POTASSIUMSODIUM IONEXCHANGED WAVEGUIDES
AND INTEGRATED OPTICAL COMPONENTS
By
AMALIA NIKOLAOS MILIOU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1991
"For a moment there was a wild lurid light
alone, visiting and penetrating all things"
Edgar Allan Poe
Copyright
1991
Amalia Nikolaos MIliou
ACKNOWLEDGEMENTS
I would like to express my deep gratitude to my advisors Dr.
Ramakant Srivastava and Dr. Ramu V. Ramaswamy for all their
guidance and constant encouragement throughout my study. I am
really thankful to them for providing to me a well equipped Photonics
Research Laboratory for completing the experimental work needed for
this project and also an atmosphere conducive to research and to the
interaction and exchange of knowledge among the group members. I
am deeply grateful to Dr. R. Srivastava whose invaluable suggestions
and critical discussions helped me immensely during the course of
this research work. I would like also to express my gratitude to Dr. R.
V. Ramaswamy whose challenging discussions stimulated my interest
in photonics. Also I would like to extend my deep appreciation to the
other members of my Ph.D committee Dr. G. Bosman, Dr. E. Thomson,
Dr. U. Kurzweg and Dr. U. Das for taking time out of their busy
schedule and being on my committee. Appreciations are also
extended to my fellow coworkers Dr. Hsing Chien Cheng and
Christopher Hussell for their assistance and stimulating discussions
and also to the other present and former members of the Photonics
group: Ron Slocumb, Kirk Lewis, Jamal Natour, Mike Pelczynski,
Hyoun S. Kim, Sang K. Han, Dr. Young Soon Kim, Dr. Song Jae Lee, Dr.
Sang Sun Lee and Dr. Chang Min Kim and fellow researchers: Huo
Zhenguang and Dr. Simon Xiaofan Cao for their informative discussions
in our weekly group meetings.
I am also grateful to many members of the department for their
assistance without which it would be very difficult for me to complete
my project; I thank each of them: James Chamblee and Jim Hales for
maintaining the Microelectronics Laboratory and providing assistance
on various processing procedures, Allan Herlinger for machining parts
necessary for this work as well as the secretaries Peggy Lee and Betty
Lachowski for their help with the administrative procedures.
Last, and certainly not the least, I shall always remain grateful to my
family for their constant love and support in all my endeavors and
especially my husband and best friend Nikola who believed in me and
shared with me the difficulties and successes of this journey.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS .................................................................................. iv
A B ST R A C T ..................................................................................................................ix
CHAPTERS
1 INTRODUCTION.................................................................................... 1
1.1. Integrated O ptics ......................................................................... 1
1.2. K+Na Ion Exchange.................................................................. 2
1.2.1. KNa Ion Exchanged Waveguides........................... 3
1.2.2. K+Na* Ion Exchanged Waveguide Devices ........... 5
1.3. Organization of the Chapters.................................... ............ 7
2 IONEXCHANGE/DIFFUSION IN GLASS AND
NUMERICAL MODELING .............................................. ........... ..... 9
2.1. In trodu action ........................................................................................ 9
2.2. Equilibrium at the Ion Source Glass Interface............... 10
2.3. Diffusion Kinetics..................................... ................. ..... ........ 17
2.3.1. Diffusion Equation..................... ............................ 17
2.3.2. Thermal Diffusion Without Applied Electric
Field................................................. ............................. 2 6
2.3.3. Diffusion With Applied Electric Field ................. 30
2.3.4. TwoStep Diffusion Process.................................... 34
2.4. Peculiarity of K Na Ion Exchange ....................................34
2.5. Solutions of the Diffusion Equation.............................. ........... 36
2.5.1. Analytical Solutions of the Diffusion Equation..... 36
2.5.2. Numerical Techniques for the Solution of the
Diffusion Equation....................................................... 39
2.6. Index Change by Ion Substitution....................................... 42
2.6.1. Polarizability and Volume Change.......................... 43
2.6.2. StressInduced Index Change................................... 44
2.7. M ixedAlkali Effect.............................. ....................................... 46
2.8. Space Charge Effect............................................................. ....... 49
2.9. Side Diffusion........................................................................... 50
3 GUIDED WAVES IN PLANAR AND CHANNEL WAVEGUIDES
NUMERICAL ANALYSIS ......................................................................... 52
3.1. Introduction ............................................................................... 52
3.2. GuidedWave Helmholtz Equation ......................................... 54
3.3. Numerical Analysis of 1D Waveguides ................................ 57
3.3.1. W KB M ethod ................................................................ 58
3.3.2. Finite Difference Method (FDM)............................... 63
3.3.3. Multilayer Stack Theory ......................................... 66
3.4. Numerical Analysis of 2D Waveguides ................................ 70
3.4.1. Effective Index Method ........................................... 70
3.4.2. Finite Difference Method........................................... 74
4 WAVEGUIDE FABRICATION ............................................................ 76
4.1. Fabrication Procedure .......................................................... 76
4.2. Planar (1D) Waveguides 77
4.2.1. Sample Preparation.................................................... 77
4.2.2. Melt Preparation ............................... .............. 79
4.2.3. Diffusion Ion Excahange ................... .................. 79
4.2.4. EndFace Polishing..................................................... 80
4.3. Channel (2D) Waveguides........................................................ 81
4.3.1. Deposition of Mask Material............... ........................ 82
4.3.2. Photolithography......... ............................................. 83
4.3.3. Wet Etching Technique................................................ 85
4.3.4. Removal of Mask.................................................. 86
4.3.5. Liftoff Technique ................ .............. .............. 86
5 CHARACTERIZATION OF WAVEGUIDES........................................ 88
5.1. Introduction ............... ............................... .................. 88
5.2. Refractive Index Profile.......................................................... 88
5.2.1. Mode Index Measurements ........................................ 89
5.2.2. Inverse WKB Method............................ ........... .. 93
5.2.3. Concentration Profile Measurements ................... 95
5.3. M ode Field Profile ...................................................................... 97
5.4. Loss Measurements ............................... ......... 100
5.4.1. Fresnel Loss.................................1...... 00
5.4.2. Mode Mismatch Loss .................................... .. 102
5.4.3. Propagation Loss......................................... .............103
5.5. Spectral Response ....................................... .......................... 104
6 MODELING OF INDEX CHANGE ........................................................ 107
6.1. Introduction .............................................. 107
6.2. Index Change by Ion Exchange ................................ 110
6.2.1. Huggins and Sun (HS) Model.................................. 112
6.2.2. Appen Model............................. ............................. 114
6.3. Volume Change, Stress and Birefringence......................115
6.3.1. Volum e Change ........................................ ............... 115
6.3.2. StressInduced Index Change.................................120
6.3.3. Total Index Change and Birefringence...............123
6.4. M ethodology................................. ............................................124
6.5. Results and Discussion................................................... ...... 128
6.6 Sum m ary............................................................ ............................ 13 5
7 DIRECTIONAL COUPLERS AS TE/TM
POLARIZATION SPLITTER AND 3 dB POWER SPLITTER.......137
7.1. Introduction ................................... ........................... 137
7.2. K Na TE/TM Polarization Splitter..................................139
7.3. TheoreticalApproach.................................................................139
7.3.1. CoupledMode Theory.......................................... 140
7.3.2. NormalMode Approach .......................... ..... 144
7.3.3. Relation between CoupledModes and
Norm alM odes................................................ .... 146
7.4. Modeling of the TE/TM Polarization Splitter .................. 150
7.4.1. Design Considerations.............................................150
7.4.2. Index Profile ........................ .... ............. 153
7.4.3. Numerical Method........................................................ 158
7.4.4. Numerical Simulation of the Device......................158
7.5. Fabrication .................................... ........ 171
7.6. Characterization.................. .......................175
7.6.1. Spectral Response Measurements.....................175
7.6.2. Intensity Profile Measurements .......................... 181
7.6.3. Comparison of Experimental Results
with Modeling ........................ ........... 181
7.7. 3 dB Power Splitter .................................................. ..................190
7.8. Sum m ary...................................................... ..............................193
8 CONCLUSION AND FUTURE WORK.......................................... 194
8.1. Conclusion............................................................... .................. 194
8.2. Future W ork................................................ ................................ 195
8.3. K+ Na+ Ion Exchange DevicesPractical
Configurations.................................. .. ......................................195
8.3.1. Asymmetric Directional Coupler............................196
8.3.2. Asymmetric Y Branch......................................197
8.4. K* Na+ Ion Exchange Devices
Novel Applications .............................................. ..................198
8.4.1. Waveguide Lasers and Amplifiers............................198
8.4.2. Nonlinear Waveguides ........................................ 199
8.4.3. OptoElectronic Integration .....................................199
REFERENCES............................................................ ...................00
BIOGRAPHICAL SKETCH .............................. ................... 213
viii
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
POTASSIUMSODIUM IONEXCHANGED WAVEGUIDES
AND INTEGRATED OPTICAL COMPONENTS
By
Amalia Nikolaos Miliou
December 1991
Chairman: Ramakant Srivastava
Cochairman: Ramu V. Ramaswamy
Major Department: Electrical Engineering
The K' Na+ ion exchange process is very attractive for
fabrication of passive integrated optical components with assured
reproducibility within specified tolerance. Main advantages are the
small diffusion rates, use of pure KNO3 molten bath, and singlemode
fibercompatible mode sizes.
A systematic theoretical and experimental study of the KNa+
ion exchange process was recently conducted in our laboratory which
resulted in the correlation of the index profile with the diffusion
theory, and a correlation of the index change with the melt
composition. A novel method was developed to determine the
ix
mobility of K+ ions, and fibercompatible buried channel waveguides
were fabricated for the first time with a twostep process.
In this work, the previous studies are extended to obtain a
detailed modeling of the index change caused by ionic substitution.
The modeling includes the limitations of the existing models and their
applicability to the K+Na ion exchange process. Moreover, a unique
systematic method for determining the compressive stress generated
in glass due to the ionic substitution is described.
To obtain guidelines for fabrication of various devices, the single
mode K+Na+ ion exchange channel waveguide has been modeled using
the effective index method and the multilayed step index
approximation and the results are correlated with the potassium
concentration measurements with the electron microprobe.
The results are finally applied to device developments. As an
example, symmetric directional couplers as TE/TM polarization
splitter is modeled, fabricated, and tested. The spectral response of
the device is measured in the 1.01.6 gm wavelength region and the
power output at 1.32 gm wavelength for both polarizations shows a
cross power ratio of greater than 18 dB. In the course of this work, a
3 dB power splitter for TE as well as for TM polarization is also
realized using a symmetrical directional coupler. The splitting ratio is
52:48 for the TE and 51:49 for the TM polarization.
CHAPTER 1
INTRODUCTION
1.1. Integrated Optics
Integrated optical communication systems consist of "active"
components such as lasers, modulators, etc. and "passive"
components such as power dividers, polarization splitters, etc. These
components are small and compact and their integration can lead to
optical circuits with durable and reliable construction with lowpower
requirements. These requirements have directed the research into
several areas: the development of substrate materials, fabrication
processes and techniques and also the development of new devices.
A variety of substrate materials such as polymers, fused silica,
sodalime silicate glass, borosilicate glass, pyrex glass, lithium niobate,
and IIIV semiconductors have been used. The processes of
fabrication include ion implantation, sputtering, chemical vapor
deposition (CVD), ion diffusion, metal indiffusion, and epitaxial layer
growth. The first four methods are used to fabricate waveguides in
glass, the metal indiffusion and ion exchange are used for LiNbO3 and
the last one for semiconductor materials.
Propagation loss of the waveguide and ease of fabrication have
been the major criteria for choosing a substrate material. From these
considerations, titanium indiffusion in LiNbO3 and ion exchange in
glass have become the most popular techniques for waveguide
fabrication, followed by a rapid development of epitaxial layer growth
of semiconductor materials. The devices developed extend from
simple branching structures to more complicated configurations such
as high speed switches and electrooptic modulators.
The majority of these devices so far has been fabricated on
LiNbO3 substrates using Ti indiffusion. The electrooptic tuning
applied is extremely useful in meeting the desired device
specifications. However, at the receiving end of an optical
communication system there is no need to modulate any information
and some inexpensive "passive" material such as glass can be
considered.
1.2. K Na' Ion Exchange
The feasibility of using binary ion exchange in glass for making
graded index optical waveguides was demonstrated almost two
decades ago [Izaw72]. Since then, significant progress has been made
toward understanding the ion exchange process and the role of the
processing conditions on the propagation characteristics of the
resulting waveguides. Ion exchange process and ionic diffusion of
various cations in glasses and the resulting characteristics have been
discussed in many excellent reviews [Dore69, Tera75, Find85,
Rama88b]. In fact, today, passive glass waveguides are considered to
be promising candidates for applications in optical communication
systems principally due to compatibility with optical fibers, low cost
of fabrication and low propagation losses.
Since the glass index cannot be tuned by application of an
external field, glass devices must be fabricated with assured
reproducibility within specified tolerances. Such reproducible
characteristics are more likely to be achieved with the K+Na+ ion
exchange process where small diffusion rates and the use of pure
KNO3 molten bath assure better control of the process. Another
advantage of the K+Na+ exchange is the relatively small refractive
index change (0.0080.009) which is very attractive for conventional
singlemode fibercompatible mode sizes. The K+Na+ exchanged
waveguides are also characterized by relatively high birefringence
(1x103), low losses, and negligible depolarization [Jack85].
1.2.1. K+ Na+ Ion Exchanged Waveguides
The first study of K+Na+ exchanged waveguides was reported by
Giallorenzi et al. [Gial73]. Since then, studies of diffusion profile of K+
Na+ ion exchanged planar waveguides in sodalime silicate [Albe85,
Yip85], BK7 [Gort86a], Pyrex [Gort86b], and semiconductordoped
glasses [Cull86] have been reported. Effects such as stressinduced
index change and birefrigence in K+ ion exchanged waveguides have
also been observed [Bran86, Tsut86, Albe87, Abou88, Tsut88].
However, in these reports, neither the waveguides nor the device
structures were optimized for fiber compatibility, and a systematic
study of the diffusion process and its correlation with the waveguide
characteristics was lacking.
Recently, we conducted a detailed study in our laboratory
permitting a deeper understanding of the role of the processing
conditions and of the substrate glass in influencing the index profile of
planar, surface and buried channel waveguides [Mili89]. The results of
this work are listed below.
1. Verification of the index profile. An electron microprobe was used
to measure the K+ concentration profile and the data were correlated
with the index profile derived from the modeindex characterization
and with the diffusion profile calculated by solving the diffusion
equation. The index profile for the case of planar waveguides was
examined in detail for sodalime silicate and BK7 glasses and was
observed to be Gaussian and ERFC, respectively, in agreement with the
earlier reports in similar glasses. The differences in the profiles in
the two glasses were attributed to the large disparity of the mobility
ratio of the two glasses. This was the first study which explained the
index profiles satisfactorily.
2. Correlation of the index change with the melt composition. The
surfaceindex change was measured as a function of the melt
composition ( by varying the KNOs/NaNO3 ratio ) and was observed to
vary nonlinearly with the melt concentration of K+ a result in sharp
contrast to the linear behavior reported earlier in another glass
[Garf68]. The observed birefringence and the surfaceindex change in
BK7 glass have now been quantitatively explained on the basis of the
stressinduced effects in the glass (see chapter 6 for details).
3. Novel method to determine the mobility of K+ ions. The mobility of
the K+ ions was estimated by determining the diffusion depth of the
planar surface waveguides fabricated by electromigration. A variation
of the ionic current was observed (an almost exponential decrease for
sodalime glass and almost no change for BK7), and the behavior was
explained for both the glasses on the basis of the mixedalkali effect
[Tera75].
4. Fabrication of buried channel waveguides with a twostep process.
Using a twostep process, buried channel waveguides for operation at
1.3 (pm were fabricated in both the glasses with nearly circular
symmetric nearfield intensity profiles. Finally, record lowloss
waveguides with insertion loss of less than 1.0 dB in 20 mm long
devices were obtained using the twostep process in BK7 glass
[Mili89].
1.2.2. K'Na+ Ion Exchanged Waveguide Devices
Passive waveguide components in glass have shown great
promise in recent years for various applications needing optical
circuitry.
Such devices fabricated by the K+Na+ ion exchange include
wavelength division multiplexers/demultiplexers (WDM), power
dividers, ring resonator and TE/TM polarization splitter at visible
wavelengths.
A single mode wavelength division multiplexer directional
coupler was demonstrated by Corning (Europe) for operation at 0.63
(pm and 0.79 (pm [McCo87]. The measured loss was 2.1 dB at 0.63 .m
for 4 p.m windows and the calculated rejection was 22 dB for a device
of 2.9 mm [McCo87]. The same function was realized with a Ybranch
wavelength multi/demultiplexer [Goto90] for 1.3 pim and 1.55 pim.
Wavelength separation was demonstrated with rather poor extinction
ratios of 5.3 and 5.7 dB at 1.3 and 1.55 p.m respectively.
Power dividers have been demonstrated both in the directional
coupler [Yip84] as well as in lxN star coupler configurations [Find82,
Haru85, Bett90a]. For the first case, the power dividing ratio was
close to unity with further control or fine tuning with the deposition of
a thin dielectric film over the coupling region [Yip84]. For the second
case, the results of the power division vary depending on the number
of branches, the presence or absence of bends, as well as on the
geometry of the junction region. The splitting ratio was observed to
be wavelength and polarization insensitive over a wavelength range of
700 to 900 nm [Bett90a]. However, no reports of operation at 1.3 pIm
or 1.55 jm are available.
A ring resonator was demonstrated in K+Na+ ion exchange
[Hond84] but due to fabrication problems the finesse of the device was
poor. Nevertheless, using the device as a sensor, it was possible to
measure wavelength deviations to 5x104 AO, and temperature
variations to 0.01 oC.
Finally, a TE/TM polarization splitter has recently been reported
[Bett90b]. The device relies on the use of birefringence to create
differences in the coupling coefficients for the two orthogonal
polarizations in an optical directional coupler. The device was
demonstrated with an operating bandwidth of 50 nm at 830 nm. The
best crosstalk performance achieved was 15 dB and 12 dB in the two
output channels and the propagation loss was 0.4 dB/cm.
1.3. Organization of the Chapters
In this dissertation a systematic theoretical and experimental
study of the K+Na+ ion exchange process for fabrication of single
mode optical waveguides is presented. Moreover, based on those
guidelines a symmetric directional coupler has been modelled,
fabricated, and tested for TE/TM polarization splitting at 1.3 rim.
In chapter 2 the ion exchange equilibrium and kinetics are
presented. The nonlinear diffusion equation is derived from the first
principles. Issues such as the contributions to index change, mixed
alkali effect, space charge effect and side diffusion are discussed.
Chapter 3 covers the propagation analysis of planar and channel
waveguides and determination of the propagation constants of the
guided modes as well as their associated transverse electric field
distributions by solving the Helmholtz equation. A comprehensive
review of the numerical techniques used in this work is also
presented.
Chapter 4 describes the fabrication procedure for the case of
planar as well as surface channel and buried waveguides.
The characterization of the waveguides is covered in chapter 5.
Mode index measurements, concentration and index profile
determination, channel waveguide mode field profile, propagation
loss, and spectral response measurements are reported in this
chapter.
In chapter 6, we discuss the modeling of the index change
caused by the ionic substitution. The modeling includes the
verification of the validity of the existing models, their limitations and
applicability to the K+Na+ ion exchange case. A unique systematic
method for determining the compressive stress generated in glass due
to the ion substitution is also described.
Chapter 7 presents the modeling, fabrication and testing of a
symmetric directional coupler used as a TE/TM polarization splitter.
The spectral response of the device is measured in the 1.01.6 pm
wavelength region. The device splits the two polarizations at 1.32 jm
wavelength with a cross power ratio of greater than 18 dB. In the
course of this work, 3 dB power splitters for TE as well as for TM
polarization are also realized using a symmetrical direction coupler.
The splitting ratio is 52:48 for the TE and 51:49 for the TM
polarization.
Finally the results are summarized in chapter 8 and an
illustration of other applications of the K+Na+ ion exchange process
and modeling for devices useful in optical communications is also
presented.
CHAPTER 2
ION EXCHANGE / DIFFUSION IN GLASS
AND NUMERICAL MODELING
2.1. Introduction
The study of ion exchange/diffusion in glass is important since it
is a simple and effective way of forming a higher index layer in glass
substrates necessary for fabrication of optical waveguides. The
diffusion mechanism essentially determines the device parameters
such as the waveguide depth and the index profile while the ion
exchange equilibrium controls the magnitude of the index change at
the surface. Ion exchange in glass, by common definition, is the
exchange of ions of the same sign and valence between a melt or a
solution and the glass framework upon immersion. The process is
reversible and stoichiometric, i.e., every ion removed is replaced by an
equivalent amount of another ionic species, but the concentration ratio
of the two counterions is not necessarily the same in both phases.
In the last two decades significant progress has been made
[Garf68, Dore69, Izaw72, Find85, Rama88b, Rama88c] toward a better
understanding of the ion exchange process and the role of the
processing conditions on the propagation characteristics of the
resulting waveguides.
In the following discussion we describe the ion exchange
equilibrium and kinetics at the glassmelt interface and how they
influence the boundary conditions for the cation diffusion in the glass.
The nonlinear diffusion equation is also derived from first principles
giving an insight of the diffusion process. Several cases are considered
and it is shown that analytical solutions of the diffusion equation can be
obtained only for a few specific diffusion conditions. Different available
numerical methods for the solution of the diffusion equation will also
be discussed and associated results will be presented. The last
sections of this chapter deal with issues related to ion
exchange/diffusion such as the contributions to index change, mixed
alkali effect, space charge effect and finally the side diffusion under
the mask. These issues have a direct impact on the final result of the
process (optical waveguide),
2.2. Equilibrium at the Ion Source Glass Interface
Upon immersion of the glass in the melt, given sufficient time,
an equilibrium condition is reached between the cations in the melt
and in the glass surface such that there are no gradients of
concentration in the exchanger. Figure 2.1 describes qualitatively the
process. At equilibrium, the ions will be distributed between the two
phases in a fixed ratio. This equilibrium can be represented by
A+B<A+B
(2.1)
BEFORE ION EXCHANGE
0 0
MOLTEN SALT
GLASS SUBSTRATE
AFTER ION EXCHANGE
MOLTEN SALT
MOLTEN SALT
GLASS WAVEGUIDE
Fig.2.1 Ion exchange process
where the bar denotes the cations on the interface inside the glass. In
a liquidsolid exchange process, the rate of ion exchange can be
limited by mass transfer of the reactants to and removal of products
from the reaction interface in the melt (i.e. source depletion), by the
kinetics of the reaction at the interface, and by the transport of
cations in the glass phase. However, if the liquidphase diffusion can
supply ample reactants and remove enough products to and from the
interface, the process is not melt mass transfer limited. Also the
surface kinetics are not likely to be a rate limiting factor since they are
much faster than the transport process in the melt and glass phase.
Transfer of cations in the melt takes place mainly via diffusion and to a
somewhat reduced extent via convection. Convection is driven by
density variations and is not expected to be a dominant mechanism
because of isothermal operation and small exchange amounts.
Convection can be enhanced by stirring the melt and thus increasing
the melt mass transfer rate. However, even in the enhanced
convection case, a region may exist near the glassmelt interface
where no convective mixing occurs because of fluid friction at the
interface. Across this boundary, all the mass transfer occurs through
diffusion. The equilibrium state of Eq.(2.1) specifies the surface
boundary condition for the diffusion process and the accompanying
surface index change. The equilibrium state for the exchange reaction
in Eq.(2.1) is governed by the equilibrium thermodynamic constant
given by Garfinkel [Garf68]
aA aB
K = (2.2)
aA aB
where a's are the thermodynamic activities of the cations. The
absolute value of K depends upon the reference functions and states
chosen to define the activities. The value of K reflects the "selectivity"
of the exchanger and it is a quantitative measure of the preference of
the exchanger for one ion over another in solution with it. Many
empirical equations have been proposed to describe the variation of
selectivity of ion exchangers with solution concentration. The most
successful of these equations is that of Rothmund and Kornfeld
[Rothl8] who suggested that in the glass phase the ratio of the
activities of the ions in an ion exchange system is given by
NAn
aB [XB (2.3)
where NA = CA / (CA + CB), (CA is the absolute concentration of the
cation A in the glass) and n = lnaiA /alnCA. For ions in silicate glasses
"n" is usually of the order of unity. It is assumed that the activities of
the cations in the exchanger are proportional to the nt power of their
concentration [Lait57, Karr62]; thus, ai = C!. Equation (2.3) has been
referred to as ntype behavior [Karr62] and, besides the cation pair
depends also on the glass composition. Garrels and Christ [Garr65]
have shown that this empirical relationship is equivalent to the regular
solution theory [Kirk61] for intermediate glass compositions. Using
the regular solution theory for the ratio of the activity coefficients in
the melt and the ntype behavior in the glass phase, the relation
between the melt concentration (NA) and the surface concentration in
the glass (NA) can be written [Garf68] as follows:
n[NA1 E (1 2NA) = n A InK (2.4)
11NAJ RT 41NA
where E is the interaction energy of the two cations in the melt
R is the gas constant (8.317 J/oK.mole)
and T is the exchange temperature (oK)
If the assumption of the regular solution behavior is valid, then a plot
of the lefthand side of Eq.(2.4) versus In[NA / (1 NA)] should give a
straight line with slope equal to "n" and intercept equal to In(l/K).
So far the only work reported in ion exchange equilibrium
between glass and molten salt for potassium ions is that of Garfinkel
[Garf68] which gives a linear relation between NK' in the melt and NK*
in the glass with n=1.2 and K=0.94. Thus, the glass exhibits almost
equal preference for the two ions at T=500 oC with sodium being
slightly favored for that particular composition of glass that does not
contain any potassium. However, the value of K is a function of
temperature and composition of the glass and the melt. Therefore, in
a study conducted in our laboratory, measuring the surface index
change as a function of the melt concentration at T=370 oC in BK7 and
sodalime silicate glasses, we observed a nonlinear behavior, Fig.2.2.
As an attempt to explain our results we compared the content of
K20 and Na20 in the two glasses as well as the difference in ionic radii
of the two ions, Table 2.1. Moreover, using the results of Fig.2.2 we
0.01
0.009  BK7
e Soda Lime
0.008
0.007
An 0.006
0.005
0.004
0.003
0.002
70 75 80 85 90 95 100
% KN03
Fig. 2.2 Index change vs percentage of KN03 in the melt
TABLE 2.1
GLASS COMPOSITION (wt %)
Composition SodaLime Silicate BK7
Si02
Na20
K20
MgO
B203
A1203
Traces
72.25
14.31
1.2
6.4
4.3
1.2
0.34
69.6
8.4
8.4
9.9
2.5
1.2
rK = 1.28 x 1010 m and
rNa = 0.95 x 1010 m
plot Eq.(2.4) for BK7 glass, Fig.2.3. As it was expected we observe a
linear behavior. From the plot we can deduce the values of n=1.3 and
K=0.0004, and since K<1 the glass has a preference in retaining
sodium for this particular temperature and composition. This result is
in good agreement with the small diffusion coefficient and index
change of the system.
2.3. Diffusion Kinetics
The index profile in ionexchanged waveguides is a replica of the
diffusion (concentration) profile which can be calculated by solving the
diffusion equation with appropriate boundary conditions.
In general, the intruding and original ions will have different
mobilities; thus as diffusion proceeds, one ion tends to outrun the
other, leading to a buildup of electrical charge. However,
accompanying this charge is a gradient in electrical potential that
slows down the faster ion and speeds up the slower one. To preserve
the electrical neutrality, the fluxes of the two ions must be equal and
opposite: the electrical potential ensures this condition in spite of the
difference in the mobilities of the two ions.
2.3.1. Diffusion Equation
Much of the recent experimental work has been devoted to
testing and confirming theoretical predictions of the NernstPlanck
model, which has replaced earlier models that employ constant inter
diffusion coefficient. The interdiffusion system described in this
work is based on the NernstPlanck equations, which are derived by
0.5
0.4
0.3
0.2
0.1
5.9 5.8 5.7 5.6
n [ NA / 1 NA
Fig. 2.3 Plot of Eq. (2.4). The slope of the line is equal
to "n" and the intercept is equal to ln(1/K)
5.5
assuming that the driving force for diffusion is the electrochemical
potential and that the interdiffusion coefficient describing the system
is concentration dependent. According to the NernstPlanck
equations the flux for each cation species is given by
Oi = Di Vi +L i zi Ci E (2.5)
alnCi
where Oi: molar flux of cation i (A or B) (moles/m2. sec)
Ci: concentration of cation i in glass (moles/m3)
Di: selfdiffusion coefficient of cation i (m2/sec)
gi: electrochemical mobility (m2/V. sec)
zi : electrochemical valence (zi =1 monovalent ions)
and E: local electric field (V/m), which consists of the space
charge induced field distribution near the diffusion boundary, Es,
which moves deeper into the substrate as time evolves, and the
externally applied field EA. The space charge field is discussed later
in this section.
The first term in Eq.(2.5) represents the ions diffused under the
chemical potential difference resulting from the concentration
gradient, while the second term represents the ions driven under the
electric potential difference or the drift term. The Di and ti are
temperature dependent [Dore64] and are assumed not to be functions
of concentration Ci and therefore independent of position and time
during the diffusion; however, this oversimplification is not justifiable
in the case of strong multialkali effect [Isar69, Day76, Char82b].
It should also be noted that the selfdiffusion coefficient varies with
glass composition and its temperature dependence below the glass
transition temperature is given by an Arrhenius type relation, Fig.2.4,
[Dore64]
Di =Do exp[ (2.6)
where Do: constant
and AH: activation energy (J/mole). The activation energy is
made up of two contributions: the Coulombic energy required to
separate positive and negative charges and the energy to squeeze an
ion through a restricted opening in the network. The Do and AH
depend on the glass composition as well as the ion pair involved in the
exchange.
Further, if it is also assumed (for simplification) that the ionic
transport (conduction) mechanism under an applied field is the same
as that for diffusion. Then the NernstEinstein relation is satisfied, i.e.
Di = kT = k _T iL (2.7)
ni q2 q
where k: Boltzmann's constant (1.38 x 1023 J/K)
ni: number of ions i per m3, i=A, B
oi: ionic conductivity (mho/m)
and q: electronic charge (Cb)
6.8
I I I I
7
7.2
7.4 1
0.00151
0.001527
0.001543
0056
0.00156
1/T (OK)
Fig. 2.4 Selfdiffusion coefficient of potassium ion in BK7
glass as a function of temperature
In most cases, however, the thermal migration of cations is thought to
be slightly different from the electric fieldinduced transport and
Eq.(2.7) is not satisfied [Beie85, Tera75]. In these cases the relation
between the selfdiffusion coefficient and the mobility is instead
written as
Di = fk i (2.8)
The above relation involves a correlation factor f whose value varies
from 0.1 to 1 depending upon the composition of the glass [Beie85,
Tera75].
It is convenient to assume that the exchange is strictly oneto
one at all times and thus the space charge can be neglected, i.e.,
CD << Co
However, there are two dissimilar species of unequal mobilities
involved in the exchange process. As a result, local imbalance in the
charge distribution (spacecharge) is created in the glass giving rise to
an electrochemical potential. The local field thus created assists the
movement of the slower ions (K+) while impeding the progress of the
faster ions (Na+), making the flow rates of the two cations equal and
maintaining the charge neutrality condition, outside the space charge
region.
From the condition of electrical neutrality for the glass, it is
required that the total ionic concentration be constant.
CA + CB + CD = Co
(2.9)
where Co : concentration of ion B in glass before diffusion.
and CD : concentration of the net depleted mobile ions (space
charge.
The diffusion equation for the incoming ion can be derived as follows;
the electric field E in Eq.(2.5) is determined via the Poisson equation
V. (eE) = p (2.10)
where the local space charge density, p, is given by
p = q (CA+ CB Co) =q CD (2.11)
For a weakly guiding case, the change in the dielectric constant, E,
due to ion substitution is very small and Eq.(2.10) can be replaced by
V. (E) = (CA + CB Co) 0 (2.12)
In most cases of interest the spacecharge effect can be neglected,
because the space charge density corresponding to the induced local
field is usually quite smaller compared to the mobile charge density.
Thus, the flux of the incoming ions into an elemental volume
*e does not represent the dielectric constant at optical frequencies,
which determines the refractive index of the medium and depends on
the composition of the glass. It is rather a low frequency dielectric
constant of the isotropic substrate.
24
approximately balances the flux of the outgoing ions, i.e., we assume
CA + CB = Co, where Co is constant. Thus,
aCA aCB (2.13)
(2.13)
at at
Since the number of ions is conserved, the continuity equations are
applicable
aCA
 V. )A
at
and (2.14)
aCB
 V. tB
at
Hence
V.((A+ )B) = 0 (2.15)
Substituting Eq.(2.5) for the two ions in Eq.(2.15), and assuming
constant selfdiffusion coefficients and mobilities, the divergence of
the field E can be expressed as
2 2 .
n (DA VCA + DB V 2CB) E.(A VCA + B VCB) (
V. E (2.16)
1A CA + B CB
The rate of change of the concentration of ion A can be derived
substituting Eqs(2.5) and (2.16) into Eq.(2.14)
aCA 2
S=n DA VCA [A E.VCA A CA V.E
at
DA 2C (DA/DB) JLA CA + 9B CB
= n DAV CA
CA CA + PB CB
A LB(CA + CB)
A E. VCA CA+C (2.17)
9A CA + 9B CB
In terms of normalized concentration NA and NB (mole fractions)
Eq.(2.17) can be rewritten as
aNA 2 (DA/DB) A NA + B NB
= n DAV NA
at 9A NA + B NB
9A 9B
E. VNA  AB (2.18)
A NA + tB NB
or
A =DV2NA E. VNA (2.19)
at
where D is the interdiffusion coefficient defined [Dore64] as
Sn D (DA/DB) 9A NA + B NB 2.20
D = n DA (2.20)
DA NA + lB NB
and g is the intermobility coefficient defined as
9L= IAB 9A A (2.21)
A NA + LB NB 1 ( 1 ) NA 1 a NA
9B
where
a= ( A) (2.22)
tB
Applying the Einstein relation, Eq.(2.7), we see that DA/ 9IA = DB / PB
and Eq.(2.20) can be simplified as
p = n DADB nDA nDA (2.23)
DA NA + DB NB 1 ( 1 A NA 1 a NA
tLB
However, it is well known that the Einstein relation does not always
hold and a modification is needed. The modified Einstein equation,
Eq.(2.8) can be applied to obtain Eq.(2.23), assuming that the
correlation factor fA = fB.
From the definition of the interdiffusion coefficient Eq.(2.20)
and intermobilitycoefficient Eq.(2.21) it is clear that D and R are
concentration dependent for a NA # 0. Equation (2.19) is the
nonlinear diffusion equation of the system. Numerical techniques,
described in section 2.5.2, have been employed to solve the nonlinear
differential equation Eq.(2.19). However, explicit analytical solutions
can also be obtained for specific cases as shown in, section 2.5.1.
2.3.2. Thermal Diffusion without Applied Electric Field
The absence of an electric current (EA = 0) gives
(2.24)
OA + OB = 0
27
In this case, E = Es, the local field due to space charge is given by
[Lili82]
Es = (DA DB)VNA (2.25)
A NA + LB NB
or
Es =fkT VNA (2.26)
q (1 a NA)
Then the nonlinear diffusion equation can be rewritten as
NA D A +NA ~.V NA)1 = V. (D VNA) (2.27)
5t (1 a NA)
Several cases are considered where simplifications of the 3D
nonlinear equation (2.27) occurs as follows [Cran75]:
(a) Two dimensional case:
Equation (2.27) can be reduced to
DNA n DA ANA a n DA aNA
( + ( (2.28)
at ax 1 a NA ax ay 1 a NA ay
The analysis of the 2D case with concentration dependent inter
diffusion coefficient is complex due to lack of boundary conditions
when masks are present [Wilk78]. Numerical solutions are usually
used for the solution of Eq.(2.28), described in section 2.5.2.
(b) One dimensional case:
Further reduction of Eq.(2.27) yields
aNA D n DA aNA
NA ( n DA N) (2.29)
at 3x 1 a NA 3x
(c) a=0, i.e.; 9A = 9B
The two ions are equally mobile. Thus Eq.(2.28) and Eq.(2.29)
become
NA n DA 2NA + NA (2.30)
at )x2 y2
and
NA n DA (2.31)
at ax2
respectively. Crank [Cran75] expects a complementary error function
solution as "a" approaches zero. An example is given in Fig.2.5 which
presents the K+ concentration profile in the BK7 glass.
(d) NA << 1
When the concentration of the incoming ion is much smaller than the
host alkali ion in the glass, Eq.(2.28) and Eq.(2.29) are reduced to
Eq.(2.30) and Eq.(2.31) respectively. This approximation is not valid
when diffusion is carried out from concentrated melts, K+Na+ process
from pure KNO3 melt, but can be used when Ag+Na+ exchange is
performed from dilute melts [Rama88b].
o eMicroprobe
... ERFC
SA Diffusion Eq.
0.8 a = 0.01
No = 0.90
0.59
N(x)
0.39
0.18 '
0 5 10 15 20 25
x [rim]
Fig. 2.5 Concentration profile of BK7 glass fitted to the
solution of the Diffusion Eq. and to a ERFC profile
(e) = 1, i.e., 4A << B (DA << DB)
Equations (2.28) and Eq.(2.29) are reduced to Eq.(2.32) and Eq.(2.33)
respectively
aNA ( n DA NA) a n DA aNA
) + ( ___) (2.32)
at ax 1 NA x By 1 NA ay
and
aNA a ( n DA NA2.33)
at ax 1 NA ax
Crank [Cran75] expects a steplike profile as "a" approaches unity.
For intermediate values of "a" between 0 and 1 the profile obtained is
Gaussianlike as in the case of K+ Na+ exchange in sodalime silicate
glass, Fig.2.6.
2.3.3. Diffusion with Applied Electric Field
The diffusion of incoming ions can be enhanced by applying an
external electric field across the substrate and this decreases the
processing time and temperature and modifies the index profile.
In the case of an applied electric field the total ionic flux, 0o,
corresponding to the electrical current density, J = F Io is
A +B = 0 OO (2.34)
where F is Faraday's constant (96,485 Cb/mole). Using Eq.(2.5), the
total field can be written as
0.8
0
0 eMicroprobe
0.7 0  auian
0
A Difuon Eq
0.6 a 0.98
N 0.8
0.5
N(x)
0.4 0
0
0.3 0
0.2
0.1 O O
0 05 10 15 20 25 30
x (Gm)
Fig. 2.6 Concentration profile of sodalime glass fitted to
the solution of Diffusion Eq. and a Gaussian profile
E=Es+EA f kT aVNA +fkT (OA + B)
q (1 a NA) q BCO ( NA) (2.35
where the applied electric field is given by
f k T (A + OB)
EA = fk (PA+4~B (2.36)
qEA C (1 a NA) (
The space charge is generally nonzero in the case of fieldassisted
diffusion if "a" is not zero. Simplification of Eq.(2.19) with applied
electric field for the case of NA << 1 or a = 0 yields
aRA 2
S= n DA V A E VNA (2.37)
St
Abouel Leil and Cooper [Abou791 have analyzed the problem of electric
field induced ion exchange in detail from the viewpoint of
strengthening the glass and compared the results with the
experimental data. Recently numerical solutions of Eq.(2.37) have
been obtained and compared also with the experimental results in the
cases of Ag+ Na+ [Chen87] and K+ Na+ waveguides [Mili89, Albe90].
Figure 2.7 presents a typical steplike index profile (a = 0) when an
external electric field is applied.
1.518 '
A Mode Indices (InWKB points)
Diffusion Eq.
1.5164 E2= 100 V/mm
t2= 82 sec
1.5148
1.5132
1.5116
1.51 I I I
8 12 16 20 24 28
x [im]
Fig. 2.7 Index profile of sodalime glass with applied field
fitted to the solution of the Diffusion Equation
2.3.4. TwoStep Diffusion Process
In order to increase the symmetry of the index profile (thereby
improve the fiberwaveguide coupling) and reduce the scattering
losses caused by the proximity of the glass surface to the waveguiding
region, a secondstep diffusion is necessary in absence of cation A in
the melt. The diffusion equation is solved numerically in this case
with the firststep index profile as the initial condition [Chen87]. In
the case of planar waveguides, the width of the diffused guides varies
as a square root of the secondstep diffusion time t2 and the depth
Xpeak, to which the guide is buried, is proportional to the product of
the applied field E2 and the time t2. Moreover, in order to achieve
symmetric index profile electric field must be applied in the second
step. An attempt to model the twostep process for a singlemode
buried channel waveguide has been made by Albert et al [Albe90].
However, the model does not include the mixedalkali effect, stress,
and the effect of side diffusion which have significant contribution to
the K Na ion exchange case. Thus, a detailed simulation of the two
step process and comparison with experimental data remains a
challenge.
2.4. Peculiarity of K+ Na' Ion Exchange
When the sizes of the exchanging ions are quite different, e.g.
the K' Na+ exchange, large stresses are built up in the glass. These
stresses slowly relax if the exchange temperature is not far below the
glass transition temperature, and this relaxation can change the short
range structure of the glass and the mobilities of the ions. However, if
the ion exchange is conducted at sufficiently low temperatures, the
structure of the glass cannot relax, thereby creating internal stresses
arising from swelling compressivee stress). Under these conditions
we are no longer dealing with a system in mechanical equilibrium.
Thus the basic transport equation Eq.(2.5), in addition to chemical
and electrical potential terms, includes a third term deriving from the
partial stress of the penetrant [Cran75]. In this case, the
thermodynamics of the irreversible processes leads to an expression
for the flux given by
i = Di VCi + +lizi Ci E + Ci vs (2.38)
alnCi
where vs is the velocity of the ions moving under strain. The velocity
Vs can be expressed as a function of the created stress. Thus on
making the assumption that the stress is proportional to the total
uptake of the penetrant ions Eq.(2.38) becomes
aln1in
i Di VCi + Ji zi Ci E + L V. o (2.39)
alnCi
where a is the stress tensor of the penetrant ions [Fris69, Wang69,
Sane78]. Moreover, the generalized diffusion equation, Eq.(2.19), is
substituted by
aNA 2 2
NA = D V2NA E. VNA [ V2 (2.40)
at
Thus in cases as K_ Na+ exchange where the swelling is
predominant, Eq.(2.40) must be used instead of Eq.(2.19). Some
researchers as will be explained in chapter 6 often include the stress
effect in the diffusion term by defining a polarization dependent self
diffusion coefficient [Yip85]. This approach is unrealistic since the
stress induced term is not proportional to the gradient of the ionic
concentration. However, the index change depends on polarization
due to anisotropic stress [Tsut88].
2.5. Solutions of the Diffusion Equation
Analytical, close form solutions are only possible for the case of
onedimensional diffusion equation with or without an applied external
field in the glass and under special conditions. However, in the two
dimensional case, there is an abundant number of different numerical
techniques that can be used for the solution of the diffusion equation.
2.5.1. Analytical Solutions of the Diffusion Equation
(a) Solution of Eq. (2.31)
Analytical solutions are available for 1D Eq.(2.31) which is valid
in the case that no external field is applied and one of the following
holds: (i) equal mobility ions (i.e., no space charge); (ii) unequal
mobilities but low concentration (i.e., small space charge); these
belong to cases (c) and (d) in the previous section.
A complete solution of Eq.(2.31) requires knowledge of the initial and
boundary conditions. The concentration NA at the glass surface is
constant during the diffusion; therefore,
At the surface,
NA(x,t=0) = 0,
NA(x=O, t) = NAo ,
for x > 0
for t 2 0
(2.41b)
At the other boundary for infinite glass thickness,
NA(, t) = 0,
for t 2 0
(2.41c)
The solution of Eq.(2.31) [Dore64, Cran75] is
NA(x,t) = NAo erfc( X
Weff
(2.42)
where erfc( x / Weff) is a complementary error function profile
defined as
ex/Weff
erfc( x / Weff) = 1 exp(s2) ds
Jo
where
Weff = 2 n DAt
(2.43)
is called the effective depth of diffusion and corresponds to that
distance from the waveguide surface (x=0) where
NA / NAo = erfc(1) = 0.157. The 1/e width of this profile is given
[Rama 86] by Wi/e = 0.64 Weff. Such index profiles have been observed
(2.41a)
in dilute Ag Na+ exchange [Lagu86a], and K Na process in BK7
glass [Mili89], as shown in Fig.2.5.
In general, the K Na+ ion exchange is carried out from
concentrated melt solutions (infinite diffusion source) and ion
exchange equilibrium results near the surface shows that NAo  1.
For finite diffusion source the concentration at the surface decreases if
the ions are not replenished during the diffusion and the index profile
becomes closer to Gaussian due to the difficulty in terminating the
diffusion exactly at the point of exhaustion.
(b) Solution of 1D in the presence of an external field
For one dimension Eq. (2.37) is reduced to
aNA n DA 2A AE aNA (2.44)
at ax2 ax
Approximate analytical solutions are obtained for the following specific
cases:
(i) For the special boundary conditions
NA = NAo, NA0 for x = (2.45a)
ax
and
NA = for x oo (2.45b)
ax
the solution of Eq.(2.44) is [Abou79]
NA(X) = + exp[vNAo +expvN Ao a (x v t)]1 (2.46)
\n DA
with the concentrationdependent diffusion front velocity
v = vo (1 a) / (1 a NAo) and Vo depending on the current density
caused by the applied field.
(ii) For low concentrations (NA << 1)
For the conditions given by Eq.(2.41) an analytical solution
[Cuch61, Malk61, Lili82] is obtained as
NA = NAo erfc(u u') + exp(4uu') erfc(u + u')} (2.47)
2
where u = and u' AE t Weff is defined by Eq.(2.43) and t is
Weff Weff
the time in seconds.
For large electric fields such that u' > 2.5 it can be shown [Rama86]
that the contribution of the second term in Eq.(2.47) can be neglected
and diffusion does not play significant role in the index profile. The
index profile is steplike with the diffusion depth given by Vo t
[Abou79], where Vo is the velocity of the diffusion front.
2.5.2. Numerical Techniques for the Solution of the Diffusion Equation
The 1D and 2D Diffusion equations can be solved numerically
by a number of well known methods: the finite difference method
(FDM) [Cran75], finite element method (FEM), the Green's function
approach [Chen87, Rama88a] etc.. In this section we describe briefly
the finite difference method (FDM) used in this work.
To illustrate the FDM technique let us consider 1D diffusion.
We first let the range in x be divided into equal intervals 6x and the
time into intervals 6t, so that the region xt is covered by a grid of
rectangles of sides 6x, 6t. Let the coordinates of the representative
grid point (x,t) be (i6x, j8t), and also denote the value of NA at the
above point as NA(i,j) where i and j are integers, Fig.2.8.
Fig.2.8 Uniform discretized grid for FDM scheme
By using the Taylor's series in the x direction and keeping t constant,
we can write
S2 x2
NA(i+l,j) = NA(ij) + 6x (A) + 1 (x)2 (a NA
ax x,,tj 2 ax2 x,,tj
+ 0 (5x)2
DNA
NA(il,j) = NA(i,j) 6x ( A)
Sx x,t,
+ 1 (5x)2 (aNA)
2 aX2 x,,tj
+ 0 (5x)2
On adding we find
S2NA NA(i+1,j) 2 NA(ij) + NA (i ,j) + (5x)2
a2x2 ,tj (8x2)
(2.48)
(2.49)
(2.50)
41
Similarly, by applying Taylor's series in the t direction, keeping x
constant, we have
8N82
_NAA 1 (8t)2 (NA
NA(ij+1) = NA(i,j) + t (aNA) + x t 2 (t2 + .. (2.51)
from which it follows that
4NA = NA(ij+1) NA(ij) + (2.52)
at xtj 8t
where the O (8x)2 and O (6t) are the terms that have been neglected
(error terms).
By substituting Eq.(2.50) and Eq.(2.52) into Eq.(2.31) we find after
slight rearrangement
NA(ij+l) = NA(ij) + r (NA(i,j) 2 NA(i,j) = NA(i+l,j) (2.53)
where r = n DA6t / (5x)2 We can use Eq.(2.53) with a chosen value of
"r" to calculate the values of NA at all points along the successive time
rows of the grid provided we are given some initial starting value at
t=0, and some conditions on each of the boundaries. This method is
called "explicit method" since one unknown value can be expressed as
a function of known values.
Crank and Nicolson [Cran75] proposed a variation of this
method, called the "implicit method", by replacing a2NA / ax2 by the
mean of its finitedifference representation on the jth and (j+l)th
time rows. Under this approximation Eq.(2.31) becomes
r NA(ilj+l) + (2 +2r)NA(ij+l) r NA(i+l,j+l)
= r NA(il,j) + (2 2r) NA(ij) + r NA(i+1,j) (2.54)
Thus if there are N internal grid points along each time row, then for
j=0 and i=1,...,N Eq.(2.54) gives N simultaneous equations for N
unknown values along the first time row expressed in terms of the
known initial values and the boundary values at i=0 and N+1. The
strong advantage of this method is its convergence and stability.
However, this method can not be used for solving the nonlinear
diffusion equation since it is impossible to convert a nonlinear
equation to a matrix which is linear in nature. In this case the first
approach is more appropriate.
2.6. Index Change by Ion Substitution
The substrate of the glass is a network of glass former, the most
common of which are SiO2 and B203, modified by other components of
the glass composition. In the binary ion exchange process, both the
cations which exchange with each other are network modifiers. As a
result, the basis structure of the glass is left unchanged and only the
refractive index of the glass is modified. The net index variation
depends on three major physical changes; namely, (1) ionic
polarizability, (2) molar volume due to the different ionic radii and (3)
43
stress created by the size mismatch upon substitution since the ion
exchange takes place below the stress relaxation temperature of the
glass.
2.6.1. Polarizability and Volume Change
The electric field of the light propagating through the glass
interacts with its polarizable ions causing displacement of the
electronic charges with respect to their nuclei, creating dipoles. Such
an interaction lowers the phase velocity of light by a factor n, the
index of refraction of the glass. For an isotropic medium consisting of
a large number of molecules of various species, the polarization is
given by
P Ni Pi = Ni ai Ei (2.55)
i i
where Pi is the dipole moment of the i molecule, Ni is the average
number per unit volume and ai is the molecular polarizability. The
Eoc0 represents the local field, given by the summation of the
macroscopic electric field E with an internal field, created by the
closely packed molecules, i.e.,
1oc = E + 1 (2.56)
3eo
The dielectric constant of the medium is defined as
=D EoE+ P (2.57)
EoE EoE
where eo is the free space permittivity and D is the electric
displacement.
Combining Eq.(2.55)Eq.(2.57) we get
e 1 = Ni ai (2.58)
e + 2 3 Eo
Equation (2.58) is known as the ClausiusMossotti relation. At optical
frequencies, Re(e) = n2, where n is the index of refraction, the
equation is called the LorentzLorentz [Lore80a, Lore80b] relation
n2 1 Ni 1i R (2.59)
n2 + 2 3 eo V
where R is defined as the molar polarizability and V the molar volume.
The molar volume determines the density of the polarized ions.
Several models exist in order to estimate the refractive index of
glass or its changes as a function of composition. These are all based
on the fact that the refractive index is an additive quantity and they do
not take into account the multialkali effect (see section 2.7). These
models are described in details in chapter 6.
2.6.2. StressInduced Index Change
The theory which relates the changes in the index of refraction
to the state of stress is based on the photoelastic effect which couples
the mechanical stress to the optical index of refraction. The state of
stress and stain in an elastic body (solid) is characterized by second
order tensors. When stress is applied the dielectric tensor changes
and in the first approximation these changes are linearly related to the
stress and strain components [Dall65]
n i
(2.60)
where A I is the index of refraction tensor,
and Pij is elements of the stress optic tensor.
medium) Pij is given by [Dall65]
Pij=
P11 P12 P12
P12 P11 P12
P12 P12 P11
0 0 0
0 0 0
0 0 0
0
0
0
P44
0
0
0 0
0 0
0 0
0 0
P44 0
0 P44
oJ is the stress tensor
For glasses isotropicc
(2.61)
where 2P44 = (P11 P12). Combining Eq.(2.60) and Eq.(2.61) we get
P11 ox + P12 (Gy + Tz)
A P11
n2 Pi
Gy + P12 (Gx + aGz)
(2.62)
oz + P12 (Gx + ay)
A '=
Ln2J]x
Forming the equation of the index of ellipsoid in the presence of
stress we get the new refractive indices and also the index differences
given by [Dall65]
Anx = C11 x + C12 (ay + oz)
Any = C11 (y + C12 (Ox + z) (2.63)
Anz = C11 oz + C12 (Ox + y)
where Ci = n3 P, is known as the photoelastic coefficient and n is
the refractive index of the unstressed medium. Equations (2.63) show
that stress transforms an isotropic medium into an anisotropic one,
causing strained glass to exhibit birefringence. This birefringence can
be easily observed in K+Na+ exchanged.
2.7. Mixed Alkali Effect
The "mixedalkali effect" is the nonlinear behavior of the
material properties on substitution of one alkali (i.e. K+) ion for
another (i.e. Na+) in alkali containing substrates, such as occurs in the
ionexchange process. Properties most affected are those associated
with alkali ion movement such as electrical conductivity, alkali
diffusion and alkali selfdiffusion coefficient [Isar69, Day76, Neum79]
which typically exhibit either a minimum or a maximum, with the
substitution of a second alkali, at a composition close to that of the
crossover of the individual alkali properties Fig.2.9(a) and Fig.2.9(b).
The mixed alkali effect increases with the difference in size between
the two alkali ions.
The alkali exchange process in glass leads to interdiffusion of
the monovalent cations in the rigid glass matrix. As the two ions of
different mobilities interdiffuse, a potential gradient is set up to
maintain electrical neutrality. When ionic activity coefficients in the
glass are constant, interdiffusion of ions A and B can be treated in
terms of an interdiffusion coefficient D defined in Eq.(2.20). When
the glass is first immersed into the molten salt the concentration of
the second alkali in the glass (NA) is very small and D = n DA, i.e. the
exchange process is controlled by the mobility of the second alkali
which is smaller than that of the first alkali. It is important to note
that although the selfdiffusion coefficient of the alkali in the glass DB
(Na+) is larger than DA (K+), its value is initially unimportant to the
exchange process, Fig.2.9(a). Ideally, the maximum initial exchange
rate occurs when the two alkali ions have the same mobility. However,
compositional changes that increase DA (K+), will increase the
exchange rate, since increasing the mobility for the less mobile ions is
more important than lowering the mobility of the more mobile ions by
the same amount.
A general consequence of adding a second alkali is a significant
reduction in the diffusion coefficient of the original alkali ion,
Fig.2.9(a). The alkali ratio corresponding to the crossover point, is
usually not found at equal concentrations of the alkali ions but is a
function of: (i) the type of alkali ions present (ii) the total alkali
concentration and (iii) the glass former. The change in alkali diffusion
48
DNa
8 
10
o (a)
11
S12
0 0.2 0.4 0.6 0.8 1.0
Alkali ratio K/K+Na
100
O
O
4 10 
o
0
I 1 \
0 0.2 0.4 0.6 0.8 1.0
Alkali ratio K/K+Na
Fig.2.9 (a) Alkali diffusion coefficient and (b)
electric conductivity vs alkali ratio
coefficient provides a basis for interpreting also the change in the
electrical conductivity. Because of the different compositional
dependence for the two alkali diffusion coefficients, the minimum in
the conductivity occurs closer to the concentration where CA = CB,
Fig.2.9(b).
Several theories have been presented in the literature trying to
explain the mixedalkali effect emphasizing the structural features of
the glass and electrodynamic interaction between ions but none of
them has been successful in explaining the phenomenon fully [Day76].
In general, however, the selection of alkali system is very
important since the diffusion coefficients vary with the overall
chemical composition. It should be possible to minimize the
difference between the the two diffusion coefficients by compositional
changes which increase the diffusion coefficient of the less mobile ion.
Such optimization of the glass composition has indeed been
undertaken recently [Ludw87]. Finally, the alkali ratio in the molten
salt bath can be controlled so that the alkali ratio at the surface does
not exceed that value where D is a maximum.
2.8. Space Charge Effect
As discussed previously, since the substrate glass is rich in
cations B, 9A ( [B when the substrate is immersed in the melt. Cation
B migrates faster than cation A, causing the space charge effect. As a
consequence an electric potential develops balancing the diffusion
process [Char80]. The induced field affects the movement of both the
ions, A (K+) and B (Na+), opposing the chemical potential driving B
(Na) and aiding that of A (K+). As a result the outdiffusion of the
sodium ions is slowed down and the movement of the potassium ions
into the glass is accelerated until an ionic equilibrium is reached. The
extent to which space charge effect modifies the index profile
depends upon the range of variation of MA and JB in the two extreme
concentrations resulting in index profiles from complimentary error
function to step index.
2.9. Side Diffusion
In the case of channel waveguide fabrication by ionexchange,
side diffusion under the mask is created by an electric field generated
by a potential difference between the nitrate bath and the metal mask
used in the process, due to metal cations (e.g. A13+) ions going into the
solution and leaving electrons behind [Walk83b]. Thus, the mask is at
negative potential with respect to the melt. If the negative potential of
the mask is greater than the negative potential of the glass with
respect to the melt then the indiffused ions will be accelerated and
the outdiffused ions will be retarded. However, since the ion
exchange is strictly onetoone the refractive index increase must also
be retarded and all excess indiffused ions not balanced by the out
diffusion will be deposited beneath the mask edge. The side diffusion
increases the waveguide's width and influences its index profile.
Moreover, in the case of Ag* exchange it was found that the induced
local field at the edges of the mask reduce the silver ions into metallic
form in that region, increasing the loss of the waveguide [Walk83b].
Anodization of the aluminum diffusion mask has been proven to be
useful and simple technique for the production of dielectric masks
[Walk83b]. Moreover, the use of a sufficiently thick (300 AO) A1203
layer of low ionic conductivity may be used to isolate the metal from
the melt, as described in chapter 4, and thus reduce the side diffusion
effect as well as the silver ion reduction.
Finally, a more complicated technique can be used to avoid side
diffusion by placing "guard rings" at the areas underneath the mask
where the accumulation of undesired ions is taking place [Clin86].
CHAPTER 3
GUIDED WAVES IN PLANAR AND CHANNEL
WAVEGUIDES NUMERICAL ANALYSIS
OF PROPAGATION
3.1. Introduction
The field of integrated optics is based primarily in the fact that
light waves can be guided in very thin layers (optical waveguides) of
optically transparent materials. Combining these waveguides together
and shaping them into appropriate configurations, a large variety of
components that perform a wide range of operations on optical waves
have been realized especially for applications in the areas of optical
communications and signal processing. The understanding of the
waveguide characteristics of both planar and channel waveguides is,
therefore, essential in the design of high performance devices.
Propagation analysis of a dielectric waveguide with a given index
profile (determined by the diffusion process) consists mainly of solving
the Helmholtz wave equation. This process yields the number of
modes supported by the waveguide, their propagation constants, and
their associated transverse electric field distributions. This analysis is
only exact for a selected number of refractive index distributions i.e.,
quadratic and exponential. However, in most practical cases the index
profile may approach a Gaussian or a complementary error function
and it is necessary to use numerical techniques. Some of the widely
used methods include
(a) WKB Method [Whit76, Chia85, Sriv87a]
(b) Matrix Method [Chil84, Ghat87]
(c) Multilayer Stack Theory [Thur86]
(d) 1D and 2D Finite Difference Method (FDM) [Ster88a, Ster88b]
(e) Variational Approach [Mats73, Mish86, Haus87b]
(f) Variational Approach with Finite Element Method (FEM) [Yeh79,
Maba81, Kosh85a, Kosh85b, Lagu86b]
(g) Perturbation Technique [Kuma83]
(h) Spectral Index Method [Mcil89, Mcil90, Ster90], and
(j) Effective Index Method (EIM) [Rama74, Hock77, Chia86,
Veld88].
Generally in all of the above numerical techniques we assume that the
index profile is invariant along the zdirection (propagation direction).
However, some integrated optical devices involve evolution of the
propagating wave along the zaxis i.e., taper velocity couplers, cross
couplers, etc. In these cases numerical techniques such the beam
propagation method (BPM) [Feit78, Feit79, Feit80], the local normal
mode approach [Marc74] and the step transition model [Marc70] are
applicable. A comprehensive review of numerical methods is
presented by Kogelnik [Koge88].
The first part of this chapter deals with the derivation of the
Helmholtz equation while the remainder discusses some of the most
used numerical approaches for both one and two dimensional
waveguides.
3.2. GuidedWave Helmholtz Equation
Wave propagation in a sourcefree, linear dielectric medium is
governed by the vector Helmholtz equations, known also as the wave
equations. Assuming a zpropagating wave with its time and z
dependence expressed as exp(jot jpz) and E, No and P independent
of z, the wave equations become [Koge88]
Vt2 Et + Vt (Et Vt InE) + (W2 Epo p2) Et = (3.1)
and
Vt2 Ht + (Vt Ine) x (Vt x Ht) + (02 E to 2) Ht = 0 (3.2)
where Et: transverse component of the electric field
Ht : transverse component of the magnetic field
co : angular frequency
S: permittivity of the medium
to : permeability of the medium
and 3 : propagation constant
The Et, Ht and E depend only on the transverse coordinates x and y
and they are also independent of time. The transverse Laplacian is
a2 92
defined in the Cartesian coordinates as Vt2 + T h e
ax2 ay2
corresponding zcomponents of the fields are derived from
V. H = 0 and V. (E E) = 0 and are given in terms of the transverse
components as follows:
j Ez = Vt. Et + Et. Vt Ine
j 3 Hz = Vt. Ht
(3.3)
(3.4)
The e, Ez and Hz are also independent of time and the z coordinate.
In each of the three regions of a step index waveguide the term Vt Ine
is equal to zero. It can also be neglected for a slowly varying index
V= << 1, and the gradient terms (second
medium, Vt Ine = V Ine = 1, and the gradient terms (second
e
term) in Eq.(3.1) and Eq.(3.2) can be ignored.
equations above can be rewritten as
Vt2 Et + (02 e to p2) Et = 0
Vt2 Ht + (0c2 E o 32) Ht = 0
Thus, the four
(3.5)
(3.6)
(3.7)
j P Ez = Vt. Et
j 1 Hz = Vt. Ht
(3.8)
We only need to solve the above equations for either Et or Ht. The
other field vector can be calculated by using Maxwell's curl equations
VxE= j o Lo0 H
and
(3.9)
VxH=j coE (3.10)
For a twodimensional surface waveguide, with the ydirection parallel
to the surface of the substrate there are two types of modes; the quasi
TE modes with the electric field primarily polarized in the ydirection
(i.e., Ex / Ey is nearly zero), and the quasiTM modes with the
magnetic field primarily polarized in the ydirection (i.e., Hx / Hy is
nearly zero). Thus, the scalar Helmholtz equation is expressed as
Vt2 + (0o2 Eo p2) F = 0 (3.11)
where F represents Ey for quasiTE modes or Hy for quasiTM modes.
In the above equation, F and e are functions of x and y only. Equation
(3.11), the scalar Helmholtz equation, can be rewritten as
D2F 22F
+ + + (n2 2) F = 0 (3.12)
ax2 Oy2
where ( = ko N
N : effective mode index
ko : free space wave vector, ko = o2 Eo Lo
and n : refractive index =
In the onedimensional waveguide (no confinement in the y
direction), the above equation can be further simplified as
2+ k2(n2 N2)F = (3.13)
ax2
Furthermore, normalization of Eq.(3.13) might be useful in
applications with arbitrary index profiles [Hock77]
a2F
+ V2 [f(x') b] F = 0 (3.14)
ax'2
where x' = x / W
W : 1 /e width of the waveguide
V : normalized frequency, V = ko W (nf n )1/2
b : normalized propagation constant, b = (N2 n3)/(n2 nQ)
nf: surface refractive index
ns: substrate refractive index
and f(x') : normalized diffusion profile. The f(0) is always 1.0
and f(1) is zero for linear and quadratic profile, 0.157 for
complementary error function profile and represents the 1/e value for
exponential and Gaussian profiles [Lagu83].
3.3. Numerical Analysis of 1D Waveguides
Two approaches are usually used to characterize a one
dimensional waveguide; the ray optics approach and the wave optics.
The ray optics or geometrical optics approach is an approximation
that is valid as long as the waveguide dimensions are much larger than
the wavelength of the light. On the other hand, the wave optics
approach is always valid. Since we are mostly interested in graded
index waveguides where the refractive index n(x) varies gradually over
the crosssection of the waveguide from maximum change to zero, we
will emphasize the field distribution of the guided waves. However,
the ray approach is simpler and has a direct application in the WKB
method, thus it will also be discussed in the following subsections.
3.3.1. WKB Method
The wellknown WKB method was first used to obtain
approximate solutions of the Schridinger equation. It can also be used
to solve the wave equation Eq.(3.13) for the modes in a multimode
graded index waveguide with slowly varying index profile n(x).
Once the effective mode indices have been determined, Eq.(3.13)
becomes an ordinary differential equation. The fields can be obtained
by solving the equation either analytically or by using a numerical
technique.
Consider the ray propagation in a gradedindex waveguide as
shown in Fig.3.1. According to the WKB method, the characteristic
equation of the mth order mode is given by [Gord66]
2 ko [n2(X) Nm]1/2 dx 2 0s 2 0c = 2 m 7 (3.15)
where xm : turning point of the mth order
2 Os : phase shift at the turning point
and 2 c : phase shift at the coverguide interface
Equation (3.15) is called the dispersion relation. The left hand side of
Eq.(3.15) represents the total transverse phase shift the ray
Total Internal
Reflection
(c = n/2
N(Xm)=Nm
n(x)
Turning Point 0s = n/4
Fig.3.1 Ray diagram for the gradedindex waveguides. The index
profile is shown in the left.
experiences in one round trip across the waveguide. This must equal
2min in order to achieve constructive interference, a condition for the
light to be guided along the length of the guide. At the turning point
Xm, the wave is propagating in the z direction with the propagation
constant equal to Pm, i.e.,
ko n(xm) = pm = ko Nm (3.16)
The phase shift at the coverguide interface equals
c = tan1 r N n (3.17)
L V n? N_
while, the phase shift at the turning point is given by
s = tan r n (3.18)
where nf: surface index
ns: substrate index
nc: cover index
and r : constant, r=l for TE modes and n? / n? > 1 for the TM
modes. Since at x=xm, the ray approaches grazing incidence with the
index discontinuity approaching zero, the phase shift for both TE and
TM modes, in this limiting case equals ( n/2) [Hock77]. It must be
pointed out, however, that the approximation 2 Os = n/2 is valid only
in the well guided regime. It has been argued [Sriv87a] that as the
mode approaches cutoff, Ps 0, and the above approximation gives
erroneous results.
The accuracy of the WKB method depends on how accurate we
can estimate the two phase shifts if no approximation is used. For the
case of a threelayered slab, step index waveguide, the approximations
s = C / 4 and Os = n / 2 are not accurate enough and the exact
expressions for the phase shifts must be used. For TE modes,
Eq(3.15) becomes
2 koW /n? Nm 2 cs 2 Oc= 2 m (3.19)
where W is the guide width. No analytical solutions for the above
equation are available.
In dealing with waveguides it is convenient to write the
dispersion relations, Eq.(3.19), in terms of the normalized frequency
V and normalized propagation constant b [Koge74] as
o V Vf(x') b dx' = s + 0c + m 7 (3.20)
The normalized parameters V, b and f(x') were defined in the previous
section.
In general, for practical cases of interest, the index difference
between the cover and the surface is rather large and it can be
represented by the asymmetry parameter "a", defined as
aTE = (n3 n@) / (nF n3) for the TE modes and aTM = (n / nf) aTM for
TM modes. Figure 3.2 shows the b versus V for TE modes and for a=
. ..... .......I........r ...... f ............. .... ............ .....
.. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .
..... .. ..
. . . . . . . . . . . . . . . . .. . . .
V
Fig. 3.2 The normalized mode index (b) vs the normalized
frequency (V) characteristics of 3layer stepindex
waveguides for m=0,1,2 and 3.
0, 1, 10, and o for a 3layered step index waveguide. The b value is
zero for modes close to cutoff and one for wellguided modes. There
is no cutoff for the fundamental mode when a=0. The effective mode
indices for TE modes are always larger than the TM modes with the
same mode number m in the case of an isotropic medium. Exceptions
occurs in birefringent media such as LiNbO3 or glasses where, due to
the stress induced by the ion exchange, the effective mode index of
the TM mode is larger than that of the corresponding TE mode.
3.3.2. Finite Difference Method (FDM)
As described earlier, the nonlinear diffusion equation can be
converted into a finite difference form and solved by iterations.
Similarly, the scalar Helmholtz equation can be converted into a finite
difference expression. However, the Helmholtz equation Eq.(3.13)
involves two unknowns; the propagation constant of the mode and the
corresponding mode field.
Using the finite difference method, the bV characteristics and
the modal fields are simultaneously determined by solving an eigen
value equation. The high accuracy of the method has been tested by
calculating the bV curves and the guided mode field profiles in the
cases of step and exponential profiles for which exact analytical
solutions are also available.
The FDM is described as follows: In FDM, the domain of interest [Xo,
Xp+1] is divided in p equal elements of length Ax' = h. The points x'
are called the grid points or nodes and h is the grid spacing.
Let us consider the first term in Eq.(3.13) which can be represented
in the form of Eq.(2.51). Thus, at x=xi
F(i+1) 2 F(i) + F(i1) 2 2
F (i+l ) 2 F(i) + F(i + k [n2(i) N2 F(i) = 0 (3.21)
(Ax)2
or
F(i1) + (k2 n2 Ax2 2) F(i) + F(i+l) = k N2 Ax2 F(i) (3.22)
where F(i) = F(x=xi). It is known that the guided field is zero away
from the waveguide region. Therefore, it is reasonable to assume that
F(0) = 0 and F(p+l) = 0 at the calculation boundaries, as shown in
Fig.3.3. For i=l to p Eq.(3.22) can be expressed as
al 1 0 FI F
la2'. F2 F2
1 = (3.23)
0 1 ap Fp_ Fp_
where ai = ki n2 (Ax)2 and Y = k0 N2 (Ax)2.
The eigenvectors of this matrix are the mode fields of the waveguide.
The eigenvalues and vectors can be solved by commercially available
subroutines.
Since we assume that the fields vanish at the calculation
boundaries, only the guided modes can be solved with this method
while for modes close to cutoff the fields may extend away from the
guided region and more points should be taken.
Compared to other methods, the finite difference method is easy
and quick and gives accurate results as long as an appropriate
0)
o =0 Fp+l =0
Xo Xi2 Xi1 Xi Xi+l Xi+2 Xp+l
Fig. 3.3 1D field profile with the assumption Fo and Fp+1 =0
using FDM.
F
scientific subroutine package is available. However, the algorithm
above does not distinguish between the polarization of the modes, and
therefore the method is not suitable to simulate devices that are
birefringent without modification.
3.3.3. Multilaver Stack Theory
The multilayer step index approximation was presented first in
[Thur86] for multichannel step index waveguide structure and later
used successfully in [Huss89] for multichannel graded index
structures. According to this method we divide the space into p
regions including two semiinfinite regions. Each region is
represented with a step index, Fig.3.4, with interfaces between
constant index regions at xi, i=1, ... ,p1. If F represents the normal
mode propagating along + z, then
F(x,z) = F(x) exp( i pz) (3.24)
where F satisfies the wave equation in each slab of constant index ni,
2Fi
Fi+ kh [n (x) Neff] Fi = 0 (3.25)
ax2
The wave equation (3.25) has trigonometric solutions if Nff < ni and
hyperbolic or exponential if Nff > n,.
I I I I I I I I I I I
I I I I I I I I I I I I
Semi
Infinite
Region
ni n2 n3
xl X2
I I I I
nn1 1n
Xn1
Fig. 3.4 Multilayer stack theory
Semi
Infinite
Region
Sai cos (ui) + bi sin(ui)
I ai cosh (ui) + bi sinh (ui)
Neff < ni
(3.26)
Neff > ni
where ui = ki ( x xi1) and ki = ko VNeifn? i=2, 3 ..., p
Now we apply the boundary conditions (BC). The fundamental BC is
that the field must go to zero at infinity. The second BC is that F (Ey
and Hz for TE and Hy and Ez for TM mode) and DF/ax are continuous
along the boundaries at any x=xi. Using these boundary conditions we
can evaluate the a and b coefficients in relation to the corresponding
coefficients of the previous slice. Thus for TE modes we get:
f ai cos (ui) + bi sin(ui)
I ai cosh (ui) + bi sinh (ui)
 ai ki sin (ui ) + bi ki cos (ui)
bi+1 ki+1= =
[ ai ki sinh (ui) + bi ki cosh (ui)
also
al= a2
Neff < ni
Neff > ni
Neff ni
(3.27)
(3.28)
Neff > ni
(3.29a)
(3.29b)
al ki = b2 k2
(3.29c)
ai = bi
an= bn (3.29d)
For TM modes, since continuity of Ez implies continuity of (1/n2)
DHy/Dx in Eqs. (3.27)(3.29), each ki needs to be replaced by (ki /n2).
For convenience in the calculations we assume ai = 1 until such time
as all the ai, bi are redetermined by normalization of the eigen
functions. The process is as follows: we start with an initial guess of N
and Eq.(3.27) and Eq.(3.28) enable us to determine ai and bi. Then
these values are checked with Eq. (3.29d) and if they are not satisfied
a new guess of Neff is made. The limitation of this method is that it is
difficult to find the number of roots and the initial guesses of the Neff.
A good solution to this problem is the use of bisection method
[Hage88] to determine the mode indices or, even better, a
combination of bisection and Newton's method [Hage88] offering a fast
convergence and highly accurate results. The main advantage of the
multilayer step index approximation is that one can determine the
field profile and the mode indices very accurately within specified
tolerance.
The method was tested for several profiles against different
numerical methods. In the case of step index profile the method gives
exact results while using other profiles the results are as good as any
other method's. However, for steep profiles, in order to get accurate
results, there is a need to use a large number of slices, which adds to
the round off error accumulated from the calculation of successive
coefficients. The way to approach the problem is by using non
uniform slicing or Aiken's method [Hage88].
3.4. Numerical Analysis of 2D Waveguides
Most guidedwave optical devices demand the solution of a two
dimensional channel waveguide. Only few of them have explicit
analytical solutions [Marc79]. Thus, numerical methods are very
important for an understanding of 2D guidedwave behavior in
rectangular geometry.
Two methods are introduced in this section. The first is the effective
index method and the second is the finite difference method.
3.4.1. Effective Index Method
The effective index method, first suggested by Knox et al
[Knox70] and later by Ramaswamy [Rama74], is relatively easy to
implement for 2D channel waveguides. This method is mainly used
to calculate the propagation constants of the waveguides. By
converting a 2D index profile into its 1D equivalent profile, the
propagation constants of the various guided modes can be evaluated
using the numerical or analytical methods introduced in the last
section for the solution of the 1D planar waveguide. In general, the 1
D equivalent index profile is much less cumbersome to solve than the
2D index profile.
The method is briefly described below : If we consider the evolution of
one transverse component of the electromagnetic field in time and
space of the form
F = F(x,y) exp[ j (pz cot)] (3.30)
then the scalar wave equation can be written as
F(x,y)+ 2F(x + k2 [n2(x,y) N2] F(x,y) = 0 (3.31)
aX2 ay2
where F is the transverse field and it is represented by Ex and Hx for
the quasiTE and the quasiTM modes respectively, ko = 2n: / X is the
wavenumber in vacuum and N is the mode index of the waveguide. In
order to find a method to determine the mode index, we express the
modal field F(x,y) as
F(x,y) = Fi(x,y) F2(y) (3.32)
where Fl(x,y) is a slowly varying function of y so that
a2F(x,y) a2 a2F27)
[F2Fx(x,y) F2(y)= F1(x,y) 2F2(y(3.33)
y2 ay2 ay2
Defining an effective index profile neff (y) such that
Lt + k2 [n2(x,y) n2ff{y)] F1 = 0 (3.34)
ax2
the original twodimensional scalar wave equation can be reduced into
a onedimensional scalar wave equation
a2F2
2+ ko2 [nff(y) N2] F2 = 0 (3.35)
3y2
This means that once the effective index profile neff(y) has been
determined by solving Eq.(3.34), the mode index N in the original two
dimensional waveguide can be obtained from a one dimensional
waveguide with index profile neff(y), by solving Eq.(3.35). To
determine the effective index profile we consider its definition
relation. At each point yi, neff(yi) is obtained by solving numerically
the one dimensional asymmetrical three layer problem for a particular
index profile. The implementation of the effective index method is
shown in Fig.3.5. The channel waveguide is sliced along the the x
direction in many slices. The thickness of the slice should be very
small in order to accommodate index variations especially in the case
where the profile is very steep, i.e., exponential and ERFC. For those
profiles a nonuniform slicing would be recommended with increasing
thickness away from the center of the waveguide. Each slice is now
considered as a three layer problem, and the 1D waveguide problem
can be solved using one of the methods described in section 3.3. The
profile constructed now from the effective indices for each slice
represents the 1D profile used in the ydirection to calculate the
mode index N with one of the available numerical techniques.
Although there are restrictions in the derivation of the effective index
method, that is, F1(x,y) should be a slowly varying field, the algorithm
73
Effective Index Method
TM mode
I I I I I I
S I i I I I
S I I I I I
I I I I I I
I I I I I I I
I I I I I I
I I I I I
I I I I I
I I I I
I I I I I
I I I I I
Y1I Yl Yi+l
I TE mode
Effective index profile
Mode index
Fig. 3.5 The procedure in the effective index method
for evaluating the mode index of 2D guides
is still valid for the first two normal modes of a weakly coupled system
provided both the modes are away from the cutoff. The effective index
method is known to fail near the cutoff [Chia86].
An important point that needs to be emphasized is that the quasiTE
mode of a channel waveguide has its polarization parallel to the surface
of the guide (ydirection). Therefore in Eq.(3.35) F2 represents the
transverse E field. However, in order to maintain the polarization in
the same direction, TM mode is used in Eq.(3.34), where F1
represents the transverse H field. Similar considerations apply for the
TM modes.
3.4.2. Finite Difference Method
The one dimensional finite difference method can be easily
extended to the two dimensional method. In this case Eq.(3.23)
becomes
di 1 0 F1 F1
1 d2 F2 F2
=T (3.36)
1
0 1 dpy _Fpxpy _Fpxpy
where di is a matrix, given by
a(i1)px+l
1
di =
0
1
a(1. 1)p,+2
0
1
1 a(i)p,+px
(3.37)
Special care should be taken for modes that are close to cutoff where
the boundaries need to be extended and the number of points must be
increased to maintain the accuracy. Nonuniform finite difference
method in this case may be appropriate since it allows to take more
points in the areas needed without increasing the total number.
CHAPTER 4
WAVEGUIDE FABRICATION
4.1. Fabrication Procedure
Several commercial glass substrates are available for ion
exchange such as sodalime silicates and borosilicate glasses. For the
experimental work reported in this study we used BK7 borosilicate
glass commercially available from Schott. The BK7 glass was chosen
due to its high optical homogeneity, excellent transmission properties
at the wavelength of interest ( 1.3 gm) and the ability to withstand
the attack of molten salt, specifically the nitrate salts. Moreover, the
sodium and potassium contents of the glass are sufficient to achieve
the desirable index change (An < 0.1 for Ag+Na+ exchange and 0.01
for K+Na+ exchange), suitable for singlemode application. Moreover,
the transition temperature of the glass is also higher than the
temperature required for the potassium or silver ion exchange
process.
The process of fabricating highperformance, lowloss optical
waveguides is described in details in the following sections.
4.2. Planar l(D) Waveguides
The procedure for fabricating one dimensional (1D) waveguides
is rather simple. It does not require any masking process since they
are fabricated on planar substrates. Onedimensional waveguides are
used for the characterization of the diffusion and waveguide
parameters such as the selfdiffusion coefficient, activation energy,
mobility ratios, and maximum index change. These parameters are
then used for the design and simulation of various devices. The
fabrication process involves the sample and melt preparations and
diffusion/ionexchange.
4.2.1. Sample Preparation
Glass substrates are first cut by a wafer saw to appropriate sizes
which depend on the different application at hand. Orientation of the
glass substrate is not important, as in the case of LiNbO3, since it is an
isotropic material. Sample cleaning is carried out in a class 100 clean
room and includes the following steps. First the sample is scrubbed
with a mildly abrasive detergent and washed with DI water. Next, the
sample is immersed sequentially into warm methanol, acetone, TCA
(1,1,1Trichloroethane), acetone and finally methanol and dried out
with a N2 gun. The samples are then hardbaked at 150 o C for three
hours before diffusion to remove any moisture remaining at the
surface.
Fig. 4.1 The diffusion setup used in the ionexchange process
4.2.2. Melt Preparation
The preparation of the melt depends on the ions used for the
exchange. The desirable characteristics which influence the choice of
the salt for a given ion are its melting point and the dissociation
temperature. Nitrate salts have some of the lowest melting
temperatures and exhibit reasonable stability. For the case of K+ Na+
ion exchange, pure KNOs is used since the maximum index change
achieved is of the order of An 0.01, suitable for singlemode
applications. Other ions can also be used for exchange such as Ag
where it was found experimentally that the optimum melt
concentration in order to achieve approximately the same index
change is 0.333 MF KNOa + 0.666 MF NaNO3 + 0.001 MF AgNO3
[Bran86]. Both ions have been proven to be an excellent choice for
ionexchange since they provide very low loss waveguides.
Great care must be taken in order to keep the melt free of
contaminations since any contamination can alter the exchange
conditions and reduce the performance of the waveguide. Therefore
only ACS grade chemicals are used. A frequent change of the salt bath
and etching of the aluminum recipient after each use is also necessary.
4.2.3. DiffusionIon Exchange
The diffusion is performed in an aluminum vessel because the
oxidized surface layer on the vessel is inert to the nitrate melt and Al
cannot easily diffuse into the sample. The experimental setup is
shown in Fig.4.1. The sample, after preheating for a few minutes right
above the surface of the melt, is placed at the bottom of the interior
vessel with a thermocouple above it to record the temperature of the
melt near the surface of the sample. Since the temperature variation
of the hot plate may be quite large in order to assure temperature
uniformity ( 1 OK) we place the Al vessel in a large salt bath that
contains KN03 + NaN03 and use a stirrer to keep the temperature of
the bath constant. A temperature control and a precise temperature
setting are very important since the waveguide characteristics strongly
depend on the diffusion temperature. A timer is used to monitor the
diffusion. Inhomogeneities in the waveguides during the diffusion
process can be avoided by assuring the substrate surface to be free of
air bubbles.
The diffusion temperature was optimized (for best control of the
diffusion rate) at 370 oC and the diffusion times depending on the
application range from 148 h. After diffusion, the sample is allowed
to cool and the residue of the solidified salt on the glass surface is
removed with warm DI water. The surface of the waveguide is then
cleaned with warm methanol and dried with a N2 gun.
4.2.4. EndFace Polishing
An important step for endfire coupling of light to the
waveguide, is the endface polishing. The end surface of the sample
should be smooth, flat, and scratch free and the edges of the sample
should be sharp since the waveguide is close to the substrate surface,
confined to few pm in depth. To prevent rounding of the waveguide
edges during polishing, two pieces of the same substrate material are
attached at the edges of the waveguide surface to be polished, using a
high temperature wax. The waveguide ends, sandwiched between the
pads, are mounted in a holder with a low temperature wax, making it
easy to remove the waveguide from the holder after polishing. The
edge of the sample/pads extends 23 mm beyond the edge of the
holder.
The polishing procedure consists of grinding the sample edges
in four different compounds with progressively decreasing particle
sizes. First, the sample is lapped using a 400 grit silicon carbide
powder and a 5.0 p.m aluminum oxide powder tracing "figure eight".
Subsequently the sample is polished using diamond paste (1 pm and
0.25 gm) on a nylon cloth, using a mechanically rotating plate until all
scratches from previous steps have disappeared. After polishing, the
wax is removed from the sample by placing it on a hot plate in a
beaker with acetone and heating it at 60 o C for 10 minutes.
4.3. Channel (2D) Waveguides
The fabrication process of surface channel (2D) waveguides is
quite similar to the planar (1D) waveguides described earlier but it
includes additional steps such as the choice of mask material and
mask deposition, photolithography, etching or liftoff and the removal
of the mask material after diffusion. Moreover, for the fabrication of
buried (2D) waveguides the two step process is needed. The second
step involves the diffusion of the sample in molten salt bath in the
absence of ion A (i.e. potassium) with or without the application of an
external field.
4.3.1. Deposition of Mask Material
The masking of the substrate is a crucial step in the process of
creating the desired waveguide pattern. The choice of the mask
material is dictated by the following; good bond to the substrate,
tolerance to the chemical attack by nitrate salt and immune to
reaction with the exchanged ions without affecting the diffusion
process, withstand high temperatures, not contaminate the melt,
dissolve easily after the diffusion and finally being able to open
windows with sharp edges with wet etching or liftoff technique
employed in our laboratory. Taking into consideration all of the above
criteria the best choice for mask is 1500 Ao Aluminum (Al) with a thin
layer of Aluminum Oxide (A1203) 300 Ao between the substrate and the
Aluminum, Fig.4.2(a). The Aluminum Oxide is used in order to
minimize the side diffusion in the waveguide [Walk83b] since a
dielectric mask is reported to have smaller electrochemical gradient
effect discussed in chapter two. For the evaporation of A1203 and Al we
use a Veeco EBeam evaporator at a vacuum of about 105 torr. The
thickness is measured by a quartz crystal thickness monitor.
4.3.2. Photolithography
After the deposition of the mask material, the mask is patterned
as illustrated in Fig.4.2. The substrate glass is spin coated at 4000
rpm for 30 seconds with a layer of positive photoresist (Shipley AZ
140017) to a thickness of approximately 0.5 pm, Fig.4.2(b). The
photoresist is subsequently hardened by softbaking the samples at 90
0 C for 30 minutes in N2 atmosphere. The photoresist patterns are
delineated by exposure to UV light ( 365 nm wavelength, intensity of
Mask
Evaporation
(a)
Aluminum
Gla A s203
Glass substrate
Photoresist
Spinning
(b)
UV Exposure
Glass substrate
SPhotoresist
140017
DarkField
< Photomask
Exposed area
Glass substrate
Development
(d)
Glass substrate
Mask Etching
(e)
Photoresist
Removal
(f)
Diffusion
(g)
Fig. 4.2 Photolithographic procedure using wet etching technique
~LL~ $U ~k~
16.5 mW/cm2) using a KarlSeuss mask aligner with provision for
constant intensity control. The dark field photomasks used in this
experiment were designed using a workstation VIA100 and generated
by an electron beam exposure system (EBES) with 0.25 pm resolution.
The photomask has several bands of the same pattern separated by
100 pm openings. Each band has twenty straight channels ranging
from 0.5 p.m to 10 [pm width in steps of 0.5 pm with 75 ptm separation
between them. The optimum exposure time for the above intensity of
the light is approximately 4.5 seconds, Fig.4.2(c). After the exposure,
the exposed photoresist is developed for 5060 seconds using Shipley
351 developer diluted 5:1 with DI water, and immediately rinsed in DI
water for 2 minutes to stop the action of the developer, Fig.4.2(d).
The exposure and development of the pattern are very critical steps in
the fabrication of high quality reproducible waveguides and precise
control of the time is absolutely necessary in order to avoid over or
under exposure or development of the pattern.
The quality of the pattern is inspected for the straightness of the
windows, the sharpness of the edges and the openings are measured
under an optical microscope. Finally, the substrate is hardbaked at
110 o C for 10 minutes in order to improve the adhesion and chemical
resistance of the image.
An important factor in the whole process is the cleanliness of
the substrate and its maintenance throughout the process. Any dirt or
micronsize dust particles can cause poor bondage between the
substrate and the photoresist patterns, making the sample unusable.
4.3.3. Wet Etching Technique
The etching process is used to dissolve the Al and A1203 through
the opening defined by the photoresist in order to open a window for
the ionexchange to take place. The PAE aluminum etchant used is
commercially available and consists of 16 parts phosphoric acid, two
parts water, one part nitric acid and one part acetic acid. Etching
time may vary from sample to sample depending on the thickness
variations along the substrate during the EBeam evaporation. To
assure reproducibility, we etch for an additional minute after the 100
jim channels are approximately 80 % transparent, Fig4.2(e). The PAE
etchant is also able to etch the alumina; however, the etching rate is
much slower. In order to estimate the etching time several samples
were tested and measured with the Sloan Dektak IIA surface
profilometer. After etching, the samples are rinsed in DI water to
stop the action of the etchant, dried with N2 gun and inspected again
under the microscope. Overetched samples tend to exhibit larger
widths than the mask openings and rougher edges while underetched
samples have dark spots or lumps in the etched regions. After
etching, the photoresist is removed with acetone, Fig.4.2(f), and the
samples are cleaned again by the standard cleaning procedure and
prebaked to harden the pattern and remove the moisture from the
sample before diffusion. Diffusion is carried out in the same way as for
planar waveguides with the ion exchange occurring through the open
windows, Fig.4.2(g).
4.3.4. Removal of Mask
After diffusion the aluminum mask is removed by immersing the
sample in the PAE etchant at 70 0 C for 1520 minutes. The alumina
can also be removed by the same etchant, however, it is preferable to
use dilute nitric acid for 30 minutes. Aluminum or alumina residue
can increase the waveguide losses substantially Subsequently the
samples are rinsed in DI water, dried and cleaned again by the
standard cleaning procedure before proceeding to polishing and
characterization.
4.3.5. Liftoff Technique
Figure 4.3 describes briefly the liftoff technique. It is an
alternative technique to wet etching which usually provides better
resolution and reproducibility of the mask pattern than wet etching.
However, there are distinct disadvantages with the liftoff procedure
which makes it quite difficult to implement in our laboratory. First, of
all, exposure and development times must be carefully controlled.
However, since both the parameters are temperature dependent this
is difficult to implement. Secondly, thorough cleaning of the
photoresist is necessary before evaporation of aluminum because
photoresist residue can cause breaks in the Al film, making the sample
unusable. Finally, the need of using HMDS to stick the photoresist to
the glass and chlorobenzene to help create the mushroom structure
[West87] for the break away of aluminum in the exposed areas adds
more steps in the already complex procedure.
Photoresist
Spinning
(a)
UV Exposure
(b)
o Photoresist
lasssbs t  140017
////////////////// HMDS
Glass substrate
+Y ClearField
Photomask
[ // ////////f/"." J////i/A
Glass substra
Development
(c)
Mask
Evaporation
(d)
Unexposed
te area
Glass substrate
< Aluminum
Glass substrate
Photoresist
Removal
(e)
Diffusion
(f0
Glass substrate
Glass substrate
Fig. 4.3 Photolithographic procedure using liftoff technique
J
 i
!
I
CHAPTER 5
CHARACTERIZATION OF WAVEGUIDES
5.1. Introduction
The ionexchanged optical waveguides are characterized by the
refractive index profile, the propagation constants of the guided
modes (mode index), the mode field profile and the propagation
losses. In addition, since the characteristics of the waveguide depend
on the wavelength, a spectral response measurement is performed to
determine the cutoff wavelength of the modes and thus control the
fabrication conditions in order to produce a singlemode waveguide
over a desirable wavelength range. For the spectral response
measurements a high power tungsten lamp is used as a source for the
monochromator. All the measurements are done with a 1.3 gim
semiconductor laser as the source. A HeNe laser (0.6328 gm) is used
for the visual alignment of the system. Descriptions of the different
methods used to characterize the waveguides are presented in the
next sections.
5.2. Refractive Index Profile
Two different approaches are used for determining the
refractive index profile. The first one is based on the optical
measurements of the mode indices of the guided modes [Tien69,
Tien70, Ulri70]. The InWKB, [Whit76, Chia85] is then used to
construct the refractive index profile from the mode indices as well as
estimate the diffusion coefficient of the waveguide. The waveguides
used in this experiment must support at least 4 or 5 guided modes, a
condition that assures higher accuracy in the constructed index
profile, Section 5.2.3. The second approach is based on the fact that
the concentration profile of the substituting ion is analogous to the
index profile and one employs analytical tools such as electron or ion
microprobe [Gial73], scanning electron microscope [Lagu86a],
secondary ion mass spectroscopy (SIMS), and atomic absorption
spectrophotometry [Chlu87] for measuring the concentration profile
directly. The two methods employed in this study are the mode index
measurements and the concentration measurements using electron
microprobe, described in the following sections.
5.2.1. Mode Index Measurements
Measurements of the mode indices are made by the prism
coupling technique shown in Fig.5.1 [Tien69, Tien70, Ulri70]. A He
Ne laser or a 1.3 pm semiconductor laser beam, (either TE or TM), is
incident on a prism whose refractive index is higher than that of the
film region of the waveguide. The polarization of the beam is
controlled by a BabinetSoleil compensator to choose either the TE or
TM polarization. The beam is focused by a large focal length lens such
that the beam waist coincides with the coupling point at the prism
base. The prism is placed on an XYZ translation stage that is mounted
on a precision (<1 min of arc) rotation table. The prism used was
Aperture ixuUc .s '
prism
Chopper
HeN / 13 Compensator 
HeNe /1.3 pJm
Laser Power
Lens Power
Lens meter
Fig. 5.1 Prism coupler measurement setup
