader is referred
to chapter 3 for a detailed explanation of this filter structure. We demonstrate the
 Dielectric Mirror 
  Corrugated Waveguide 
0.2
0 L
1000
1
0.8
0.6
U
o 0.4
0.2
0
153
1200 1400 1600 1800 2000
Wavelength (nm)
2200
 Dielectric Mirror I
 Corrugated Waveguide
I
'I
~l~
/ I ~ ~ I'
I 
33
1533.5
I'
I'
II
I I\ N
A ,
1534
Wavelength (nm)
1534.5
1535
Figure 1.8. Comparison of the reflection spectra of a dielectric endface mirror and a
corrugated waveguide. a) shows that the reflection band of a corrugated waveguide is
very narrow compared to the spectra of the dielectric mirror. b) shows an expanded
view of the corrugated waveguide spectra.
experimental results of the proposed filter in a very practical and mature material
system: LiNbO3.
A reader familiar with distributed feedback (DFB) lasers will recognize the ASFP
filter as a commonly used structure for low threshold DFB lasers. The structure is
the same, except the filter does not have a gain medium and uses the electrooptic
effect for tunability. However, requirements the two devices have on the resonant
structure are very different. This will be addressed in section 3.3.3.
This work proposes that the ASFP structure be used as a passive filter ("passive"
refers to the lack of a gain medium). Furthermore, it is shown how the filter can
be tuned by a simple electrode scheme, utilizing the linear electrooptic effect. The
modeling of this device as well as the experimental results are presented.
1.3 A Novel Waveguide Technology
The ASFP filter proposed in this work requires two highreflectance mirrors. For
decades, high reflectance dielectric mirrors have been available. Corrugated waveg
uides have been studied for over two decades, but highreflectance, lowloss corrugated
waveguides have yet to be demonstrated in a material system suitable for making the
tunable ASFP filter.
In light of the failure of conventional methods to produce high reflectance corru
gated waveguides, such as etching gratings into the waveguide or buffer layer, a new
waveguide geometry is proposed and developed. Figure 1.9 shows the transverse cross
section of the waveguide. It consists of a conventional gradedindex channel waveg
uide with an overlay of higher refractive index material. A grating in this material
(figure 1.10) can be used to create a considerably more perturbation to the waveguide
mode than conventional corrugated waveguides without producing excessive scatter
ing loss. This is necessary to achieve high reflectance and low loss. Of course there
Gradedindex
waveguide
N,
Highindex
overlay
,
Figure 1.9. Transverse cross section of a gradedindex channel waveguide with an
overlaying film of refractive index near or higher than the core of the waveguide.
Gradedindex
waveguide
R
nclad
Highindex
grating overlay
/i
nsubI
Figure 1.10. Longitudinal cross section of a DBR corrugated waveguide
a highindex overlay.
made from
T
 r
is always a tradeoff between the perturbation and radiation loss, but the benefit of
using this waveguide geometry is that more perturbation can be achieved with less
radiation loss. A detailed description of our waveguide geometry will be presented.
Commercially viable materials and waveguide technologies are used for our ex
perimental demonstration. LiNbO3 is chosen as the substrate material because it is
currently the material system primarily used in the industry for stateoftheart pas
sive IOC devices. Waveguides are made by either annealed proton exchange (APE)
or titanium indiffusion. Silicon is chosen as the overlay not only because of its high
refractive index, but also because it is opaque to visible light, is easily deposited by
ebeam evaporation, is easily etched by reactive ion etching (RIE), and can screen
pyroelectrically generated surface charges. We will cover these benefits in detail.
A somewhat startling discovery was made during the course of the investigation
and characterization of the structure: when a planar film of highindex silicon with
thickness large enough to support multiple planar modes is deposited on the surface
of a singlemode channel waveguide in LiNbO3, the resulting channel waveguide may
still support only a single mode. Furthermore, in contrast to planar waveguides alone,
a comparably small amount of optical power is drawn up into the silicon layer, the
mode index increases by a small amount, and the propagation loss remains low. This
is exactly what is required to make good corrugated waveguides: low propagation
and mismatch losses with very little change in the mode field profile.
Rib waveguides consisting of planar films with thicknesses capable of supporting
multiple modes have already been shown to produce singlemode waveguides by Soref
et al. [73]. However, the prescribed concept in this dissertation goes one step further
as it applies to waveguides of much lower refractive index than the planar layer.
Furthermore, the proposed waveguide geometry is much more practical because it is
applicable to gradedindex channel waveguides made by APE or titanium indiffusion
which have proved to be commercially viable technologies.
It will be shown in that commonlyused theoretical estimations of the propagation
constants and mode field profiles cannot be applied to this structure and hence, do
not support these claims. Perhaps this may explain why such a simple structure has
not been investigated to date. Complete waveguide characterization measurements
including mode index, loss, and field profile are presented. All of the experimental
measurements made in this work support these claims. In addition, some very unique
transmission measurements of DBR corrugated waveguides are presented that can not
be explained without this new insight.
This new waveguide geometry is used to demonstrate the highest reflection in
LiNbO3 corrugated waveguides reported to date. The results are in excellent agree
ment with theory. The proposed filter structure is also demonstrated using this new
technology.
1.4 Chapter Organization
Chapter 2 covers most of the numerical modeling used in this work for device
simulation. In particular, two transfer matrix formalisms that are useful in solving
the optical fields in planar multiple layered structures are presented. The benefits
of each approach and their application to lasers, other FabryPerot structures, and
waveguides are then discussed. A brief description of the spectral index method is
given. The derivation of these formalisms from Maxwell's equations is used to develop
the notation and identify all the approximations, as well as some common mistakes.
A detailed discussion of DBR FabryPerot devices and an indepth proposal of
the new wavelength selective tunable filter follows in chapter 3. Calculations, with
reasonably achievable waveguide parameters without neglecting loss, are presented
for the proposed structure. The device is shown to exhibit the characteristics neces
sary for a denselyspaced WDM system only for structures with lowloss corrugated
waveguides.
To achieve good tunable ASFP filters, highreflection lowloss corrugated waveg
uides must be made in a material system that exhibits a large electrooptic effect,
such as, LiNb03. In chapter 4, first we describe many of the previous attempts to
make high reflectance corrugated waveguides, most of which yielded mediocre results.
Then, we propose a new waveguide geometry in LiNb03 that overcomes several fun
damental limitations. This geometry is applicable to several other material systems
as well.
Chapter 5 illustrates the details of the various fabrication and processing tech
niques required to fabricate the devices. These techniques include reactive ion etching
(RIE), waveguide fabrication by APE, grating lithography and transfer of the grating
to the waveguide, electrode fabrication, endface polishing, and endface DBR deposi
tion.
A detailed experimental analysis of the new waveguide geometry is presented in
chapter 6. The mode index, mode field profiles, and loss are measured. These results
are in agreement with the theory, in support of the claims made in this work, resulting
in a clear understanding of the unique characteristics of the geometry as applied to
our devices.
Chapter 7 presents the experimental results of several corrugated waveguide filters
made with the new waveguide geometry. The results of a simple corrugated waveguide
reflector is found to be in excellent agreement with the theory and exhibits the
highest reflection from a corrugated waveguide in LiNbO3 reported to date. With the
fundamental building blocks in place, the proposed ASFP structure is demonstrated
and compared with theory.
19
Finally, chapter 8 summarizes the results. The scope of this work cannot begin to
cover all of the potential uses and enhancements to this new cuttingedge technology.
It seems that many other uses for the new highindex overlay technology exist. An
outline of future work, some of which is already under way, is also described in this
chapter.
CHAPTER 2
THEORY OF OPTICAL FIELDS IN MULTILAYER STRUCTURES
This chapter covers the theory used throughout most of this work to calculate
reflection and transmission spectra for dielectric mirrors and DBR corrugated waveg
uides, and the waveguide mode characteristics. This chapter does not present any
new theory or derivation, but it approaches the subject from a very unique and sim
plistic angle. To develop the notation and justify all of the assumptions made, we
start from Maxwell's equations.
Most textbooks cover reflection of TE and TM waves independently. Here, we
use a formalism that is used in only a few textbooks [40] to describe both cases
with a common set of equations. The angle of incidence and polarization is factored
out in a way that makes the equations take the form of those of normal incidence.
Most references use only one of two matrix methods [40, 22], but, depending on the
problem at hand, one formalism is generally more convenient than the other. This
will be demonstrated by several examples in section 2.7. Section 2.7.4 will show that
both matrix formalisms can be applied to waveguide devices by substituting the local
normal mode indices for the refractive indices in the previously derived equations.
Finally, the theory of the spectral index method to solve for waveguides with two
dimensional cross sections is presented in section 2.8. The spectral index method is
useful for calculating the local normal mode indices which can then be used in the
matrix methods for calculating the reflection and transmission spectra of corrugated
waveguide structures.
2.1 Basic Equations
We begin with Maxwell's equations along with the constitutive equations for
isotropic media:
V. D p (2.1)
V B = 0 (2.2)
OB
Vx E (2.3)
Vx H = J + (2.4)
D = eE (2.5)
B = pH (2.6)
J = oE (2.7)
where E is the electric field strength [V/m], D is the electric displacement [C/m2],
H is the magnetic field strength [A/m], J is the electric current density [A/m2], B is
the magnetic flux density [T], p is the electric charge density [C/m3], a is the electric
conductivity [1/Om], p is the permeability [H/m], and e is the permittivity [F/m].
Furthermore, we add that
e = Creo (2.8)
p = PrPo (2.9)
1
co = 2 (2.10)
POC2
where e, and pr are the relative permittivity and permeability, respectively (dimen
sionless), co is the permittivity of free space (8.8541853 x 1012F/m), Po is the per
meability of free space (47r x 10H/m), and c is the speed of light in free space
(2.997925 x 108m/s).
In an isotropic, homogeneous medium of no free charges, we may rewrite Maxwell's
equations in terms of two field variables
V.E= 0 (2.11)
V H =0 (2.12)
V x E = (2.13)
OE
Vx H = oE + E (2.14)
at
Now taking the curl of equation 2.13 using the vector identity
V x (V x A) = V(V A) V2A
and substituting equations 2.11 and 2.14, we have
0 OVE 02E
V x (V x E) = V(V E) V2E = V2E = pV x H = paO tpe 2
or
2E 02E
V2E = 0 (2.15)
In the same manner, by taking the curl of equation 2.14 we obtain
OH 02H
V2H  =0 (2.16)
Equations 2.15 and 2.16 are wave equations for the E and H fields. The solutions
for the E and H fields are of the same form. The solution of equations 2.15 and 2.16
for a plane wave traveling in the r direction is
E = Eoei(krwt) (2.17)
and
H = Hoei(krwt) (2.18)
where k is the propagation vector, r is the position vector of an arbitrary point with
respect to a reference, and w is the angular frequency of the wave. Some authors
prefer to use ei(wtk'r). Generally, the results between the two formulations may be
compared by replacing i with i and visa versa. This will not be the case in equation
2.69. Equations 2.17 and 2.18 do not represent physical quantities because they are
complex, but they are a general solution to the pair of wave equations 2.15 and 2.16.
However, physical quantities may be obtained by taking the real part of the complex
solutions. The results by doing such will also be a solution to the wave equations
because they are linear allowing the real and imaginary parts of the solutions to be
separated. We could take the real part here and obtain
R(E) = R(Eo) cos(k r wt) '(Eo) sin(k r wt)
but it is most often easier to carry the complex expression through the arithmetic and
take the real part after manipulation when a physical quantity is desired. However,
when using them in a relation which is not linear with respect to E or H such as
the power relationship, the complex quantities may not be used unless an adequate
formulation for complex numbers is used.
Substituting equation 2.17 into equation 2.15 we find that
k2 = Ik2 = w21 + iwA,0 (2.19)
The phase velocity of the wave is given by v = w/k and the velocity of light in free
space is given by c = 1//po. Combining these equations with equations 2.19, 2.8,
and 2.9 we obtain the square of the complex refractive index n:
2 c2k2 .r )2
nr r + i = (nr + ii2 (2.20)
where nr and ni are the real and imaginary parts of the complex refractive index,
respectively. nr is referred to as simply the refractive index and ni (often denoted by
k) is known as the extinction coefficient. With the free space propagation constant
being
w 27r
ko= (2.21)
c Ao
equation 2.17 becomes
E = Eoei(nk'krwt) = Eoenikokre i(nrkokrwt) (2.22)
The first exponent on the right hand side of equation 2.22 represents absorption (or
gain) while the second represents phase.
The wave equations, although derived from Maxwell's equations, do not represent
all of the information useful for determining the fields. They only help us to define
a certain class of solutions, namely the transverse propagating waves. To determine
a solution of Maxwell's equations, equations 2.17 and 2.18 must be substituted into
equations 2.11 2.14. Doing this, a simplified form of Maxwell's equations for trans
verse propagating waves is obtained:
k E = 0 (2.23)
k H=0 (2.24)
k x E = pwH (2.25)
k x H = (cw + io)E (2.26)
From these equations, we see that E, H and the direction of propagation form a
mutually orthogonal set. From equation 2.25 we see that the magnitudes of the field
vectors are related by
E =7 HI (2.27)
where, using equations 2.20, 2.10, and 2.9,
Sn o n(2.28)
k n eo n
The quantity q] is the wave impedance; the impedance of free space 7ro = 377P.
The right hand side of equation 2.28 makes the assumption that Pr = 1 at optical
frequencies.
Next, we discuss the Poynting vector S. It is defined by
S = Ex H (2.29)
and is the energy current density [W/m2]. Equation 2.29 may be derived from
Maxwell's equations and the relation describing the loss of electromagnetic energy
due to Joule heating, fy Jf E dr, and making the argument that rate of electromag
netic energy decreasing within a volume must be equal to that consumed by Joule
heating plus that which is leaving the volume. The Poynting vector "points" in the
direction of energy flow. The above definition for the Poynting vector is not limited
to time varying fields. It is important to realize, as stated earlier that in calculating
S, both E and H must be physical quantities, i.e. real, and they must be the total
resultant of all the waves of a given frequency at the point of interest. As previously
illustrated, however, it is more convenient to work with complex quantities to rep
resent time varying fields. Extracting the time dependence from equations 2.17 and
2.18 and expressing them as
E = (Eoek.r)eiwt = Eceiwt (2.30)
and
H = (Hoeikr)eit = Heiwt (2.31)
it may be shown that the time average of the energy flow is
(S) = 1 R(E, x H*) (2.32)
which is in the direction of k. In doing this, we have simplified the calculation of
the energy flow of time varying fields in terms of the complex timeinvariant field
amplitudes. It must be remembered that the complex field amplitudes are used in
equation 2.32 while only their real part is used in equation 2.29. For a single wave
traveling in direction k, equation 2.32 may be further reduced using equations 2.25,
2.27, and the identity
A x (B x C) = B(A C) C(A B)
resulting in
(S) = r IEc2 1k (2.33)
the scalar intensity is then
I = n EcE* (2.34)
2 7,
where Ec is the complex scalar electric field amplitude. From equation 2.22, we find
that
1 n Eo 12 e2nikokr= Ieakr (2.35)
Io is the intensity at the reference point, Ir = 0, and
47rn,
ao  (2.36)
is the absorption coefficient. Two important points must be remembered about the
calculation of intensity:
In deriving equations 2.33 2.35, we assumed that there was a single wave
propagating in the direction k. Therefore, when calculating intensities resulting
from fields traveling in different directions, equation 2.32 must be used.
I oc nr (amplitude)2 should be remembered rather than simply I oc (amplitude)2
which is often quoted. If the later is used when comparing intensities present
in two different media, the result will be in error.
These two important points must be remembered when calculating reflectance or
transmittance, which is done in the next section.
Figure 2.1. k vector relationships with light incident on a single surface.
2.2 Reflection and Transmission From a Single Boundary
Figure 2.1 shows the k vectors from the incident ki, reflected kr, and transmitted,
kt plane waves. The boundary conditions for the fields may be found from Maxwell's
equations. These state that the tangential components of E and H are continuous.
These conditions must be satisfied at any point on the boundary for all values of time.
This implies that the exponential terms of the incident, reflected, and transmitted
waves must be equal for all values of time. Thus, all the waves have the same
frequency w and
ki Arb = kr Arb = kt. Arb (2.37)
where Arb is the position vector between any two points in the boundary. This
assures us that all the waves have the same phase at any point on the boundary at
any time t. From equation 2.37 we find that
6, = Or (2.38)
and
no sin 0i = nl sin 0t (2.39)
which is Snell's law.
S E E r E Hkr
E E H H E
H k7no kno
E E
H
(a) kt t (b) t
Figure 2.2. Conventions for the positive directions of the E and H field vectors for
the (a) TE or spolarized waves and (b) TM or ppolarized waves.
Figure 2.2 shows the convention we use to define the positive directions of the
electric and magnetic field vectors for the cases when E is perpendicular to the plane
of incidence (referred to as TE or spolarized) (a) and when E lies within the plane
of incidence (referred to as TM or ppolarized) (b). The tangential components of
the field vectors, and 7, are related to the field amplitudes E and H by
8= E and 7=H cos (TE) (2.40)
for the TE case and
= Ecos and7= H (TM) (2.41)
for the TM case. 0 is the angle between the normal fi and the propagation vector k
in each medium.
Now we introduce a constant by which we will factor out the polarization and angle
dependence to simplify further derivations. This is the modified optical admittance
y which connects the two tangential components ET and HT by
y = 7//
(2.42)
where, for the two polarizations,
n n
yTE =cos0 and YTM = (2.43)
77o 1o cos 0
It should be noted that y is the modified optical admittance for a single wave and
not the total field.
To solve for the reflectivity and the transmissivity, we simply need to meet the
boundary conditions with the resultant tangential fields. Noting the sign conventions
in figure 2.2 we have
Ei + r = st (2.44)
and
Hi r = Ht (2.45)
Solving equations 2.44, 2.45, and 2.42, then comparing field amplitudes using 2.38,
2.40, and 2.41 we find that the reflectivity r is
r = Yo = E, (2.46)
Ei yo +y E+
Similarly, but comparing with equation 2.39 instead of equation 2.38 we find that the
transmissivity t is
t= 2yo which is E for the TM case (2.47)
Si o + yi E T
Using equation 2.41 we find that
Et 2yo cos 0i
Sfor the TM case
E= yo + Yi cos Ot
which is more commonly used as the transmissivity. We, however, will use the tan
gential form in equation 2.47 because it is more convenient to use since it represents
both polarizations and the angular dependence has been factored out. These equa
tions are particularly easy to remember since they appear exactly the same as those
for normal incidence. Furthermore, we will see that they are more consistent with
the power relations.
The resultant tangential electric and magnetic fields in the incident medium are
i+r = (1 +r)
and
Hi Hr = yo( r)S
and for the transmitting region are tSi and y1t8, respectively. Using equation 2.32,
we find the intensity in each region. When we equate these we get
1 rr i(Y) (r Yr*) = ft
r (Yo) r () )
which is similar to the power rule, 1 R = T where R is the reflectance and T is the
transmittance, except for the last term on the left hand side. This difficulty may be
avoided by assuming that the incident medium is nonabsorbing, making F(yo) = 0.
This is true for most practical circumstances. Now we have the reflectance and
transmittance:
R = rr* = (2.48)
and
(yo) 4(yo+y)(yoy(1) (249
R(o) (Yo + Yi)(Yo + Yv)*
Notice that the definition we use for t results in a very simple equation for T, unlike
many textbooks that redefine T by making power arguments based on using a finite
area of the incident beam versus the resulting area of the transmitted beam as is done
in many textbooks. The dependence of this area on angle was factored out in the
original relations for y, thus illustrating another nicety of this uncommon formalism.
noBoundary a
ni d
Boundary b
n2
Figure 2.3. A single dielectric planar layer on a substrate.
2.3 Characteristic Matrix of a Thin Film
Now we extend the above analysis to the case of a thin film as in figure 2.3.
Multiple reflections exist between the two surfaces of the film and, if the film thick
ness is smaller than the coherence length of the incident light, they interfere. One
way to solve this problem is to set up an equation which takes the sum of all of
the reflections taking the phase of each reflected wave into account. The following,
however, is a much more straightforward approach which may easily be extended
to accommodate multilayer structures. This procedure rederives the reflectivity and
transmissivity for the whole structure without using the results for a simple single
interface system. In doing this, we will solve for the fields in all regions simultaneously.
In any medium, only two waves can exist: forward traveling and reverse traveling.
These are what would be the resultants of many reflections if it were calculated by
the former procedure.
We begin by denoting waves traveling in the direction of incidence (forward) with
a + superscript and those traveling in the reverse direction with a superscript.
Using the same convention for the direction of the E and H vectors, their tangential
components at the lower boundary b are
9b = 9U + Slb
Hb = y91 y11b
(2.50)
Solving in terms of the resultants Lb and b we get
E = I('b/1+b)
1
lb = ( b/Y1 + b) (2.51)
1
Hb = Y1L= 1 +(bY19b)
1
'Hb = yigf = ('b Yl1b)
The tangential fields at the same lateral position on the top boundary, a, may be
found from equations 2.51 by considering difference in phase 6 between the top and
bottom interfaces at the same instant where
6 = konidcos 01 (2.52)
Note that the cosine factor is present because we are comparing the phase of the
waves at the same horizontal position between the top and bottom interfaces. The
fields at the top interface now become
Ea = 1(lb/y + Eb)e'6
1
Sa = 1(b/1 + cb)e"
1
H = =2 ('b Ylb)eib
so that
Sa = a + 1a = Lb COS 6 isin 6
Yi
'Ha ='H + la= iylb sin 6 + b COS 6
which can be written in matrix form as
S ,a [ cos 6 isin(6)/y1 1 5
LHa iyI sin 6 cos J (b
The 2 x 2 matrix on the right hand side of equation 2.53 is called the characteristic
or transfer matrix of the film. If we define the optical admittance for the assembly
to be
Y = Ha/Sa (2.54)
the task of determining the reflectance becomes the same as for the single boundary
case so that
Yo Y
r = Y(2.55)
yo +Y
and
R= (YoY) ( Y)* (2.56)
Using equation 2.54 and assuming Sb is a single propagating wave, ie. the last layer,
equation 2.53 becomes
a 1 = [ cos6 isin(6)/yl [ 1 ] b (2.57)
Y iyj sin 6 cos 6 Y2
The characteristic matrix may be further simplified from 2 x 2 form to a 1 x 2 matrix
by letting
B cos 6 isin(6)/y [1 (2.58)
C iyj sin 6 cos 6 y2
so that Y = C/B. This notation will be useful in the next section.
2.4 Characteristics of a Multilayer Structure
Now we will consider the general thin film structure of figure 2.4. The thin film
results obtained above may be generalized to obtain the characteristics of an assembly
of thin films by recognizing that the tangential fields at boundaries b and c may be
no Incident medium
n2 n d2d
c Multilayer Structure
n1 di m
nm Substrate
Figure 2.4. A general multilayer thinfilm structure.
related in the same way that those of boundaries a and b were in section 2.3. That
is,
[ b COS 62 i Sin(62)/y2 (2.59)
[ b ] 2iy2 sin 62 cos 62 7Hc
where
6k = konkdk cos Ok (2.60)
Combining 2.53 with 2.59 and using 2.54 and 2.42, the solution becomes
ea [ = [ C ] =
E cos 6 i sin(61)/yi cos 62 i sin(62)/Y2 1 ]c (2.61)
iyj sin 61 cos 61 iy2 sin 62 cos 62 Y3 J
and continuing the process, we see that the characteristic matrix for the whole as
sembly of figure 2.4 is
B = i cos 6k i sin(6k) Ik 1 (262)
C iyk sin bk cos 6k m 22
Some care must be taken when calculating the phase factor when absorption or
total internal reflection is involved. The angles may be complex and are found by
Snell's law:
nk sin Ok = no sin 00o (2.63)
Since the complex arcsine function is often unavailable on computers, we calculate
6k by
k = kodk /n no sin2 (0
(2.64)
As we have assumed, no and 0o are real and for absorption, nk is in the first quadrant
of the complex plane. The square root is then in the first or third quadrants. 6k must
be in the first quadrant because the phase must be increasing and the amplitude
exponentially decreasing in the direction of k in order for the situation to be physically
possible. Similarly, if medium k produces gain, we find that 6k must be in the fourth
quadrant. In either case the + sign in equation 2.64 should be chosen since most
computer algorithms produce positive real parts when calculating the complex square
root. However, it is a good idea to verify if this is indeed the case. The same value
for cos Ok should be used for both 6k and Yk. That is,
nj nj sin2 O0
cos k = n (2.65)
nk
The reflection coefficient may be found using equation 2.56 with Y being the
optical admittance for the multilayer assembly. With Y = C/B,
(yoBC (yoBC\ (2.66)
R= yoB +C yoB+C)
To find the transmittance T and absorptance A, we note that the intensity normal
to the last interface m and the first interface a are
Im = gmVy' ) = 1W m m
2 2
and
la = If(BC*)8m*
2
la is the intensity normal to the interface actually entering the assembly. This is
related to the incident intensity 1i by
a = (1 R)I
so that
2(1 R)m
A 2(1 R)
The transmittance is
Im R(ym)(1 R) 4yo((Ym)
Ii R(BC*) (yoB + C)(yoB + C)* (2.67)
The absorptance is
I( (ym) 4yo(BC* ym)
A=1RT=(1R) 1 (Y) 4 (y(BC* (2.68)
R(BC*)J (yoB + C)(yoB + C)*
The phase change on reflection can be found from equation 2.55 to be
Sarctan (iyo(B*C C*) (269)
where it is important to note the signs of the numerator and denominator to assure
the correct quadrant since the arctangent only has a range within the first and fourth
quadrants.
2.5 Alternative Matrix Formulation
Next we describe an alternative formulation to the solution of a multilayer struc
ture given in section 2.4. This is the approach used in reference [22]. It will be shown
in the next section that both formalisms have much to offer in understanding and
calculation of multilayer structures. The previous formalism was a set of equations
calculating the total electric and magnetic fields at each interface. This formulation,
however, will calculate the forward and reverse electric fields on each side of each
interface.
At boundary a of figure 2.3, the total fields are
Sa = 0a + + a = g= a + Sa
RHa = ( a) = Y1i(gl+a + gh)
Rearranging these equations, we have
SO+a = 1la S l+a +agla
ta ta
where
Yo Yi 2yo
ra= and ta= 2 (2.70)
yo + Y1 Yo + Y1
Written in matrix form,
[ J 1 ra 1 J (2.71)
The electric fields on each side of film a are related by
where 61 = konlidcosOt. Written in matrix form,
p+ \ii n0 S +
[a l ] (2.72)
the fields throughout a multilayer structure may be calculated by repetitive use of
equations 2.71 and 2.72equation 2.71 for each interface and equation 2.72 for each
film.
Examples illustrating the benefits and uses of each matrix formalism will be given
in section 2.7. First, the relationship between the two formalisms will be given in
section 2.6.
2.6 Connection Between the Two Formalisms
Both matrix formalisms derived in sections 2.4 and 2.5 are useful for solving fields
in planar multilayer dielectric structures. A connection matrix will be derived in this
section showing that they equivalent. Even though they are equivalent, the knowledge
of both of them facilitates deriving different sets of basic analytic expressions. This
will become clear in section 2.7 as the advantages of each formalism will be discussed
for a variety of cases.
In the first formalism (section 2.4), the total and '7 fields were determined
at each interface and a transfer matrix was formed by multiplying a set of 2 x 2
matrices together, each matrix representing a film. We will call this the film method
because each matrix contains only information about one film and is not dependent
on adjacent films.
In the second formalism (section 2.5), where the forward and reverse tangential
electric fields (S+ and 8) were determined at each interface, a transfer matrix was
formed by multiplying a set of 2 x 2 matrices together for each interface and each
film. We will call this the interface/film method.
We start with the tangential fields at boundary a. They are given by
Sa = '0+a+ 8 F
Ha = yO(+a a)
In matrix form, this becomes
Ea iI (2.73)
Ha O 0 O 6a6
By inverting the matrix, we get
Eoa \a (2.74)
,%a 2 1 1 H/a
If M is the transfer matrix for a structure with boundaries a through m such that
Ea] = M (2.75)
then the use of relations 2.73 and 2.74 can be used to make equation 2.75 become
[ = 1 [ 1 1 1 M 1 Y[ (2.76)
[o O 2 1 [Ys Ys Y sm
where s denotes the last media on the right hand side of the last boundary m.
Equation 2.76 is in the same form as equations 2.71 and 2.72 showing that the two
formalisms are connected by the two transformation matrices in equation 2.76.
2.7 Applications of the Formalisms to Real Structures
This section will use each formalism to derive some basic relations in a very
straightforward way. Depending on the problem at hand, it will be shown that
in some cases the interface/film method is preferred, and in another case, the film
method is preferred.
In general, the film method is better for computer calculation of the fields of a
structure. This is because it is composed of a stack of independent 2 x 2 film matrices.
For a periodic structure, a matrix may be generated for a single period and taken
to the power N where N is the number of periods. As will be shown below, this
yields a very simple expression for the reflectance of a periodic A/4 structure. The
interface/film method is better suited for calculation of resonance conditions. These
will now be discussed in detail.
2.7.1 Reflectance from Periodic QuarterWave Structures
A periodic stack of dielectric films is of particular interest because it can be used
for a high reflectance mirror. The optimum thicknesses for the films, in this case, is
A/4 where A = Ao/ni is the center wavelength of the reflection band. (Due to the
asymmetry of the reflection band A is not the true center, but it does correspond
to the maximum reflectance.) Either the film or the interface/film methods can be
used to solve for the reflection, but the film method yields an analytic solution for
the reflection at the center wavelength in a very straightforward fashion.
If the periodic structure in figure 2.5 is nonabsorbing and the thickness of each
layer is AO, using equation 2.62, we find that
i = Ys
~ [ [Y2
no n1 n2 n1 n2 n1 n2 n1 n2 ns
So o
a b m
Figure 2.5. Dielectric mirror made from a periodic quarterwave stack.
[ (_y)N 1
(Y2 JOySJ
(2.77)
so that
B= Y2 C=y Y and y==ys )2N (2.78)
Y1 Y2 B Y2
and using equation 2.56,
y(_2)2N 1
r = )" 1 (2.79)
Yo (2 2 )2N +
This equation is very useful for the design of high reflectance dielectric mirrors. It
allows a straightforward determination of the number of periods and ratio of the
indices required for a desired reflectance. Furthermore, it shows that if y, > yo,
better reflectance is obtained if yi > y2.
2.7.2 Basic Laser Relationships
So far, the multilayer matrix formulations derived in sections 2.4 and 2.5 have
been used to solve problems in which there are incident, reflected and transmitted
waves. For the case of the laser this is not the case. However, the formalism is general
and, as will now be shown, is applicable to the laser as well. For reasons that should
soon be apparent, the interface/film method is more convenient for calculation of the
no n1 n2
'F= 0
a b
Figure 2.6. Singlefilm laser.
conditions required for lasing. (If it is not obvious, it is suggested that the reader
try to derive these conditions using the film formalism.) To demonstrate this, we
start with the singlefilm layer laser in figure 2.6. The interface/film method requires
the multiplication of three matricesone for interface a, one for film 1, and one for
interface b. Using equations 2.71 and 2.72 we have
[oa 1 [ ra e'e 0 1 1 r, 2b
[Oa tatb ra 1 0 eii rb 1 2b (2.80)
multiplying, we get
6(+,a 1 [ ei6 + rarbe'61 raei6l + rbe'i6' 1 r 1(281
Soa tatb raei6 + rbei61 ei6l + rarbe'6'
Radiation is emitted from the laser and propagates away from the device to infinity
in regions 1 and 2. The laser cannot produce incoming waves from infinity. Therefore,
the boundary conditions for the problem are o = 0 and  = 0. Using these
boundary conditions in equation 2.81 we find that
e~61 + rarbe'61 = 0 (2.82)
Rearranging, we have the basic gain relationship:
rarbe2i51 = 1 (2.83)
This relationship is more fundamental than those given in most textbooks because it
contains both magnitude and phase information in one simple relation. To see this,
we separate the amplitude reflection coefficients into magnitude and phase, and 61
into real and imaginary parts:
Ira ei(O.+7) Irbl e ibe261e2i61r = 1 (2.84)
where 61 = 61r + i61i. The 7r phase shift is introduced by considering the negative
sign. It is important to consider that both ra and rb are considered from left to right.
The negative sign in equation 2.83 is the result of defining ra with the incident wave
in medium 0. Even though this is not the case, the equations still hold provided the
consistency of the definitions is observed and not intermittently changed. Defining
ra this way causes a 7r phase shift. For this equation to hold, both the magnitude
and phase components must equal one:
Ira Irb e2651 = 1 and ei(261r+a+b+7) = 1 (2.85)
Using equations 2.36 and 2.48 with the magnitude part, it may be easily shown that
1 1
g = n (2.86)
g = 2d RaRb
where g = a is the intensity gain coefficient. The phase portion shows us that
261,r + a + Ob + T = m27r where m = 0, 1,2,... (2.87)
The last two equations correspond to those found most often in textbooks and are
equivalent to equation 2.83. They offer a more conceptual view because they are
easily derived by arguing that the round trip gain equals one and the round trip
optical path length is a multiple of A. It should be noted that in extreme cases,
the gain affects the reflectance of the mirrors and the gain should be found through
a
x nl d
b
n2
Figure 2.7. Singlefilm waveguide.
iteration of equations 2.86 and 2.87. However, this may usually be neglected because
the gain is almost always very small compared to the real part of the refractive index.
2.7.3 Basic Waveguide Relationships
The multilayer matrix formalisms developed in sections 2.4 and 2.5 have been
used so far for solving for the fields for a structure with traveling wave components
perpendicular to the boundaries. However, even though reflection coefficients and
other relations that conceptually imply traveling waves were used, it was never ac
tually assumed that they were traveling. This section will demonstrate how general
the two formalisms really are by applying them to a singlefilm waveguide.
Figure 2.7 shows a waveguide of a single film. The tangential field in media 0 is
given by
go = So&eiox + EoaeiKO~ (2.88)
where ,i = niko cos 0i. Assuming total internal reflection (ni and n2 must also be
real), Snell's law becomes
n1
sin0o = sin 0 > 1 (2.89)
no
and equations 2.88 and 2.89 can be linked by the trigonometric law
cos 0o = /J sin2 20 (2.90)
It is convenient, but not necessary, to choose the + sign. This makes Ko positive
and imaginary which, in turn, causes one component in equation 2.88 to increase
towards infinity for negative x. This cannot represent physical boundary conditions.
Approaching the other side of the waveguide in the same way, we find that the
boundary conditions are
0+ = 0 = 0 (2.91)
This is the same set of boundary conditions as for the laser! Therefore, the gain
relationship (equation 2.83) applies also to the waveguide. However, it is interpreted
somewhat differently. With the waveguide, Iral = Irbl = 1 and g = a = 0 (which
was already assumed). The characteristic equation (equation 2.87) is the same.
This example illustrated the use of the matrix formalism to solve for modes of
a lossless planar waveguide. However, it was not necessary to assume the lossless
case. It can be shown that with loss, the basic resonance equation (equation 2.83)
still holds and the solution requires both traveling and evanescent components in the
solutions of the side regions. With loss, equation 2.83 cannot be separated into two
independent relations. However, as with the case of the laser, if the imaginary part of
the refractive index is small compared to the real part, the solution can be separated
and approximated by the magnitude and phase relationships. It is also interesting to
note that equation 2.83 is useful for solving surface plasmon modes.
2.7.4 Application to Corrugated Waveguides
This section will show how the matrix formalism can be used to solve for corru
gated waveguides. The result will be a very simple, yet effective model for corrugated
x
a
Figure 2.8. Single mode waveguide with an abrupt step.
waveguide devices. The applications of the formalisms thus far have been exact. Due
to the added complexity of this problem, we are now required to make some approx
imations, but the approximate theory presented here will yield accurate solutions for
the devices of interest in this work.
We will apply the theory developed by Burns and Milton [12, 44, 81]. The problem
becomes very simple for single mode waveguides (as is the case in this work). We
start with a single mode waveguide with an abrupt step at boundary a (figure 2.8).
The solution of the TE electric field can be expressed as
Ey = +(z)P(x, z)ei(z)z + (z)P(x, z)eio(z)z (2.92)
where .+(z) and 9(z) are the complex amplitudes of the forward and reflected
waves, respectively, P(x, z) is the local normal mode profile at position z, f3(z) is the
mode propagation constant. We will assume negligible scattering of the energy from
the guided mode into the radiation modes. The boundary conditions on each side
of the step in figure 2.8 require that the tangential electric and magnetic fields be
continuous. Therefore, at z = a
EPoeiPoa + o Poei~a = +PieI31a (2.93)
and
o PoeiPa o&o Poeia = +PIe (2.94)
By eliminating +Piresila in these two equations, we get
S6o 03o 01 No N (2.95)
SOT 3 "+ 31 No+N,
where Ni = Oi/ko are the mode indices. The solution for the reflection at an abrupt
transition (equation 2.95) is in the same form as the reflection from a planar interface
(equation 2.46). There is really nothing mysterious here because equations 2.93
and 2.94 are analogous to equations 2.44 and 2.45, respectively. In fact, by assuming
both forward an reverse propagating waves in all regions, the whole matrix theory for
multiple layers can be applied to multiple step waveguides (a corrugated waveguide
being one example) by substituting the mode indices for the local normal modes for
the individual film indices. This is very useful as most of the results presented thus
far are applicable to grating devices. Furthermore, the same computer programs that
are used to calculate the reflectance, transmission, and absorption spectra of planar
multilayer structures can be used to calculate the behavior of waveguide devices, in
particular, waveguide FabryPerot interferometers and lasers. As long as there is low
loss (light is not scattered into radiation modes), and there is no mode conversion
between guided modes (in multimode waveguides) this simulation method is very
accurate.
The modeling of DBR corrugated waveguides is analogous to the periodic quarter
wave structure analyzed in section 2.7.1. Taking advantage of our ability to directly
substitute mode indices for film indices we can use equation 2.79 to find the reflectance
of a first order grating at the center wavelength:
NN, NH)
where No is the mode index of the input waveguide to the grating, Ns is the mode
index of the output waveguide of the grating, NL is the mode index of halfperiods
exhibiting the lower mode index, NHis the mode index of the other halfperiods, and
M is the number of grating periods. This shows that to obtain high reflectance, we
require a large difference in mode index, AN = NH NL, between each halfperiod
of the grating, or large number of periods.
2.8 Spectral Index Method
The channel waveguide structures presented in this work require extensive nu
merical techniques to adequately solve for their propagation constants and field pro
files [62, 63, 47, 60]. The spectral index method represents a much less rigorous and
computer intensive method and has been shown to give very accurate results [43]. It
involves a simple variational technique that utilizes the FFT rather than large ma
trix eigenvalue problems. However, the spectral index method is limited to simple rib
waveguide structures with square homogeneous stripes on top of a planar structure
varying only in the depth direction. Even though this does not represent the type
of structures of interest in this work, it will be found useful in section 4.7 to gain
qualitative insight.
Figure 2.9 shows the general case of a dual rib waveguide structure for which we
want to solve for the mode index and electric field profile. The ribs are separated by
2S, the widths of the ribs are W1 and W2, their heights are H1 and H2, and the planar
guiding layer has a thickness of D. The refractive index of each region is also shown
in the figure. Region 1 is the cladding (usually air), region 2 is the guiding layer,
W, W2
n H .
n :5H
n2 S S D
n3
Figure 2.9. Dual rib waveguide cross section. The dashed line represents the effective
movement of the boundary by the GoosHinchen shift.
region 3 is the substrate, and regions 4 and 5 are stripes of possibly different indices.
The principle of the spectralindex technique is to express the solution in the regions
below the ribs in terms of a Fourier transform in the x direction and approximate the
solution in the rib in terms of sines and cosines with a field of zero in the cladding.
These two solutions are linked together by requiring continuity and using Parseval's
formula to obtain a simple dispersion relation. The method is outlined as follows.
Assuming a source free media (lossless), equation 2.15 reduces to
02E
V2E pC a2 = 0 (2.97)
For a z propagating wave in a structure invariant with z, equation 2.17 reduces to
E(x, y, z) = Eo(x, y)ei(zWzt) (2.98)
After substituting equation 2.98 into equation 2.97, dropping the common terms,
and using k2 = w2/1e (equation 2.19 in a source free media), we get a scalar wave
equation for the solution of the electric field in each region
02E (2E
+ + (k2 )32)E = 0 (2.99)
_x2 + a2
where we now will use E to represent a polarization of Eo(x, y).
In deriving equation 2.15, equation 2.11 assumed that Ve = 0 (homogeneous
media). Therefore, equation 2.99 must be applied to each individual medium with
the proper boundary conditions met at each interface. Some methods, such as the
beam propagation method (BPM) and finite difference methods, give approximate
solutions by using equation 2.99 with an inhomogenious index, n(x,y), inserted
(k(x, y) = 2irn(x,y)/A). This approximation is usually acceptable for structures
where the refractive index varies slowly. This is not the case in rib waveguides, but
the approximations sometimes yield acceptable results.
If the refractive index of the cladding, ni, is much smaller than n2, n4, and n5
as it is with an air cladding, the electric field may be approximated to be zero on
the interface if the boundary is moved slightly using the GoosHiinchen shift. These
boundaries are shown by the dotted lines in figure 2.9. The new dimensions are given
by
2k2
W',2 = W1,2 + (2.100)
D'= D + q (2.101)
and
H2 = H1,2 + q (2.102)
where for TE modes p = q = 1, and for TM modes p = kl/k4,5 and q = kl/kj.
Applying the Fourier transform with respect to x on the wave equation (2.99), we
have
022
2Y +{k2(y)s2 32}8 = 0 (2.103)
where 8(s, y) is the Fourier transform of E(x, y). The solution is similar to that of a
slab waveguide with a spectral index, n, = n2 s2/ko.
The solution in the rib regions may be expressed as E(x, y) = F(x)G(y) where
SAi cos(si(x + C1)) Ci WT/2 < x < CI + W7/2
F(x) = A2cos(s2( + C2)) W /2 < x < C2 + W2/2 (2.104)
0 elsewhere
and
Ssin(y(Hy)) C W /2 < x < CI + W{/2
G(y) = sin(y2(Hy)) C2 W2/2 < x < C2 + W2/2 (2.105)
10 elsewhere
with
C, = S + W1/2 (2.106)
C2 = S + W2/2 (2.107)
71 = ki s ,2 (2.108)
7'2 = k5 s2 /2 (2.109)
si = r/Wl (2.110)
S2 = r/W2 (2.111)
k4,5 are the propagation constants of the respective ribs, /3 is the propagation constant
of the mode, and the + sign in equation 2.104 represents the fundamental mode while
the sign represents the first higherorder mode.
Using Parseval's formula, it may be shown that the spectral solution below the
rib and the realspace solution within the rib may be linked together by [43]
0EE dx = E ds (2.112)
oo ay 27 oo 9y
This represents a dispersion relation of which there are three unknowns: A1, A2,
and 0/. One of the field amplitudes is arbitrary while the other may be used to
determine the correct field profile. Since /3 depends on the field profile variationally,
an extremum in 3 will be observed when the correct value for A1/A2 is found. Thus,
the dispersion relation, 2.112, must be used in conjunction with an optimization
routine.
2.9 Summary of the Numerical Methods
This chapter presented a simple method to calculate the characteristics of corru
gated waveguide filters and lasers. In doing so, one first must determine the local
normal mode propagation constants. The spectral index method was presented for
this purpose. Next an appropriate matrix formalism should be chosen to solve the
problem at hand. The film matrix method was shown to be the best choice for
determining reflection and transmission spectra (such as a mirror or filter). The
interface/film method was shown to be the best choice for determining resonance
conditions (such as for a waveguide or laser). Analytical solutions were derived for
simple cases of these. However, to solve general problems, computer calculation of
the matrices may be required.
CHAPTER 3
A NEW WAVELENGTH SELECTIVE FABRYPEROT FILTER
This chapter introduces the concepts of a new tunable wavelength selective filter.
Most IOC filter technologies require a very long device length or the use of the
acoustooptic effect (chapter 1). The device we present has the potential of being very
short in length (r1cm) and utilizes the electrooptic effect. Furthermore, it has the
potential of being tuned to over 50 wavelength channels with bandwidths comparable
to that available with modernday electronics. The chapter starts by reviewing some
of the basic concepts of dielectric mirrors and the FabryPerot interferometer before
the new device structure is introduced. Computer programs implementing the theory
of section 2.4 were used for this purpose. The new waveguide FabryPerot device that
is characterized by a single transmission peak with a very large free spectral range is
then proposed.
3.1 Thin Film Dielectric FabryPerot Filters
3.1.1 Characteristics of a Thin Film Dielectric Mirror
We begin by reviewing some of the properties of dielectric mirrors. Figure 3.1(a)
shows an alternating dielectric stack typical of bulkoptic dielectric mirrors. It was
stated in section 2.7.1 that this periodic structure can be used as a mirror if the
thickness of each layer is A/4. Using equation 2.79 with the following values used for
the refractive indices, no = 1, n8 = 2.14, nL = 1.2, nH = 2.2, and with 4 periods, we
get R = rr* = 0.985 at the center wavelength. This is more reflectance than can be
achieved by uncoated silver or aluminum mirrors.
i

no
'R
a)
1
0.8
0.6
0.4
0.2
0 I
500
Dielectric Stack
nH nL nH nL nH nL nH nL
1000 1500 2000
Wavelength
Figure 3.1. Simple DBR consisting of a multilayer dielectric stack (a), and its re
flectance and transmission spectra (b). The parameters are no = 1, n8 = 2.14,
nL = 1.2, nH = 2.2, and 4 periods.
A computer program is required to solve the matrix equations 2.62 for values
of the reflection at wavelengths other than the center wavelength. The reflection
spectra in figure 3.1(b) shows the result. In this case, the refractive index difference
between successive layers, An = 1. This large index difference not only contributes
to a high reflectance with a small number of periods but also makes the full width
half maximum (FWHM) of the reflection peak very broad (803nm).
3.1.2 Characteristics of a Thin Film Dielectric FabryPerot Filter
When two mirrors are separated by a cavity, they form a FabryPerot interferom
eter. Such a device is shown in figure 3.2(a) using two of the mirrors just described
T
ns
2500
1200 1400 1600 1800
Wavelength
T
2000 2200 2400
Figure 3.2. FabryPerot filter consisting of two multilayer dielectric stack DBRs
separated by a short waveguide cavity (a), reflectance spectra of the structure with
a cavity length of 101im (b). The parameters for each mirror are no = 1, n, = 2.14,
nL = 1.2, nH = 2.2, and 4 periods.
in section 3.1.1. In this case, they are separated by a waveguide cavity 10/tm in
length. Even with this very short cavity length, figure 3.2(b) shows that there are
many transmission peaks in the reflection band. To utilize the electrooptic effect to
make a tunable filter, the cavity length must be increased, but this causes the peaks
to become too close together and the reduced free spectral range will cause crosstalk
in a WDM system.
The reflection spectra in figure 3.2(b) was calculated with the methods presented
in chapter 2 with one more approximation. The structure was calculated as a planar
Lc 
i
0 IW
1000
multilayer structure with the mode index of the waveguide (shown as the cavity in
figure 3.2(a)) substituted in for the refractive index of the cavity. This was already
shown to be a good approximation in section 2.7.4. Here, we go one step further
by assuming the overlap between the mode field of the waveguide and the reflected
wave by the endface mirror is quite good. This approximation becomes better as the
total thickness of the mirror is decreased because the mode field is dispersed less by
the shorter structure. To make a thin mirror with high reflection, we require AN to
be large. A 4 period mirror with nL = 1.2, nH = 2.2, and a center wavelength A =
1534nm, is 1.98/pm thick. The field divergence within these dimensions is negligible.
These values were only used to demonstrate the concept. For the mirrors used in this
work, nH = 3.5 and nL = 1.56 (which correspond the the refractive index of silicon
and A1203), and about the same reflection as for the mirror just illustrated can be
achieved in just 2 periods. Therefore, the total mirror thickness is reduced to just
0.82/m which makes the approximation made above is even more applicable to our
structures.
3.2 DBR Corrugated Waveguide FabryPerot Filters
3.2.1 Characteristics of a DBR Corrugated Waveguide
In contrast to an endface mirror, a DBR corrugated waveguide has many (typically
thousands) periods and the mode index difference between the peaks and valleys is
very small (typically AN ~ 104). The structure is illustrated in figure 3.3 (a).
According to the theory in section 2.7.4, the solution of the reflection and transmission
spectra of this structure is essentially the same as the dielectric mirror with the mode
index difference used in place of the refractive index of a film.
To demonstrate the characteristics of such a grating, we find that to obtain about
the same reflection from a lossless grating structure as we did for the dielectric mirror
Waveguide Grating
i .n T
R Ins
no
a)
0 L,
1533
1533.5 1534 1534.5 1535
Wavelength
Figure 3.3. DBR corrugated waveguide (a), reflectance spectra (b). The parameters
used in the calculation are NH = 2.1406 and NL = No = N, = 2.14 and 10,000
periods. Loss of 0.35dB/cm is also included in the calculation.
in section 3.1.1 with a mode indices of NH = 2.1406 and NL = No = N, = 2.14 for
the peaks and valleys, respectively (AN = 6 x 104), we require 10,000 periods. For
a center wavelength of A = 1534nm, this device is 3.6mm long, and for the lossless
case equation 2.96 predicts that R = 98.5%.
We expect there to be some loss in the waveguide. Lowloss APE:LiNb03
and Ti:LiNbO3 waveguides typically exhibit 0.10.2dB/cm loss. The loss in dB,
a[dB/cm], is related to the absorption coefficient, a[1/cm] by
a[cm_ a[dB/cm] (3.1)
10 log e
According to equation 2.36, these values for loss correspond to the imaginary part
of the refractive index, ni, (in this case we are really talking about the mode index)
between 2.81 x 10' and 5.6 x 107. When there is loss, we need to use the computer
calculation of the matrices even for the reflection at the center wavelength. Figure 3.3
(b) shows the reflection spectra of this corrugated waveguide structure with a loss
of 0.35dB/cm (n, = 1 x 106). A somewhat higher loss was used because we expect
the grating to induce some additional loss on the waveguide. The peak reflection
is reduced from 98.5% for the lossless case to 97.5% for the case where we have
0.35dB/cm loss.
The distinct differences between the characteristics of the corrugated waveguide
reflector and a dielectric mirror are that the corrugated waveguide can have more than
three orders of magnitude shorter bandwidth (FWHM is 3.4A in this case), and a
much longer length with orders of magnitude more grating periods is required for high
reflection. The difficulty in achieving high reflection from a corrugated waveguide is
maintaining low loss when the grating is formed. In this example, we showed that a
very low loss of 0.35dB/cm already has a noticeable effect on the peak reflectance.
This issue will be discussed many times throughout the rest of this dissertation.
Cavity
R
a)
0.8
S0.6
g 0.4
0.2 
0
1533 1533.5 1534 1534.5 1535
b) Wavelength (nm)
Figure 3.4. FabryPerot filter consisting of two DBR corrugated waveguides separated
by a short waveguide cavity (a), reflectance spectra of the structure (b).
3.2.2 Characteristics of a DBR Corrugated Waveguide FabryPerot Filter
Figure 3.4 (a) shows a FabryPerot interferometer formed by introducing a short
cavity or a quarterwave shift in the center of the grating. Figure 3.4 (b) shows the
reflectance spectrum of the device using two gratings with the same parameters as
were used in section 3.2.1 and with a quarterwave shift between them. A single
transmission peak is observed in the center of the reflection band. Only one peak is
observed because the width of the reflection peak is narrower than the free spectral
range.
The free spectral range (FSR) of a FabryPerot interferometer is given by
AA2sr = (3.2)
2nL,
where A0 is the free space wavelength, n is the refractive index of the cavity, and
Lc is the length of the cavity. A FabryPerot made with a mirror with a FWHM
of 3.4A and a center wavelength of 1534nm will exhibit at most one transmission
peak if Lc < 1.6mm. However, if this device were to be used as a filter, interchannel
crosstalk would still be a problem because the stopband itself is narrow and the
sidelobes exhibit extremely high transmission.
This device can make a good single frequency laser because there is only one
transmission peak in the reflection band. All the other FabryPerot peaks are in the
sidelobes where equation 2.86 cannot be satisfied. The sidelobes have little detrimen
tal effect on its performance (the only detrimental effect is that spontaneous emission
is allowed to pass through. However, a filter is adversely affected by the sidelobes as
light is allowed to pass through them almost as well as the FabryPerot transmission
peak.
3.3 Proposal: Asymmetric FabryPerot Filter
In this section, we propose the asymmetric FabryPerot (ASFP) filter that takes
advantage of the characteristics of both the dielectric mirror and the corrugated
waveguide DBR reflector. Asymmetric refers to the use of two mirrors differing vastly
in their reflection spectra. The DBR mirrors described in sections 3.1.1 and 3.2.1 are
two such examples where the one in section 3.1.1 has a reflection band FWHM over
2300 times that of section 3.2.1.
Corrugated Waveguide DBR
R T
Figure 3.5. Asymmetric FabryPerot filter consisting of a DBR corrugated waveguide
with a dielectric mirror deposited on the endface.
3.3.1 Passive ASFP Filter
We recall that the FabryPerot made from dielectric mirrors had a broad stop
band, but many transmission peaks within that stopband. The FabryPerot made
from corrugated waveguide reflectors had the opposite scenario where the stopband
was extremely narrow and there was only one transmission peak. A somewhat novel
approach would be to combine the favorable characteristics of both types of DBRs
as shown in figure 3.5. The dielectric stack deposited on the endface of a waveg
uide will provide a very broad stopband while the waveguide grating will provide
rejection of unwanted FabryPerot transmission peaks. When they are fabricated
in cascade, forming an asymmetric FabryPerot filter (ASFP filter), they provide a
single, very narrow transmission peak with an artificially 1 high free spectral range
(figure 3.6). The phase shift between the grating and the DBR is crucial to assure
that a transmission peak lies within the stopband of the grating.
In the calculations, typical values for the dimensions and indices of the two DBRs
were used and loss was included. The filter characteristics (shown in dB in figure 3.7)
are suitable for a wavelength channel with a bit rate of about 1 Gb/s and an inter
channel crosstalk of about 14 dB. The solid curve was calculated assuming a grating
'The reason why this FSR is called "artificial" is because there are still many transmission peaks
within the stopband as the phase condition required for interference is met. However, the reflection
of the grating reflector is so low that little interference takes place and the peaks are not visible in
the graph.
61
0.8
0.6
c 0.4
0.2
0
500 1000 1500 2000 2500
a) Wavelength
0.6 . . .
0.5
o 0.4
0.3
0.2
0.1
0
), 1533 1533.5 1534 1534.5 1535
U) Wavelength
Figure 3.6. Transmission spectra of an asymmetric FabryPerot filter (a); expanded
view of the transmission peak (b). The parameters used for the dielectric mirror were
the same as those used for figure 3.1(b) and those used for the corrugated waveguide
were the same as those use for figure 3.3(b).
0
0.05 A 635 MHz
5 
10
10.25 A 3.2 GHz
15
25
30
1533.5 1534 1534.5
Wavelength (nm)
Figure 3.7. Transmission spectra of an ASFP filter for a corrugated waveguide loss
of .35dB/cm (solid) and 3.5dB/cm (dotted).
loss of .35 dB/cm. However, the dotted curve shows that if the grating loss is 3.5
dB/cm, the filter response is broadened and the magnitude of the transmission peak
is comparable to the sidelobes making the device useless for a filter. Hence, the most
important technology for the realization of this and similar devices is the fabrication
of high reflectance waveguide grating mirrors with low loss. There is some redun
dancy in this statement in that a high reflectance waveguide grating mirror implicitly
implies low loss.
3.3.2 Tuning an ASFP filter
The device proposed in section 3.3.1 has the potential of being tuned to many
channels in a very simple and straightforward manner by utilizing the linear electro
optic effect with an appropriate electrode scheme (figure 3.8). The electrooptic effect
is used to change the mode index of the corrugated waveguide and the cavity (if one is
included). Small changes in the mode index are sufficient to shift the peak to another
wavelength channel. Unlike many other electrooptic devices, the total phase shift
DBR
4 pair
Electrodes Waveguide Grating
Electrodes 10k Periods
Z+ LiNbO
Z+ LiNbO3
Device Length: 3.6 mm
Figure 3.8. Tunable ASFP filter.
can be much less than 7r. Therefore, the total device length can be very short. For
the device we are discussing, the length is merely 3.6mm.
To determine the amount of change in the mode index realizable, we recall that
the refractive index of the extraordinary wave in LiNbO3 is dependent upon the DC
electric field along the optical axis (z), expressed as
ne(Ez) = ne er33Ez (3.3)
2
were ne is the extraordinary refractive index without applied field, Ez is the ap
plied field, and r33 is the appropriate Pockel's coefficient which, for LiNbO3, r33 =
30.8 x 1012m/V. LiNbO3 breaks down with electric fields greater than 2223kV/mm,
and hence Ez must be somewhat less than this. With these applied fields, the ex
traordinary index change An = 3.3 x 103. The change in mode index is related to
the change in refractive index by AN = FAn where F is called the fill factor. The
fill factor is basically the normalized overlap integral between the DC (modulating)
electric field and the optical field profiles. Electrode configurations that result in
r > 0.5 are easily achieved experimentally. The mode index change is further limited
S 635 MHz FWHM
5
15
20
25 
30 I .* * * *
1533.2 1533.6 1534 1534.4 1534.8
Wavelength (nm)
Figure 3.9. Transmission spectra of a Tunable ASFP filter for several values of the
mode index change.
by the breakdown of air, but either a polymer or an oxide overlay can help eliminate
this problem. All things considered, the upper limit for the mode index change is
about AN = 103.
To determine the range of tunability of the device, several transmission curves
with varying mode index change are plotted in figure 3.9. If the maximum mode
index change is AN = 103, the device is capable of tuning to 57 channels spaced at
3.2 GHz apart with a crosstalk of 14 dB. These results may be improved with lower
loss, higher reflection, and sidelobe suppression of the grating DBR. Furthermore, a
longer, but weaker, grating may be used to increase the number of tunable channels.
The number of channels can also be increased by fabricating an array of these
tunable filters with varying grating periods. Wideband prefilters, such MachZehnder
interferometers, can be used to distribute the groups of wavelength channels to the
appropriate ASFP filters, or if power is not an issue, a simple star coupler can be
used to distribute the power to each ASFP filter.
In contrast to most other wavelength selective filters, the ASFP filter is much less
complex, extremely short (less than 4 mm), and tunable to many channels with a
single low power controller. Relatively few technologies are yet to be developed and
successful development will make this filter amenable to large scale production.
3.3.3 Tunable DFB laser
As we recall from section 1.2, the concept of using the ASFP structure as a
wavelength selective filter is new, but the device structure is not. Many practical
DFB lasers use the same resonator structure, but the purpose is different. The
purpose of the corrugated waveguide is essentially the same for the laser as it is for the
filter: produce narrowband reflection capable of selecting a single transmission peak.
The purpose of the endface mirror is different. The filter requires the broadband
mirror to produce a very broad stopband with very small transmission sidelobes. An
endface mirror is useful for DFB lasers because it usually causes the transmission
spectra to be asymmetric with a dominant transmission peak [74]. The use of a
single grating causes a DFB laser to exhibit dual wavelength operation at the two
dominate transmission sidelobes (shown in figure 3.3). An alternative method to
achieving single longitudinal mode operation is to use a quarterwave shifted grating,
which is essentially the same as the structure in figure 3.4, but this is generally
more difficult to fabricate. Transmission sidelobes of the FabryPerot structure are
of no consequence with the laser since, when laser action begins, the spontaneous
emission in these sidelobes is greatly dwarfed by the high power laser emission from
the dominant mode.
The ASFP filter functions as a DFB or DBR laser if enough gain is introduced
to the cavity and/or grating to overcome the losses. The conditions for lasing can
be determined by equations 2.86 and 2.87 if the gain is only in a cavity, but if there
is gain in the grating, an iterative calculation of the matrix method is necessary to
calculate the gain required for lasing.
The DFB and DBR lasers do not require a high quality grating to lase. Such lasers
have been experimentally demonstrated with erbium doped LiNbO3 [23]. The lasers
were made with a 35mm erbium doped cavity and DBR corrugated waveguides 30mm
and 16mm in length. The 30mm grating exhibited a maximum of 80% reflection and
the 16mm grating exhibited a maximum of 55%. The device with the 30mm grating
was shown to lase even without an endface mirror (the LiNbO3/air interface exhibits
R = 14.3%).
The ASFP laser can be tuned in the same way as its filter counterpart. The
wavelength shifts (shown in figure 3.9) are the same, but the linewidth of the laser is
much narrower. The theory in chapter 2 would predict infinitesimally small linewidths
for the laser, but several broadening mechanisms will produce a linewidth, probably
on the order of 1MHz.
The ASFP filter and laser complement each other. Both can be fabricated on the
same chip and designed to have the same tunability with the same applied voltage.
3.3.4 Summary
In this chapter, we presented the basics of endface dielectric and corrugated waveg
uide reflectors. Their application to a FabryPerot filter showed that neither arrange
ment was sufficient to act as a WDM filter alone, but when the benefits of both of
them are exploited in a cascaded arrangement, forming an asymmetric FabryPerot
filter, a much more useful filter is possible. This filter was then shown to have
almostideal characteristics for many WDM applications: reduced complexity; ex
tremely short length (less than 4 mm); tunability to many channels with just a single
DC controller (low power); and amenability to large scale production. Most other
filters reported to date lack one or more of these characteristics. The same structure
can be used to make a laser (which has been reported elsewhere very recently). This
new filter has the potential of becoming an ideal companion device with a tunable
laser because they are fabricated with mostly the same steps, can be integrated to
gether on the same chip, and have the same tuning ranges with the same applied
voltage.
It was shown that the filter is much more demanding on the quality of the grating.
There are few reports of good DBR corrugated waveguidesnone in LiNbO3. The
next chapter introduces a new waveguide geometry that has the potential to achieve
good corrugated waveguides for the first time in LiNbO3.
CHAPTER 4
A NOVEL WAVEGUIDE GEOMETRY
This chapter proposes a novel waveguide geometry that provides a potential solu
tion to an ageold problem: producing high reflectance DBR corrugated waveguides.
It is amenable to many existing waveguide technologies including those that are cur
rently used and most viable in the industry. In addition, it will be shown that it
offers a number of other benefits as well.
4.1 The Need for Better DBR Corrugated Waveguides
Presently, the only commercially viable use of reflection corrugated waveguides is
for DFB lasers. This is because the material systems used for DFB lasers is excep
tionally amenable to making good gratings (the reasons are presented in section 4.3),
and often high reflection is not required when the optical gain is very high. Con
sequently, as the loss in the grating decreases, the reflection improves. This is why
DFB lasers (lasers that have the grating in the active region) are generally better
than DBR lasers which use a passive grating outside the active region. Other devices
and material systems lack these conveniences.
The device proposed in section 3.3 is only one example where a good DBR cor
rugated waveguide is required. Describing all of the needs for such components is
analogous to defining all the uses of a bulkoptical dielectric mirror. With significant
technological advances, the proposed DBR corrugated waveguide can become a basic
building block for a broad array of IOCs, sharing in the success of directional couplers
and Ybranches.
The characteristics of corrugated waveguide mirrors are clearly unique. First, they
have a very narrow reflection band (~ 1A). This is useful for narrow bandpass filters
such as the one proposed in section 3.3 and others proposed in references [33, 25, 26,
27]. The fact that they do not require the use of an endface make them very attractive
for IOC devices. Many gratings with varying periods may be placed anywhere on the
chip. Generally, with endface mirrors, all waveguides exiting the same facet as the
ones with mirrors must be polished perpendicular to the waveguide. This prevents
the use of one of the most common methods to prevent unwanted backreflections:
angle polishing.
4.2 Corrugated Waveguide Technologies (Previous Work)
Much research has been devoted to periodic waveguide structures ever since their
first few demonstrations in DFB lasers [36, 86, 87] and filters [16, 19]. Since then, the
use of corrugated waveguides has excelled, providing low threshold and high power
single longitudinal semiconductor lasers. However, passive devices have not shared
that success.
Ironically, there have been many reports of high reflectance corrugated waveg
uides, but many of them used ambiguous measurement techniques that are not de
pendable. There have been several claims of almost 100% reflectivity, even 99%, but
very few reports support the claims with indisputable results. To the best of our
knowledge, the best indisputable results to date were produced in silica waveguides
by Adar et al. [2]. They measured 69% of the light back into a fiber using a 2 x 2
fiber coupler. Their measurement was calibrated by butt coupling the fiber to a ma
terial of known reflectance. Furthermore, they made a reasonable claim that the light
undergoes two fiberwaveguide coupling losses that could total a value near 1.6dB.
Therefore, it is quite reasonable that their DBR corrugated waveguide produced re
flectance > 95%.
4.2.1 Investigations of DBR Corrugated Waveguides
This section briefly summarizes the most important results of the other efforts on
passive DBR corrugated Waveguides, from their first demonstration to present. Many
of the results reported were not properly substantiated by indisputable measurement
techniques.
Most of the early investigations were carried out on planar glass waveguides [19,
67, 84, 30, 10]. They all used a prism coupler (section 6.3) to measure the reflected
power and reported better than 80% reflection. An absolute determination of the
amount of light that is coupled in and out of the waveguide is generally impossible.
If they were able to accomplish this, the details were not reported. Therefore, the
measurements are somewhat ambiguous.
A planar DBR corrugated waveguide was demonstrated in InGaAsP/InP in ref
erence [4]. 99% reflectivity was reported, but not measured. Only the transmission
was measured and they used R = 1 T to calculate the reflection. This assumes
there is no loss. The FabryPerot device, later made by the same group, indicated
that loss was indeed a problem [5]. Ridge DBR corrugated waveguides have been
reported in GaAs/AlGaAs as well [83].
There have been several studies of higher order gratings [69, 68, 64, 65, 9, 10].
Shinozaki et al. [69, 68] reported DBR reflectors in LiNbO3 with periods of 10/tm
and above, using proton exchange. One was used for a marginally improved second
harmonic generation (SHG) device and another was used for feedback of both an
external cavity laser and SHG.
Polarization independent DBR corrugated waveguides have also been reported [1].
This was achieved by etching trenches along the sides of the waveguides to practically
eliminate the straininduced birefringence. The two transmission peaks coincided
when the two orthogonal mode indices were equal.
Low loss Bragg reflectors have also been reported. A technique to measure the
scattering loss from the surface roughness of a corrugated waveguide was demon
strated by Lee et al. [38]. The transmission characteristics of a Si3N4 channel cor
rugated waveguide were observed with and without an indexmatching fluid on the
grating surface. The index matching fluid eliminates the scattering loss as well as
the reflection peak. By comparing the passband response of the sample for the
two cases, they determined that the incoherent scattering loss of their structure was
~ 0.2dB/cm. Reflection measurements were not reported.
4.2.2 Passive DBR Corrugated Waveguide Devices
This section summarizes most of the passive DBR corrugated waveguide devices
that have been demonstrated. None have proven to be quite practical yet. However,
with the improvement of waveguide grating technology, many of these devices may
gain acceptance.
There are several reports of FabryPerot interferometers of two DBR corrugated
waveguides separated by a cavity (figure 3.4). A planar corrugated waveguide in
glass with a wedge etched out of the center, was used to demonstrate a filter that
is tunable by positioning the beam [34]. A quarterwave shifted grating resonator
with a 7dB transmission peak 1A wide in the center of the narrow reflection band
(AA ~ 10A) was demonstrated in InGaAsP/InP [5]. The most impressive result is
found in reference [28]. They used a Si3N4 channel waveguide for their quarterwave
shifted grating resonator. Their measurement setup was similar to the one shown
in figure 6.3, but they did not report the absolute reflectivity of the device. An
electrooptically tunable FabryPerot filter using waveguide gratings has also been
demonstrated [48, 49].
Other corrugated waveguide devices previously demonstrated have been the res
onant optical reflector [53], the folded directional coupler [3], and the narrowband
channeldropping filter [39] (see figure 1.6). None of these exhibited commercially
acceptable passive characteristics, but the resonant optical reflector was used as an
externalcavity reflector of a laser. 5mW of optical power and a linewidth of 135kHz
was observed. Simple corrugated waveguides have been used for the same purpose
to achieve a 1MHz linewidth [54].
4.3 Difficulties in Achieving High Reflectance
The theoretical calculations of a corrugated waveguide in section 3.2.1 indicate
that it is possible to achieve 97.5% reflection in a corrugated waveguide of 10,000
periods, a AN = 6 x 104, and a loss of 0.35dB/cm. This corresponds to a grating
length of only 3.59mm. Even higher reflection can be expected for a larger number
of periods, larger AN, or lower loss.
Unfortunately, many difficulties have prevented the realization of such high reflec
tion corrugated waveguides in any material. The best results of previous work (section
4.2) have produced gratings of better than 95% reflection in glass waveguides. Re
ports of reflection from corrugated waveguides in LiNbO3 and related electrooptic
materials are very scarce. Perhaps this is because it is more difficult to make a high
reflectance corrugated waveguide in LiNbO3 than in other common materials used
for IOC technology. Several factors significantly complicate the fabrication of high
reflection DBR corrugated waveguides in LiNbO3. These will now be discussed.
First, the chemical reactivity of LiNbO3 is poor. Device shaping using aqueous
acid and other solutions is almost impossible [58]. Solutions of HF and HNO3 have
generally been useful only for identifying domain polarity and crystal defects. Reac
tive Ion Etching (RIE) is usually ineffective because the etch rate is very low and the
process causes damage to the crystal (see section 5.1.1). Since etching of LiNbO3 by
wetetching, RIE, ion milling, or by other means, have generally proven far inferior
to the etching technology of Si02, semiconductors, polymers, and other materials,
gratings in the LiNb03 crystal itself cannot easily be made deep. Therefore, making
waveguide gratings by etching LiNb03 directly is not a promising approach for high
reflectance devices.
Second, most practical lowloss waveguides in LiNb03 have mode indices very
close to the substrate index. This fundamentally limits the mode index difference,
AN, that can be achieved by etching a grating directly on LiNb03. Etching com
pletely through the waveguide creates very high AN at the expense of high scattering
losses.
Third, LiNbO3 has a relatively high refractive index (compared to glass and poly
mers). This poses two problems: conventional holography is more complicated, and
perturbation of the optical field by a corrugated buffer layer is lower. These will now
be explained in more detail.
Holography is more difficult due to more intense reflections from the bottom
surface of the substrate. Glass and polymers have lower refractive index, so that the
reflection from substrateair interfaces is about a third less than in LiNbO3. Even
more important is that their is a multitude of suitable index matching fluids available
for glass and polymers to match the substrate to a prism, absorber, or other beam
diversion element. Index matching fluids for LiNb03 are usually highly absorptive in
the visible and UV wavelength range and are thus much more reflective than LiNbO3.
Most buffer layers used for LiNbO3 waveguide devices have a low refractive index.
(Usually SiO2 is used which has a refractive index of about 1.45 at A = 1.5ym, whereas
the index of LiNbO3 is about 2.14.) The field profile of the guided wave in LiNbO3
exhibits a very small evanescent tail in such a buffer layer. As a consequence, a
periodical perturbation in the buffer layer thickness produces little perturbation to
the optical field. According to equation 2.96, this is needed to achieve high reflection
within a reasonable number of periods. For the same reasons, polymers generally do
not produce good corrugated waveguides on LiNbOa.
4.4 Proposal: Channel Waveguides With HighIndex Overlays
A high refractive index film on top of a planar or channel waveguide (figure 1.9),
in the place of a lower refractive index buffer layer, can be used as a medium in which
a high perturbation grating can be formed. When the refractive index of the film is
greater than the mode index, the solution of the field in the film is no longer evanes
cent and the film becomes part of the guiding media. It will be shown in section 4.7
that this overlay causes the guided mode index to become more dependent on the
thickness of the film. This is useful in making high reflectance gratings (figure 4.1 (a))
because the difference in the local normal mode index between the peaks and valleys
of the corrugated waveguide (figure 4.1 (b)) can be very significantmuch more so
than by etching a grating on a lowindex buffer layer or the waveguide itself.
This may initially appear to present detrimental effects. First, if a significant
portion of the optical mode is guided in the overlay, the electrooptic effect is reduced
since most overlay materials would exhibit little electrooptic effect. Second, for
relatively thick overlays with very high refractive index the optical field is expected
to be drawn out of the lowindex material and yield a smaller mode size. This would
present problems with coupling to other waveguides, particularly fibers. Third, a
Highindex
NH NL grating overlay
nsub
y
Figure 4.1. Longitudinal cross section of a gradedindex waveguide with an overlaying
film grating (a); local normal mode index profile along the propagation direction (b).
i
R
a)
N
NH
NL
b)
............ .............. I ............. ...... ........ .............. ..............
significant portion of the field in the overlay may result in increased loss, especially
due to radiation. However, these problems do not necessarily exist. Some common
misconceptions of similar structures will be addressed in section 4.6.
The term "overlay" as opposed to "buffer layer" is used because the intent is
to perturb the guided optical mode rather than buffering the field at the surface.
Since the field in the overlay is significant, in practical devices, a buffer layer should
be used on top of the "overlay" to reduce surface scattering and provide spacing
from electrodes. In discussions that follow, the term "mode index" is used to refer
to the local normal mode index which is the mode index of a zinvariant (along
the propagation direction) waveguide with a transverse cross section identical to the
waveguide in question at a particular local position.
There is some similarity between this proposal and the work in reference [2]. Fig
ure 4.2 (a) shows the structure they used. In that work, they deposited a higherindex
material (Si3N4) on the vertical walls of a glass grating. Their reasoning was that cov
ering the vertical groove walls would form highly reflecting interfaces perpendicular
to the guide axis. However, a more precise interpretation would be the same as that
for our structure. That is, the highindex material helps create a high modeindex
difference between the two halfperiods of the waveguide. As shown in figure 4.2, the
former interpretation led to a comparatively complicated structure where the mode
index variation in the propagation direction has spikes. Because they are so thin,
and the optical mode overlap in that region is small, they can be expected to produce
little perturbation to the mode. However, the thin layer of Si3N4 in the valleys of
the grating can be expected to create a significant perturbation because it resides
directly on the surface of the guiding layer and there is significant overlap.
The solid curve in figure 4.2 (c) represents the mode index profile in the propaga
tion direction if the Si3N4 were placed only in the valley of the grating. The dotted
Si3N4
SB,PGlass
T
R
N
b) __
N
c)
y
Figure 4.2. Corrugated waveguide structure used by Adar et al. [2] (a); Local normal
mode index profile of the structure along the propagation direction (b); Local normal
mode index profile of the structure with Si3N4 only in the valleys of the grating (solid)
and with no Si3N4 at all (dotted) (c).
curve represents the mode index profile of just the glass grating without adding Si3N4.
Comparing the two, we see that by putting Si3N4 in the valley, the mode index profile
becomes inverted with a much larger differential, improving the reflection.
When considering such extreme changes in the local normal modes associated
with the structure in figure 4.2, one must also consider some assumptions that were
made with the step transition method we used to derive equation 2.96. The paraxial
approximation requires that the overlap between the fields on each side of a discon
tinuity be good. This is not so for the spikes in the mode index profile of figure 4.2
(b). Consequently, it is expected that these spikes will have relatively little effect on
the mode field as it propagates through the structure. The main perturbation comes
from the film of Si3N4 in the valleys of the grating. More rigorous calculations are
necessary to determine their effect exactly, but this is beyond the scope of this work.
With this insight, the structure in figure 4.2 (a) is determined to be much more
complicated than necessary due to the presence of the glass/Si3N4 grating. Only a
grating of Si3N4 is needed, or even beneficial.
4.5 Benefits of Silicon as a HighIndex Overlay On LiNbO.
We found silicon to be very suitable to demonstrate our concept of a high refractive
index overlay. There are many reports in the literature of devices made from deposited
films of LiNb03 on silicon for non waveguide applications, but there are few reports
that would relate its use as a waveguide device. These few relevant references are
discussed briefly below.
4.5.1 Previous Reports of Waveguides with Silicon Overlays
The work of reference [70] suggests that by putting a semiconductor film on the
surface of a LiNbO3 waveguide device, the pyroelectricallyinduced bound surface
charges can be screened out as they are being generated by slow thermal variations.
The capacitive loading of the electrodes causes the potential of the electrodes to
change much less than the the potential of the rest of the z surface. The potential
difference between the electrode and the surrounding area leads to significant drift.
The addition of a semiconductor film on the surface to reduce the potential difference
has been used to reduce drift in experimental devices [52, 51]. These references state
that a "thin semiconductor film" was used, and do not specify the material.
Silicon was used in reference [66] for the electrodes of a parallelplate electro
optic waveguide modulator. 0.231Lm of RF sputtered lithium niobate films were
sandwiched between a 0.165pm layer single crystal silicon on a sapphire substrate
and a 0.181/m layer of vapordeposited hydrogenated amorphous silicon. The device
loss was calculated to be about 1dB, but the loss of the experimental device was not
reported.
Most research on optical waveguides on silicon substrates has concentrated on
the use of deposited layers on oxidized silicon substrates. However, some recent work
has focused on silicononinsulator (SOI) waveguide structures. Moderately lowloss
silicon waveguides (2 dB/cm) have been reported [18]. These studies have shown
that ridge waveguides with very thick films of silicon can be singlemoded. A further
discussion will follow in section 4.6.
The emission properties (coupling of the guided modes into radiation modes) of
corrugated waveguides with a metal or semiconductor coating was studied in refer
ence [7]. To overcome the problem of the unwanted emission of light from a grating
to cladding orders, they used a highly reflecting barrier on one side of the corrugated
waveguide, with and without a buffer layer spaced between them. The purpose of
this reflecting layer was to redirect the light from the unwanted orders back into the
substrate. Their conclusion was that the best emission properties could be achieved
by using a highly reflecting metal with a buffer layer. A structure using an aluminum
grating was demonstrated to couple out 81% of power from the waveguide.
4.5.2 Additional Benefits of Silicon on LiNbOa
The only previouslyreported benefit of a silicon overlay on a waveguide device
as it pertains to this work is its use towards the screening of the bound charges gen
erated by the pyroelectric effect [70]. For the purposes of our proposed structure,
silicon exhibits a number of other benefits as well. Separate sections of this disserta
tion will discuss each of the following benefits in detail. The high refractive index of
silicon (~ 3.5, appendix A) is suitable to achieve a large AN corrugated waveguide
(section 4.7). The use of silicon greatly simplifies the fabrication process because it
is easily deposited by ebeam evaporation (section 5.5), it is easily etched by a num
ber of welldeveloped processes (section 5.1.1), and its high absorption in the visible
spectral region makes it very useful for greatly reducing substrate backreflections
during holography (section 5.3). Based on our studies, there is no appreciable dif
ference in loss between waveguides with and without relatively thick layers of silicon
(section 6.2.2).
4.6 Misconceptions of Guiding in ThickFilm Structures
The purpose of this section is to alleviate some common misconceptions of guid
ing in channel waveguides. This is necessary because,in cases such as the proposed
waveguide geometry in figure 1.9, there is a vast contrast between the results of
some standard theories used to solve channel waveguides, namely the effective in
dex method [61], and more accurate analysis. In these cases, only rigorous compu
tational methods, that do not utilize the separation of variables when solving the
twodimensional wave equation, can yield accurate results. Even though this has
been shown several years ago, it is not commonly known, perhaps because an elegant
analytical technique to adequately solve these types of problems is not available and
the former, simplified theory is more often used in textbooks. For the purpose of this
section, rather than using elaborate methods to draw the same conclusions as pre
viously reported for simpler structures, we will describe those conclusions in detail,
comparing with the use of the effective index method; and then use the similarity
between the structures to argue that the same principles apply to our structure.
Richard Soref et al. investigated single mode siliconbased ridge waveguides [73].
It was shown that, although a planar silicon waveguide becomes double moded with
thicknesses greater than 0.3pm, contrary to analysis of the effective index method,
singlemode ridge waveguides can be made with the planar portion several microm
eters thick! This implies that by removing the ridge portion of a single mode guide,
the remaining planar waveguide can support multiple modes. This seems to be a
contradiction; but the fallacy here is that in one case we have one dimensional con
finement, and in the other we have two dimensional confinement. They cannot be
related in such simple terms.
It is sometimes assumed that for a singlemode ridge waveguide, its crosssectional
dimensions must be close to the thickness of a singlemode planar waveguide of the
same material. Perhaps this erroneous assumption stems from the misuse of the
effective index method for solving 2D structures. The analysis in reference [29]
shows that the effective index method has a tendency to overestimate the mode
indicesespecially near cutoff. Therefore, it predicts the presence of modes below
their actual cutoff. Even so, if the results of reference [29] are accepted to be true
for the general case, we get a false impression that the number of modes predicted
by the effective index method is accurate within 1 or 2.
To demonstrate that even this assumption is incorrect, figure 4.3 shows a sim
ple ridge waveguide structure that has been previously shown to exhibit only one
x
W=.65m
n2=3.5 )H=.65gm D=.52gm
n3=1.45
Figure 4.3. Simple ridge waveguide structure used to illustrate a case were the effec
tive index method gives the wrong number of modes. This waveguide is single mode,
but the effective index method predicts five modes.
mode [73]. This structure is analyzed by the effective index method [61, 37] by
determining the mode indices of the waveguide at each position, y, as if it were a
vertically confined planar waveguide with those dimensions. In this case, there are
only two different regions with film thicknesses of D and H. They are depicted in
figure 4.4 (a) and (b), respectively. The equivalent horizontal confinement is a planar
waveguide with an index profile derived from the mode indices of the vertical confin
ing waveguide at each position, y. In this case, it is a symmetric planar waveguide
(figure 4.4 (c)).
Using the parameters given in figure 4.3 at a wavelength of 1.3p/m, we find that
each of the planar waveguides in figure 4.4 (a) and (b) support three TE modes
with approximate normalized propagation constants [35] of bH = .93, .73, .42 and
bD = 0.9, 0.6, 0.2, respectively, where
b = (4.1)
nf ns
H
n, D n W 14.
n2 2
n33 ND NH ND
a) b) c)
Figure 4.4. The effective index method is applied to the waveguide structure in
figure 4.3 by finding the mode indices of asymmetric planar waveguides with film
thicknesses of D (a) and H (b). These mode indices are applied to a symmetric
laterally confined planar waveguide (c). The mode indices of the waveguide depicted
in (c) are generally accepted to be equivalent to the mode indices of the structure in
figure 4.3.
n, is the substrate index (n3) and nf is the film index (n2). The corresponding mode
indices are NH = 3.36, 2.95, 2.31, and ND = 3.30,2.68, 1.86, respectively. By paring
these numbers together to form the horizontally confined waveguide in figure 4.4 (c),
we get the normalized parameters of v = 2.0, 3.9,4.3 where
2Wr 2N2
v = W/NH ND (4.2)
Using the normalized bv diagram in reference [35], we find that the first value for v
supports one mode while the other two each support two modes. In all, the effective
index method predicts five modes for this structure. However, reference [73] shows
that this structure supports only a single mode! In fact, structures several times
larger support only one mode. In these cases, the effective index method would
predict even more modes.
The waveguides of interest in this work (figure 1.9) are quite different to the one
just illustrated (figure 4.3). However, some similarity is apparent when the structure
of figure 1.9 is viewed upside down as shown in figure 4.5. Instead of having a rect
angular ridge of the same refractive index as the planar layer, the proposed structure
