Potassium-sodium ion-exchanged waveguides and integrated optical components

Material Information

Potassium-sodium ion-exchanged waveguides and integrated optical components
Miliou, Amalia Nikolaos, 1961- ( Dissertant )
Srivastava, Ramakant ( Thesis advisor )
Ramaswamy, Ramu V. ( Thesis advisor )
Bosman, Gijs ( Reviewer )
Thomson, Ewen M. ( Reviewer )
Phillips, Winfred M. ( Degree grantor )
Lockhart, Madelyn M. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
University of Florida
Publication Date:
Copyright Date:
Physical Description:
x, 213 leaves : ill. ; 29 cm.


Subjects / Keywords:
Alkalies ( jstor )
Diffusion index ( jstor )
Directional couplers ( jstor )
Electric fields ( jstor )
Electrons ( jstor )
Ions ( jstor )
Light refraction ( jstor )
Optical waveguides ( jstor )
Waveguides ( jstor )
Wavelengths ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Electrooptical devices ( lcsh )
Integrated optics ( lcsh )
Optical wave guides ( lcsh )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


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 single-mode fiber-compatible mode sizes. A systematic theoretical and experimental study of the K+-Na+ ion exchange process was recently conducted in our laboratory which resulted int he 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 mobility of K+ ions, and fiber-compatible buried channel waveguides were fabricated for the first time with a two-step 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. More over, 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 1.0-1.6um wavelength region and the power output at 1.32 um 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.
Thesis (Ph. D.)--University of Florida, 1991.
Includes bibliographical references (leaves 200-212).
General Note:
General Note:
Statement of Responsibility:
by Amalia Nikolaos Miliou.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AJH2820 ( NOTIS )


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"For a moment there was a wild lurid light
alone, visiting and penetrating all things"

Edgar Allan Poe



Amalia Nikolaos MIliou


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 co-workers 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.



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

A B ST R A C T ..................................................................................................................ix


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

1.1. Integrated O ptics ......................................................................... 1
1.2. K+-Na Ion Exchange.................................................................. 2
1.2.1. K-Na Ion Exchanged Waveguides........................... 3
1.2.2. K+-Na* Ion Exchanged Waveguide Devices ........... 5
1.3. Organization of the Chapters.................................... ............ 7

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. Two-Step 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. Stress-Induced Index Change................................... 44
2.7. M ixed-Alkali Effect.............................. ....................................... 46
2.8. Space Charge Effect............................................................. ....... 49
2.9. Side Diffusion........................................................................... 50

NUMERICAL ANALYSIS ......................................................................... 52

3.1. Introduction ............................................................................... 52
3.2. Guided-Wave Helmholtz Equation ......................................... 54
3.3. Numerical Analysis of 1-D 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 2-D 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 (1-D) Waveguides 77
4.2.1. Sample Preparation.................................................... 77
4.2.2. Melt Preparation ............................... .............. 79
4.2.3. Diffusion- Ion Excahange ................... .................. 79
4.2.4. End-Face Polishing..................................................... 80
4.3. Channel (2-D) 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. Lift-off 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. Stress-Induced 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


7.1. Introduction ................................... ........................... 137
7.2. K Na TE/TM Polarization Splitter..................................139
7.3. TheoreticalApproach.................................................................139
7.3.1. Coupled-Mode Theory.......................................... 140
7.3.2. Normal-Mode Approach .......................... ..... 144
7.3.3. Relation between Coupled-Modes and
Norm al-M 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 Devices-Practical
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. Opto-Electronic Integration .....................................199

REFERENCES............................................................ ...................00

BIOGRAPHICAL SKETCH .............................. ................... 213


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



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 single-mode

fiber-compatible mode sizes.

A systematic theoretical and experimental study of the K-Na+

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

mobility of K+ ions, and fiber-compatible buried channel waveguides
were fabricated for the first time with a two-step 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 1.0-1.6 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.



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 low-power

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,

soda-lime silicate glass, borosilicate glass, pyrex glass, lithium niobate,

and III-V 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


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 (0.008-0.009) which is very attractive for conventional

single-mode fiber-compatible mode sizes. The K+-Na+ exchanged
waveguides are also characterized by relatively high birefringence
(1x10-3), 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 soda-lime silicate [Albe85,
Yip85], BK7 [Gort86a], Pyrex [Gort86b], and semiconductor-doped
glasses [Cull86] have been reported. Effects such as stress-induced
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 mode-index 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 soda-lime 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
surface-index 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 surface-index change in

BK7 glass have now been quantitatively explained on the basis of the
stress-induced 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
soda-lime glass and almost no change for BK7), and the behavior was

explained for both the glasses on the basis of the mixed-alkali effect


4. Fabrication of buried channel waveguides with a two-step process.

Using a two-step process, buried channel waveguides for operation at
1.3 (pm were fabricated in both the glasses with nearly circular

symmetric near-field intensity profiles. Finally, record low-loss

waveguides with insertion loss of less than 1.0 dB in 20 mm long

devices were obtained using the two-step process in BK7 glass


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


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
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 Y-branch
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 5x10-4 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

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
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 1.0-1.6 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


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




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 glass-melt 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




0 0







Fig.2.1 Ion exchange process

where the bar denotes the cations on the interface inside the glass. In

a liquid-solid 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 liquid-phase 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 glass-melt 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

a-B [X-B (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 n-type 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 n-type 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)
11-NAJ RT 41-NA

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 left-hand 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
soda-lime 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


0.009 --- BK7
-e- Soda Lime


An 0.006




70 75 80 85 90 95 100
% KN03

Fig. 2.2 Index change vs percentage of KN03 in the melt



Composition Soda-Lime Silicate BK7





















rK = 1.28 x 10-10 m and

rNa = 0.95 x 10-10 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 ion-exchanged 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 Nernst-Planck
model, which has replaced earlier models that employ constant inter-
diffusion coefficient. The inter-diffusion system described in this
work is based on the Nernst-Planck equations, which are derived by






-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)


assuming that the driving force for diffusion is the electrochemical
potential and that the inter-diffusion coefficient describing the system
is concentration dependent. According to the Nernst-Planck
equations the flux for each cation species is given by

Oi = Di Vi +L i zi Ci E (2.5)

where Oi: molar flux of cation i (A or B) (moles/m2. sec)
Ci: concentration of cation i in glass (moles/m3)
Di: self-diffusion 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 multi-alkali effect [Isar69, Day76, Char82b].

It should also be noted that the self-diffusion coefficient varies with
glass composition and its temperature dependence below the glass

transition temperature is given by an Arrhenius type relation, Fig.2.4,


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
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 Nernst-Einstein relation is satisfied, i.e.

Di = kT = k _T iL (2.7)
ni q2 q

where k: Boltzmann's constant (1.38 x 10-23 J/K)
ni: number of ions i per m3, i=A, B
oi: ionic conductivity (mho/m)

and q: electronic charge (Cb)





-7.4 1




1/T (OK)

Fig. 2.4 Self-diffusion 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 field-induced transport and

Eq.(2.7) is not satisfied [Beie85, Tera75]. In these cases the relation

between the self-diffusion 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,

It is convenient to assume that the exchange is strictly one-to-

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 (space-charge) 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

From the condition of electrical neutrality for the glass, it is

required that the total ionic concentration be constant.

CA + CB + CD = Co


where Co : concentration of ion B in glass before diffusion.
and CD : concentration of the net depleted mobile ions (space

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 space-charge 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.


approximately balances the flux of the outgoing ions, i.e., we assume

CA + CB = Co, where Co is constant. Thus,

aCA aCB (2.13)
at at

Since the number of ions is conserved, the continuity equations are

-- V. )A

and (2.14)
-- V. tB

V.((A+ )B) = 0 (2.15)

Substituting Eq.(2.5) for the two ions in Eq.(2.15), and assuming
constant self-diffusion 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-

= n DAV CA

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

A =DV2NA- E. VNA (2.19)

where D is the inter-diffusion coefficient defined [Dore64] as

Sn D (DA/DB) 9A NA + B NB 2.20
D = n DA (2.20)

and g is the inter-mobility coefficient defined as

9L= IAB 9A A (2.21)
A NA + LB NB 1 ( 1 ) NA 1- a NA


a= ( A) (2.22)

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

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 inter-diffusion coefficient Eq.(2.20)

and inter-mobility-coefficient 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


OA + OB = 0


In this case, E = Es, the local field due to space charge is given by

Es = (DA DB)VNA (2.25)

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 3-D
nonlinear equation (2.27) occurs as follows [Cran75]:
(a) Two dimensional case:
Equation (2.27) can be reduced to

( + -( (2.28)
at ax 1 a NA ax ay 1 a NA ay

The analysis of the 2-D 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)

NA n DA 2NA + NA (2.30)
at )x2 y2

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 e-Microprobe
...- ERFC
SA Diffusion Eq.
0.8- a = 0.01
No = 0.90



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)

aNA ( n DA NA) a n DA aNA
) + -( ___) (2.32)
at ax 1- NA x By 1 NA ay

aNA a ( n DA NA2.33)
at ax 1- NA ax

Crank [Cran75] expects a step-like profile as "a" approaches unity.
For intermediate values of "a" between 0 and 1 the profile obtained is

Gaussian-like as in the case of K+ Na+ exchange in soda-lime 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 e-Microprobe
0.7 0 -- auian
A Difuon Eq

0.6 a 0.98
N 0.8


0.4 0

0.3 0


0.1 O O

0 05 10 15 20 25 30

x (Gm)

Fig. 2.6 Concentration profile of soda-lime 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 field-assisted
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)

Abou-el 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 step-like 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




1.51 I I I
8 12 16 20 24 28
x [im]

Fig. 2.7 Index profile of soda-lime glass with applied field
fitted to the solution of the Diffusion Equation

2.3.4. Two-Step Diffusion Process
In order to increase the symmetry of the index profile (thereby

improve the fiber-waveguide coupling) and reduce the scattering
losses caused by the proximity of the glass surface to the waveguiding
region, a second-step diffusion is necessary in absence of cation A in

the melt. The diffusion equation is solved numerically in this case

with the first-step 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 second-step 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 two-step process for a single-mode
buried channel waveguide has been made by Albert et al [Albe90].
However, the model does not include the mixed-alkali 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

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)

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

i -Di VCi + Ji zi Ci E + L V. o (2.39)

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)

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

one-dimensional 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 1-D 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


At the other boundary for infinite glass thickness,

NA(-, t) = 0,

for t 2 0


The solution of Eq.(2.31) [Dore64, Cran75] is

NA(x,t) = NAo erfc( X


where erfc( x / Weff) is a complementary error function profile

defined as

erfc( x / Weff) = 1 exp(-s2) ds


Weff = 2 n DAt


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


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 1-D 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

(i) For the special boundary conditions

NA = NAo, NA0 for x = (2.45a)

NA = for x -oo (2.45b)

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 concentration-dependent 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)

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 step-like 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 1-D and 2-D 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 1-D diffusion.
We first let the range in x be divided into equal intervals 6x and the

time into intervals 6t, so that the region x-t 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

NA(i-l,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
a-2x2 ,tj (8x2)





Similarly, by applying Taylor's series in the t direction, keeping x
constant, we have

_NA-A 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 finite-difference representation on the jth and (j+l)th
time rows. Under this approximation Eq.(2.31) becomes

-r NA(i-lj+l) + (2 +2r)NA(ij+l) r NA(i+l,j+l)

= r NA(i-l,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)


stress created by the size mismatch upon substitution since the ion

exchange takes place below the stress relaxation temperature of the


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)

The dielectric constant of the medium is defined as

=D EoE+ P (2.57)

where eo is the free space permittivity and D is the electric


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 Clausius-Mossotti relation. At optical
frequencies, Re(e) = n2, where n is the index of refraction, the

equation is called the Lorentz-Lorentz [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 multi-alkali effect (see section 2.7). These

models are described in details in chapter 6.

2.6.2. Stress-Induced 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


where A -I is the index of refraction tensor,

and Pij is elements of the stress optic tensor.

medium) Pij is given by [Dall65]


P11 P12 P12
P12 P11 P12
P12 P12 P11
0 0 0
0 0 0
0 0 0


0 0
0 0
0 0
0 0
P44 0
0 P44-

oJ is the stress tensor

For glasses isotropicc


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)


oz + P12 (Gx + ay)

A '=

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 "mixed-alkali 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
ion-exchange process. Properties most affected are those associated
with alkali ion movement such as electrical conductivity, alkali
diffusion and alkali self-diffusion 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 inter-diffusion of

the monovalent cations in the rigid glass matrix. As the two ions of
different mobilities inter-diffuse, a potential gradient is set up to
maintain electrical neutrality. When ionic activity coefficients in the
glass are constant, inter-diffusion of ions A and B can be treated in
terms of an inter-diffusion 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 self-diffusion 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


-8 -


o (a)


0 0.2 0.4 0.6 0.8 1.0
Alkali ratio K/K+Na



4 10 -


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,

Several theories have been presented in the literature trying to
explain the mixed-alkali 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 out-diffusion 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 ion-exchange,

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 in-diffused ions will be accelerated and
the out-diffused ions will be retarded. However, since the ion
exchange is strictly one-to-one the refractive index increase must also

be retarded and all excess in-diffused 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].



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) 1-D and 2-D 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,
Generally in all of the above numerical techniques we assume that the
index profile is invariant along the z-direction (propagation direction).
However, some integrated optical devices involve evolution of the
propagating wave along the z-axis 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

3.2. Guided-Wave Helmholtz Equation

Wave propagation in a source-free, linear dielectric medium is
governed by the vector Helmholtz equations, known also as the wave
equations. Assuming a z-propagating 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)

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 z-components 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



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

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




j P Ez = Vt. Et

j 1 Hz = Vt. Ht


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



VxH=j coE (3.10)

For a two-dimensional surface waveguide, with the y-direction 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 y-direction
(i.e., Ex / Ey is nearly zero), and the quasi-TM modes with the
magnetic field primarily polarized in the y-direction (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 quasi-TE modes or Hy for quasi-TM 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 one-dimensional waveguide (no confinement in the y-
direction), the above equation can be further simplified as

-2+ k2(n2 N2)F = (3.13)

Furthermore, normalization of Eq.(3.13) might be useful in
applications with arbitrary index profiles [Hock77]

+ V2 [f(x') b] F = 0 (3.14)

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 1-D 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 cross-section 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 well-known 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
Consider the ray propagation in a graded-index 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 cover-guide 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
(c = n/2



Turning Point 0s = n/4

Fig.3.1 Ray diagram for the graded-index 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 cover-guide 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 three-layered 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
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 ............. .... ............ .....

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .

..... .. ..

. . . . .. .


Fig. 3.2 The normalized mode index (b) vs the normalized
frequency (V) characteristics of 3-layer step-index
waveguides for m=0,1,2 and 3.

0, 1, 10, and o for a 3-layered step index waveguide. The b value is
zero for modes close to cutoff and one for well-guided 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 b-V 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 b-V 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(i-1) 2 2
F (i+l ) 2 F(i) + F(i- + k [n2(i) N2 F(i) = 0 (3.21)

F(i-1) + (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 eigen-vectors of this matrix are the mode fields of the waveguide.

The eigen-values and vectors can be solved by commercially available
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


o =0 Fp+l =0

Xo Xi-2 Xi-1 Xi Xi+l Xi+2 Xp+l

Fig. 3.3 1-D field profile with the assumption Fo and Fp+1 =0
using FDM.


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 semi-infinite regions. Each region is

represented with a step index, Fig.3.4, with interfaces between
constant index regions at xi, i=1, ... ,p-1. 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,

Fi+ kh [n (x) Neff] Fi = 0 (3.25)

The wave equation (3.25) has trigonometric solutions if Nff < ni and

hyperbolic or exponential if Nff > n,.



ni n2 n3

xl X2


nn-1 1n


Fig. 3.4 Multilayer stack theory


Sai cos (ui) + bi sin(ui)

I ai cosh (ui) + bi sinh (ui)

Neff < ni


Neff > ni

where ui = ki ( x- xi-1) and ki = ko VNeif-n? 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)


al= a2

Neff < ni

Neff > ni

Neff ni



Neff > ni



al ki = b2 k2


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

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 2-D Waveguides

Most guided-wave 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 2-D guided-wave 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 2-D channel waveguides. This method is mainly used

to calculate the propagation constants of the waveguides. By
converting a 2-D index profile into its 1-D 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 1-D planar waveguide. In general, the 1-
D equivalent index profile is much less cumbersome to solve than the
2-D 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 quasi-TE and the quasi-TM 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)

the original two-dimensional scalar wave equation can be reduced into
a one-dimensional scalar wave equation

2+ ko2 [nff(y) N2] F2 = 0 (3.35)

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 non-uniform 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 1-D 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 1-D profile used in the y-direction 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


Effective Index Method

TM mode


S I i I I I
Y1-I 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 2-D 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 quasi-TE

mode of a channel waveguide has its polarization parallel to the surface
of the guide (y-direction). 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)


di 1 0 F1 F1
1 d2 F2 F2

=T (3.36)


0 1 dpy _Fpxpy _Fpxpy

where di is a matrix, given by

di =


a(1. 1)p,+2


1 a(i-)p,+px


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.



4.1. Fabrication Procedure

Several commercial glass substrates are available for ion

exchange such as soda-lime 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 single-mode application. Moreover,

the transition temperature of the glass is also higher than the

temperature required for the potassium or silver ion exchange
The process of fabricating high-performance, low-loss optical

waveguides is described in details in the following sections.

4.2. Planar l(-D) Waveguides

The procedure for fabricating one dimensional (1-D) waveguides

is rather simple. It does not require any masking process since they

are fabricated on planar substrates. One-dimensional waveguides are

used for the characterization of the diffusion and waveguide
parameters such as the self-diffusion 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

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

(1,1,1-Trichloroethane), acetone and finally methanol and dried out
with a N2 gun. The samples are then hard-baked at 150 o C for three

hours before diffusion to remove any moisture remaining at the

Fig. 4.1 The diffusion setup used in the ion-exchange 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 single-mode

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

ion-exchange 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. Diffusion-Ion 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 set-up 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 1-48 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. End-Face Polishing
An important step for end-fire coupling of light to the

waveguide, is the end-face 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 2-3 mm beyond the edge of the
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 (2-D) Waveguides

The fabrication process of surface channel (2-D) waveguides is

quite similar to the planar (1-D) waveguides described earlier but it
includes additional steps such as the choice of mask material and
mask deposition, photolithography, etching or lift-off and the removal
of the mask material after diffusion. Moreover, for the fabrication of
buried (2-D) 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 lift-off 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 E-Beam evaporator at a vacuum of about 10-5 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
1400-17) to a thickness of approximately 0.5 pm, Fig.4.2(b). The

photoresist is subsequently hardened by soft-baking 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


-Gla A s203
Glass substrate


UV Exposure

Glass substrate


<- Photomask

--Exposed area

Glass substrate


Glass substrate

Mask Etching



Fig. 4.2 Photolithographic procedure using wet etching technique

~LL~ $U ~-k~

16.5 mW/cm2) using a Karl-Seuss mask aligner with provision for
constant intensity control. The dark field photomasks used in this
experiment were designed using a workstation VIA-100 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 50-60 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 hard-baked 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
micron-size 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 ion-exchange 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 E-Beam 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. Over-etched samples tend to exhibit larger
widths than the mask openings and rougher edges while under-etched
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 15-20 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

4.3.5. Lift-off Technique
Figure 4.3 describes briefly the lift-off 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 lift-off 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.


UV Exposure

o Photoresist
lasssbs t -- 1400-17
////////////////// HMDS

Glass substrate

+Y Clear-Field
[ // ////////f/"." J////i/A

Glass substra



te area

Glass substrate

<- Aluminum

Glass substrate



Glass substrate

Glass substrate

Fig. 4.3 Photolithographic procedure using lift-off technique


| i





5.1. Introduction

The ion-exchanged 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 cut-off wavelength of the modes and thus control the

fabrication conditions in order to produce a single-mode 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 He-Ne 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 Babinet-Soleil 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 '

HeN / 13 Compensator |
He-Ne /1.3 pJm
Laser Power
Lens Power
Lens meter

Fig. 5.1 Prism coupler measurement setup

Full Text
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