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Analysis and design of cleaved edge lithium niobate waveguide resonators
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Analysis and design of cleaved edge lithium niobate waveguide resonators
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Content
ANALYSIS AND DESIGN OF CLEAVED EDGE
LITHIUM NIOBATE WAVEGUIDE RESONATORS
by
Satsuki Takahashi
||||||||||||||||||||||||||||||||{
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
August 2012
Copyright 2012 Satsuki Takahashi
ii
DEDICATION
To my mother and father, Tomoko and Tatsuo.
iii
ACKNOWLEDGMENTS
Perhaps it is a rite of passage when endeavoring to solve a problem, that there is
most often initially nothing but the overwhelming feeling of blindness. Such was
a feeling that I encountered numerous times during the feat of this dissertation,
and I am sure to have continued aimlessly running around in circles, if it wasn't
for my advisor, Professor William H. Steier, who could, and often did, nudge me in
the right direction. I am sincerely grateful for his role as an advisor, mentor and
teacher. He gives the necessary independence and
exibility for one to feel more
condent, and is always accessible. I feel very fortunate to have been given the
opportunity to work with him. There could not have been anyone better.
I would also like to express my gratitude to the research members of Professor
Steier's group that I had the privilege of working with. They are: Dr. Bipin Bhola,
Dr. Greeshma Gupta, Hari Mahalingam, Dr. Thanh Le, Professor Sang-Shin Lee,
Dr. Yoo Seung Lee, Nutthamon Suwanmonkha, and Dr. Andrew Yick.
I am thankful for the time and patience of my qualifying exam and dissertation
committees. They are: Professor Andrea Armani, Professor Atul Konkar, Dean
John O'Brien, Professor Alexander Sawchuk and Professor Armand Tanguay.
I am grateful to the members of Professor Tanguay's group for their support
iv
and help. They are: Dr. Michelle Hauer and Dr. Patrick Nasiatka.
I am eternally grateful to Professor Aluizio Prata Jr., for all that I have learned
of electromagnetic theory comes from him. In fact, the section on the analysis of
the electromagnetic wave in a uniform, homogeneous, anisotropic material with its
crystalline axis tilted with respect to the z-axis, was taken directly from one of his
homework assignments.
And nally I would like to express my deepest appreciation for the patience
and support of my parents. They have willingly provided me the opportunity to
continue my education, for which I know is the best gift I have ever received. Their
perseverance and methodical nature is that which I aspire to obtain.
v
TABLE OF CONTENTS
Dedication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii
Acknowledgments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii
List of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii
Abstract : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiii
Chapter 1 INTRODUCTION 1
1.1 Current Technologies for Electro-Optic Resonant Modulators . . . 2
1.1.1 Polymer Resonant Modulators . . . . . . . . . . . . . . . . 2
1.1.2 Electro-Optic Crystal Disk Resonant Modulators . . . . . 3
1.1.3 Electro-Optic Crystal Fabry-Perot Resonant Modulators . 4
Chapter 2 LITHIUM NIOBATE WAVEGUIDES 7
2.1 Titanium Indiused Lithium Niobate Waveguides . . . . . . . . . 7
2.2 Cleavage Planes of Lithium Niobate . . . . . . . . . . . . . . . . . 14
Chapter 3 THEORETICAL ANALYSIS OF AN ANISOTROPIC
WAVEGUIDE 18
3.1 Rectangular Waveguide Filled with Anisotropic
Medium: C-axis Parallel to Z-Axis . . . . . . . . . . . . . . . . . 19
3.2 Wave in Uniform, Homogeneous, Anisotropic Material: C-Axis Tilted
with Respect to Z-Axis . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Slab Waveguide in Homogeneous Anisotropic Material: C-Axis Tilted
with Respect to Z-Axis . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Slab Waveguide Innite in X-Direction . . . . . . . . . . . 35
3.3.2 Slab Waveguide Innite in Y-Direction . . . . . . . . . . . 40
3.4 Eective Index Method of Homogeneous, Anisotropic Dielectric Wave-
guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 Wave in Homogeneous, Anisotropic Dielectric Waveguide . . . . . 60
3.6 Wave in Inhomogeneous, Anisotropic Dielectric Waveguide: C-axis
Tilted with Respect to Z-Axis . . . . . . . . . . . . . . . . . . . . 67
Chapter 4 FABRY-PEROT RESONANT MODULATOR 69
4.1 Fabry-Perot Resonator . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Fabry-Perot Resonator with Modulation . . . . . . . . . . . . . . 75
vi
Chapter 5 THEORETICAL ANALYSIS OF THE ELECTRO-OPTIC
EFFECT FOR CLEAVED EDGE WAVEGUIDES 82
Chapter 6 EXPERIMENTAL ANALYSIS OF WAVEGUIDE
RESONATORS 118
6.1 Waveguide Fabrication Procedure . . . . . . . . . . . . . . . . . . 118
6.2 Index Distribution and Eective Index Measurement . . . . . . . 122
6.3 Z-Cut LiNbO
3
Waveguide Resonator . . . . . . . . . . . . . . . . 125
6.4 Cleaved Edge X-Cut LiNbO
3
Waveguide Resonator . . . . . . . . 135
6.4.1 Waveguide Facet Tilt . . . . . . . . . . . . . . . . . . . . . 135
6.4.2 Cleaved Edge . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4.3 Resonance Condition . . . . . . . . . . . . . . . . . . . . . 151
Chapter 7 EQUILATERAL TRIANGLE RESONATOR 159
Chapter 8 CONCLUSION 167
Bibliography 170
vii
LIST OF FIGURES
2.1 Refractive index of the ordinary wave in LiNbO
3
for wavelength
= 0:4 5 m, using a modied Sellmeier equation [65]. . . . . . 8
2.2 Refractive index of the extraordinary wave in LiNbO
3
for wavelength
= 0:4 5 m, using a modied Sellmeier equation [65]. . . . . 9
2.3 Example of a titanium indiused waveguide in x-cut LiNbO
3
. The
titanium indiuses into LiNbO
3
and creates a region of higher index,
which is used as a waveguiding core region. . . . . . . . . . . . . 9
2.4 Activation energies and diusion coecients for ordinary and ex-
traordinary modes of x-cut, y-cut and z-cut LiNbO
3
[7]. . . . . . . 11
2.5 Calculated diusion depths for diusion temperatures of 1050
C and
1080
C, given the activation energies and diusion coecients of
Figure 2.4, and a diusion time of 8 hours. . . . . . . . . . . . . . 12
2.6 Refractive indices of LiNbO
3
and titanium indiused LiNbO
3
for
optical telecommunications operating wavelengths of 1.31 m and
1.55 m, given the diusion depths of Figure 2.5, and titanium
thickness of 90 nm [14]. . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 LiNbO
3
viewed down the crystalline axis [1]. The crystal structure
contains both hexagonal and rhombohedral unit cells. . . . . . . . 15
2.8 The crystal structure of LiNbO
3
[1], as a projection upon the plane
of the (0001) basis. Oxygen ions are located at dierent planes on
the z-axis, forming a screw-like fashion. . . . . . . . . . . . . . . . 16
3.1 Unit vector rotation by angle about the
^
x
0
axis, according to the
right hand rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
viii
3.2 Unit vector rotation by angle about the
^
x
0
axis, according to the
right hand rule, in relation to a cleaved edge titanium indiused
LiNbO
3
waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Anisotropic dielectric slab waveguide innite in the x-direction. . 36
3.4 Anisotropic dielectric slab waveguide innite in the y-direction. . . 41
3.5 Anisotropic dielectric waveguide with unit vector rotation about the
^
x
0
axis, according to the right hand rule. . . . . . . . . . . . . . . 55
3.6 Eective index method assuming rst that the anisotropic dielectric
slab is innite in the x-direction, then innite in the y-direction. . 56
3.7 Eective index method assuming rst that the anisotropic dielectric
slab is innite in the y-direction, then innite in the x-direction. . 58
3.8 Marcatili's method assumes that the elds in the shaded regions
are negligible, and involves matching boundary conditions between
regions 1 & 2, regions 1 & 3, regions 1 & 5, and regions 1 & 4. . . 63
4.1 Fabry-Perot resonator consists of a medium sandwiched between two
partially transmitting mirrors, such that an electromagnetic wave
bounces back and forth within. . . . . . . . . . . . . . . . . . . . 70
4.2 Example of a Fabry-Perot waveguide resonator conguration. . . . 72
4.3 Example of the re
ected intensity response from a Fabry-Perot res-
onator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Best achievable nesse with respect to propagation loss for a critically-
coupled Fabry-Perot resonator, assuming that one mirror re
ectivity
is equal to unity, operating wavelength = 1:55 m, the refractive
index of the medium is for the extraordinary mode, n
e
= 2:14856,
and FSR = 10 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Schematic for the desired orientation of a titanium indiused z-
cut LiNbO
3
Fabry-Perot resonant modulator, in relation to the RF
electric eld and the optical wave. . . . . . . . . . . . . . . . . . . 79
ix
4.6 Example of modulation performed on a resonance curve. . . . . . 80
5.1 Possible electrode placement for the straight and tilted waveguides
with cleaved edges. The y
s
- and z
s
-directions are parallel and per-
pendicular to the straight waveguide axis. They
t
- andz
t
-directions
are parallel and perpendicular to the tilted waveguide axis. The
yellow rectangles represent electrodes. . . . . . . . . . . . . . . . . 86
5.2 Unit vector rotation of angles
s
and
t
about the ^ x axis, according
to the right hand rule. They
s
- andz
s
-directions are parallel and per-
pendicular to the straight waveguide axis. They
t
- andz
t
-directions
are parallel and perpendicular to the tilted waveguide axis. . . . 87
5.3 Unit vector rotation of angle
f
about the
^
x
t
axis, according to the
right hand rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Unit vector rotation with angle
f
about the ^ y axis according to
right-hand rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Unit vector rotation with angle
f
about the
^
z
0
axis according to
right-hand rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6 Unit vector rotation with angle
f
about the
^
x
m
axis according to
right-hand rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1 Indiusion setup up high temperature quartz tube furnace, quartz
crucibles, deionized water for increasing water vapor in tube, argon
and oxygen gas with
ow meters. . . . . . . . . . . . . . . . . . . 121
6.2 Schematic of the eective index measurement. . . . . . . . . . . . 123
6.3 Eective index measurement setup. The input laser source at =
1:3 m is re
ected o of a mirror, passed through a polarizer and
coupled into a rutile prism. The sample is pushed up against one of
the prism facets with an air piston such that an air gap is minimized
between the waveguide and the prism facet. A germanium detector
is placed at the output . . . . . . . . . . . . . . . . . . . . . . . . 124
x
6.4 Eective index measurement of extraordinary wave for z-cut tita-
nium indiused LiNbO
3
slab waveguide. Two modes identied with
eective indices 2.15132 and 2.15598. . . . . . . . . . . . . . . . . 126
6.5 Schematic and fabricated titanium indiused z-cut LiNbO
3
wave-
guide Fabry-Perot resonator. . . . . . . . . . . . . . . . . . . . . 127
6.6 Block diagram of experimental setup of titanium indiused z-cut
LiNbO
3
waveguide Fabry-Perot resonator, for resonance measure-
ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 Transmitted output of the TE-like and TM-like modes of a titanium
indiusedz-cut LiNbO
3
waveguide Fabry-Perot resonator, withFSR
values of 9.7 GHz and 10.2 GHz, respectively. . . . . . . . . . . . 130
6.8 Transmitted output of the TE-like mode of a titanium indiused
z-cut LiNbO
3
waveguide Fabry-Perot resonator, with a FSR value
of 9.7 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.9 Transmitted output of the TM-like mode of a titanium indiused
z-cut LiNbO
3
waveguide Fabry-Perot resonator, with a FSR value
of 10.2 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.10 Re
ected output of the TE-like and TM-like modes of a titanium in-
diusedz-cut LiNbO
3
waveguide Fabry-Perot resonator, with FSR
values of 9.7 GHz and 10.2 GHz, respectively. . . . . . . . . . . . 133
6.11 Re
ected output of the TM-like mode of a titanium indiusedz-cut
LiNbO
3
waveguide Fabry-Perot resonator, with aFSR value of 10.2
GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.12 An integrated device: Titanium indiused LiNbO
3
waveguide res-
onator ber pigtailed with epoxy NOA61. . . . . . . . . . . . . . 136
6.13 Power re
ection coecient vs. tilt of mirror facet. . . . . . . . . . 137
6.14 Schematic of cleaving process. LiNbO
3
is rst saw cut leaving a
thickness of 100m to be cleaved. One side of the sample is adhered
to a glass slide with epoxy NOA61 or bonding wax, and remains xed
to a stage. The other half of the sample is lifted with a manual linear
stage until cleavage occurs. . . . . . . . . . . . . . . . . . . . . . 138
xi
6.15 Setup of LiNbO
3
cleaving process . . . . . . . . . . . . . . . . . . 140
6.16 FESEM image of cleave on the +z face, along they-axis of LiNbO
3
. 141
6.17 Light microscope image of cleave on the +z face, along the y-axis
of LiNbO
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.18 FESEM image of cleave on thez face, along they-axis of LiNbO
3
. 143
6.19 Light microscope image of cleave on thez face, along the y-axis
of LiNbO
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.20 Light microscope image of cleave on thez face, 30
with respect
to the y-axis of LiNbO
3
. . . . . . . . . . . . . . . . . . . . . . . . 146
6.21 Damaged ion-implanted LiNbO
3
as further evidence of obtainable
cleaved planes in an equilateral triangle geometry . . . . . . . . . 147
6.22 Dominant LiNbO
3
cleavage plane occurs at an angle of = 32:75
with respect to the z-axis . . . . . . . . . . . . . . . . . . . . . . 148
6.23 FESEM image of cleave on thez face, along the x-axis of LiNbO
3
149
6.24 Light microscope image of cleave on the +x face, tilted by 33
with
respect to the z-axis of LiNbO
3
. . . . . . . . . . . . . . . . . . . 150
6.25 Laser re
ection experiment to conrm if the cleaved edge of LiNbO
3
is perpendicular to the top surface. . . . . . . . . . . . . . . . . . 152
6.26 Two waveguide designs for LiNbO
3
resonant modulators with cleaved
edges. The placement of the waveguide is shown by the thick blue
line. For the extraordinary case, the waveguide must be tilted at an
angle of approximately 2
from the z-axis. . . . . . . . . . . . . . 153
6.27 Titanium indiused LiNbO
3
waveguide resonator with cleaved edges
coated with 2 nm of gold. An optical ber coupled input and 10
lens to collect transmitted power are shown. . . . . . . . . . . . . 156
xii
6.28 Resonances of cleaved edgex-cut titanium indiused LiNbO
3
wave-
guide resonator. MeasuredFSR values are approximately 13.5 GHz
and 14.3 GHz for the ordinary and extraordinary modes, respec-
tively. The theoretical values are 13.8 GHz and 14.2 GHz, respec-
tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.1 Schematic of the equilateral triangle resonator . . . . . . . . . . . 160
7.2 Fabricated equilateral triangle resonator. The accuracy of the cleav-
ing process is20 m, which results in waveguide misalignment. 163
7.3 Fabricated equilateral triangle on a pedestal. . . . . . . . . . . . 164
7.4 Normalized power for a mode oset laterally from the dened core
waveguide region. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.5 Dependence of nesse on a cleaved facet re
ectance, given the other
two cleaved facets possess re
ectances of unity. . . . . . . . . . . . 165
xiii
ABSTRACT
An electro-optic resonant modulator consists of a dielectric material that responds
to an applied electric eld by way of amplitude modulation, as an optical signal
propagates through and resonates within. This work focuses on the analysis and
fabrication of the resonator. The resonator is of a Fabry-Perot type, employing a
titanium indiused lithium niobate waveguide as the optical guide. The resonator
is an integrated device and able to be ber pigtailed, to be used as a remote device.
A z-cut lithium niobate resonator with dielectric mirror coatings was fabricated
with a nesse of 14, for the TM-like mode at an operating wavelength of 1.55 m.
The cleavage planes of lithium niobate are investigated and incorporated in the nal
design of the resonator. The cleaved edges eliminate the need for the edge polishing
that is used for the mirrors of the Fabry-Perot resonant cavity. The desired cleaved
edges require the use of ax-cut lithium niobate substrate, with diering orientations
of the titanium indiused waveguide for the ordinary and extraordinary modes.
This dissertation presents a theoretical analysis of the electromagnetic wave in a
medium with o-diagonal components in the permittivity tensor, as is necessary
for the cleaved edge resonator geometry. A theoretical analysis of the electro-optic
xiv
eect for the cleaved edge geometry is also presented. A cleaved edge titanium
indiused lithium niobate resonator was fabricated, and resonances were observed
at 13.5 GHz and 14.3 GHz for the ordinary and extraordinary modes, respectively
with 2 nm gold coated facets, at an operating wavelength of 1.3 m. Given the
useable cleavage planes of z-cut lithium niobate, an equilateral triangle resonator
is proposed. The theoretical and experimental results contained in this dissertation
validate the feasibility and viability of a Fabry-Perot resonator as a modulating
medium.
1
Chapter 1
INTRODUCTION
Electro-optic modulators operate on a principle known as the Pockels eect, which
is also termed as the electro-optic eect. When a voltage is applied to a medium pos-
sessing electro-optic characteristics, the index of refraction of the medium changes.
If an optical wave is travelling through a medium possessing electro-optic qualities,
and if a voltage or an electric eld is applied to this medium, the optical wave will
be aected, by the change in the refractive index due to the applied electric eld.
In such a situation, the optical wave accumulates a phase change. This particular
eect will be used for the optical resonant modulator of this dissertation.
The ideal resonant modulator possesses characteristics of high sensitivity, low
switching power, fast operating speed, photo-stability and low loss. It is desirable
to use a material possessing a large electro-optic coecient, as modulation per-
formance is correlated to the electro-optic properties of a material. Also, a long
interaction length between the applied eld and the optical wave will enhance the
modulation performance.
The electro-optic resonator studied in this dissertation will be a Fabry-Perot
2
type, with a titanium indiused lithium niobate (LiNbO
3
) cleaved edge waveguide,
operating at optical communication wavelengths.
1.1 Current Technologies for Electro-Optic Resonant Mod-
ulators
There are several competing technologies for electro-optic resonant modulators.
Use of photolithography has enabled device makers to develop modulators with a
length of less than 100 m [46]. In designing an electro-optic resonant modulator
it is important to consider the material properties and limitations. The three
main types of resonant modulators are: polymer resonant modulators, electro-optic
crystal disk resonant modulators, and electro-optic crystal Fabry-Perot resonant
modulators. The following subsections will describe each.
1.1.1 Polymer Resonant Modulators
The electro-optic polymer resonant modulator contains an electro-optic polymer,
which is a chemically engineered material with an enhanced electro-optic coecient.
When applied with an electric eld of 80-125 V/m and heated to a point just below
the glass transition temperature, the molecules of the polymer are aligned with the
direction of electric eld. Once the molecules are aligned, and with the electric eld
still applied, the polymer is cooled to room temperature. Once the polymer has
3
cooled to room temperature, the molecules are locked into the desired alignment. In
the electro-optic polymer community, this process is termed electro-optic \poling."
Reported high values of electro-optic coecients in poled polymers [41], [54], make
them suitable candidates for an optical modulator.
Although, not of the resonant type, IPITEK has developed a polymer electro-
optic modulator with a device area of 3.6 cm
2
with a minimum detection limit of 70
mV/m
p
Hz up to 12 GHz [61]. The device consisted of a Mach-Zehnder structure.
A 70m 150m microring resonator with a detection limit of 100 mV/m up to
550 MHz has also been reported [54].
The disadvantages of polymers are their photo- and thermal-instability with
time [41]. Studies have shown that inert environments help improve the threshold
of breakdown [56], however more stable polymers are still in need of development.
1.1.2 Electro-Optic Crystal Disk Resonant Modulators
Electro-optic dielectric crystals, such as lithium niobate (LiNbO
3
), and lithium
tantalate (LiTaO
3
), are commonly used to form optical disk electro-optic resonators.
Reported disk diameters are on the order of a few millimeters [23]. These resonators
operate with the optical wave mode, also known as the whispering-gallery mode,
travelling around the periphery of the disk. When an external eld is applied to
the resonator, the response can be seen on the resonance curve of the optical wave
coupled out of the disk.
4
A LiNbO
3
disk modulator has been reported, with a device area of 34 mm
2
,
with a sensitivity of 0.13 mV/m
p
Hz at 6.7 and 13.4 GHz [47]. Although, the
device size is much larger than the polymer resonant modulator reported in Section
1.1.1, LiNbO
3
disk resonators have the ability to obtain a higher nesse (upwards
of several hundred) when compared to a Fabry-Perot resonant modulator [32]. As
will be described in Chapter 4, a cavity with a high nesse, means the optical wave
exhibits a low propagation loss, and the cavity possesses a narrow bandwidth of
operation.
Due to the fact that the optical wave is travelling along the periphery of the
disk, it is usually conned to a small area. If there is a desire to increase the power
of the optical wave, which is directly related to increasing the sensitivity, the power
density could exceed the power-handling threshold of the material and damage the
resonator.
The preferred method of coupling to a disk resonator is with a prism. Although
the prism can be small, due to the fact that it resides outside of the resonator with
its own platform, it prevents the disk from being an integrated device.
1.1.3 Electro-Optic Crystal Fabry-Perot Resonant Modulators
An electro-optic dielectric crystal resonant modulator of the Fabry-Perot type, con-
sists of an electro-optic medium with partially transmitting mirrors placed perpen-
dicular to the propagation direction of the optical wave. The optical wave inside,
5
traverses back and forth between the mirrors creating a standing wave, or a reso-
nance.
Fabry-Perot resonators typically achieve a lower nesse value compared to a
disk resonator [6], [55]. A lower nesse implies a larger bandwidth. The advantage
of the Fabry-Perot resonator is that it can be coupled with an optical ber, thus it
is an integrated device incorporating a waveguide, resonator and modulator. Since
Fabry-Perot waveguides can be designed to be larger than that achievable by a
resonant disk modulator, the threshold for power-handling breakdown is greater.
Previously, a LiNbO
3
optical waveguide Fabry-Perot modulator has been re-
ported with a nesse,F , equal to 4, at an operating wavelength of = 0.81m. A
reported value of titanium indiused LiNbO
3
waveguide propagation loss of around
0.1 dB/cm at = 1:15 m [45], is at least 20 times smaller than the propagation
loss in polymer waveguides [41].
This dissertation will focus on the analysis and fabrication of a titanium indif-
fused LiNbO
3
optical waveguide Fabry-Perot modulator with cleaved facets. Tita-
nium indiused LiNbO
3
waveguides are theoretically analyzed. Cleavage planes of
LiNbO
3
are studied and incorporated in the resonator as the mirror facets. The
modulation eect is theoretically analyzed followed by the design of the waveguide.
Experimental analyses of az-cut LiNbO
3
waveguide resonator with dielectric coated
mirrors, and a x-cut LiNbO
3
waveguide resonator with gold coated cleaved edge
mirrors are performed. Finally an equilateral triangle resonator is proposed with
6
the use of cleaved LiNbO
3
edges.
7
Chapter 2
LITHIUM NIOBATE WAVEGUIDES
2.1 Titanium Indiused Lithium Niobate Waveguides
The eld of integrated optics is giving rise to miniaturized light processing devices.
Waveguides with anisotropic lms and substrates have been used in mode con-
verters [49], [51], distributed-feedback lasers [9], [29], and polarization mode lters
[50]. Thin lm waveguides have been created using out-diusion [26], and by in-
diusion [48]. This dissertation will rely on the use of a titanium indiused LiNbO
3
waveguide.
The refractive index of LiNbO
3
has been examined previously [65]. The modied
Sellmeier equation of LiNbO
3
is,
n
2
1 =
A
2
2
B
+
C
2
2
D
+
E
2
2
F
,
(2.1)
where, for the ordinary wave the Sellmeier coecients are: A = 2.6734, B = 0.01764,
C = 1.2290, D = 0.05914, E = 12.614, and F = 474.6. For the extraordinary wave
the Sellmeier coecients are: A = 2.9804, B = 0.02047, C = 0.5981, D = 0.0666,
E = 8.9543, and F = 416.08.
8
0 1 2 3 4 5
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
Wavelength ( μm)
Refractive Index (Ordinary)
Figure 2.1: Refractive index of the ordinary wave in LiNbO
3
for wavelength = 0:4 5
m, using a modied Sellmeier equation [65].
The refractive index for the ordinary and extraordinary waves are shown in
Figures 2.1 and 2.2 for wavelengths from 0.4 m to 5 m.
A titanium strip with a typical thickness of 90 nm is deposited on a LiNbO
3
substrate with electron beam metal deposition, and the strip width is dened pho-
tolithographically. The LiNbO
3
sample is placed in a quartz tube furnace of ap-
proximately 1050
C - 1100
C, where indiusion of the titanium into LiNbO
3
takes
place. The indiused region possesses an increased refractive index and thus may
be used for light guidance, as in an optical waveguide. An example drawing of a
titanium indiused LiNbO
3
waveguide is shown in Figure 2.3.
There are several functions that describe diusion. For the case of a long diu-
9
0 1 2 3 4 5
2
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
Wavelength ( μm)
Refractive Index (Extraordinary)
Figure 2.2: Refractive index of the extraordinary wave in LiNbO
3
for wavelength
= 0:4 5 m, using a modied Sellmeier equation [65].
x
z
y
Figure 2.3: Example of a titanium indiused waveguide in x-cut LiNbO
3
. The titanium
indiuses into LiNbO
3
and creates a region of higher index, which is used as a waveguiding
core region.
10
sion time, such that the titanium to be indiused has been completely exhausted
from the surface, the function describing the diusion into the substrate is a Gaus-
sian function [30],
f (x) =exp
x
2
D
2
x
,
(2.2)
where,
D
x
= 2
p
(D
x
t) : (2.3)
D
x
is the diusion depth in the x-direction,D
x
is the diusion coecient in the
x-direction, and t is the diusion time.
The diusion also takes place along the width of the waveguide and the com-
monly used function to describe the diusion is with the error function [21],
g (y) =
1
2
ferf
W
2D
y
1 +
2y
W
+ erf
W
2D
y
1
2y
W
g
,
(2.4)
where W is the initial titanium strip width, andW=2yW=2. Also,
D
y
= 2
q
(D
y
t) ; (2.5)
whereD
y
is the diusion depth in they-direction, andD
y
is the diusion coecient
in the y-direction.
The diusion coecientsD
x
andD
y
follow Arrhenius law, such that,
D
x;y
=D
x;y
0
exp
E
x:y
0
kT
,
(2.6)
11
X – cut Y – cut Z – cut
Ordinary Extraordinary Ordinary Extraordinary Ordinary Extraordinary
Activation
Energy (eV)
2.65 2.4 1.85 2.31 2.34 2.53
Diffusion
Coefficient
(cm
2
/s)
2.2x 10
-2
2.0 x 10
-3
2.4 x 10
-5
9.0 x 10
-4
2.2 x 10
-3
8.5 x 10
-3
Figure 2.4: Activation energies and diusion coecients for ordinary and extraordinary
modes of x-cut, y-cut and z-cut LiNbO
3
[7].
where,
E
x;y
0
= activation energies in the x and y directions (eV) ; (2.7)
k = Boltzmann constant = 8:617332 10
5
(eV=K) ; (2.8)
T = diusion temperature (K) : (2.9)
Reported values of the activation energies and diusion coecients of x-cut, y-
cut and z-cut LiNbO
3
are shown in Figure 2.4 [7]. Figure 2.5 shows the calculated
diusion depths for a time of 8 hours at 1050
C and 1080
C.
The diusion process can be ne-tuned with changes in diusion temperature,
diusion time and thickness of the initial titanium strip, to obtain the needed
mode size. If the titanium has not completely diused into the LiNbO
3
substrate,
a rough surface may result, leading to surface scattering. However, oxidation of the
surface is believed to create a smooth, unblemished dielectric lm [18]. Thus, an
oxygen quench sequence near the end of the diusion cycle may be desired. It has
12
1050 °C (1323 K) 1080 °C (1353 K)
Diffusion depth (μm) Diffusion depth (μm)
Ordinary Extraordinary Ordinary Extraordinary
X - cut 4.43 4 5.7 5
Y - cut 4.92 4.07 5.89 5.09
Z - cut 5.46 4.68 6.86 5.98
Figure 2.5: Calculated diusion depths for diusion temperatures of 1050
C and 1080
C,
given the activation energies and diusion coecients of Figure 2.4, and a diusion time
of 8 hours.
been reported that a lossy surface guide can appear and is attributed to the out-
diusion of Li
2
O. The presence of water vapor in the environment during diusion
can suppress the out-diusion [24]. It has been proposed that the presence of the
added hydrogen in the crystal suppresses the lithium mobility and prevents the
out-diusion to occur. Thus often times during the diusion process, water vapor
is increased in the furnace.
Since optical telecommunication is performed at wavelengths of 1.31 m and
1.55m where the optical ber has the least amount of attenuation, the refractive
indices for these values, as well as the refractive indices with titanium indiusion,
are shown in Figure 2.6. The refractive index changes are assumed to be, 0:5310
2
and 1:110
2
for the ordinary and extraordinary wave at 1.31m, and 0:4910
2
and 1:1 10
2
for the ordinary and extraordinary wave at 1.55 m [14]. The
13
1.31 μm 1.55 μm
Refractive
Index
(n)
Relative
Permittivity
(ε
r
)
Permittivity
(ε = ε
0
ε
r
)
x 10
-11
F/m
Refractive
Index
(n)
Relative
Permittivity
(ε
r
)
Permittivity
(ε = ε
0
ε
r
)
x 10
-11
F/m
LiNbO
3
Ordinary
2.21997 4.89281 4.33210 2.21111 4.88901 4.32873
LiNbO
3
Extraordinary
2.14512 4.60154 4.07420 2.13756 4.56916 4.04554
Ti:LiNbO
3
Ordinary
2.22527 4.95183 4.38435 2.21601 4.91070 4.34793
Ti:LiNbO
3
Extraordinary
2.15612 4.64885 4.11609 2.14856 4.61631 4.08728
Figure 2.6: Refractive indices of LiNbO
3
and titanium indiused LiNbO
3
for optical
telecommunications operating wavelengths of 1.31 m and 1.55 m, given the diusion
depths of Figure 2.5, and titanium thickness of 90 nm [14].
assumed diusion depths for the purpose of future calculations and fabrication,
are 6 m and 5 m and for the ordinary and extraordinary waves, for a titanium
strip width of 8 m, titanium strip thickness of 90 nm, diusion temperature of
1080
C and diusion time of 8 hours [7]. The diusion depth of the ordinary mode
is slightly larger than the extraordinary mode, and thus this correlates to the index
change being larger for the extraordinary mode as the mode is more conned [30].
Of utmost importance for optical devices is the waveguide propagation loss. A
reported value of propagation loss for titanium indiused LiNbO
3
is approximately
0.3 dB/cm for a 1 cm long waveguide operating at = 1.32m [3]. Another report
indicates a propagation loss of 0.09 dB/cm for the HE
11
mode at = 1.15 m
14
[45]. Thus due to the
exibilities in fabrication, dierent losses may be obtained,
however they are low enough to allow titanium indiused LiNbO
3
waveguides to
be a useful medium in optical communications.
2.2 Cleavage Planes of Lithium Niobate
LiNbO
3
possesses optical, piezoelectric, electro-optic, elastic, photoelastic, and pho-
torefractive properties that have been advantageously used in many communica-
tions devices. LiNbO
3
is a ferroelectric material, has a trigonal crystal structure,
and is a member of the 3m point group.
The crystal structure includes both hexagonal and rhombohedral unit cells
[1],[62], as shown in Figure 2.7. The structure of LiNbO
3
contains planar sheets
of oxygen atoms in a distorted hexagonal construction. In Figure 2.8, it can be
seen that the screw-like fashion of the oxygen atoms do not lie on top of each other
along the three-fold axis. Mapping along the direction of the crystalline axis, at
the center of the octahedral structure is an alternating pattern of a lithium atom,
a niobium atom and a vacant site. The crystal structure has mirror symmetry
about three planes that are 60
apart. Thus, the crystallographic axes (a
1
;a
2
;a
3
)
represent a hexagonal unit cell and are 120
apart. The three crystal physical axes
(x, y, z) are dened by the piezoelectric eect, an IEEE standard.
It has been reported that cleaved surfaces of LiNbO
3
produce repeatable and
useable waveguide edges [27]. The natural cleavage planes of LiNbO
3
are (01
12); (
1012)
15
a
2
a
1
Li Nb O
Figure 2.7: LiNbO
3
viewed down the crystalline axis [1]. The crystal structure contains
both hexagonal and rhombohedral unit cells.
16
a
3
a
2
a
1
z
z + 1/6 z + 1/3
z+1/2
z + 2/3 z + 5/6
Figure 2.8: The crystal structure of LiNbO
3
[1], as a projection upon the plane of the
(0001) basis. Oxygen ions are located at dierent planes on the z-axis, forming a screw{
like fashion.
17
and (1
102) [16]. The cleaved planes are a result of vacancy sites. The cleavage plane
along they-axis can be utilized for az-cut LiNbO
3
waveguide resonant modulator.
All three cleavage planes will be utilized for the z-cut LiNbO
3
equilateral triangle
resonator of Chapter 7.
There is also a dominant cleavage plane at an angle of = 32:75
with respect to
thez-axis [25]. This particular plane was identied as the plane parallel to a plane
of vacant octahedral sites, sandwiched between a plane of Li atoms on one side and
Nb atoms on the opposing side. The plane was hypothesized to be a cleavable plane
because of the large separation between the atoms, and thus the force between the
atoms was assumed to be weak [25]. This cleavage plane will be utilized for the
tilted waveguide resonant modulator of Chapter 6.
A typical orientation of a LiNbO
3
modulator is that the crystalline axis (c-axis)
coincides with the z-axis. However, if one intends to use the cleaved edge of the
waveguide, one nds that the c-axis is not parallel to the z-axis. The theoretical
analysis of the tilted c-axis is described in the following chapter.
18
Chapter 3
THEORETICAL ANALYSIS OF AN ANISOTROPIC
WAVEGUIDE
Titanium indiused LiNbO
3
is considered an inhomogeneous and anisotropic ma-
terial. The purpose of this chapter is to analyze how a wave propagates in an
anisotropic material. The medium is rst considered to be boundless, as in a bulk
material. Then, the medium is bounded in one dimension, such as in a slab wave-
guide. Finally, the eld inside a waveguide is considered with approximations on
the o-diagonal terms of the permittivity tensor.
Anisotropic materials have an interesting permittivity characteristic in that the
permittivity value is dierent for each of the Cartesian crystallographic axes. Since
Maxwell's equations are typically written with the use of the permittivity constant,
, the theoretical analysis will mostly utilize this constant. However, in the eld of
optics, the prevalent use of the refractive index,n, and especially the measured data
values of it are more accessible in the literature. Thus, mostly in terms of example
calculations, the refractive index may be substituted in for the permittivity with
the following relation,
19
=
0
n
2
; (3.1)
The double overbar is used to represent the dyadic nature, and
0
is the free-space
permittivity constant with a value of 8.85410
12
F/m.
Along a specic crystallographic axis, the permittivity value may also vary. This
is the case for an inhomogeneous material, as in the titanium indiused material.
Due to the fact that the titanium may indiuse into the LiNbO
3
substrate in the
transverse and lateral directions, the inhomogeneity is in two-dimensions. Further-
more, since LiNbO
3
is a uniaxial anisotropic material, the permittivity tensor in
this analysis will represent a uniaxial medium.
3.1 Rectangular Waveguide Filled with Anisotropic
Medium: C-axis Parallel to Z-Axis
This section will contain the analysis of a metallic rectangular waveguide lled with
an anisotropic medium. The importance of this analysis is to understand how the
elds behave in a simplied condition, in that the boundary conditions of a metallic
wall impose the tangential elds to be zero.
20
In the time harmonic form, Maxwell's equations are as follows [52]:
r
~
E =j!
~
B
~
M ; (3.2)
r
~
H =j!
~
D +
~
J ; (3.3)
r
~
D =P
e
; (3.4)
r
~
B =P
m
; (3.5)
where,
~
E = electric eld intensity (V/m) ,
~
H = magnetic eld intensity (A/m) ,
~
D = electric
ux density (C/m
2
) ,
~
B = magnetic
ux density (Wb/m
2
or T) ,
~
J = electric current density (A/m
2
) ,
~
M = magnetic current density (V/m
2
) ,
P
e
= electric charge density (C/m
3
) ,
P
m
= magnetic charge density (Wb/m
3
) .
The constitutive parameters are,
= permittivity (F/m) ,
= permeability (H/m) ,
e
= electric conductivity (S/m) ,
21
and the constitutive relations are,
~
D =
~
E ; (3.6)
~
B =
~
H ; (3.7)
~
J =
e
~
E : (3.8)
In a non-magnetic, source-free medium (
~
M =
~
J = P
e
= P
m
= 0), Maxwell's
equations reduce to,
r
~
E =j!
~
H ; (3.9)
r
~
H =j!
~
E ; (3.10)
r
~
D = 0 ; (3.11)
r
~
B = 0 : (3.12)
In Cartesian coordinates the permittivity tensor has the general form,
=
xx
^ x^ x +
xy
^ x^ y +
xz
^ x^ z +
yx
^ y^ x +
yy
^ y^ y +
yz
^ y^ z +
zx
^ z^ x +
zy
^ z^ y +
zz
^ z^ z : (3.13)
In an isotropic material, all o-diagonal terms of the permittivity tensor equal zero
(
xy
=
yx
=
yz
=
zy
= 0), and the diagonal terms are all equivalent (
xx
=
yy
=
zz
), which leads to a linear relationship between
~
E and
~
D. However, in an
anisotropic material there may no longer be a linear relationship between
~
E and
~
D.
It is instructive to rst consider a rectangular waveguide lled with an anisotropic
material with only diagonal elements in the permittivity tensor [5]. This analysis
22
will help in understanding the specic form of the waveguide modes in an anisotropic
material, and thus may be used to relate to the waveguide modes in an anisotropic
material with o-diagonal elements.
The permittivity dyadic in Cartesian coordinates is
=
11
^ x^ x +
22
^ y^ y +
33
^ z^ z : (3.14)
The crystalline axis is aligned with the ^ z direction. Assuming that the time-
harmonic electromagnetic elds are propagating toward the +^ z direction, the elds
are,
~
E =
~
E
0
e
zz
e
j!t
; (3.15)
~
H =
~
H
0
e
zz
e
j!t
; (3.16)
where
z
is the propagation constant and ! is the radian frequency, and
~
E
0
= ^ xE
x0
+ ^ yE
y0
+ ^ zE
z0
; (3.17)
~
H
0
= ^ xH
x0
+ ^ yH
y0
+ ^ zH
z0
: (3.18)
The term,
e
j!t
; (3.19)
is implicitly assumed in all solutions of the time-harmonic electromagnetic elds
and is subsequently omitted.
Using Equation (3.9) and the vector identity,
r
~
E = ^ x
@E
z
@y
@E
y
@z
+ ^ y
@E
x
@z
@E
z
@x
+ ^ z
@E
y
@x
@E
x
@y
,
(3.20)
23
and assuming that =
0
= 410
7
H/m, which is true for the dielectric medium
considered in this dissertation, the following set of equations can be obtained,
H
x
=
1
j!
0
@E
z
@y
z
E
y
,
(3.21)
H
y
=
1
j!
0
z
E
x
+
@E
z
@x
,
(3.22)
H
z
=
1
j!
0
@E
y
@x
+
@E
x
@y
.
(3.23)
Similarly using Equation (3.10) the following equations can be obtained,
E
x
=
1
j!
11
@H
z
@y
+
z
H
y
,
(3.24)
E
y
=
1
j!
22
z
H
x
+
@H
z
@x
,
(3.25)
E
z
=
1
j!
33
@H
y
@x
@H
x
@y
.
(3.26)
Solving for E
x
;E
y
;H
x
; and H
y
in terms of E
z
and H
z
gives,
E
x
=
1
!
2
0
11
+
2
z
j!
0
@H
z
@y
+
z
@E
z
@x
,
(3.27)
E
y
=
1
!
2
0
22
+
2
z
z
@E
z
@y
j!
0
@H
z
@x
,
(3.28)
H
x
=
j
!
2
0
22
+
2
z
!
22
@E
z
@y
+j
z
@H
z
@x
,
(3.29)
H
y
=
j
!
2
0
11
+
2
z
!
11
@E
z
@y
j
z
@H
z
@x
.
(3.30)
Using Equations (3.23), (3.26) - (3.30) the following equations can be derived,
1
g
2
2
@
2
H
z
@x
2
+
1
g
2
1
@
2
H
z
@y
2
+
z
j!
1
g
2
1
1
g
2
2
@
2
E
z
@x@y
+H
z
= 0 ; (3.31)
11
g
2
1
@
2
E
z
@x
2
+
22
g
2
2
@
2
E
z
@y
2
+
z
j!
@
2
H
z
@y
2
+
z
j!
1
g
2
1
1
g
2
2
@
2
H
z
@x@y
+
33
E
z
= 0 ; (3.32)
24
where,
g
2
1
=!
2
0
11
+
2
z
; (3.33)
g
2
2
=!
2
0
22
+
2
z
: (3.34)
In a uniaxial medium
11
=
22
thus, g
1
= g
2
and Equations (3.31) and (3.32)
become,
@
2
H
z
@x
2
+
@
2
H
z
@y
2
+g
2
H
z
= 0 ; (3.35)
@
2
E
z
@x
2
+
@
2
E
z
@y
2
+
33
11
g
2
E
z
= 0 : (3.36)
Equations (3.35) and (3.36) can be solved using the separation of variables method
[52]. Assuming a waveguide with perfectly conducting rectangular cross-section
walls of dimensions 0 x a and 0 y b, and applying the boundary condi-
tions,
E
y
=E
z
= 0 at x = 0 and x =b ; (3.37)
E
x
=E
z
= 0 at y = 0 and y =a ; (3.38)
the following solution for the TE mode can be derived,
H
z
=A
mn
cos
m
b
x
cos
n
a
y
e
zz
m = 0; 1; 2::: ;n = 0; 1; 2::: (3.39)
2
z
=!
2
0
11
m
b
x
2
n
a
y
2
m = 0; 1; 2::: ;n = 0; 1; 2::: (3.40)
and the subsequent solutions for eld E
x
; E
y
; H
x
and H
y
can be found through
25
(3.27) - (3.30). Similarly applying the boundary conditions
E
y
=H
z
= 0 at x = 0 and x =b ; (3.41)
E
x
=H
z
= 0 at y = 0 and y =a ; (3.42)
the following solution for the TM mode can be derived,
E
z
=B
mn
sin
m
b
x
sin
n
a
y
e
zz
m = 0; 1; 2::: ;n = 0; 1; 2::: (3.43)
2
z
=!
2
0
11
11
33
m
b
x
2
+
n
a
y
2
m = 0; 1; 2::: ;n = 0; 1; 2:::
(3.44)
and the subsequent solutions for eld E
x
; E
y
; H
x
and H
y
can be found through
(3.27) - (3.30).
It is important to note the two dierent propagation constants that arise for the
TE and TM modes. The electromagnetic modes are sinusoidal functions, much like
one would expect for an isotropic dielectric-lled rectangular waveguide [4]. Thus,
knowing the sinusoidal solution, one may use this to postulate that the elds in a
anisotropic medium with o-diagonal terms may also have electromagnetic modes
that are sinusoidal functions.
3.2 Wave in Uniform, Homogeneous, Anisotropic Material:
C-Axis Tilted with Respect to Z-Axis
It is now of interest to analyze the electromagnetic elds in a bulk, homogeneous
and anisotropic material. The theory of a plane wave propagating in an anisotropic
26
media with only diagonal terms in the permittivity tensor has been derived pre-
viously [8]. In the analysis, it was found that two particular waves propagate in
the material, and they are termed the ordinary and the extraordinary wave. The
extraordinary wave propagates, such that the propagation vector,
z
, and the direc-
tion of energy propagation, given by the Poynting vector
^
S, do not coincide. This
particular element of the extraordinary wave will be of importance when designing
the cleaved edge resonator.
As will be recognized in the following chapters, the cleaved plane will be of
utmost importance for the resonant modulator investigated in this dissertation.
The particular cleavage plane will orient the crystalline axis (c-axis), such that it
is tilted with respect to the waveguide axis. In this section, the eect of a tilted
c-axis will be studied for the bulk material. The theoretical analysis of the plane
wave propagating in a bulk material will be revisited, and considered with a slight
modication of the permittivity tensor.
From Section 3.1 the permittivity dyadic of an anisotropic, uniaxial and bire-
fringent dielectric crystal in the Cartesian coordinate system is,
=
11
^
x
0 ^
x
0
+
11
^
y
0 ^
y
0
+
33
^
z
0 ^
z
0
: (3.45)
The primed coordinates in this section, will represent the crystallographic axes of
the dielectric crystal. If the uniaxial crystal is rotated about
^
x
0
by an angle
according to the right hand rule, then the unprimed coordinates can designate the
27
ˆ z
ˆ' z
ˆ y
ˆ' y
Figure 3.1: Unit vector rotation by angle about the
^
x
0
axis, according to the right hand
rule.
new unit vector directions of ^ x, ^ y and ^ z as shown in Figure 3.1.
Although the waveguide structure is not analyzed in this section, it is benecial
to consider the scenario in which this may be important. The rotation of the
^
y
0
and
^
z
0
directions can be clearly seen in a waveguide conguration as in Figure 3.2,
which will be a recurring structure throughout this dissertation. Thus ^ z represents
the direction along the waveguide axis.
The primed coordinates are rotated to the unprimed coordinates with the fol-
lowing relations,
^
x
0
= ^ x ; (3.46)
^
y
0
= ^ y cos + ^ z sin ; (3.47)
^
z
0
=^ y sin + ^ z cos ; (3.48)
Plugging in the values of
^
x
0
;
^
y
0
and
^
z
0
, the permittivity dyadic becomes,
=
xx
^ x^ x +
yy
^ y^ y +
yz
^ y^ z +
zy
^ z^ y +
zz
^ z^ z ; (3.49)
28
x, x’
y’
z’
z
y
Figure 3.2: Unit vector rotation by angle about the
^
x
0
axis, according to the right hand
rule, in relation to a cleaved edge titanium indiused LiNbO
3
waveguide.
where,
xx
=
11
; (3.50)
yy
= (
33
11
) sin
2
+
11
; (3.51)
yz
= (
11
33
)
sin 2
2
,
(3.52)
zy
=
yz
; (3.53)
zz
= (
33
11
) cos
2
+
11
: (3.54)
Assuming that the time-harmonic electromagnetic elds are propagating toward
the +^ z direction like in Equations (3.15) and (3.16), and with the use of Maxwell's
29
equations and the vector identity in Equation (3.20), it can be shown that,
~
z
~
E =j!
0
~
H ; (3.55)
~
z
~
H =j!
~
E ; (3.56)
~
z
~
H = 0 ; (3.57)
~
z
~
D = 0 : (3.58)
Taking the cross product of ~
z
with Equation (3.55) leads to,
~
z
~
z
~
E
=j!
0
~
z
~
H
; (3.59)
and with some algebraic manipulation the equation becomes,
^
z
^
z
I
!
2
0
2
z
~
E = 0 : (3.60)
In matrix form this is,
2
6
6
6
6
6
6
4
0 0 0
0 0 0
0 0 1
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
E
x
E
y
E
z
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
E
x
E
y
E
z
3
7
7
7
7
7
7
5
!
2
0
2
z
2
6
6
6
6
6
6
4
xx
0 0
0
yy
yz
0
zy
zz
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
E
x
E
y
E
z
3
7
7
7
7
7
7
5
= 0 : (3.61)
Equation (3.61) contains three equations,
E
x
!
2
0
2
z
xx
E
x
= 0 ; (3.62)
E
y
!
2
0
2
z
(
yy
E
y
+
yz
E
z
) = 0 ; (3.63)
!
2
0
2
z
(
zy
E
y
+
zz
E
z
) = 0 : (3.64)
30
There are two solutions that can be obtained; one is termed the ordinary wave and
the other is termed the extraordinary wave. Setting,
E
y
= 0 ; (3.65)
E
z
= 0 ; (3.66)
yields the ordinary wave solution. The propagation constant is,
o
z
=j!
p
0
11
: (3.67)
The superscript `o' corresponds to the ordinary wave. The electric eld is then,
~
E
o
= ^ xE
o
x
e
o
z
z
: (3.68)
Using Equation (3.55) the magnetic eld is found to be,
~
H
o
= ^ y
o
z
j!
0
E
o
x
e
o
z
z
: (3.69)
Next, the extraordinary wave solution can be found by setting
E
x
= 0 : (3.70)
This yields a relationship between E
y
and E
z
, which is,
E
e
z
=
(
33
11
) sin (2)
2 [(
33
11
) cos
2
+
11
]
E
e
y
: (3.71)
The propagation constant for the extraordinary wave is,
e
z
=j!
r
0
11
33
(
33
11
) cos
2
+
11
.
(3.72)
31
The superscript `e' corresponds to the extraordinary wave. The electric eld is
then,
~
E
e
=
^ yE
e
y
+ ^ z
1
2
(
33
11
) sin (2)
(
33
11
) cos
2
+
11
E
e
y
e
e
z
z
: (3.73)
Using Equation (3.55) the magnetic eld is
~
H
e
=^ x
e
z
j!
0
E
e
y
e
e
z
z
: (3.74)
Knowing the solutions of the electric and magnetic elds, it is now possible to
observe the direction of the power
ow of the ordinary and extraordinary waves.
The average power for a time-harmonic electromagnetic wave is given by the Poynt-
ing vector
~
S [44], which is,
~
S =
1
2
~
E
~
H
.
(3.75)
Assuming that the medium is lossless, the propagation vectors are,
o
z
=j
o
z
; (3.76)
e
z
=j
e
z
: (3.77)
For the ordinary wave the Poynting vector is,
~
S
o
= ^ z
1
2
o
z
!
0
E
o
x
E
o
x
: (3.78)
For the extraordinary wave the Poynting vector is,
~
S
e
=
1
2
e
z
!
0
E
e
y
E
e
y
^ z ^ y
E
e
z
E
e
y
.
(3.79)
32
The direction of the Poynting vector is given by the unit vector of
~
S. For the
ordinary wave the Poynting vector is in the +^ z direction and thus the propagation
direction is parallel to the Poynting vector direction. However for the extraordinary
wave, the Poynting vector direction has a component in the +^ z direction and a
component in the^ y direction. To further analyze the Poynting vector, one can
obtain the unit Poynting vector for the extraordinary wave as,
^
S
e
=
^ z ^ y
E
e
z
=E
e
y
q
1 +
E
e
z
=E
e
y
2
.
(3.80)
For the extraordinary wave, the direction of the power
ow is not parallel to
the propagation direction. This has been previously termed as a walk-o condition
[10]. The angle
e
between the extraordinary wave Poynting vector direction and
the propagation direction is given by,
cos
e
= ^ z
^
S
e
; (3.81)
e
= arccos
0
@
1
q
1 +
E
e
z
=E
e
y
2
1
A .
(3.82)
Assuming values of n
11
= 2:21601;n
33
= 2:14856 and = 57:25
for titanium
indiused LiNbO
3
at = 1:55 m, the following values can be obtained,
11
=
0
n
2
11
= 4:34793 10
11
F/m ; (3.83)
33
=
0
n
2
33
= 4:08728 10
11
F/m ; (3.84)
E
e
z
E
e
y
=0:03 ; (3.85)
e
= 1:76
: (3.86)
33
Returning to the unit Poynting vector, the extraordinary wave is thus propa-
gating along the +^ z direction, with its power propagation direction tilted at 1:76
toward the +^ y direction from the +^ z axis . The importance of this section is the
recognition of the walk-o eect for an extraordinary wave in a bulk anisotropic
uniaxial dielectric crystal. Eventually a waveguide will be formed in this bulk ma-
terial, and it is important to note that from this section, that the eect of walk-o
is a bulk material property rather than a waveguiding property.
The walk-o eect will be incorporated in a resonant structure. In order for
resonance to occur in a resonator, the phase fronts which are normal to the propaga-
tion direction, must be parallel to the mirror end facets. This has been proven for a
wave propagating in the +^ z direction. This condition must also now be proven for a
wave propagating in the^ z direction, and furthermore it is necessary to understand
the direction of the Poynting vector for the reverse propagation direction.
The same rotation about the +
^
x
0
is taken however an additional rotation is
added. Thus, the coordinate rotation is modied to be,
^
x
0
= ^ x ; (3.87)
^
y
0
= ^ y cos ( +) + ^ z sin ( +) ; (3.88)
^
z
0
=^ y sin ( +) + ^ z cos ( +) ; (3.89)
34
The following trigonometric identities can be used for this analysis,
sin ( +) = sin ; (3.90)
cos ( +) = cos ; (3.91)
sin [2 ( +)] = sin (2) : (3.92)
The electromagnetic elds are now considered to be propagating in the^ z direction
and thus the elds have an exponential term with the following form,
e
zz
: (3.93)
The entire derivation is equivalent to the case when the electromagnetic elds
were propagating in the +^ z direction, except that the +^ y and +^ z directions are
replaced with^ y and^ z directions, respectively. This results in an ordinary wave
with its propagation vector and Poynting vector in the^ z direction, and an ex-
traordinary wave with its propagation vector in the^ z direction and its Poynting
vector with a component in the^ z direction and a component in the^ y direction.
The conclusion is that if the waveguide end is a perfect mirror facet with no loss,
the wave propagating in the +^ z direction would impinge upon the perfect mirror,
re
ect and then propagate in the^ z direction for both the ordinary wave and the
extraordinary wave. Furthermore, the Poynting vector direction would also retrace
upon itself for the incoming and the re
ected wave. This idea will be revisited when
designing the resonator with cleaved edges.
35
3.3 Slab Waveguide in Homogeneous Anisotropic Material:
C-Axis Tilted with Respect to Z-Axis
For a wave propagating in a slab waveguide, there is connement in one direction,
and no connement in the other transverse direction. The slab waveguide is an
important case to consider as it will enhance the understanding of the elds with
simply two boundary conditions. It will also be useful for the analysis of the eective
index method of Section 3.4. The slab waveguides to be studied will contain a
permittivity tensor with o-diagonal elements.
3.3.1 Slab Waveguide Innite in X-Direction
The analysis of this section involves the slab waveguide that is innite in the x-
direction as shown in Figure 3.3.
The derivation of the symmetric slab waveguide loosely follows a previous anal-
ysis [39]. The electromagnetic elds are assumed to be propagating in the +^ z
direction as in Equations (3.15) and (3.16). Assuming a lossless medium the prop-
agation vector is,
z
=j
z
: (3.94)
The homogeneous anisotropic material considered is a uniaxial dielectric crystal
with the c-axis not parallel to the z-axis. The permittivity dyadic is given by
Equation (3.49). The use of Equation (3.10) and the vector identity Equation
36
x, x’
y’
z’
z
y
2b
Figure 3.3: Anisotropic dielectric slab waveguide innite in the x-direction.
37
(3.20), and assuming no variation along the ^ x -axis, the following equations can be
obtained,
H
x
=
1
j!
0
@E
z
@y
+j
z
E
y
,
(3.95)
H
y
=
z
!
0
E
x
,
(3.96)
H
z
=
1
j!
0
@E
x
@y
,
(3.97)
@H
z
@y
+j
z
H
y
=j!
xx
E
x
; (3.98)
j
z
H
x
=j!
yy
E
y
+j!
yz
E
z
; (3.99)
@H
x
@y
=j!
yz
E
y
+j!
zz
E
z
: (3.100)
Two conditions arise from the above three equations; one is TE-like whilst the other
is TM-like. The TM-like solution is identical to a symmetric isotropic slab [37]. For
the TE-like solution, which will be covered in this section, the solution of H
x
can
be determined.
To do so, rstly,E
y
andE
z
are expressed in terms ofH
x
using Equations (3.122)
and (3.123) as,
E
y
=
j
!
yz
@H
x
@y
j
z
zz
H
x
; (3.101)
E
z
=
j
!
yy
@H
x
@y
j
z
yz
H
x
; (3.102)
where,
=
yy
zz
2
yz
: (3.103)
38
Then by using Equation (3.95), the following equation for H
x
can be written,
@
2
H
x
@
2
y
2j
z
yz
yy
@H
x
@y
+
!
2
0
yy
2
z
zz
yy
H
x
= 0 : (3.104)
The solution to the second order dierential equation of H
x
is,
H
x
= [A
1
cos (y) +B
1
sin (y)]e
jy
e
jzz
; (3.105)
where,
=
z
yz
yy
,
(3.106)
=
2
yy
!
2
0
yy
2
z
1=2
: (3.107)
In order to analyze the boundary conditions of the electromagnetic elds, it is
easier to observe the cosine and sine components of H
x
individually. Knowing the
solution ofH
x
, yields the solution ofE
x
andE
z
. For reasons that will be explained
in a future section, for this derivation, it is only necessary to consider a symmetric
slab waveguide, as opposed to the more general asymmetric case. The asymmetric
case requires an additional boundary condition. One set of the assumed form of
the elds is then,
H
x
=
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
A cos (b)e
jb
e
(yb)
e
j(yb)
e
jzz
yb
A cos (y)e
jy
e
jzz
jyjb
A cos (b)e
jb
e
(y+b)
e
j(y+b)
e
jzz
yb
(3.108)
39
where,
s
=
s
yy
s
zz
s
yz
2
; (3.109)
=
z
s
yz
s
yy
,
(3.110)
=
"
s
s
yy
2
2
z
!
2
0
s
yy
#
1=2
; (3.111)
and the superscript `s' designates that the component belongs to the substrate
medium. Taking the derivative of H
x
leads to,
@H
x
@y
=
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
A ( +j) cos (b)e
jb
e
(+j)y
e
(j)b
e
jzz
yb
A [ sin (y) +j cos (y)]e
jy
e
jzz
jyjb
A ( +j) cos (b)e
jb
e
(+j)y
e
(+j)b
e
jzz
yb
(3.112)
The other set of the assumed form of the elds is,
H
x
=
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
A sin (b)e
jb
e
(yb)
e
j(yb)
e
jzz
yb
A sin (y)e
jy
e
jzz
jyjb
A sin (b)e
jb
e
(y+b)
e
j(y+b)
e
jzz
yb
(3.113)
and taking the derivatives lead to,
@H
x
@y
=
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
A ( +j) sin (b)e
jb
e
(+j)y
e
(j)b
e
jzz
yb
A [ cos (y) +j cos (y)]e
jy
e
jzz
jyjb
A ( +j) sin (b)e
jb
e
(+j)y
e
(+j)b
e
jzz
yb
(3.114)
40
The two sets of H
x
with their derivatives are used to match the elds E
x
and
E
z
at the boundaries y =b and y =b. This leads to two eigenvalue equations,
yy
s
sin (b)
s
yy
cos (b) = 0 ; (3.115)
yy
s
cos (b) +
s
yy
sin (b) = 0 : (3.116)
The two eigenvalue equations resulted from the separation of the two forms of the
H
x
eld: the cosine and the sine function. Thus the two eigenvalue equations can
be combined into one equation as,
tan (b) =
2
s
yy
s
yy
(
s
)
2
2
(
yy
)
2
()
2
2
s
yy
2
,
(3.117)
which can be used to solve the propagation constant,
z
, of the guided mode.
Equation (3.117) is the equation for a symmetric anisotropic TM slab waveguide.
3.3.2 Slab Waveguide Innite in Y-Direction
The analysis in this section involves an asymmetric slab waveguide that is innite in
they-direction as shown in Figure 3.4. The asymmetry arises from the fact that the
upper cladding and lower substrate regions are dierent mediums. The derivation
of the symmetric slab waveguide was analyzed previously [40]. The derivation is
extended to analyze the asymmetric slab waveguide case.
As in subsection 3.3.1, the electromagnetic elds are assumed to be propagating
in the +^ z direction in a lossless medium. Using the vector identity Equation (3.20),
assuming no variation along the ^ y - axis, using the permittivity dyadic given by
41
x, x’
y’
z’
z
y
2d
Figure 3.4: Anisotropic dielectric slab waveguide innite in the y-direction.
42
Equation (3.49) and Maxwell's Equations (3.9) and (3.10), the following six equa-
tions can be derived,
H
x
=
1
j!
0
@E
y
@z
; (3.118)
H
y
=
1
j!
0
@E
x
@z
@E
z
@x
; (3.119)
H
z
=
1
j!
0
@E
y
@x
; (3.120)
@H
y
@z
=j!
xx
E
x
; (3.121)
@H
x
@z
@H
z
@x
=j! (
yy
E
y
+
yz
E
z
); (3.122)
@H
y
@x
=j! (
yz
E
y
+
zz
E
z
); (3.123)
where the values of the dyadic permittivity tensors are dened in Equations (3.50)
- (3.54).
The solution of the electromagnetic elds in Section 3.1 for the rectangular
anisotropic dielectric with diagonal permittivity tensor is important, in that it will
allow one to assume the eld solutions for the slab waveguide as a sine and cosine
function. Thus, the assumed solution of the elds in a lossless slab waveguide core
has two types: even and odd. The odd solutions are as follows,
E
xo
=A
xo
cos (x)e
jzz
; (3.124)
E
yo
=A
yo
sin (x)e
jzz
; (3.125)
E
zo
=A
zo
sin (x)e
jzz
; (3.126)
where is the transverse propagation constant in the core region and the subscript
43
`o' indicates an odd mode solution. The even mode solutions in the core are,
E
xe
=A
xe
sin (x)e
jzz
; (3.127)
E
ye
=A
ye
cos (x)e
jzz
; (3.128)
E
ze
=A
ze
cos (x)e
jzz
; (3.129)
where the subscript `e' indicates an even mode solution. It is important to remember
that the following solution will contain ordinary and extraordinary modes as in
Section 3.2, which will be designated with a superscript. Thus the total elds are a
combination of the extraordinary and ordinary elds, in which each have even and
odd modes, respectively,
~
E =
~
E
o
o
+
~
E
o
e
+
~
E
e
o
+
~
E
e
e
; (3.130)
~
H =
~
H
o
o
+
~
H
o
e
+
~
H
e
o
+
~
H
e
e
: (3.131)
Carefully analyzing the eld solutions for the extraordinary and ordinary modes
will yield relationships between the constant coecients A
o
xo
and A
o
yo
in terms of
A
o
zo
, and A
e
xo
and A
e
yo
in terms of A
e
zo
. Similarly, A
o
xe
and A
o
ye
can be described in
terms of A
o
ze
, and A
e
xe
and A
e
ye
can be described in terms of A
e
ze
. Using equation
(3.119) and (3.121) results in,
1
j!
@
2
E
x
@
2
z
+
@
2
E
z
@x@z
=j!
11
E
x
; (3.132)
Plugging in the odd solutions E
xo
and E
zo
into Equation (3.132), one obtains,
A
xo
=
j
z
(
o
)
2
A
zo
; (3.133)
44
where,
(
o
)
2
=!
2
0
11
2
z
: (3.134)
The ordinary wave relationship and the extraordinary wave relationship of the
coecients for the odd modes are,
A
o
xo
=
j
z
o
A
o
zo
; (3.135)
A
e
xo
=
j
z
e
(
o
)
2
A
e
zo
; (3.136)
where,
(
e
)
2
=!
2
33
2
z
sin
2
+
33
11
cos
2
: (3.137)
Plugging in the even solutions E
xe
and E
ze
into Equation (3.132), one obtains,
A
xe
=
j
z
(
o
)
2
A
ze
; (3.138)
and the subsequent ordinary wave relationship and the extraordinary wave rela-
tionship of the coecients of the even modes are,
A
o
xe
=
j
z
o
A
o
ze
; (3.139)
A
e
xe
=
j
z
e
(
o
)
2
A
e
ze
: (3.140)
Using Equations (3.118) and (3.120) results in,
@H
x
@z
@H
z
@x
=j!
(
33
11
) sin
2
+
11
E
y
+ (
11
33
)
sin 2
2
; (3.141)
Plugging in the odd solutions E
yo
and E
zo
and even solutions E
ye
and E
ze
into
45
Equation (3.141) yield similar relationships between the coecients,
A
o
yo
=U
o
y
A
o
zo
; (3.142)
A
e
yo
=
!
2
0
11
(
o
)
2
U
o
y
A
e
zo
; (3.143)
A
o
ye
=U
o
y
A
o
ze
; (3.144)
A
e
ye
=
!
2
0
11
(
o
)
2
U
o
y
A
e
ze
; (3.145)
where,
U
o
y
= cot : (3.146)
It is now instructive to analyze the upper cladding region. The assumed elds
in the cladding region xd are,
E
x
=B
x
e
&x
e
jzz
; (3.147)
E
y
=B
y
e
&x
e
jzz
; (3.148)
E
z
=B
z
e
&x
e
jzz
; (3.149)
where & is the transverse propagation constant in the cladding region.
Though for the titanium indiused waveguide the upper cladding medium is
air, for generality, the cladding medium is assumed to be an anisotropic uniaxial
dielectric crystal with thec-axis not parallel to thez-axis. The permittivity dyadic
in the upper cladding is,
=
xx
^ x^ x +
yy
^ y^ y +
yz
^ y^ z +
zy
^ z^ y +
zz
^ z^ z ; (3.150)
46
It is assumed that the cladding region has the same tilt angle, between thec-axis
and the z-axis, as the core region. Similar to the method used to solve for the
relationship of the coecients in the core region, one can solve for the relationship
of the coecient in the cladding region. Thus, the corresponding relationships are,
B
o
xo
=
j
z
&
o
B
o
zo
; (3.151)
B
e
xo
=
j
z
&
e
(&
o
)
2
B
e
zo
; (3.152)
B
o
xe
=
j
z
&
o
B
o
ze
; (3.153)
B
e
xe
=
j
z
&
e
(&
o
)
2
B
e
ze
; (3.154)
B
o
yo
=U
o
y
B
o
zo
; (3.155)
B
e
yo
=
!
2
0
11
(&
o
)
2
U
o
y
B
e
zo
; (3.156)
B
o
ye
=U
o
y
B
o
ze
; (3.157)
B
e
ye
=
!
2
0
11
(&
o
)
2
U
o
y
B
e
ze
: (3.158)
where,
(&
o
)
2
=
2
z
!
2
0
11
; (3.159)
(&
e
)
2
=
2
z
sin
2
+
33
11
cos
2
!
2
0
33
: (3.160)
It will be useful dene two constants,
U
e
y
=
!
2
0
11
(
o
)
2
U
o
y
,
(3.161)
V
e
y
=
!
2
0
11
(&
o
)
2
U
o
y
.
(3.162)
47
The analysis of the boundary conditions for the core and upper cladding can
be separated into the odd and even mode case. The rst analysis will consider the
odd mode solutions. Knowing Equation (3.119) and the odd mode eld solutions,
H
yo
in the core and upper cladding regions are,
H
core
yo
=
1
j!
0
[(j
z
A
o
xo
+
o
A
o
zo
) cos
o
x + (j
z
A
e
xo
+
e
A
e
zo
) cos
e
x]e
jzz
;
(3.163)
H
uclad
yo
=
1
j!
0
(j
z
B
o
xo
&
o
B
o
zo
)e
&
o
x
] + (j
z
B
e
xo
&
e
B
e
zo
)e
&
e
x
e
jzz
: (3.164)
Knowing Equation (3.120) and the odd mode eld solutions, H
zo
in the core and
upper cladding regions are,
H
core
zo
=
1
j!
0
o
U
o
y
A
o
zo
cos
o
x +
e
U
e
y
A
e
zo
cos
e
x
e
jzz
; (3.165)
H
uclad
zo
=
1
j!
0
&
o
U
o
y
B
o
zo
e
&
o
x
+&
e
V
e
y
B
e
zo
e
&
e
x
e
jzz
: (3.166)
The boundary conditions for the slab waveguide require the tangential elds
E
y
; E
z
; H
y
and H
z
to be continuous across the boundary x = d. The resulting
four equations that satisfy the boundary conditions for the odd modes are,
U
o
y
B
o
zo
e
&
o
d
+V
e
y
B
e
zo
e
&
e
d
=U
o
y
A
o
zo
sin
o
d +U
e
y
A
e
zo
sin
e
d ; (3.167)
B
o
zo
e
&
o
d
+B
e
zo
e
&
e
d
=A
o
zo
sin
o
d +A
e
zo
sin
e
d ; (3.168)
2
z
&
o
B
o
zo
&
o
B
o
zo
e
&
o
d
+
&
e
2
z
(&
o
)
2
B
e
zo
&
e
B
e
zo
e
&
e
d
=
2
z
o
A
o
zo
+
o
A
o
zo
cos
o
d +
e
2
z
(
o
)
2
A
e
zo
+
e
A
e
zo
cos
e
d ;
(3.169)
48
&
o
U
o
y
B
o
zo
e
&
o
d
+&
e
V
e
y
B
e
zo
e
&
e
d
=
o
U
o
y
A
o
zo
cos
o
d
e
U
e
y
A
e
zo
cos
e
d : (3.170)
The previous four equations can be rewritten in matrix form.
2
6
6
6
6
6
6
6
6
6
6
4
a
1
b
1
c
1
d
1
e
1
f
1
g
1
h
1
i
1
j
1
k
1
l
1
m
1
n
1
o
1
p
1
3
7
7
7
7
7
7
7
7
7
7
5
| {z }
P
1
2
6
6
6
6
6
6
6
6
6
6
4
A
o
zo
A
e
zo
B
o
zo
B
e
zo
3
7
7
7
7
7
7
7
7
7
7
5
= 0 ; (3.171)
49
where,
a
1
=U
o
y
sin
o
d ; (3.172)
b
1
=U
e
y
sin
e
d ; (3.173)
c
1
=U
o
y
e
&
o
d
; (3.174)
d
1
=V
e
y
e
&
e
d
; (3.175)
e
1
= sin
o
d ; (3.176)
f
1
= sin
e
d ; (3.177)
g
1
=e
&
o
d
; (3.178)
h
1
=e
&
e
d
; (3.179)
i
1
=
2
z
o
o
cos
o
d ; (3.180)
j
1
=
2
z
e
(
o
)
2
e
cos
e
d ; (3.181)
k
1
=
2
z
&
o
+&
o
e
&
o
d
; (3.182)
l
1
=
2
z
&
e
(&
o
)
2
+&
e
e
&
e
d
; (3.183)
m
1
=
o
U
o
y
cos
o
d ; (3.184)
n
1
=
!
2
11
o
(
o
)
2
U
o
y
cos
e
d ; (3.185)
o
1
=&
o
U
o
y
e
&
o
d
; (3.186)
p
1
=&
e
V
e
y
e
&
e
d
: (3.187)
The solution for matrix Equation (3.171) involves solving the determinant of
50
matrix P
1
and setting it equal to zero. The determinant is,
detP
1
=a
1
[f
1
(k
1
p
1
l
1
o
1
)g
1
(j
1
p
1
l
1
n
1
) +h
1
(j
1
o
1
k
1
n
1
)]
b
1
[e
1
(k
1
p
1
l
1
o
1
)g
1
(i
1
p
1
l
1
m
1
) +h
1
(i
1
o
1
k
1
m
1
)]
+c
1
[e
1
(j
1
p
1
l
1
n
1
)f
1
(i
1
p
1
l
1
m
1
) +h
1
(i
1
n
1
j
1
m
1
)]
d
1
[e
1
(j
1
o
1
k
1
n
1
)f
1
(i
1
o
1
k
1
m
1
) +g
1
(i
1
n
1
j
1
m
1
)] = 0 :
(3.188)
The determinant can be reduced to,
(
o
F sin
o
d +&
o
G cos
o
d) (
11
&
e
G sin
e
d +
11
e
F cos
e
d)
+!
2
11
(
o
)
2
+
11
(&
o
)
2
2
U
o
y
2
cos
o
d sin
e
d = 0 ;
(3.189)
where,
F =
&
o
U
o
y
2
!
2
0
11
; (3.190)
G =
o
U
o
y
2
+!
2
0
11
: (3.191)
Similarly for the even mode solutions, its determinant reduces to,
(
o
F cos
o
d&
o
G sin
o
d) (
11
&
e
G cos
e
d +
11
e
F sin
e
d)
+!
2
11
(
o
)
2
+
11
(&
o
)
2
2
U
o
y
2
cos
e
d sin
o
d = 0 :
(3.192)
The lower substrate medium x d is also assumed to be an anisotropic
uniaxial dielectric crystal with thec-axis not parallel to thez-axis. The permittivity
51
dyadic in the lower substrate is,
=
xx
^ x^ x +
yy
^ y^ y +
yz
^ y^ z +
zy
^ z^ y +
zz
^ z^ z : (3.193)
It is assumed that the dierence between thec-axis and thez-axis in the substrate
region has the same angle , as the core region.
To represent the lower substrate layer the eld solutions are,
E
x
=C
x
e
x
e
jzz
; (3.194)
E
y
=C
y
e
x
e
jzz
; (3.195)
E
z
=C
z
e
x
e
jzz
; (3.196)
where is the transverse propagation constant in the substrate region. The same
method used heretofore can be used to solve for the boundary condition equations
52
at the x =d interface. The relationship between the coecients is as follows,
C
o
xo
=
j
z
o
C
o
zo
; (3.197)
C
e
xo
=
j
z
e
(
o
)
2
C
e
zo
; (3.198)
C
o
xe
=
j
z
o
C
o
ze
; (3.199)
C
e
xe
=
j
z
e
(
o
)
2
C
e
ze
; (3.200)
C
o
yo
=U
o
y
C
o
zo
; (3.201)
C
e
yo
=
!
2
0
11
(
o
)
2
U
o
y
C
e
zo
; (3.202)
C
o
ye
=U
o
y
C
o
ze
; (3.203)
C
e
ye
=
!
2
0
11
(
o
)
2
U
o
y
C
e
ze
; (3.204)
and
(
o
)
2
=
2
z
!
2
0
11
; (3.205)
(
e
)
2
=
2
z
sin
2
+
33
11
cos
2
!
2
0
33
: (3.206)
The boundary conditions are that the tangential eldsE
y
; E
z
; H
z
and H
z
must
be continuous for the boundary at x =d. In the same manner as in the even
mode solutions, the determinant of the odd mode solutions can be reduced to,
(
o
M sin
o
d
o
G cos
o
d) (
11
e
G sin
e
d
11
e
M cos
e
d)
+!
2
0
11
(
o
)
2
+
11
(
o
)
2
2
U
o
y
2
cos
o
d sin
e
d = 0 :
(3.207)
53
Also, the determinant of the even mode solutions is then,
(
o
M cos
o
d +
o
G sin
o
d) (
11
e
G cos
e
d
11
e
M sin
e
d)
+!
2
0
11
(
o
)
2
+
11
(
o
)
2
2
U
o
y
2
cos
e
d sin
o
d = 0 ;
(3.208)
where,
M =
o
U
o
y
2
!
2
0
11
: (3.209)
In summary, the determinantal equations for the even and odd modes atx =d,
namely Equations (3.189) and (3.192), and at x =d, namely Equations (3.207)
and (3.208) must be satised for a guided wave to exist.
One can further note that the determinantal equations reduce to the isotropic
asymmetric slab waveguide case, assuming that,
=
o
=
e
; (3.210)
& =&
o
=&
e
; (3.211)
=
o
=
e
; (3.212)
=
11
=
33
; (3.213)
=
11
=
33
; (3.214)
=
11
=
33
: (3.215)
The odd mode determinantal equations become,
FG (
cosd sind) ( cosd +
sind) = 0 ; (3.216)
MG ( cosd sind) ( cosd + sind) = 0 ; (3.217)
54
and the even mode determinantal equations become,
FG (
sind + cosd) ( sind
cosd) = 0 ; (3.218)
MG ( sind + cosd) ( sind cosd) = 0 : (3.219)
Combining the odd and even mode solutions, namely Equations (3.217) and (3.219),
one can obtain the TE and TM asymmetric isotropic slab waveguide determinantal
solutions. They are,
tand =
( +&)
(
2
&)
,
(3.220)
tand =
(& +)
(
2
2
&)
.
(3.221)
If the symmetric isotropic slab eigenvalue equations are needed, the asymmetric
form may be used with the substitution of =& and =. Then the TE and TM
symmetric isotropic slab waveguide eigenvalue equations are,
tand =
2&
(
2
&
2
)
,
(3.222)
tand =
2&
(
2
2
2
&
2
)
.
(3.223)
The asymmetric and symmetric eigenvalue equations are used to calculate the prop-
agation constant,
z
, of the guided mode.
55
x, x’
y’
z’
z
y
Figure 3.5: Anisotropic dielectric waveguide with unit vector rotation about the
^
x
0
axis,
according to the right hand rule.
3.4 Eective Index Method of Homogeneous, Anisotropic
Dielectric Waveguide
As shown in the previous sections, the propagation constant of the guided mode,
z
, can be obtained for the anisotropic slab waveguides. Furthermore, the eect
of the crystalline axis tilted with respect to the z-axis has been accounted for. In
order to relate the slab waveguides to an indiused waveguide as shown in Figure
3.5, we may use the eective index method [28]. This method considers the 2-D
geometry into separate 1-D geometries with a dominant eld.
For example, if the eective index for E
x
eld is desired, one may consider
E
x
E
y
, and analyze the 1-D geometries, which are the slab waveguide solutions.
56
y
x
n
3
n
3
n
3
n
2
n
1
n
2
n
2
n
2
n
2
y
x
n
eff 3,x
n
eff 1,x
n
eff 2,x
Figure 3.6: Eective index method assuming rst that the anisotropic dielectric slab is
innite in the x-direction, then innite in the y-direction.
The process is shown in Figure 3.6
The method divides the 2-D geometry into four separate 1-D geometries. The
rst three 1-D geometries are symmetric slab waveguides that are innite in the
x-direction. The eective index is calculated for each of the slab waveguides. Since
E
x
is considered the dominant eld, it is necessary to remember that the eect of
the crystalline axis tilt is of no consequence. The ordinary mode eld results and
thus the calculation may be handled as the isotropic case with the permittivity
given by
xx
for each of the regions. To calculate the eective index of each of the
rst three 1-D geometries, the symmetric isotropic TE eigenvalue Equation (3.222),
is used.
Once the eective indices for the three regions are calculated, the fourth 1-D
57
geometry considers a slab waveguide that is innite in the y-direction. The E
x
eld is still considered dominant, and thus, the eect of the crystalline axis tilt
is of no consequence so the slab is isotropic with
xx
. The asymmetric isotropic
TM eigenvalue Equation (3.221) is used. The asymmetry arises from the indiused
waveguide structure with diering upper cladding and lower substrate. The up-
per cladding is typically air, or a dielectric buer layer used for the placement of
electrodes.
If the eective index for E
y
eld is desired, one may consider E
y
E
x
, and
in the same manner, analyze the four slab waveguide geometries. There are two
methods considered here.
The rst method considers the rst three 1-D geometries as symmetric slab
waveguides that are innite in the x-direction. Since E
y
is the dominant eld, the
crystalline axis tilt will have an eect in the calculation. The eective indices are
calculated with the symmetric anisotropic TM eigenvalue Equation (3.117).
Once the eective indices for the three regions are calculated, the fourth 1-D
geometry considers a slab waveguide that is innite in they-direction. The approx-
imation in this analysis considers just accounting for the eect of the crystalline
axis tilt for the rst three 1-D geometries and considers the slab as isotropic for the
fourth slab waveguide. Thus the eective index is calculated with the asymmetric
isotropic TE eigenvalue Equation (3.220). Again, the asymmetry arises from the
dierence between the upper and lower substrates for an indiused waveguide.
58
y
x
n
3
n
3
n
3
n
2
n
1
n
2
n
2
n
2
n
2
y
x
n
eff 2,y
n
eff 1,y
n
eff 3,y
Figure 3.7: Eective index method assuming rst that the anisotropic dielectric slab is
innite in the y-direction, then innite in the x-direction.
The second method considers the rst three 1-D geometries as slab waveguides
that are innite in the y-direction. In this case all four of the determinantal equa-
tions, Equations (3.189), (3.192), (3.207), and (3.208), must be satised for each of
the three slab waveguides.
Once the eective indices for the three regions are calculated, the fourth 1-D
geometry considers a slab waveguide that is innite in the x-direction. Since the
eect of the crystalline axis tilt was accounted for in the rst three slab waveguides,
the fourth slab waveguide is consider isotropic with the calculated eective indices.
In this case to calculate the eective index for the fourth region, the symmetric
isotropic TM eigenvalue Equation (3.223) is used. The process is shown in Figure
3.7.
As an example the wavelength is considered to be = 1:55 m, with the
59
following core permittivity tensor in units of F/m,
c
=
2
6
6
6
6
6
6
4
c
xx
0 0
0
c
yy
c
yz
0
c
zy
c
zz
3
7
7
7
7
7
7
5
; (3.224)
where,
c
xx
= 4:34793 10
11
; (3.225)
c
yy
= 4:16356 10
11
; (3.226)
c
yz
= 1:18591 10
12
; (3.227)
c
zy
=
c
yz
; (3.228)
c
zz
= 4:27165 10
11
; (3.229)
and the substrate permittivity tensor,
s
=
2
6
6
6
6
6
6
4
s
xx
0 0
0
s
yy
s
yz
0
s
zy
s
zz
3
7
7
7
7
7
7
5
; (3.230)
where,
s
xx
= 4:32873 10
11
; (3.231)
s
yy
= 4:12842 10
11
; (3.232)
s
yz
= 1:28846 10
12
; (3.233)
s
zy
=
s
yz
; (3.234)
s
zz
= 4:24585 10
11
: (3.235)
60
Assuming a waveguide width of 8 m and a depth of 6 m , the eective index
of the fundamental E
x
mode is calculated to be 2.2123. The eective index of the
fundamental E
y
mode for the rst method is calculated to be 2.1642 and for the
second method is calculated to be 2.1640. Thus the diering values for the eective
index of the E
y
mode is minimal for the two methods.
3.5 Wave in Homogeneous, Anisotropic Dielectric Wave-
guide
Section 3.4 considered the two-dimensional geometry as one-dimensional slab waveg-
uides. It is possible to consider the two-dimensional geometry from the outset, with
assumptions of the eld solutions. These assumptions cannot be made blindly and
thus the previous analyses of the solutions were necessary to make a concise guess
of the eld solutions.
When the crystalline axis lies at an angle in the yz plane, mode coupling
occurs and the normal modes of the waveguide are hybrid modes [10]. The theo-
retical analysis of such a waveguide is dicult, and typically requires a computer
analysis. In order to fully understand the elds, however, one can study the wave-
guide assuming that the o diagonal terms of the permittivity tensor are small.
The derivation shown here is similar to Marcatili's method for anisotropic dielec-
tric waveguide with no o-diagonal terms in the permittivity tensor [36], [53]. It
61
is analyzed again here so that it may be compared to the eective index method
developed in Section 3.4 for a permittivity tensor with o-diagonal components.
The elds are assumed to be propagating in the +^ z direction with a propagation
term,
e
jzz
; (3.236)
The permittivity tensor is assumed to have no o-diagonal components and thus it
is,
=
2
6
6
6
6
6
6
4
xx
0 0
0
yy
0
0 0
zz
3
7
7
7
7
7
7
5
; (3.237)
Maxwell's equations and the permittivity tensor lead to the following six equations,
H
x
=
1
j!
0
@E
z
@y
+j
z
E
y
,
(3.238)
H
y
=
1
j!
0
j
z
E
x
+
@E
z
@x
,
(3.239)
H
z
=
1
j!
0
@E
y
@x
@E
x
@y
,
(3.240)
E
x
=
1
j!
xx
@H
z
@y
+j
z
H
y
,
(3.241)
E
y
=
1
j!
yy
j
z
H
x
+
@H
z
@x
,
(3.242)
E
z
=
1
j!
zz
@H
y
@x
@H
x
@y
.
(3.243)
62
The elds H
x
, H
y
, H
z
and E
z
can be written in terms of E
x
and E
y
. Then,
H
z
=
1
j!
0
@E
x
@y
@E
y
@x
,
(3.244)
H
x
=
1
!
0
z
@
2
E
x
@x@y
@
2
E
y
@x
2
!
2
0
yy
E
y
,
(3.245)
H
y
=
1
!
0
z
@
2
E
x
@y
2
@
2
E
y
@x@y
+!
2
0
xx
E
x
,
(3.246)
E
z
=
1
j
z
zz
xx
@E
x
@x
+
yy
@E
y
@y
.
(3.247)
The previous equations can be used to obtain equations with onlyE
x
andE
y
terms.
xx
zz
@
2
E
x
@x
2
@
2
E
x
@y
2
+
2
z
E
x
!
2
0
xx
E
x
=
yy
zz
zz
@
2
E
y
@x@y
,
(3.248)
yy
zz
@
2
E
y
@y
2
@
2
E
y
@x
2
+
2
z
E
y
!
2
0
yy
E
y
=
xx
zz
zz
@
2
E
x
@x@y
.
(3.249)
Marcatili's method involves pre-supposing the solution of the elds in all ve of
the regions outlined in Figure 3.8. Region 1 is the core region. Region 3, 4 and 5 are
the lower substrate regions. Region 2 is the upper cladding region. The shadowed
regions are assumed to have elds that are so small that they are negligible.
The assumed elds are then,
E
x;y
=N
x;y
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
cos (k
x
x +) cosk
y
y 1
cos (k
x
b +) cosk
y
ye
c(xb)
2
cos (k
x
x +) cosk
y
ae
y(ya)
3
cos cosk
y
ye
sx
4
cos (k
x
x +) cosk
y
ae
y(y+a)
5
(3.250)
63
1
3
5
2
4
y
x
Figure 3.8: Marcatili's method assumes that the elds in the shaded regions are negligible,
and involves matching boundary conditions between regions 1 & 2, regions 1 & 3, regions
1 & 5, and regions 1 & 4.
Using the assumed elds and the requirement for the boundary conditions that
the tangential elds are continuous, the following equations can be derived for the
boundaries between region 1 and region 2, between region 1 and region 3, and
between region 1 and region 4, respectively,
c
xx
c
zz
xx
zz
[k
x
tan (k
x
a +)
c
]
N
y
N
x
2
=
"
!
2
0
c
yy
yy
!
2
0
yy
k
2
x
#
yy
zz
!
2
0
c
yy
k
2
y
!
2
0
yy
k
2
y
k
x
tan (k
x
a +)
c
yy
c
zz
c
,
(3.251)
s
yy
s
zz
yy
zz
[k
y
tan (k
y
b)
y
]
N
x
N
y
2
=
!
2
0
(
s
xx
xx
)
!
2
0
xx
k
2
x
xx
zz
!
2
0
s
xx
k
2
x
!
2
0
xx
k
2
x
k
y
tan (k
y
b)
s
xx
s
zz
y
,
(3.252)
64
s
xx
s
zz
xx
zz
[k
x
tan +
s
]
N
y
N
x
2
=
"
!
2
0
s
yy
yy
!
2
0
yy
k
2
y
#
yy
zz
!
2
0
s
yy
k
2
y
!
2
0
yy
k
2
y
k
x
tan +
s
yy
s
zz
s
,
(3.253)
where the superscript `c' and `s' represents the upper cladding and lower substrate
regions.
It can now be observed that if E
x
is the dominant eld, then N
x
N
y
and
if E
y
is the dominant eld, then N
y
N
x
. Thus observing the three equations
from the boundary conditions, for E
x
as the dominant eld, the right hand side of
Equation (3.251) and (3.253) should go to zero. By symmetry the left hand side
of Equations (3.252) should also go to zero. An additional approximation assumes
that in comparison the values of k
x
;k
y
and k
z
are small,
!
2
0
c
yy
k
2
y
!
2
0
yy
k
2
y
c
yy
yy
,
(3.254)
!
2
0
s
xx
k
2
x
!
2
0
xx
k
2
x
s
xx
xx
,
(3.255)
!
2
0
s
yy
k
2
y
!
2
0
yy
k
2
y
s
yy
yy
.
(3.256)
With these approximate conditions, the following eigenvalue equations can be de-
rived,
tan (k
y
b) =
y
k
y
,
(3.257)
tan (k
x
a) =
zz
k
x
(
s
zz
c
+
c
zz
s
)
c
zz
s
zz
k
2
x
2
zz
c
s
.
(3.258)
65
Similarly if E
y
is considered the dominant eld then the right hand side of
Equation (3.252) should also go to zero, and by symmetry the left hand side of
Equations (3.251) and (3.253) should also go to zero. These conditions result in
the following eigenvalue equations,
tan (k
y
b) =
zz
y
s
zz
k
y
,
(3.259)
tan (k
x
a) =
k
x
(
c
+
s
)
k
2
x
s
c
.
(3.260)
If we return to the Equations (3.248) and (3.249), and assume that the anisotropy
factors on the right hand side of the equation are small and negligible, the left hand
side of the equations must also equal to zero. Then for theE
x
eld, one can obtain
the following equations,
2
z
=!
2
0
xx
k
2
y
xx
zz
k
2
x
,
(3.261)
2
c
=
c
zz
c
xx
!
2
0
(
xx
c
xx
)
xx
zz
k
2
x
,
(3.262)
2
s
=
s
zz
s
xx
!
2
0
(
xx
s
xx
)
xx
zz
k
2
x
,
(3.263)
2
y
=!
2
0
(
xx
s
xx
)k
2
y
xx
zz
s
xx
s
zz
k
2
x
.
(3.264)
remembering the fact that the relationship with
2
z
diers for the dierent regions.
66
Similarly for the E
y
eld, the equations are,
2
z
=!
2
0
yy
k
2
x
yy
zz
k
2
y
,
(3.265)
2
c
=!
2
0
yy
c
yy
k
2
x
yy
zz
c
yy
c
zz
k
2
y
,
(3.266)
2
y
=
s
zz
s
yy
!
2
0
yy
s
yy
yy
zz
k
2
y
,
(3.267)
2
s
=!
2
0
yy
s
yy
k
2
x
yy
zz
s
yy
s
zz
k
2
y
.
(3.268)
To nd the propagation constant
z
for theE
x
eld, one rst needs to calculate
the value of k
x
using the eigenvalue equation, then knowing k
x
, solve for k
y
and
then calculate
z
. Similarly for the E
y
eld, k
y
is rst determined, then k
x
, and
then
z
can be obtained.
In comparison to the example of Section 3.4, the same permittivity tensor is
used with the o-diagonal terms set to zero. Assuming a waveguide width of 8 m
and a depth of 6 m, the eective index is calculated to be 2.2128 for the E
x
eld
and 2.1648 for the E
y
eld.
The solution that results from the eective index method diers from Marcatili's
method by 5 10
4
for E
x
, and by 6 10
4
and 8 10
4
for the rst and second
method respectively, for E
y
. Although the o-diagonal terms of the permittivity
tensor are small in comparison to the diagonal terms, the eect of the o-diagonal
terms aect the solution of the eective index on the average of 6 10
4
. Since the
smallest refractive index change for titanium indiused LiNbO
3
is approximately
an order of magnitude larger, the discrepancy is still minimal.
67
3.6 Wave in Inhomogeneous, Anisotropic Dielectric Wave-
guide: C-axis Tilted with Respect to Z-Axis
The previous analyses considered a homogenous medium, however, an indiused
waveguide will result in an inhomogeneous medium. Thus the previous analyses
serve as an approximation to the waveguide mode solution for an inhomogeneous
medium.
Due to the complex geometry of an indiused region, the analysis must be
diverted and conducted with the use of a numerical simulation for an arbitrary
permittivity tensor to represent the anisotropy, and an arbitrary permittivity prole
to represent the inhomogeneity. The result is a matrix formulation of the vector
electromagnetic elds, solved as an eigenvalue problem. Since the anisotropy and
inhomogeneity results in large matrices, it requires a long computation time. It has
also been studied previously and thus is not performed here, but will be covered
brie
y to complete the theoretical analysis.
One popular approach is with the use of the nite-dierence method [11]. This
process involves substituting in dierence operators, for the dierentiation opera-
tors of the partial dierential equations of Maxwell's equations. Another popular
method is with the use of the nite-element method [15], [31], [57], which invokes a
variational principle or an equation used to approximate the solution of the partial
dierential equations of Maxwell's equations.
68
The inhomogeneity is described by the diusion prole in the lateral and trans-
verse directions, and depends on the concentration of titanium, the diusion coef-
cients, the diusion time and temperature as discussed in Section 2.1. The nu-
merical simulation would entail calculating the propagation constant for an index
distribution given by combination of the Gaussian and error functions.
The next chapter describes the use of a titanium indiused lithium niobate
waveguide analyzed in this chapter, as a resonant modulator.
69
Chapter 4
FABRY-PEROT RESONANT MODULATOR
4.1 Fabry-Perot Resonator
A typical Fabry-Perot resonator is shown in Figure 4.1. An optical cavity is formed
with two parallel partially transmitting mirrors separated by a distance, L. As
depicted in the Figure 4.1, an incident wave of wavelength (or frequencyf) with
an amplitude a
1
, strikes the partially transmitting mirror, Mirror 1. The wave
is then partially transmitted and partially re
ected. The transmitted amplitude is
jt
1
a
1
, while the re
ected amplitude isr
1
a
1
. The transmitted wave continues toward
Mirror 2 and acquires a total phase of [20],
=
!n2L cos
c
,
(4.1)
where, ! is the radian frequency, n is the refractive index of the medium, c is
the speed of light with a value of 299,792,458 m/s, and is the angle between the
propagation direction of the incident wave and the axis perpendicular to the face
of the mirror.
70
Figure 4.1: Fabry-Perot resonator consists of a medium sandwiched between two partially
transmitting mirrors, such that an electromagnetic wave bounces back and forth within.
At Mirror 2, the wave is again partially transmitted and partially re
ected. The
transmitted wave is (jt
2
) (jt
1
)a
1
e
j=2
and the re
ected wave is (r
2
) (jt
1
)a
1
e
j=2
.
The re
ected wave continues back toward Mirror 1 and accumulates a phase of=2.
The total elds exiting on the left side of Mirror 1 can be summed to represent the
total re
ected wave. Similarly the elds exiting the right side of Mirror 2 can be
summed to represent the total transmitted wave [20].
After summing the terms for the transmitted and re
ected waves, the transmit-
ted intensity can be written as,
I
t
I
i
=
(1r
2
1
) (1r
2
2
)e
2L
(1r
1
r
2
e
2L
)
2
+ 4 (r
1
r
2
e
2L
) sin
2
(=2)
; (4.2)
71
and similarly the re
ected intensity is [13],
I
t
I
i
=
"
r
1
r
2
e
2L
2
+ 4r
1
r
2
e
2L
sin
2
(=2)
(1r
1
r
2
e
2L
)
2
+ 4 (r
1
r
2
e
2L
) sin
2
(=2)
#
; (4.3)
where,
=
!n
eff
2L
c
,
(4.4)
=
4n
eff
L
,
(4.5)
and,
I
i
= incident intensity (W/m
2
) ,
I
t
= transmitted intensity (W/m
2
) ,
I
r
= re
ected intensity (W/m
2
) ,
L = cavity length (m) ,
n
eff
= eective index of the medium ,
= propagation loss (dB/cm) .
As it will be pertinent to the optical device of this dissertation, the refractive
index, n, in Figure 4.1 has been changed to an eective refractive index, n
eff
,
to account for a waveguide structure in the cavity. The design of a Fabry-Perot
resonant waveguide is shown in Figure 4.2.
It is important to remember that for one round trip (from Mirror 1 to Mirror
2 and back to Mirror1), the total length is 2L. One can realize from Figure 4.1,
that one round trip is a multiple of 2 in phase. Knowing the phase that is
72
Figure 4.2: Example of a Fabry-Perot waveguide resonator conguration.
accumulated for one round trip, one can write,
=
!n
eff
2L
c
,
(4.6)
=
4fn
eff
L
c
,
(4.7)
=m 2
,
(4.8)
where m is an integer, and f is the frequency of the input wave.
From the equation for, the frequency spacing between successivem values can
be derived. The frequency separation of the transmission-to-transmission/re
ection-
to-re
ection peaks, is called the free spectral range, FSR, as shown in Figure 4.3.
The loaded quality factorQ
L
of the cavity can also be determined [46]. Q
L
is a
measure of the selectivity of the resonance, and can be dened as,
Q
L
=
p
r
1
r
2
e
2L
2Ln
eff
(1r
1
r
2
e
2L
)
.
(4.9)
The full-width half max,
1=2
, is the separation of the o-resonance frequencies,
when the ratio of re
ected (or transmitted) to incident intensity is equal to a half,
73
Figure 4.3: Example of the re
ected intensity response from a Fabry-Perot resonator.
and,
1=2
=
f
Q
L
.
(4.10)
With the parameters dened heretofore, the nesse,F , of the cavity can also be
determined. The nesse illustrates the relationship between the free spectral range
and the full-width half max as,
F =
FSR
1=2
.
(4.11)
It can be seen that there is an inverse relationship between bandwidth and
sensitivity of a resonator. This limitation is important to understand for modu-
lators, in that, to achieve a broader bandwidth means the slope of the resonance
curve becomes shallower. In terms of modulation, a shallower curve corresponds
to decreased sensitivity. The achievable nesse of the cavity will depend upon the
propagation loss, as shown in Figure 4.4. The condition R
2
= 1:0, implies that the
cavity is operating in the re
ection mode, where no power is transmitted through.
74
Propagation Loss
(dB/cm)
Finesse
0.1
195
R
1
= 0.984, R
2
= 1.0
0.2
97
R
1
= 0.968, R
2
= 1.0
0.3
65
R
1
= 0.953, R
2
= 1.0
0.6
33
R
1
= 0.908, R
2
= 1.0
Figure 4.4: Best achievable nesse with respect to propagation loss for a critically-coupled
Fabry-Perot resonator, assuming that one mirror re
ectivity is equal to unity, operating
wavelength = 1:55 m, the refractive index of the medium is for the extraordinary
mode, n
e
= 2:14856, and FSR = 10 GHz.
75
Thus, if the interest is to make a Fabry-Perot cavity with a nesse of approximately
200, the stringent requirements of achieving a propagation loss of 0.1 dB/cm and
the second mirror re
ectivity of 1.0 must be met.
In the next section, the use of the Fabry-Perot resonance for modulation will be
described. One can imagine that if the optical wave is tuned to a particular spot
(termed the bias point) on the resonance slope, a modulator would then consist of
changing the parameters of the resonant cavity such that the bias point moves up
and down along this slope. The change of the bias point can be correlated to a
change in the detected intensity and thus be used as a modulator.
4.2 Fabry-Perot Resonator with Modulation
This section considers a simple Fabry-Perot resonator utilizing a z-cut titanium
indiused LiNbO
3
waveguide. The derivation here does not apply to the cleaved
edge waveguide, which is an x-cut sample, but is a useful conguration to study
due to its simplicity. The electro-optic eect for the cleaved edge waveguide is
analyzed in Chapter 5. The resonator discussed in this section was fabricated and
the experimental details are described in Section 6.3.
As discussed in the introduction, the electro-optic eect is an observed change
in the refractive index of a material when an electric eld is applied. This eect
has been used to create many devices such as Mach-Zehnders, periodically-poled
devices, and multi-mode interferometers [34], [58], [59].
76
Due to its large electro-optic coecient, LiNbO
3
is a commonly chosen material
for modulators. It has been reported that polymers can obtain larger electro-optic
coecients than LiNbO
3
, however the reported material propagation loss of 2 - 4
dB/cm makes it an infeasible medium for a Fabry-Perot cavity [41].
The analysis of the electro-optic eect is performed by observing the changes in
the index ellipsoid. Since LiNbO
3
is a negative uniaxial anisotropic material, the
value of the index of refraction in a given axis of the crystal will be dierent. The
refractive index tensor for an anisotropic material with only diagonal terms is,
n =
2
6
6
6
6
6
6
4
n
2
11
0 0
0 n
2
22
0
0 0 n
2
33
3
7
7
7
7
7
7
5
; (4.12)
The index ellipsoid can then be written as [63],
x
2
n
2
11
+
y
2
n
2
22
+
z
2
n
2
33
= 1 : (4.13)
In the presence of an electric eld, the index ellipsoid becomes,
1
n
2
11
+r
1k
E
k
x
2
+
1
n
2
22
+r
2k
E
k
y
2
+
1
n
2
33
+r
3k
E
k
z
2
+ 2yzr
4k
E
k
+ 2zxr
5k
E
k
+ 2xyr
6k
E
k
= 1 ;
(4.14)
where k = 1; 2; 3 for the electro-optic coecient dyadic,
r, or for the electric eld
polarization k = x;y;z. The above equation fully describes the response of an
electro-optic material with an applied electric eld. The electro-optic coecient
77
dyadic,
r, for LiNbO
3
is given by [63],
r =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 r
22
r
13
0 r
22
r
13
0 0 r
33
0 r
51
0
r
51
0 0
r
22
0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
; (4.15)
where,
r
12
=r
61
=r
22
; (4.16)
r
23
=r
13
; (4.17)
r
42
=r
51
: (4.18)
For LiNbO
3
, the measured values are [63],
r
22
= 3:4 pm/V ; (4.19)
r
13
= 8:6 pm/V ; (4.20)
r
33
= 30:8 pm/V ; (4.21)
r
51
= 28 pm/V : (4.22)
For LiNbO
3
in the presence of an electric eld, the index ellipsoid becomes,
1
n
2
11
r
22
E
y
+r
13
E
z
x
2
+
1
n
2
11
+r
22
E
y
+r
13
E
z
y
2
+
1
n
2
33
+r
33
E
z
z
2
+ 2yzr
51
E
y
+ 2zxr
51
E
x
+ 2xyr
61
E
x
= 1 ;
(4.23)
78
where n
2
22
=n
2
11
for a uniaxial dielectric crystal.
If the applied electric eld is
~
E = ^ zE
z
then the index ellipsoid becomes,
1
n
2
11
+r
13
E
z
x
2
+
1
n
2
11
+r
13
E
z
y
2
+
1
n
2
33
+r
33
E
z
z
2
= 1 : (4.24)
The values n
11
and n
33
are the ordinary and extraordinary refractive indices, re-
spectively.
Observing Equation (4.24), an alternate refractive index can be dened. The
alternate refractive index includes the change due to the electric eld as follows,
1
n
2
x
=
1
n
2
o
+r
13
E
z
n
x
=n
o
1
2
r
13
n
3
o
E
z
; (4.25)
1
n
2
y
=
1
n
2
o
+r
13
E
z
=) n
y
=n
o
1
2
r
13
n
3
o
E
z
; (4.26)
1
n
2
z
=
1
n
2
e
+r
33
E
z
n
z
=n
e
1
2
r
33
n
3
e
E
z
; (4.27)
assuming that r
13
n
2
o
E
z
1 and r
33
n
2
e
E
z
1:
Sincer
33
is the largest electro-optic coecient for LiNbO
3
, it is then desirable to
polarize the optical wave such that it is aected by the extraordinary refractive in-
dex . In this way, one can obtain a larger modulation response in comparison to the
ordinary wave case, where it possesses a lower electro-optic coecient. Figure 4.5
shows the orientation for the maximum modulation response of the z-cut LiNbO
3
crystal in relation to the applied electric eld, E
z
. The polarization of the optical
wave is in the ^ z direction. The polarization was chosen so as to obtain a wave that
will be aected by the extraordinary refractive index. The -z-axis is noted because
79
-z
-x
y
Applied E field
Optical
wave
E
z
Figure 4.5: Schematic for the desired orientation of a titanium indiused z-cut LiNbO
3
Fabry-Perot resonant modulator, in relation to the RF electric eld and the optical wave.
titanium indiusion for z-cut LiNbO
3
is typically performed on the -z face due to
the ferroelectric nature, and the ease of domain reversal on the +z face.
The eect of the change in refractive index can be seen as a phase change in
the optical wave. The phase term exp (j
z
L) of the optical wave, where
z
=
2n
,
(4.28)
and L is the length, is modied to be,
exp (j
z
L) = exp
j
2
n
e
1
2
r
33
n
3
e
E
z
L
;
= exp
j
2
n
e
L
r
33
n
3
e
E
z
L
: (4.29)
From Equation (4.29), it can be seen that the added phase term is (=)r
33
n
3
e
E
z
L.
Thus to obtain a phase shift of , the added phase term is set equal to . The
80
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P
o
P
o
RF
E
M
RF
λ
r
i
I
I
Figure 4.6: Example of modulation performed on a resonance curve.
necessary electric eld for a phase shift of is then,
E
z
=
r
33
n
3
e
L
,
(4.30)
where is the overlap factor between the optical and electric eld [2],[46].
Figure 4.6 shows a typical example of the modulated response of a resonant
cavity. If one were interested in modulating a signal with a resonant cavity, a bias
point along the slope of the resonance curve should be chosen. The maximum
sensitivity occurs, when the bias point is chosen at the maximum slope of the
curve. In terms of the parameters discussed in Section 4.1, it is necessary to nd
the maximum ofd (I
t
=I
i
)=d. As the RF electric eld is applied to the electro-optic
cavity, the optical wave inside the resonator will experience a phase change. This is
81
relatable to a shift in the resonance, and can be detected as the dierence between
the bias point and the modulated intensity seen on the detector.
The Fabry-Perot resonator contains resonances at integer values of the FSR as
was seen in Section 4.1. If an RF carrier wave, f
RF
, modulates the optical wave in
the resonant cavity, the resulting optical sub carrier frequencies are
opt
f
RF
. If
f
RF
=FSR, then the RF carrier wave appears as a modulated optical sub-carrier
at the rstFSR. If then, the RF carrier wave contains modulation side bands, the
side bands will contribute as optical base band modulation about each sub-carrier
frequency [12]. It is then possible to operate the Fabry-Perot resonator at integer
values of the FSR, however the placement of the electrodes diers for the desired
frequencies.
The analysis of the electro-optic eect from this section will be revisited in
Chapter 5 where the cleaved edge waveguide geometry is considered.
82
Chapter 5
THEORETICAL ANALYSIS OF THE ELECTRO-OPTIC
EFFECT FOR CLEAVED EDGE WAVEGUIDES
This chapter will revisit the analysis of the electro-optic eect from Section 4.2, for
the case of the cleaved edge waveguide geometry. Thus, the most general form of
the index ellipsoid and the electro-optic coecients will be needed.
In the presence of an applied electric eld, the general form of the index ellipsoid
of a dielectric crystal is [64],
1
n
2
x
+r
11k
E
k
x
2
+
1
n
2
y
+r
22k
E
k
y
2
+
1
n
2
z
+r
33k
E
k
z
2
+ 2xyr
12k
E
k
+ 2yzr
23k
E
k
+ 2zxr
13k
E
k
= 1 :
(5.1)
where k = 1; 2; 3 or in Cartesian coordinates k =x;y;z. The general form for the
83
electro-optic coecient tensor has 27 terms,
r
ijk
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
r
111
r
112
r
113
r
221
r
222
r
223
r
331
r
332
r
333
r
231
r
232
r
233
r
321
r
322
r
323
r
131
r
132
r
133
r
311
r
312
r
313
r
121
r
122
r
123
r
211
r
212
r
213
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: (5.2)
Assuming a lossless, and optically inactive medium, a symmetric relationship
between the coecients allows one to reduce the tensor components with contracted
indices. Thus,
r
1k
=r
11k
; (5.3)
r
2k
=r
22k
; (5.4)
r
3k
=r
33k
; (5.5)
r
4k
=r
23k
=r
32k
; (5.6)
r
5k
=r
13k
=r
31k
; (5.7)
r
6k
=r
12k
=r
21k
: (5.8)
The general form of the electro-optic coecient tensor with contracted indices
84
is then,
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
r
11
r
12
r
13
r
21
r
22
r
23
r
31
r
32
r
33
r
41
r
42
r
43
r
51
r
52
r
53
r
61
r
62
r
63
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: (5.9)
where for LiNbO
3
it is known that because of its 3m symmetry class the electro-
optic coecient becomes,
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
0 r
22
r
13
0 r
22
r
13
0 0 r
33
0 r
51
0
r
51
0 0
r
22
0 0
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: (5.10)
If one considers the waveguide design for the cleaved end faces, a new electro-
optic coecient tensor must be calculated with the inclusion of the tilted axes. To
analyze this, it is easier to return to the general tensor form for LiNbO
3
without
85
the contracted indices. The electro-optic tensor is then,
r
ijk
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
0 r
112
r
113
0 r
222
r
223
0 0 r
333
0 r
232
0
0 r
322
0
r
131
0 0
r
311
0 0
r
121
0 0
r
211
0 0
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: (5.11)
From the analysis of the resonator with cleaved edges, there are two dierent
waveguide orientations for the resonance of the TM-like and the TE-like mode, that
are oset by an angle
e
. Thus, the waveguides are designed such that the axis
of the waveguides are parallel to the respective Poynting vectors,
^
S
o
and
^
S
e
, for
the ordinary and extraordinary wave. A waveguide aligned parallel to its Poynting
vector will conne the light in the waveguide and also satisfy the resonance condition
that the propagation vector be perpendicular to the waveguide edge facet. Since
the Poynting vector of the extraordinary wave is at an angle with respect to the
propagation direction, the waveguide must be tilted. For the ordinary wave, a tilt
is not necessary. Figure 5.1 shows a possible conguration of electrode placement
on the cleaved edge waveguides.
86
Cleaved-Edge Waveguide Design
Top View of Waveguide
Case 1: Ordinary wave
Case 2: Extraordinary wave
z
s
Top View of Waveguide
y
s
Tilted waveguide
cleaved facet
cleaved facet
cleaved facet
cleaved facet
x
s
x
t
z
t
y
t
c-axis
c-axis
Figure 5.1: Possible electrode placement for the straight and tilted waveguides with
cleaved edges. The y
s
- and z
s
-directions are parallel and perpendicular to the straight
waveguide axis. The y
t
- and z
t
-directions are parallel and perpendicular to the tilted
waveguide axis. The yellow rectangles represent electrodes.
87
ˆ z
ˆ
t
z
ˆ y
ˆ
t
y
s
ˆ
s
y
ˆ
s
z
s
t
t
Figure 5.2: Unit vector rotation of angles
s
and
t
about the ^ x axis, according to the
right hand rule. The y
s
- and z
s
-directions are parallel and perpendicular to the straight
waveguide axis. The y
t
- and z
t
-directions are parallel and perpendicular to the tilted
waveguide axis.
Now, it will be useful to calculate the eect of a coordinate transformation on the
electro-optic coecients, if one is interested in placing electrodes on the waveguides.
Figure 5.2 shows the two angles that are needed to describe the two orientations
of the waveguides. The angle,
s
, represents the coordinate transformation for the
TM-like mode or the straight waveguide. The superscript `s' is used to describe
all that pertains to the straight waveguide coordinate transformation. Similarly,
the angle,
t
, represents the coordinate transformation for the TE-like mode or the
tilted waveguide. The superscript `t' is used to describe all that pertains to the
tilted waveguide coordinate transformation.
The ^ y and ^ z axis are the crystallographic coordinates. The coordinates with
superscript `s' are the coordinates to describe the TM-like mode of the cleaved
edge waveguide, where ^ z
s
is parallel to the cleaved edge, and ^ y
s
is perpendicular to
88
the cleaved edge and parallel to the TM-like mode waveguide axis. The coordinates
with superscript `t' are the coordinates to describe the TE-like mode, where ^ z
t
and
^ y
t
are the axes that are perpendicular and parallel to the TE-like mode waveguide
axis, respectively.
The unit vector transformations from the crystallographic coordinates to the
straight waveguide coordinates is a rotation about the ^ x axis according to the
right-hand rule,
^
x
s
= ^ x ; (5.12)
^
y
s
= ^ y cos
s
^ z sin
s
; (5.13)
^
z
s
= ^ y sin
s
+ ^ z cos
s
; (5.14)
where it will be useful to dene,
a
22
= cos
s
; (5.15)
a
23
= sin
s
: (5.16)
Similarly, the unit vector transformations from the crystallographic coordinates
to the tilted waveguide coordinates is a rotation about the ^ x axis according to the
right-hand rule,
^
x
t
= ^ x ; (5.17)
^
y
t
= ^ y cos
t
^ z sin
t
; (5.18)
^
z
t
= ^ y sin
t
+ ^ z cos
t
; (5.19)
89
where it will be useful to dene,
b
22
= cos
t
; (5.20)
b
23
= sin
t
: (5.21)
It is the necessary to calculate the values of the electro-optic coecients in the
straight waveguide coordinates, r
s
ijk
, and in the tilted waveguide coordinates, r
t
ijk
.
The general form for the electro-optic tensor, r
ijk
, for LiNbO
3
in crystallographic
coordinates can be written as,
r
ijk
=r
112
^ x^ x^ y +r
113
^ x^ x^ z +r
222
^ y^ y^ y +r
223
^ y^ y^ z +r
333
^ z^ z^ z +r
232
^ y^ z^ y
+r
322
^ z^ y^ y +r
131
^ x^ z^ x +r
311
^ z^ x^ x +r
121
^ x^ y^ x +r
211
^ y^ x^ x :
(5.22)
Using the transformation denitions from the crystallographic coordinates to the
straight waveguide coordinates, r
s
ijk
can be calculated to be,
r
s
ijk
=r
s
112
^
x
s
^
x
s
^
y
s
+r
s
113
^
x
s
^
x
s
^
z
s
+r
s
222
^
y
s
^
y
s
^
y
s
+r
s
223
^
y
s
^
y
s
^
z
s
+r
s
332
^
z
s
^
z
s
^
y
s
+r
s
333
^
z
s
^
z
s
^
z
s
+r
s
232
^
y
s
^
z
s
^
y
s
+r
s
233
^
y
s
^
z
s
^
z
s
+r
s
322
^
z
s
^
y
s
^
y
s
+r
s
323
^
z
s
^
y
s
^
z
s
+r
s
131
^
x
s
^
z
s
^
x
s
+r
s
311
^
z
s
^
x
s
^
x
s
+r
s
121
^
x
s
^
y
s
^
x
s
+r
s
211
^
y
s
^
x
s
^
x
s
;
(5.23)
90
where,
r
s
112
=r
112
a
22
+r
113
a
23
; (5.24)
r
s
113
=r
112
a
23
+r
113
a
22
; (5.25)
r
s
222
=r
222
a
3
22
+r
223
a
2
22
a
23
+r
232
a
2
22
a
23
+r
322
a
2
22
a
23
+r
333
a
3
23
; (5.26)
r
s
223
=r
222
a
2
22
a
23
+r
223
a
3
22
r
232
a
22
a
2
23
r
322
a
22
a
2
23
+r
333
a
22
a
2
23
; (5.27)
r
s
332
=r
222
a
22
a
2
23
+r
223
a
3
23
r
232
a
2
22
a
23
r
322
a
2
22
a
23
+r
333
a
2
22
a
23
; (5.28)
r
s
333
=r
222
a
3
23
+r
223
a
22
a
2
23
+r
232
a
22
a
2
23
+r
322
a
22
a
2
23
+r
333
a
3
22
; (5.29)
r
s
232
=r
222
a
2
22
a
23
r
223
a
22
a
2
23
+r
232
a
3
22
r
322
a
22
a
2
23
+r
333
a
22
a
2
23
; (5.30)
r
s
233
=r
222
a
22
a
2
23
r
223
a
2
22
a
23
r
232
a
2
22
a
23
+r
322
a
3
23
+r
333
a
2
22
a
23
; (5.31)
r
s
322
=r
222
a
2
22
a
23
r
223
a
22
a
2
23
r
232
a
22
a
2
23
+r
322
a
3
22
+r
333
a
22
a
2
23
; (5.32)
r
s
323
=r
222
a
22
a
2
23
r
223
a
2
22
a
23
+r
232
a
3
23
r
322
a
2
22
a
23
+r
333
a
2
22
a
23
; (5.33)
r
s
131
=r
131
a
22
r
121
a
23
; (5.34)
r
s
311
=r
311
a
22
r
211
a
23
; (5.35)
r
s
121
=r
121
a
22
+r
131
a
23
; (5.36)
r
s
211
=r
211
a
22
+r
311
a
23
: (5.37)
In matrix form the electro-optic coecient tensor rotated about the ^ x-axis by an
91
angle
s
is then,
r
s
ijk
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
0 r
s
112
r
s
113
0 r
s
222
r
s
223
0 r
s
332
r
s
333
0 r
s
232
r
s
233
0 r
s
322
r
s
323
r
s
131
0 0
r
s
311
0 0
r
s
121
0 0
r
s
211
0 0
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: (5.38)
Similarly using the coordinate transformation from the crystallographic coor-
dinates to the tilted waveguide coordinates, the electro-optic coecient tensor be-
comes,
r
t
ijk
=r
t
112
^
x
t ^
x
t ^
y
t
+r
t
113
^
x
t ^
x
t ^
z
t
+r
t
222
^
y
t ^
y
t ^
y
t
+r
t
223
^
y
t ^
y
t ^
z
t
+r
t
332
^
z
t ^
z
t ^
y
t
+r
t
333
^
z
t ^
z
t ^
z
t
+r
t
232
^
y
t ^
z
t ^
y
t
+r
t
233
^
y
t ^
z
t ^
z
t
+r
t
322
^
z
t ^
y
t ^
y
t
+r
t
323
^
z
t ^
y
t ^
z
t
+r
t
131
^
x
t ^
z
t ^
x
t
+r
t
311
^
z
t ^
x
t ^
x
t
+r
t
121
^
x
t ^
y
t ^
x
t
+r
t
211
^
y
t ^
x
t ^
x
t
;
(5.39)
92
where,
r
t
112
=r
112
b
22
+r
113
b
23
; (5.40)
r
t
113
=r
112
b
23
+r
113
b
22
; (5.41)
r
t
222
=r
222
b
3
22
+r
223
b
2
22
b
23
+r
232
b
2
22
b
23
+r
322
b
2
22
b
23
+r
333
b
3
23
; (5.42)
r
t
223
=r
222
b
2
22
b
23
+r
223
b
3
22
r
232
b
22
b
2
23
r
322
b
22
b
2
23
+r
333
b
22
b
2
23
; (5.43)
r
t
332
=r
222
b
22
b
2
23
+r
223
b
3
23
r
232
b
2
22
b
23
r
322
b
2
22
b
23
+r
333
b
2
22
b
23
; (5.44)
r
t
333
=r
222
b
3
23
+r
223
b
22
b
2
23
+r
232
b
22
b
2
23
+r
322
b
22
b
2
23
+r
333
b
3
22
; (5.45)
r
t
232
=r
222
b
2
22
b
23
r
223
b
22
b
2
23
+r
232
b
3
22
r
322
b
22
b
2
23
+r
333
b
22
b
2
23
; (5.46)
r
t
233
=r
222
b
22
b
2
23
r
223
b
2
22
b
23
r
232
b
2
22
b
23
+r
322
b
3
23
+r
333
b
2
22
b
23
; (5.47)
r
t
322
=r
222
b
2
22
b
23
r
223
b
22
b
2
23
r
232
b
22
b
2
23
+r
322
b
3
22
+r
333
b
22
b
2
23
; (5.48)
r
t
323
=r
222
b
22
b
2
23
r
223
b
2
22
b
23
+r
232
b
3
23
r
322
b
2
22
b
23
+r
333
b
2
22
b
23
; (5.49)
r
t
131
=r
131
b
22
r
121
b
23
; (5.50)
r
t
311
=r
311
b
22
r
211
b
23
; (5.51)
r
t
121
=r
121
b
22
+r
131
b
23
; (5.52)
r
t
211
=r
211
b
22
+r
311
b
23
: (5.53)
In matrix form the electro-optic coecient tensor rotated about the ^ x-axis by an
93
angle
t
is then,
r
t
ijk
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
0 r
t
112
r
t
113
0 r
t
222
r
t
223
0 r
t
332
r
t
333
0 r
t
232
r
t
233
0 r
t
322
r
t
323
r
t
131
0 0
r
t
311
0 0
r
t
121
0 0
r
t
211
0 0
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
: (5.54)
The electro-optic coecients of LiNbO
3
are known values and repeated again
here [63],
r
12
=r
61
=r
22
; (5.55)
r
23
=r
13
; (5.56)
r
42
=r
51
: (5.57)
and,
r
22
= 3:4 pm/V ; (5.58)
r
13
= 8:6 pm/V ; (5.59)
r
33
= 30:8 pm/V ; (5.60)
r
51
= 28 pm/V : (5.61)
94
Relating the known values of the electro-optic coecients in contracted form
with those in the non-contracted form, for the crystallographic coordinates, one
obtains,
r
112
=r
22
; (5.62)
r
113
=r
13
; (5.63)
r
222
=r
22
; (5.64)
r
223
=r
13
; (5.65)
r
333
=r
33
; (5.66)
r
232
=r
51
; (5.67)
r
322
=r
51
; (5.68)
r
131
=r
51
; (5.69)
r
311
=r
51
; (5.70)
r
121
=r
22
; (5.71)
r
211
=r
22
: (5.72)
For the cleaved edge waveguide geometry,
s
= 32:75
and
t
= 30:99
, assum-
95
ing
e
= 1:76
, as calculated in Section 3.2. This results in,
a
22
= 0:8410 ; (5.73)
a
23
= 0:5410 ; (5.74)
b
22
= 0:8573 ; (5.75)
b
23
= 0:5149 : (5.76)
96
The resulting values for the electro-optic coecients in units of pm/V are then,
r
s
112
= 1:7928 r
t
112
= 1:5134 ; (5.77)
r
s
113
= 9:0722 r
t
113
= 9:1230 ; (5.78)
r
s
222
= 31:6185 r
t
222
= 30:7900 ; (5.79)
r
s
223
=2:3874 r
t
223
=1:5957 ; (5.80)
r
s
332
=7:4446 r
t
332
=7:5887 ; (5.81)
r
s
333
= 33:6850 r
t
333
= 33:6211 ; (5.82)
r
s
232
= 13:9288 r
t
232
= 15:0350 ; (5.83)
r
s
233
= 3:0503 r
t
233
= 2:4001 ; (5.84)
r
s
322
= 13:9288 r
t
322
= 15:0350 ; (5.85)
r
s
323
= 3:0503 r
t
323
= 2:4001 ; (5.86)
r
s
131
= 25:3884 r
t
131
= 25:7538 ; (5.87)
r
s
311
= 25:3884 r
t
311
= 25:7538 ; (5.88)
r
s
121
= 12:2878 r
t
121
= 11:5022 ; (5.89)
r
s
211
= 12:2878 r
t
211
= 11:5022 : (5.90)
In order to fully understand the eect of an applied electric eld on an electro-
optic crystal, it is instructive to observe the change in the index ellipsoid. The index
ellipsoid allows one to visualize the dierent values of the refractive index with
regard to a coordinate axis, for a uniaxial dielectric crystal. The index ellipsoid
97
equation is used to calculate the change in refractive index due to an applied electric
eld. Assuming either the straight waveguide or tilted waveguide transformation
and the subsequent electro-optic coecients that arise, and using Equation (5.1),
the index ellipsoid for an arbitrary applied electric eld is,
"
1
n
t;s
x
2
+r
t;s
112
E
t;s
y
+r
t;s
113
E
t;s
z
#
x
t;s
2
+
"
1
n
t;s
y
2
+r
t;s
222
E
t;s
y
+r
t;s
223
E
t;s
z
#
y
t;s
2
+
"
1
n
t;s
z
2
+r
t;s
332
E
t;s
y
+r
t;s
333
E
t;s
z
#
z
t;s
2
+ 2x
t;s
y
t;s
r
t;s
121
E
t;s
x
+ 2y
t;s
z
t;s
r
t;s
232
E
t;s
y
+ 2y
t;s
z
t;s
r
t;s
233
E
t;s
z
+ 2z
t;s
x
t;s
r
t;s
131
E
t;s
x
= 1 ;
(5.91)
where the superscript `s' and `t' represent values for the straight and tilted wave-
guide coordinates respectively.
In order to modulate an electro-optic crystal, there are a couple typical place-
ments of the electrodes. Furthermore, depending on the orientation of the electric
eld, the dierent values of the electro-optic tensor may result in dierence in mod-
ulation strength. It is vital to notice that the largest electro-optic coecient is
r
333
for both the straight and the tilted waveguide. In the typical z-cut LiNbO
3
waveguide, ther
333
coecient appears as a modulation eect for the extraordinary
wave, when the dielectric crystal has an applied electric eld in the ^ z direction as
was described in Section 4.2. Thus, one can postulate that the tilted waveguide
geometry (which exhibits an extraordinary wave resonance), may have a similar
modulation eect whereby the r
t
333
coecient may appear. To analyze this, the
98
index ellipsoid for the tilted waveguide is observed when a ^ z polarized electric eld,
E
t
z
, is present. In this case, the index ellipsoid becomes,
1
(n
t
x
)
2
+r
t
113
E
t
z
x
t
2
+
"
1
n
t
y
2
+r
t
223
E
t
z
#
y
t
2
+
1
(n
t
z
)
2
+r
t
333
E
t
z
z
t
2
+ 2y
t
z
t
r
t
233
E
t
z
= 1 :
(5.92)
To observe the change in refractive index, it is necessary to rotate the coordi-
nates such that the mixed terms of the index ellipsoid disappear. This is also called
matrix diagonalization. In order to eliminate the mixed terms, one must rotate to
the coordinates
x
f
;y
f
;z
f
by the following relations,
x
t
=x
f
; (5.93)
y
t
=y
f
cos
f
z
f
sin
f
; (5.94)
z
t
=y
f
sin
f
+z
f
cos
f
; (5.95)
as shown in Figure 5.3. The rotation is about the
^
x
t
axis according to the right-hand
rule. Then the index ellipsoid equation becomes,
C
1
x
f
2
+
C
2
cos
2
f
+C
3
sin
2
f
+ 2r
t
233
E
t
z
sin
f
cos
f
y
f
2
+
C
2
sin
2
f
+C
3
cos
2
f
2r
t
233
E
t
z
sin
f
cos
f
z
f
2
+
C
2
sin
f
cos
f
+C
3
sin
f
cos
f
+ 2r
t
233
E
t
z
cos
2
f
y
f
z
f
+
C
2
sin
f
cos
f
+C
3
sin
f
cos
f
2r
t
233
E
t
z
cos
2
f
z
f
y
f
= 1 ;
(5.96)
99
ˆ
t
z
ˆ
f
z
ˆ
t
y
ˆ
f
y
f
f
Figure 5.3: Unit vector rotation of angle
f
about the
^
x
t
axis, according to the right
hand rule.
where,
C
1
=
1
(n
t
x
)
2
+r
t
113
E
t
z
; (5.97)
C
2
=
1
n
t
y
2
+r
t
223
E
t
z
; (5.98)
C
3
=
1
(n
t
z
)
2
+r
t
333
E
t
z
: (5.99)
There is a particular angle
f
, which will allow the mixed-axes term to disap-
pear, leaving only the principle axes. This can be found by observing solely the
mixed-axes terms and equating them to zero, as below,
2
"
1
n
t
y
2
+r
t
223
E
t
z
#
sin
f
cos
f
+ 2
1
(n
t
z
)
2
+r
t
333
E
t
z
sin
f
cos
f
+ 2r
t
233
E
t
z
cos
2
f
sin
2
f
= 0 : (5.100)
100
The resulting angle
f
, is then,
tan
2
f
= 2r
t
233
E
t
z
"
1
n
t
y
2
1
(n
t
z
)
2
+
r
t
223
r
t
333
E
t
z
#
1
;
=
2r
t
233
E
t
z
n
t
y
2
(n
t
z
)
2
(n
t
z
)
2
n
t
y
2
+ (r
t
223
r
t
333
)E
t
z
n
t
y
2
(n
t
z
)
2
.
(5.101)
The
y
f
2
term of Equation (5.96) can then be rewritten,
"
1
n
t
y
2
+r
t
223
E
t
z
#
cos
2
f
+
1
(n
t
z
)
2
+r
t
333
E
t
z
sin
2
f
+ 2r
t
233
E
t
z
sin
f
cos
f
=
"
1
n
t
y
2
+r
t
223
E
t
z
#
"
1
n
t
y
2
1
(n
t
z
)
2
+
r
t
223
r
t
333
E
t
z
#
sin
2
f
+ 2r
t
233
E
t
z
sin
f
cos
f
;
=
1
n
t
y
2
+r
t
223
E
t
z
2r
t
233
E
t
z
sin
2
f
tan (2
f
)
cos
t
sin
f
;
=
1
n
t
y
2
+r
t
223
E
t
z
+r
t
233
E
t
z
tan
f
: (5.102)
Similarly the (z
t
)
2
term becomes,
"
1
n
t
y
2
+r
t
223
E
t
z
#
sin
2
f
+
1
(n
t
z
)
2
+r
t
333
E
t
z
cos
2
f
2r
t
233
E
t
z
sin
f
cos
f
=
1
(n
t
z
)
2
+r
t
333
E
t
z
r
t
233
E
t
z
tan
f
: (5.103)
The resulting equation for the index ellipsoid with the elimination of the mixed
coordinate terms is then,
1
(n
t
x
)
2
+r
t
113
E
t
z
x
t
2
+
"
1
n
t
y
2
+r
t
223
E
t
z
+r
t
233
E
t
z
tan
f
#
y
t
2
+
1
(n
t
z
)
2
+r
t
333
E
t
z
r
t
233
E
t
z
tan
f
z
t
2
= 1 ; (5.104)
101
where
f
is given by the value obtained in Equation (5.101) .
It is now possible to observe each of the square-bracketed terms individually.
The (x
t
)
2
term can be rewritten to represent a refractive index value for the x
f
coordinate.
1
n
f
x
2
=
1
(n
t
x
)
2
+r
t
113
E
t
z
; (5.105)
=
1 +r
t
113
(n
t
x
)
2
E
t
z
(n
t
x
)
2
.
(5.106)
Then, using the binomial approximation and assuming that r
t
113
(n
t
x
)
2
E
t
z
1,
n
f
x
2
=n
t
x
h
1 +r
t
113
n
t
x
2
E
t
z
i
1
2
; (5.107)
n
f
x
=n
t
x
1
2
r
t
113
n
t
x
3
E
t
z
: (5.108)
In the same manner the following refractive index values can be obtained for the
y
f
and z
f
coordinates, assuming that r
t
223
n
t
y
2
E
t
z
+r
t
233
n
t
y
2
E
t
z
tan
f
1, and
r
t
333
(n
t
z
)
2
E
t
z
r
t
233
(n
t
z
)
2
E
t
z
tan
f
1,
n
f
y
=n
t
y
1
2
n
t
y
3
E
t
z
r
t
223
+r
t
233
tan
f
; (5.109)
n
f
z
=n
t
z
1
2
n
t
z
3
E
t
z
r
t
333
r
t
233
tan
f
: (5.110)
It is now possible to compare the dierence in performance between the tilted
geometry and the typicalz-cut conguration as shown in Figure 4.5. The same unit
vector transformation that has been used to transform the electro-optic coecients
can be used to calculate the refractive index in the tilted waveguide coordinates.
102
By the usual nomenclature n
e
=n
33
which is the refractive index along the z-axis.
For the cleaved edge waveguide, the tilted waveguide coordinates were dened as a
unit vector transformation by an angle
t
= 30:99
. Then, assuming = 1:55 m,
the value for n
t
33
can be calculated,
n
11
= 2:21601 ; (5.111)
n
33
= 2:14856 ; (5.112)
11
=
0
n
2
11
= 4:34793 10
11
; (5.113)
33
=
0
n
2
33
= 4:08728 10
11
; (5.114)
t
33
=
t
zz
= (
33
11
) cos
2
t
+
11
= 4:15638 10
11
; (5.115)
n
t
z
=n
t
33
=
s
t
33
0
= 2:16665 : (5.116)
Similarly,
t
22
=
yy
= (
33
11
) sin
2
t
+
11
= 4:27883 10
11
; (5.117)
n
t
y
=n
t
22
=
s
t
22
0
= 2:19833 : (5.118)
Assuming the applied electric eld is E
t
z
= 10
6
V/m, then
t
=3:91 10
4
rad from Equation (5.101). The refractive index change is then,
n
f
z
=n
t
z
1:71 10
4
; (5.119)
which shows a modulation response that is on par with the typical z-cut congu-
103
ration with,
n
z
=n
e
1:53 10
4
: (5.120)
Thus the cleaved edge waveguide can be used as a modulator with the extraordinary
mode, with a comparable performance to the typicalz-cut conguration, even with
the transformed electro-optic and refractive index components.
For the TM-like mode, the waveguide is oriented such that the waveguide axis is
perpendicular to the cleaved face. The resulting coordinate transformation would
be to transform the crystallographic axes to the straight waveguide coordinates. In
this case, the ordinary mode is resonant and hence, the electrodes must be oriented
such that the applied electric eld has an
^
x
s
component. In such a case the index
ellipsoid becomes,
1
(n
s
x
)
2
(x
s
)
2
+
"
1
n
s
y
2
#
(y
s
)
2
+
1
(n
s
z
)
2
(z
s
)
2
+ 2x
s
y
s
r
s
121
E
s
x
+ 2x
s
z
s
r
s
131
E
s
x
= 1 :
(5.121)
There are two mixed coordinate terms containing all three axes and thus at a
minimum, two rotations will be needed to minimized the mixed coordinate terms.
The rst rotation will be in the (x
s
;z
s
) plane as shown in Figure 5.4. The rotation
occurs about the ^ y axis according to the right-hand rule. The unit vector relations
104
ˆ
s
z
ˆ' z
ˆ' x
f
ˆ
s
x
f
Figure 5.4: Unit vector rotation with angle
f
about the ^ y axis according to right-hand
rule.
are,
x
s
=x
0
cos
f
z
0
sin
f
; (5.122)
y
s
=y
0
; (5.123)
z
s
=x
0
sin
f
+z
0
cos
f
: (5.124)
With the rotated x
s
and z
s
unit vectors, the index ellipsoid equation becomes,
1
(n
s
x
)
2
cos
2
f
+
1
(n
s
z
)
2
sin
2
f
+ 2r
s
131
E
s
x
sin
f
cos
f
(x
0
)
2
+
1
n
s
y
2
(y
0
)
2
+
1
(n
s
x
)
2
sin
2
f
+
1
(n
s
z
)
2
cos
2
f
+ 2r
s
131
E
s
x
sin
f
cos
f
(z
0
)
2
+
1
(n
s
x
)
2
sin
f
cos
f
+
1
(n
s
z
)
2
sin
f
cos
f
+ 2r
s
131
E
s
x
cos
2
f
x
0
z
0
+
1
(n
s
x
)
2
sin
f
cos
f
+
1
(n
s
z
)
2
sin
f
cos
f
2r
s
131
E
s
x
sin
2
f
z
0
x
0
+ 2r
s
121
E
s
x
cos
f
x
0
y
0
+ 2r
s
121
E
s
x
sin
f
z
0
y
0
= 1 : (5.125)
It is now possible to eliminate the mixed-coordinate terms by determining the angle
105
f
which causes the mixed-coordinate term to equal zero. This occurs when,
1
(n
s
x
)
2
sin
f
cos
f
+
1
(n
s
z
)
2
sin
f
cos
f
+r
s
131
E
s
x
cos
2
f
sin
2
f
= 0 ; (5.126)
which results in,
tan
2
f
=
2r
s
131
E
s
x
(n
s
x
)
2
(n
s
z
)
2
(n
s
z
)
2
(n
s
x
)
2
,
(5.127)
Assuming that the rotation has eliminated some of the mixed coordinate terms, it
is useful to rewrite the index ellipsoid to the following form,
A
0
11
(x
0
)
2
+A
0
22
(y
0
)
2
+A
0
33
(z
0
)
2
+ 2A
0
12
x
0
y
0
+ 2A
0
32
z
0
y
0
= 1 ; (5.128)
where,
A
0
11
=
1
(n
s
x
)
2
cos
2
f
+
1
(n
s
z
)
2
sin
2
f
+ 2r
s
131
E
s
x
sin
f
cos
f
; (5.129)
A
0
22
=
1
n
s
y
2
,
(5.130)
A
0
33
=
1
(n
s
x
)
2
sin
2
f
+
1
(n
s
z
)
2
cos
2
f
+ 2r
s
131
E
s
x
sin
f
cos
f
; (5.131)
A
0
12
=r
s
121
E
s
x
cos
f
; (5.132)
A
0
32
=r
s
121
E
s
x
sin
f
: (5.133)
It is instructive to notice that as long as one continues to rotate about an axis,
mixed coordinate terms will continuously appear. However, one can rotate the axes
such that the mixed coordinate terms become a minimum value. A minimum value
of approximate 10
8
for the coecient of the mixed coordinate term is reasonable
[35].
106
ˆ
f
y
ˆ' x
ˆ
f
x
f
f
ˆ' y
Figure 5.5: Unit vector rotation with angle
f
about the
^
z
0
axis according to right-hand
rule.
The second coordinate transformation occurs in the (x
0
;y
0
) plane as shown in
Figure 5.5. The rotation occurs about the x
0
axis according to the right-hand rule.
The axes relations are,
x
0
=x
f
cos
f
y
f
sin
f
; (5.134)
y
0
=x
f
sin
f
+y
f
cos
f
; (5.135)
z
0
=z
f
: (5.136)
Rotating about the x' axis leads to the index ellipsoid equation,
A
0
11
cos
2
f
+A
0
22
sin
2
f
+ 2A
0
12
sin
f
cos
f
x
f
2
+
A
0
11
sin
2
f
+A
0
22
cos
2
f
2A
0
12
sin
f
cos
f
y
f
2
+A
0
33
z
f
2
+
A
0
11
sin
f
cos
f
+A
0
22
sin
f
cos
f
+ 2A
0
12
cos
2
f
x
f
y
f
+
A
0
11
sin
f
cos
f
+A
0
22
sin
f
cos
f
2A
0
12
sin
2
f
y
f
x
f
+ 2A
0
32
cos
f
z
f
y
f
+ 2A
0
32
sin
f
z
f
x
f
= 1 : (5.137)
107
The term containing the mixed coordinates x
f
and y
f
can be eliminated with the
proper angle
f
. Thus, setting the mixed coordinate terms to zero one obtains,
A
0
11
sin
f
cos
f
+A
0
22
sin
f
cos
f
+A
0
12
cos
2
f
sin
2
f
= 0 ; (5.138)
which is satised when,
tan
2
f
=
2A
0
12
A
0
11
A
0
22
.
(5.139)
Rewriting the index ellipsoid will allow one to see the remaining coecients. The
index ellipsoid is now,
A
f
11
x
f
2
+A
f
22
y
f
2
+A
f
33
z
f
2
+ 2A
f
32
z
f
y
f
+ 2A
f
31
z
f
x
f
= 1 ; (5.140)
where,
A
f
11
=A
0
11
cos
2
f
+A
0
22
sin
2
f
+ 2A
0
12
sin
f
cos
f
; (5.141)
A
f
22
=A
0
11
sin
2
f
+A
0
22
cos
2
f
2A
0
12
sin
f
cos
f
; (5.142)
A
f
33
=A
0
33
; (5.143)
A
f
32
=A
0
32
cos
f
; (5.144)
A
f
31
=A
0
32
sin
f
: (5.145)
One could continue rotating coordinates to obtain smaller mixed coordinate terms,
however with two rotations, the mixed coordinate terms are typically on the order
of 10
8
and deemed small enough.
108
It is now possible to compare the straight waveguide modulation eect of the
ordinary wave with the typicalz-cut LiNbO
3
conguration with an applied eld in
the x
s
and x direction for the straight waveguide and the typical z-cut waveguide,
respectively.
It was found that the coordinates (x
s
;y
s
;z
s
) could be obtained with a rotation
angle
s
= 32:75
. The subsequent permittivity values are,
s
x
=
11
= 4:34793 10
11
; (5.146)
s
y
= (
33
11
) sin
2
t
+
11
= 4:26872 10
11
; (5.147)
s
z
= (
33
11
) cos
2
t
+
11
= 4:16356 10
11
: (5.148)
Then, the refractive index values are,
n
s
x
=
r
s
x
0
= 2:21601 ; (5.149)
n
s
y
=
r
s
y
0
= 2:19573 ; (5.150)
n
s
z
=
r
s
z
0
= 2:16852 : (5.151)
Since r
s
131
= 25:3884 10
12
pm/V, r
s
121
= 12:2878 10
12
pm/V, and assuming
109
that E
s
x
= 10
6
V/m, one obtains,
f
=2:82 10
3
rad ; (5.152)
A
0
11
= 0:203636871 ; (5.153)
A
0
22
= 0:207415939 ; (5.154)
A
0
33
= 0:212653579 ; (5.155)
A
0
12
= 1:228775 10
5
; (5.156)
A
0
32
=3:465155 10
8
: (5.157)
Then also,
f
=3:251 10
3
rad ; (5.158)
A
f
11
= 0:203636831 ; (5.159)
A
f
22
= 0:207415979 ; (5.160)
A
f
33
= 0:212653579 ; (5.161)
A
f
32
=3:47 10
8
; (5.162)
A
f
31
= 1:04 10
10
: (5.163)
The magnitude of the coecients for the mixed coordinate terms are less than
5 10
8
and thus negligible. For the straight waveguide geometry, the ordinary
mode is resonant. Thus the change in the refractive index in the x
f
direction is of
110
interest. Each of the coecients is relatable to the refractive indices as follows,
A
f
11
=
1
n
f
x
2
,
(5.164)
A
f
22
=
1
n
f
y
2
,
(5.165)
A
f
33
=
1
n
f
z
2
.
(5.166)
Returning to the previous example, one nds that,
n
f
x
= 2:2160106 ; (5.167)
which results in basically no index change. It is then a clear contrast between the
modulation that was achievable for the extraordinary mode. Thus, if one were to
consider using the cleaved edges of LiNbO
3
for a resonator, it is only useful to use
the extraordinary mode when modulating.
It is now possible to compare the cleaved edge straight waveguide modulation
eect of the ordinary mode with that of the typical z-cut LiNbO
3
waveguide. For
the z-cut waveguide, the same procedure ensues except that refractive indices are
not rotated. In this case,
n
x
= 2:21601 ; (5.168)
n
y
= 2:21601 ; (5.169)
n
z
= 2:14856 : (5.170)
111
Sincer
131
=r
51
= 28 pm/V andr
121
=r
61
=3:4 pm/V, and again assuming that
E
x
= 10
6
V/m, one obtains,
f
=2:16 10
3
rad ; (5.171)
A
0
11
= 0:203636883 ; (5.172)
A
0
22
= 0:203636943 ; (5.173)
A
0
33
= 0:216623066 ; (5.174)
A
0
12
=3:4 10
6
; (5.175)
A
0
32
= 6:8 10
9
: (5.176)
Then also,
f
= 0:781 rad ; (5.177)
A
f
11
= 0:203633512 ; (5.178)
A
f
22
= 0:203640313 ; (5.179)
A
f
33
= 0:216623066 ; (5.180)
A
f
32
= 4:8 10
9
; (5.181)
A
f
31
= 4:8 10
9
: (5.182)
The magnitudes of the mixed coordinate coecients are less than 10
8
and thus
negligible. In this case one nds that,
n
f
x
= 2:21602866 ; (5.183)
112
which results in an index change of approximately 1:9 10
5
. Thus, using the
typical z-cut orientation, a modulation can be observed, which is in contrast to
the cleaved edge design. In comparison to the extraordinary mode, the modulation
eect is one order of magnitude less, and hence as assumed, the modulation of the
extraordinary mode results in the best achievable conguration.
Up to this point, the analyses for the electrode congurations involved an ap-
plied eld in a direction orthogonal to the waveguide axes. This is realizable by
positioning electrodes either above or below, or to the sides of the indiused wave-
guide. It is not feasible to position electrodes such that the applied eld is in the
direction of the waveguide axis because this would entail positioning electrodes di-
rectly on the mirror surface. However, if the resonant modulator were to contain
no electrodes and were to perform as a detector in free-space, one would need to
know the eect of an applied electric eld along the axis of the waveguide. Thus
to complete the analyses, an applied electric eld along the axis of the waveguide
will be studied.
Assuming now that the waveguide conguration can be either for the straight
or tilted waveguide the index ellipsoid in the presence of an applied electric eld in
the ^ y direction results in,
1
(n
m
x
)
2
+r
m
112
E
m
y
(x
m
)
2
+
"
1
n
m
y
2
+r
m
222
E
m
y
#
(y
m
)
2
+
1
(n
m
z
)
2
+r
m
332
E
m
y
(z
m
)
2
+ 2y
m
z
m
r
m
232
E
m
y
= 1 ;
(5.184)
113
f
z
m
y
f
y
f
f
m
z
Figure 5.6: Unit vector rotation with angle
f
about the
^
x
m
axis according to right-hand
rule.
where m = s;t, which corresponds to either the straight or tilted waveguide con-
guration. There is one mixed coordinate term and thus a rotation about the x
m
axis according to the right hand rule is necessary to minimize the mixed coordinate
term. The axes rotation is shown in Figure 5.6. The axes relations are,
x
m
=x
f
; (5.185)
y
m
=y
f
cos
f
z
f
sin
f
; (5.186)
z
m
=y
f
sin
f
+z
f
cos
f
: (5.187)
Rotating about the x
m
axis leads to the index ellipsoid equation,
1
(n
m
x
)
2
+r
m
112
E
m
y
x
f
2
+
"
1
n
m
y
2
+r
m
222
E
m
y
#
y
f
cos
f
z
f
sin
f
2
+
1
(n
m
z
)
2
+r
m
332
E
m
y
y
f
sin
f
+z
f
cos
f
2
+ 2
y
f
cos
f
z
f
sin
f
y
f
sin
f
+z
f
cos
f
r
m
232
E
m
y
= 1 :
(5.188)
This equation is very similar to the index ellipsoid that resulted in the applied eld
114
E
t
z
. Thus, following the same procedure the angle
f
that results in eliminating
the mixed-coordinate term is,
tan
2
f
=
2r
m
232
E
m
y
n
m
y
2
(n
m
z
)
2
(n
m
z
)
2
n
m
y
2
+ (r
m
222
r
m
332
)E
m
y
(n
m
z
)
2
n
m
y
2
.
(5.189)
One can also derive the eect of the applied electric eld on the refractive index
with the following relations using the binomial approximation and assuming that,
r
m
112
(n
m
x
)
2
E
m
y
1 ; (5.190)
r
m
222
n
m
y
2
E
m
y
+r
m
232
n
m
y
2
E
m
y
tan
f
1 ; (5.191)
r
m
332
(n
m
z
)
2
E
m
y
r
m
232
(n
m
z
)
2
E
m
y
tan
f
1 ; (5.192)
so that,
n
f
x
=n
m
x
1
2
r
m
112
(n
m
x
)
3
E
m
y
; (5.193)
n
f
y
=n
m
y
1
2
n
t
y
3
E
m
y
r
m
222
+r
m
232
tan
f
; (5.194)
n
f
z
=n
m
z
1
2
(n
m
z
)
3
E
m
y
r
m
332
r
m
232
tan
f
: (5.195)
It is now possible to see the eect of an applied electric eld in the y direction
on the extraordinary mode refractive index for the tilted waveguide, by assigning
m =t and assuming that,
E
t
y
= 10
6
V/m ; (5.196)
n
t
y
= 2:19833 ; (5.197)
n
t
z
= 2:16665 ; (5.198)
115
then,
f
=2:482 10
3
; (5.199)
n
f
z
=n
t
z
+ 3:8403 10
5
: (5.200)
Recalling the result from an applied eld in the z
t
direction from Equation
(5.120), whereby the change in the extraordinary index was found to be1:53
10
4
, and comparing with the result from an applied eld in the y
t
direction from
Equation (5.200), one can see that any additional eld in the y
t
direction degrades
the modulation eect.
In fact there is a certain condition when the modulation eect is completely
nulled. This occurs when,
E
t
z
E
t
y
=
r
t
332
r
t
232
tan
f
y
r
t
333
r
t
233
tan
f
z
,
(5.201)
where
f
y
and
f
z
are the angles for the coordinate transformations when there
exists an applied y
t
eld and an appliedz
t
eld, respectively. Though these angles
depend on the strength of the applied electric eld, for the sake of the example
considered here, the values of the angles are small. Thus for the example outlined
in this section, one can approximate that the modulation eect is nulled when
E
t
z
E
t
y
0:226 : (5.202)
To complete the analysis, the eect of an applied eld in the x
t
direction must
also be considered for the tilted waveguide. Previously examined was the eect of
116
an applied eld in the x
s
direction. The same results may be used except that the
components will be determined for the tilted waveguide geometry rather than the
straight waveguide geometry. Thus using Equation (5.127), the angle needed to
rotate the coordinates in order to eliminate the mixed coordinate terms is,
tan
2
f
=
2r
t
131
E
t
x
(n
t
x
)
2
(n
t
z
)
2
(n
t
z
)
2
(n
t
x
)
2
,
(5.203)
Assuming E
t
x
= 10
6
V/m one obtains,
f
=2:744 10
3
rad : (5.204)
Of most importance when analyzing the eect of the extraordinary index is the
value of A
f
33
, the equation of which was calculated previously and repeated again
here,
A
f
33
=
1
(n
t
x
)
2
sin
2
f
+
1
(n
t
z
)
2
cos
2
f
+ 2r
t
131
E
t
x
sin
f
cos
f
; (5.205)
A
f
33
=
1
n
f
z
2
.
(5.206)
Finally one nds for this example that,
n
f
z
= 2:16665 : (5.207)
Thus, there is basically no index change for the applied eld of E
t
x
= 10
6
V/m.
By way of example, the change in the extraordinary index for the tilted waveguide
resonator, due to an applied electric eld for each of the Cartesian axes (x
t
;y
t
;z
t
)
was calculated.
117
In summary, if the tilted waveguide resonator is to be used as a free-space
modulator, the incoming electric eld would need to be divided into three compo-
nents oriented along the tilted waveguide axes (x
t
;y
t
;z
t
) such that the electric eld
becomes
E
t
x
;E
t
y
;E
t
z
. The strength of the electric eld components dictates the
modulation eect. The modulation eect is then a sum of the eect of each of the
components. The total change in the refractive index is given by the equation,
n
f
z
=n
t
z
+ n
t
z
; (5.208)
where,
n
t
z
= n
t
z;x
+ n
t
z;y
+ n
t
z;z
; (5.209)
represents the change in the refractive index due to an applied eld in each of the
x
t
;y
t
; and z
t
directions respectively. From the previous analyses, the following can
be found,
n
t
z;x
=
"
(n
t
z
)
2
sin
2
f
+ (n
t
x
)
2
cos
2
f
+ 2r
t
131
E
t
x
(n
t
x
)
2
(n
t
z
)
2
sin
f
cos
f
(n
t
x
)
2
(n
t
z
)
2
#
1=2
n
t
z
;
(5.210)
n
t
z;y
=
1
2
n
t
z
3
E
t
y
r
t
332
r
t
232
tan
t
y
; (5.211)
n
t
z;z
=
1
2
n
t
z
3
E
t
z
r
t
333
r
t
233
tan
t
z
: (5.212)
This concludes the analysis of an arbitrary applied electric eld and its eect on
the extraordinary refractive index for the tilted LiNbO
3
waveguide.
118
Chapter 6
EXPERIMENTAL ANALYSIS OF WAVEGUIDE
RESONATORS
6.1 Waveguide Fabrication Procedure
This chapter covers the experimental analysis of titanium indiused LiNbO
3
wave-
guide resonators. The fabrication procedure of a titanium indiused LiNbO
3
wave-
guide is as follows:
1. A 90 nm layer of titanium is deposited onto the LiNbO
3
substrate by e-beam
evaporation (at a pressure of 10
7
Torr) with a crystal quartz deposition
thickness monitor.
2. The LiNbO
3
sample is saw cut to a manageable size of approximately 2 cm
2 cm.
3. Photoresist is spun directly on top of the titanium.
4. The waveguide is dened photolithographically with a photomask, and by
exposing the photoresist to UV light.
119
5. The substrate is placed in the photoresist developer solution, leaving behind
the photoresist that denes the waveguide.
6. The substrate is placed in Transene titanium etchant TFTN to remove all
unwanted titanium and leaving only the titanium strip below the undeveloped
photoresist. (Etch rate: 10
A/sec at 70
C or 50
A/sec at 85
C.)
7. The remaining photoresist is removed with the photoresist stripper solution.
8. The sample is placed in a high-temperature quartz tube furnace and heated
to the desired temperature (1050-1100
C) at a set rate (4-6
C per minute).
The titanium is diused into the LiNbO
3
, creating the high index region
for wave guidance. The extent of the diusion time depends on the desired
diusion depth (typically 6-9 hours for pure LiNbO
3
[2].) Argon/O
2
passes
through an H
2
O bubbler and fed into the furnace to create an inert and humid
environment. (Typically Ar is fed through the quartz tube from the beginning
of the heating process at a rate of 1 liter/min followed by O
2
at about the
same rate during the temperature descent).
An alternative fabrication procedure eliminates the need for titanium etchant.
The alternative procedure is as follows:
1. The LiNbO
3
sample is saw cut to a manageable size of approximately 2 cm
2 cm.
120
2. A layer of photoresist is spun onto the LiNbO
3
substrate.
3. The waveguide is dened photolithographically with a photomask, and by
exposing the photoresist to UV light. The sample is placed in photoresist
developer solution, and the slotted region denes the waveguide region. No
photoresist remains in the waveguide region.
4. A 90 nm layer of titanium is deposited onto the LiNbO
3
substrate by e-beam
evaporation (at a pressure of 10
7
Torr) with a crystal quartz deposition
thickness monitor.
5. The substrate is placed in the photoresist stripper solution, lifting o the
undesired photoresist and the titanium layer above it. The titanium strip
waveguide remains.
6. The sample is placed in a high-temperature quartz tube furnace and heated
to the desired temperature (1050-1100
C) at a set rate (4-6
C per minute).
The titanium is diused into the LiNbO
3
, creating the high index region
for wave guidance. The extent of the diusion time depends on the desired
diusion depth (typically 6-9 hours for pure LiNbO
3
[2].) Argon/O
2
passes
through an H
2
O bubbler and fed into the furnace to create an inert and humid
environment. (Typically Ar is fed through the quartz tube from the beginning
of the heating process at a rate of 1 liter/min followed by O
2
at about the
same rate during the temperature descent).
121
Furnace
Argon Oxygen
DI Water
1000 mL
Flow meters
Crucibles
Location: USC
Figure 6.1: Indiusion setup up high temperature quartz tube furnace, quartz crucibles,
deionized water for increasing water vapor in tube, argon and oxygen gas with
ow meters.
The high-temperature quartz tube furnace and titanium indiusion setup are
shown in Figure 6.1. As shown in Figure 2.3 the indiused location, having an
index of refraction larger than its surroundings, will be the guide for the optical
wave.
122
6.2 Index Distribution and Eective Index Measurement
By using a rutile coupling prism and a precision rotating stage, the eective index of
a titanium indiused LiNbO
3
mode can be determined. A titanium indiused slab
waveguide was fabricated on a LiNbO
3
substrate, to measure the eective refractive
index. It is necessary that the coupling prism possesses refractive indices that are
larger than the eective index of the guided wave. For rutile, the ordinary and
extraordinary refractive indices at 1.3 m are 2.48376 and 2.74706, respectively.
As shown in Figure 6.2, the input laser source is oriented at an angle
a
with
respect to one of the prism facets. By Snell's law the light is transmitted into the
prism as long as the angle is less than the condition for total internal re
ection at
the prism facet. The light continues to propagate inside the prism until it reaches
the prism facet that is in contact with the indiused sample. The angle
p
is larger
than the critical angle and thus, total internal re
ection occurs. The light then
propagates to the third facet and exits the prism. A detector is placed at the
location where the light beam exits.
Even though total internal re
ection is satised at the prism-waveguide inter-
face, at a particular angle a phase match condition may occur such that the evanes-
cent tail couples into the slab waveguide. The relationship between the eective
refractive index of the waveguide and the angle
p
is as follows [22],
n
eff
=n
p
sin
p
: (6.1)
123
a
p
p
n
p
n
a
n
f
Figure 6.2: Schematic of the eective index measurement.
The particular angle, at which the prism is oriented with respect to the input laser
source, can then be related to the value for the eective index of the guided mode,
such that,
a
= arcsin
n
p
n
a
sin (
p
p
)
: (6.2)
In terms of the detected output light, a decrease in detected intensity should appear
when the input light source is coupled into the titanium indiused waveguide mode.
The experimental setup is shown in Figure 6.3. The input laser source is re-
ected o of a mirror, passed through a polarizer and coupled into a rutile prism.
The sample is pushed up against one of the prism facets with an air piston such that
an air gap is minimized between the waveguide and the prism facet. A detector is
placed at the output.
124
Laser
Mirror
Polarizer
Piston
Prism
Detector
Sample
Figure 6.3: Eective index measurement setup. The input laser source at = 1:3 m is
re
ected o of a mirror, passed through a polarizer and coupled into a rutile prism. The
sample is pushed up against one of the prism facets with an air piston such that an air
gap is minimized between the waveguide and the prism facet. A germanium detector is
placed at the output
125
The results for the eective index measurement of a z-cut titanium indiused
LiNbO
3
are shown in Figure 6.4. Two TM-like modes were identied in the slab
waveguide with eective indices 2.15132 and 2.15598. The TM asymmetric isotropic
slab waveguide determinantal equation, Equation (3.221), may be used to postulate
the eective core thickness of the slab waveguide. Though this equation does not
consider an indiused index prole, it remains as a method for comparison. With a
slab core thickness equal to 6.4m, the solution provides two modes with eective
indices, 2.1496 and 2.1558. Thus in comparison to the experimental results, the
values dier by 1:7 10
3
and by 1:8 10
4
, indicating that the indiused slab is
comparable to a homogeneous slab of thickness 6:4 m with an index dierence of
1:1 10
2
between the lower substrate of LiNbO
3
and the titanium indiused core.
6.3 Z-Cut LiNbO
3
Waveguide Resonator
In this section, a z-cut LiNbO
3
waveguide resonator with polished edges will be
described. The achievable nesse, F , was approximately 14 for the TM-like mode.
A titanium indiused LiNbO
3
waveguide with a length of 7 mm, was fabricated
using the procedure outlined in Section 6.1, with titanium etchant. Assuming
n
11
=n
o
= 2:21601 and n
33
=n
e
= 2:14856, which are the refractive index values
for titanium indiused LiNbO
3
at = 1:55m, the corresponding theoreticalFSR
values for the TE-like and TM-like modes are 9.7 GHz and 10 GHz, respectively.
The samples were polished and coated with dielectric mirrors by Silicon Light-
126
θ
a
(deg)
39.6 40.3 41 38.9
Detected Power (μW)
0.450
0.452
0.454
0.456
0.458
0.460
0.462
0.464
Figure 6.4: Eective index measurement of extraordinary wave for z-cut titanium indif-
fused LiNbO
3
slab waveguide. Two modes identied with eective indices 2.15132 and
2.15598.
127
7/28/09
7 mm
1 mm
1 mm
1.5 mm
4 mm
Ti-indiffused
waveguide
Output Mirror
R = 0.98
Input Mirror
R = 0.91
y
z
x
1 mm
Figure 6.5: Schematic and fabricated titanium indiused z-cut LiNbO
3
waveguide Fab-
ry-Perot resonator.
128
wave Technology, Inc., with re
ectances of 0.91 and 0.98 for the input and output
mirrors, respectively. In order to obtain a uniform deposition of the dielectric mir-
rors, two cover pieces of LiNbO
3
were adhered to the top surface with epoxy. The
1 mm thick cover pieces were cut to dimensions of 1.5 mm 11 mm, as can be
seen in Figure 6.5.
The experimental setup to observe the resonance response is shown in Figure
6.6. A NewFocus tunable laser operating at 1:55 m was used, and controlled
by a function generator with a sawtooth signal of 50 Hz with 3 VPP to obtain a
tunability of 30 GHz. The output ber of the tunable laser was connected to a ber
polarization controller, followed by a ber optic circulator. The ber optic circulator
output was ber butt-coupled to the titanium indiused waveguide resonator. The
transmitted light from the resonator was collected with a 20 lens, passed through
a polarizer to conrm the polarization of the input light, then passed through an iris
and detected with a photodetector. The back-re
ected light of the resonator was
coupled into the ber optic circulator and detected with a detector and observed
on an oscilloscope. The function generator was also connected to the oscilloscope
as a reference for the FSR measurement.
Without a polarization controller, the TE-like and TM-like modes simultane-
ously appeared at the output as shown in Figure 6.7, with FSR values of 9.7 GHz
and 10.2 GHz, respectively. Subsequently with a polarization controller, one can
observe the individual modes as shown in Figures 6.8 and 6.9, for the TE-like and
129
Tunable
Laser
Fiber Optic
Circulator
Ti:LiNbO
3
Resonator
20x
Lens
Detector
Fiber
Polarization
Controller
Detector
Oscilloscope
Function
Generator
Polarizer
Iris
Figure 6.6: Block diagram of experimental setup of titanium indiused z-cut LiNbO
3
waveguide Fabry-Perot resonator, for resonance measurement.
TM-like mode, respectively.
With the use of an optical circulator, the re
ected intensity may also be ob-
served. Again, without the polarization controller, both TE-like and TM-like modes
were present as shown in Figure 6.10. With a polarization controller, the re
ected
TM-like mode was observed and the output is shown in Figure 6.11. In observ-
ing the output with no polarization controller, the TM-like mode nesse, F , was
approximately 14.
The experimental FSR values of 9.7 GHz and 10.2 GHz conrm well with the
theoretical values of 9.7 GHz and 10 GHz, for the TE-like and TM-like mode,
respectively. An improvement in the nesse could be achieved by optimized fab-
rication to obtain low-loss waveguides. As a proof of concept that the resonator
can be an integrated device, the resonator was ber pigtailed with epoxy NOA61
130
10.2 GHz
9.7 GHz
0
1
1549.88 1550 1550.12
Wavelength (nm)
Zero line saw tooth signal
Zero line output light
Figure 6.7: Transmitted output of the TE-like and TM-like modes of a titanium indiused
z-cut LiNbO
3
waveguide Fabry-Perot resonator, with FSR values of 9.7 GHz and 10.2
GHz, respectively.
131
9.7 GHz
0
1
1549.88 1550 1550.12
Wavelength (nm)
Zero line saw tooth signal
Zero line output light
Figure 6.8: Transmitted output of the TE-like mode of a titanium indiusedz-cut LiNbO
3
waveguide Fabry-Perot resonator, with a FSR value of 9.7 GHz.
132
10.2 GHz
0
1550.12 1550 1549.88
Wavelength (nm)
1
Zero line saw tooth signal
Zero line output light
Figure 6.9: Transmitted output of the TM-like mode of a titanium indiused z-cut
LiNbO
3
waveguide Fabry-Perot resonator, with a FSR value of 10.2 GHz.
133
9.7 GHz
10.2 GHz
0
1
1550.12 1550 1549.88
Wavelength (nm)
Zero line saw tooth signal
Zero line output light
Figure 6.10: Re
ected output of the TE-like and TM-like modes of a titanium indiused
z-cut LiNbO
3
waveguide Fabry-Perot resonator, with FSR values of 9.7 GHz and 10.2
GHz, respectively.
134
10.2 GHz
0
1
1550.12 1550 1549.88
Wavelength (nm)
Zero line saw tooth signal
Zero line output light
Figure 6.11: Re
ected output of the TM-like mode of a titanium indiusedz-cut LiNbO
3
waveguide Fabry-Perot resonator, with a FSR value of 10.2 GHz.
135
as shown in Figure 6.12.
6.4 Cleaved Edge X-Cut LiNbO
3
Waveguide Resonator
6.4.1 Waveguide Facet Tilt
The eect of a tilted mirrored facet is important when fabricating a Fabry-Perot
resonator. The required polishing necessary for the mirror ends may cause rounding
of the edges. Since the waveguide is indiused on the top surface, rounding of the
edge can greatly aect the loss. Shown in Figure 6.13 is the drop in the power
re
ection coecient due to an increase in tilt angle.
The calculation is fully described in [38], wherein the assumption is that if
a tilt is present, the incoming and re
ecting waves are oset by a phase factor.
The overlap between the two waves can be directly correlated to the loss due to a
tilt. Incorporating cleaved edges in a waveguide resonator eliminates the need for
polishing.
6.4.2 Cleaved Edge
As described in Section 2.2, there are useful cleaved edges of LiNbO
3
that can be
utilized for various geometries of a waveguide resonant modulator. The process of
cleaving LiNbO
3
is shown in Figure 6.14.
The cleaving process begins after the diusion of titanium into the LiNbO
3
sub-
136
Figure 6.12: An integrated device: Titanium indiused LiNbO
3
waveguide resonator ber
pigtailed with epoxy NOA61.
137
Figure 6.13: Power re
ection coecient vs. tilt of mirror facet.
138
Saw cut
ADT resin blade (with 20 micron diamonds)
Blade thickness: 6 mil (0.1524 mm)
Feed rate: 0.3 mm/sec
Spindle speed: 25 - 30 krpm
Water pressure: 0.7 Liter/min
Lithium
Niobate
1 mm
Adhered to glass slide with
NOA61 Epoxy (UV cured)
This end stays fixed to a stage.
Separate manual linear stage
lifts sample upwards
Break occurs at cleavage plane
Cleaved portion
100 μm left for cleaving
y
x
z
Cleaved portion
Lithium
Niobate
Lithium
Niobate
Figure 6.14: Schematic of cleaving process. LiNbO
3
is rst saw cut leaving a thickness
of 100 m to be cleaved. One side of the sample is adhered to a glass slide with epoxy
NOA61 or bonding wax, and remains xed to a stage. The other half of the sample is
lifted with a manual linear stage until cleavage occurs.
139
strate, the process of which is described in Section 6.1. Even though commercially
available LiNbO
3
substrates have thicknesses of 500 m or 1 mm, a precise and
clean cleaved edge can only be obtained for a thicknesses of at most 200 m. The
best achievable cleaved surfaces were obtained with thicknesses of approximately
100 m.
After one has decided the cleaved edge location, the sample is cut with an ADT
resin-bond dicing blade with a 6 mil (152 m) blade thickness and diamond grit
size of 20 m. The feed rate is kept low at 0.3 mm/sec so as not to cause any
sheering stress on the substrate. Deionized water is continuously applied to the
substrate whilst sawing with a pressure of 0.7 liter/min, so as to prevent the blade
from overheating, and to remove unwanted debris. The cut depth is programmed
so as to leave approximately 100 m uncut, from the surface of the indiusion.
Part of the sample is then adhered to a glass slide with UV-curable NOA61 epoxy
or mounting wax, leaving the desirable cleaved edge suspended in air as shown in
Figure 6.15. On the free
oating portion of the sample, force is applied upwards
with a separate manual linear stage to create a break at the cleavage plane. Due
to the fact that the dicing blade creates a cut width of approximately 150 m the
cleaved portion creates a lip from the rest of the substrate. Typically the cleaved
portion is centered in the middle of the cut. However if the portion of the sample
that is free
oating in air is too small, the cleaved edge is not centered in the middle
of the cut, and the subsequent lip of the free
oating piece is larger.
140
LiNbO
3
sample
Micrometer
stage
Figure 6.15: Setup of LiNbO
3
cleaving process
141
Lithium Niobate top surface
+z face
Lithium Niobate top surface
+z face
Cleaved edge
Cleaved edge
Saw cut edge face
Lithium
Niobate
1 mm
y
x
z
Figure 6.16: FESEM image of cleave on the +z face, along the y-axis of LiNbO
3
.
Various samples of LiNbO
3
have been cleaved in dierent orientations, so as
to identify the useable cleaved edges and conrm the cleavage plane locations as
described in section 2.2. FESEM and light microscope images were taken to analyze
the edges. For the FESEM images, 2 nm of Au was sputtered directly on the cleaved
surface so as to create a conductive surface.
A supposed cleavage edge is parallel to the y-axis. Figures 6.16 and 6.17 show
the cleaved edge parallel to they-axis, on the +z face of LiNbO
3
. Figures 6.18 and
6.19 show the cleaved edge parallel to the y-axis, on thez face of LiNbO
3
.
It is instructive to note the visible terraces on the cleaved surface. The terraces
142
10x
20x
Cleaved edge
Cleaved edge
Saw cut
edge face
Saw cut
edge face
Lithium
Niobate
1 mm
y
x
z
Figure 6.17: Light microscope image of cleave on the +z face, along they-axis of LiNbO
3
.
143
Lithium Niobate top surface
-z face
Lithium Niobate top surface
-z face
Lithium Niobate top surface
-z face
Cleaved edge
Cleaved edge
Cleaved edge
Terraces
Saw cut edge face
Lithium
Niobate
1 mm
y
x
z
Figure 6.18: FESEM image of cleave on thez face, along the y-axis of LiNbO
3
.
144
10x
20x
Cleaved edge
Cleaved edge
Saw cut
edge face
Saw cut
edge face
Lithium
Niobate
1 mm
y
x
z
Figure 6.19: Light microscope image of cleave on thez face, along they-axis of LiNbO
3
.
145
appear approximately 20 m from the top surface. Since the indiused waveguide
only occupies a depth of, at most, approximately 10 m from the top surface, the
terraces are of no consequence.
Regardless of whether the cleaved edge occurs on the +z orz face, the FE-
SEM images show that the cleaved face is perpendicular to the top face, when the
cleaved edge is parallel to they-axis. To conrm this fact, a laser surface re
ection
measurement was conducted, and will be explained later in this section. All of the
cleavage planes analyzed in this section were discussed in Section 2.2.
The structure of LiNbO
3
also contains two other cleavage planes that are ori-
ented 60
with respect to the y-axis, or 30
with respect to the x-axis. These two
cleaved edges along with the cleaved edge in parallel with the y-axis, allow for a
possible cleaved edge equilateral triangle resonator. To analyze the cleaved edges
a light microscope image of the cleaved surface was taken as shown in Figure 6.20.
In the microscope image, there are faint terraces that are visible at approximately
50m from the top surface of the cleaved edge. However, cleaved edges of a thick-
ness of 200 m produced more visible terraces at approximately 20 m from the
top surface of the cleaved edge. Mirrored about the x-axis, another cleavage plane
exists at 30
from thex-axis, which produced similar cleavage planes as the images
in Figure 6.20.
The cleavage planes for an equilateral triangle structure were identied in a
LiNbO
3
ion implantation experiment, where a dose of 4.5E16 of hydrogen ions was
146
Cleaved Edge
100 μm thickness
y
x
Reference edge
30°
Optical 5x
Optical 20x
Cleaved edge
-z face
Cleaved edge
-z face
+z face
z
Figure 6.20: Light microscope image of cleave on thez face, 30
with respect to the
y-axis of LiNbO
3
147
Figure 6.21: Damaged ion-implanted LiNbO
3
as further evidence of obtainable cleaved
planes in an equilateral triangle geometry
applied to create a sacricial layer, in hopes of subsequent removal of a thin (approx-
imately 1 m layer of LiNbO
3
.) The high dosage damaged the LiNbO
3
substrate
and cracks could be seen that followed the cleavage plane directions. Conrmation
of these cracks was further evidence that these cleavage planes may be easily ob-
tainable. The image of the damaged LiNbO
3
substrate with ion implantation is
shown in Figure 6.21. Debris can be seen on the image due to the cracking of the
substrate.
Also, in Section 2.2, a dominant cleavage plane was mentioned to occur at an
148
y
- x
z
32.75
Lithium
Niobate
Figure 6.22: Dominant LiNbO
3
cleavage plane occurs at an angle of = 32:75
with
respect to the z-axis
angle of = 32:75
with respect to the z-axis. The drawing of which is shown
in Figure 6.22. A thickness of 100 m of a z-cut LiNbO
3
substrate was cleaved
on thez face and parallel to the x-axis to observe this dominant cleavage plane.
Subsequently, as shown in the FESEM image of Figure 6.23, the cleavage plane was
tilted with respect to the z-axis, and later conrmed to be approximately 33
.
The same cleavage plane can be observed in an x-cut LiNbO
3
sample. To
conrm this dominant cleavage plane, a 1 mmx-cut LiNbO
3
substrate was cleaved
following the same procedure as Figure 6.14. The orientation of the cleaved edge
with respect to the x, y and z axes of LiNbO
3
is shown in Figure 6.24, along
with the light microscope images at 5 and 20 magnication. The noticeably
149
Lithium Niobate top surface
-z face
Cleaved edge
Lithium Niobate top surface
-z face
Cleaved edge face
Saw cut edge face
Top View
Side View
Lithium
Niobate
1 mm
x
y
z
Figure 6.23: FESEM image of cleave on thez face, along the x-axis of LiNbO
3
150
Cleaved edge
+x face
Optical 5x
Optical 20x
Cleaved edge
+x face
Cleaved Edge
100 μm thickness
z
y
Reference edge
33°
-x face
x
x
Figure 6.24: Light microscope image of cleave on the +x face, tilted by 33
with respect
to the z-axis of LiNbO
3
smooth cleaved surface is perpendicular to the top surface, and veried with the
laser re
ection experiment.
The laser re
ection experiment shown in Figure 6.25 was used to conrm if the
cleaved surface is perpendicular to the top surface of the substrate. The sample
to be tested was mounted on a rotation stage with azimuthal and polar controls.
A HeNe laser was shone onto the polished top surface of the LiNbO
3
substrate.
An iris was used to reduce the spot size of the HeNe laser to approximately 300
m. The re
ected light was located on the iris and the position of the sample was
151
calibrated so as to direct the re
ected light directly back toward the center of the
iris. The substrate was then rotated by 90
and the re
ected light was observed.
If the re
ected light was coincident with the incident light, the cleaved surface was
proven to be perpendicular to the top surface.
All of the imaged cleavage planes were analyzed with the laser re
ection exper-
iment. Those that were hypothesized to have a cleavage plane perpendicular to the
top surface were proven correct to the degree of accuracy of the experiment, which
was approximately 1
. The z-cut sample of Figure 6.22 which exhibits the tilted
cleaved facet was also conrmed to have an angle of approximately = 33
.
6.4.3 Resonance Condition
If one were to consider making a resonator whereby thec-axis is not parallel to the
z-axis of LiNbO
3
, the placement of the waveguide is crucial. Figure 6.26 illustrates
the proposed orientations of the resonator.
For the Fabry-Perot resonator, it is necessary to keep the wave propagation
vector perpendicular to the cleaved edge, otherwise, a standing wave will not occur
in the cavity. For the ordinary wave case, nothing unusual arises in the placement
of the waveguide since the propagation vector and the Poynting vector are in the
same direction along ^ z.
However, for the extraordinary wave, one sees that even though the propagation
vector satises the requirement to be perpendicular to the cleaved edge, the power
152
LiNbO
3
HeNe
Laser
HeNe
Laser
Iris
Sample
Rotation stage
Iris
Figure 6.25: Laser re
ection experiment to conrm if the cleaved edge of LiNbO
3
is
perpendicular to the top surface.
153
Cleaved-Edge Waveguide Design
y
Top View of Waveguide
z
c-axis
ˆ
e
S
57.25
ˆ
e
2
e
Case 1: Ordinary wave
Case 2: Extraordinary wave
y
Top View of Waveguide
z
c-axis
ˆ
o
S
ˆ
o
and
Tilted waveguide
57.25
cleaved facet
cleaved facet
cleaved facet
cleaved facet
x
x
Figure 6.26: Two waveguide designs for LiNbO
3
resonant modulators with cleaved edges.
The placement of the waveguide is shown by the thick blue line. For the extraordinary
case, the waveguide must be tilted at an angle of approximately 2
from the z-axis.
154
travels in a direction tilted from the z-axis. If the waveguide was positioned along
thez-axis, the power would leak out of the waveguide in the direction of the Poynt-
ing vector, and cause the waveguide to be lossy. Moreover, the resonance condition
would be violated, and a resonance would not be obtained. If the waveguide is
oriented at an angle,
e
, such that the waveguide is parallel to the Poynting vector
direction,
^
S, then the power remains conned to the waveguide whilst simultane-
ously satisfying the requirement for the propagation vector to be perpendicular to
the cleaved edge. Assumingn
11
=n
o
= 2:22568 andn
33
=n
e
= 2:15647 (refractive
index values for titanium indiused LiNbO
3
at = 1:31 m),
e
is equal to 1:76
,
where the Poynting vector possesses a +^ z and a +^ y component.
Two titanium indiused LiNbO
3
waveguides were fabricated and their edges
were cleaved. One waveguide was oriented parallel to the z-axis, whilst the other
waveguide was tilted by 2
relative to the z-axis as shown in Figure 6.26. The
waveguide oriented parallel to thez-axis will exhibit resonance only for the ordinary
wave, which is provided by an input light polarized along thex-axis. The waveguide
oriented at a 2
tilt will exhibit resonance only for the extraordinary wave, which is
provided by an input light polarized perpendicular to thex-axis and parallel to the
cleaved facet. The cleaved facets were coated with 2 nm of gold to serve as mirrors.
The resonance was observed with a setup similar to that shown in Figure 6.6.
The input light source was a NewFocus tunable laser source centered at 1.31 m.
A sawtooth waveform of 50 Hz and 8 VPP tuned the laser light source by ap-
155
proximately 56 GHz. The input laser source was fed into a polarization controller,
which oriented the light in either a TM-like, which in this case refers to light polar-
ized parallel to thex-axis, or a TE-like polarization, which refers to light polarized
perpendicular to the x-axis and parallel to the cleaved facet. The input light was
then fed into a circulator which provided access to the light travelling in the di-
rection opposite to the input direction. The input light was then butt-coupled to
the cleaved edge facet with an optical ber at the input of the waveguide. The
transmitted light was collected with a 10 lens, and detected on an IR camera or a
photodetector. A polarizer at the output conrmed the polarization of the output
light. The back-re
ected light was also sent to a photodetector, whose amplitude
was monitored on an oscilloscope. The function generator was also connected to
the oscilloscope to display the sawtooth driving signal. An image of the gold coated
cleaved edge titanium indiused LiNbO
3
waveguide resonator is shown in Figure
6.27.
If a resonance occurs, it is expected that at a certain wavelength, the back-
re
ected light should exhibit a drop in transmission. The fabricated waveguide was
4.9 mm in length. The corresponding theoreticalFSR value for the ordinary mode
is 13.8 GHz, whilst the correspondingFSR for the extraordinary mode is 14.2 GHz.
The measured FSR values were approximately 13.5 GHz and 14.3 GHz for
the ordinary and extraordinary modes, respectively, as shown in the oscilloscope
traces in Figure 6.28. For the waveguide oriented parallel to the z-axis, or termed
156
Fiber input
Resonant waveguides
Figure 6.27: Titanium indiused LiNbO
3
waveguide resonator with cleaved edges coated
with 2 nm of gold. An optical ber coupled input and 10 lens to collect transmitted
power are shown.
157
the straight waveguide, the resonance was observed when the input polarization
was oriented in the ^ x direction. For the waveguide with the 2
tilt, the resonance
was observed when the input light was polarized parallel to the cleaved facet, thus
having a combination of both y and z components.
The purpose of this experiment was a preliminary justication of the resonance
condition for a cleaved edge titanium indiused LiNbO
3
waveguide resonator. Thus,
to enhance the resonance performance, the gold mirrors should be replaced with
dielectric mirrors, as was used in the z-cut titanium indiused LiNbO
3
resonator.
Once the resonance is enhanced, it may be useful as an electro-optic modulator.
158
Figure 6.28: Resonances of cleaved edge x-cut titanium indiused LiNbO
3
waveguide
resonator. Measured FSR values are approximately 13.5 GHz and 14.3 GHz for the
ordinary and extraordinary modes, respectively. The theoretical values are 13.8 GHz and
14.2 GHz, respectively.
159
Chapter 7
EQUILATERAL TRIANGLE RESONATOR
In this chapter, connement of an optical wave by total internal re
ection will be
studied. The particular structure incorporates a titanium indiused waveguide in
LiNbO
3
with cleaved facets. The optical wave may be conned in the resonator if
the wave satises the condition of total internal re
ection at the interface by Snell's
law.
i
sin
1
n
2
n
1
; (7.1)
where
i
is the angle of incidence, n
1
is the refractive index of the medium within
which the wave travels before incurring the boundary, andn
2
is the refractive index
of the surrounding medium, or the medium on the other side of the boundary.
Assuming an operating wavelength of = 1:31 m, the refractive index value
for titanium indiused LiNbO
3
for the ordinary and extraordinary waves are n
o
=
2:22568 and n
e
= 2:15647. Thus the minimum angles needed for total internal
160
y
x
z
x
cleaved facets
cleaved facet
Ti-indiffused
waveguide
LiNbO
3
60°
Figure 7.1: Schematic of the equilateral triangle resonator
re
ection are,
o
i
26:70
; (7.2)
e
i
27:63
; (7.3)
where the superscripts `o' and `e' stand for the ordinary and extraordinary mode,
respectively. A simple structure that will satisfy the condition of total internal
re
ection is an equilateral triangle. The apex angle of 60
requires the angle of
incidence to be 30
at the boundaries. Thus, if a waveguide mode exists within
an equilateral triangle structure, the condition for total internal re
ection would
be satised at every boundary, resulting in a resonant structure. The equilateral
triangle resonator is shown in Figure 7.1
In Section 6.4, the cleavage planes of z-cut LiNbO
3
were analyzed and it was
161
found that an equilateral structure could be cleaved. There exists cleavage planes
that are oriented 60
with respect to the y-axis, or 30
with respect to the x-axis.
These two cleaved edges along with the cleaved edge parallel to the y-axis, allow
for a possible cleaved edge equilateral triangle resonator.
Titanium indiused waveguides became necessary in the incorporation of this
resonator due to the fact that without a waveguiding mechanism, beam divergence
for a resonator becomes much too signicant to ignore. Also, the waveguide provides
a good visual reference as to where the optical wave should travel. The cleaved edge
limits the useable height from the top surface to approximately 20m, which is of no
consequence for an indiused waveguide because of the limited depth of indiusion.
However, if an indiused waveguide is not available, the input light is constrained
to a beam waist of approximately 20 m in the lateral direction. If one considers
the input beam as a Gaussian beam, the angular beam spread can be calculated
[64].
The beam spot size at a distance, z is,
!
o;e
(z) =!
o
1 +
z
2
(z
o;e
0
)
2
1=2
; (7.4)
where `o' and `e' stands for the ordinary and extraordinary mode, respectively, and
z
0
is the confocal parameter given by,
z
o;e
0
=
!
2
o
n
o;e
,
(7.5)
and !
o
is the minimum beam waist.
162
Thus, assuming !
o
= 20 m, at a distance of z = 10 mm, the beam waist
becomes,
!
o
= 96 m ; (7.6)
!
e
= 99 m : (7.7)
A roundtrip length of a resonator can vary depending on the feasibility of cleaving
small samples. It was found that one can easily cleave an equilateral triangle
resonator with the smallest round trip length of approximately 10 mm. The increase
in beam waist with distance is detrimental to the performance of a resonator.
The divergence of the optical beam can be mitigated with the use of an indiused
waveguide. The waveguide also serves as a reference to the precise cleavage planes of
the equilateral triangle. Unfortunately, with the cleaving process developed for this
dissertation, the accuracy of the cleaved edge location is20m. Shown in Figure
7.2 is a titanium indiused waveguide at one of the cleaved facets with a misaligned
cleaved edge. Figure 7.4 shows the amount of power coupled into the waveguide if
the cleaved edge has an oset from the ideal cleaved location. A normalized power
of unity describes the situation where there is no oset between the re
ected optical
mode and the waveguide. The model considers two modes with lateral diusion
proles given by Equation (2.4) and their intensity overlap as a function of lateral
displacement from the waveguide, assuming a temperature of 1080
C, a diusion
time of 8 hours, a titanium strip width of 8m and a strip height of 90 nm. As can
163
cleaved facet
Ti-indiffused
waveguide
Figure 7.2: Fabricated equilateral triangle resonator. The accuracy of the cleaving process
is20 m, which results in waveguide misalignment.
be seen from the gure, an error of 20 m contributes to a large loss of the optical
intensity. Furthermore, if both of the other cleaved facets were perfect mirrors
and aligned perfectly, as described with re
ectances equal to unity, the eect of a
decreased re
ectance from the third cleaved facet can be related to the nesse, as
shown in Figure 7.5. Thus in order to establish the equilateral triangle resonator
as a feasible device, better control of the cleaving process must be obtained.
Much like the electro-optic crystal disk resonator of Section 1.1.2, the preferred
164
Figure 7.3: Fabricated equilateral triangle on a pedestal.
165
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Offset (micrometer)
Normalized Overlap
Figure 7.4: Normalized power for a mode oset laterally from the dened core waveguide
region.
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
35
40
Reflectance
Finesse
Figure 7.5: Dependence of nesse on a cleaved facet re
ectance, given the other two
cleaved facets possess re
ectances of unity.
166
method of coupling to the equilateral triangle resonator will be with a prism. The
disadvantage is that the device fails to be an integrated device, as the prism resides
outside of the structure. Nevertheless, the equilateral triangle resonator may still
be a useful device to study as it eliminates the need for polished end mirrors and
can be used as an electro-optic modulator.
167
Chapter 8
CONCLUSION
The work of this dissertation was aimed at understanding and employing a Fabry-
Perot optical resonator to ultimately be used as an electro-optic modulator. The
titanium indiused lithium niobate waveguide was fabricated and incorporated in
two dierent types of resonant geometries. A titanium indiused z-cut lithium
niobate waveguide resonator, with dielectric mirror coatings with re
ectances of
91% and 98%, was fabricated with a nesse of 14 for the TM-like mode, at an
operating wavelength of 1.55 m. The resonator proved to be an integrated device
with the use of ber pigtailing.
An alternate geometry was analyzed with the use of the cleavage planes of
lithium niobate. The cleaved edges eliminated the need for the mirror facet polish-
ing that is necessary for Fabry-Perot resonators. The desired cleaved edges require
the use of an x-cut lithium niobate substrate, with diering orientations of the
titanium indiused waveguide for the ordinary and extraordinary modes. For reso-
nance of the extraordinary mode, the waveguide was tilted by 2
relative to ordinary
mode waveguide.
168
The cleaved edge waveguide required an electromagnetic wave to propagate in
a medium where the c-axis was not parallel to the z-axis, which is the direction
normal to the propagating phase fronts. In order to fully understand the electro-
magnetic eld within the cleaved edge waveguide geometry, the theoretical analysis
began with an investigation of a rectangular waveguide lled with an anisotropic
dielectric medium with the c-axis parallel to the z-axis. Understanding the form
of the elds within this waveguide is directly applicable to a dielectric waveguide,
as they are similar. The electromagnetic wave was analyzed in a uniform, homoge-
neous, anisotropic material with the c-axis tilted with respect to the z-axis. This
analysis led to the understanding that the extraordinary mode possesses a prop-
agation vector that is not parallel to the Poynting vector. Thus, the waveguide
for the extraordinary mode resonator was oriented at an angle with respect to the
propagation vector. The slab waveguide was then analyzed for the anisotropic ma-
terial with thec-axis tilted with respect to thez-axis, and related to a homogeneous
dielectric waveguide. The eective index method was analyzed and incorporated a
permittivity tensor with o-diagonal components.
A cleaved edge titanium indiused x-cut lithium niobate resonator was fabri-
cated, and operated at a wavelength of 1.31m. Resonances were observed at 13.5
GHz and 14.3 GHz for the ordinary and extraordinary modes, respectively. As a
preliminary experiment, the cleaved facets were coated with 2 nm of gold to serve
as the mirrors. A more ecient resonator should possess dielectric mirror coatings.
169
Finally, an equilateral triangle resonator was proposed, with the use of the
cleavage planes for z-cut lithium niobate. The tolerance of the preciseness of the
cleaved locations for this resonator made the present cleaving method inadequate.
Thus, the idea was presented for future work, as the cleaving method may develop
and become more precise. To conclude, the Fabry-Perot resonators presented in
this dissertation are a feasible alternative to the available resonators.
170
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Abstract (if available)
Abstract
An electro-optic resonant modulator consists of a dielectric material that responds to an applied electric field by way of amplitude modulation, as an optical signal propagates through and resonates within. This work focuses on the analysis and fabrication of the resonator. The resonator is of a Fabry-Perot type, employing a titanium indiffused lithium niobate waveguide as the optical guide. The resonator is an integrated device and able to be fiber pigtailed, to be used as a remote device. A z-cut lithium niobate resonator with dielectric mirror coatings was fabricated with a finesse of 14, for the TM-like mode at an operating wavelength of 1.55 micrometers. The cleavage planes of lithium niobate are investigated and incorporated in the final design of the resonator. The cleaved edges eliminate the need for the edge polishing that is used for the mirrors of the Fabry-Perot resonant cavity. The desired cleaved edges require the use of a x-cut lithium niobate substrate, with differing orientations of the titanium indiffused waveguide for the ordinary and extraordinary modes. This dissertation presents a theoretical analysis of the electromagnetic wave in a medium with off-diagonal components in the permittivity tensor, as is necessary for the cleaved edge resonator geometry. A theoretical analysis of the electro-optic effect for the cleaved edge geometry is also presented. A cleaved edge titanium indiffused lithium niobate resonator was fabricated, and resonances were observed at 13.5 GHz and 14.3 GHz for the ordinary and extraordinary modes, respectively with 2 nm gold coated facets, at an operating wavelength of 1.3 micrometers. Given the useable cleavage planes of z-cut lithium niobate, an equilateral triangle resonator is proposed. The theoretical and experimental results contained in this dissertation validate the feasibility and viability of a Fabry-Perot resonator as a modulating medium.
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Asset Metadata
Creator
Takahashi, Satsuki
(author)
Core Title
Analysis and design of cleaved edge lithium niobate waveguide resonators
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
07/06/2012
Defense Date
06/08/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
anisotropic waveguides,electro-optic modulators,equilateral triangle resonators,OAI-PMH Harvest,optical resonators,waveguide resonators
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Steier, William Henry (
committee chair
), Armani, Andrea M. (
committee member
), Sawchuk, Alexander A. (Sandy) (
committee member
)
Creator Email
satsuki.takahashi@gmail.com,satsukit@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-52984
Unique identifier
UC11289925
Identifier
usctheses-c3-52984 (legacy record id)
Legacy Identifier
etd-TakahashiS-921.pdf
Dmrecord
52984
Document Type
Dissertation
Rights
Takahashi, Satsuki
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
anisotropic waveguides
electro-optic modulators
equilateral triangle resonators
optical resonators
waveguide resonators