TAPERED VELOCITY COUPLERS AND DEVICES:
A TREATISE
By
HYOUN SOO KIM
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1994
To
My Parents
ACKNOWLEDGMENTS
I would like to express my deep gratitude to my advisor Prof. Ramu V.
Ramaswamy for his guidance, encouragement, and support throughout my graduate
study. His high standard in academic achievement really inspired me to do my very best
for the completion of this work. In addition, through his discipline and leadership I have
learned a lot more than the scientific knowledge required in the integrated optics field.
I would like to thank Prof. T. Nishida, D. Tanner, M. Uman, and P. Zory for their
participation on my supervisory committee. I also would like to express my thanks to Dr.
Figueroa whose enthusiasm was what initially got me interested in studying in this field.
I thank my fellow graduate students, particularly, Dr. Young Soon Kim, Dr. Sang
Sun Lee, Dr. Hsing Chien Cheng, Dr. Amalia Miliou, Sang Kook Han, Chris Hussel, S.
Muthu, J. Natour, K. Lewis, Young Soh Park, Maj. Mike Grove, Capt. Craig Largent, and
fellow researchers Dr. Simon Cao, Dr. Sanjai Sinha, and Dr. K. Thyagarajan for many
stimulating and interesting discussions. Thanks are extended to my friends, Dr. Sung
Jong Choi, Dr. Chang Yong Choi, Kwang Rip Hyun, Min Jong Yeo, Duck Hyun Chang,
Dong wook Suh, who have provided many unforgettable memories throughout all the
years I spent in Gainesville.
I am greatly indebted to my parents and sisters for their endless love, patience,
and support during all the years of my life. Last, but not least, I would like to give my
special thanks to my wife for her support and sacrifice and to my lovely two daughters for
their encouragement.
TABLE OF CONTENTS
ACKNOW LEDGM ENTS ........................................................................................... iii
A B STRA CT ......................................................................................................................... vi
CHAPTERS
ONE INTRODUCTION .................................................................................... 1
TWO NORMAL MODE THEORY
AND ADIABATIC THEOREM............................................................ 11
2.1 Definition of Normal Modes........................................................... 11
2.2 Characteristics of Normal Modes
in Fivelayer Waveguide Structure ............................................. 18
2.3 Adiabatic Theorem in Optical Devices .............................................. 28
THREE TAPERED, BOTH IN INDEX AND IN DIMENSION,
VELOCITY COUPLER IN Ti:LiNbO3 ............................................. 33
3.1 Single Mode Channel Guide:
Normal Mode Analysis and Field Profile ................................. 34
3.2 Tapered, Both in Index and Dimension, Velocity Coupler................... 40
3.3 Step Transition Model: Power Flow in the TVC.................................... 50
3.4 Theoretical and Experimental Results ............................................... 53
FOUR TAPERED, BOTH IN DIMENSION AND IN INDEX
VELOCITY COUPLER SWITCH .................................... ........... .. 62
4.1 Principles of Operation ................................................................... 63
4.2 Theory and Experiments ................................................................... 66
4.3 Sum m ary ............................................................................................ 69
FIVE TAPERED VELOCITY COUPLER
USING SEGMENTED WAVEGUIDES.................................................. 71
5.1 Proton Exchanged Periodically Segmented
W aveguide in LiNbO3 ................................... .................................. 74
5.2 Modeling of Proton Exchanged Periodically Segmented
W aveguides in LiNbO3............................................... ................ 76
5.3 Periodically Segmented Waveguides in AlGaAs/GaAs
and Their Application to Tapered Velocity Coupler ................................ 84
SIX AN MQWSQW TAPERED WAVEGUIDE INTERCONNECT ................. 94
6.1 Introduction ......................................................................................... 94
6.2 Tapered Waveguide Interconnect Using IILD........................................ 95
6.3 Modeling of Impurity Induced Layer Disordered MQW and
Analysis of Tapered Waveguide Interconnect.................................. 97
6.4 Optimization of the Taper Modal Evolution......................................... 104
6.5 Conclusion .................................................................................... 111
SEVEN CONCLUSIONS AND SUMMARY ......................................................... 112
REFEREN CES.......................................................................................................... 115
BIOGRAPHICAL SKETCH .................................................................................... 120
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
TAPERED VELOCITY COUPLERS AND DEVICES:
A TREATISE
By
Hyoun Soo Kim
April 1994
Chairman: Ramu V. Ramaswamy
Major Department: Electrical Engineering
A polarization independent device is highly desirable for use in singlemode fiber
optical communication systems. Tapered velocity coupler (TVC) is expected to play an
important role since its operation is polarization independent as well as wavelength
insensitive. Thus far, TVC has received little attention primarily because of the unusually
long device length required for complete power transfer. In this dissertation we establish
that a TVC with an acceptable device length for integration can be indeed realized and
integrated by tapering in index as well as in dimension.
We demonstrate, for the first time, that complete power transfer can be achieved
in a tapered, both in index and in dimension, velocity coupler in Ti:LiNbO3 with device
length reduced to one quarter of that of conventional TVC. The coupler is analyzed by
use of step transition model in conjunction with local normal modes of the grade index
TVC, overcoming the deficiency of the fivelayer step index model.
We further demonstrate a Ti:LiNbO3 digital optical switch with the smallest
voltage length product reported to date, namely, 7.2 Vcm for TM and 24 Vcm TE mode
with a 15 dB cross talk.
In an effort to extend the tapered, both in index and in dimension, velocity coupler
concepts to step index compound semiconductor waveguides, we introduce proton
exchanged periodically segmented (PEPS) waveguides. PEPS waveguides in LiNbO3 are
first studied theoretically and experimentally. The mode index of PEPS waveguides
increases linearly and saturates finally with increase of duty cycle. Next, segmented
waveguides in AlGaAs/GaAs are characterized in terms of propagation loss and modal
size with respect to duty cycle. These segmented waveguides will be utilized in the
development of step index tapered velocity couplers.
Finally, we present an application for TVC as an optical interconnect. In
particular, a tapered waveguide interconnect between a single quantum well (SQW) laser
and a multiquantum well (MQW) modulator is presented. Using vertical coupling
between SQW guiding layers and a tapered, both in index and dimension, MQW layer,
the tapered interconnect is modified for complete power transfer from SQW laser to
MQW modulator.
CHAPTER ONE
INTRODUCTION
Tapered velocity couplers (TVC) play an important role in optical communication
systems. In particular, they are useful as digital optical switches for optical signal
processing applications as well as optical interconnects within an integrated optical chip
interconnecting a laser and a modulator. The present dissertation is a treatise on the subject
of TVC.
Tapered velocity couplers consist of two waveguides separated by a constant gap
but with at least one of the waveguides tapered along the direction of propagation. Several
cases of such couplers are illustrated in Fig. 1.1. The primary advantage of a TVC is that
its behavior is predicated upon the evolution of the fundamental and/or the first order
normal mode along the longitudinal direction of the coupler. Gradual tapering of the guides
in the coupler that meets the adiabatic invariance condition prevents mode conversion
between the local normal modes as well as into radiation modes so that the optical power in
a local normal mode at the input remains unchanged while the mode propagates along the
coupler with the evolution of its field profile. In addition, it is possible to physically
transfer power from one of the waveguides at the input of the TVC to another at the output,
provided the local normal modes of the TVC, both at the input and the output, approximate
that of the individual guides. The adiabatic condition can be met over a wide range of
parameters governing the device rather than at discrete values or positions as is the case
with the interferometric device.
The concept of the TVC was first suggested by Cook [Co55] for applications at
microwave frequencies and it was analyzed by Fox [Fo55] and Louisell [Lo55] using local
normal modes. Later, Wilson and Teh [Wi73, Wi75] followed by considering the
(a)
(c)
Fig. 1.1 Schematics of couplers. (a) Directional coupler. (b) Tapered velocity
coupler [Wi75]. (c) Optical fiber tapered veloctiy coupler [Ro91].
(e)
Fig. 1.1continued (d) Tapered, both in dimension and in coupling, velocity
coupler for digital optical switch [Xi93]. (e) Tapered, both
in index and in dimension, velocity coupler [Ki93a]. The
gradual index change is qualitatively illustrated by the
varying degree of the shadow in the waveguide regions.
_ _
~
tolerance aspect of 5layer TVC in terms of the coupled mode theory and experimentally
demonstrated complete power transfer between waveguides. Milton and Burns [Mi75]
established the criteria for the adiabatic operation and complete power transfer in a planar
step index TVC. Using the step transition model [Mar70], they considered mode
conversion between approximated local normal modes under the assumption of weak
coupling. Analytic solutions for the mode amplitudes in the TVC [Sm75, Sm76] were
derived under the assumptions of a constant coupling coefficient and linearly varying
propagation constants, yielding a simple analytic expression for the coupler efficiency as
well as quantitative criteria for the coupler length. In their recent work [Ki89a], Kim and
Ramaswamy achieved complete power transfer between the channel waveguides of the
TVC, thus demonstrating the adiabatic regime; however, theoretical analysis using the step
transition model and the beam propagation method did not correlate well with the
experimental results. TVCs have been considered and their characteristics analyzed for the
applications as power dividers [Ca90] and fibertofiber couplers [Ro91].
The underlying concept of the TVC can be best understood by considering the
adiabatic condition (Eq. 1.1) in a tapered velocity coupler and another inequality (Eq. 1.2)
for the complete power transfer, identified by Milton and Burns [Mi75] based on a five
layer slab waveguide step transition model,
A~PT1AZT
 <1.5 (1.1)
(A T/2)2
> 80 (1.2)
so that
ApT AZT 2 213
(1.3)
where APT is the total phase difference between the uncoupled, individual guided modes of
the coupler over the taper length of AZT and K is the coupling coefficient. The adiabatic
condition Eq. 1.1 must be satisfied to prevent undesired normal mode conversion.
Equation 1.2 must also be satisfied to achieve at least 95% of power transfer. And Eq. 1.3
represents the design limitation on the device length where the first two conditions are
satisfied. Although the above relations are derived by writing the local normal modes in
terms of uncoupled modes to arrive at the adiabatic condition, which, as stated before, is
valid only for the weak coupling case, certain conclusions can be drawn from Eq. 1.1. To
begin with, the adiabatic condition can be satisfied by a slow enough taper (small
APT/AZT) and strong coupling (large K), which can be achieved by a small taper angle and
a small gap. However, for small K, ApT/AZT has to be extremely small which means that
in order to meet the above condition, the device length has to become prohibitively large
and mode conversion between the local normal modes is more likely. However, in this
case, exclusive excitation of an individual waveguide at the input and the sorting of the
mode power entirely into either of the individual waveguides at the output of the 2x2 TVC
is easily achieved. On the other hand stronger coupling helps meet the adiabatic condition
more easily while facilitating a shorter length device. The major problem in this case is that
the two normal mode excitation is unavoidable and as a result, the concentration of power
in one of the guides either at the input or at the output can not be realized (see Eq. 1.2).
Furthermore, the coupled mode theory is no longer applicable. Clearly, to achieve shorter
length devices and localization of energy into one of waveguides both at the input and the
output of the TVC, larger value of K should be incorporated in the middle, permitting a
relatively large APT/AZT at the center while decreasing K at both ends. This can be
achieved by tapering at least one channel waveguide dimension from a narrow to a larger
value while gradually increasing the index at the same time.
We demonstrate the concept using Ti:LiNbO3 channel waveguides where we take
advantage of the fact that the surface index of the waveguide is proportional to erf(w/2dy)
where w is the Ti strip width prior to diffusion and dy is the effective diffusion width. We
model this device by first converting the 2D graded index profile of the five layer TVC
structure into a 1D profile by using the effective index method [Ra74, Fur74]. Field
profiles and mode indices of the local normal modes of the 1D index profile is then
calculated using the multilayer stack analysis and finally using the step transition model
[Mar70, Mi75, Ya78], we calculate the overlap integral between the field profiles of the
local normal modes successively at each section of the TVC to arrive at the power flow
along the coupler. The results show that for a taper length in the interaction region, of only
4.8 mm, i. e., about one fourth of the length indicated in Eq. 1.3, a APTAZT = 51.36 is
sufficient to achieve a cross talk of better than 15 dB due to primarily, the considerably
reduced coupling at both the input and output. Therefore, the inequality in Eq. 1.3, which
imposes the minimum device length for a given tapered structure, can be rewritten as
ApTAZT > 50, thus, facilitating a smaller device length. The analysis, in general, is
applicable to any tapered (dimension, index, coupling, or any combination of the three)
velocity coupler. We further show that use of a step transition model with a five layer slab
model or an equivalent step index waveguide, representing each graded index channel
waveguide section of TVC, leads to erroneous results making the use of the local normal
modes of the composite graded index channel waveguide TVC structure imperative.
Since strong coupling between two waveguides occurs in the middle of TVCs,
normal mode theory must be used for any type of numerical analysis of the TVC. We
introduce the normal mode theory in chapter two and show its application to numerical
analysis of 5layer step index TVC. In addition, the adiabatic theorem in optical zvariant
devices is also described. Qualitative explanation of power flow in zvariant structures is
presented, using the adiabatic theorem in conjunction with the normal mode theory. In
chapter three, we consider a novel 2x2, tapered velocity coupler, analyze the structure and
demonstrate excellent agreement with experimental results. The discrepancies reported in
our previous work [Ki89a] arise mainly due to an approximation of a graded index channel
waveguide TVC as a 5layer step index TVC with a constant, equivalent refractive index
for the tapered guide. This approximation has been widely used [Si87, Sy89] in the case
of many zvariant devices due to the relative ease of the numerical computation and this
approach provides reasonable agreement with experimental results for conventional tapers
as well. However, the above approximation is inappropriate for our coupler which
incorporates tapered index as well as tapered dimension along its length. Furthermore,
approximate representation of the local normal modes as the superposition of the modes of
the uncoupled guides [Mi75] is also inappropriate since the TVC often involves strongly
coupled guides. The model presented here corrects the previous deficiencies and predicts
accurately the behavior of the TVC, using the local normal modes of the composite, five
layer graded index, channel waveguide structure [Ki93a]. Although we must use local
normal modes to describe the strongly coupled guides, we facilitate single mode excitation
at the input and sorting of the power at the output, by tapering both the dimension and
index.
As an application, we proposed and demonstrated [Ra93] a novel digital optical
switch using the TVC in Ti:LiNbO3, which exhibits the smallest voltagelength product
reported in the literature to date. The superior performance of our digital optical switch is
mainly due to the short device length of the TVC in Ti:LiNbO3, using the tapered, both in
index and in dimension, channel waveguide. Numerical analysis has also been performed
and presented in chapter four, confirming the operation of the switch in the adiabatic
regime.
Extension of this technique to proton exchanged waveguides poses a problem since
the index change in protonexchange waveguides does not vary with waveguide width. On
the other hand, index tapering together with dimension tapering in semiconductor
waveguides is not impossible; however it requires elaborate pregrowth processing such as
the delineation of desired dielectric patterns on substrate, so called selective growth
technique [Az81, Ka85], or epitaxially grown spacer and mask layers and two steps of
preferential etching through photolithographically defined windows on substrate, namely,
epitaxial growth through shadow mask [De90, V191]. Both techniques give rise to
composition and thickness change in epitaxially grown ridge waveguides with respect to
the width of the mask. Kato et al. [Ka92] demonstrated integration of a multiquantum
well (MQW) DFB laser and modulator, using selective growth technique. However, it
should be noted that the composition and thickness of the epitaxially grown ridge
waveguides are also subject to a filling factor [Ga90] of the patterned substrate, which is
defined by ratio of area of mask opening to that of the whole substrate. Since compositions
and thicknesses of all the devices are correlated through the filling factor of the substrate,
modification of one device affects all other devices on the chip. Therefore, we believe both
techniques are inappropriate for integration of several devices in one chip.
As an alternative, we propose the use of segmented waveguides, presented in
chapter five for the fabrication of tapered, both in index and in dimension, proton
exchanged channel waveguides in LiNbO3 or semiconductor ridge waveguides. While the
current effort was in progress, several researchers in Israel, Weissman and Hardy [We92,
We93] and Eger et al. [Eg93], have taken the initiative simultaneously to conduct a
theoretical and analytical study on segmented waveguides. Periodically segmented channel
waveguides consisting of an array of high refractive index regions surrounded by lower
index regions, have been used as gratings recently, to achieve phasematched second
harmonic generation (SHG), particularly in KTiOPO4 [Bi90, Po90] and LiNbO3 [We89,
Ca92b]. The high refractive index regions are responsible for both domain inversion and
waveguiding where as in LiNbO3, the segmented waveguides act as a grating for the
guided wave, usually, in a proton exchanged waveguide. Using segmented waveguides,
efficient SHG was obtained and thus we may achieve remarkably good waveguiding in
spite of the segmentation [Bi90]. Waveguiding characteristics of segmented waveguides
show that the effective index of the propagating mode depends on the duty cycle of the
segmentation. Therefore, tapering both in duty cycle and width of segmented waveguides
results in tapered, both in index and dimension, channel waveguides in protonexchange
waveguides as well as semiconductor waveguides. As part of a preliminary study, we
investigated the effective indices of segmented waveguides with different duty cycles using
planar segmented protonexchange waveguides in LiNbO3 [Th94]. Both the experimental
and the theoretical results show that the effective index linearly increases with duty cycle up
to 70 % of duty cycle, then increases sublinearly, and finally saturates at the effective index
value of the continuous protonexchange waveguide. Measured effective indices of first
two modes for different periods with a duty cycle remained unchanged. Segmented
waveguides in semiconductors were fabricated using epitaxially grown AlGaAs/GaAs
sample and then characterized by the investigation of the variation of propagation loss, and
near field intensity profile on different duty cycles. The ability to control the effective index
of the mode by changing the duty cycle of the segmented waveguide enables us to design
and fabricate the tapered, both in index and in dimension, velocity coupler in
AlGaAs/GaAs.
So far, horizontal coupling between a tapered and a straight channel waveguide in a
substrate plane have been utilized for power transfer in the TVCs in Ti:LiNbO3 and
AlGaAs/GaAs. We have also extended our concept by considering vertical coupling
between epitaxially grown single quantum well (SQW) guiding layers and tapered multi
quantum well (MQW) layers. Tapering index and dimension of the MQW was performed
by tapered impurity induced layer disordering (IILD) [La81, Si92, Ki93b] of the MQW
which modifies their equivalent refractive index profile as the MQW region in such a way
that the index profile is gradually increased along the beam propagation direction.
Equivalent refractive indices for MQW regions before and after disordering were
determined for numerical analysis as those of AlxGalxAs layers which have the same
bandgap as the respective MQW regions [Ki93b]. With the equivalent indices of the
tapered MQW regions, adiabatic characteristics of the tapered interconnects [Si92, Ki93b]
were successfully analyzed by use of the step transition model; results are presented in
10
chapter six. We also present a modified tapered interconnect which incorporates efficient
vertical coupling between an SQW guiding layer and a tapered MQW waveguide. The
modification of the tapered interconnect results in a better coupling from an external laser to
SQW guiding layers with complete power transfer from the SQW guiding layers to the
MQW layer within a reasonable taper length, and concentration of the mode in the MQW
region for higher efficiency in modulation.
CHAPTER TWO
NORMAL MODE THEORY
AND ADIABATIC THEOREM
Tapered velocity coupler (TVC) structures are not easily amenable to analytical
solutions. Often computer simulations are the only means even to achieve physical
understanding of the wave propagation in such complex structures. The normal mode
theory should be used rather than coupled mode theory to analyze a TVC structure by
computer simulation since strong coupling between two waveguides must occur for
efficiency power transfer in TVCs. To gain a general insight into the normal modes, we
present various characteristics of normal modes in symmetric or asymmetric fivelayer step
index waveguides. Aidiabatic theorem in optical devices is discussed in conjunction with
the normal mode theory. Qualitatively explanations of output power profiles in Ybranch
and cross couplers are to be presented in this chapter using the adiabatic theorem and the
normal mode theory.
2.1 Definition of Normal Modes
A fivelayer directional coupler considered in this section consists of two planar
waveguides and a gap of finite thickness. There are two guiding layers (A and B) of
thickness D1 and D2, respectively, and a lower refractive index material of thickness T is
placed between the two guiding layers. The cross section of a fivelayer directional coupler
is shown in Fig. 2.1. As is the case with conventional directional couplers, the refractive
indices ni and n2 of the guiding layers A and B are chosen to be higher than that of the
cover (nc), the gap (ng), and substrate (ns).
cover
x=Oj
z
D2 waveguide B
T gap
D1 waveguide A
substrate
Fig. 2.1 A cross sectional view of fivelayer step index waveguide structure.
For the sake of simplicity, we restrict our attention to TE mode waveguide
solutions, which by definition have the electric field polarized along yaxis. For the
conventional fivelayer directional coupler, we can write the fields in each layer as follows:
Eys = As exp(ksx)
Eyl = Al sin(klx) + B1 cos(klx)
Eyg = Ag exp(kgx) + Bg exp(kgx)
Ey2 = A2 sin(k2x) + B2 cos(k2x)
Eye = Ac exp(kcx)
(x <0)
(0 < x D1)
(DI < x DI+T)
(DI+T < x < DI+D2+T)
(x 2 DI+D2+T)
where ko = 2m'/ is the free space propagation constant,
ks= k N2n2,
kl=ko n?N2,
kg=ko N2n2,
k2 = ko0/nN2, and
kc =k0oN2n2.
X represents the free space wavelength of light and N is called the effective index of a mode
or simply mode index. N equals P/ko, where 0 is the propagation constant of the mode.
The most general TE mode expression in each layer may, unlike the above
equations, be represented as
Eyq = Aq exp(jkqx) + Bq exp(jkqx) (2.2)
where
(2.1)
kq=k nN2 q = 1, 2, s, g, c.
Equation 2.2 is valid, in general, for any fivelayer structure regardless of the magnitude of
each layer's refractive index. Under guiding conditions, the mode index N is always larger
than ns and ne and we obtain exponentially decaying solutions that vanish at infinity. For
unguided leaky or radiation modes, the solutions grow exponentially. Under the
assumption, ni, n2 > ns, ng, nc, a more specific representation in the form of Eq. 2.2 and
the resulting fivelayer dispersion relation presented by Yajima [Ya78] are obtained. We
are interested in only the guided mode, not the leaky or radiation mode. The properly
chosen refractive indices (ni, n2 > ns, ng, nc) for guiding an optical wave and a finite value
of the electric field component Ey(x), i. e., approaching zero, at infinity (x = oo or oo)
allow us to write Eq. 2.2 into a set of equations given by Eq. 2.1 for each of the five
layers.
Hx(x) and Hz(x), the other components of the TE mode, are given by Maxwell's
equation as follows:
k0N
Hx = o Ey
Hz= ( E (2.3)
where o is the angular frequency and Ig is the permeability.
Application of Maxwell's equation at each of the four boundaries assures the
continuity of the tangential component at each boundary. These boundary conditions
demand that Ey (and automatically thereby Hx) and DEy/Ox (and thereby Hz) be continuous
across the layer boundaries at x = 0, D1, DI+T, and DI+D2+T. For Ey(x) to be continuous
at each boundary, we require
(2.4)
A1 sin(kiDI) + B1 cos(kiD1)
= Ag exp(kg D1) + Bg exp(kg Di) (2.5)
Ag exp(kg(Di+T)) + Bg exp(kg(Di+T))
= A2 sin(k2 (DI+T)) + B2 cos(k2 (DI+T)) (2.6)
A2 sin(k2(Dl+D2+T)) + B2 cos(k2(DI+D2+T))
= Ac exp(kc (DI+D2+T)) (2.7)
Imposing the continuity of EEy//x at each boundary, we obtain
ks As = ki Al (2.8)
kl (A1 cos(kiDI) BI sin(klD1))
= kg (Ag exp(kg Di) Bg exp(kg Di)) (2.9)
kg (Ag exp(kg(Di+T)) Bg exp(kg(Di+T))
= k2 (A2 cos(k2 (DI+T)) + B2 sin(k2 (DI+T))) (2.10)
k2 (A2 cos(k2(Dl+D2+T)) B2 sin(k2(Dl+D2+T)))
= kc Ac exp(kc (DI+D2+T)) (2.11)
From the above eight equations (Eqs. 2.4 2.11), we want to obtain a value of N,
which is the mode index of the fivelayer structure, and relations between the eight
amplitude coefficients. Matrix representation can be used to simplify the above equations
by relating each amplitude coefficient as follows:
MIA = A) (2.12)
(AM ) (Ag (2.13)
M2 B1 =Bg9
M3 =(A 2) (2.14)
M4 (A2)=M5 Ac (2.15)
(4B2
where M1 and M5 are 2x1 matrices while M2, M3, and M4 are 2x2 matrices. All the matrix
elements can be easily obtained from Eqs. 2.4 2.11 with a little bit of algebra; so we omit
explicit analytic expressions of these elements. Substitution of Eqs. 2.12 2.15 yields
M5 Ac = M4 M3 M2 M1 As = M As (2.16)
where M = M4 M3 M2 M1 and size of M is 2x1. Let us define [Q]ij as an ith row and jth
column element of a matrix Q. According to Eq. 2.16, we have
Ac = ( 1[M / M5111) As (2.17)
Ac= ([M]21 / [M521) As (2.18)
However, a value of Ac is uniquely defined for a given value of As, since the field
distribution of a mode has a unique shape. Therefore,
M]II [M]21 = [M5]11 / [M5121 (2.19)
Equation 2.19 consists of the known constants such as D1, D2, T, ko and the
unknown variables, which are ks, kl, kg, k2, kc. However, wave vectors of each layer can
be easily determined as long as we know the mode index N (see Eq. 2.1). Therefore Eq.
2.19 can be interpreted as a nonlinear solving Eq. 2.19. Once N is determined, eight
coefficients as well as the five wave vectors at each layer can be obtained by considering
the normalization equation making the mode power unity. A normalized power equation is
usually stated as follows:
P= 1= I E (x) dx (2.20)
2cotof0
So far, we have presented the general expression (Eq. 2.2) of field distribution
along the xaxis normal to the beam propagation direction (+zaxis) and a simplified set of
equations (Eq. 2.1) under the assumption of appropriately chosen indices (ni, n2 > ns, ng,
nc). And then by considering boundary conditions, a set of eight equations (Eqs. 2.4 
2.11) has been obtained. By the consideration of one more equation (Eq. 2.20), that is the
normalization of mode power, we obtained nine equations for nine unknowns, which are
eight amplitude coefficients of Eq. 2.1 and the mode index N. Now we propose a
methodology for handling the numerical problem. We use the bisection method for solving
Eq. 2.19; representation of which is modified as follows:
F(N) = [M] 11/ [M]21 [M511 / [M5121 = 0 (2.21)
where a value of N of a given mode should lie in the region of min(nl, n2) > N > max(ns,
ng, nc). While N decrease from min(nl, n2) to max(ns, ng, nc) in small, discrete steps,
F(N) would change its sign between two discrete values of N. Since the function F(N) is
continuous, it is obvious that the change in sign of F(N) between two consecutive points
guarantees at least one solution of F(N) = 0. Let us evaluate the function F(N) at the mid
point of the two adjacent points and compare the sign. We then choose a smaller interval
for the solution of F(N) = 0 where the sign changes. We can continue this process of
intervalhalving to determine a smaller and smaller interval within which the solution for
F(N) = 0 must lie. Since the solution obtained by this method would be the largest value of
N satisfying Eq. 2.21, we call this the fundamental mode index, denote by NO. Further
decreasing of N by small, discrete steps and following the above procedure, we obtain the
second largest solution. It is called the first order mode index (Ni). When N decreases to
max(ns, ng, nc), all possible solutions of Eq. 2.21 would have been found at every sign
change by the bisection method.
2.2 Characteristics of Normal Modes
in Fivelayer Waveguide Structures
As numerical examples, we will consider two cases, one symmetric case and the
other asymmetric. As a first example, we consider a symmetric structure with ns = ng = nc
= 2.16 and ni = n2 = 2.2, D1 = D2 = 3 Jm, X = 1.3 pim and the gap between the two
planar waveguides T = 2 gLm. This specific waveguide structure supports only four guided
modes. The Ey(x) components of the four TE modes of this fivelayer waveguide structure
are shown in Figs. 2.2 (a)(d). All the field distributions are normalized by assuming each
mode to have a unity power. Same as in the threelayer waveguide, the number of zero
crossings of Ey(x) coincides with the mode number. In addition, even mode has a
symmetric field distribution and add mode an antisymmetric distribution. The effective
indices of each mode are NO = 2.19411, N1 = 2.19398, N2 = 2.17773, N3 = 2.17680.
For the purpose of reference, we calculate the mode index of symmetric threelayer
waveguide which consists of substrate, film, and cover layer and indices of each layer are
ns = 2.16, nf =2.2, and nc =2.16, respectively. The thickness of film layer is 3 gim. This
structure is nothing but one of the two waveguides waveguidee A or B) in our fivelayer
structure with a infinity gap (T = oo). Using a wellknown threelayer dispersion equation
[Ko74], we have (NO)A = (NO)B = 2.19404 and (NI)A = (N1)B = 2.17729. To avoid the
confusion we discriminate between fivelayer normal mode and threelayer individual mode
by the notation. That is, Ni is the ith mode index of the fivelayer normal mode and (Ni)A
is the ith uncoupled threelayer individual mode in waveguide A. We recognize the
obvious fact
NO = (NO)A + 80/2
N1 =(No)A 50/2
N2 = (N1)A + 1/2
N3 = (N1)A 81/2 (2.22)
0.4
0.2
0
0.2
0.4
2 0 2 4
x (nm)
(a)
 Fundamental
 First order
I I I \' I
 
 
I n II ,, I
6 8 10
Fig 2.2 Normalized field distribution of normal modes. D1 = 3 gim, D2 = 3 ptm, T
= 2 Lm, X = 1.3 gpm, ns = ng = nc = 2.16, and nl = n2 =2.2. (a) The
fundamental (solid curve) and the first order (dotted curve) normal mode.
0.4
0.2
0
0.2
0.4
2 0 2 4
x (pam)
(b)
Fig. 2.2continued.
 second order
 Third order
6 8 10
(b) The second order (solid curve) and the third order (dotted
curve) normal mode.
where
80 = NO N1
81 = N2 N3
This trend is plotted in Fig. 2.3 where NO, N1, N2, and N3 are shown as a function
of thickness T. As T increases, 80 decreases and with the result NO and N1 converge into
the value of (NO)A. The same is true in the case of 81. So N2 and N3 becomes (N1)A. It
means that as the separation between waveguide A and B becomes large, No and N1
become degenerate and equal (N0)A, and each waveguide gives no influence to the other
waveguide. On the other hand, 80 and 81 become larger and larger as the gap (T) continues
to decrease.
So far, we have investigated the various characteristics of a symmetric fivelayer
directional coupler. We turn our attention to an asymmetric structure. Our asymmetric
structure has the same dimensions and refractive indices as the previous symmetric
structure has except for D2 = 2 pm. In Fig. 2.4, we present the electric field distribution of
the first four modes of asymmetric structure. In order to understand these field
distributions, a comparison between individual mode indices of each waveguide is needed.
In this case, we have (N0)A > (No)B > (N1)A > (N1)B. According to this order, we can
imagine a rough field distribution without exact computation. (N0)A corresponds to No
and a field distribution of TEo (the fundamental mode of TE in fivelayer structure)
essentially consists of that of (TEo)A (the fundamental mode of TE mode in waveguide A)
and a small lobe in waveguide B. Likewise, (N0)B corresponds to N1 and the field
distribution of TE1 is composed of that of (TEo)B and a small lobe in waveguide A, and so
on. Also in asymmetric structures the number of zerocrossing of a mode is the mode
number. Let us consider a more asymmetric case. If we set D2 = 0.5 pm and all the other
dimensions and refractive indices are same as before, (Ni)A and (Ni)B should be calculated
2.2
2.19
2.18
2.17
2
T (gm)
Fig. 2.3 Normal mode indices as a function of T.
SN
.NI
* 3^ ====== ~
0.6
0.4
0.2
0
0.2
2 0 2 4
x (im)
(a)
 Fundamental
.... First order
6 8 10
Fig. 2.4 Normalized field distribution of normal modes. D1 = 3 gpm, D2 = 2 gim,
T = 2 jm, = 1.3 gm, ns = ng = nc = 2.16, and nl = n2 = 2.2. (a) The
fundamental (solid curve) and the first order (dotted curve) normal mode.
0.4
0.2
0
0.2
0.4
2 0 2 4
x (pm)
(b)
Fig. 2.4continued.
Second order
 Third order
6 8 10
(b) The second order (solid curve) and the third order (dotted
curve) ormal mode.
by the threelayer dispersion equation and their magnitude comparison is as follows: (NO)A
>(N1)A > (NO)B. Therefore the field distribution of TEO essentially consists of (TEO)A
with a perturbation in waveguide B. A field distribution of TE1 is composed of that of
(TE1)A and a small lobe in waveguide B, and for TE2 field distribution, the filed of (TEO)B
and a little perturbation in waveguide A can be matched. The computer calculated field
distributions for the TEO, TE1, and TE2 modes are shown in Fig. 2.5.
Now we describe the general characteristics of a normal mode field distribution in
fivelayer structures, based upon previous results and many other simulations we have
done.
The general characteristics of symmetric directional couplers are
1. The number of zerocrossing of the guided mode corresponds to the mode number.
2. Power confinement factor of both individual waveguides are identical. Even mode
filed distribution is symmetric while odd mode distribution is antisymmetric.
3. The number (Np) of peak points in the power profile is
Np = 2 (Int(i/2) + 1) (2.23)
where i is mode number and Int(x) integer function.
The general characteristics of asymmetric directional couplers are
1. The number of zerocrossings of the guided mode again corresponds to the mode
number.
2. Depending upon the asymmetry, power confinement factor of one waveguide,
having the maximum peak power point, is larger than that of the other waveguide.
3. An approximate field distribution can be drawn by considering the field distribution
of the corresponding individual mode in the uncoupled guide and the number of
zerocrossings as well as the smoothness and continuity of a field.
 Fundamental
 First order
2 0 2 4
x (wm)
(a)
6 8 10
Fig. 2.5 Normalized field distribution of normal modes. D1 = 3 gm, D2 = 0.5
gIm, T = 2 pm, X = 1.3 gm, ns = ng = nc = 2.16, and nl = n2 = 2.2. (a)
The fundamental (solid curve) and the first order (dotted curve) normal
mode.
0.4
0.2
0
0.2
0.4
Second order
I I Im' I I
2 0 2 4
x (pm)
(b)
6 8 10
Fig. 2.5continued. (b) The second order (solid curve) normal mode
0.6
0.4
0.2
0
0.2
2.3 Adiabatic Theorem in Optical Devices
It is possible to predict qualitatively the output power profile in zvariant devices
without complicated numerical calculations by utilizing the adiabatic theorem in conjunction
with the normal mode theory. In this section, we consider, first, the adiabatic theorem,
apply it to Ybranch and cross coupler structures, and qualitatively discuss the field
distributions using normal modes.
The adiabatic theorem states that motion in some dynamic state with slowly varying
parameters has some invariable quantities, called adiabatic invariants. A few papers [Bu75,
Ya78, Bu80] discussing the behavior of an asymmetric Ybranch waveguide have used the
adiabatic theorem. For the normal mode propagation process in the Ybranch waveguide,
the state is the propagation mode in the fivelayer waveguide with slowly varying
separation (distance between the branches) being the external parameter. The adiabatic
invariants are the mode number and the mode energy. This means that the mode
propagation in the branching waveguide keeps its initial state as a local normal mode which
is defined as the normal mode of the coupled structure that is evolving along the length of
tapers and that essentially no energy transfer between the local normal mode occurs, as
long as the change in the waveguide parameter, i. e., the separation, varies very slowly
along the propagation direction. If the gradualness of the slowly varying parameter in the
Ybranch, which is determined by the angle between two branches, is sufficiently small for
the adiabatic invariants, namely, the mode number and the mode power, to maintain their
initial values, then the adiabatic condition is satisfied. Supposing that the widths of the
branches differ, the fundamental mode at the input would keep its power as the
fundamental local normal mode at the output as long as the adiabatic condition is met.
Therefore, at distance far away from the branch, the entire output power would physically
be found in the wider branch. Of course, the first order mode at the input should come out
of the narrower guide at the output. A schematic of Ybranch is shown in Fig. 2.6 (a) for
the fundamental mode at the input and Fig. 2.6 (b) for the first order mode, where the
Fig. 2.6 A schematic of field evolution along the Ybranch.
(a) For the fundamental normal mode input.
(b) For the first order normal mode input
adiabatic property is illustrated. These results were verified experimentally by Yajima
[Ya73].
For the purpose of qualitative discussion, we now apply the adiabatic theorem to
cross coupler, which has asymmetric input branches and symmetric output branches
[Si87]. A schematic of the cross coupler is shown in Fig. 2.7 (a). Let us assume that the
cross coupler meets the adiabatic condition, namely, the angle between the branches is
sufficiently small enough for the normal mode power to be conserved in the same mode
along the coupler. Widths of all four branches are appropriately chosen for single mode
propagation, while the width of center region is wide enough to have two modes
propagate. Supposing that we excite the narrower branch of the input side using a single
lobe input beam, which is essentially the first order normal mode in the fivelayer structure
consisting of the two asymmetric guides, the gap in between the guides, and the two outer
claddings. As the first order mode in the input side propagates toward the center region
and separation becomes smaller, the adiabatic condition requires that the energy remain in
the first order mode. This implies that the field distribution has to gradually evolve from an
essentially singlelobe distribution to twolobe distribution with a phase shift of nt between
them. This corresponds to the first order mode of the doublemoded center region, and the
power profiles at the center region should now have two equal lobes. Since the output
branches are symmetric, the input power is equally divided between the two branches as
the beam propagates toward the output. Finally we have equal power profiles with the field
amplitudes exhibiting a t phase difference at the output branches. In Fig. 2.7 (b) and (c),
we present the schematic representation of the evolution of the fields along the cross
coupler for both the first order normal mode and the fundamental normal mode input,
respectively. For the fundamental mode input, we excite a singlelobe input beam in the
wider input branch. A small lobe with the field amplitude whose phase is the same as that
of the lobe in the wider branch is built up in the narrower branch as the fundamental mode
(a)
rcC
Fig. 2.7 A schematic of field evolutions along the cross coupler.
(a) Structure of an asymmetric cross coupler.
(b) For the first order normal mode input.
(c) For the fundamental normal mode input.
~~c:
32
approaches the center region. In this case, we have equally divided power in both the
symmetric output branches with their field amplitude in phase.
So far we have discussed the normal mode theory and the adiabatic theorem in five
layer stepindex waveguides. These concepts are extended further in the next chapter for
numerical analysis of graded index channel waveguide TVC.
CHAPTER THREE
TAPERED, BOTH IN INDEX AND IN DIMENSION,
VELOCITY COUPLER IN Ti:LiNbO3
In Ti:LiNbO3 waveguides, width of a Ti strip before its diffusion governs not only
the waveguide dimension but also determines the peak index change at the center of
diffusion profile. Within single mode regime, the peak index change at center of the Ti
strip monotonically increases and then saturates at the planar waveguide value with
increase of strip width. Therefore, a tapered Ti strip yields naturally a channel waveguide
with its both index and dimension tapered so that it can be used in a tapered velocity
coupler for reducing length of the coupler.
In 1989, Kim and Ramaswamy [Ki89a] reported, for the first time, the realization
of a tapered velocity coupler (TVC) using Ti diffused channel waveguides in LiNbO3.
By virtue of the fact that tapering exists both in index and in dimension of the channel
waveguides of the TVC, we were able to reduce a device length as much as onequarter
of minimum length possible as predicted by Milton and Burs [Mi75] while maintaining
the coupler in the adiabatic regime. Although complete power transfer was achieved
while satisfying the adiabatic condition, the theoretical model that used in [Ki89a] was
inadequate in predicting the behavior of the TVC.
In this chapter, we analyze the structure by using local normal modes of the entire
structure and extend the concept by considering a carefully designed tapered velocity
coupler (TVC) that is tapered both in index and in dimension which meets the adiabatic
invariance condition with sufficiently strong coupling between the fundamental modes of
individual guides in the center region of the coupler while permitting individual
excitation at the input end and sorting of the modes at the output end. This approach
helps reduce the device length considerably by permitting much higher taper angle. We
model a TVC that consist of one tapered and another straight, graded index waveguide,
by using normal modes of the entire, composite TVC structure [Ki93a]. The analytical
results are in excellent agreement with experimental results in a TVC fabricated in
Ti:LiNbO3, substantiating the possibility of a shortlength TVC. In particular, we show
that the representation of the local normal modes as the superposition of the modes of the
uncoupled guides leads to erroneous results, as the avoidance of mode conversion
between the local normal modes in a reasonably short length TVC invariably involves
strong coupling between the guides and that the use of actual local normal modes of the
TVC structure under consideration is imperative in the modeling to accurately describe
the device.
3.1. Single Mode Channel Guide:
Normal Mode Analysis and Field Profile
Before describing the TVC, we present the results of the numerical analysis and
the experimental verification of the normal mode index and the associated field profile
for a single mode, straight channel waveguide. The results are used later to describe the
characteristics of local normal mode of the composite tapered velocity coupler. The 2D
refractive index profile of an individual Ti diffused LiNbO3 channel waveguide, in
general, can be expressed as
n(x,y) = nb + An(T,t,w) G(x) Erf(y) (3.1)
where nb is the bulk index, T and t are diffusion temperature and diffusion time,
respectively, w is the titanium strip width, G(x) is the normalized Gaussian function with
G(0)=1 and Erf(y) is a normalized linear combination of the error functions with Erf(0) =
1, as well. The exact index distribution, substantiated by the experimental results of
Fukuma et al. [Fu78], qualitatively illustrated in Fig. 3.1 (a), is given by
f(y+w/2 erfy+w/2
n(x,y) = nb + An exp((x/dx)2) I y y (3.2)
2 erf Qw'2)
y
where x represents the depth direction and y is measured along the width of the titanium
strip, from its midpoint. The propagation direction is assumed to be along the z axis.
Furthermore,
nb = bulk index'of LiNbO3,
An = dn 2 erf( 2) (3.3)
de in dx dy
dn/dc = rate of change of index with concentration,
t = Ti strip thickness,
w = Ti strip width,
T = diffusion temperature in C,
t = diffusion time in hours,
dx =2 D xt, and
dy = 2 /Dyt.
The validity of the above expression have been verified by a number of researchers
[Ki89b, Ko82, Su87]. For our operating condition, we have nb = 2.2195, substrate index
(no) for quasi TE mode, and dn/dc = 0.5 at X = 1.32 gpm. Titanium film of thickness t =
800 A was deposited over a z cut LiNbO3 crystal with an ebeam evaporator and a strip
width of w = 4 pLm was delineated using standard photolithography techniques and wet
2D index profile : n(x,y)
n(x,y)
k '7
S.7
n(x,y)
Gradedindex
Gaussian
profile
Approximated
stepindex
Gaussian
profile
Fig. 3.1 A schematic of graded index profile of a Ti:LiNbO3 channel
waveguide. (a) A schematic cross section of a Ti:LiNbO3 channel
waveguide. The gradual index change is qualitatively illustrated
by varying degree of the shadow in the diffused region. (b)
Graded index profile along the depth direction and the stair case
approximation of the profile
m
1\7B~
(etching process. The diffusion was carried out at a temperature T = 1025 'C for t
= 6 hours in a wet oxygen atmosphere to minimize Li out diffusion and to ensure the
crystal remains fully oxidized through the entire diffusion process. The end faces of the
crystals are polished to facilitate near field measurements. For our fabrication condition,
Dx = Dy = 1.2 x 104 pm2/sec, dx = dy = 2.684 9m.
Very briefly, we outline the analytical approach including the effective index (EI)
method and the multilayer stack analysis, as we use them extensively in the subsequent
sections for the analysis of the TVC structure. To convert the two dimensional index
profile into one dimensional profile, we use the El method. Figure 3.1 (b) illustrates the
graded index, Gaussian profile G(x), at a given discrete position along the yaxis. The
Gaussian profile can be conveniently approximated by a staircase, stepindex profile with
its surface index at that position along y and is given by
ns(y) = n(0,y) = nb + An(w) Erf(y) (3.4)
We numerically solve the approximated step index profile at a given position y by using
the multilayer stack analysis [Hu90, Th87]. By considering a multilayer infinite slab in
the y direction, with r2 layers and two semiinfinite regions above and below,
representing the surface and the substrate, we can represent each ith layer with
appropriate index ni and thickness ti = xi xi1, in the x direction, thus approximating the
Gaussian profile. The field amplitudes, for example for the quasi TE mode (Ey, hx, hz), at
each ith layer (slab) can be written as
ai cos(ui) + bi sin(ui), if N 5 ni (
i ai cosh(ui) + bi sinh(ui), if N > ni )
where
ul = k1(x1 x),
ui = ki(x xi1), i = 2,3,...,r,
ki= 2 nk2,
N = p / ko, effective mode index,
ko = free space propagation constant and
3 = propagation constant of the mode.
We apply the appropriate boundary conditions at each interface of the slab. For the quasi
TE mode, the field amplitude F corresponding to ey(x) and its derivative dF/dx are
continuous at r1 interfaces. Thus, we have 2(r1) equations. In addition, we have two
equations in the region 1 and r, where we have F1 = exp(x), for x < xl, resulting in al =
bl and Fr = exp(x) for x > Xr, with ar = br. Thus, we have 2(r+l) unknowns viz., al,
a2, ., ar, bl, b2, ..., br and3. In addition, we have also an equation for the normalization
of the field intensity.
f 2t F2 dx= 1 (3.6)
By solving successively for the effective mode index N at each discrete position
y, for the index profile G(x), we can solve for the effective index profile N(y). As
always, this value is bounded by the surface index at that point ns(y) and the substrate
index nb, at each position y. Figure 3.2 shows the converted 1D index profile N(y),
calculated from the 2D index profile n(x,y) for a 4 pim wide Ti:LiNbO3 channel
waveguide. The dotted line curve represents the converted 1D profile N(y) nb. The
figure also shows the surface index profile ns(y) nb, indicated by the solid line. For the
purpose of reference, the 4 tpm wide Ti strip is also shown.
The above index profile N(y) in Fig. 3.2 is solved once again by using the
multilayer stack analysis, now for the quasi TM mode ey(y), so that the polarization of the
original quasi TE mode field remains consistent. The normalized intensity (square of the
0.01
0.008
S0.006
0.004
0.002 .*"'" \
10 5 0 5 10
y (wm)
Fig. 3.2 Converted 1D index profile (N(y)nb, dotted curve) and actual surface
index profile (ns(y)nb, solid curve) of a 4 g m wide channel
waveguide.
electric field) profile of the fundamental mode is illustrated in Fig. 3.3 by the solid line.
The near field intensity profile was measured by launching the light from an 1.32 ptm
laser diode into a single mode fiber which was then used to excite the quasi TE mode in 4
p.m wide guide. The output was collected with an objective and focused on to a
germanium detector placed at the image plane through a 10 pm pin hole. The output was
scanned parallel to the substrate plane across the y direction at the peak intensity position
in the x (depth) direction and is shown by the dotted line. As seen from the figure,
excellent agreement is obtained between the theoretical analysis and the experimental
result. The calculated fundamental mode index No (= 3/ko) for the 4 pgm wide guide is
2.2205 at 1.32 p.m.
Once we were satisfied with the accuracy of the channel waveguide
representation, we varied strip width of titanium from 1 p.m to 12 p.m and calculated the
effective mode indices No, N1, and N2 of the first three modes of the channel waveguide.
These normal mode indices of the individual channel waveguides are plotted in Fig. 3.4
where the maximum surface index n(0,0) at the surface (dotted line) is also included.
Note that the fundamental mode is cut off at about 2 gpm. The guide remains single
moded till around 6 pum at which point the first order mode begins to propagate. No
varies from 2.2195 to 2.2218 as w is changed from 2 to 6 p.m. Although the cutoff
wavelengths of the guides in the TVC which are strongly coupled, will be different, these
results, nevertheless, are useful in the preliminary design of TVC. When the gap in the
tapered coupler device becomes quite large, the normal modes approach the modes of the
uncoupled case and these results are directly applicable.
3.2 Tapered. Both in Index and in Dimension.
Velocity Coupler
Figure 3.5 (a) illustrates the schematic sketch of the delineated Ti strip waveguide
patterns, on a z LiNbO3 substrate prior to diffusion of the tapered velocity structure. It
consists of a 7800 pim long, uniform Ti strip of width 4 pm and an equally thick, 2 ptm to
41
1.2 . .... . iprimnt
W=4.  Experiment
1 Theory
4 
r: 0.8
a 0.6
0.4
0
Z 0.2
0 . ......,........I '
10 5 0 5 10
y (pim)
Fig. 3.3 Theoretical (solid curve) and experimental (dashed curve) intensity
profiles of the quasi TE fundamental mode for a Ti:LiNbO3 graded
index, 4 tim wide waveguide.
 Mode index
 Maximum surface index
/f
*
0 2 4 6 8
Width of Waveguide
10
(pmn)
. N2
12
12
Fig. 3.4 Quasi TE mode indices (solid curve) for the first three modes and
maximum surface index (dashed curve) of Ti:LiNbO3 channel
waveguides.
2.2255
2.2245
2.2235
2.2225
2.2215
2.2205
2.2195
x
Fig. 3.5 Schematics of LiNbO3 sample. (a) with its delineated pattern and
before Ti diffusion on z surface and (b) with index profile with
coordinate system after diffusion.
6 p.m wide, 4800 p.m long, tapered Ti strip separated by a constant gap. Only the straight
channel is extended to the input end to prevent input light from coupling into the tapered
channel. The diffused index profiles are qualitatively illustrated in Fig. 3.5 (b) and the
gradual increase in the index of the tapered guide with w, is illustrated by the darker
region. The zaxis is chosen to coincide with the beginning of the tapered guide, but as
before, the xy origin is located at the center of the 4pm wide straight guide. At the input,
the index of the straight guide is larger than that of the tapered guide and vice versa at the
output; as a result, the modes are well guided (except for the 2 pm guide at the input) and
weakly coupled permitting excitation of the individual waveguides both at the input and
output. At the center, where the guides are of equal width and hence identical, they are
strongly coupled permitting a larger AP3T/AZT.
To solve for the field profile of TVC at any point along its length, we must
consider the evolution of the mode profile along the taper. To accomplish that we need
the knowledge of the normal mode index N and the field Ey(x,y) at any point along the
taper, by treating it as if it is a uniform directional coupler, infinite in extent in the z
direction, consisting of two graded index guides of constant widths, one corresponding to
a straight guide and the other to the width of the tapered guide at that point, separated by
the same gap. The eigen values (N and ey(y)) obtained through the El method, are the
solutions of the eigen mode at that point, referred to as the local normal mode.
In order to solve for the eigen values of the local normal mode, we assume the
composite index profile of the structure at any point is the superposition of the index
profiles of the straight 4 p.m channel and the tapered channel width (between 2 pm and 6
p.m) at that point For the purpose of discussion, we will identify the diffused waveguides
by referring to the strip width prior to diffusion. As before, the converted 1D index
profile i.e., the effective index change N(y) nb of the composite index profile for the
quasi TE mode is obtained by the use of El method and is plotted in Fig. 3.6. Three
curves are shown, one at each end of the tapered coupler and a third one at the mid point
Output end
0.004
S Middle of
0.003 the coupler
U 0.002 :, Input end
S0.001
0
I
5 0 5 10 15
y(pm)
Fig. 3.6 Effective index change (N(y)nb) at the input end (solid curve), middle
of the coupler (dashed curve), and the output end (dotted line).
of the coupler. It is interesting to note, the perturbation of the effective index profile at
the input end is rather small, since the index of the narrower guide is smaller where as at
the output end, the wider tapered guide exhibits a larger index change and hence
influences the profile more strongly. As expected, the profile is symmetrical at the
midpoint of the coupler. The location of the Ti strips, of both the straight and the tapered
channel waveguide, are shown in the figure. In addition, the darkness (shadow) of the
strip is indicative of the amount of index change due to Ti in diffusion and depth of the
strip has no physical meaning. Darker shadow represents stronger waveguides.
As before, the 1D effective index profile of the composite structure was solved
by using multilayer stack analysis, now for the quasi TM mode, so that we are consistent.
The fundamental and the first order local normal mode indices, namely, No and N1, for
the quasi TE mode as well as the intensity profiles of these modes were calculated as a
function of the width of the tapered guide, for various gaps. The normalized intensity
profile at the input of the device at z = 0 is illustrated in Fig. 3.7. The theoretical
intensity profile of the uncoupled 4 p.m channel waveguide replotted from Fig. 3.3 is also
shown in the same figure. Clearly, the fundamental mode of the composite structure
nearly coincides with that of the individual, straight 4 gpm channel waveguide, thus
facilitating sole excitation of the straight channel at the input. The overlap with the
fundamental mode is better than 97% while the overlap with the first order mode,
although exists, is extremely small. The field profiles of both fundamental and first order
local normal modes along the length of the coupler are later used in the step transition
model to study the evolution of the modes and the power flow along the TVC.
It can be clearly seen in Fig. 3.7 that the first order local normal mode at the input
can not be properly represented by the superposition of the two modes of the uncoupled
guides due to considerably shifted peak position from the center of the 4 p.m wide
channel waveguide and large asymmetry of both the lobes in the intensity profile. In
addition, investigation of the normal mode field profiles confirms the fact that it is
1 S
S Input to TVC
S....... Fundamental of TVC
S0.8 Firstorder of TVC
S 0.6
0.4
Z 0.2
10 5 0 5 10 15 20
y (pm)
Fig 3.7 Normalized intensity profiles of the fundamental mode (dotted curve),
the first order mode (dashed curve) at the input end (z=0), and the
profile (solid curve) of 4 pm wide channel guide prior to input to the
TVC.
inappropriate to describe the local normal modes, especially the first order modes, in the
case of small gap (g = 2, 3, and 4 gim) couplers in terms of the superposition of the
uncoupled modes.
In Fig. 3.8, the fundamental mode index No of the uncoupled, uniform guide viz.,
2.2205 is indicated by the dashed, straight line. The other dashed curve in the figure
represents the mode index No of the uncoupled, tapered guide, replotted from Fig. 3.4, in
the single mode regime (W = 2 to 6 im ). The two dashed curves cross when the guides
become identical, i.e., when their widths equal 4 jim. The figure also shows the mode
indices of the fundamental (No) and the first order (N1) mode as the width of the tapered
guides is varied from 2 to 6 jim with the gap between the straight guide and the tapered
guide being the parameter. The gap was also varied from 2 to 6 pm in steps of 1 p.m. As
seen from the figure, No of the composite structure at the input section is close to that of
the wider, straight channel guide which supports most of the energy and increases
monotonically with increasing gap width; the situation reverses at the output end where
the tapered guide width is now larger and supports most of the energy in the structure. If
the tapering is slow enough and the adiabatic invariance condition is met, the evolution of
the fundamental mode is unaffected although physically, now different guides both at the
input and output support most of the energy in the coupler. As a result, the energy of the
fundamental mode in the straight section at the input now arrives at the tapered guide at
the output. The switching occurs around the central region of the coupler. As seen from
the figure, the tapering of the guide seems to affect the mode index No of the fundamental
mode of the structure the most while N1 is affected very little. At the input and output of
the device, for large gaps, both No and N1 are nearly equal to the mode indices of the
uncoupled guides. As the gap decreases, the guides become strongly coupled, the
perturbation becomes quite strong, which reaches a maximum at the midpoint of the
coupler. No increases monotonically with decreasing gap width where as N1 undulates
although the deviations appear to be rather small. While N1 at the input end is very close
2.24
g = 2.m
N 3pm
0 4,5,6pm
2.239
2pm
o 3g3m
S2.238     4,5,6pm
2.237 L I I
2 3 4 5 6
Width of the Tapered Waveguide (gLm)
Fig. 3.8 Local normal mode indices (solid curves) along the TVC for different
gaps and mode indices (dashed curves) for the uncoupled channel
waveguides of the TVC.
to that of the tapered guide, independent of the gap width, No is higher for smaller values
of the gap width. It is clear, by increasing the length of the input section further, it is
possible to make No approach closer to that of the straight guide; but our aim is to keep
the device length as small as possible, while permitting nearly sole excitation of
waveguides is possible at either end, and simultaneously allowing power transfer via the
adiabatic condition. It is interesting to note that the value of N1 is independent of the gap
width for two specific tapered guide widths, around 2.8 gpm and 4.6 gim. It appears that
the bunching of the curves is merely a coincidence, since Ni's at both the extremes viz.,
zero and infinite gap, are almost equal (Fig. 3.4). For example, NO of a 4 gim guide as
seen from Fig. 3.4, equals N1 for a TVC consisting of 4 and 4.6 pim wide guides with an
infinite gap (Fig. 3.8). In addition, as seen from Fig. 3.4, it almost equals N1 of an 8.6
pm wide, that is the case of a TVC with zero gap between the two guides. Thus N1, for
all the values of the gap in between is nearly equal to each other.
3.3 Step Transition Model: Power Flow in the TVC
Now that we have, based on the one dimensional effective index profile N(y), an
accurate description of the local normal modes indices No,1(=Po,1/k) and their field
profiles Eyo,yl(y) of a uniform directional coupler of constant guide widths (one fixed and
another variable) and a constant gap, the step transition model [Mar70] and the enhanced
step transition model [Mi75] can now be applied to calculate the power transfer between
local normal modes. This model approximates the gradual, continuous increase in the
width of the tapered guide by the stair case structure consisting of a series of small,
piecewise continuous but abrupt steps (Fig. 3.9 (a)). While coupling between local
normal modes between the sections occur at each step discontinuity, no coupling is
assumed to occur within a given section. We assume two local normal modes (Fig. 3.9
(b)) on either side of such a step discontinuity in the sections labeled 0 and 1. The
fundamental and the first order modes, illustrated on either side, are also designated by 0
and 1 respectively. The first digit of the mode nomenclature identifies the mode while
0 side
U 9~
4  I.
Overlap integral :
I, = edy y, = 00, 10, 01, or 11
Fig. 3.9 Illustrations for step transition model. (a) Stair case approximation of
the TVC structure. (b) Normal mode field profiles in the two sections
at the step discontinuity and the overlap integral across the
discontinuity.
1 side
the second, represents the section. Thus, for example, 10 would signify the first order
mode on side 0. Although the step discontinuity is rather small, for the purposes of
illustration, the mode shapes have been exaggerated in Fig. 3.9 (b). We summarize the
results below following the treatment in Ref.[Mi75].
we write the general expression for the guided normal mode at any point z along
the coupler is
Ey = Ak(z) Eyk(y) exp(ia(z)) (3.7)
where
Ak(z)= Field amplitude (real) of the guided mode k = 0 or 1
Eyk(y)= Normalized field distribution of the guided mode k = 0 or 1
a(z) = pz+ (
p = Propagation constant
= Arbitrary phase constant
By considering the continuity of the transverse field components hxk(y) and Eyk(y) across
the discontinuity, we can calculate the transmitted field amplitude. By normalizing the
local normal mode amplitudes for unit power, the overlap integral between the mode
fields is then,
I = f edy y,8 = 00, 10, 01, or 11 (3.8)
It follows, the complex transmission coefficient for the jth mode (j = 0 or 1) across the
discontinuity is given by [Mi75]
Ajl exp(ajl) = Cij AiO exp(aio) + CjjAjo exp(ajo)
(3.9)
where Cij is the coupling coefficient between the ith order mode on the incident (0) side
to the jth order on the transmitted side and Cjj is the coupling coefficient between the jth
modes on either side. Therefore, the real and imaginary parts of Eq.2.9 represent the
amplitude and the phase transmission coefficient Aji and ajl and describe the transmitted
mode j (0 or 1) in terms of the input modes.
Ajl = Cij Aio cos(aioajl) + CjjAjo cos(ajoajl) (3.10a)
and
Cij Aio sin(ao) + Cjj Ajo sin(ajo)
tan ail= Cij Aio cos(aio) + Cjj Ajo cos(ajo) (3.1b)
where the coupling coefficient between the transmitted jth mode and the incident ith
mode is given by
Ci 2 ij PjO+l jOj (3.10c)
= jo+Pj1 Pio+Pil JIoi0ijiji
In Eq. 3.10c, the coupling coefficient Cjj is obtained by substituting j for i. Equations
3.10a3.10c and their counterparts for the ith mode describe the mode conversion and the
radiation loss in the tapered waveguide structure approximated by a piecewise continuous
staircase structure. Cij (and Cjj) in Eq. 3.10c can easily be evaluated with the knowledge
of py's. Equations 3.10a and 3.10b can be numerically solved by iterative means for both
aji and Ajl. Since we are interested in the transmitted amplitude on side 1, we will drop
the subscript 1 and use Aj, with j = 0 and 1, to represent the amplitudes of the
fundamental and the first order mode respectively.
3.4 Theoretical and Experimental Results
Since we excite only the fundamental mode at the input z=0, the initial conditions
for the mode amplitudes are A00(z=0)=Ao=1 and Ao1(z=0)=AI=0 corresponding to the
normalized power in the modes P0(0)=1 and PI(0)=0. The 2 to 6 Rim taper was
approximated by one hundred staircase steps of size 0.04 im. The distance between
consecutive steps is 0.048 mm and the resulting taper angle is therefore 0.0480. Figure
3.10 shows the mode amplitudes A0 and A1 for various gaps, as the input beam
propagates along the coupler. (A0)2+(A1)2, which represents the guided mode power, is
almost equal to unity for all the five cases so that the calculated radiation loss is to be less
than 1% of the input power. If the adiabatic regime is defined as less than 5% conversion
of input mode power into other modes, the TVCs with g = 2, 3, 4 gim can be classified as
meeting the adiabatic invariance condition. As seen from Eq. 1.1, the mode conversion
becomes severe with increasing gap width, that is, as the coupling becomes weak.
Oscillation of A0,1 within the adiabatic regime where (A0)2+(A1)2 = 1, can be clearly
seen in Al, although it is not so obvious in A0, for g =2 or 3 plm. It is caused by
interference between the two normal modes [Mi75] and has a period of 2i/(0oP 1). From
A0 and Al at the output with phase constant ao and al, we obtain the output intensity
profile as
Iout(y) = AoEoeiao + A leleial I
= A'e~(y) + A'ec(y) + 2AoA leo(y)El(y) cos(azoal) (3.11)
where Ai is the amplitude ratio at the output, ai is the phase constant at the output and ei
is the normalized field profile for the local ith (i=0,1) mode at the output.
Output intensity profiles obtained from Eq. 3.11 are shown in Figs. 3.11 (a)(c)
with solid lines. Locations of the guides and their widths at the output are illustrated by
the rectangular boxes under the horizontal axis. The degree of darkness of the
rectangular box once again indicates the amount of index change as illustrated before in
Fig. 3.6. The output intensity profiles in Figs. 3.11 (a) and (b) are almost identical with
that of the fundamental local normal mode at the output. The output intensity profile for
1 ... .
A0   2,3,4m
A
0.8 A
~ A1 = 6gm
0.6
S0.4 5m
0.2
4gm
3pm
0 12.m
2 3 4 5 6
WIDTH OF THE TAPERED WAVEGUIDE
Fig. 3.10 Amplitude ratios of the fundamental and first order local normal mode
along the TVC as a function of the width of the tapered channel for
various gaps.
1.2 . . I . . I .. .. .
S Graded index g = 2pm
1  Step index
0 Experiment
0.8
S 0.6
0.4
o /
0
0.2 / \
10 5 0 5 10 15 20
y(tm)
Fig. 3.11 Normalized output intensity profiles calculated using the graded index
model (solid curve) and 5layer step index model (dashed curve)
compared with the measured intensity profile (open square) for (a) g =
2 pm.
1.2 . . I . . 1
Graded index
1  Step index
S Experiment
5 0.8
1 0.6
0.4 
o0
0.2 /
/
0 0.
10 5 0 5
y(ptm)
10 15 20
Fig 3.11continued. (b) g = 3 urm.
1  Step index
S Experiment
0.8
0.6 i
0.4 /
0.2 7
10 5 0 5
y(pm)
Fig. 3.11continued. (c) g = 5 pm.
15 20
g = 2 Rpm shows large mode width as well as asymmetry due to the strong coupling. In
addition, the peak position does not coincide with the center of the tapered guide but
shifted to the straight guide so that the representation of the local normal modes as the
superposition of the modes of the uncoupled guides [Mi75] is inappropriate for this and
other strongly coupled case. As we move our attention to the weakly coupled cases,
namely, the large gap TVC, we observe that the amount of the power within the straight
guide reduces, until the adiabatic condition breaks down. For the case of g = 5, small
lobe on the straight guide due to mode conversion can be recognized. With the 3 cases
considered here, although A3rAZT (= 51.36) is constant, we are able to achieve both the
adiabatic and nonadiabatic operation, by varying the gap and hence the coupling
coefficient. Equations (1.1)(1.3) are not applicable for describing the behavior of our
couplers, which employs both the tapered index and tapered dimension. For a given gap,
an order of magnitude difference in coupling constant (K) exists along its length and thus
violating the constant coupling constant assumption under which conditions Eqs. (1.1)
(1.3) were derived.
To verify our theoretical predictions, we fabricated several tapered velocity
couplers shown in Fig. 3.5 with the same fabrication parameters and procedures as we did
for 4 Rim wide strip channel waveguide. The measured output intensity profiles for each
of the coupler are presented in Figs. 3.11 (a)(c). Excellent agreements are obtained for
all three cases, which are g = 2, 3, and 5 Rim. As expected, there is no mode conversion
for strongly coupled cases, viz. g = 2 and 3 jim so that the theoretical output intensity
profiles of the fundamental mode in these two cases agree very well with the measured
intensity profiles. In Fig. 3.11 (a), slight deviation of the experimental results from the
theoretical predictions based on graded index model especially within the straight guide
and gap regions, is due to the fact that approximated composite index profile as the
superposition of the individual index profiles yields larger than actual index values in
these regions for the couplers with small gap. With g = 5 jim, we can observe the mode
power conversion due to the weak coupling, and also the excellent agreements even in the
straight and gap regions by which our approximated composite index profile is proven to
be very accurate for couplers of large gap. The extinction ratios, which is defined as
10log(Po/P1) where Pi is the power of the ith local normal mode at the output, are 17.0
dB, 15.2 dB, and 7.7 dB for g = 2, 3, and 5 pm, respectively. The insertion losses for
these three devices were measured to be less than 3.5 dB.
For the purpose of illustration, we compare the numerical results using the
popular [Sy89] equivalent index slab model. In this model, the TVC is approximated as a
step index 5layer structure with a fixed refractive index (ng) for the guiding layers, one
of which is tapered with thickness increasing from 2 p.m to 6 pm. The other guiding
player is a 4 p.m thick straight guiding layer separated by constant gap from the tapered
layer. Refractive index (ncl) for the cladding layers is assumed to equal no (=2.2195),
which is the same as the bulk substrate index (nb) in the graded index model. To
determine ng, we follow the procedure successfully used by Suchosky and Ramaswamy
[Su87] for modeling a constant width, variable index tapered waveguide in Ti:LiNbO3,
by determining a equivalent step index slab waveguideby comparing the field profiles.
First we assume a hypothetical 3layer slab waveguide with a 4 pim thick guiding layer
with the cladding layers of index no (=2.2195). Then we calculate the intensity profiles
of the fundamental mode for various refractive indices for the guiding layer. The
calculated intensity profiles are compared to that of the graded index channel waveguide
of 4 pgm wide in Fig. 3.3. Over 99% overlap was achieved for a guiding layer index of
2.2215 which we use as the guiding layer index ng of the 5layer step index model. Upon
finding ng, to determine the output intensity profile, we follow the same procedure as we
did with the graded index profile (N(y)). This involves evaluating 1) the local normal
mode indices and field profiles, 2) overlap integrals between the local normal modes
across the steps, 3) the Ai's and ai's along the coupler with the input condition A0 = 1
and A1 = 0, and finally, 4) the output intensity profiles. The results are shown in Figs.
61
3.11 (a)(c) as dashed lines. Analytical results of the step index model are indeed quite
different from that of the graded index model especially in describing the intensity profile
over the straight guide region of the coupler. The experimental results clearly
demonstrates the inappropriateness of the step index model.
CHAPTER FOUR
TAPERED, BOTH IN DIMENSION AND IN INDEX
VELOCITY COUPLER SWITCH
Tapered Velocity Couplers (TVC) are attractive candidates for optical signal
processing applications, as their behavior is predicated upon the evolution of a normal
mode along the longitudinal direction of the coupler. As a result, the tapered velocity
couplers exhibit polarization independent behavior and they are insensitive to wavelength
within the limits imposed by the adiabatic condition. On the contrary, interferometric
devices like conventional directional coupler devices, MachZehnder interferometers, and
two mode interference devices (BOA), depend on the precise phase relationships between
the interfering modes. Consequently, these interferometric devices have to meet strict
fabrication tolerances.
A number of switches using the modal evolution, known as digital optical
switches with step like response rather than the conventional sine squared response of the
interferometric counterpart, have been demonstrated. The first and the foremost was by
Silberberg [Si87], where the switching was accomplished at the output of the symmetrical
arms of an intersecting, 2x2, twomode cross coupler in zcut LiNbO3 which was made
asymmetrical by the application of an external d.c. bias. In this case, the voltage length
product was 135 Vcm for TM mode with a cross talk of 15 dB. Another cross coupler
digital switch on xcut LiNbO3 with a better voltage length product (114 Vcm with a
cross talk of 17 dB) was demonstrated later [Mc91]. Ybranch digital switches on
LiNbO3 (a voltage length product of 68.4 Vcm with 14 dB) [Gr90, Thy89], in
InP/GaInAsP [Cav91, Vi92], and shaped Ybranch digital switches (60 Vcm with 15 dB)
on LiNbO3 have also been demonstrated.
Recently, Xie et al. [Xi92] reported a 2x2 digital optical switch in InGaAsP/InP
using both the tapered dimension and tapered coupling between step index, channel
waveguides of the TVC. This structure was analytically examined previously [Sy89] for
the case of LiNbO3. However, this device shows poor extinction ratio.
In this chapter, we propose and demonstrate a novel digital switch using tapered
velocity coupler in Ti:LiNbO3, taking an advantage of short operating length of the TVC.
The schematic of our switch is shown in Fig. 4.1. A straight channel (5 p.m wide) is
separated by a uniform gap (4pm) from the tapered channel whose width increases from
3 pim to 5 jim, with the width of the straight channel remaining the same, namely, 5 ipm
[Ra93]. Application of positive and negative voltages between the electrodes switches
the states at the output. We achieved 15 dB extinction ratio with 50 V swing for TE
mode and 15 V swing for TM mode in a 2.4 mm long device length, yielding the
smallest voltage length products, reported to date, of 24 Vcm for TE and 7.2 Vcm for TM
mode, respectively
4.1 Principles of Operation
To understand the switching mechanism, consider the following: Without any
applied voltage, the propagation constants of both channels are identical at the output,
that is, the phase matching condition occurs at the output (solid line in Fig. 4.2); light
launched into the straight channel at the input should be equally divided between the
channels at the output if the adiabatic condition is ensured. As we increase the voltage so
as to decrease the refractive index of the straight channel and increase that of the tapered
channel, the phase matching point would shift towards the central region of the switch
(dashed lines in Fig. 4.2). Then the power launched into the straight channel should be
transferred to the tapered channel in the central region, close to the phase matching point
and exit out of the tapered channel at the output. On the other hand, with increased
reverse bias, phase matching does not occur along the entire device length (dotted lines in
Fig. 4.2) and the power would remain in the straight channel.
Wti=3 gm
Wto = Ws = 5 m
L = 2400 pm
G=4gm
Fig. 4.1 A schematic of the tapered velocity digital optical switch
The hatched area illustrates the electrode structure.
65
P
unbiased 
+V biased 
V biased .
U 00
0 0
 
0 
=
c I 
Propagation Length
Fig. 4.2 Propagation constants of both the straight and tapered channel at
three bias voltages.
Operation of this device can also be understood from another point of view, using
the normal mode theory. The light launched through the straight channel is coupled into
the fundamental local normal mode at the input (z=0) and propagates without mode
conversion along the device. Although mode evolves along the TVC, the mode remains
as the fundamental mode of the composite structure as long as the adiabatic condition is
satisfied; however the output intensity profile can be switched the output between the
tapered and straight channel waveguide by the application of bias voltage of appropriate
polarity. Without the bias voltage, the fundamental local normal mode at the output has a
symmetric profile so that the power on each channel waveguide is same. With forward
bias, refractive index of the tapered channel increases. Energy of the fundamental local
normal mode at the output becomes concentrated in the tapered waveguide with increased
forward bias voltage. Similarly, it follows that the power of the output intensity profile
tends to concentrate in the straight channel with the reverse bias voltage.
4.2 Theory and Experiments
We believe that the 5layer step index approximation [Si87, Sy89] with a constant
equivalent index guiding layer regardless of the variation of the thickness of the taper is
not appropriate for analysis of gradedindex Ti:LiNbO3 channel waveguide tapered
velocity coupler [Ki93a]. Primary reason is that the width of the tapered Tistrip before
its diffusion governs not only the physical dimension of the guiding region but also the
absolute index change resulting from the Ti indiffusion [Fu78]. In addition, strong
coupling between the channels does not allow the use of the conventional coupled mode
theory for the description of the physical power transfer between the waveguides [Ki93a].
For the same reason, the superposition of the individual modes [Mi75] for an
approximated expression for the local normal modes is not adequate on account of the
strong coupling in the power transfer region dictated by the adiabatic condition.
For the theoretical analysis, we assume the following fabrication parameters:
800A of Ti diffused at 1025 oC for 6 hours. The coupler is divided into large number of
segments along its length by approximating the taper by a piecewise continuous staircase
structure. Each segment consists of two straight channels; the width of one channel is 4
plm and the width of the other is determined by that of the tapered channel at that point.
A 2D index profile for a cross section of a segment is derived from the superposition of
the index profiles of the two straight channels, resulting in the Gaussian profile along the
depth direction. To convert the 2D index profile into a 1D index profile, the graded
index Gaussian profile along the depth direction at a given discrete position along the
width is approximated by a staircase stepindex profile. For the approximated stepindex
profile at each position, we can obtain the effective index 1D profile, in the conventional
manner, using multilayer stack theory [Hu90, Th87]. Once again, we apply the multi
layer stack theory to the resultant 1D effective index profile to obtain the mode indices
and field profiles of the first two normal modes at a given segment along the propagation
direction. Following the standard procedure of the step transition model [Mi75], we can
evaluate mode power conversion along the coupler by calculating overlap integral across
the abrupt step between the two adjacent segments. We found that no mode conversion
occurs along the taper, thus satisfying the adiabatic condition. The refractive index
change due to the bias voltage was also taken into account in our calculation. Detailed
theoretical analysis of the gradedindex channel waveguide TVC is presented in chapter
three [Ki93a], with substantiating the previously reported experimental results [Ki89a],
demonstrating, for the first time, complete power transfer between channel waveguides of
the TVC.
Evolution of the calculated intensity profile of the fundamental normal mode in
the substrate plane of the switch is shown in Fig. 4.3 for three biased states. In all the
three cases the adiabatic condition is satisfied. Therefore, the fundamental local normal
mode coupled at the input does not suffer any mode conversion and evolves along as the
fundamental mode of the structure with its mode power unchanged and switching is
accomplished with the forward and reverse bias.
 V biased
unbiased
+V biased
Fig. 4.3 Modal evolutions for three bias voltage.
The switch was fabricated on z cut LiNbO3 using the same fabrication conditions
assumed in the theoretical analysis. 2000 A of SiO2 buffer layer was sputtered over the
device. 2.4 mm long aluminum electrodes were delineated over the Tidiffused TVC
waveguides by photolithography and wet etching technique. The switching behavior was
investigated by characterizing the output intensity profile in the substrate plane at each
bias voltage. Figure 4.4 (a) shows the output intensity profiles at X=1.32 pm for the three
cases for TE mode which uses the rl3 coefficient. In unbiased case, almost equal power
division were achieved. With 50V swing, we were able to observe digital switching
with 15 dB cross talk. For TM mode which uses r33 coefficient (almost three times
bigger than r13), we were also able to observe the digital switching in 15 V swing with
better than 15 dB cross talk (Fig. 4.4 (b)). This translates to a voltage length product of
7.2 Vcm for TM and 24 Vcm for TE mode. This is the shortest voltage length product
for a digital switch reported so far. This has been accomplished by the use of tapered,
both in index and in index, velocity coupler [Ki93a] where weak coupling both at input
and output of the TVC due to increased index difference assures concentration of energy
in one of the waveguides at the ends while strong coupling in the center region enables
the adiabatic condition to be achieved for secured operation of the switch.
4.3 Summary
In summary, we have presented a novel digital optical switch using tapered, both
in dimension and in index, velocity coupler, which shows smallest voltage length product
among those of the reported digital optical switches. Numerical results for the graded
index channel waveguide coupler has been introduced, which can be applied to all z
variant structure with the 2D gradedindex profile.
Fig. 4.4 Output intensity profiles for (a) TE and (b) TM
polarization.
The black areas illustrates the waveguide regions.
CHAPTER FIVE
TAPERED VELOCITY COUPLER
USING SEGMENTED WAVEGUIDES
Recently, segmented waveguides consisting of a periodic array of high refractive
index regions surrounded by lower index regions (see Fig. 5.1 (a)), have received
considerable attention for applications in efficient second harmonic generation in KTP. In
the case of LiNbO3 [Ca92a, Ca92b, Li89] and LiTaO3 [Bi90, La93, Ma93, Va90], the
segmentation acts as a grating, orthogonal to the direction of the guided wave propagating
along another waveguide, such as proton exchanged waveguides (Fig. 5.1 (b)). Such
segmented waveguides in Fig. 5.1 (a) act as gratings for achieving quasi phase matching in
nonlinear interactions since the periodic segmentation also leads to periodic domain
reversal, i. e., reversal of optic axis.
Besides as grating structures, segmented waveguides by themselves are also
interesting since the effective index of the propagating mode can be controlled by simply
varying the duty cycle of the segmentation. The ability to control the effective index of the
mode by changing the duty cycle of the segmented waveguide can be used for the efficient
design of zvariant waveguide devices such as mode expanders, polarization converters,
wavelength filters, tapered velocity couplers, etc.
In Ti:LiNbO3, tapering of the Ti strip width before its diffusion yields tapering in
index as well as in dimension after diffusion due to side diffusion. A tapered, both in
dimension and in index, velocity coupler in Ti:LiNbO3 can be obtained directly by the
single diffusion of Ti. But ionexchanged waveguides in glass [Ra88] and proton
exchanged waveguides [Go89] in LiNbO3 give a fixed surface refractive index for specific
fabrication parameters (e.g. exchange time and temperature, annealing time and
segmented waveguide
. ...V
domain inversion
yi\~
waveguide
Fig. 5.1 Top view of segmented waveguides (a) in KTP and (b) in LiNbO3 with
gratings orthogonal to the propagating wave.
temperature) regardless of channel widths. To achieve a taper, annealing has been used
which requires gradual variation of annealing temperature along the propagation direction
[Le90]. However, one can get tapered, both in index and in dimension, channel
waveguides even in these channel waveguides by using segmented waveguides. It is also
known that index tapering in a semiconductor channel waveguide needs elaborate
fabrication techniques, such as selective growth on patterned substrates [Ka92] or on
shadow masked substrate [Co92]. But segmented waveguides make it possible to fabricate
tapered, both in index and in dimension, channel waveguides and couplers in
semiconductor without elaborate processing steps.
Segmented waveguides in KTP with step index segments and vertical walls have
been analyzed using a lamellar grating analysis; it was shown that a step index segmented
waveguide can be represented by an equivalent uniform step index planar waveguide with a
film index equal to the weighted average of the high and low index values [Li92].
Segmented waveguides have also been analyzed using the BPM method [We92, We93]
and the coupled mode theory [We93]. Very recently, reflection and transmission
characteristics and mode field profiles as well as second harmonic generation involving
QPM interaction in KTP segmented waveguides have been measured and modeled with a
linearly graded refractive index variation [Eg93]. Although the validity of the proposed
model as applied to KTP has been demonstrated by the above comparison, to date no direct
measurement of effective indices of the various modes and its variation with the duty cycle
and period of segmentation have been reported in the literature.
In this chapter we present experimental and theoretical results on the propagation
characteristics of planar proton exchanged periodically segmented (PEPS) waveguides in
lithium niobate. Prism coupling measurements of the variation of the effective index of the
modes of the waveguide as a function of the duty cycle and period of segmentation for
different annealing times are presented. Since the PEPS waveguide is a graded index
segmented waveguide, it is assumed that they can be modeled by an equivalent zinvariant
graded index waveguide. Using the measured effective index values, we show that the
PEPS waveguide can be represented by an equivalent graded index planar waveguide with
a Gaussian refractive index distribution with the peak index change varying almost linearly
with duty cycle for small duty cycles and saturating at large duty cycles, with its depth
independent of the duty cycle. Results obtained using our model are in excellent agreement
with the measured values for single as well as multimode PEPS waveguides [Th94].
Proton exchange waveguides exhibit a graded index profile where as compound
semiconductor waveguides exhibit a step index profile. Segmentation can be used to vary
the effective index. Variation in propagation loss and mode size with respect to duty cycles
in GaAs/AlGaAs is presented. Tapered, both in dimension and in index, velocity couplers
in GaAs/AlGaAs will be proposed in this chapter using tapered segmented waveguides.
5.1 Proton Exchanged Periodically Segmented
Waveguides in LiNbO3
A 50 nm tantalum mask for a PEPS planar waveguide with a period of 10 gpm and a
duty cycle varying from 0.15 to 0.9 was patterned on a pair of Z+ cut lithium niobate
substrates using standard photolithographic techniques. In parallel, in another Z+ lithium
niobate substrate, similar mask for a planar PEPS waveguide with a period of 10 gim with a
constant duty cycle of 0.5 was also patterned. Proton exchange of all the substrates was
then carried out in pyrophosphoric acid at a temperature (Te) of 200 C for a period (te) of
1 hour. The waveguides were subsequently annealed for three different annealing times (ta
= 3, 5, and 8 hours) at a temperature (Ta) of 300 'C (see Fig. 5.2). Recent studies have
demonstrated that the effect of the ambient conditions during annealing on the propagation
characteristics of the waveguide is relatively insignificant [Lo92].
Since proton exchange creates a high index region close to the surface, the annealed
proton exchanged segmented waveguide sample has a refractive index grating with varying
duty cycle on its surface. With a laser beam incident on the surface of the substrate we
could observe various diffracted orders in the reflected beam. Measurements of the angles
Proton exchanged
region for guiding
Z+ LiNb03
Te =200 C
te = 1 hour
Ta = 300 C
ta = 3, 5, and 8 hours
L = length of segmentation
A = period
L
Duty Cycle A
(01.0)
Fig. 5.2 A schematic of proton exchanged periodically segmented waveguides in LiNbO3
with fabrication conditions.
Propa
direct
nation
ion
V
of diffraction of the various orders were consistent with the period of 10 pm of the periodic
grating structure. The intensity of diffraction was also observed to vary with the duty cycle
of the grating. Such measurements could be used to estimate the surface index change by
measuring the intensity of the various orders for different polarization directions.
Prism coupling measurements with a HeNe laser operating at 632.8 nm were
carried out for different duty cycles after each annealing. The sample exhibited very strong
scattering up to 2 hours of annealing and no mode measurements were possible for these
annealing times. Indeed for short annealing times we have also observed strong diffraction
of the light incident in the prism into +1 and 1 orders. By properly choosing the angle of
incidence (and hence the phase vector) in the prism and the grating vector (= 2m7t/A, A
being the period of the segmentation ) of the segmented waveguide we have observed that
we can indeed excite any individual waveguide mode with any of the diffracted orders.
The scattering from the waveguides reduced with increased annealing times due to
reduced index change in the diffused regions. Mode angle measurements for various
modes and for different duty cycles were carried out using a prism coupling arrangement
for annealing times greater than 3 hours. Open circles in Figs. 5.3 (a), (b), and (c) show
the measured variation of the effective index of the propagating modes of the waveguide as
a function of duty cycle for ta = 3 5, and 8 hours.
5.2 Modeling of Proton Exchanged Periodically
Segmented Waveguides in LiNbO3
Annealed proton exchange waveguides have a graded refractive index profile and
various analytical models describing the profile are already available [Ca92b, Go89, Ni91,
Vo89, Za93]. These include complimentary error function profile [Vo89], hyperbolic
tangent profile function [Ca92b], Gaussian profile [Go89, Za93] and a generalized
Gaussian function [Ni91]. All of the above mentioned profiles have very similar behavior
and except for the hyperbolic tangent profile need numerical methods for
2.26
2.25
2.24
2.23
2.22
2.21
2.2
0.2
0.4
0.6
0.8
Duty Cycle
(a)
Fig. 5.3 Variation of mode indices of PEPS waveguide with respect to duty cycle
for (a) 3hour annealed sample (Circles represent measured values and
solid curves correspond to calculated variation of an equivalent zinvariant
graded index waveguide with Gaussian index distribution),
I I I I
STa=3000C
ta=3hrs.
 dx=1.Om TMo
TTM
IIM,
TM1
0.2 0.4 0.6 0.8
Duty Cycle
(b)
(b) 5hour annealed sample, and
I I I I
Ta=3000C
ta=5hrs.
dx=1.2pu m
2.26
2.25
2.24
2.23
2.22
2.21
2.2
Fig. 5.3continued
2.26 I a
Ta=300C
2.25 ta=8hrs.
2.24 dx= 1.4pm
" 2.23
S2.22
2.21 
2.2
0.2 0.4 0.6
Duty Cycle
(c)
TM2 .
0.8
Fig. 5.3continued. (c) 8hour annealed sample.
 I
estimating the mode indices. Our fabrication (Te = 200 OC and te = 1 hour) and annealing
(Ta = 300 oC and ta = 3 to 10 hours) conditions correspond to thin waveguides, which can
be approximated very well by a Gaussian refractive index profile [Go89, Za93]. Thus we
model the planar PEPS waveguide by an equivalent zinvariant graded index waveguide
with the following Gaussian refractive index distribution:
n(x) = ns + An exp( x2 / dx2) (5.1)
where ns is the substrate index, An is the peak index change and dx is the diffusion depth.
For duty cycles and annealing times for which the waveguide supports two modes, both
An and dx are uniquely determined. Fitting to the measured effective indices was
performed by a numerical evaluation of the mode indices of the modeled waveguide. While
fitting with the measured effective indices at different duty cycles, we found, as expected,
that the depth of the equivalent waveguide is independent of the duty cycle (within
experimental errors of the measured effective indices) while the peak index change
increases almost linearly with the duty cycle for low duty cycles. Hence the same depth
was assumed even for the region where the waveguide supports just a single mode. Our
findings regarding the depth independence of the equivalent waveguide with the duty cycle
are also consistent with the equivalent waveguide model of step index segmented
waveguide [Li92] and equivalent waveguide models used to represent tapered diffused
waveguides [Su87].
In Figs. 5.3 (a)(c) we have also plotted the calculated effective mode index (solid
curves) of the fitted equivalent graded index planar waveguide with equivalent thicknesses
(dx) of 1.0 pim, 1.2 p.m and 1.4 pim corresponding to ta of 3, 5, and 8 hours, respectively.
As can be seen the agreement between the measured and fitted effective indices is very
good. Figure 5.4 shows the corresponding variation of surface index change of the
equivalent graded index waveguide with the duty cycle. As the duty cycle increases, the
surface index change of the equivalent waveguide increases almost linearly with the duty
cycle (for small duty cycles) and as should be expected, tends to saturate at large duty
cycles. In addition, as the annealing time increases, the depth of the equivalent waveguide
also increases. Our estimated surface index change and the depth of the equivalent
waveguide are consistent with those of nonsegmented annealed proton exchange
waveguides as obtained in recent research [Ca92b, Go89, Pu93].
To determine the dependence of the effective index of the modes of the segmented
waveguide on the period of segmentation, we made prism coupling measurements on the
PEPS waveguide sample with a period of 10 plm and a constant duty cycle of 0.5. By
changing the angle between the direction of propagation of the guided light and the
direction of segmentation, we could generate different periods of segmentation without
changing the duty cycle. Figure 5.5 corresponds to a typical measured variation for TMO
and TM1 modes for 5 hours of annealing and show that the effective index is independent
of the period of segmentation. These experimental results confirm that the effective index
of the propagating mode is independent of the period of segmentation and depends only on
the duty cycle [Li92].
Figures 5.3 and 5.4 provide conclusive evidence that the PEPS waveguide can be
represented by an equivalent zinvariant graded index planar waveguide with a Gaussian
refractive index profile with the peak An value increasing almost linearly at low duty cycles
and saturating as the duty cycle approaches unity while the corresponding diffusion depth
is independent of duty cycle for a given annealing condition (i.e., given Ta and ta).
In summary, we have presented the characterization of proton exchanged periodic
segmented waveguide in lithium niobate. Variation of effective indices for different modes
for different annealing times as a function of duty cycle and period of segmentation have
been presented. It is shown that the PEPS waveguide can be modeled by a zinvariant
graded index waveguide with a Gaussian refractive index distribution.
0.1
0.08
, 0.06
0.04
0.02
0
0.2
0.4
0.6
0.8
Duty Cycle
Fig. 5.4 Variation of An at the surface of the equivalent graded index waveguide
with duty cycle
I t I3hrs. d=l.Op
_ ta=3hrs.
  t=5hrs" dxlhO
 ta=8hrs. m_
a
^ ^ ^ .. . .......
. . 
I I I I
2.26
2.25
S2.24
S2.23
2.22
2.21
2.2
9 10 11 12
Period
13 14 15 16
(pmn)
Fig. 5.5 Measured variation of the effective indices of the TMO and TM1 modes
with period of segmentation of a PEPS waveguide with a duty cycle of
0.5
I I I I I I
T =300C
a
 t =5hrs.
a
TM
O O O O O 0
TM1
0
I I I I I I I 
5.3 Periodically Segmented Waveguides
in AlGaAs/GaAs and their Application to
Tapered Velocity Couplers
In PEPS planar waveguides in LiNbO3 shown in Fig. 5.2, the segmented
waveguides consist of discontinuous proton exchanged guiding regions along the beam
propagation direction. Segmented waveguides in AlGaAs/GaAs have been realized by the
use of a continuous guiding layer but with a segmented ridge, cladding region. As shown
in Fig. 5.6 (a), GaAs guiding layer is sandwiched by two Al0.15Ga0.85As cladding layers.
The upper cladding, however, is a segmented ridge structure along the beam propagation
direction, giving rise to effects of segmentation as well as horizontal confinement of the
mode. Figure 5.6 (b) and (c) illustrate the front and side view of the final segmented
waveguide structure, respectively.
To fabricate the segmented waveguide structure, however, we require two etching
steps. A ridge waveguide was defined by standard photolithography and wet etching
technique as shown in Fig. 5.7 (a). Without removing photoresist (PR) on the ridge, the
sample was exposed to UV light to delineate the segmented PR patterns on the ridge using
the appropriate mask pattern. After removing the exposed part of the remaining PR on the
ridge, the sample was dipped into etching solution again. The second etching yielded not
only segmentation of the ridge but also further etching outside the ridge as shown in Fig.
5.7 (b). About 0.7 and 0.5 lpm were etched out by the first and second etching process,
respectively. Diluted phosphoric acid (1H3P04:1H202:10H20) was used for the etching
solution, which provides an equal etching rate for both GaAs and A10.15Ga0.85As layers,
viz., 0.4 p.m/min at room temperature.
As shown in Fig. 5.6 (a), final structure of the segmented waveguide is a ridge
GaAs channel waveguide with segmented Al0.15Ga0.85As strip which gives high and low
mode index regions along the GaAs ridge according to the thickness of the strip. Front and
side views of the segmented waveguide are shown in Fig. 5.6 (b) and (c), respectively. It
should be noted that there are two key parameters which must be optimized. These are
GaAs substrate
(a)
Periodically segmented waveguide in A1GaAs/GaAs (a) Schematic of the
final structure.
Fig. 5.6
AklsGao.8 s
(b)
L
Duty Cycle =L
A
(c)
Fig. 5.6continued (b) Front and (c) Side view of the segmented waveguide.
0.5 im
0.8 Rm
A.15Gao.85As
Photoresist
/\
GaAs substrate
(a)
Photoresist
GaAs substrate
(b)
Illustration for two step etching process.
Sample after (a) the first etching and (b) the second etching.
0.7
Fig. 5.7
GaAs thickness (tl) outside the ridge and Al0.15Ga0.85As thickness (t2) in the low mode
index region. First ti needs to be smaller than 0.9 pgm so as to obtain well confined,
channel waveguide modes by suppressing guided mode outside the ridge. Second, t2 must
be larger than 0.4 pgm so that the segmented waveguides do not suffer large propagation
loss due to large difference in mode size between high and low mode index region.
AlGaAs/GaAs segmented waveguides with different duty cycles illustrated in Fig.
5.6 were characterized by measuring the propagation loss and near field intensity profiles.
Propagation loss was measured by the cut back method and the results are shown in Fig.
5.8. Propagation loss of segmented waveguides with high and low duty cycles is less than
that of 0.5 duty cycle segmented waveguides [Li92, We93]. This is to be expected since
segmented waveguides with duty cycle of 0 and 1 correspond to straight channel, GaAs
ridge waveguides with 0.5 and 1 ipm thick AlGaAs strip loading, respectively, and as such
these should have the smallest propagation loss compared to segmented waveguide with a
duty cycle of 0.5.
Variation in mode size with respect to duty cycle was obtained by near field
intensity measurements. No significant variation in modal depth with respect to duty cycle
was observed in Fig 5.9 (a). This is understandable because the effective index in the
central region is hardly affected by varying the height of cladding from 0.5 to 1.0 p.m since
the effect due to decreased asymmetry is negligible. On the other hand, noticeable variation
in modal width with respect to duty cycle was obtained, as presented in Fig. 5.9 (b). As
can be seen in Fig. 5.9 (b), minimum modal width occurs at duty cycle of 0.5. Using the
effective index method, modal width can be calculated by considering a hypothetical
symmetric threelayer waveguide whose effective indices for guiding and cladding layers
are those of ridge and outside ridge region, respectively. The hypothetical waveguide
certainly has minimum modal width with changing effective index of guiding layer [Ko74].
We believe that change in duty cycle of the segmented waveguides results in change in
89
5 I II
i 
0
0
3 
o o \
& 1 i I I lt)
0 0.2 0.4 0.6 0.8 1
Duty Cycle
Fig. 5.8 Variation of propagation loss of segmented waveguides with respect to
duty cycle. Circles represent measured values and solid curve
corresponds to fitting.
3 I II 1Ii
S2.8
S2.6 0o u o o o
2.4
o
2.2
2 I I I  I I ,
0 0.2 0.4 0.6 0.8 1
Duty Cycle
(a)
Fig. 5.9 Variation in mode size with respect to duty cycle. (a) Modal depth
variation withespect to duty cycle. Circles represent measured values
and solid curve corresponds to fitting.
Fig. 5.9continued
0.2 0.4 0.6 0.8 1
Duty Cycle
(b)
(b) Modal width variation with respect to duty cycle.
effective index of the ridge and thus minimum of the modal width should be able to be
observed.
Realization of segmentation in GaAs which has a very high refractive index (n =
3.3) is indeed difficult since a careful balance between the nonguiding planar section and
low propagation loss ridge section has to be arrived at. In addition, we must ensure that
segmentation of the ridge is still sufficiently strong enough to affect the propagation
constant without causing too much propagation loss. Using the segmented waveguides
that has been developed thus far, there are new opportunities to fabricate a tapered, both in
index and in dimension, velocity coupler (TVC) in AlGaAs/GaAs in the future. Proposed
TVC will consist of two ridge waveguides one of which, for example, is a straight ridge
waveguide with constant width (Wst) and duty cycle of 0.6 along the beam propagation
direction. The other waveguide can be a waveguide tapered in width (from Wti to Wto,
Wti
cycle would effectively taper the index so that there is a weak coupling between the two
ridges at input and output due to large difference both in dimension and in index of each of
the ridge. By employing tapered duty cycle together with tapered width of the ridge, we
expect that complete power transfer can be achieved with a reasonable device length.
Fabrication of the proposed tapered velocity coupler, verification of complete power
transfer, and its application to digital optical switch still remain to be investigated.
Duty Cycle =
0.2
Duty Cycle =
1.0
Fig. 5.10 Top view of the proposed tapered velocity coupler with both dimension and
duty cycle tapered along the beam propagation direction.
