Title Page
 Table of Contents
 Normal mode theory and adiabatic...
 Tapered, both in index and in dimension,...
 Tapered, both in dimension and...
 Tapered velocity coupler using...
 An MQW-SQW tapered waveguide...
 Coclusions and summary
 Biographical sketch

Title: Tapered velocity couplers and devices
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00082375/00001
 Material Information
Title: Tapered velocity couplers and devices a treatise
Physical Description: vii, 120 leaves : ill. ; 29 cm.
Language: English
Creator: Kim, Hyoun Soo, 1964-
Publication Date: 1994
Subject: Directional couplers -- Mathematical models   ( lcsh )
Optical wave guides -- Mathematical models   ( lcsh )
Adiabatic invariants   ( lcsh )
Optoelectronic devices -- Mathematical models   ( lcsh )
Electrical Engineering thesis Ph.D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1994.
Bibliography: Includes bibliographical references (leaves 115-119).
Statement of Responsibility: by Hyoun Soo Kim.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082375
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 001975597
oclc - 31798924
notis - AKF2427

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
        Page vii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Normal mode theory and adiabatic theorem
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
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        Page 24
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        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    Tapered, both in index and in dimension, velocity coupler in Ti:LiNbO3
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
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        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
    Tapered, both in dimension and in index velocity coupler switch
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
    Tapered velocity coupler using segmented waveguides
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
    An MQW-SQW tapered waveguide interconnect
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
    Coclusions and summary
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
    Biographical sketch
        Page 120
        Page 121
        Page 122
Full Text








My Parents


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.


ACKNOW LEDGM ENTS ........................................................................................... iii

A B STRA CT ......................................................................................................................... vi


ONE INTRODUCTION .................................................................................... 1

AND ADIABATIC THEOREM............................................................ 11

2.1 Definition of Normal Modes........................................................... 11
2.2 Characteristics of Normal Modes
in Five-layer Waveguide Structure ............................................. 18
2.3 Adiabatic Theorem in Optical Devices .............................................. 28

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

VELOCITY COUPLER SWITCH .................................... ........... .. 62

4.1 Principles of Operation ................................................................... 63
4.2 Theory and Experiments ................................................................... 66
4.3 Sum m ary ............................................................................................ 69

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


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



Hyoun Soo Kim

April 1994

Chairman: Ramu V. Ramaswamy
Major Department: Electrical Engineering

A polarization independent device is highly desirable for use in single-mode 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 five-layer 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 multi-quantum 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.


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



Fig. 1.1 Schematics of couplers. (a) Directional coupler. (b) Tapered velocity
coupler [Wi75]. (c) Optical fiber tapered veloctiy coupler [Ro91].


Fig. 1.1-continued (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 5-layer 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 fiber-to-fiber 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,

-- --<1.5 (1.1)

(A T/2)2
> 80 (1.2)

so that

ApT AZT 2 213


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 2-D graded index profile of the five layer TVC

structure into a 1-D profile by using the effective index method [Ra74, Fur74]. Field

profiles and mode indices of the local normal modes of the 1-D 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 5-layer step index TVC. In addition, the adiabatic theorem in optical z-variant

devices is also described. Qualitative explanation of power flow in z-variant 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 5-layer 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 z-variant 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


As an application, we proposed and demonstrated [Ra93] a novel digital optical

switch using the TVC in Ti:LiNbO3, which exhibits the smallest voltage-length 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


Extension of this technique to proton exchanged waveguides poses a problem since

the index change in proton-exchange 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 multi-quantum

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 phase-matched 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 proton-exchange

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 proton-exchange 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 proton-exchange 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


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 AlxGal-xAs 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


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.


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 five-layer step

index waveguides. Aidiabatic theorem in optical devices is discussed in conjunction with

the normal mode theory. Qualitatively explanations of output power profiles in Y-branch

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 five-layer 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 five-layer 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).




D2 waveguide B

T gap

D1 waveguide A


Fig. 2.1 A cross sectional view of five-layer 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 y-axis. For the

conventional five-layer 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 N2-n2,

kl=ko n?-N2,
kg=ko N2-n2,

k2 = ko0/n-N2, and

kc =k0oN2-n2.

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)



kq=k n-N2 q = 1, 2, s, g, c.

Equation 2.2 is valid, in general, for any five-layer 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 five-layer 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


Hx(x) and Hz(x), the other components of the TE mode, are given by Maxwell's

equation as follows:

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


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 five-layer 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)

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 i-th row and j-th
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)

So far, we have presented the general expression (Eq. 2.2) of field distribution

along the x-axis normal to the beam propagation direction (+z-axis) 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

interval-halving 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 Five-layer 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 five-layer 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 three-layer 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 three-layer

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 five-layer

structure with a infinity gap (T = oo). Using a well-known three-layer 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 five-layer normal mode and three-layer individual mode

by the notation. That is, Ni is the i-th mode index of the five-layer normal mode and (Ni)A

is the i-th uncoupled three-layer 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)






-2 0 2 4
x (nm)

-- 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.






-2 0 2 4
x (pam)

Fig. 2.2-continued.

-- second order
--------- Third order

6 8 10

(b) The second order (solid curve) and the third order (dotted
curve) normal mode.


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 five-layer

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 five-layer 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 zero-crossing 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





T (gm)

Fig. 2.3 Normal mode indices as a function of T.



* 3^ ====== ~






-2 0 2 4
x (im)

- 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.






-2 0 2 4
x (pm)

Fig. 2.4-continued.

Second order
--------- Third order

6 8 10

(b) The second order (solid curve) and the third order (dotted
curve) ormal mode.

by the three-layer 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

five-layer structures, based upon previous results and many other simulations we have


The general characteristics of symmetric directional couplers are

1. The number of zero-crossing 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 zero-crossings of the guided mode again corresponds to the mode


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

zero-crossings as well as the smoothness and continuity of a field.

- Fundamental
--------- First order

-2 0 2 4
x (wm)

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






Second order
I I Im' I I

-2 0 2 4
x (pm)

6 8 10

Fig. 2.5-continued. (b) The second order (solid curve) normal mode






2.3 Adiabatic Theorem in Optical Devices

It is possible to predict qualitatively the output power profile in z-variant 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 Y-branch 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 Y-branch waveguide have used the

adiabatic theorem. For the normal mode propagation process in the Y-branch waveguide,

the state is the propagation mode in the five-layer 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

Y-branch, 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 Y-branch 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 Y-branch.
(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


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 five-layer 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 single-lobe distribution to two-lobe distribution with a phase shift of nt between

them. This corresponds to the first order mode of the double-moded 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 single-lobe 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



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.



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 step-index waveguides. These concepts are extended further in the next chapter for

numerical analysis of graded index channel waveguide TVC.


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 one-quarter

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 short-length 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 2-D

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)

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.


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 e-beam evaporator and a strip
width of w = 4 pLm was delineated using standard photolithography techniques and wet

2-D index profile : n(x,y)

------------------------------------k '7





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



(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 10-4 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 y-axis. The

Gaussian profile can be conveniently approximated by a staircase, step-index 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 r-2 layers and two semi-infinite regions above and below,

representing the surface and the substrate, we can represent each i-th layer with

appropriate index ni and thickness ti = xi xi-1, in the x direction, thus approximating the

Gaussian profile. The field amplitudes, for example for the quasi TE mode (Ey, hx, hz), at

each i-th 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 )


ul = k1(x1 x),

ui = ki(x xi-1), 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 r-1 interfaces. Thus, we have 2(r-1) 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 1-D index profile N(y),

calculated from the 2-D index profile n(x,y) for a 4 pim wide Ti:LiNbO3 channel

waveguide. The dotted line curve represents the converted 1-D 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





0.002 .*"'" \

-10 -5 0 5 10
y (wm)

Fig. 3.2 Converted 1-D index profile (N(y)-nb, dotted curve) and actual surface
index profile (ns(y)-nb, solid curve) of a 4 g m wide channel

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


1.2 . .... .- iprimnt
W=4. -------- Experiment
-1 Theory

-4 -
r: 0.8

a 0.6

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


0 2 4 6 8
Width of Waveguide


. N2


Fig. 3.4 Quasi TE mode indices (solid curve) for the first three modes and
maximum surface index (dashed curve) of Ti:LiNbO3 channel









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 z-axis 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 1-D 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

S Middle of
0.003 the coupler

U 0.002 :, Input end



-5 0 5 10 15

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 1-D 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 -----First-order of TVC

S 0.6


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

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

g = 2.m
N 3pm
0 4,5,6pm

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

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)


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 j-th mode (j = 0 or 1) across the

discontinuity is given by [Mi75]

Ajl exp(-ajl) = Cij AiO exp(-aio) + CjjAjo exp(-ajo)


where Cij is the coupling coefficient between the i-th order mode on the incident (0) side

to the j-th order on the transmitted side and Cjj is the coupling coefficient between the j-th
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(aio-ajl) + CjjAjo cos(ajo-ajl) (3.10a)

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 j-th mode and the incident i-th

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

3.10a-3.10c and their counterparts for the i-th 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/(0o-P 1). From

A0 and Al at the output with phase constant ao and al, we obtain the output intensity

profile as

Iout(y) = AoEoe-iao + A lele-ial I
= A'e~(y) + A'ec(y) + 2AoA leo(y)El(y) cos(azo-al) (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 i-th (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
0.8 A
-~ A1 = 6gm

S0.4 5m

0 12.m
2 3 4 5 6

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

S 0.6

o /

0.2 / \

-10 -5 0 5 10 15 20

Fig. 3.11 Normalized output intensity profiles calculated using the graded index
model (solid curve) and 5-layer 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 -
0.2 /
0 0.
-10 -5 0 5

10 15 20

Fig 3.11-continued. (b) g = 3 urm.

1 - -Step index
S Experiment

0.6 i

0.4 /

0.2 7

-10 -5 0 5

Fig. 3.11-continued. (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 non-adiabatic 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 i-th 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 5-layer 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 waveguide-by comparing the field profiles.
First we assume a hypothetical 3-layer 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 5-layer 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.


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.


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, Mach-Zehnder 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, two-mode cross coupler in z-cut 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 x-cut LiNbO3 with a better voltage length product (114 Vcm with a

cross talk of 17 dB) was demonstrated later [Mc91]. Y-branch digital switches on

LiNbO3 (a voltage length product of 68.4 Vcm with 14 dB) [Gr90, Thy89], in

InP/GaInAsP [Cav91, Vi92], and shaped Y-branch 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

Fig. 4.1 A schematic of the tapered velocity digital optical switch
The hatched area illustrates the electrode structure.


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 5-layer 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 graded-index Ti:LiNbO3 channel waveguide tapered

velocity coupler [Ki93a]. Primary reason is that the width of the tapered Ti-strip before

its diffusion governs not only the physical dimension of the guiding region but also the

absolute index change resulting from the Ti in-diffusion [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 2-D 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 2-D index profile into a 1-D index profile, the graded-

index Gaussian profile along the depth direction at a given discrete position along the

width is approximated by a staircase step-index profile. For the approximated step-index

profile at each position, we can obtain the effective index 1-D profile, in the conventional

manner, using multi-layer stack theory [Hu90, Th87]. Once again, we apply the multi-

layer stack theory to the resultant 1-D 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 graded-index 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


+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 Ti-diffused 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 2-D graded-index profile.

Fig. 4.4 Output intensity profiles for (a) TE and (b) TM
The black areas illustrates the waveguide regions.


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 z-variant 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 ion-exchanged 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



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 z-invariant

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

Duty Cycle- A


Fig. 5.2 A schematic of proton exchanged periodically segmented waveguides in LiNbO3
with fabrication conditions.




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 He-Ne 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












Duty Cycle


Fig. 5.3 Variation of mode indices of PEPS waveguide with respect to duty cycle
for (a) 3-hour annealed sample (Circles represent measured values and
solid curves correspond to calculated variation of an equivalent z-invariant
graded index waveguide with Gaussian index distribution),

- dx=1.Om TMo



0.2 0.4 0.6 0.8
Duty Cycle


(b) 5-hour annealed sample, and

dx=1.2pu m








Fig. 5.3-continued

2.26 I a
2.25 ta=8hrs.

2.24 dx= 1.4pm

" 2.23


2.21 -

0.2 0.4 0.6
Duty Cycle


TM2 .


Fig. 5.3-continued. (c) 8-hour 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 z-invariant 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 z-invariant 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 z-invariant

graded index waveguide with a Gaussian refractive index distribution.



, 0.06








Duty Cycle

Fig. 5.4 Variation of An at the surface of the equivalent graded index waveguide
with duty cycle

I t I-3hrs. d=l.Op
_- ta=3hrs.
- - t=5hrs" dxlhO
--------- ta=8hrs. m_

^ ^ ^ .. .-- .......

.- -. --









9 10 11 12

13 14 15 16

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

T =300C
- t =5hrs.

O O O O O 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


Periodically segmented waveguide in A1GaAs/GaAs (a) Schematic of the
final structure.

Fig. 5.6

AklsGao.8 s


Duty Cycle -=L


Fig. 5.6-continued (b) Front and (c) Side view of the segmented waveguide.

0.5 im

0.8 Rm




GaAs substrate



GaAs substrate


Illustration for two step etching process.
Sample after (a) the first etching and (b) the second etching.


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 three-layer 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


5 I II

i -
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


S2.6 0o u o o o


2 I I I -- I I ,
0 0.2 0.4 0.6 0.8 1
Duty Cycle

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. 5.9-continued

0.2 0.4 0.6 0.8 1
Duty Cycle

(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


Realization of segmentation in GaAs which has a very high refractive index (n =

3.3) is indeed difficult since a careful balance between the non-guiding 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,

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 =

Duty Cycle =

Fig. 5.10 Top view of the proposed tapered velocity coupler with both dimension and
duty cycle tapered along the beam propagation direction.

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