Title Page
 Table of Contents
 List of Figures
 Theory of optical fields in multilayer...
 A new wavelength selective fabry-perot...
 A novel waveguide geometry
 Device fabrication techniques
 Experimental results of the new...
 Experimental results of grating...
 Conclusions and future work
 Biographical sketch

Title: novel tunable filter for wavelength division multiplexed communication systems
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00082356/00001
 Material Information
Title: novel tunable filter for wavelength division multiplexed communication systems
Series Title: novel tunable filter for wavelength division multiplexed communication systems
Physical Description: Book
Creator: Hussell, Christopher Parke,
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Bibliographic ID: UF00082356
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: oclc - 33628675
alephbibnum - 002056059

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
        Page v
        Page vi
    Table of Contents
        Page vii
        Page viii
        Page ix
    List of Figures
        Page x
        Page xi
        Page xii
        Page xiii
        Page xiv
        Page xv
        Page xvi
        Page xvii
        Page 1
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        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
    Theory of optical fields in multilayer structures
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
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        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
    A new wavelength selective fabry-perot filter
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
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        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
    A novel waveguide geometry
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
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        Page 86
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        Page 88
    Device fabrication techniques
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
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        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
    Experimental results of the new waveguide geometry
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
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        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
    Experimental results of grating filters in LiNbO3
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
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    Conclusions and future work
        Page 156
        Page 157
        Page 158
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    Biographical sketch
        Page 180
        Page 181
Full Text







Copyright 1995


Christopher Parke Hussell

To my loving wife, Mary.


Many people have assisted me with my work at the University of Florida and it

would be impossible to acknowledge all of them. However, I would like to acknowledge

several individuals in particular.

First, I want to thank Dr. Ramu V. Ramaswamy for his hard work and devotion

to his students in providing funds, well-equipped laboratory facilities, and a stable

working environment conducive for research. His devotion towards generating funds

for the Photonics Research Laboratory and the University is truly commendable.

His success in doing so allowed me to take a novel concept and buy the necessary

equipment for its demonstration in a very reasonable time frame. His willingness to

share his broad knowledge of this and related areas proved to be essential for the

completion of this work.

I would also like to thank Dr. Robert Tavlykaev for constantly providing helpful

discussions. I found him to be very knowledgeable of the current as well as past

literature. This has made him an invaluable resource for me.

Very conducive discussions with Dr. K. Thyagarajan helped me to have a deeper

understanding of the problem at hand. I found him to have a very broad theoretical

knowledge. He was always willing to assist.

I am grateful to Dr. Sanjai Sinha for assisting me with MOCVD growth. I also

found him to be very helpful in providing me with ideas. In fact, when looking for

the best materials to use for a dielectric mirror, he suggested using silicon. Not long

after that, I formulated the idea of using silicon as a buffer layer for DBR grating


I am indebted to Hsing Chien Cheng. He was very eager on countless occasions to

help me with theoretical modeling, fabrication, and measurement. I greatly respect

his hard work and pleasant attitude.

I want to thank the professors in the area: Dr. Peter Zory, and Dr. Ramakant

Srivastava. Their work at the University is commendable.

I want to thank Dr. Scott Samson for reviewing the rough draft of this dissertation.

His comments helped me to convey the information in this work more effectively.

Also, I appreciate my other colleagues, particularly, Wuhong Li, Suning Xie, Dave

Marring, Kirk Lewis, Ron Slocumb, Dr. Amalia Miliou, Jamal Natour, Dr. Sang K.

Han, Dr. Hyoun S. Kim, Dr. Song Jae Lee, Dr. Young Soon Kim, Dr. Sang Sun

Lee. Through personal discussions and group meetings, these people have been very


Several members of the department have also provided me with necessary as-

sistance in their areas of expertise. The machinist, Allan Herlinger, professionally

modeled several optical measurement devices for me. Keith Rambo helped me to

plan the installation of the RIE, provided me with spare parts, and assisted me with

finding vendors of vacuum equipment. The Microelectronics Laboratory managers,

James Chamblee, Tim Vaught, Steve Schein, and Jim Hales, provided me with two

well-maintained laboratories.

The brotherhood in Gainesville at the Campus Church of Christ have warmly

opened their homes to me and my family constantly from the time we first moved

into the area. These people, far too many to list here, have kept me in their prayers

and offered much moral support.

My father, Lewis, and mother, Rosalee, have been supportive to me towards

fulfilling my dreams since childhood. Their love and support continue to encourage

me to attain higher goals. I am truly grateful for their financial support. I love them

very much.

My greatest appreciation goes to my wife, Mary. She has unceasingly supported

me in keeping the home and family in order. She is very devoted and giving to our

lovely children, Ashley and Kristin, who take after their mother in beauty. She has

endured much and keeps on smiling. Truly, she has made it all worth while. My love

for her is unending. My children truly bring joy to my life.


ACKNOWLEDGEMENTS ............................ iv

LIST OF FIGURES .................... .......... x

ABSTRACT .................................... xvi


1 INTRODUCTION ............................... 1

1.1 M otivation . . . . . . . . . . . . . . . . 1
1.1.1 Bandwidth of Optical Fiber .................. 2
1.1.2 Practical WDM Communication Systems ........... 4
1.1.3 Wavelength Selective lOCs ................... 6
1.2 A New Wavelength Selective Tunable Filter ............. 11
1.3 A Novel Waveguide Technology .................... 14
1.4 Chapter Organization ........................... 17


2.1 Basic Equations .............................. 21
2.2 Reflection and Transmission From a Single Boundary ........ 27
2.3 Characteristic Matrix of a Thin Film ................. 31
2.4 Characteristics of a Multilayer Structure ............... 33
2.5 Alternative Matrix Formulation .................... 36
2.6 Connection Between the Two Formalisms ............... 37
2.7 Applications of the Formalisms to Real Structures . . . . ... 39
2.7.1 Reflectance from Periodic Quarter-Wave Structures . . 39
2.7.2 Basic Laser Relationships . . . . . . . . . . 40
2.7.3 Basic Waveguide Relationships . . . . . . . . 43
2.7.4 Application to Corrugated Waveguides . . . . . ... 44
2.8 Spectral Index Method .......................... 47
2.9 Summary of the Numerical Methods . . . . . . . . . 51


3.1 Thin Film Dielectric Fabry-Perot Filters . . . . . . . . 52
3.1.1 Characteristics of a Thin Film Dielectric Mirror . . ... 52
3.1.2 Characteristics of a Thin Film Dielectric Fabry-Perot Filter 53
3.2 DBR Corrugated Waveguide Fabry-Perot Filters . . . . . ... 55

3.2.1 Characteristics of a DBR Corrugated Waveguide . . ... 55
3.2.2 Characteristics of a DBR Corrugated Waveguide Fabry-Perot
F ilter . . . . . . . . . . . . . . . . 58
3.3 Proposal: Asymmetric Fabry-Perot Filter . . . . . . . . 59
3.3.1 Passive ASFP Filter .... ................... 60
3.3.2 Tuning an ASFP filter ..................... 62
3.3.3 Tunable DFB laser ....................... 65
3.3.4 Sum m ary ............................ 66

4 A NOVEL WAVEGUIDE GEOMETRY . . . . . . . . . 68

4.1 The Need for Better DBR Corrugated Waveguides . . . . ... 68
4.2 Corrugated Waveguide Technologies (Previous Work) . . . ... 69
4.2.1 Investigations of DBR Corrugated Waveguides . . . .... 70
4.2.2 Passive DBR Corrugated Waveguide Devices . . . .... 71
4.3 Difficulties in Achieving High Reflectance . . . . . . . . 72
4.4 Proposal: Channel Waveguides With High-Index Overlays . . 74
4.5 Benefits of Silicon as a High-Index Overlay On LiNbO3 . . ... 78
4.5.1 Previous Reports of Waveguides with Silicon Overlays . . 78
4.5.2 Additional Benefits of Silicon on LiNbO3 . . . . . ... 80
4.6 Misconceptions of Guiding in Thick-Film Structures . . . .... 80
4.7 Properties of LiNbO3 Waveguides With Varying Silicon Overlay Thick-
ness . . . . . . . . . . . . . . . . . . 85
4.8 Sum m ary . ... .... .... ... .... .... .. .. .. .. 88


5.1 Reactive Ion Etching .......................... 89
5.1.1 Cl2 Reactive Ion Etching . . . . . . . . . . 93
5.1.2 02 Plasma Ashing ....................... 96
5.2 Waveguide Fabrication ......................... 100
5.3 Grating Lithography and Transfer . . . . . . . . . . 103
5.4 Polishing . .. . . . . . . . . . . . . . . 107
5.5 Endface DBR Fabrication ....................... 108
5.6 Sum m ary ... .. .. ... .... .... .... .. .. ... . 110


6.1 Near Field Profile Measurement .
6.1.1 Measurement Technique .
6.1.2 Field Profile Results ...
6.2 Loss Measurement ........
6.2.1 Waveguide Device Reflection
6.2.2 Loss Results .........
6.3 Mode Index Measurement ....
6.3.1 Prism Coupler Technique. .
6.3.2 Prism Coupler Results . .
6.4 Summary .............

. . . . . .
. . . . . .
. . . . . .
. . . . . .
and Transmission
. . . . . .
. . . . . .
. . . . . .

. . . . o .



7.1 Waveguide Grating Results ........................ 129
7.1.1 The First Sample: APE:LiTaOs . . . . . . . . 131
7.1.2 The First APE:LiNbO3 Sample ................ 132
7.1.3 The First Ti:LiNbO3 Sample .................. 136
7.1.4 Short Period DBR Corrugated Waveguide . . . . ... 137
7.1.5 The Best Reflection Measurements . . . . . . . 137
7.2 ASFP Filter Results .......................... 140
7.2.1 Results of ASFP #1 ...................... 140
7.2.2 ASFP Filter Results with Endface-Deposited DBR Mirror on
ASFP # 2 .... ... .... ... .... .. .. ... .. 144
7.3 Prospects for Improvement ........................ 146
7.3.1 Improvement of Grating Fabrication . . . . . . . 147
7.3.2 Optimization of APE:LiNbO3 Waveguides . . . . ... 151
7.3.3 Optimization of the Silicon Thickness . . . . . . . 151
7.3.4 Optimize Structure With Buffer Layer . . . . . . 152
7.4 Sum m ary ................................ 152

8 CONCLUSIONS AND FUTURE WORK . . . . . . . . . 156

8.1 Conclusions . . . . .. . . . . . . . . . . 156
8.2 Future W ork ................................ 158
8.2.1 Detailed Study of Waveguides With High-Index Overlays . 159
8.2.2 Optimization of Grating Performance . . . . . . . 160
8.2.3 High Performance lOCs .................... 161


A DISPERSION OF SILICON ................ .... ... 166


REFERENCES ................................... 173

BIOGRAPHICAL SKETCH ............................ 180


1.1 Attenuation of a low-loss optical fiber (reference [45]). . . . . 3

1.2 Simplified schematic of a WDM distribution system . . . . . 4

1.3 Simplified schematic of a WDM ring network. . . . . . . 5

1.4 Polarization insensitive acoustically tunable optical filter. . . . 7

1.5 Cascaded Mach-Zehnder interferometer. . . . . . . . . 9

1.6 Resonant grating assisted directional coupler filter. . . . . . 9

1.7 Proposed Tunable Wavelength Selective Filter. . . . . . .. 12

1.8 Comparison of the reflection spectra of a dielectric endface mirror and
a corrugated waveguide. a) shows that the reflection band of a corru-
gated waveguide is very narrow compared to the spectra of the dielec-
tric mirror. b) shows an expanded view of the corrugated waveguide
spectra. . . . . . . . . . . . . . . . .. . 13

1.9 Transverse cross section of a graded-index channel waveguide with an
overlaying film of refractive index near or higher than the core of the
w aveguide .. . . . . . . . . . . . . . . . . 15

1.10 Longitudinal cross section of a DBR corrugated waveguide made from
a high-index overlay. ......................... 15

2.1 k vector relationships with light incident on a single surface. ... . 27

2.2 Conventions for the positive directions of the E and H field vectors for
the (a) TE or s-polarized waves and (b) TM or p-polarized waves. . 28

2.3 A single dielectric planar layer on a substrate. . . . . . ... 31

2.4 A general multilayer thin-film structure. . . . . . . .. 34

2.5 Dielectric mirror made from a periodic quarter-wave stack. .. . . 40

2.6 Single-film laser.. ............................. 41

2.7 Single-film waveguide. ........................ 43

2.8 Single mode waveguide with an abrupt step. . . . . . ... 45

2.9 Dual rib waveguide cross section. The dashed line represents the ef-
fective movement of the boundary by the Goos-Hiinchen shift. . . 48

3.1 Simple DBR consisting of a multilayer dielectric stack (a), and its
reflectance and transmission spectra (b). The parameters are no = 1,
ns = 2.14, nL = 1.2, nH = 2.2, and 4 periods . . . . . .... 53

3.2 Fabry-Perot filter consisting of two multilayer dielectric stack DBRs
separated by a short waveguide cavity (a), reflectance spectra of the
structure with a cavity length of 10pm (b). The parameters for each
mirror are no = 1, n, = 2.14, nL = 1.2, nH = 2.2, and 4 periods. . 54

3.3 DBR corrugated waveguide (a), reflectance spectra (b). The parame-
ters used in the calculation are NH = 2.1406 and NL = No = N, = 2.14
and 10,000 periods. Loss of 0.35dB/cm is also included in the calculation. 56

3.4 Fabry-Perot filter consisting of two DBR corrugated waveguides sepa-
rated by a short waveguide cavity (a), reflectance spectra of the struc-
ture (b) . . . . . . . . . . . . . . . . . 58

3.5 Asymmetric Fabry-Perot filter consisting of a DBR corrugated waveg-
uide with a dielectric mirror deposited on the endface . . . .... 60

3.6 Transmission spectra of an asymmetric Fabry-Perot filter (a); expanded
view of the transmission peak (b). The parameters used for the dielec-
tric mirror were the same as those used for figure 3.1(b) and those used
for the corrugated waveguide were the same as those use for figure 3.3(b). 61

3.7 Transmission spectra of an ASFP filter for a corrugated waveguide loss
of .35dB/cm (solid) and 3.5dB/cm (dotted). . . . . . .. 62

3.8 Tunable ASFP filter. .......................... 63

3.9 Transmission spectra of a Tunable ASFP filter for several values of the
mode index change ............................. 64

4.1 Longitudinal cross section of a graded-index waveguide with an over-
laying film grating (a); local normal mode index profile along the prop-
agation direction (b) ............................ 75

4.2 Corrugated waveguide structure used by Adar et al. [2] (a); Local nor-
mal mode index profile of the structure along the propagation direction
(b); Local normal mode index profile of the structure with Si3N4 only
in the valleys of the grating (solid) and with no Si3N4 at all (dotted) (c). 77

4.3 Simple ridge waveguide structure used to illustrate a case were the
effective index method gives the wrong number of modes. This waveg-
uide is single mode, but the effective index method predicts five modes. 82

4.4 The effective index method is applied to the waveguide structure in
figure 4.3 by finding the mode indices of asymmetric planar waveguides
with film thicknesses of D (a) and H (b). These mode indices are
applied to a symmetric laterally confined planar waveguide (c). The
mode indices of the waveguide depicted in (c) are generally accepted
to be equivalent to the mode indices of the structure in figure 4.3. . 83

4.5 Viewing the proposed structure (figure 1.9) upside down, we see some
similarity between it and the structure in figure 4.3. In this case, the
rectangular ridge is replaced by a lower refractive index strip load with
larger cross section ............................. 84

4.6 High-index strip-loaded waveguide structure used for spectral index
calculations . . . . . . . . . . . . . . . . 85

4.7 Dependence of the mode index of the device in figure 4.6 on silicon
strip thickness for several strip widths. . . . . . . . .. 86

5.1 Schematic illustration of the voltage drop across a plasma discharge. 90

5.2 RIE machine used for etching the devices in this work. . . . . 91

5.3 Diagram of the RIE system pictured in figure 5.2. . . . . .... 92

5.4 A trench 3pm wide by 9pm deep, etched in GaAs. . . . . .... 94

5.5 A 1/m period grating etched into GaAs by Cl2 RIE. . . . .... 94

5.6 Schematic of a multilayer mask that can be used for very deep RIE or
thick lift-off, a) shows the structure just after developing the imaging
layer and b) shows the completed structure. . . . . . .. 97

5.7 Picture of the multilayer photoresist mask from an optical microscope. 98

5.8 SEM picture the result of etching GaAs using a multilayer structure
without the SiO2 layer and non-optimized 02 etching parameters. . 100

5.9 Standard fabrication process for annealed proton exchanged LiNbO3
waveguides. . . . . .. . . . . . . . . . .. . 102

5.10 Diagram of the grating lithography setup. . . . . . . ... 104

5.11 Plot of equation 5.1. ......................... 105

5.12 Setup used to verify the period and measure the chirp of the gratings. 105

5.13 SEM picture of an etched grating on a silicon substrate . . . .

5.14 Picture of two stages of an LiNbO3 endface polish. After polishing
with 0.25/m diamond paste (a), and after using a colloidal alumina
polishing compound (b,c). The LiNbO3 substrate (bottom) with a
thin silicon layer on top of it, followed by a 2/pm thick layer of SiO2.
The silicon layer is barely visible in the picture. Above the SiO2 layer
is a thin layer of wax, which is thick enough to be visible only in (c),
that does not polish well. A silicon polishing block is on the top. The
picture was taken by an optical microscope with the contrast enhanced
to show any scratches. ........................ 109

5.15 Diagram of the sample holder/shutter assembly (a), and picture (b) of
the e-beam deposition system ....................... 110

5.16 Calculated and measured transmission of two dielectric mirrors de-
posited by the system shown in figure 5.15 (a) onto glass microscope
slides, and calculated reflection of the same structure (b). . . .... 111

6.1 Near field measurement setup. . . . . . . . . . .. 114

6.2 Contour plots of the optical intensity profiles for the fiber (upper left)
and waveguides of several mask widths with a 1775A silicon overlay. 120

6.3 Experimental setup used to measure reflection, transmission, and loss
of waveguide device samples. . . . . . . . . . . ... 121

6.4 Loss measurements of z-cut APE:LiNbO3 waveguides proton exchanged
in pyrophosphoric acid at 200C for 71 minutes, and annealed for 7
hours at 3500C for two sample lengths. . . . . . . . ... 123
6.5 Loss measurements of z-cut APE:LiNbO3 waveguides proton exchanged
in pyrophosphoric acid at 200C for 71 minutes, and annealed for 7
hours at 350C for three thicknesses of silicon: none, 4120A, and 8200A. 124

6.6 Schematic diagram of the prism coupler. . . . . . . ... 125

6.7 Measurements of the mode index of APE:LiNbO3 waveguides with
silicon overlays of different thicknesses. . . . . . . . .. 127

7.1 Transmission spectra of a 5pm wide APE:LiTaO3 waveguide on the
first DBR corrugated waveguide sample (DBR #1), fabricated with
the high-index overlay technology. . . . . . . . . . . 132

7.2 Transmission spectra of an 8.5/tm waveguide on the first APE:LiNbO3
DBR corrugated waveguide sample (DBR #2). . . . . . .. 133


7.3 Exact solution of the mode index of the first three modes of a planar
waveguide with a cladding index of 1.0, a film index of 3.5 (silicon), and
a substrate index of 2.14 for varying film thicknesses at a wavelength
of 1534nm .. . . . . . . . . . . . . . . . . 134

7.4 The grating not only causes coupling between the forward and re-
verse guided modes, but also between the forward guided mode and
reverse radiation modes. The planar high-index overlay allows for one-
dimensional radiation with mode indices above those of the channel
w aveguide .. . . . . . . . . . . . . . . . . 135

7.5 Transmission spectra of the TM mode of a 5.5/pm waveguide on the
first Ti:LiNbO3 DBR corrugated waveguide sample (DBR #3). . . 136

7.6 Transmission spectra of a 10pm waveguide on a DBR corrugated waveg-
uide sample with a shorter period. After these measurements, the
silicon grating was removed and a new one fabricated on the sample
(D BR # 4). . . . . . . . . . . . . . . .. . 137

7.7 Reflection spectra of a 10ym waveguide on an APE:LiNbO3 DBR cor-
rugated waveguide sample (DBR #4). . . . . . . . ... 138

7.8 Calculated and measured reflection and transmission of a DBR cor-
rugated waveguide with ~ 10,300 periods, AN = 2.8 x 10-4, A =
358.55nm, and a 50:50 duty cycle (DBR #5). . . . . . .. 139

7.9 SEM picture of a grating over an APE:LiNbO3 waveguide. .. . . 139

7.10 Structure of sample ASFP #1. The cavity is of sufficient length to
add electrodes. .. ....... .... ... .... .. .. ... 141

7.11 Reflection spectra of the sample configuration depicted in figure 7.10. 142

7.12 An external dielectric mirror was index matched and butt-coupled to
the device of figure 7.10. ....................... 142

7.13 Reflection spectra of the sample with an external butt-coupled mirror
as depicted in figure 7.12. ...................... 143

7.14 Measured and calculated transmission of the device in figure 7.12. . 143

7.15 ASFP #1 after fabricating the electrodes. . . . . . . ... 144

7.16 Two sets of measured results for OV and 30V for the device in figure 7.15.145

7.17 A dielectric mirror was deposited on the end of a device as shown. . 145

7.18 Calculated and measured reflection spectra for the device in figure 7.17.146

7.19 SEM picture of the grating profile produced on a silicon substrate by
R IE . . . . . . . . . . . . . . . . .. . 147

7.20 Calculated reflection spectra of a DBR corrugated waveguide with a
30:70 duty cycle. ........................... 148

7.21 A grating with a curved longitudinal mode index profile (a) was used
to calculate the effect of a non-square grating on the reflectance spectra
(b ) . . . . . . . . . . . . . . . . . . . 149

7.22 SEM picture of an etched grating on a silicon substrate produced by
under exposure ............................... 149

7.23 Reflection spectra of a chirped DBR corrugated waveguide. .. . . 150

8.1 Conceptual drawing of an ASFP directional coupler. . . . .... 162

8.2 A tunable Er:LiNbO3 single longitudinal mode laser. . . . .... 162

8.3 Application of high index overlay strips to reduce radiation loss at
waveguide bends. ........................... 163

8.4 Conceptual drawing of a WDM Mux/Demux Chip. . . . .... 164

A.1 Match between the measured and theoretical transmission through an
8050A thick silicon film on a glass microscope slide after iteration of
the dispersion curves ............................ 167

A.2 Comparison between the known dispersion curves of single crystal sil-
icon and the e-beam deposited 8050A film . . . . . . ... 168

A.3 Match between the measured and theoretical transmission through a
4210A thick silicon film on a glass microscope slide using the e-beam
Si dispersion curves in figure A.2. . . . . . . . . .. 168

A.4 Match between the measured and theoretical transmission through a
4180A thick silicon film on a glass microscope slide using the e-beam
Si dispersion curves in figure A.2. The poor result is attributed to
oxidation of the Si film caused by evaporating SiO2 immediately prior
to the Si evaporation ............................ 169

A.5 Comparison of the dispersion curves of single crystal silicon, uncontam-
inated e-beam deposited silicon, and contaminated e-beam deposited
silicon . . . . . . . . . . . . . . . . . . 170

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Christopher Parke Hussell

August 1995

Chairman: Dr. Ramu V. Ramaswamy
Major Department: Electrical and Computer Engineering

Two essential integrated optical components of wavelength division multiplexed

(WDM) systems are tunable lasers and tunable wavelength selective filters, which we

address in this dissertation within a common framework by proposing a new concept

that has commercial potential.

Conventional Fabry-Perot filters lack the free spectral range (FSR) required in

WDM systems. A new approach to achieve transmission filters with large FSR,

tunable over many channels using the linear electro-optic effect, is proposed. By

simply adding gain to the waveguide, this configuration is extendible as a tunable,

single longitudinal mode laser The most significant obstacle in realizing these filters

is achieving high-reflectance corrugated waveguides-an essential consideration for

achieving a useful filter.

The primary contribution of this work is the novel approach used to achieve

high-reflectance corrugated waveguides. The concept involves channel waveguides

with planar films of higher refractive index on top. This influences the propagation

constant of the channel waveguide mode. However, the film does not become the

primary guiding layer (the majority of the guided power remains in the lower-index

material) even for a considerably thick film. A somewhat startling finding is that

planar films with thicknesses capable of supporting multiple planar guided modes

do not necessarily have a measurable affect on the loss and mode field profile of the

channel waveguide. The effect on the propagation constant is small, but measurable.

This new technology forms the basis of an ideal geometry for creating corrugated

waveguide devices.

These new concepts are applied and demonstrated in a material system in which

it is difficult to achieve high reflectance from a corrugated waveguide-LiNbO3. We

choose this material not only to prove the robustness of the new technology, but

because it is one of the few material systems in which integrated optical circuit prod-

ucts are commercially available. Preliminary structures exhibit the highest reflection

reported to date in LiNbO3 corrugated waveguides (80%) with an order of magni-

tude shorter length (3.7mm) than that has recently been reported for devices with

comparable reflection. The results of the proposed wavelength selective filter are in

excellent agreement with the theory.



Our motivation for this dissertation begins with a discussion of how wavelength

division multiplexing in optical fiber communication systems can be used to meet the

growing need for high-bandwidth services. Most of the components in these commu-

nication systems such as tunable laser sources, external modulators, star couplers,

fiber amplifiers, and detectors are already commercially available. Several concepts

for the remaining crucial component, the tunable wavelength selective filter, have also

been reported; however, they are either unrealizable for commercial applications or

inadequate in meeting the needs of practical WDM systems. We propose a new filter

device that can meet the needs of even the most complex WDM system. However, the

primary difficulty of this filter is that it requires a totally new technology: the fabri-

cation of a high reflectance corrugated waveguide in an electro-optic material. This

problem is addressed and a novel waveguide geometry that overcomes the difficulties

inherent to most other waveguide geometries associated with high reflectance corru-

gated waveguides is proposed. Section 1.4 presents the organization of the ensuing

chapters of this dissertation.

1.1 Motivation

This information age that we are living in now continues to demand more and more

communications bandwidth. However, the bandwidth of typical residential telephone

services has remained virtually the same for decades. Integrated Services Digital

Network (ISDN) offers a modest improvement in digital transmission rates, but much

higher bandwidth is required in order to move into a new era of communications along

with the continuing advancements of computer hardware that is propelling us into

this explosive information age.

There is great demand for broadband distribution networks such as CATV, but

consumers also require services that can only be provided by two-way, routable net-

works. These broadband communication networks are the key to the future of many

recently emerging technologies such as video on demand, video teleconferenceing, re-

mote medical imaging, home shopping, and virtual reality networking, just to name

a few. These technologies will further enable us to meet in a virtual world to discuss

and share ideas, to conduct business transactions, and to embark towards new lev-

els of entertainment, all in the comfort of our homes. Recently, all the processing,

storage, and imaging technology required to do these things has become available at

affordable prices. However, these services will be severely limited until the realization

of Broadband ISDN.

It is inevitable that a broadband communications era is on the horizon. The

question is, "What is the most efficient way to realize broadband communication


1.1.1 Bandwidth of Optical Fiber

The question just posed is easy to answer because conventional optical fiber offers

the bandwidth we need-much more than what is presently being utilized. To deter-

mine the bandwidth of an optical fiber, we recall that the bandwidth B is related to

the spectral width AA by

B=AAf= (1.1)



1000 1200 1400 1600 1800 2000
Wavelength (nm)

Figure 1.1. Attenuation of a low-loss optical fiber (reference [45]).

where c and A are the speed of light and its wavelength in vacuum, respectively. The

theoretical maximum binary signaling efficiency is

R 2 bits
- (1.2)
B Hz sec

Nationwide fiber-optic networks are based upon single wavelength transmission

at 1.3 or 1.55[m with bit rates of about 565 Mb/s. This data rate is high enough for

more than 7600 multiplexed voice or about 8 video channels, and equations 1.1 and 1.2

suggest that this transmission rate can be contained within a spectral width less than

0.003nm when A = 1.55[/m. Typical systems, however, require more bandwidth due

to laser chirp, but this can be avoided by using external-cavity modulators.

Figure 1.1 shows that the fiber provides low loss regions around the wavelengths

of 1.3/tm and 1.5/im that are about 100nm wide. In principle, the fiber has more

than 10 THz of bandwidth to offer in the 1.51tm region. That is theoretically enough

bandwidth for about 300,000 high quality uncompressed video transmissions (70Mb/s

each). Since the operating bandwidth of low-cost electronics is presently far below

100 GHz, it appears that the only available means of utilizing more of the fiber

bandwidth is to move towards wavelength division multiplexing (WDM).

Ch 1 L' Narrow Bandpass Z ,
Ch 2 L 2 Wavelength Filter
Ch 2- LD 2| 7Star

ChM m LD I Controller Chx
----------------- --------------------------j

Figure 1.2. Simplified schematic of a WDM distribution system.

1.1.2 Practical WDM Communication Systems

Figure 1.2 shows a simple WDM distribution system. First, many electrical chan-

nels are multiplexed together using a scheme such as time division multiplexing. Each

of those are used to externally or internally modulate a laser for one of the wavelength

channels. Each laser output becomes an individual optical channel. The optical chan-

nels are multiplexed together by a star coupler and then distributed. Each receiver
has the ability to select the desired optical channel using a narrow bandpass tunable

optical filter with high free spectral range. After detection, the electrical signal is

then further demultiplexed for the user.

Recently, a couple of groups have successfully demonstrated the feasibility of such

systems. In 1990, H. Toba et al. [79] from NTT Laboratories in Japan demonstrated a

100 channel WDM transmission system at 622 Mb/s over 50 km. They have since ex-

tended their system to 128 channels over 480 km using erbium doped fiber amplifiers

(EDFAs) [50]. A. Chraplyvy et al. [14] from AT&T Bell Laboratories demonstrated

a WDM system transmitting over 0.3 Tb/s through 150 km of fiber. That is enough

bandwidth for almost 5000 uncompressed high quality video transmissions, which is


WDM Mux/Demux Chip

------------ -------y-----
|Tunable LD O utput'
Input t Output
---------- ---I 11------- -- ---- *--*Otu

ANe )k -Router

Figure 1.3. Simplified schematic of a WDM ring network.

almost as many video channels as today's systems offer for voice conversations. By

using newly developed video compression schemes, this figure can easily be increased

by one or two orders of magnitude. Although these experiments were conducted at

a wavelength of 1.55/m, they utilized conventional 1.3pm zero dispersion fiber. This

means that the existing fiber network can be dramatically enhanced by simply re-

placing transmitter, receiver, and repeater hardware, instead of laying thousands of

miles of new fiber.

These concepts are expected to be applied to ring networks where each subscriber

can transmit and receive data on a network. Figure 1.3 shows a simplified diagram of

an optical WDM ring network which is then routed to a wide area network (WAN).

The expanded view in the inset of figure 1.3 shows a conceptual view of the optical

hardware of one of the subscribers to the ring. In this case, each subscriber has a

tunable laser diode and a WDM multiplexing/demultiplexing chip. The tunable laser

allows the subscriber to transmit information on any of the wavelength channels. The

purpose of the WDM multiplexing/demultiplexing chip is threefold: extract all the

optical power of only the desired channel from the fiber optic bus, route the rest of the

channels back onto the bus for other subscribers, and retransmit a new signal from

the tunable laser tuned to the same wavelength channel as the filter. In this simplified

conceptual system, to prevent interference of signals from different sources, it would

be necessary for the laser and filter to be tuned to the same frequency channel and

for the filter to extract most of the input light in that channel (perhaps with -15dB

or less transmitted).

Many essential components of WDM systems such as tunable semiconductor laser

diodes, external modulators, waveguide couplers, detectors, and EDFAs are com-

mercially available. However, commercially viable technologies for filters of densely

spaced WDM systems are yet to be reported. As will be discussed below, the filters

used in previous systems [79, 50, 14] do not meet all of the requirements of a practical


1.1.3 Wavelength Selective IOCs

So far, we have presented a motivation for WDM as a realistic means of obtaining

very high bandwidth communication systems. Crucial to the realization of WDM in

commercial communication systems is the advent of practical wavelength selective

integrated optical circuit (IOC) filters. Many wavelength selective IOC filter designs

have been reported for WDM applications, but few are adequate for densely spaced

WDM. Typically, the devices are very long, have rather broad pass-band spectra,

require high power, and often involve complex fabrication processes. The remainder of

Acoustic Transducer TE-TM Splitter
T-TM Splitter Acoustic Bea




Figure 1.4. Polarization insensitive acoustically tunable optical filter.

our motivation will discuss some of the most common designs of wavelength selective

IOC filters reported in the literature. The pitfalls of these devices as they apply

to densely spaced WDM systems will be discussed. Then the goals of this research

work will be presented in section 1.2 as it presents a proposal for a new wavelength

selective IOC filter more suitable for densely spaced WDM systems.

Acoustically Tunable Optical Filter

Figure 1.4 shows a diagram of a polarization insensitive acoustically tunable op-

tical filter (ATOF) [71]. The two waveguide polarizations are first separated by a

polarization splitter and fed through two identical waveguides that are under the

influence of a surface acoustic wave (SAW). Within a narrow band of wavelengths,

the difference between the propagation constants of the TE and TM modes equals

the propagation constant of the SAW and the two modes can then couple.

The ATOF is often regarded as one of the most promising filters for WDM sys-

tems because it can easily be made polarization insensitive, it is tunable over more

than 100nm, and it can route many channels at one time by creating a SAW with

multiple frequency components. However, the ATOF is not suitable for systems such

as those in the most notable demonstrations of WDM systems [79, 50, 14]. This is

primarily because the passband is too broad (~'nm). Equation 1.1 shows that Inm

bandwidth corresponds to a 125 GHz channel at A = 1.55Lm. This is theoretically

enough bandwidth for a channel capacity of over 200 Gb/s. Since even very expen-

sive electronic and electro-optic hardware presently cannot achieve these rates, the

ATOF would result in wasted bandwidth. The ATOF makes up for this deficiency

by having a very broad tuning range, but currently available tunable semiconductor

lasers and laser arrays (excluding external cavity lasers) can only operate within a

wavelength range of only a few nanometers. Therefore, a filter with a very narrow

passband is needed to more fully utilize the band of wavelengths within the tuning

range of commercial lasers.

Other disadvantages of the ATOF are that it is several centimeters long and

requires high power to generate the surface acoustic wave (SAW), typically more

than 100mW. In turn, the requirement for heat dissipation can make it difficult to

maintain stable operation. Recently, significant advances have been made to maintain

temperature stability in a commercial device [80].

Cascaded Filters

The systems demonstrated in references [79, 50, 14] used cascaded tunable band-

pass filters. A cascade of three filters was used in reference [14] to achieve a 0.65nm

full width half maximum (FWHM) passband. This provided enough bandwidth for

a bit rate of over 100 Gb/s, although only 20 Gb/s was achieved.

The filter cascade used in references [79, 50] had a much narrower bandwidth

of 0.04nm (5 GHz), which corresponded to the bandwidth of the frequency shift

keyed (FSK) optical signal they used. This filter consisted of a cascade of seven

Mach-Zehnder filters. Figure 1.5 shows a diagram of a similar device with only three

Thin Film Heater

Figure 1.5. Cascaded Mach-Zehnder interferometer.

Figure 1.6. Resonant grating assisted directional coupler filter.

Mach-Zehnder filters. The total device length was about 21cm which required a chip
size of 5 x 6cm2. Although it was not mentioned, it is reasonable to assume that the

filter was fabricated around the perimeter of the chip. The polarization dependence

of each Mach-Zehnder filter was compensated using a laser trimming technique. Not
only is this a very long device which would be very difficult and costly to mass

produce, but each of the 7 Mach-Zehnder stages must be individually controlled to

achieve wavelength selection.

Resonant Optical Directional Coupler

The resonant optical directional coupler (RODC) proposed by Kazarinov et al. [33]

is shown in figure 1.6 is a modification of the original proposal by Haus et al. [26].

The structure consists of two waveguide Fabry-Perot interferometers with zero-length

cavities (quarter-wave shifted gratings) coupled to a center waveguide that acts as

the signal transmission bus. Light couples out of the upper grating waveguide only

at wavelengths for which the Fabry-Perot waveguide is resonant. The second Fabry-

Perot waveguide below the center waveguide permits a complete power transfer to

the filtered output of the top waveguide at the resonant frequency if both Fabry-

Perot waveguides are resonant at the same wavelength. Without the second resonant

waveguide, only half of the power at the resonant wavelength would be coupled to

the filtered output channel with a quarter of it being transmitted to the unfiltered

output and the other quarter being reflected to the source.

The theoretical device transmission characteristic is very narrow (~ .01nm), but

has yet to be adequately demonstrated. The first attempt at demonstrating a sim-

ilar device without the lower resonant waveguide was reported in reference [39] in

GaAs/AlGaAs. The theoretical response of the RODC indicates that it is well suited

for WDM applications, but there are two fundamental limitations for which no solu-

tions have been reported: it requires excellent DBR corrugated waveguides and has

a narrow stopband (typically ~ 2nm).

The new innovations in waveguide grating fabrication and the concepts that are

presented in this dissertation may prove to be invaluable towards circumventing the

limitations of the RODC and ultimately realizing an electro-optically tunable RODC

as well as other Fabry-Perot filters that are suitable for WDM. The filter we propose

is a Fabry-Perot waveguide structure that can be used in the RODC to extend its

stopband (see section 8.2.3). Section 1.3 will introduce a new waveguide geometry

well suited to make high reflection corrugated waveguides in commercially viable

electro-optic materials such as LiNbO3.

1.2 A New Wavelength Selective Tunable Filter

A wavelength selective filter most suitable for densely spaced WDM systems

should have a very narrow passband (~ 0.01nm), a very broad stopband (> 100nm),

and a high extinction ratio (> 14dB). The narrow passband is adequate for moderate

bit rates (< 2 Gb/s) and enables the fiber bandwidth to be utilized more efficiently

as the wavelength channels are densely spaced. The broad stopband and high extinc-

tion ratio allows a receiver with a discriminant signal detection scheme to isolate one

of the wavelength channels without crosstalk. Other desirable aspects of a tunable

wavelength selective filter are the ability to operate at high speed with low electrical

power, a simple biasing scheme, and few controllers. In order for the device to be

commercially practical, it should be as compact as possible and be manufactured with

standard microelectronics technology. Most wavelength selective filters reported to

date, including all of those described in section 1.1.3, lack one or more of the above

requirements. We propose a new filter scheme that has the potential to satisfy all

these requirements.

Figure 1.7 shows the proposed filter structure. It consists of a corrugated waveg-

uide that exhibits high reflectance in a narrow wavelength spectrum in cascade with

a broadband high reflectance mirror deposited on the endface. The reflection spectra

of these two mirrors is shown in figure 1.8. The reflection band of the corrugated

waveguide is shown as a sharp spike in figure 1.8 (a). Comparing the width of the

reflection bands of the dielectric mirror from the plot in figure 1.8 (a) with that of the

corrugated waveguide in the expanded view in figure 1.8 (b), we see that the reflec-

tion band of the dielectric mirror is over three orders of magnitude wider than that

of the corrugated waveguide. These two mirrors, fabricated in cascade constitute an

asymmetric Fabry-Perot (ASFP) interferometer. We use the term "Asymmetric" to

Corrugated Waveguide

AR Coating

Figure 1.7. Proposed Tunable Wavelength Selective Filter.

refer to the use of two mirrors with vastly different reflection spectra as opposed to

the conventional case where two similar mirrors are used.

The two conventional cases of waveguide Fabry-Perot filters, where similar mirrors

are used on both ends, are not adequate for WDM. For the case where two dielectric

mirrors are separated by a cavity, there exist many transmission peaks. The use of

two corrugated waveguides separated by a cavity may exhibit a single transmission

peak, but the stopband is very narrow. It will be shown that the proposed struc-

ture in figure 1.7 utilizes the desirable characteristics of both mirrors: a very broad

stopband contributed by the dielectric endface mirror, and the selection of only one

transmission peak by the narrow reflection band of the corrugated waveguide mirror.

The resulting transmission spectra consists of a single transmission peak with an

extremely high free spectral range (FSR). It will be shown that by using the electro-

optic effect via an appropriate electrode scheme, the transmission peak can be shifted

to accommodate many wavelength channels in a WDM system. The reader is referred
to chapter 3 for a detailed explanation of this filter structure. We demonstrate the

-- Dielectric Mirror |
- - Corrugated Waveguide |


0 L



o 0.4



1200 1400 1600 1800 2000
Wavelength (nm)


-- Dielectric Mirror I
- Corrugated Waveguide



/ I ~ ~ I'
I- -



I I\ N
A ,

Wavelength (nm)



Figure 1.8. Comparison of the reflection spectra of a dielectric endface mirror and a
corrugated waveguide. a) shows that the reflection band of a corrugated waveguide is
very narrow compared to the spectra of the dielectric mirror. b) shows an expanded
view of the corrugated waveguide spectra.

experimental results of the proposed filter in a very practical and mature material

system: LiNbO3.

A reader familiar with distributed feedback (DFB) lasers will recognize the ASFP

filter as a commonly used structure for low threshold DFB lasers. The structure is

the same, except the filter does not have a gain medium and uses the electro-optic

effect for tunability. However, requirements the two devices have on the resonant

structure are very different. This will be addressed in section 3.3.3.

This work proposes that the ASFP structure be used as a passive filter ("passive"

refers to the lack of a gain medium). Furthermore, it is shown how the filter can

be tuned by a simple electrode scheme, utilizing the linear electro-optic effect. The

modeling of this device as well as the experimental results are presented.

1.3 A Novel Waveguide Technology

The ASFP filter proposed in this work requires two high-reflectance mirrors. For

decades, high reflectance dielectric mirrors have been available. Corrugated waveg-

uides have been studied for over two decades, but high-reflectance, low-loss corrugated

waveguides have yet to be demonstrated in a material system suitable for making the

tunable ASFP filter.

In light of the failure of conventional methods to produce high reflectance corru-

gated waveguides, such as etching gratings into the waveguide or buffer layer, a new

waveguide geometry is proposed and developed. Figure 1.9 shows the transverse cross

section of the waveguide. It consists of a conventional graded-index channel waveg-

uide with an overlay of higher refractive index material. A grating in this material

(figure 1.10) can be used to create a considerably more perturbation to the waveguide

mode than conventional corrugated waveguides without producing excessive scatter-

ing loss. This is necessary to achieve high reflectance and low loss. Of course there



Figure 1.9. Transverse cross section of a graded-index channel waveguide with an
overlaying film of refractive index near or higher than the core of the waveguide.




grating overlay



Figure 1.10. Longitudinal cross section of a DBR corrugated waveguide
a high-index overlay.

made from

- r

is always a tradeoff between the perturbation and radiation loss, but the benefit of

using this waveguide geometry is that more perturbation can be achieved with less

radiation loss. A detailed description of our waveguide geometry will be presented.

Commercially viable materials and waveguide technologies are used for our ex-

perimental demonstration. LiNbO3 is chosen as the substrate material because it is

currently the material system primarily used in the industry for state-of-the-art pas-

sive IOC devices. Waveguides are made by either annealed proton exchange (APE)

or titanium indiffusion. Silicon is chosen as the overlay not only because of its high

refractive index, but also because it is opaque to visible light, is easily deposited by

e-beam evaporation, is easily etched by reactive ion etching (RIE), and can screen

pyroelectrically generated surface charges. We will cover these benefits in detail.

A somewhat startling discovery was made during the course of the investigation

and characterization of the structure: when a planar film of high-index silicon with

thickness large enough to support multiple planar modes is deposited on the surface

of a single-mode channel waveguide in LiNbO3, the resulting channel waveguide may

still support only a single mode. Furthermore, in contrast to planar waveguides alone,

a comparably small amount of optical power is drawn up into the silicon layer, the

mode index increases by a small amount, and the propagation loss remains low. This

is exactly what is required to make good corrugated waveguides: low propagation

and mismatch losses with very little change in the mode field profile.

Rib waveguides consisting of planar films with thicknesses capable of supporting

multiple modes have already been shown to produce single-mode waveguides by Soref

et al. [73]. However, the prescribed concept in this dissertation goes one step further

as it applies to waveguides of much lower refractive index than the planar layer.

Furthermore, the proposed waveguide geometry is much more practical because it is

applicable to graded-index channel waveguides made by APE or titanium indiffusion

which have proved to be commercially viable technologies.

It will be shown in that commonly-used theoretical estimations of the propagation

constants and mode field profiles cannot be applied to this structure and hence, do

not support these claims. Perhaps this may explain why such a simple structure has

not been investigated to date. Complete waveguide characterization measurements

including mode index, loss, and field profile are presented. All of the experimental

measurements made in this work support these claims. In addition, some very unique

transmission measurements of DBR corrugated waveguides are presented that can not

be explained without this new insight.

This new waveguide geometry is used to demonstrate the highest reflection in

LiNbO3 corrugated waveguides reported to date. The results are in excellent agree-

ment with theory. The proposed filter structure is also demonstrated using this new


1.4 Chapter Organization

Chapter 2 covers most of the numerical modeling used in this work for device

simulation. In particular, two transfer matrix formalisms that are useful in solving

the optical fields in planar multiple layered structures are presented. The benefits

of each approach and their application to lasers, other Fabry-Perot structures, and

waveguides are then discussed. A brief description of the spectral index method is

given. The derivation of these formalisms from Maxwell's equations is used to develop

the notation and identify all the approximations, as well as some common mistakes.

A detailed discussion of DBR Fabry-Perot devices and an in-depth proposal of

the new wavelength selective tunable filter follows in chapter 3. Calculations, with

reasonably achievable waveguide parameters without neglecting loss, are presented

for the proposed structure. The device is shown to exhibit the characteristics neces-

sary for a densely-spaced WDM system only for structures with low-loss corrugated


To achieve good tunable ASFP filters, high-reflection low-loss corrugated waveg-

uides must be made in a material system that exhibits a large electro-optic effect,

such as, LiNb03. In chapter 4, first we describe many of the previous attempts to

make high reflectance corrugated waveguides, most of which yielded mediocre results.

Then, we propose a new waveguide geometry in LiNb03 that overcomes several fun-

damental limitations. This geometry is applicable to several other material systems

as well.

Chapter 5 illustrates the details of the various fabrication and processing tech-

niques required to fabricate the devices. These techniques include reactive ion etching

(RIE), waveguide fabrication by APE, grating lithography and transfer of the grating

to the waveguide, electrode fabrication, endface polishing, and endface DBR deposi-


A detailed experimental analysis of the new waveguide geometry is presented in

chapter 6. The mode index, mode field profiles, and loss are measured. These results

are in agreement with the theory, in support of the claims made in this work, resulting

in a clear understanding of the unique characteristics of the geometry as applied to

our devices.

Chapter 7 presents the experimental results of several corrugated waveguide filters

made with the new waveguide geometry. The results of a simple corrugated waveguide

reflector is found to be in excellent agreement with the theory and exhibits the

highest reflection from a corrugated waveguide in LiNbO3 reported to date. With the

fundamental building blocks in place, the proposed ASFP structure is demonstrated

and compared with theory.


Finally, chapter 8 summarizes the results. The scope of this work cannot begin to

cover all of the potential uses and enhancements to this new cutting-edge technology.

It seems that many other uses for the new high-index overlay technology exist. An

outline of future work, some of which is already under way, is also described in this



This chapter covers the theory used throughout most of this work to calculate

reflection and transmission spectra for dielectric mirrors and DBR corrugated waveg-

uides, and the waveguide mode characteristics. This chapter does not present any

new theory or derivation, but it approaches the subject from a very unique and sim-

plistic angle. To develop the notation and justify all of the assumptions made, we

start from Maxwell's equations.

Most textbooks cover reflection of TE and TM waves independently. Here, we

use a formalism that is used in only a few textbooks [40] to describe both cases

with a common set of equations. The angle of incidence and polarization is factored

out in a way that makes the equations take the form of those of normal incidence.

Most references use only one of two matrix methods [40, 22], but, depending on the

problem at hand, one formalism is generally more convenient than the other. This

will be demonstrated by several examples in section 2.7. Section 2.7.4 will show that

both matrix formalisms can be applied to waveguide devices by substituting the local

normal mode indices for the refractive indices in the previously derived equations.

Finally, the theory of the spectral index method to solve for waveguides with two-

dimensional cross sections is presented in section 2.8. The spectral index method is

useful for calculating the local normal mode indices which can then be used in the

matrix methods for calculating the reflection and transmission spectra of corrugated

waveguide structures.

2.1 Basic Equations

We begin with Maxwell's equations along with the constitutive equations for
isotropic media:
V. D p (2.1)

V B = 0 (2.2)
Vx E (2.3)

Vx H = J + (2.4)

D = eE (2.5)

B = pH (2.6)

J = oE (2.7)

where E is the electric field strength [V/m], D is the electric displacement [C/m2],
H is the magnetic field strength [A/m], J is the electric current density [A/m2], B is
the magnetic flux density [T], p is the electric charge density [C/m3], a is the electric
conductivity [1/Om], p is the permeability [H/m], and e is the permittivity [F/m].
Furthermore, we add that
e = Creo (2.8)

p = PrPo (2.9)
co = 2 (2.10)
where e, and pr are the relative permittivity and permeability, respectively (dimen-
sionless), co is the permittivity of free space (8.8541853 x 10-12F/m), Po is the per-
meability of free space (47r x 10-H/m), and c is the speed of light in free space
(2.997925 x 108m/s).

In an isotropic, homogeneous medium of no free charges, we may rewrite Maxwell's

equations in terms of two field variables

V.-E= 0 (2.11)

V H =0 (2.12)

V x E = (2.13)

Vx H = oE + E (2.14)
Now taking the curl of equation 2.13 using the vector identity

V x (V x A) = V(V A) V2A

and substituting equations 2.11 and 2.14, we have

0 OVE 02E
V x (V x E) = V(V E) V2E = -V2E = -pV x H = -pa-O tpe -2

2E 02E
V2E- = 0 (2.15)

In the same manner, by taking the curl of equation 2.14 we obtain

OH 02H
V2H -- =0 (2.16)

Equations 2.15 and 2.16 are wave equations for the E and H fields. The solutions

for the E and H fields are of the same form. The solution of equations 2.15 and 2.16

for a plane wave traveling in the r direction is

E = Eoei(k-r-wt) (2.17)


H = Hoei(k-r-wt) (2.18)

where k is the propagation vector, r is the position vector of an arbitrary point with

respect to a reference, and w is the angular frequency of the wave. Some authors

prefer to use ei(wt-k'r). Generally, the results between the two formulations may be

compared by replacing i with -i and visa versa. This will not be the case in equation

2.69. Equations 2.17 and 2.18 do not represent physical quantities because they are

complex, but they are a general solution to the pair of wave equations 2.15 and 2.16.

However, physical quantities may be obtained by taking the real part of the complex

solutions. The results by doing such will also be a solution to the wave equations

because they are linear allowing the real and imaginary parts of the solutions to be

separated. We could take the real part here and obtain

R(E) = R(Eo) cos(k r wt) '(Eo) sin(k r wt)

but it is most often easier to carry the complex expression through the arithmetic and

take the real part after manipulation when a physical quantity is desired. However,

when using them in a relation which is not linear with respect to E or H such as

the power relationship, the complex quantities may not be used unless an adequate

formulation for complex numbers is used.

Substituting equation 2.17 into equation 2.15 we find that

k2 = Ik|2 = w21 + iwA,0 (2.19)

The phase velocity of the wave is given by v = w/k and the velocity of light in free

space is given by c = 1//-p-o-. Combining these equations with equations 2.19, 2.8,

and 2.9 we obtain the square of the complex refractive index n:
2 c2k2 .r )2
nr -r + i- = (nr + ii2 (2.20)

where nr and ni are the real and imaginary parts of the complex refractive index,

respectively. nr is referred to as simply the refractive index and ni (often denoted by

k) is known as the extinction coefficient. With the free space propagation constant

w 27r
ko= (2.21)
c Ao
equation 2.17 becomes

E = Eoei(nk'k-r-wt) = Eoe-nikokre i(nrkok-r-wt) (2.22)

The first exponent on the right hand side of equation 2.22 represents absorption (or

gain) while the second represents phase.

The wave equations, although derived from Maxwell's equations, do not represent

all of the information useful for determining the fields. They only help us to define

a certain class of solutions, namely the transverse propagating waves. To determine

a solution of Maxwell's equations, equations 2.17 and 2.18 must be substituted into

equations 2.11 2.14. Doing this, a simplified form of Maxwell's equations for trans-

verse propagating waves is obtained:

k E = 0 (2.23)

k H=0 (2.24)

k x E = pwH (2.25)

k x H = -(cw + io)E (2.26)

From these equations, we see that E, H and the direction of propagation form a

mutually orthogonal set. From equation 2.25 we see that the magnitudes of the field

vectors are related by

|E =7 |HI (2.27)

where, using equations 2.20, 2.10, and 2.9,

Sn o n(2.28)
k n eo n

The quantity q] is the wave impedance; the impedance of free space 7ro = 377P.

The right hand side of equation 2.28 makes the assumption that Pr = 1 at optical


Next, we discuss the Poynting vector S. It is defined by

S = Ex H (2.29)

and is the energy current density [W/m2]. Equation 2.29 may be derived from

Maxwell's equations and the relation describing the loss of electromagnetic energy

due to Joule heating, fy Jf E dr, and making the argument that rate of electromag-

netic energy decreasing within a volume must be equal to that consumed by Joule

heating plus that which is leaving the volume. The Poynting vector "points" in the

direction of energy flow. The above definition for the Poynting vector is not limited

to time varying fields. It is important to realize, as stated earlier that in calculating

S, both E and H must be physical quantities, i.e. real, and they must be the total

resultant of all the waves of a given frequency at the point of interest. As previously

illustrated, however, it is more convenient to work with complex quantities to rep-

resent time varying fields. Extracting the time dependence from equations 2.17 and

2.18 and expressing them as

E = (Eoek.r)e-iwt = Ece-iwt (2.30)


H = (Hoeik-r)e-it = He-iwt (2.31)

it may be shown that the time average of the energy flow is

(S) = 1 R(E, x H*) (2.32)

which is in the direction of k. In doing this, we have simplified the calculation of

the energy flow of time varying fields in terms of the complex time-invariant field

amplitudes. It must be remembered that the complex field amplitudes are used in

equation 2.32 while only their real part is used in equation 2.29. For a single wave

traveling in direction k, equation 2.32 may be further reduced using equations 2.25,

2.27, and the identity

A x (B x C) = B(A C) C(A B)

resulting in

(S) = r IEc2 1k (2.33)

the scalar intensity is then

I = n EcE* (2.34)
2 7,
where Ec is the complex scalar electric field amplitude. From equation 2.22, we find


1 n |Eo 12 e-2nikok-r= Ie-ak-r (2.35)

Io is the intensity at the reference point, I|r| = 0, and
ao -- (2.36)

is the absorption coefficient. Two important points must be remembered about the

calculation of intensity:

In deriving equations 2.33 2.35, we assumed that there was a single wave

propagating in the direction k. Therefore, when calculating intensities resulting

from fields traveling in different directions, equation 2.32 must be used.

I oc nr (amplitude)2 should be remembered rather than simply I oc (amplitude)2

which is often quoted. If the later is used when comparing intensities present

in two different media, the result will be in error.

These two important points must be remembered when calculating reflectance or

transmittance, which is done in the next section.

Figure 2.1. k vector relationships with light incident on a single surface.

2.2 Reflection and Transmission From a Single Boundary

Figure 2.1 shows the k vectors from the incident ki, reflected kr, and transmitted,

kt plane waves. The boundary conditions for the fields may be found from Maxwell's

equations. These state that the tangential components of E and H are continuous.

These conditions must be satisfied at any point on the boundary for all values of time.

This implies that the exponential terms of the incident, reflected, and transmitted

waves must be equal for all values of time. Thus, all the waves have the same

frequency w and

ki Arb = kr Arb = kt. Arb (2.37)

where Arb is the position vector between any two points in the boundary. This

assures us that all the waves have the same phase at any point on the boundary at

any time t. From equation 2.37 we find that

6, = Or (2.38)


no sin 0i = nl sin 0t (2.39)

which is Snell's law.

S E E r E Hkr
H k7no kno


(a) kt t (b) t

Figure 2.2. Conventions for the positive directions of the E and H field vectors for
the (a) TE or s-polarized waves and (b) TM or p-polarized waves.

Figure 2.2 shows the convention we use to define the positive directions of the
electric and magnetic field vectors for the cases when E is perpendicular to the plane
of incidence (referred to as TE or s-polarized) (a) and when E lies within the plane
of incidence (referred to as TM or p-polarized) (b). The tangential components of
the field vectors, and 7-, are related to the field amplitudes E and H by

8= E and 7-=H cos (TE) (2.40)

for the TE case and
= Ecos and7-= H (TM) (2.41)

for the TM case. 0 is the angle between the normal fi and the propagation vector k
in each medium.
Now we introduce a constant by which we will factor out the polarization and angle
dependence to simplify further derivations. This is the modified optical admittance
y which connects the two tangential components ET and HT by

y = 7-//


where, for the two polarizations,

n n
yTE =-cos0 and YTM = (2.43)
77o 1o cos 0

It should be noted that y is the modified optical admittance for a single wave and

not the total field.

To solve for the reflectivity and the transmissivity, we simply need to meet the

boundary conditions with the resultant tangential fields. Noting the sign conventions

in figure 2.2 we have

Ei + r = st (2.44)


Hi r = Ht (2.45)

Solving equations 2.44, 2.45, and 2.42, then comparing field amplitudes using 2.38,

2.40, and 2.41 we find that the reflectivity r is

r = Yo = E, (2.46)
Ei yo +y E+

Similarly, but comparing with equation 2.39 instead of equation 2.38 we find that the

transmissivity t is

t-= 2yo which is E for the TM case (2.47)
Si o + yi E T

Using equation 2.41 we find that

Et 2yo cos 0i
Sfor the TM case
E= yo + Yi cos Ot

which is more commonly used as the transmissivity. We, however, will use the tan-

gential form in equation 2.47 because it is more convenient to use since it represents

both polarizations and the angular dependence has been factored out. These equa-

tions are particularly easy to remember since they appear exactly the same as those

for normal incidence. Furthermore, we will see that they are more consistent with
the power relations.
The resultant tangential electric and magnetic fields in the incident medium are

i+r = (1 +r)

Hi Hr = yo( r)S

and for the transmitting region are tSi and y1t8, respectively. Using equation 2.32,
we find the intensity in each region. When we equate these we get

1 rr i(Y) (r Yr*) = ft
r (Yo) r () )

which is similar to the power rule, 1 R = T where R is the reflectance and T is the
transmittance, except for the last term on the left hand side. This difficulty may be
avoided by assuming that the incident medium is non-absorbing, making F(yo) = 0.
This is true for most practical circumstances. Now we have the reflectance and
R = rr* = (2.48)


(yo) 4(yo+y)(yoy(1) (249
R(-o) (Yo + Yi)(Yo + Yv)*
Notice that the definition we use for t results in a very simple equation for T, unlike
many textbooks that redefine T by making power arguments based on using a finite
area of the incident beam versus the resulting area of the transmitted beam as is done
in many textbooks. The dependence of this area on angle was factored out in the
original relations for y, thus illustrating another nicety of this uncommon formalism.

noBoundary a
ni d
Boundary b

Figure 2.3. A single dielectric planar layer on a substrate.

2.3 Characteristic Matrix of a Thin Film

Now we extend the above analysis to the case of a thin film as in figure 2.3.

Multiple reflections exist between the two surfaces of the film and, if the film thick-

ness is smaller than the coherence length of the incident light, they interfere. One

way to solve this problem is to set up an equation which takes the sum of all of

the reflections taking the phase of each reflected wave into account. The following,

however, is a much more straightforward approach which may easily be extended

to accommodate multilayer structures. This procedure rederives the reflectivity and

transmissivity for the whole structure without using the results for a simple single-

interface system. In doing this, we will solve for the fields in all regions simultaneously.

In any medium, only two waves can exist: forward traveling and reverse traveling.

These are what would be the resultants of many reflections if it were calculated by

the former procedure.

We begin by denoting waves traveling in the direction of incidence (forward) with

a + superscript and those traveling in the reverse direction with a superscript.

Using the same convention for the direction of the E and H vectors, their tangential

components at the lower boundary b are

9b = 9U + Slb

Hb = y91 -y11b


Solving in terms of the resultants Lb and -b we get

E = I('b/1+b)
lb = (- b/Y1 + b) (2.51)
-Hb = Y1L= 1 +(bY19b)
'H-b = -yigf = -('b Yl1b)

The tangential fields at the same lateral position on the top boundary, a, may be
found from equations 2.51 by considering difference in phase 6 between the top and
bottom interfaces at the same instant where

6 = konidcos 01 (2.52)

Note that the cosine factor is present because we are comparing the phase of the
waves at the same horizontal position between the top and bottom interfaces. The
fields at the top interface now become

Ea = 1(lb/y + Eb)e-'6
Sa = 1(--b/1 + cb)e"

H = =2 ('b Ylb)eib

so that

Sa = a + 1a = Lb COS 6 i-sin 6
'Ha ='H + la= -iylb sin 6 + b COS 6

which can be written in matrix form as

S ,a [ cos 6 -isin(6)/y1 1 5
LHa -iyI sin 6 cos J (b

The 2 x 2 matrix on the right hand side of equation 2.53 is called the characteristic
or transfer matrix of the film. If we define the optical admittance for the assembly
to be
Y = Ha/Sa (2.54)

the task of determining the reflectance becomes the same as for the single boundary
case so that
Yo Y
r = Y-(2.55)
yo +Y
R= (Yo-Y) ( -Y)* (2.56)

Using equation 2.54 and assuming Sb is a single propagating wave, ie. the last layer,
equation 2.53 becomes

a 1 = [ cos6 -isin(6)/yl [ 1 ] b (2.57)
Y -iyj sin 6 cos 6 Y2

The characteristic matrix may be further simplified from 2 x 2 form to a 1 x 2 matrix
by letting
B cos 6 -isin(6)/y [1 (2.58)
C -iyj sin 6 cos 6 y2
so that Y = C/B. This notation will be useful in the next section.

2.4 Characteristics of a Multilayer Structure

Now we will consider the general thin film structure of figure 2.4. The thin film
results obtained above may be generalized to obtain the characteristics of an assembly
of thin films by recognizing that the tangential fields at boundaries b and c may be

no Incident medium

n2 n d2d
c Multilayer Structure
n1 di m
nm Substrate

Figure 2.4. A general multilayer thin-film structure.

related in the same way that those of boundaries a and b were in section 2.3. That
[ b COS 62 -i Sin(62)/y2 (2.59)
[ b ] 2-iy2 sin 62 cos 62 7Hc
6k = konkdk cos Ok (2.60)

Combining 2.53 with 2.59 and using 2.54 and 2.42, the solution becomes

ea [ = [ C ] =

E cos 6 -i sin(61)/yi cos 62 -i sin(62)/Y2 1 ]c (2.61)
-iyj sin 61 cos 61 -iy2 sin 62 cos 62 Y3 J
and continuing the process, we see that the characteristic matrix for the whole as-
sembly of figure 2.4 is
B = i cos 6k -i sin(6k) Ik 1 (262)
C -iyk sin bk cos 6k m 22
Some care must be taken when calculating the phase factor when absorption or
total internal reflection is involved. The angles may be complex and are found by
Snell's law:
nk sin Ok = no sin 00o (2.63)

Since the complex arcsine function is often unavailable on computers, we calculate
6k by

k = kodk /n no sin2 (0


As we have assumed, no and 0o are real and for absorption, nk is in the first quadrant

of the complex plane. The square root is then in the first or third quadrants. 6k must

be in the first quadrant because the phase must be increasing and the amplitude

exponentially decreasing in the direction of k in order for the situation to be physically

possible. Similarly, if medium k produces gain, we find that 6k must be in the fourth

quadrant. In either case the + sign in equation 2.64 should be chosen since most

computer algorithms produce positive real parts when calculating the complex square

root. However, it is a good idea to verify if this is indeed the case. The same value

for cos Ok should be used for both 6k and Yk. That is,

nj- nj sin2 O0
cos k = ----n-- (2.65)
The reflection coefficient may be found using equation 2.56 with Y being the

optical admittance for the multilayer assembly. With Y = C/B,

(yoB-C (yoB-C\ (2.66)
R= yoB +C yoB+C)
To find the transmittance T and absorptance A, we note that the intensity normal

to the last interface m and the first interface a are

Im = -gmVy' ) = 1W m m
2 2

la = If(BC*)8m*
la is the intensity normal to the interface actually entering the assembly. This is

related to the incident intensity 1i by

a = (1 R)I

so that
2(1 R)m
A 2(1 R)

The transmittance is
Im R(ym)(1 R) 4yo((Ym)
Ii R(BC*) (yoB + C)(yoB + C)* (2.67)
The absorptance is
I( (ym) 4yo(BC* ym)
A=1-R-T=(1-R) 1 -(Y) 4 (y(BC* (2.68)
R(BC*)J (yoB + C)(yoB + C)*
The phase change on reflection can be found from equation 2.55 to be
Sarctan (iyo(B*C C*) (2-69)

where it is important to note the signs of the numerator and denominator to assure
the correct quadrant since the arctangent only has a range within the first and fourth

2.5 Alternative Matrix Formulation

Next we describe an alternative formulation to the solution of a multilayer struc-
ture given in section 2.4. This is the approach used in reference [22]. It will be shown
in the next section that both formalisms have much to offer in understanding and
calculation of multilayer structures. The previous formalism was a set of equations
calculating the total electric and magnetic fields at each interface. This formulation,
however, will calculate the forward and reverse electric fields on each side of each
At boundary a of figure 2.3, the total fields are

Sa = 0a + + a = g= a + Sa

RHa = ( a) = Y1i(gl+a + gh)

Rearranging these equations, we have

SO+a = 1la S l+a -+agla
ta ta

Yo Yi 2yo
ra=- and ta= 2 (2.70)
yo + Y1 Yo + Y1
Written in matrix form,

[ J 1 ra -1 J (2.71)

The electric fields on each side of film a are related by

where 61 = konlidcosOt. Written in matrix form,

p+ \-ii n0 S +
[a l ] (2.72)

the fields throughout a multilayer structure may be calculated by repetitive use of

equations 2.71 and 2.72-equation 2.71 for each interface and equation 2.72 for each
Examples illustrating the benefits and uses of each matrix formalism will be given

in section 2.7. First, the relationship between the two formalisms will be given in

section 2.6.

2.6 Connection Between the Two Formalisms

Both matrix formalisms derived in sections 2.4 and 2.5 are useful for solving fields

in planar multilayer dielectric structures. A connection matrix will be derived in this

section showing that they equivalent. Even though they are equivalent, the knowledge

of both of them facilitates deriving different sets of basic analytic expressions. This

will become clear in section 2.7 as the advantages of each formalism will be discussed

for a variety of cases.

In the first formalism (section 2.4), the total and '7 fields were determined
at each interface and a transfer matrix was formed by multiplying a set of 2 x 2
matrices together, each matrix representing a film. We will call this the film method
because each matrix contains only information about one film and is not dependent
on adjacent films.
In the second formalism (section 2.5), where the forward and reverse tangential
electric fields (S+ and 8-) were determined at each interface, a transfer matrix was
formed by multiplying a set of 2 x 2 matrices together for each interface and each
film. We will call this the interface/film method.
We start with the tangential fields at boundary a. They are given by

Sa = '0+a+ 8 F

Ha = yO(+a a)

In matrix form, this becomes

Ea i-I (2.73)
Ha O -0 O -6a6
By inverting the matrix, we get

Eoa \a (2.74)
,%a -2 1 --1 H/a
If M is the transfer matrix for a structure with boundaries a through m such that

Ea] = M (2.75)

then the use of relations 2.73 and 2.74 can be used to make equation 2.75 become

[ = 1 [ 1 1 1 M 1 Y[ (2.76)
[o O 2 1 [Ys -Ys Y -sm
where s denotes the last media on the right hand side of the last boundary m.
Equation 2.76 is in the same form as equations 2.71 and 2.72 showing that the two
formalisms are connected by the two transformation matrices in equation 2.76.

2.7 Applications of the Formalisms to Real Structures

This section will use each formalism to derive some basic relations in a very
straightforward way. Depending on the problem at hand, it will be shown that
in some cases the interface/film method is preferred, and in another case, the film
method is preferred.
In general, the film method is better for computer calculation of the fields of a
structure. This is because it is composed of a stack of independent 2 x 2 film matrices.
For a periodic structure, a matrix may be generated for a single period and taken
to the power N where N is the number of periods. As will be shown below, this
yields a very simple expression for the reflectance of a periodic A/4 structure. The
interface/film method is better suited for calculation of resonance conditions. These
will now be discussed in detail.

2.7.1 Reflectance from Periodic Quarter-Wave Structures

A periodic stack of dielectric films is of particular interest because it can be used
for a high reflectance mirror. The optimum thicknesses for the films, in this case, is
A/4 where A = Ao/ni is the center wavelength of the reflection band. (Due to the
asymmetry of the reflection band A is not the true center, but it does correspond
to the maximum reflectance.) Either the film or the interface/film methods can be
used to solve for the reflection, but the film method yields an analytic solution for
the reflection at the center wavelength in a very straightforward fashion.
If the periodic structure in figure 2.5 is nonabsorbing and the thickness of each
layer is AO, using equation 2.62, we find that

i = Ys
~ [ [Y2

no n1 n2 n1 n2 n1 n2 n1 n2 ns

So o--

a b m

Figure 2.5. Dielectric mirror made from a periodic quarter-wave stack.

[ (_y)N 1

so that

B= Y2 C=y Y- and y=-=ys )2N (2.78)
Y1 Y2 B Y2

and using equation 2.56,
y(_2)2N 1
r = )" 1 (2.79)
Yo (2 2 )2N +

This equation is very useful for the design of high reflectance dielectric mirrors. It

allows a straightforward determination of the number of periods and ratio of the

indices required for a desired reflectance. Furthermore, it shows that if y, > yo,

better reflectance is obtained if yi > y2.

2.7.2 Basic Laser Relationships

So far, the multilayer matrix formulations derived in sections 2.4 and 2.5 have

been used to solve problems in which there are incident, reflected and transmitted

waves. For the case of the laser this is not the case. However, the formalism is general

and, as will now be shown, is applicable to the laser as well. For reasons that should

soon be apparent, the interface/film method is more convenient for calculation of the

no n1 n2

'F-= 0

a b

Figure 2.6. Single-film laser.

conditions required for lasing. (If it is not obvious, it is suggested that the reader
try to derive these conditions using the film formalism.) To demonstrate this, we
start with the single-film layer laser in figure 2.6. The interface/film method requires
the multiplication of three matrices-one for interface a, one for film 1, and one for
interface b. Using equations 2.71 and 2.72 we have

[oa 1 [ ra e-'e 0 1 1 r, 2b
[Oa tatb ra 1 0 eii rb 1 2b (2.80)
multiplying, we get

6(+,a 1 [ e-i6 + rarbe'61 raei6l + rbe'i6' 1 r 1(281
Soa tatb rae-i6 + rbei61 ei6l + rarbe-'6'

Radiation is emitted from the laser and propagates away from the device to infinity
in regions 1 and 2. The laser cannot produce incoming waves from infinity. Therefore,
the boundary conditions for the problem are o = 0 and - = 0. Using these
boundary conditions in equation 2.81 we find that

e-~61 + rarbe'61 = 0 (2.82)

Rearranging, we have the basic gain relationship:

rarbe2i51 = 1 (2.83)

This relationship is more fundamental than those given in most textbooks because it
contains both magnitude and phase information in one simple relation. To see this,
we separate the amplitude reflection coefficients into magnitude and phase, and 61
into real and imaginary parts:

I|ra ei(O.+7) Irbl e ibe-261e2i61r = 1 (2.84)

where 61 = 61r + i61i. The 7r phase shift is introduced by considering the negative
sign. It is important to consider that both ra and rb are considered from left to right.
The negative sign in equation 2.83 is the result of defining ra with the incident wave
in medium 0. Even though this is not the case, the equations still hold provided the
consistency of the definitions is observed and not intermittently changed. Defining
ra this way causes a 7r phase shift. For this equation to hold, both the magnitude
and phase components must equal one:

I|ra Irb e-2651 = 1 and ei(261r+-a+b+7) = 1 (2.85)

Using equations 2.36 and 2.48 with the magnitude part, it may be easily shown that

1 1
g = n (2.86)
g = 2d RaRb

where g = -a is the intensity gain coefficient. The phase portion shows us that

261,r + a + Ob + T = m27r where m = 0, 1,2,... (2.87)

The last two equations correspond to those found most often in textbooks and are
equivalent to equation 2.83. They offer a more conceptual view because they are
easily derived by arguing that the round trip gain equals one and the round trip
optical path length is a multiple of A. It should be noted that in extreme cases,
the gain affects the reflectance of the mirrors and the gain should be found through


x nl d


Figure 2.7. Single-film waveguide.

iteration of equations 2.86 and 2.87. However, this may usually be neglected because

the gain is almost always very small compared to the real part of the refractive index.

2.7.3 Basic Waveguide Relationships

The multilayer matrix formalisms developed in sections 2.4 and 2.5 have been

used so far for solving for the fields for a structure with traveling wave components

perpendicular to the boundaries. However, even though reflection coefficients and

other relations that conceptually imply traveling waves were used, it was never ac-

tually assumed that they were traveling. This section will demonstrate how general

the two formalisms really are by applying them to a single-film waveguide.

Figure 2.7 shows a waveguide of a single film. The tangential field in media 0 is

given by

go = So&e-iox + EoaeiKO~ (2.88)

where ,i = niko cos 0i. Assuming total internal reflection (ni and n2 must also be

real), Snell's law becomes
sin0o = sin 0 > 1 (2.89)

and equations 2.88 and 2.89 can be linked by the trigonometric law

cos 0o = /J sin2 20 (2.90)

It is convenient, but not necessary, to choose the + sign. This makes Ko positive

and imaginary which, in turn, causes one component in equation 2.88 to increase

towards infinity for negative x. This cannot represent physical boundary conditions.

Approaching the other side of the waveguide in the same way, we find that the

boundary conditions are

0+ = 0 = 0 (2.91)

This is the same set of boundary conditions as for the laser! Therefore, the gain

relationship (equation 2.83) applies also to the waveguide. However, it is interpreted

somewhat differently. With the waveguide, Iral = Irbl = 1 and g = -a = 0 (which

was already assumed). The characteristic equation (equation 2.87) is the same.

This example illustrated the use of the matrix formalism to solve for modes of

a lossless planar waveguide. However, it was not necessary to assume the lossless

case. It can be shown that with loss, the basic resonance equation (equation 2.83)

still holds and the solution requires both traveling and evanescent components in the

solutions of the side regions. With loss, equation 2.83 cannot be separated into two

independent relations. However, as with the case of the laser, if the imaginary part of

the refractive index is small compared to the real part, the solution can be separated

and approximated by the magnitude and phase relationships. It is also interesting to

note that equation 2.83 is useful for solving surface plasmon modes.

2.7.4 Application to Corrugated Waveguides

This section will show how the matrix formalism can be used to solve for corru-

gated waveguides. The result will be a very simple, yet effective model for corrugated



Figure 2.8. Single mode waveguide with an abrupt step.

waveguide devices. The applications of the formalisms thus far have been exact. Due
to the added complexity of this problem, we are now required to make some approx-
imations, but the approximate theory presented here will yield accurate solutions for
the devices of interest in this work.
We will apply the theory developed by Burns and Milton [12, 44, 81]. The problem
becomes very simple for single mode waveguides (as is the case in this work). We
start with a single mode waveguide with an abrupt step at boundary a (figure 2.8).
The solution of the TE electric field can be expressed as

Ey = +(z)P(x, z)e-i(z)z + -(z)P(x, z)eio(z)z (2.92)

where .+(z) and 9-(z) are the complex amplitudes of the forward and reflected
waves, respectively, P(x, z) is the local normal mode profile at position z, f3(z) is the
mode propagation constant. We will assume negligible scattering of the energy from
the guided mode into the radiation modes. The boundary conditions on each side
of the step in figure 2.8 require that the tangential electric and magnetic fields be

continuous. Therefore, at z = a

EPoe-iPoa + o Poei~a = +Pie-I31a (2.93)


o Poe-iPa o&o Poeia = +PIe- (2.94)

By eliminating +Pire-sila in these two equations, we get

S6o 03o 01 No N (2.95)
SOT 3 "+ 31 No+N,

where Ni = Oi/ko are the mode indices. The solution for the reflection at an abrupt

transition (equation 2.95) is in the same form as the reflection from a planar interface

(equation 2.46). There is really nothing mysterious here because equations 2.93

and 2.94 are analogous to equations 2.44 and 2.45, respectively. In fact, by assuming

both forward an reverse propagating waves in all regions, the whole matrix theory for

multiple layers can be applied to multiple step waveguides (a corrugated waveguide

being one example) by substituting the mode indices for the local normal modes for

the individual film indices. This is very useful as most of the results presented thus

far are applicable to grating devices. Furthermore, the same computer programs that

are used to calculate the reflectance, transmission, and absorption spectra of planar

multilayer structures can be used to calculate the behavior of waveguide devices, in

particular, waveguide Fabry-Perot interferometers and lasers. As long as there is low

loss (light is not scattered into radiation modes), and there is no mode conversion

between guided modes (in multimode waveguides) this simulation method is very


The modeling of DBR corrugated waveguides is analogous to the periodic quarter-

wave structure analyzed in section 2.7.1. Taking advantage of our ability to directly

substitute mode indices for film indices we can use equation 2.79 to find the reflectance

of a first order grating at the center wavelength:


where No is the mode index of the input waveguide to the grating, Ns is the mode

index of the output waveguide of the grating, NL is the mode index of half-periods

exhibiting the lower mode index, NHis the mode index of the other half-periods, and

M is the number of grating periods. This shows that to obtain high reflectance, we

require a large difference in mode index, AN = NH NL, between each half-period

of the grating, or large number of periods.

2.8 Spectral Index Method

The channel waveguide structures presented in this work require extensive nu-

merical techniques to adequately solve for their propagation constants and field pro-

files [62, 63, 47, 60]. The spectral index method represents a much less rigorous and

computer intensive method and has been shown to give very accurate results [43]. It

involves a simple variational technique that utilizes the FFT rather than large ma-

trix eigenvalue problems. However, the spectral index method is limited to simple rib

waveguide structures with square homogeneous stripes on top of a planar structure

varying only in the depth direction. Even though this does not represent the type

of structures of interest in this work, it will be found useful in section 4.7 to gain

qualitative insight.

Figure 2.9 shows the general case of a dual rib waveguide structure for which we

want to solve for the mode index and electric field profile. The ribs are separated by

2S, the widths of the ribs are W1 and W2, their heights are H1 and H2, and the planar

guiding layer has a thickness of D. The refractive index of each region is also shown

in the figure. Region 1 is the cladding (usually air), region 2 is the guiding layer,

W, W2
n H .

n :5H
n2 S S D


Figure 2.9. Dual rib waveguide cross section. The dashed line represents the effective
movement of the boundary by the Goos-Hinchen shift.

region 3 is the substrate, and regions 4 and 5 are stripes of possibly different indices.
The principle of the spectral-index technique is to express the solution in the regions
below the ribs in terms of a Fourier transform in the x direction and approximate the
solution in the rib in terms of sines and cosines with a field of zero in the cladding.
These two solutions are linked together by requiring continuity and using Parseval's
formula to obtain a simple dispersion relation. The method is outlined as follows.
Assuming a source free media (lossless), equation 2.15 reduces to
V2E pC a2 = 0 (2.97)

For a z propagating wave in a structure invariant with z, equation 2.17 reduces to

E(x, y, z) = Eo(x, y)ei(z-Wzt) (2.98)

After substituting equation 2.98 into equation 2.97, dropping the common terms,
and using k2 = w2/1e (equation 2.19 in a source free media), we get a scalar wave
equation for the solution of the electric field in each region

02E (2E
+ + (k2 )32)E = 0 (2.99)
_x2 + -a2

where we now will use E to represent a polarization of Eo(x, y).

In deriving equation 2.15, equation 2.11 assumed that Ve = 0 (homogeneous

media). Therefore, equation 2.99 must be applied to each individual medium with

the proper boundary conditions met at each interface. Some methods, such as the

beam propagation method (BPM) and finite difference methods, give approximate

solutions by using equation 2.99 with an inhomogenious index, n(x,y), inserted

(k(x, y) = 2irn(x,y)/A). This approximation is usually acceptable for structures

where the refractive index varies slowly. This is not the case in rib waveguides, but

the approximations sometimes yield acceptable results.

If the refractive index of the cladding, ni, is much smaller than n2, n4, and n5

as it is with an air cladding, the electric field may be approximated to be zero on

the interface if the boundary is moved slightly using the Goos-Hiinchen shift. These

boundaries are shown by the dotted lines in figure 2.9. The new dimensions are given

W',2 = W1,2 + (2.100)

D'= D + q (2.101)

H2 = H1,2 + q (2.102)

where for TE modes p = q = 1, and for TM modes p = kl/k4,5 and q = kl/kj.

Applying the Fourier transform with respect to x on the wave equation (2.99), we

2Y +{k2(y)s2- 32}8 = 0 (2.103)
where 8(s, y) is the Fourier transform of E(x, y). The solution is similar to that of a
slab waveguide with a spectral index, n, = n2 s2/ko.

The solution in the rib regions may be expressed as E(x, y) = F(x)G(y) where
SAi cos(si(x + C1)) -Ci WT/2 < x < -CI + W7/2
F(x) = A2cos(s2( + C2)) W- /2 < x < -C2 + W2/2 (2.104)
0 elsewhere
Ssin(y(Hy)) -C W- /2 < x < -CI + W{/2
G(y) = sin(y2(Hy)) -C2 W2/2 < x < -C2 + W2/2 (2.105)
10 elsewhere

C, = S + W1/2 (2.106)

C2 = S + W2/2 (2.107)

71 = ki s ,2 (2.108)

7'2 = k5 s2 /2 (2.109)

si = r/Wl (2.110)

S2 = r/W2 (2.111)

k4,5 are the propagation constants of the respective ribs, /3 is the propagation constant
of the mode, and the + sign in equation 2.104 represents the fundamental mode while
the sign represents the first higher-order mode.
Using Parseval's formula, it may be shown that the spectral solution below the
rib and the real-space solution within the rib may be linked together by [43]

0EE dx = -E ds (2.112)
-oo ay 27 -oo 9y
This represents a dispersion relation of which there are three unknowns: A1, A2,
and 0/. One of the field amplitudes is arbitrary while the other may be used to
determine the correct field profile. Since /3 depends on the field profile variationally,
an extremum in 3 will be observed when the correct value for A1/A2 is found. Thus,
the dispersion relation, 2.112, must be used in conjunction with an optimization

2.9 Summary of the Numerical Methods

This chapter presented a simple method to calculate the characteristics of corru-

gated waveguide filters and lasers. In doing so, one first must determine the local

normal mode propagation constants. The spectral index method was presented for

this purpose. Next an appropriate matrix formalism should be chosen to solve the

problem at hand. The film matrix method was shown to be the best choice for

determining reflection and transmission spectra (such as a mirror or filter). The

interface/film method was shown to be the best choice for determining resonance

conditions (such as for a waveguide or laser). Analytical solutions were derived for

simple cases of these. However, to solve general problems, computer calculation of

the matrices may be required.


This chapter introduces the concepts of a new tunable wavelength selective filter.

Most IOC filter technologies require a very long device length or the use of the

acousto-optic effect (chapter 1). The device we present has the potential of being very

short in length (r1cm) and utilizes the electro-optic effect. Furthermore, it has the

potential of being tuned to over 50 wavelength channels with bandwidths comparable

to that available with modern-day electronics. The chapter starts by reviewing some

of the basic concepts of dielectric mirrors and the Fabry-Perot interferometer before

the new device structure is introduced. Computer programs implementing the theory

of section 2.4 were used for this purpose. The new waveguide Fabry-Perot device that

is characterized by a single transmission peak with a very large free spectral range is

then proposed.

3.1 Thin Film Dielectric Fabry-Perot Filters

3.1.1 Characteristics of a Thin Film Dielectric Mirror

We begin by reviewing some of the properties of dielectric mirrors. Figure 3.1(a)

shows an alternating dielectric stack typical of bulk-optic dielectric mirrors. It was

stated in section 2.7.1 that this periodic structure can be used as a mirror if the

thickness of each layer is A/4. Using equation 2.79 with the following values used for

the refractive indices, no = 1, n8 = 2.14, nL = 1.2, nH = 2.2, and with 4 periods, we

get R = rr* = 0.985 at the center wavelength. This is more reflectance than can be

achieved by uncoated silver or aluminum mirrors.








0 I

Dielectric Stack

nH nL nH nL nH nL nH nL

1000 1500 2000

Figure 3.1. Simple DBR consisting of a multilayer dielectric stack (a), and its re-
flectance and transmission spectra (b). The parameters are no = 1, n8 = 2.14,
nL = 1.2, nH = 2.2, and 4 periods.

A computer program is required to solve the matrix equations 2.62 for values

of the reflection at wavelengths other than the center wavelength. The reflection

spectra in figure 3.1(b) shows the result. In this case, the refractive index difference

between successive layers, An = 1. This large index difference not only contributes

to a high reflectance with a small number of periods but also makes the full width

half maximum (FWHM) of the reflection peak very broad (803nm).

3.1.2 Characteristics of a Thin Film Dielectric Fabry-Perot Filter

When two mirrors are separated by a cavity, they form a Fabry-Perot interferom-

eter. Such a device is shown in figure 3.2(a) using two of the mirrors just described




1200 1400 1600 1800


2000 2200 2400

Figure 3.2. Fabry-Perot filter consisting of two multilayer dielectric stack DBRs
separated by a short waveguide cavity (a), reflectance spectra of the structure with
a cavity length of 101im (b). The parameters for each mirror are no = 1, n, = 2.14,
nL = 1.2, nH = 2.2, and 4 periods.

in section 3.1.1. In this case, they are separated by a waveguide cavity 10/tm in

length. Even with this very short cavity length, figure 3.2(b) shows that there are

many transmission peaks in the reflection band. To utilize the electro-optic effect to

make a tunable filter, the cavity length must be increased, but this causes the peaks

to become too close together and the reduced free spectral range will cause crosstalk

in a WDM system.

The reflection spectra in figure 3.2(b) was calculated with the methods presented

in chapter 2 with one more approximation. The structure was calculated as a planar

-Lc -


0 IW

multilayer structure with the mode index of the waveguide (shown as the cavity in

figure 3.2(a)) substituted in for the refractive index of the cavity. This was already

shown to be a good approximation in section 2.7.4. Here, we go one step further

by assuming the overlap between the mode field of the waveguide and the reflected

wave by the endface mirror is quite good. This approximation becomes better as the

total thickness of the mirror is decreased because the mode field is dispersed less by

the shorter structure. To make a thin mirror with high reflection, we require AN to

be large. A 4 period mirror with nL = 1.2, nH = 2.2, and a center wavelength A =

1534nm, is 1.98/pm thick. The field divergence within these dimensions is negligible.

These values were only used to demonstrate the concept. For the mirrors used in this

work, nH = 3.5 and nL = 1.56 (which correspond the the refractive index of silicon

and A1203), and about the same reflection as for the mirror just illustrated can be

achieved in just 2 periods. Therefore, the total mirror thickness is reduced to just

0.82/m which makes the approximation made above is even more applicable to our


3.2 DBR Corrugated Waveguide Fabry-Perot Filters

3.2.1 Characteristics of a DBR Corrugated Waveguide

In contrast to an endface mirror, a DBR corrugated waveguide has many (typically

thousands) periods and the mode index difference between the peaks and valleys is

very small (typically AN ~ 10-4). The structure is illustrated in figure 3.3 (a).

According to the theory in section 2.7.4, the solution of the reflection and transmission

spectra of this structure is essentially the same as the dielectric mirror with the mode

index difference used in place of the refractive index of a film.

To demonstrate the characteristics of such a grating, we find that to obtain about

the same reflection from a lossless grating structure as we did for the dielectric mirror

Waveguide Grating

i .n T
R Ins


0 L,

1533.5 1534 1534.5 1535

Figure 3.3. DBR corrugated waveguide (a), reflectance spectra (b). The parameters
used in the calculation are NH = 2.1406 and NL = No = N, = 2.14 and 10,000
periods. Loss of 0.35dB/cm is also included in the calculation.

in section 3.1.1 with a mode indices of NH = 2.1406 and NL = No = N, = 2.14 for

the peaks and valleys, respectively (AN = 6 x 10-4), we require 10,000 periods. For

a center wavelength of A = 1534nm, this device is 3.6mm long, and for the lossless

case equation 2.96 predicts that R = 98.5%.

We expect there to be some loss in the waveguide. Low-loss APE:LiNb03

and Ti:LiNbO3 waveguides typically exhibit 0.1-0.2dB/cm loss. The loss in dB,

a[dB/cm], is related to the absorption coefficient, a[1/cm] by

a[cm-_ a[dB/cm] (3.1)
10 log e

According to equation 2.36, these values for loss correspond to the imaginary part

of the refractive index, ni, (in this case we are really talking about the mode index)

between 2.81 x 10-' and 5.6 x 10-7. When there is loss, we need to use the computer

calculation of the matrices even for the reflection at the center wavelength. Figure 3.3

(b) shows the reflection spectra of this corrugated waveguide structure with a loss

of 0.35dB/cm (n, = 1 x 10-6). A somewhat higher loss was used because we expect

the grating to induce some additional loss on the waveguide. The peak reflection

is reduced from 98.5% for the lossless case to 97.5% for the case where we have

0.35dB/cm loss.
The distinct differences between the characteristics of the corrugated waveguide

reflector and a dielectric mirror are that the corrugated waveguide can have more than

three orders of magnitude shorter bandwidth (FWHM is 3.4A in this case), and a

much longer length with orders of magnitude more grating periods is required for high

reflection. The difficulty in achieving high reflection from a corrugated waveguide is

maintaining low loss when the grating is formed. In this example, we showed that a

very low loss of 0.35dB/cm already has a noticeable effect on the peak reflectance.

This issue will be discussed many times throughout the rest of this dissertation.






g 0.4

0.2 -

1533 1533.5 1534 1534.5 1535
b) Wavelength (nm)

Figure 3.4. Fabry-Perot filter consisting of two DBR corrugated waveguides separated
by a short waveguide cavity (a), reflectance spectra of the structure (b).

3.2.2 Characteristics of a DBR Corrugated Waveguide Fabry-Perot Filter

Figure 3.4 (a) shows a Fabry-Perot interferometer formed by introducing a short

cavity or a quarter-wave shift in the center of the grating. Figure 3.4 (b) shows the

reflectance spectrum of the device using two gratings with the same parameters as

were used in section 3.2.1 and with a quarter-wave shift between them. A single

transmission peak is observed in the center of the reflection band. Only one peak is

observed because the width of the reflection peak is narrower than the free spectral


The free spectral range (FSR) of a Fabry-Perot interferometer is given by

AA2sr = (3.2)

where A0 is the free space wavelength, n is the refractive index of the cavity, and

Lc is the length of the cavity. A Fabry-Perot made with a mirror with a FWHM

of 3.4A and a center wavelength of 1534nm will exhibit at most one transmission

peak if Lc < 1.6mm. However, if this device were to be used as a filter, inter-channel

crosstalk would still be a problem because the stopband itself is narrow and the

sidelobes exhibit extremely high transmission.

This device can make a good single frequency laser because there is only one

transmission peak in the reflection band. All the other Fabry-Perot peaks are in the

sidelobes where equation 2.86 cannot be satisfied. The sidelobes have little detrimen-

tal effect on its performance (the only detrimental effect is that spontaneous emission

is allowed to pass through. However, a filter is adversely affected by the sidelobes as

light is allowed to pass through them almost as well as the Fabry-Perot transmission


3.3 Proposal: Asymmetric Fabry-Perot Filter

In this section, we propose the asymmetric Fabry-Perot (ASFP) filter that takes

advantage of the characteristics of both the dielectric mirror and the corrugated

waveguide DBR reflector. Asymmetric refers to the use of two mirrors differing vastly

in their reflection spectra. The DBR mirrors described in sections 3.1.1 and 3.2.1 are

two such examples where the one in section 3.1.1 has a reflection band FWHM over

2300 times that of section 3.2.1.

Corrugated Waveguide DBR

R --T

Figure 3.5. Asymmetric Fabry-Perot filter consisting of a DBR corrugated waveguide
with a dielectric mirror deposited on the endface.

3.3.1 Passive ASFP Filter

We recall that the Fabry-Perot made from dielectric mirrors had a broad stop-

band, but many transmission peaks within that stopband. The Fabry-Perot made

from corrugated waveguide reflectors had the opposite scenario where the stopband

was extremely narrow and there was only one transmission peak. A somewhat novel

approach would be to combine the favorable characteristics of both types of DBRs

as shown in figure 3.5. The dielectric stack deposited on the endface of a waveg-

uide will provide a very broad stopband while the waveguide grating will provide

rejection of unwanted Fabry-Perot transmission peaks. When they are fabricated

in cascade, forming an asymmetric Fabry-Perot filter (ASFP filter), they provide a

single, very narrow transmission peak with an artificially 1 high free spectral range

(figure 3.6). The phase shift between the grating and the DBR is crucial to assure

that a transmission peak lies within the stopband of the grating.

In the calculations, typical values for the dimensions and indices of the two DBRs

were used and loss was included. The filter characteristics (shown in dB in figure 3.7)

are suitable for a wavelength channel with a bit rate of about 1 Gb/s and an inter-

channel crosstalk of about -14 dB. The solid curve was calculated assuming a grating
'The reason why this FSR is called "artificial" is because there are still many transmission peaks
within the stopband as the phase condition required for interference is met. However, the reflection
of the grating reflector is so low that little interference takes place and the peaks are not visible in
the graph.




c 0.4


500 1000 1500 2000 2500
a) Wavelength

0.6 . . .


o 0.4




), 1533 1533.5 1534 1534.5 1535
U) Wavelength

Figure 3.6. Transmission spectra of an asymmetric Fabry-Perot filter (a); expanded
view of the transmission peak (b). The parameters used for the dielectric mirror were
the same as those used for figure 3.1(b) and those used for the corrugated waveguide
were the same as those use for figure 3.3(b).

0.05 A 635 MHz
-5 -

10.25 A 3.2 GHz



1533.5 1534 1534.5
Wavelength (nm)

Figure 3.7. Transmission spectra of an ASFP filter for a corrugated waveguide loss
of .35dB/cm (solid) and 3.5dB/cm (dotted).

loss of .35 dB/cm. However, the dotted curve shows that if the grating loss is 3.5

dB/cm, the filter response is broadened and the magnitude of the transmission peak

is comparable to the sidelobes making the device useless for a filter. Hence, the most

important technology for the realization of this and similar devices is the fabrication

of high reflectance waveguide grating mirrors with low loss. There is some redun-

dancy in this statement in that a high reflectance waveguide grating mirror implicitly

implies low loss.

3.3.2 Tuning an ASFP filter

The device proposed in section 3.3.1 has the potential of being tuned to many

channels in a very simple and straightforward manner by utilizing the linear electro-

optic effect with an appropriate electrode scheme (figure 3.8). The electro-optic effect

is used to change the mode index of the corrugated waveguide and the cavity (if one is

included). Small changes in the mode index are sufficient to shift the peak to another

wavelength channel. Unlike many other electro-optic devices, the total phase shift

4 pair

Electrodes Waveguide Grating
Electrodes 10k Periods

Z+ LiNbO

Z+ LiNbO3

Device Length: -3.6 mm

Figure 3.8. Tunable ASFP filter.

can be much less than 7r. Therefore, the total device length can be very short. For
the device we are discussing, the length is merely 3.6mm.
To determine the amount of change in the mode index realizable, we recall that
the refractive index of the extraordinary wave in LiNbO3 is dependent upon the DC
electric field along the optical axis (z), expressed as

ne(Ez) = ne er33Ez (3.3)
were ne is the extraordinary refractive index without applied field, Ez is the ap-
plied field, and r33 is the appropriate Pockel's coefficient which, for LiNbO3, r33 =
30.8 x 10-12m/V. LiNbO3 breaks down with electric fields greater than 22-23kV/mm,
and hence Ez must be somewhat less than this. With these applied fields, the ex-
traordinary index change An = 3.3 x 10-3. The change in mode index is related to
the change in refractive index by AN = FAn where F is called the fill factor. The
fill factor is basically the normalized overlap integral between the DC (modulating)
electric field and the optical field profiles. Electrode configurations that result in
r > 0.5 are easily achieved experimentally. The mode index change is further limited

S 635 MHz FWHM



-25 -

-30 I .* * * *
1533.2 1533.6 1534 1534.4 1534.8
Wavelength (nm)

Figure 3.9. Transmission spectra of a Tunable ASFP filter for several values of the
mode index change.

by the breakdown of air, but either a polymer or an oxide overlay can help eliminate

this problem. All things considered, the upper limit for the mode index change is

about AN = 10-3.

To determine the range of tunability of the device, several transmission curves

with varying mode index change are plotted in figure 3.9. If the maximum mode

index change is AN = 10-3, the device is capable of tuning to 57 channels spaced at

3.2 GHz apart with a crosstalk of -14 dB. These results may be improved with lower

loss, higher reflection, and sidelobe suppression of the grating DBR. Furthermore, a

longer, but weaker, grating may be used to increase the number of tunable channels.

The number of channels can also be increased by fabricating an array of these

tunable filters with varying grating periods. Wideband prefilters, such Mach-Zehnder

interferometers, can be used to distribute the groups of wavelength channels to the

appropriate ASFP filters, or if power is not an issue, a simple star coupler can be

used to distribute the power to each ASFP filter.

In contrast to most other wavelength selective filters, the ASFP filter is much less

complex, extremely short (less than 4 mm), and tunable to many channels with a

single low power controller. Relatively few technologies are yet to be developed and

successful development will make this filter amenable to large scale production.

3.3.3 Tunable DFB laser

As we recall from section 1.2, the concept of using the ASFP structure as a

wavelength selective filter is new, but the device structure is not. Many practical

DFB lasers use the same resonator structure, but the purpose is different. The

purpose of the corrugated waveguide is essentially the same for the laser as it is for the

filter: produce narrowband reflection capable of selecting a single transmission peak.

The purpose of the endface mirror is different. The filter requires the broadband

mirror to produce a very broad stopband with very small transmission sidelobes. An

endface mirror is useful for DFB lasers because it usually causes the transmission

spectra to be asymmetric with a dominant transmission peak [74]. The use of a

single grating causes a DFB laser to exhibit dual wavelength operation at the two

dominate transmission sidelobes (shown in figure 3.3). An alternative method to

achieving single longitudinal mode operation is to use a quarter-wave shifted grating,

which is essentially the same as the structure in figure 3.4, but this is generally

more difficult to fabricate. Transmission sidelobes of the Fabry-Perot structure are

of no consequence with the laser since, when laser action begins, the spontaneous

emission in these sidelobes is greatly dwarfed by the high power laser emission from

the dominant mode.

The ASFP filter functions as a DFB or DBR laser if enough gain is introduced

to the cavity and/or grating to overcome the losses. The conditions for lasing can

be determined by equations 2.86 and 2.87 if the gain is only in a cavity, but if there

is gain in the grating, an iterative calculation of the matrix method is necessary to

calculate the gain required for lasing.

The DFB and DBR lasers do not require a high quality grating to lase. Such lasers

have been experimentally demonstrated with erbium doped LiNbO3 [23]. The lasers

were made with a 35mm erbium doped cavity and DBR corrugated waveguides 30mm

and 16mm in length. The 30mm grating exhibited a maximum of 80% reflection and

the 16mm grating exhibited a maximum of 55%. The device with the 30mm grating

was shown to lase even without an endface mirror (the LiNbO3/air interface exhibits

R = 14.3%).

The ASFP laser can be tuned in the same way as its filter counterpart. The

wavelength shifts (shown in figure 3.9) are the same, but the linewidth of the laser is

much narrower. The theory in chapter 2 would predict infinitesimally small linewidths

for the laser, but several broadening mechanisms will produce a linewidth, probably

on the order of 1MHz.

The ASFP filter and laser complement each other. Both can be fabricated on the

same chip and designed to have the same tunability with the same applied voltage.

3.3.4 Summary

In this chapter, we presented the basics of endface dielectric and corrugated waveg-

uide reflectors. Their application to a Fabry-Perot filter showed that neither arrange-

ment was sufficient to act as a WDM filter alone, but when the benefits of both of

them are exploited in a cascaded arrangement, forming an asymmetric Fabry-Perot

filter, a much more useful filter is possible. This filter was then shown to have

almost-ideal characteristics for many WDM applications: reduced complexity; ex-

tremely short length (less than 4 mm); tunability to many channels with just a single

DC controller (low power); and amenability to large scale production. Most other

filters reported to date lack one or more of these characteristics. The same structure

can be used to make a laser (which has been reported elsewhere very recently). This

new filter has the potential of becoming an ideal companion device with a tunable

laser because they are fabricated with mostly the same steps, can be integrated to-

gether on the same chip, and have the same tuning ranges with the same applied


It was shown that the filter is much more demanding on the quality of the grating.

There are few reports of good DBR corrugated waveguides-none in LiNbO3. The

next chapter introduces a new waveguide geometry that has the potential to achieve

good corrugated waveguides for the first time in LiNbO3.


This chapter proposes a novel waveguide geometry that provides a potential solu-

tion to an age-old problem: producing high reflectance DBR corrugated waveguides.

It is amenable to many existing waveguide technologies including those that are cur-

rently used and most viable in the industry. In addition, it will be shown that it

offers a number of other benefits as well.

4.1 The Need for Better DBR Corrugated Waveguides

Presently, the only commercially viable use of reflection corrugated waveguides is

for DFB lasers. This is because the material systems used for DFB lasers is excep-

tionally amenable to making good gratings (the reasons are presented in section 4.3),

and often high reflection is not required when the optical gain is very high. Con-

sequently, as the loss in the grating decreases, the reflection improves. This is why

DFB lasers (lasers that have the grating in the active region) are generally better

than DBR lasers which use a passive grating outside the active region. Other devices

and material systems lack these conveniences.

The device proposed in section 3.3 is only one example where a good DBR cor-

rugated waveguide is required. Describing all of the needs for such components is

analogous to defining all the uses of a bulk-optical dielectric mirror. With significant

technological advances, the proposed DBR corrugated waveguide can become a basic

building block for a broad array of IOCs, sharing in the success of directional couplers

and Y-branches.

The characteristics of corrugated waveguide mirrors are clearly unique. First, they

have a very narrow reflection band (~ 1A). This is useful for narrow bandpass filters

such as the one proposed in section 3.3 and others proposed in references [33, 25, 26,

27]. The fact that they do not require the use of an endface make them very attractive

for IOC devices. Many gratings with varying periods may be placed anywhere on the

chip. Generally, with endface mirrors, all waveguides exiting the same facet as the

ones with mirrors must be polished perpendicular to the waveguide. This prevents

the use of one of the most common methods to prevent unwanted backreflections:

angle polishing.

4.2 Corrugated Waveguide Technologies (Previous Work)

Much research has been devoted to periodic waveguide structures ever since their

first few demonstrations in DFB lasers [36, 86, 87] and filters [16, 19]. Since then, the

use of corrugated waveguides has excelled, providing low threshold and high power

single longitudinal semiconductor lasers. However, passive devices have not shared

that success.

Ironically, there have been many reports of high reflectance corrugated waveg-

uides, but many of them used ambiguous measurement techniques that are not de-

pendable. There have been several claims of almost 100% reflectivity, even 99%, but

very few reports support the claims with indisputable results. To the best of our

knowledge, the best indisputable results to date were produced in silica waveguides

by Adar et al. [2]. They measured 69% of the light back into a fiber using a 2 x 2

fiber coupler. Their measurement was calibrated by butt coupling the fiber to a ma-

terial of known reflectance. Furthermore, they made a reasonable claim that the light

undergoes two fiber-waveguide coupling losses that could total a value near 1.6dB.

Therefore, it is quite reasonable that their DBR corrugated waveguide produced re-

flectance > 95%.

4.2.1 Investigations of DBR Corrugated Waveguides

This section briefly summarizes the most important results of the other efforts on

passive DBR corrugated Waveguides, from their first demonstration to present. Many

of the results reported were not properly substantiated by indisputable measurement


Most of the early investigations were carried out on planar glass waveguides [19,

67, 84, 30, 10]. They all used a prism coupler (section 6.3) to measure the reflected

power and reported better than 80% reflection. An absolute determination of the

amount of light that is coupled in and out of the waveguide is generally impossible.

If they were able to accomplish this, the details were not reported. Therefore, the

measurements are somewhat ambiguous.

A planar DBR corrugated waveguide was demonstrated in InGaAsP/InP in ref-

erence [4]. 99% reflectivity was reported, but not measured. Only the transmission

was measured and they used R = 1 T to calculate the reflection. This assumes

there is no loss. The Fabry-Perot device, later made by the same group, indicated

that loss was indeed a problem [5]. Ridge DBR corrugated waveguides have been

reported in GaAs/AlGaAs as well [83].

There have been several studies of higher order gratings [69, 68, 64, 65, 9, 10].

Shinozaki et al. [69, 68] reported DBR reflectors in LiNbO3 with periods of 10/tm

and above, using proton exchange. One was used for a marginally improved second

harmonic generation (SHG) device and another was used for feedback of both an

external cavity laser and SHG.

Polarization independent DBR corrugated waveguides have also been reported [1].

This was achieved by etching trenches along the sides of the waveguides to practically

eliminate the strain-induced birefringence. The two transmission peaks coincided

when the two orthogonal mode indices were equal.

Low loss Bragg reflectors have also been reported. A technique to measure the

scattering loss from the surface roughness of a corrugated waveguide was demon-

strated by Lee et al. [38]. The transmission characteristics of a Si3N4 channel cor-

rugated waveguide were observed with and without an index-matching fluid on the

grating surface. The index matching fluid eliminates the scattering loss as well as

the reflection peak. By comparing the passband response of the sample for the

two cases, they determined that the incoherent scattering loss of their structure was

~ 0.2dB/cm. Reflection measurements were not reported.

4.2.2 Passive DBR Corrugated Waveguide Devices

This section summarizes most of the passive DBR corrugated waveguide devices

that have been demonstrated. None have proven to be quite practical yet. However,

with the improvement of waveguide grating technology, many of these devices may

gain acceptance.

There are several reports of Fabry-Perot interferometers of two DBR corrugated

waveguides separated by a cavity (figure 3.4). A planar corrugated waveguide in

glass with a wedge etched out of the center, was used to demonstrate a filter that

is tunable by positioning the beam [34]. A quarter-wave shifted grating resonator

with a 7dB transmission peak 1A wide in the center of the narrow reflection band

(AA ~ 10A) was demonstrated in InGaAsP/InP [5]. The most impressive result is

found in reference [28]. They used a Si3N4 channel waveguide for their quarter-wave

shifted grating resonator. Their measurement setup was similar to the one shown

in figure 6.3, but they did not report the absolute reflectivity of the device. An

electro-optically tunable Fabry-Perot filter using waveguide gratings has also been

demonstrated [48, 49].

Other corrugated waveguide devices previously demonstrated have been the res-

onant optical reflector [53], the folded directional coupler [3], and the narrow-band

channel-dropping filter [39] (see figure 1.6). None of these exhibited commercially

acceptable passive characteristics, but the resonant optical reflector was used as an

external-cavity reflector of a laser. 5mW of optical power and a linewidth of 135kHz

was observed. Simple corrugated waveguides have been used for the same purpose

to achieve a 1MHz linewidth [54].

4.3 Difficulties in Achieving High Reflectance

The theoretical calculations of a corrugated waveguide in section 3.2.1 indicate

that it is possible to achieve 97.5% reflection in a corrugated waveguide of 10,000

periods, a AN = 6 x 10-4, and a loss of 0.35dB/cm. This corresponds to a grating

length of only 3.59mm. Even higher reflection can be expected for a larger number

of periods, larger AN, or lower loss.

Unfortunately, many difficulties have prevented the realization of such high reflec-

tion corrugated waveguides in any material. The best results of previous work (section

4.2) have produced gratings of better than 95% reflection in glass waveguides. Re-

ports of reflection from corrugated waveguides in LiNbO3 and related electro-optic

materials are very scarce. Perhaps this is because it is more difficult to make a high

reflectance corrugated waveguide in LiNbO3 than in other common materials used

for IOC technology. Several factors significantly complicate the fabrication of high

reflection DBR corrugated waveguides in LiNbO3. These will now be discussed.

First, the chemical reactivity of LiNbO3 is poor. Device shaping using aqueous-

acid and other solutions is almost impossible [58]. Solutions of HF and HNO3 have

generally been useful only for identifying domain polarity and crystal defects. Reac-

tive Ion Etching (RIE) is usually ineffective because the etch rate is very low and the

process causes damage to the crystal (see section 5.1.1). Since etching of LiNbO3 by

wet-etching, RIE, ion milling, or by other means, have generally proven far inferior

to the etching technology of Si02, semiconductors, polymers, and other materials,

gratings in the LiNb03 crystal itself cannot easily be made deep. Therefore, making

waveguide gratings by etching LiNb03 directly is not a promising approach for high

reflectance devices.

Second, most practical low-loss waveguides in LiNb03 have mode indices very

close to the substrate index. This fundamentally limits the mode index difference,

AN, that can be achieved by etching a grating directly on LiNb03. Etching com-

pletely through the waveguide creates very high AN at the expense of high scattering


Third, LiNbO3 has a relatively high refractive index (compared to glass and poly-

mers). This poses two problems: conventional holography is more complicated, and

perturbation of the optical field by a corrugated buffer layer is lower. These will now

be explained in more detail.

Holography is more difficult due to more intense reflections from the bottom

surface of the substrate. Glass and polymers have lower refractive index, so that the

reflection from substrate-air interfaces is about a third less than in LiNbO3. Even

more important is that their is a multitude of suitable index matching fluids available

for glass and polymers to match the substrate to a prism, absorber, or other beam

diversion element. Index matching fluids for LiNb03 are usually highly absorptive in

the visible and UV wavelength range and are thus much more reflective than LiNbO3.

Most buffer layers used for LiNbO3 waveguide devices have a low refractive index.

(Usually SiO2 is used which has a refractive index of about 1.45 at A = 1.5ym, whereas

the index of LiNbO3 is about 2.14.) The field profile of the guided wave in LiNbO3

exhibits a very small evanescent tail in such a buffer layer. As a consequence, a

periodical perturbation in the buffer layer thickness produces little perturbation to

the optical field. According to equation 2.96, this is needed to achieve high reflection

within a reasonable number of periods. For the same reasons, polymers generally do

not produce good corrugated waveguides on LiNbOa.

4.4 Proposal: Channel Waveguides With High-Index Overlays

A high refractive index film on top of a planar or channel waveguide (figure 1.9),

in the place of a lower refractive index buffer layer, can be used as a medium in which

a high perturbation grating can be formed. When the refractive index of the film is

greater than the mode index, the solution of the field in the film is no longer evanes-

cent and the film becomes part of the guiding media. It will be shown in section 4.7

that this overlay causes the guided mode index to become more dependent on the

thickness of the film. This is useful in making high reflectance gratings (figure 4.1 (a))

because the difference in the local normal mode index between the peaks and valleys

of the corrugated waveguide (figure 4.1 (b)) can be very significant-much more so

than by etching a grating on a low-index buffer layer or the waveguide itself.

This may initially appear to present detrimental effects. First, if a significant

portion of the optical mode is guided in the overlay, the electro-optic effect is reduced

since most overlay materials would exhibit little electro-optic effect. Second, for

relatively thick overlays with very high refractive index the optical field is expected

to be drawn out of the low-index material and yield a smaller mode size. This would

present problems with coupling to other waveguides, particularly fibers. Third, a

NH NL grating overlay


Figure 4.1. Longitudinal cross section of a graded-index waveguide with an overlaying
film grating (a); local normal mode index profile along the propagation direction (b).




............ .............. I ............. ...... ........ .............. ..............

significant portion of the field in the overlay may result in increased loss, especially

due to radiation. However, these problems do not necessarily exist. Some common

misconceptions of similar structures will be addressed in section 4.6.

The term "overlay" as opposed to "buffer layer" is used because the intent is

to perturb the guided optical mode rather than buffering the field at the surface.

Since the field in the overlay is significant, in practical devices, a buffer layer should

be used on top of the "overlay" to reduce surface scattering and provide spacing

from electrodes. In discussions that follow, the term "mode index" is used to refer

to the local normal mode index which is the mode index of a z-invariant (along

the propagation direction) waveguide with a transverse cross section identical to the

waveguide in question at a particular local position.

There is some similarity between this proposal and the work in reference [2]. Fig-

ure 4.2 (a) shows the structure they used. In that work, they deposited a higher-index

material (Si3N4) on the vertical walls of a glass grating. Their reasoning was that cov-

ering the vertical groove walls would form highly reflecting interfaces perpendicular

to the guide axis. However, a more precise interpretation would be the same as that

for our structure. That is, the high-index material helps create a high mode-index

difference between the two half-periods of the waveguide. As shown in figure 4.2, the

former interpretation led to a comparatively complicated structure where the mode

index variation in the propagation direction has spikes. Because they are so thin,

and the optical mode overlap in that region is small, they can be expected to produce

little perturbation to the mode. However, the thin layer of Si3N4 in the valleys of

the grating can be expected to create a significant perturbation because it resides
directly on the surface of the guiding layer and there is significant overlap.

The solid curve in figure 4.2 (c) represents the mode index profile in the propaga-

tion direction if the Si3N4 were placed only in the valley of the grating. The dotted





b) __


Figure 4.2. Corrugated waveguide structure used by Adar et al. [2] (a); Local normal
mode index profile of the structure along the propagation direction (b); Local normal
mode index profile of the structure with Si3N4 only in the valleys of the grating (solid)
and with no Si3N4 at all (dotted) (c).

curve represents the mode index profile of just the glass grating without adding Si3N4.

Comparing the two, we see that by putting Si3N4 in the valley, the mode index profile

becomes inverted with a much larger differential, improving the reflection.

When considering such extreme changes in the local normal modes associated

with the structure in figure 4.2, one must also consider some assumptions that were

made with the step transition method we used to derive equation 2.96. The paraxial

approximation requires that the overlap between the fields on each side of a discon-

tinuity be good. This is not so for the spikes in the mode index profile of figure 4.2

(b). Consequently, it is expected that these spikes will have relatively little effect on

the mode field as it propagates through the structure. The main perturbation comes

from the film of Si3N4 in the valleys of the grating. More rigorous calculations are

necessary to determine their effect exactly, but this is beyond the scope of this work.

With this insight, the structure in figure 4.2 (a) is determined to be much more

complicated than necessary due to the presence of the glass/Si3N4 grating. Only a

grating of Si3N4 is needed, or even beneficial.

4.5 Benefits of Silicon as a High-Index Overlay On LiNbO.

We found silicon to be very suitable to demonstrate our concept of a high refractive

index overlay. There are many reports in the literature of devices made from deposited

films of LiNb03 on silicon for non waveguide applications, but there are few reports

that would relate its use as a waveguide device. These few relevant references are

discussed briefly below.

4.5.1 Previous Reports of Waveguides with Silicon Overlays

The work of reference [70] suggests that by putting a semiconductor film on the

surface of a LiNbO3 waveguide device, the pyroelectrically-induced bound surface

charges can be screened out as they are being generated by slow thermal variations.

The capacitive loading of the electrodes causes the potential of the electrodes to

change much less than the the potential of the rest of the z surface. The potential

difference between the electrode and the surrounding area leads to significant drift.

The addition of a semiconductor film on the surface to reduce the potential difference

has been used to reduce drift in experimental devices [52, 51]. These references state

that a "thin semiconductor film" was used, and do not specify the material.

Silicon was used in reference [66] for the electrodes of a parallel-plate electro-

optic waveguide modulator. 0.231Lm of RF sputtered lithium niobate films were

sandwiched between a 0.165pm layer single crystal silicon on a sapphire substrate

and a 0.181/m layer of vapor-deposited hydrogenated amorphous silicon. The device

loss was calculated to be about 1dB, but the loss of the experimental device was not


Most research on optical waveguides on silicon substrates has concentrated on

the use of deposited layers on oxidized silicon substrates. However, some recent work

has focused on silicon-on-insulator (SOI) waveguide structures. Moderately low-loss

silicon waveguides (2 dB/cm) have been reported [18]. These studies have shown

that ridge waveguides with very thick films of silicon can be single-moded. A further

discussion will follow in section 4.6.

The emission properties (coupling of the guided modes into radiation modes) of

corrugated waveguides with a metal or semiconductor coating was studied in refer-

ence [7]. To overcome the problem of the unwanted emission of light from a grating

to cladding orders, they used a highly reflecting barrier on one side of the corrugated

waveguide, with and without a buffer layer spaced between them. The purpose of

this reflecting layer was to redirect the light from the unwanted orders back into the

substrate. Their conclusion was that the best emission properties could be achieved

by using a highly reflecting metal with a buffer layer. A structure using an aluminum

grating was demonstrated to couple out 81% of power from the waveguide.

4.5.2 Additional Benefits of Silicon on LiNbOa

The only previously-reported benefit of a silicon overlay on a waveguide device

as it pertains to this work is its use towards the screening of the bound charges gen-

erated by the pyroelectric effect [70]. For the purposes of our proposed structure,

silicon exhibits a number of other benefits as well. Separate sections of this disserta-

tion will discuss each of the following benefits in detail. The high refractive index of

silicon (~ 3.5, appendix A) is suitable to achieve a large AN corrugated waveguide

(section 4.7). The use of silicon greatly simplifies the fabrication process because it

is easily deposited by e-beam evaporation (section 5.5), it is easily etched by a num-

ber of well-developed processes (section 5.1.1), and its high absorption in the visible

spectral region makes it very useful for greatly reducing substrate backreflections

during holography (section 5.3). Based on our studies, there is no appreciable dif-

ference in loss between waveguides with and without relatively thick layers of silicon

(section 6.2.2).

4.6 Misconceptions of Guiding in Thick-Film Structures

The purpose of this section is to alleviate some common misconceptions of guid-

ing in channel waveguides. This is necessary because,in cases such as the proposed

waveguide geometry in figure 1.9, there is a vast contrast between the results of

some standard theories used to solve channel waveguides, namely the effective in-

dex method [61], and more accurate analysis. In these cases, only rigorous compu-

tational methods, that do not utilize the separation of variables when solving the

two-dimensional wave equation, can yield accurate results. Even though this has

been shown several years ago, it is not commonly known, perhaps because an elegant

analytical technique to adequately solve these types of problems is not available and

the former, simplified theory is more often used in textbooks. For the purpose of this

section, rather than using elaborate methods to draw the same conclusions as pre-

viously reported for simpler structures, we will describe those conclusions in detail,

comparing with the use of the effective index method; and then use the similarity

between the structures to argue that the same principles apply to our structure.

Richard Soref et al. investigated single mode silicon-based ridge waveguides [73].

It was shown that, although a planar silicon waveguide becomes double moded with

thicknesses greater than 0.3pm, contrary to analysis of the effective index method,

single-mode ridge waveguides can be made with the planar portion several microm-

eters thick! This implies that by removing the ridge portion of a single mode guide,

the remaining planar waveguide can support multiple modes. This seems to be a

contradiction; but the fallacy here is that in one case we have one dimensional con-

finement, and in the other we have two dimensional confinement. They cannot be

related in such simple terms.

It is sometimes assumed that for a single-mode ridge waveguide, its cross-sectional

dimensions must be close to the thickness of a single-mode planar waveguide of the

same material. Perhaps this erroneous assumption stems from the misuse of the

effective index method for solving 2-D structures. The analysis in reference [29]

shows that the effective index method has a tendency to overestimate the mode

indices-especially near cutoff. Therefore, it predicts the presence of modes below

their actual cutoff. Even so, if the results of reference [29] are accepted to be true

for the general case, we get a false impression that the number of modes predicted

by the effective index method is accurate within 1 or 2.

To demonstrate that even this assumption is incorrect, figure 4.3 shows a sim-

ple ridge waveguide structure that has been previously shown to exhibit only one



n2=3.5 )H=.65gm D=.52gm


Figure 4.3. Simple ridge waveguide structure used to illustrate a case were the effec-
tive index method gives the wrong number of modes. This waveguide is single mode,
but the effective index method predicts five modes.

mode [73]. This structure is analyzed by the effective index method [61, 37] by

determining the mode indices of the waveguide at each position, y, as if it were a

vertically confined planar waveguide with those dimensions. In this case, there are

only two different regions with film thicknesses of D and H. They are depicted in
figure 4.4 (a) and (b), respectively. The equivalent horizontal confinement is a planar
waveguide with an index profile derived from the mode indices of the vertical confin-

ing waveguide at each position, y. In this case, it is a symmetric planar waveguide

(figure 4.4 (c)).
Using the parameters given in figure 4.3 at a wavelength of 1.3p/m, we find that
each of the planar waveguides in figure 4.4 (a) and (b) support three TE modes
with approximate normalized propagation constants [35] of bH = .93, .73, .42 and
bD = 0.9, 0.6, 0.2, respectively, where

b = (4.1)
nf ns

n, D n W 14.
n2 2

n33 ND NH ND

a) b) c)

Figure 4.4. The effective index method is applied to the waveguide structure in
figure 4.3 by finding the mode indices of asymmetric planar waveguides with film
thicknesses of D (a) and H (b). These mode indices are applied to a symmetric
laterally confined planar waveguide (c). The mode indices of the waveguide depicted
in (c) are generally accepted to be equivalent to the mode indices of the structure in
figure 4.3.

n, is the substrate index (n3) and nf is the film index (n2). The corresponding mode

indices are NH = 3.36, 2.95, 2.31, and ND = 3.30,2.68, 1.86, respectively. By paring

these numbers together to form the horizontally confined waveguide in figure 4.4 (c),

we get the normalized parameters of v = 2.0, 3.9,4.3 where
2-Wr -2-N2
v = W/NH ND (4.2)

Using the normalized b-v diagram in reference [35], we find that the first value for v

supports one mode while the other two each support two modes. In all, the effective
index method predicts five modes for this structure. However, reference [73] shows
that this structure supports only a single mode! In fact, structures several times
larger support only one mode. In these cases, the effective index method would

predict even more modes.
The waveguides of interest in this work (figure 1.9) are quite different to the one
just illustrated (figure 4.3). However, some similarity is apparent when the structure
of figure 1.9 is viewed upside down as shown in figure 4.5. Instead of having a rect-
angular ridge of the same refractive index as the planar layer, the proposed structure

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