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Doctoral Thesis

Analysis, optimization, and synthesis of planar integratedlightwave circuits for WDM applications

Author(s): Spühler, Michael-Martin

Publication Date: 2000

Permanent Link: https://doi.org/10.3929/ethz-a-003856734

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Diss. ETH No. 13438

Analysis, Optimization, and Synthesis of

Planar Integrated Lightwave Circuits

forWDM Applications

A dissertation submitted to the

Swiss Federal Institute of Technology Zurich

for the degree of

Doctor of Technical Sciences

presented by

Michael-Martin Spühler

Dipl. El.-Ing. ETH/HTL

born April 30, 1969

citizen of Zürich and Wasterkingen ZH, Switzerland

accepted on the recommendation of

Prof. Dr. Werner Bächtold. examiner

Prof. Dr. René Dändliker. co-examiner

Dr. Gian-Luca Bona, co-examiner

2000

Contents

Zusammenfassung 1

Abstract 3

1. Introduction 5

1.1. Motivation 6

1.2. Analysis of Optical Devices 6

1.3. Optimization With Evolutionary Algorithms 6

1.4. Analysis and Optimization of Planar Lightwave Circuits.. 7

2. Analysis of Planar Waveguide Structures With the

Beam Propagation Method 9

2.1. Introduction 9

2.2. Theory of the Finite Difference Beam PropagationMethod 10

2.3. Mode Solving Using the Imaginary-Distance Beam

Propagation Method 12

2.3.1. Overview 12

2.3.2. Theory 13

2.3.3. The Optimal Parameters 16

2.3.4. Examples 19

2.4. Discussion 23

3. Design and Implementation of a Short Spot-Size

Converter on S1ON/S1O2 25

3.1. Introduction to Spot-Size Converters 25

3.2. Design 27

3.3. Implementation 31

3.4. Measurements 32

3.5. Summary 38

4. A Study of the Optimization Behavior of Evolutionary

Algorithms 39

4.1. Introduction ..39

4.2. The Spot-Size Converter 40

4.3. Structure Representation 42

4.4. The Evolutionary Algorithm 43

4.5. Results 47

4.6. Post Processing of the Evolution 50

4.7. Discussion 55

Contents

5. A Design and Simulation Concept for Planar

Integrated Lightwave Circuits 57

5.1. Introduction 57

5.2. The Concept of the Forward Solver 59

5.3. Details of the Forward Solver 61

5.3.1. The Waveguide Description 61

5.3.2. The Geometry Description 62

5.3.3. The Semantic Analyzer and the Netlist 63

5.3.4. The Scattering Matrix Compilation 66

5.3.5. The Waveguide Library and the WaveguideDatabase 68

5.3.6. The Eigenmode Solver 69

5.3.7. The Calculation of the Coupling Coefficients 69

5.3.8. Elementary Scattering Matrices 71

5.4. Examples 73

5.4.1. Resonant Couplers 73

5.4.2. Add—Drop Filters Using Ring Resonator Devices 76

5.5. Discussion ..79

6. An Optimization Concept for Planar Integrated

Optics ...,81

6.1. Introduction .81

6.2. The Optimizer Strategy 82

6.3. The Fitness Definition 83

6.4. Mutation Operators 84

6.5. Example: A Resonant Coupler Add-Drop Device 85

6.6. Discussion 89

7. Conclusions and Outlook 91

References 93

Acknowledgments 107

Curriculum Vitae 109

Publications ...Ill

Zusammenfassung

In Kommunikationssystemen werden immer häufiger

integriert-optische Elemente eingesetzt. Diverse Wellenleiter¬

technologien und -Herstellungsverfahren wurden in den letzten

Jahren entwickelt. Dieser Trend wird bestimmt noch einige Zeit

anhalten.

Die Entwicklung von optisch integrierten Schaltungs¬elementen ist eine anspruchsvolle Aufgabe. Der Computernimmt auch in diesem Gebiet eine immer wichtigere Rolle ein.

Dies nicht nur für die notwendigen Simulationen, sondern

insbesondere auch für die Optimierung neuer Strukturen. Da die

Konkurrenz gross ist, muss der Entwickler über Analyse- und

Optimierungswerkzeuge verfügen, welche es ihm erlauben,kurze Entwicklungszeiten zu erreichen.

In der vorliegenden Arbeit wurde gezeigt, dass die Anwendungunkonventioneller Entwicklung^- und Optimierungsprozedurenes erlaubt, Lösungen zu finden, welche vorher unbekannt waren.

Evolutionäre Algorithmen stellten sich als sehr effizientes

Werkzeug heraus.

Als Grundlage für die Simulation von optischen Strukturen

wurde eine verbesserte Methode zur Berechnung von Grund¬

moden und auch Moden höherer Ordnung von beliebigen Wellen¬

leiterkonfigurationen entwickelt. Diese Technik ist dann sehr

sinnvoll, wenn BPM (Beam Propagation Method) zur Simulation

von optischen Wellenleiterstrukturen benutzt wird. Sie erweist

sich als sehr genau und reduziert gleichzeitig die Rechenzeit in

den meisten Fällen.

Um die Verluste beim Übergang von der Glasfaser auf den

Chip zu reduzieren, wurde ein neuartiger Modenkonverter

entwickelt. Er besteht aus einem nicht-periodisch segmentiertenWellenleiter mit unregelmässigem Breitenverlauf. Die Struktur

ist nur 100 bis 140 um lang und reduziert die Verluste um mehr

als 2 dB pro Übergang. Zudem hat die Struktur sehr geringeReflexionen. Die Synthese des Modenkonverters wurde mit der

Hilfe eines effizienten genetischen Algorithmus gemacht.

Das interne Verhalten des für den Modenkonverter benutzten

Optimierungsalgorithmus wurde daraufhin analysiert. Es war

1

2 Zusammenfassung

damit möglich, den impliziten Parallelismus des Algorithmus zu

zeigen, indem gezielt die Evolution von Mustern, die

Subpopulationen bildeten, beobachtet wurden. Die Erkenntnisse

die daraus entstanden wurden dazu verwendet, ein Qualitäts-mass zu definieren, welches es erlaubt, die Optimierung zu

beobachten und zu beurteilen. Es ist damit möglich, die

Evolution laufend auf deren Effizienz zu prüfen.

Im letzten Teil dieser Arbeit wurde ein neues Analyse- und

Optimierungskonzept für planare integrierte Optik entwickelt.

Es basiert auf der Streumatrix Methode und beinhaltet drei

Repräsentationsebenen für die Strukturen. Der daraus

entstandene Vorwärtslöser ist sehr genau und schnell und bildet

die Grundlage für einen Optimierungsalgorithmus. Auch hier

wurde eine evolutionäre Technik verwendet, welche allerdings,wegen der geometrischen Natur der einzelnen Strukturen, einigeRestriktionen beinhaltet.

Als Produkt dieser Arbeit entstand eine benutzerfreundliche

Microsoft Windows® Anwendung. Sie enthält alle in dieser

Dissertation entwickelten Komponenten. Das Programm erlaubt

BPM Simulationen und Modenberechnungen und integriert auch

die entwickelten CAD Funktionen für planare optisch-integrierte

Schaltungen.

Abstract

The domain of integrated optics for communication systems is

a field that is growing rapidly. Many waveguide technologies and

fabrication techniques have been developed in the last few years.

This development will probably continue in the future.

Designing integrated lightwave circuits in this context is a

very demanding task. The computer becomes more and more

important in this field. Not only for the necessary simulations

but also for the optimization of new structures. Since the

competition is very intense, the designer must utilize analysisand optimization tools to shorten the design cycle.

In this work it was shown that using non-conventional designand optimization procedures, it is possible to find solutions that

were not known before. Evolutionary optimization procedures

proved to be very efficient for this task.

As a base element for the simulation of integrated opticalstructures an improved method to extract fundamental and

higher-order eigenmodes of arbitrarily shaped waveguides was

developed. This technique is very useful, when using BPM (Beam

Propagation Method) for the simulation of optical waveguidestructures. The results are very accurate, while the calculation

time is reduced in most cases.

For the reduction of the coupling losses at the interface

between optical fibers and integrated lightwave circuits, a novel

spot-size converter was designed. It has the form of a non-

periodic segmented waveguide with irregular tapering. This

structure is only 100 to 140 urn long, reduces the coupling losses

more than 2 dB per interface and has very low reflections. The

synthesis of the spot-size converter was completed with the helpof a breeder genetic algorithm.

The intrinsic behavior of the algorithm used for the

optimization of the spot-size converter was analyzed. It was

possible to demonstrate the implicit parallelism of the algorithm

by observing the evolution of patterns included in the optimized

population. Using this insight a new evolution quality figure was

proposed. This figure was used to observe and judge the

evolution process with respect to its efficiency.

3

4 Abstract

In the last part of this thesis, a new concept for the analysisand optimization of planar integrated lightwave circuits was

developed. It is based on a scattering matrix approach and

includes three representation levels of the structures. The

resulting forward solver is very accurate and fast. Based on this

forward solver an optimizer was built. It is based on an

evolutionary procedure with some restrictions because of the

geometrical nature of the individuals.

As a result of this work a user friendly Microsoft Windows®

based application was developed. All components that were

developed and used during this thesis are included. The

application allows BPM simulations as well as eigenmodecalculations, and integrates the developed CAD functionality for

planar integrated lightwave circuits.

1. Introduction

The rapid development of the information technology, the

global activity of the companies as well as the growing privatedesire for information require an increasing data transmission

capacity of the communication networks. With the invention of

the semiconductor laser and the introduction of optical fibers as

transmission medium for large distances, new doors were

opened. Only optical fibers have the necessary bandwidth for

modern wide-area networks (WAN). The available bandwidth of

the optical fibers, is orders of magnitude higher than the

capacity of today's electronic systems. In order to take advantageof the whole capacity of an optical fiber, one wavelength is not

sufficient and therefore multiple-wavelength systems (calledwavelength division multiplexed (WDM) systems) have been

proposed [1J. In WDM-systems the output of several lasers are

multiplexed onto one single-mode fiber [2]. The parallel channels

can be transmitted over the fiber without any interference.

Because of the possibility to amplify optical signals without

having to convert them to electronic signals, the idea of an ail-

optical network becomes reachable. Thus, the data is processedoptically wherever possible. The routing and multiplexing of the

data has to be solved in the optical domain, too. The data rate at

each electrical must be reduced, thus the higher rates would be

processed optically. To achieve this goal of optical processing and

routing, very selective optical filters are needed to separate the

individual channels in a WDM system. Today, filters are

implemented as integrated lightwave circuits. A crucial filter

device is the add-drop device [3] which performs the extraction

and insertion of one single wavelength without perturbing the

others, or demultiplexers which separate all channels at once [4,5, 6]. The advantage of add-drop filters is that a system needs

only to extract the channel which contains the required data

while the others are not affected. In this way the multiplexernode needs not to detect and decode all data that is transmitted

through the optical fiber.

o

6 Introduction

1.1. Motivation

Many problems are yet unsolved in the domain of opticaldevices. Actual optical engineering tools lack the efficiency or the

generality that would be necessary to design the optical

integrated devices. In this work several aspects are treated

within the context of WDM communication. Especially the

optimization problem is treated in some detail to show new

methods in optical engineering.

1.2. Analysis of Optical Devices

For the analysis of integrated optical devices, one of the

standard methods used is the beam propagation method (BPM)[7]. This method is able to calculate the propagation of the

optical field through a dielectric waveguide. In most implemen¬tations it is assumed that the beam propagates very close to the

optical axis and that radiation occurs only with small angles.Additionally only small differences in the refractive index may

occur in the propagation direction. These assumptions

theoretically restrict the applicability of the method. In practiceit can be shown that very accurate results may be obtained even

if the above conditions are not well met. In Chapter 2 an

introduction to BPM is given together with the introduction of a

mode solver using BPM [8, 9], and in Chapter 3 it is shown that

BPM is an accurate tool, even for complex structures. As an

example a novel spot-size converter [10] is demonstrated.

1.3. Optimization With Evolutionary Algorithms

Evolutionary algorithms [11, 12, 13, 14] are optimizationmethods which mimick the natural evolution process. When

applying evolutionary algorithms a thorough understanding of

the optimization behavior is required. Only if this process is well

understood, can the procedure be successfully applied to real

problems [10, 15, 16, 17]. It is necessary to have enoughinformation about the problem to optimize, to find an

appropriate representation scheme to be used by the

optimization algorithm. However, a precise idea of the possiblesolutions is not required. This knowledge can be acquired while

the optimizer is pursuing its way.

Introduction 7

In Chapter 3 the solution of the mentioned example, the mode

converter [10], is presented. The resulting structures from the

optimizer are very robust against fabrication tolerances and

other factors such as the cleaving position, and have better

performance than other structures to date.

In Chapter 4 an analysis of the optimization behavior [13] is

presented using again the example of the spot-size converter

introduced in Chapter 3. The insight gained in this study can be

used in many optimization problems in future.

The goals of any optimizing strategy are, first, a small overall

optimizing time, and second, to find a solution that is as close as

possible to the global optimum. Evolutionary algorithms help us

to approach both objectives.

It is not always favorable to find the real global optimum. This

solution might not be very robust against fabrication tolerances

and can possibly include a number of calculation inaccuracies. To

judge if a solution is robust against small variations of the

parameters, we can observe the population during the evolution.

If the population contains many individuals which have nearlythe same fitness and are close to each other in the parameter

space, then the optimization has very likely found a robust

solution.

1.4. Analysis and Optimization of Planar

Lightwave Circuits

In the real engineering world, almost every design procedureends up with an optimization task. In electronics many CAD

systems exist to help engineers to simulate and optimize the

circuits. This is partially true for optical lightwave circuits [18,

19, 20], but the optimization part is not yet well established.

Only very few optimizations are possible and those are usually

parameter optimizations. A topology optimization does not exist

(this is also the case in the field of electronic network analysis

tools). The Chapters 5 and 6 treat a simulation and optimization

strategy that is a big step towards real topology optimization. As

a special feature this system includes an inverse problem solver

for optical integrated lightwave circuits.

8 Introduction

Every inverse problem solver consists of two parts: the forward

solver and a strategy to optimize the result. In Chapter 5 the

forward solver for optical integrated lightwave circuits is

described. It is based on a scattering matrix approach. The

particularity of this implementation of the forward solver is that

it operates on three representation levels. Each of them having a

different level of abstraction. This enables an optimizer to use

information that would not be available in a standard implemen¬tation. The resulting forward solver proves to be very accurate

and robust. It can be used as a stand-alone simulation tool for

many applications.

In Chapter 6 the optimizer part of the system, which is based

on an evolutionary approach, is presented. It is able to transform

the topology of the lightwave circuits in order to obtain possiblebetter structures. Because the system can access information on

different levels of abstraction, it is also possible to optimize on

these different levels. The applicability of this concept is

demonstrated using two examples.

2. Analysis of Planar Waveguide Structures

With the Beam Propagation Method

The beam propagation method (BPM) has evolved to a

standard numerical analysis tool of integrated optics. The

method has an intuitive approach to the problem of propagatingelectromagnetic fields in dielectric waveguide structures because

it uses a scheme where the transversal scattering problem and

the longitudinal field propagation are solved separately (this is

called a split-step method). Therefore it is easy to implement the

method on any computer. This is the reason why it is so popularand many research groups are interested in developing improvedBPM-codes.

Nevertheless separate mode solvers have been required to

obtain eigenmodes that can be used as starting fields for the

BPM simulations. These eigenmodes obtained from sophisticatedsolvers may be very precise but they don't match the simulation

context (discretization of the waveguide structure and

propagation mechanism) of BPM. It would be useful to obtain

mode profiles that behave as 'real' eigenmodes in a BPM

analysis. In the second part of this chapter a technique is

presented to obtain eigenmodes (fundamental and higher order)using a special parameter set in the BPM analysis. The

calculated eigenmodes really behave as eigenmodes within the

BPM simulations when the same grid is used. They are

propagated along the real axis through the waveguide structure.

2.1. Introduction

The beam propagation method (BPM) is presently one of the

most widely used tools for the investigation of complex photonicstructures such as non-uniform waveguides, optical junctions or

directional couplers. BPM was introduced in the field of

underwater acoustics [21] and seismology in 1973 as the

parabolic equation (PE) method [22] before it was adapted to

optical problems by Feit and Fleck [7] in 1978. Since then a great

variety of different BPM solvers have been presented, includingFFT-BPM (fast Fourier transform BPM) [7, 23]. FE-BPM (finiteelement BPM) [24, 25] and FD-BPM (finite difference BPM) [26,

9

10 Analysis of planar waveguide structures

27, 28, 29, 30]. Both unidirectional and bidirectional [31, 32, 33]BPM codes as well as full vectorial codes were developed [27, 34,

35, 36, 37, 38, 39].

When using the beam propagation method, it is usuallyassumed that the light propagates along an optical axis and that

scattering occurs within small angles only. In this case the

computational cost can be reduced. When dealing with complexstructures like Y-junctions or bends several techniques exist to

overcome the problem of non-straight propagation. These include

conformai mapping [40, 41, 42, 43] and other grid transformation

techniques [44, 45, 46].

It is possible to use BPM as mode solver. Using the imaginary-distance BPM eigenmodes and effective indices of complex

waveguide structures can be determined. The corresponding

theory is presented in Section 2.3.

2.2. Theory of the Finite Difference Beam

Propagation Method

In this section an introduction to the finite difference BPM is

given. The propagation of the electromagnetic waves in an

inhomogeneous medium is governed by the vector wave equationfor the electric field

AE(x,y,z)-n2(x,yiz)k2E(x,y,z) = 0, (2.1)

where k = m-sfs^jUQ and n(x,y,z) is the refractive index of the

medium. In the paraxial limit, the Helmholtz equation can be

reduced to the paraxial wave equation [47] which is

-

., cE(x,y) û'Ei.x.y) à2E(x,y). r . .o ,ir. . /n oX

2jknQ —V^~ = j^- + ±^- + k n(x,y,zy -

n0 E(x,y) , (2.2)œ ex oy

L J

where w0 = ßfk is the reference index and ß is the propagation

constant of the fundamental mode of the waveguide. For

multimode waveguide systems the reference index can be chosen

as the mean value of the propagation constants of the modes.

The advantage of the paraxial approximation is the reduced

numerical effort because of the simplified operators. On the

other hand, the assumption is made that the field propagates

Analysis of planar waveguide structures 11

close to the optical axis. Several wide-angle codes exist [48, 49] to

overcome this problem where it is absolutely necessary.

The equation (2.2) can be rewritten as

.

dEj — = HE,ôz

(2.3)

where

H = -

2n0k L

•Q. "io o

+

ox" cy

+ (n2 -n2)kJ (2.4)

In FD-BPM (finite difference BPM), the partial differential

equation is replaced by the finite difference approximation,which yields

â2 E E(x - Ax,y,z) - 2E(x,y,z) + E(x + Ax, y,z)

ck" Ax~

c^E E(x,y - Ay,z) - 2E(x,y,z) + E(x,y + Ay,z)(2.5)

0' Ay2

Using these equations it is possible to calculate the

propagation of the electric or magnetic field through an arbitrary

waveguide structure. For this purpose the equation (2.3) is

discretized in z -direction. The equation may then be written in

an explicit, implicit or mixed form. The general equation is

E - E" » n-

= aTIEAz

n+\ (1 -a)HEn, (2.6)

where En represents the known field at any position z and En+l

the yet unknown field at the position z + Az. a represents the

implicitness parameter where a = 0 means explicit, a-\ means

implicit and a = 0.5 represents the Crank-Nicolson case.

The resulting sparse matrix can be inverted using different

standard methods [50, 51].

Because of the limited computational domain, appropriate

boundary conditions are to be used to avoid reflections and other

effects at the boundaries. Several efficient formulas have been

developed in past [52, 53, 54. 55].


2.3. Mode Solving Using the Imaginary-DistanceBeam Propagation Method

When replacing the propagation step by an imaginary value, it

is possible to extract eigenmodes for arbitrary shaped

waveguides. Higher-order modes can be computed when applyingthe sequential mode extraction [8, 56, 57]. This technique has

been used for several applications already [58, 59]. In this

section we present an extension to the standard method of

eigenmode-extraction using the imaginary-distance beam

propagation method. We show that higher-order propagationmodes of arbitrary shaped waveguide structures can directly be

extracted by propagating the field along the imaginary axis when

the parameters are chosen in an appropriate manner. This

method requires a starting guess of the propagation constant of

the eigenmodes. In many cases this value can be determined

using fast approximate techniques such as the effective index

method [60]. Additionally, the approximate mode shape may be

introduced as starting condition and can further accelerate the

extraction of the eigenmode. The overall number of propagation

steps needed to extract multiple eigenmodes is then significantlysmaller than in the case when extracting the modes sequentiallywith the method described in [57].

2.3.1. Overview

The imaginary-distance beam propagation method in its

original implementation has been successfully used to extract

fundamental modes [8, 34, 56, 58, 61] and higher-order modes

[57, 59, 62] of arbitrary shaped waveguide structures. The modes

are extracted sequentially starting with the lowest mode by

propagating the field along the imaginary axis, and for higher-order modes all previously calculated lower modes are subtracted

periodically during the propagation. This means that to compute

a higher mode, the complete set of lower modes is required. As a

consequence, the precision and stability of the solution are

dependent on the accuracy of all previous eigenmodes.

Additionally, when the interval between the effective indices

becomes small, it turns out to be very difficult and cumbersome

to extract further modes. Consequently many propagation stepsare required.


The new technique presented here is able to directly extract a

higher-order propagation mode by choosing an appropriate set of

parameters for the propagation. This has several advantages:

1. It allows the computation of a particular mode directly,without much knowledge of all other modes.

2. A higher selectivity is reached between two modes with

similar effective indices. Therefore, even the fundamental

mode is obtained more rapidly with a smaller number of

propagation steps.3. When extracting a set of modes, the standard procedure of

calculating the modes sequentially is further accelerated byusing the set of parameters introduced in this chapter.

In the following section we first present the theory behind this

technique. In Section 2.3.3 we show how the optimal parameterscan be found. Finally, we present some examples of waveguidestructures and eigenmodes calculated with the method.

2.3.2. Theory

The following development is done for the electric field

component E. The operators and results would be exactly the

same for the magnetic field component H. We start with the

scalar approximation of the field propagation equation (2.3) (as it

was developed in section 2.2)

j^- = GE (2.7)dz

where E = E(x,y,z) is the electric field distribution, z the

propagation axis and G the operator given in (2.8).

G= —J-^~+^T + k2(n(x,yy -w02)L (2.8)2/?0,

where w0 represents the reference refractive index, k is the

vacuum wave number, «(v,y) is the index distribution of the

waveguide. In the case of standard beam propagation z is a real

value. For the imaginary-distance beam propagation method, z

is substituted by jz, which yields the new equation


^= GE. (2.9)

dz

The general solution of (2.9) is a simple exponential function of

the operator G

E(x,y,z) = E(x,y,0)e6-', (2.10)

where E(x,y,0) now represents a starting field profile before any

propagation. This starting field can be freely chosen and can

always be represented by a weighted sum of eigenmodes of the

index distribution n(x,y)

E(x,y,0) = jrV'^V,}'), (2.11)

where E{,)(x,y) represents the eigenmode /' and c{,) is the

corresponding weighting factor. In general, the weighting factors

c{'] are complex numbers but this theory deals with real values

only, thus it is not dealing with evanescent field contributions.

We can now introduce equation (2.11) into equation (2.10) and

obtain

£(x,y,z)= IV'^%^^ (2.12)

where G{,) is now the operator for only one mode /', which may be

simplified by removing the derivatives in x and y, and where

the index distribution n(x,y) reduces to a single value «^

representing the effective index of the mode /

ôf"^(«)2~4 (2'l3)

Applying a generalized Crank-Nicolson scheme to the equation(2.9) we obtain

Ëjizï ~.A= a qE)i i+(\-a) GEn, (2.14)

Az

where En and Einl are the field distributions at the steps n and

n + l respectively, and a is the implicitness parameter as it was

introduced in section 2.2 (a = l means fully implicit). Solving(2.14) for EmX yields


En ii

l + (l-q)GAz

1-aGAzE,„. (2.15)

Using (2.12) it is possible to write

'l + (l-^)G('W«i-i

~~ 2-ii=\

l~aG(,)AzE]'] (2.16)

where E^ is the portion of the eigenmode i included in the field

En, and E^ =c{')E(,). Inspecting (2.16) we can interpret the term

within brackets as an amplification factor A(,) for each

eigenmode / propagating through the waveguide

1 + (1-ûOG(,)AzA -

l-aG{*Az(2.17)

a) b)

Figure 2.1. Graphs of the amplification factor A containing a pole and a

zero. In a) the parameters are chosen such that the fundamentalmode is near the pole. All other modes have a smaller

amplification. In b) another pai am eter set sets the mode 2 near

the pole. Higher as well as loiver modes have smaller

amplification in this case. Theiefore, the mode 2 can be

extracted. In both graphs the vertical lines represent the

amplifications of different modes alien the parameters a, nQ

and Az remain constant.

The function (2.17) is shown in Figure 2.1 and has the three

parameters a, n() and Az.These parameters can be chosen to


place one individual eigenvalue n^i into (or near) the pole of

(2.17). In this case, the corresponding eigenmode is much more

amplified compared to all others.

2.3.3. The Optimal Parameters

As stated in the previous section, the amplification factor

(2.17) has three parameters a, n0 and Az.In this section we will

develop the optimal parameter set to directly obtain any mode of

a given waveguide structure. To obtain a particular mode i it is

necessary to place the corresponding effective index at or near

the pole of the amplification factor A{!). Thus, the denominator of

A{,) of equation (2.17) is required to become zero which leads to

the equation

aG(,)Az = l. (2.18)

If we assume that the step size Az has to be positive, then, as

a consequence, G(0 (see (2.13)) must also be positive. The

condition

"o<"# (2-19)

results.

The best performance can be expected if both neighboringmodes (modes adjacent to mode /) have an equal (relativelysmall) amplification

Av-T)

In this case it is guaranteed that no other mode has a strongeramplification than the desired one. Solving equation (2.20) for a

positive Az yields

te-ZAzJlzm, (2.21)4/

with

f = a(l-a)G{'-"G(l l) and g-(la- 1)(g(M) + G('+1)). (2.22)

A\nl)A (,-])

= -Al (2.20)


As we will further show, the parameter a is restricted to the

range ]0.5,1[. From equation (2.17) we can see that if we choose

a to be 0 no pole is present. If a~\ then Az would become

infinite according to (2.21). To get a positive value for Az, a

must be larger than 0.5. In general we can assume that a is not

really a free parameter. It must be chosen such that the solver

provides the most stable results. When a is near 0.5 the step size

Az is already quite large (e.g., Az > 103) which usually produces

poor results. When a is near 1 (mostly implicit) the best results

have been observed but in this case the step size Az increases to

even larger values. A good tradeoff between the accuracy of the

solver and the mode selectivity is to use a value for a between

0.99 and 0.999.

From (2.13) and (2.17) it is easy to see that if n^ =n0, the

amplification A{m) =1. For simplicity, to get the mode i, we can

set n^-n^ + An (where An is an offset) and then use (2.21) to

obtain Az. Experience shows that An = (1[---y)(n{e'i)g -n^) is a good

range. The desired result is obtained when the eigenmode is not

directly on the pole and not too far away either.

Only these parameters ensure that the desired mode has a

higher amplification than all others and that the two adjacentmodes have the same smaller amplification.

To obtain Az out of (2.21), n^ must be replaced by an

estimate of the effective index of the mode m. The accuracy of

the value should be better than 1/3 of the effective index

difference between the wanted mode i and the nearest adjacentmode. If this condition is not met, either the convergence will be

very slow or a wrong eigenmode will come up. If, due to an

inaccurate estimation of the effective indices, two eigenmodesencounter nearly the same maximal amplification, no

convergence may occur at all.

After every propagation step Az the resulting field should be

normalized in power. The effective index of the field distribution

can be calculated as follows [63]


e'sk

^k2n2(x,y)\E('\x,y)c?E(n(x,y)

âx

âE{'\x,y)

âydx dy

§E('\x,yi\dxdy. (2.23)

The effective index given by equation (2.23) is only correct, if

the field distribution corresponds to an eigenmode of the

waveguide structure. Therefore, an accurate value of the

effective index is only available once the field solution has

converged.

The discretization of the waveguide structure has a majorimpact on the effective index value obtained with this method.

Especially structures with very thin layers less accurate values

may result when using a uniform grid. Usually sophisticatedinterface- or boundary conditions (e.g., Hadley's transparent

boundary conditions [52]) fail for imaginary (nonphysical)propagation. Therefore only a boundary condition like E = 0 may

be used, while the calculation window has to be chosen largeenough to prevent mode guiding due to the boundary conditions.

The starting field must contain a certain amount of the mode

to be calculated. In non-symmetrical waveguides this condition is

not very critical. On the other hand, in symmetrical waveguidestructures it is important that the start field contains parts with

the same symmetry as the searched mode. When the fraction of

the mode to be calculated is very small in the start field, the

method may take longer to converge. If an estimate of the mode

profile is available, the convergence can be very quick.

Additionally to the parameter set defined above it is possibleto subtract previously found waveguide modes (as in the

sequential mode extraction) and even get much faster

convergence.

The method presented here has the advantage that all a prioriknowledge available can be introduced to further accelerate the

mode calculation. Therefore, if all modes are needed, the

sequential extraction is still applicable and the calculation is

much faster when using the parameter set (a , ;?0 and Az ) that

was developed in this section.


2.3.4. Examples

As a demonstration and verification example we show the TM

mode extraction of a 5 jim by 6 um rectangular bar waveguidewith a core and cladding refractive index of 1.5 and 1.45

respectively. For this waveguide, approximate effective indices to

be used as starting values may easily be calculated with the

effective index method. In order to demonstrate the behavior of

the mode extraction we choose as starting field a superposition of

all propagating modes of the structure. For each mode the

optimal parameters are determined and used for the calculation.

Figure 2.2 shows the fraction of each mode through the

propagation. It is clearly visible that the desired eigenmode is

rapidly dominant over all the others which finally become

negligible. This works even for very high order modes near

cutoff. In Figure 2.3 the corresponding field distributions are

shown.

As a more complex benchmark example we chose the same

geometry that was used in [57]. There a directional coupler with

asymmetric waveguides was calculated with the sequential

technique. In Figure 2.4 the fundamental and second lowest

mode are shown, as it was calculated with the method presentedin [57]. After less than 10 steps both eigenmodes have already

converged, comparing to the more than 100 steps required in

[57]. The fundamental mode was not subtracted after each

propagation step and the same start field - a sum of two

gaussians of equal amplitude centered at both waveguides - was

used.

20 Analysis of pianar waveguide structures

sa

a)M

hhhhhhhh s7 !

0

o —' O N " N

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1 0

09

08

07

06

0.5

04

.03

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01

00

30

&25

fi 20

o

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O <NCI O

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S-3

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W4 >•—i *ÇH i^ ^ *^

H H H H H H

(N r-H

H

.3

O O <-*

O T-H O© tS <~-<

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s-s

10

09

08

07

06

05

04

03

02

0.1

00

30

f*5CO

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30 ;

25,

20

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10

5! I.

o o <-+ —< o es *-<

O r-i O -h (N O CS

hhhhhhhh-s;

(N ^2 o oo

o M

NON(N w

U, U [h [H L., L< £—< L, r^

ä;g

Figure 2.2. Using the parameters of Table 2.1 all propagating modes were

extracted. Each graph a) - h) shows vertically the fraction \c \

(see eqs. (2.11) and (2.12)) of every mode through the

propagation. The last column (residuals) shows the part of any

spurious field that has no correlation with any guided mode. On

a horizontal line the sum is always 1.0. In general the desired

mode becomes dominant after a few steps. From h) it can be

concluded that the mode TM 12 is not guided. The mode does

not converge at all and the residuals rise due to non guided,contributions. Therefore, the highest propagating mode is

TM21.

Analysis of planar waveguide structures jù X

Table 2 1 Parameters used for the mode extraction

Mode netf

a nn Az Figs 2.2 and 2 3

TMOO 1.4919 0.99 1.4851 3429 4 a)TM10 1.4818 0.99 1.4796 10784.1 b)TMOl 1.4787 0.99 1.4719 2340.6 c)TM11 1.4686 0.99 1.4667 10337 6 d)TM20 1.4658 0.99 1.4611 3446 8 e)TM02 1.4588 0.99 1.4549 5365 9 f)TM21 1.4532 0.99 1.4510 10155.6 g)TM12 1.4500 0.99 1.4486 15434 6 h) (Fig. 2.2 only)

Bi

5

-5

-5 0 5 Y |>m]

a)

TM 10 JE?:

5

-5

0 5 \ [um]

b)

TM20

/-"-N ^

! flti^ ^ -,r^

\

: t i;

\"

l1*O^

I

5

5 X[nm] -5 0 5 X[ixm]

e)

Ei

5

TM21

/""~^ ^~^\i y Ä '^

,f N

« S V

^

•« f^

u_ i \

-5 0 5 Y[pm]

g)

TMOl

-5 0 5 X[\im]

C)

TM02

aSÉsÊ?^ë^m

y

5 0 5 Xjurn)

f)

Figure 2 3 Propagating hvbnd EH modes as they were found with the

imaginary-distance BPM (7s(

- field of the TM approximation)

22 Anahsis of planar waveguide structures

b)

Figure 2.4. a) Normalized fundamental mode (E - field), b) normalizedsecond lowest order mode of a directional coupler structure Thisstructure is the same as in [5f. For both eases the effective indexand Die field distribution comerged after only 7 propagationsteps. The start field was the sum of two gaussians of equalamplitude centered at the centers of the two waveguides for bothcases. Inset plots shou the fast cornergence of the effective index.


2.4. Discussion

BPM is an excellent method to simulate the propagation of

electromagnetic fields through dielectric waveguide structures.

The limitations and the applicability are to be checked for every

case [64]. Together with the imaginary-distance propagation it is

possible to build a complete simulation platform around one sole

numerical method, however, one must be aware of the possiblenumerical instabilities of the mode solving method presented in

this chapter. Especially if the starting field contains parts of

evanescent modes (with imaginary propagation constants) the

mode solver may have a poor convergence due to the evanescent

parts which cause oscillatory contributions to the solution.

In many cases the presented eigenmode calculation method is

very fast, i.e., requiring little simulation time to converge. For

nearly symmetric directional couplers or poor starting field

conditions the convergence may be slower than with other

methods. In the second example presented in this chapter, the

simulation time was reduced by about a factor 10 in comparisonto the standard sequential technique even though the calculation

time required for one imaginary propagation step may be higher.

A further step would be to automate the mode calculation by

generating a guess of the eigenmodes and by calculating an

approximated effective index. Determining the eigenmodes of

arbitrary shaped waveguide structures would then be possiblewithout any user interaction.

"UTS' »i* ^ -*u*f ! v

*

3. Design and Implementation of a Short

Spot-Size Converter on SiON/Si02

In WDM systems [2, 4], various integrated optical devices have

to be developed and optimized towards a minimal loss. Such

devices are, e.g., couplers [65, 66], splitters [67, 68], filters (add-

drop-devices) [69. 70], multiplexers and demultiplexers [5, 71],and spot-size converters [10].

In this chapter transitions from fibers to integrated waveguidestructures are addressed. A spot-size converter [10], which is

optimized using a genetic algorithm is proposed [13]. The

implementation is described in this chapter. Measurements are

also shown.

In Chapter 4 the optimization algorithm used for the spot-sizeconverter is analyzed in more detail. The analysis gives insightto such algorithms and their intrinsic behavior. Only a full

understanding of these optimization techniques ensures

successful applications of the genetic algorithm.

3.1. Introduction to Spot-Size Converters

At the interface of integrated optical waveguide structures and

fibers, the problem of fiber-to-chip coupling arises because of the

mismatch between the spot-sizes. Similar losses are encountered

when the light is coupled onto the chip and when coupled from

the chip into the fiber. Planar optical lightwave circuits on silica

[4] with waveguides matched to single-mode fibers show

negligible loss for butt fiber-to-chip coupling, for certain

conditions. When using small effective refractive index contrasts

[2] or implementing oversized single-mode waveguides [72, 73]losses are small. However, the low refractive-index contrast

associated with such designs leads to bent waveguide structures

having radii of more than 15 mm. This considerably hinders a

further device miniaturization in devices such as phased array

waveguide multiplexers or ring resonators. Increasing the index

contrast allows us to reduce the bonding radius, but at the

expense of a reduced butt fiber-to-chip coupling efficiency. Byproperly designing and integrating spot-size or mode converters,

25

26 Short spot-size converters

this drawback can be overcome. Spot-size converters are

implemented to reduce the coupling losses when butt coupling

single-mode fibers to an integrated waveguide structure or to a

laser diode [74. 75, 76, 77, 78, 79, 80, 81, 82, 83]. The advantage

of device miniaturization or optical integration would be lost by

using long converters. Thus, compact spot-size converters are

required from a system point of view.

Several approaches to spot-size converters have been reportedin literature. Most of them implement combined laterally and

vertically tapered waveguide structures. Such structures have

been implemented on InP/InGaAsP [76, 77, 80], InGaAsP [79],

InP [81, 83], and InGaALWInAlAs [75], as well as on SiOa/SiON

[78]. They are designed to convert the field shape almost

perfectly into the fiber mode such structures. However, the

waveguide structures are difficult to integrate and requireadditional fabrication steps. Another technique uses a taperedfiber to reduce the coupling loss [82]. In industrial applicationssuch structures are not very practical. To integrate the spot-size

converters with the whole waveguide topology in only one step,

planar structures are required. Planar spot-size converters were

presented using periodic or quasiperiodic segmented waveguideseither with [84, 85] or without [86, 87] lateral tapering. Recently,another type of converter using a grating structure [88] was

reported. Such designs transform the mode shape to obtain the

fiber mode quite well, but the structures become very long (fromseveral hundred micrometers up to several millimeters).

In this chapter another approach to planar spot-size converters

is presented where the structures are very short, using a general

nonperiodic segmented waveguide approach with irregularlateral tapering [12]. Since these structures are short, small

losses result. Negligible reflections occur due to the

nonperiodicity and the low Fresncl reflections. The converters

were designed and implemented in a SiCVSiON materials

system [89, 90], as used for the fabrication of passive optical

add/drop wavelength division multiplexing filters [1]. For this

materials system, simple manufacturing procedures are

available.

Short spot-size converters 27

3.2. Design

The spot-size converter structure was designed using an

evolutionary optimization procedure (see also Chapter 4). Such

evolutionary algorithms prove to be very efficient especially in

the case of demanding combinatorial optimization problems, e.g.,

when having a large set of discrete parameters. The algorithmitself relies on a collective learning process within a population of

potential solutions comparable to the process of natural

evolution. The process of evolution is mimicked by using genetic

reproduction operators such as selection, crossover and

mutation. In this optimization scheme, better individuals

inherently have higher reproduction probabilities and will

therefore survive longer. Because evolutionary algorithms do not

necessarily require a well prepared starting point, they are well

suited for true synthesis tasks finding novel, unexpectedsolutions.

Segment S} Segment SN

L

14 14 118 8 0 6 0 0 0 0 5 5 5 5 03 0 0 0 0 0 0 4 4 0 0

»is.

Figure 3.1. The representation of the spot-size converters for the optimizer.The converter is divided into segments S{ to SK, which can

have any discrete value (within given limits) for the width. The

length of the segments is Ls = 3 :um.

For the purpose of the evolutionary optimization procedure the

structure is divided into a number of segments of equal length.Each segment represents a short waveguide with a certain width

that is coded by an integer between 0 and 21, where, for

example, 4 means the width of the original waveguide of 3 am.

Therefore a finite number of discrete values, in steps of 0.75 am.

can be created by the optimizer. A segmentation is obtained


when the width is zero. Figure 3.1 shows this representation for

spot-size converters.

The representation was chosen to allow the total length of the

converter generated by the optimizer to be variable [12]. A

comprehensive description and analysis of the evolutionary

optimization procedure used here has been presented in [13]. A

possible resulting converter structure is shown in Figure 3.2.

Figure 3.2. A possible implementation of the spot-size converter. For

visualization purposes the upper cladding is omitted. The lower-

cladding is at Y < 0 (see also Figure 3.3 for more details on the

waveguide structure). A mode traveling in the + Z-direction is

widened.

The initial waveguide structure is a buried ridge waveguide as

shown in Figure 3.3. The waveguide structure was optimized for

small losses, reduced bending radii, and to be single-mode at

1550 nm wavelength. Upper and lower claddings are made of

Si02 with a refractive index of 1.45. The core consists of SiON

with a refractive index of 1.50. At 1550 nm, this results in a highlateral effective refractive index step of 0.02. When the

waveguide is segmented, the residual base layer - with the same

refractive index as the central core - remains. That layer

considerably hinders a full vertical expansion of the mode.


Figure 3.3. Schematic representation of the waveguide geometry and

refractive indices at X = 1550 nm.

The field simulations were performed with a semi-vectorial 3D

finite difference beam propagation method (3D FD-BPM). The

coupling losses were calculated using an overlap-integralbetween the field at the output of the spot-size converter and the

fiber mode, including the radiation losses with an appropriatenormalization [12]. The evolutionary algorithm proved to be very

efficient in solving such problems. Good results were obtained

(less than 1.5 dB loss) after only about 1000 optimization steps.The best result of the optimizer (after 1132 optimization steps)was then optimized with respect to the segment length Ls.

An optimum for the coupling losses was found at a segment

length Ls of 2.7 am. The resulting structure is slightly shorter

than the original one and therefore has smaller radiation losses.

The final structure is shown in Figure 3.4, together with the

evolution of the coupling efficiency through the converter. A

minimal theoretical loss of 1.3 dB can be achieved with this

structure. Inspecting the optimized converter structure shown in

Figure 3.4, two different converter sections may be identified: A

first section (between 0 am and around 60 am) without

segmentation, i.e., only with changes in the waveguide width and

a second consecutive section, where true segmentation takes

place.

30 Short spot-size convert eis

20 40 60 80 100 120 140 160 180 200

Propagation distance / [jim]

/

go-Tjffo-o >"£ aoapoil

40 60 80 100 120

Propagation distance Z [|im"|

140 160 180 200

40 60 80 100 120 140

Propagation distance Z [fim]

180 200

Figure 3.4. Structure resulting from the evolutionary algorithm with

contour plots of the propagating field (top and center) The

converter structure starts at the position 0/urn The theoretical

mode mismatch (overlap between the propagating mode and the

fundamental fiber mode) is reduced from 3 3 dB to 1 3 dB

(bottom) The best conversion is achieved at the distance of140 urn, marked with a vertical line Ou. ing to the smooth curve

the structure may be cut at any position between 120jam and

150 jiim without a significant degradation of performance

Looking at the actual field pattern within the structure, this

classification is additionally underpinned by a different mode

treatment considering the two converter sections. The first

converter part performs a significant field expansion m the

horizontal direction whereas the segmentation m the second part

provides a vertical enlargement of the mode profile This sort of

functional partitioning is characteristic for all well performingcoupler topologies. The last periodic part of the converter was


lengthened to show that the further guidance of the mode

provides only small changes in coupling efficiency. Thus, the

structure may be cut with a large tolerance of about 30 u.m

without degrading the performance more than 0.1 dB (Figure

3.4).

Reflections are estimated to be very low because of the small

difference in the effective refractive index between segmentedand non-segmented waveguides of about 0.02.

The theoretical direct butt coupling loss was calculated to be

3.3 dB. The performance with a spot-size converter is a couplingloss of 1.3 dB, which results in a coupling improvement of 2 dB

as shown in Figure 3.4.

3.3. Implementation

The SiON waveguide structures are deposited by PECVD

(plasma enhanced chemical vapor deposition) onto thermallyoxidized silicon wafers. Channel waveguides are formed byreactive ion etching. The single mode waveguide design is based

on the demand to obtain low-loss bends for radii as small as

1 mm. Small bending radii allow optical components, e.g.,

resonant couplers, to be designed on a smaller chip size and

complete optical building blocks to be integrated on one chip. To

meet this requirement the lateral effective index contrast of the

waveguide was chosen to be 0.02, resulting in a width of 3 am for

single-mode operation [89].

For a precise characterization of the modeled spot-sizeconverter, straight waveguides of 1.5 cm length were fabricated.

Every second waveguide ended with a converter structure. The

chip was diced using a wafer saw at the end of the converter and

the end facet polished to optical quality.

A direct comparison between a spot-size converter and an

adjacent conventional butt-coupled straight waveguide was

performed, thereby avoiding the influence of fabrication

inhomogeneities across the chip.


>l-IMlOMNHHHh-

140 um

Figure 3.5. Optical microscope image of a 140-pm-long spot-size converter

structure as developed in photoresist. The rounded comers

visible in the zoomed section of the structure are due to the

lithographic process. The segment length is 2.7p.m.

In the zoomed section of the optical microscope image shown in

Figure 3.5, the waveguide segments are rounded due to the

lithographic process. This effect can even be advantageousbecause small, sharp corners are difficult to overgrow

conformally in PECVD. Experimentally, we found no negative

impact on the performance of the converter. To estimate the

influence of geometrical errors different versions of converter

designs were implemented, ranging from simple tapered

segments to those having sharp or rounded corners. All these

variations showed similar results. A theoretical analysis of

waveguide roughness and small-scale inhomogeneities [91, 92,

93] shows that such effects have little influence on the

eigenmode of the waveguide.

3.4. Measurements

Figure 3.6 shows a block diagram of the measurement setup.The light emitted from a 1550-nm laser source is guided throughan isolator to prevent the light from returning into the laser. The

single-mode fiber is butt-coupled directly into a straightwaveguide with a converter on the other end of every other


waveguide. After coupling into a fiber, the light strikes a

detector. The signal is measured with a power meter.

Laser Source ->- Isolator

^)r

Converter

HHfMlSingle-mode Fibers

V —1É1ÉÉI

Power Detector ->- Power Meter1111111

Figure 3.6. Schematic of the measurement setup. The waveguide chip is

mounted on a translation stage which, permits the measurement

of adjacent waveguides. For reference purposes waveguides with

and without couplers arc implemented alternately.

The results of the measurements are shown as a comparisonbetween waveguides with and without a spot-size converter. As

shown in Figure 3.6, converters are placed on only one side of the

chip. The improvement of the coupling efficiency is greater than

2 dB per interface (see Figure 3.7).

Only small variations between consecutive waveguides were

measured. The devices have no significant polarization

dependence, which was to be expected because of the very small

difference in TE and TM mode shapes in addition to the small

converter length. No measurable difference between TE and TM

field expansion was observed. A very low wavelength dependencyof less than 0.1 dB in the range from 1.50 to 1.60 jim is expectedfrom simulations. Below the wavelength of 1.50 urn the originalwaveguide is no longer guaranteed to be single-mode.


-1

O

Ph

ë-4

öCO

•5 -

PÜ

-6 -

-7

1 1 1 !— —

,• •

* ** ^

~M

*^ *

*

-

with coupler

+

.

-_t_- . -

+.-

+_ i

4- +

-

[

-L 1 _

without coupler

f , , ,

12 3 4 5 6 7 8

Sample

Figure 3.7. Comparison, between the coupling losses with and without spot-size converter. The improvement with converter is greater than

2 dB per interface, which agr-ees well luith theoretical

predictions. The vertical axis shows only a, relative difference.No absolute calibration was made. The solid and the dashed

horizontal line represent the mean value of the four best

structures with and without coupler respectively.

Figure o.<

1

0.9, n

CO

•3 0.8

4 0.7

-0.6

PÜ 0.5

a

P.0.4

o

Ü0.3

13^ 0.2

0.1h

-15

—with coupler

- without coupler

-5 0 5

X Offset [pm]

10 15 20

Measured horizontal fiber alignment sensitivity with and

without, spot-size converter. Using a, converter the horizontal

(and also the vertical) alignment is slightly less critical, because

of the widened field shape. In both cases the fiber is butt-coupledto the chip.


One important factor is the horizontal and vertical alignmentsensitivity when coupling a fiber to the chip. The alignment is

slightly less sensitive with the converter (see Figure 3.8) as is

expected because of the decrease of the numerical aperture when

using a spot-size converter. The coupled power as a function of

the gap size between the fiber and the chip shows interference

effects in the first 20 urn with only small variations in the order

of 5% of the coupling efficiency. The Z -alignment for butt-

coupling is less sensitive than for coupling to a lensed fiber.

A further improvement of the coupling efficiency may be

achieved when using an index-matching oil. The difference was

measured to be 0.19 dB (the theoretical value is 0.18 dB) when

an index-matching oil of n = 1.5 is used. Therefore, when usingindex-matching oil, the improvement of the coupling efficiency is

further increased.

Single-mode Fiber

Figure 3.9. Setup for near-field measurements. At the output of the

converter a lens images the field intensity into a CCD camera.

The chip used for this measurement contained converter

structures shifted along the uavcginde with respect to each

other.

The propagating field intensity is compared to the simulations

with near-field measurements. For this purpose a series of

identical converters displaced along the waveguide at intervals

of several microns are used (see Figure 3.9). This allows us to

determine the field shape at several positions in the mode

converter on a single chip. To measure the near field, a

microscope lens was used to image the field intensity into an

36 Shoit spot-size convert eis

infrared CCD camera as shown m Figure 3.9. All elements are

specified for a wavelength of 1 55 (am

Simulated Measuied

^3 0

-4-Z= 15 um

-5 0 5 -5 0 5X[umJ a) X[|um]

-5 0 e) -5 0 S

X^m] b) X$m\

"5 X[?ml 5

c)"5 xSm] 5

X[um] d) X[um]

-5 0 5 -5 0 5X[um] e)

v

X[um]

Figure 3 10. The calculated intensit\ distribution (left) is compared with the

measured near field (right) through the converter structure The

intensities agree lerv uell with the simulations The fields are

shown for Z = 15, 65, 95, 135 urn and at the end of the

converter from a) to e), respecta ely

As shown m Figure 3 10 the measured near field intensity

compares very well with the calculated field profiles through the

converter.


The reflections due to the spot-size converter were estimated

to be very low. A reflection measurement using a position-sensitive mterferometric tool (HP 8504B) with a resolution of

40 am confirms this estimate. Figure 3.11 shows a reflection

measurement performed on a double converter structure. The

maximum reflection level induced by a converter structure is

found to be about -40 dB, which is more than 20 dB smaller than

the Fresnel reflection that would occur if a lensed fiber were

used to couple to the waveguide. This measurement shows that

BPM is an appropriate tool to simulate such structures. This

assumption proves to be valid because the reflections are so

small.

[dB]

-10

-20

-30

-40

-50

-60

-70

-80

-90

START 73 5 mm 1300 nm n=1 5 STOP 74 167 mm

Figure 3.11. Reflection measurement on a double converter structure at

1300 nm This uavelength was used because of the higher

precision The second best optimized structure was used for this

measurement The maximum reflection of one structure is about

-40 dB A picture of the structure is overlaid to show the

positions of the reflections Following the inset reflection scheme,the slight asymmetry in the reflection signal is caused by an

asymmetrical optical excitation of the structure

When taking into account a loss of 0.5 dB/cm of the

waveguides, the absolute butt-coupling loss without spot-sizeconverter is 3.7 dB, whereas the coupling loss with spot-sizeconverter is only 1.6 dB, both using index-matching oil. These

eai

\

I I

"

Incoming light

<— «_D «D 4J> «pDistnbutc drcfle rtion

A

(\

\ f\f\A*A,

AAi 1/\»

yv Y *j \j1/vA/V


values compare well with theoretical calculated couplingefficiencies of 3.3 dB and 1.3 dB, respectively. The main source of

discrepancy may come from a residual mode mismatch of the

calculated and real waveguide mode with respect to the optical

single-mode fiber. Nevertheless the spot-size converter results in

an improvement of 2.1 dB per interface.

3.5. Summary

A compact spot-size converter was optimized with an

evolutionary algorithm and implemented on high refractive-

index contrast Si02/SiON material as nonperiodically segmented

planar waveguide structure. Planar structures on the chosen

material are very cost-effective and may be fabricated using a

simple single-step lithography process. The evolutionary

optimization results in a converter length of less than 140 \xm,

which is very short compared to earlier designs. The fabrication

tolerances are larger than in classical designs. Measured

coupling losses and field shapes show very good agreement

compared to BPM simulations. A measured improvement of the

coupling efficiency of more than 2 dB per interface compared to

direct-butt coupling was obtained.

4. A Study of the Optimization Behavior of

Evolutionary Algorithms

In the Chapter 3, an evolutionary algorithm was successfullyapplied to the synthesis of an integrated spot-size converter.

Evolutionary algorithms proved to be well suited for the solution

of very complex problems having strongly nonlinear cost

functions defined over the solution space. They in most cases

work faster than other optimization techniques such as random

search or the Monte-Carlo Method because of their parallelsearch mechanisms, also referred to as intrinsic or implicit

parallelism [94. 95].

To obtain an understanding of the intrinsic behavior of the

optimization, the optimization of the spot-size converter that is

implemented as a non-periodic segmented waveguide structure

is further analyzed.

In this chapter, after a analysis of the intrinsic behavior of the

optimizer, an observation method is proposed, introducing an

evolution quality figure. This figure is used to visualize and to

qualify the evolution of the algorithm. Based on this figure a

termination condition is suggested.

4.1. Introduction

Evolutionary algorithms have been applied to solve very

complex problems [96, 97, 98]. Especially for problems includinga large number of discrete variables without any predefinedneighborhood relationship and an associated nonlinear, unsteadycost function [12], these methods show an efficient behavior and

do not stick in local optima as certain other optimizationtechniques. However, special care has to be taken when choosingof an appropriate representation and the optimizationparameters. A general overview for genetic algorithms used in

electromagnetics is given in [99]. In the previous chapter, an

evolutionary algorithm based on a breeder genetic algorithmscheme was applied to the design of an optical spot-sizeconverter. An approach using non-monotonic lateral taperingand non-periodic segmentation was used. In order to obtain all

39

40 A study of evolutionary algorithms

possible solutions, the chosen representation is kept as generalas possible. The corresponding analysis method is able to

evaluate the performance of all possible structures that can arise

during the optimization process.

If the evaluation of the 'cost' or 'fitness' value takes a

considerable amount of computation time one would be

interested to see at an early stage, whether a good repre¬

sentation and appropriate parameters have been chosen.

Statistically available information concerning a final state of a

population's evolution does not correctly represent the

optimizer's potential for a further improvement. What is missingso far is a tool that enables the possibility of optimizersupervision. Therefore, based on the data obtained during an

optimization run a post processing' procedure is applied. There

an evolution quality figure is defined which is used to indicate

the progress and the termination point of the optimizer. Such a

technique is useful, when the supervision time is negligiblecompared to the total computation time of the optimization.

4.2. The Spot-Size Converter

Spot-size converters are used to reduce the coupling loss when

connecting a single-mode fiber to the integrated waveguidestructure. Different approaches to mode converters have been

reported. The horizontal expansion of the field is achieved with a

lateral tapering of the integrated waveguide. Vertical expansioncan either be obtained with a vertical tapering of the waveguide[81] or by implementing a periodic [84, 85, 87] or non-periodic[12] segmentation. x\ simplified explanation of the effect of a

periodic segmentation is an averaging of the effective indices in

the segmented and not segmented waveguide parts that gives a

broader fundamental mode profile.

To enable simple manufacturing vertical tapering is not

included in this design because of the required additional

fabrication steps which would result in higher costs and more

elaborate equipment.

The structure is integrated on silicon [89], with a waveguide of

3am width and 1.94 um height (see Figure 3.3). The core

material consists of SiON with a refractive index of 1.50. The

upper and lower claddings are made of Si02 with a refractive

A study of evolutionary algorithms 41

index of 1.45. This results in a refractive index contrast of about

0.02. A waveguide structure with a relatively high index contrast

has the advantage of allowing small bending radii. This

facilitates

devices.

the further miniaturization of optical integrated

15

10

a

-10

Air

-15

Core

Upper Cladding /^gjj^X £„

Lower Cladding

Silicon Substrate

¥,

15 10 0

X[iim]

a)

0Y [Urn]

b)

Figure 4.1. a) The fundamental mode o\ the integrated waveguide. The

structure of the waveguide is also shown h) The Gaussian

shaped eigenmode of the fiber together with the view of a large

integrated waveguide. Because of the field parts overlapping

into the air or silicon an optimal mode conversion cannot be

obtained.

The narrow eigenmode intensity profile of the integrated

waveguide has to be adiabatically converted into a field

distribution as close as possible to the eigenmode of the single-mode fiber (see Figure 4.1). A residual base layer which has a

height of 0.64 um (see Figure 3.3) and the same refractive index

as the central core exists. Such a layer considerably hinders the

vertical expansion of the propagating field. This layer is also

present where the waveguide is segmented. Therefore it is not

possible to obtain an optimal eigenmode of the fiber using such a

design. Furthermore the total vertical extent of the waveguidestructure is not large enough to allow an adiabatic conversion of

the field into the eigenmode of the fiber. As shown in Figure 4.1

b), parts of the fiber mode overlap into the air and into the

silicon, even if we allow verv large waveguides.


The eigenmode of the integrated waveguide was calculated

using the imaginary distance beam propagation method [8]. An

example of such a planar spot-size converter structure was

shown in Figure 3.2.Ltoi

4.3. Structure Representation

The representation of a general converter structure is chosen

to allow different values for the total length of the converter and

arbitrary variations of the segmentation of the rib. Therefore the

length and the width of the structure are discrotized. Now a

converter can be represented as a fixed length string of multi¬

valued bits {£r..S^}. Each bit S, indicating the property of one

segment. Every segment can have a value from -1 up to Nw,

where Nw represents the code for the largest possible segment

width. A zero value means that the ridge of the waveguide is

omitted. A special value (—1) is used as don't care representing a

segment of zero length and zero width, e.g., segments with this

value are non-existent in the final converter and are used to

allow structures with a variable length when using fixed lengthmulti-valued bit strings (see Figure 4.2). The width Wt of the

segment number / is then calculated as

don't care, S, =-1

^ = io, S, =0 , (4.1)

(S,-\)AW+W0> S,>0

where W() is the smallest width of the waveguide and AW

represents the step size. The length of the segment Ll of the

segment / can be determined according to

L,=\'

,

4.2)'

[Ls, S,>0'

where Ls stands for the length of a segment. A converter can

therefore have the maximal length of A\t Ls.


,BitS, Bit Sf

1 0 0 4 4 0 2 0-1-10 2 -1 2 2 0 8 8 -i -i o o|TJT|o 0 6

Lv — 3 urn«—-

8 8 64 4 0 2 0 0 2 2 2 01 0 0 0 0 1 1 0 0

Figure 4.2. Representation example of a converter structure (top). The

physical structure is obtained by removing the don't care's prior

to the simulation (bottom).

4.4. The Evolutionary Algorithm

An evolutionary algorithm uses a population where each

individual represents one point m the search space. Each point of

the search space may be reachable. No physically invalid

parameter combination may be generated by the optimizer. To

ensure this, problem related constraints and rules are included.

For the case of the mode converter, the values of W0 and Ls must

be chosen correctly in order to guarantee that all possiblestructures are realizable. These two values do not have any

impact on the general representation of a converter structure for

the evolutionary algorithm.


/ : maximal number of iterations;

Initialize Population P(0);

Evaluate Population P(0);

i=0;

While (i<l and termination condition == False)

{

Select two different individuals indi, inch, out of'P(i);

Crossovetfindi, ind2) => offspring!, offspring.?

Mutate(offspringi);

Mutate(offsprings) ;

if (Valid(offspringi)) Evaluate fitness ofoffspringi;

elsefitness ofoffspringi = 0.0;

if (Valid(qffspring2)) Evaluatefitness ofoffspring?;

elsefitness ofoffspring?. = 0.0;

Sort(Population P(i) u { offsprings offsprings };

Reject worst two => P(i+1);

i=i+l;

Figure 4.3. Pseudo-code of the breeder genetic algorithm. The functionValidate checks if the genotype of the individual (as an

argument of the function) is already contained in the

population.

The evolutionary algorithm used here is based on a breeder

genetic algorithm scheme [11] and is shown in Figure 4.3.

A heuristic is used to initialize the population generatingindividuals containing concatenated blocks with LB segments of

constant waveguide widths. Block lengths LB are set to values

between LBmm and LBmx.. The corresponding value of the width is

chosen according to given probabilities Ps :

Pn

P>o

= the segment is set to don't care,

= the segment is set to zero width, and

= the segment is set to any width from 1 to N.

After generating and evaluating all Np initial individuals, they

are sorted according to their fitness.


Two bit strings are selected from the population, the

probability of selection being an increasing function of fitness.

Here the so called 'roulette wheel selection' [100] is used.

Two offsprings are generated by exchanging a part of the bit

string between the two parent structures. The part to be

exchanged is determined by selecting two positions on the stringwith a uniform probability. After this 'crossover' operation, the

two offsprings are mutated by reinitializing the bit string at

every location with a very small probability.

Figure 4.4. Simulated field propagation through a converter. Left of the

dashed line the width of the original waveguide is shown. The

propagation steps are 0.3pm and the discretization in X and

Y are 0.25pm. A horizontal slice of the waveguide is

superposed. The expansion of the propagating field is clearlyvisible.

The propagating field through the converter structure is

calculated using a 3D FD-BPM (finite difference beam

propagation method) program. An example of a propagating field

through a converter structure is shown in Figure 4.4. The fitness

of a structure is calculated after each propagation step of 0.3 urn

using an overlap-integral between the propagating field and the

fundamental mode of the fiber. The integral has the followingform:


F(z) =

\fra(xiy)z)%{x,y)dA

JJi % (x,y) f dA |J| Tr (x,y) f dA(4.3)

where X¥I is the fundamental mode of the waveguide, T^ the

fundamental mode of the fiber and vVa the propagating field

through the converter (see Figure 4.1). By using the double

integral J |jxr,/(x,y)|"c//l as normalization factor, the radiation

A

losses are taken into account. The initial field x¥l for the BPM

simulation is calculated once prior to the optimization. The

highest value obtained along the structure is returned as the

overall fitness. A possible fitness function F(z) through such a

converter is shown in Figure 4.5. The final converter structure is

cut at the location achieving the highest fitness value. Therefore,

a physically fabricated converter may be significantly shorter

than the structure defined by its multi-valued bit string (verticalline in Figure 4.5).

180

Propagation Distance [jam]

HhH11" \\ "I

Figure 4.5. The fitness evolution through the converter is shown here. The

real structure will be cut at the position where the highest

fitness is obtained. Therefore the implemented converter is

usually considerably shorter than the total structure. The fitnessis calculated after each BPM propagation step. The best fitness

ever encountered (here at about 110 urn, shown by the vertical

line) is retained as the overall fitness of the converter.


The algorithm differs from a conventional genetic algorithm in

the following elements:

Every new individual is checked whether it is alreadycontained in the population. Allowing no duplicates avoids

premature convergence and is advantageous for the algorithms

finishing behavior because the population never fully converges.

In [101] the authors also reported a large improvement of the

effectiveness allowing only one copy of any bit-string at any time.

Only better individuals than the worst contained in the

population are inserted, e.g., a strict breeding is done. New

individuals are immediately available for the next recombination

step. This strategy is therefore a kind of multi-point hill climbingwhere only better individuals are introduced into the population.The best points found are kept and the search is kept as diverse

as possible.

The use of the combined crossover/mutation operator takes

into account that in a population with a high diversity, the main

effect of disrupting the structures will be due to the use of

crossover. The more the population converges to a narrow fitness

distribution the more a local search will be carried out where the

main effect will be due to the mutation operation which

reintroduces new information at randomly chosen locations in

order to achieve a fine tuning of the converged population.

Offsprings are immediately available for recombination. The

algorithm therefore has the opportunity to exploit a promisingindividual as soon as it is created.

This algorithm is described in more detail in [97] and has also

been applied to other problems.

4.5. Results

For the runs of the algorithm the population consists of 100

structures. The following parameters were taken for the

optimization: N = 70 segments of A\ = 3 urn length, a minimum

segment width of W0=1.0 urn and width steps of AW =0.5 um.

This representation reflects the manufacturing conditions for

such structures. The block length range for the initialization is

defined by LBmm =3 and LB = 7. The width-coding range for the


S, is [-1. 40], giving a solution space of 42 =424-10in possible

converter structures Some of the real converters may be

identical - due to the don't cares - and therefore the physicalsolution space is slightly smaller but still exceptionally largeThe propagating step size for the BPM simulations was chosen to

be 0.3 urn and a grid with Ax = 0.25 am and Ay = 0.25 urn was

used.

Simulated overall coupling losses of about 1 3 dB were

obtained which represents a theoretical improvement of the

coupling efficiency of 2 dB This was achieved by evaluating10350 individuals requiring only one structure out of 10107 to be

calculated. Other optimization methods require a much highernumber of evaluations to achieve a similar result

Figure 4 6 Microscope photograph of a spot-size conter ter The resist is

shown here The corners are clearly rounded due to the

photolithography process and in part due to the restricted

resolution of the optical microscope Indeed, this effect has no

impact on the perform an ce of the converter [10, 15]

Two resulting structures have been implemented on a silicon

substrate and were optically characterized An example of an

implemented spot-size converter is shown in Figure 4 6 In

general, the genetic algorithm will not find the absolute globaloptimum in the search space This would even not necessarily be

a good achievement, because the very optimum might not be

very robust against fabrication tolerances and could be achieved


due to a number of numerical side effects. Solutions found by the

algorithm turned out to be very robust.

i l ' ' l l

0 2000 4000 6000 8000 10000 12000 14000

Evolution Steps

Figure 4.7. The évolution of the population is shown. The top curve shows

the best fitness as a function of the evolution steps. The lower

line represents the lower fitness boundary of the population. For

every evaluated individual a dot is plotted. When a structure

has a lower performance than the lower fitness boundary it is

not inserted into the population

The evolution of the optimization is shown in Figure 4.7. The

upper line represents the fitness of the best individual and the

lower line is the fitness of the worst converter contained in the

population. For each evaluation a dot is plotted. As shown in

Figure 4.7, bad individuals may be created at every stage of the

optimization.

Compared to the length of a conventionally designed converter

(usually 800 urn up to several 1000 urn) these optimizedstructures tend to be very short (generally less than 150 urn).

Thus the requirements are met with the additional benefit that

the radiation and absorption losses are lower in shorter

structures. The error introduced into the calculation by the fact

that BPM does not take reflections into account is very small.

The reflections were measured to be m the order of -40 dB. This

small value can be explained by the non-periodicity of the

structure and the relatively small refractive index difference of

0.02.


4.6. Post Processing of the Evolution

Evolutionary optimization procedures also provide overall

information about a possible solution strategy. One of the main

differences between classical optimization procedures such as

Monte Carlo or simple hill-climbing methods and evolutionary

optimization procedures is their parallel search mechanism. As

is demonstrated later, any successful converter contains

substructures that may be important for optimal performance. In

our procedure it is possible to keep track of such substructures

during the evolution. To obtain the corresponding data of the

traces, substructures of 10 segments length were compared. If no

more than 3 segments of that substructure differ from one

individual to an other the individuals are considered to be part of

one trace. The position index and the fitness of all individuals

taking part of a trace are stored. Three different types of traces

can be distinguished:

1. Traces from the initial population: Are substructures of the

initial population still contained in the population later in the

evolution process?2. Backward, traces from fitness steps: What is the history of the

substructures contained in the best individual of the

population?3. Backward traces from the final population: Are there

substructures that still remain in evolution but have not yet

contributed to the best individual?

These three types of traces will be discussed in detail.

(1) Traces from the initial population: During evolution,

substructures reproduce in order to generate new individuals

with a possibly higher fitness levels than the previous ones

that contain the same substructures. If an individual has a

lower fitness than the worst one in the population, it will be

eliminated immediately. Once a substructure is no longercontained in any individual of the population that

substructure dies out. The probability of reintroduction of the

same substructure by crossover and mutation is very low. The

first type of traces may show how long the initial patternssurvive. In the example shown in Figure 4.8 a major part of


the initial substructures die out m the first 1000-2000

evolution steps. Only two such examples (number 16 or 300)are shown. Some of them remain longer (172, 100 and 60) and

only one initial substructure is really successful and takes

part in several fitness steps (228).

0.75

0.7

0.65

S 06

0 55

0

"'% 1000 2000 3000 4000 5000 6000 7000 8000

Evolution Steps

Figure 4.8. Destiny of the initial population. The stars mark the creation

time of a substructure and the dots show, where the

substructures fall out of the population.

(2) Backward traces from fitness steps: Most of the substructures

will be mixed by crossover during evolution to form other

arrangements that never existed before. They can be traced

back, when fitness steps occur during the evolution, or at the

end of the optimization. This second type of trace is capable of

showing the history of a substructure that has produced an

increase of the fitness. They indicate where a substructure

first occurred, and if multiple substructures coexist in the

population and evolve parallel. A certain competition between

these structures may therefore be revealed. Some

substructures will temporarily be at the top of the

population's fitness ranking, while other substructures are

successful at another time. In Figure 4.9 two backward traces

from fitness steps are shown. The structure according to

Figure 4.9 a) causes a singular fitness increment whereas the

structure of Figure 4.9 b) has been successful at an earlier

stage.


» «)!- 1

s-

6

fi CD "f^*'

.*

LO, /

o

/"2000 4000

Evolution Steps

6000 2000 4000

Evolution Steps

6000

Figure 4.9. Examples of backward traces from fitness steps. Several good,

substructures are created during the optimization, see a) as an

example. They are represented in sub-populations that evolve in

parallel. One or the other will contribute to a, fitness step at a,

certain stage. There is a, competition between the different sub-

populations.

(3) Backward traces from the final population: It is possible and

desirable that new substructures are generated during the

evolution process. Such structures may survive a certain time

in the evolution process, but they are seldom at the top of the

population's fitness ranking. By tracing back substructures

from the final (or intermediate) population it is possible to see

i « » - -

ji.i 4 ——H- -l

d—•

^

ggppw^""*^"'i * * i -f

Fitness 0.6

0.7-

imess 0.6r / /

J-l

\ f ; ./m f Lfi /o

JO j

(

2000 4000 6000

Evolution Steps

a)

2000 4000 6000

Evolution Steps

b)

Figure 4. 10. To observe if there are still different sub-populations in the

actual or final, population, a trace back to earlier stages of the

population's evolution may be created. By doing so, it is possibleto observe how the evolution of sub-populations takes place.

Therefore the parallelism in the evolution is clearly visible. For

these examples, the backward traces are shown for a populationat 7300 evolution steps.


if this competition still takes place. Figure 4.10 shows that

some substructures may be included in the population but

have never produced a fitness improvement so far (Figure4.10 a)). Other substructures participated in fitness steps

earlier (e.g., Figure 4.10 b)) and may become successful againat a later stage of the evolution process.

The three types of traces show that a parallel optimization of

different structures takes place in the evolutionary process as

implemented. Each substructure may be interpreted as a part of

a sub-population containing this unique substructure. When

crossover takes place, the combination of substructures may

produce very successful new substructures as a possible basis for

a new sub-population. From the trace type (3) we can see

whether or not there are still latent substructures that may

become the fittest later. When the whole population is mainlyconstructed with only one or a few significant substructures, the

parallel search mechanism is mostly lost and the evolution

process is no longer efficient. Only a very high mutation rate

could produce other substructures. Thus, the process converges

very slowly much like a random search.

When applying evolutionary algorithms to complex and

lengthy optimization processes, a criterion to decide whether the

optimizer is efficient or not and at what time it may be

terminated would be useful. To get an idea of the progress of the

optimizer the user may observe or supervise certain variables

during the optimization. In small problems such supervising maynot be necessary, because the result is obtained rapidly.

An evolution quality figure used to represent the capability of

the population to produce further more successful structures

when continuing the optimization would be useful. Such a figure

may be defined using the number of sub-populations togetherwith the fitness of the best representing of each sub-population,normalized using the temporal maximum fitness in the whole

population.

c'w=t4?^w- {iA)

where Cp(n) is the evolution quality figure after n optimization

steps, F(n) is the temporal maximum fitness. Nsl\n) the number


of sub-populations and F,SF(n) the fitness of the most successful

représentant of the z'th sub-population. Figure 4.11 shows the

value of that evolution figure during the optimization. To get the

sub-populations, structure parts of 10 segments length were

compared. If any pattern of 10 segments in length is exactlyincluded in more than one individual, these individuals form a

sub-population.

30

Initialization Phase

o

I Evolution Phase

Terminal Phase

11LWTLL X.

Termination

0 2000 4000 6000 8000 10000 12000 14000

Evolution Steps

Figure 4.11. Value of the evolution figure during the optimization. Four

phases may be distinguished.

When observing this evolution quality figure, four phases of

the evolution process as a possible interpretation are

distinguishable. (1) Initialization phase: during this phase the

optimizer rapidly eliminates structures that are not resistant

enough to survive the evolution. The number of sub-populationsis more or less steadily decreasing during this phase. This phasemust not be confused with the generation of the initial

population. (2) Evolution phase: during this phase as many new

sub-populations are generated than are eliminated. This is the

most productive phase of the optimization. (3) Terminal phase:

during this phase all but a few sub-populations are eliminated.

The best sub-population begins to dominate. At the end of the

terminal phase the corresponding sup-population includes over

80% of the total population. (4) Termination of the evolution:


When the previous phase has ended, the evolution process may

be terminated. The termination condition has been reached.

When continuing the evolution process, because of the physical

background of this problem type, we can assume that no

significant increase of the fitness will be obtained.

Traditional methods for obtaining information about the

optimizer's state usually include values such as the fitness

evolution, fitness spreading in the population or mean values of

obtained fitnesses during the optimization. Such values do not

represent the diversity of the individuals and therefore the

optimizer's potential for further improvement. An evolution

quality figure as described here may help gain a view of the

inside state of the population.

4.7. Discussion

An evolutionary optimization procedure has been applied to a

problem in integrated optics, namely a non periodically

segmented waveguide structure for spot-size conversion. This

example shows that optimization of a very large and complex

problem is possible within a reasonable number of evaluated

structures.

The analysis of the evolution process demonstrated the

parallel search mechanism of our procedure. A supervisionmethod was presented by defining an evolution quality figure.Most of the statistically available information concerning a final

state of a population's evolution (e.g., the decreasing spread of

fitness values) usually does not accurately represent the

optimizer's potential for a further improvement. From the

viewpoint of the few but still competing patterns this soberingprospect may be reassessed into a promising one.

The evolution process shown in Figure 4.11 represents a

successful optimization. In less successful cases the evolution

phase may be shorter or even missing. Several reasons may

result in such a behavior: a poor choice of the initial population,a non-appropriate definition of the genotype, which hinders the

definition of substructures by taking neighboring segments, or

other effects, such as a bad mutation rate. It is then more

difficult to determine the terminal phase. In any case, if one

substructure starts to dominate the population, the terminal


phase has started. This condition is very easy to implement in an

optimization, but it is of course not sufficient for a correct

judgment of the optimization process.

5. A Design and Simulation Concept for

Planar Integrated Lightwave Circuits

When designing integrated optical filters there are two main

criteria to fulfill. First, the filter characteristics have to be met.

And second, the design and fabrication costs should be held low.

While the filter characteristics have to be met without

compromises, the factor of cost is always to be minimized.

The overall cost of a product is a combination of the designcosts and the fabrication costs. Depending on the production

volume, the weighting of the two factors varies in a wide range.

The following two chapters present a new approach of a designand optimization platform which can potentially reduce both cost

contributions. The design costs are reduced because the system

is able to operate autonomously with minimal user interaction,

and find solutions that are as cost-effective as possible.

The whole system is composed of a forward solver partdescribed in this chapter and a optimizer part detailed in

Chapter 6.

5.1. Introduction

Several implementations of optical filters with well known

design methods exist. Three of the most convenient structures

are waveguide grating filters [71], resonant coupler (cascadedMach-Zehnder) filters [70], and cascaded ring-resonators [102].For all three types, straight forward filter design methods exist

and can be applied to obtain the required design parameters

[103, 104]. In Figure 5.1 these three types of filters are shown

together with some design information.

The problem of all these structures is the chip space they

require. For all types, the required chip space is linearlydependent on the filter selectivity or the free spectral range that

is to be achieved.

57

58 Design and simulation concept for integrated lightwave circuits

a) Star Coupler

Waveguide Grating Filter

[71, 105,106]

Star Coupler T^esi§n Parameters:

- Number of waveguides- Waveguide spacing- Phase shifts

b)Add-

In /Directional Delay Lines

Coupler

Resonant Coupler Structure

[3, 69, 70, 107]

P Design Parameters:

Through- Number of stages- Coupling coefficients

- Delay line lengths

Coupled King Resonators

Directional Couplers [102J

c)In -

VKfa1 rvOut Design Parameters:

Drop -—A

s. ,«*M UA —Add

-- Number of rings- Coupling coefficients

Delay Lines - Ring radii (resonance

wavelengths)

Figure 5.1. Three types of optical filters for which well known designmethods exist, a) waveguide grating filter, b) resonant couplerstructure (cascaded Mach-Zehnder interferometers, FIR Filter),c) coupled ring resonator array (IIR Filter).

Approaches using ring resonators [108, 109], should be more

suited for compact designs (see examples in Figure 5.2). For such

structures, a general design method does not exist. For every

new topology, the calculation method has to be developed almost

from scratch. The work presented in this chapter directly allows

the filters to be calculated and optimized.

Design and simulation concept for integrated lightwave circuits 59

a) b)

Figure 5.2. Two examples of filters composed of ring resonators which are

more compact than standard designs. The couplers are

encircled, a) Triple-coupler ring-based waveguide resonator

[108], b) compound triple-coupler ring resonator [109].

A system, where the user can just enter the required filter

characteristics and then the system would design the most

compact filter that meets the given requirements would be very

powerful. Such a system would solve the inverse problem for the

optical filter design. It has to be composed of a forward solver

and a general optimizer to solve the inverse problem.

5.2. The Concept of the Forward Solver

The general forward solver relies on several software

components and several design representation schemes. There

are three representation formats: 1) the geometry description, 2)the semantic or functional description and 3) the netlist. Figure5.3 shows the architecture of the forward solver.

The reason to include the three representation levels into the

forward solver is to enable an optimizer to act on geometrical

structures, the simplest representation. Users of the system can

enter any of the three representations. To make the transition

between the three abstraction levels, there are two functions: 1)the semantic analyzer and 2) the netlist generator. The

implementation of these components is described in separate

sections.

Design and simulation concept for integrated lightwave circuits

InputForward Solver

Output

IncreasingAbstraction

Level

Figure 5.3. Architecture of the forward solver including three levels of

representation. They can be seen as different levels ofabstraction, of the same structure containing more and more

information about its functionality.

Waveguide

DescriptionGeometryDescription

Functional

Description

Netlist Generator

Netlist

Description

Scattering-Matrix Compilation

WaveguideDatabase

Scattering-MatrixDescription (internal)

Scattering-Matrix Analysis

Results

Figure 5.4. Flow diagram of the forward solver. Sec text for a description ofthe different steps.


To obtain a transfer function of a given structure, the

geometry is first transformed into a functional description, then

into a netlist. Together with the waveguide description, it is now

possible to obtain a scattering matrix description of the filter.

Using this scattering matrix the transfer function may be

computed. Any data that can be generally used across different

structures is stored in a waveguide database to allow rapidaccess for subsequent calculations. Such information includes

effective indices, eigenmodes and coupling coefficients. The flow

diagram of this procedure is depicted in Figure 5.4.

5.3. Details of the Forward Solver

5.3.1. The Waveguide Description

Any symmetric waveguide may be described as a superpositionof layers and laterally limited bricks, which constitute the

waveguide's cross section. Figure 5.5 shows an example of a

waveguide structure composed of the two elements. It is

assumed that the main guiding layer or brick has the highestrefractive index. From the waveguide description the eigenvaluesand eigenmodes as well as coupling coefficients can be directlycomputed.

BrickLayer

Figure 5.5. Sample of a waveguide structure composed of layers and bricks.

In the waveguide description, a standard waveguide width is

defined. To obtain other widths, the width of all bricks is

changed accordingly. For this reason, the defined waveguidewidth does not have to correspond to the width of any specific


brick allowing the user to introduce fabrication dependent

changes.

5.3.2. The Geometry Description

Almost any practical filter topology can be represented as a

concatenation of straight and bent waveguides. Additionallytapers can adapt for different waveguide widths. This means

that with only three generic elements (see Table 5.1) it is

possible to construct virtually any planar lightwave circuit.

To maintain generality, it is important that the base

description consists of only few, very simple building blocks.

In the geometry definition, absolute coordinates are given for

all elements. No connectivity check is done at all. The user has

the sole responsibility of defining geometric structures that are

correctly convertible into a functional description.

Table 5.1. Definition of the geometry elements used to define any filter

topology. The general variables X]; Yx, X7, and Y2 define the

start and end points of the element.

Element Description

Straight (A7,, F,,Ay, Y7, W)

L

W

BendLY,, F,,X2, Y7,W.R,a)

If R^O then the sign of a

defines the bending direction.

Taper (X, Y,. X, F„ Wv W?)

w,

Figure 5.6 shows an example of a simple geometric definition

composed of straights, bends and tapers.


Figure 5.6. Sample of a waveguide geometry composed of straights, bends

and tapers.

5.3.3. The Semantic Analyzer and the Netlist

The geometric description does not contain any information

about functionality of the structure. A human user could

probably easily distinguish between elements such as couplersand Y-branches by simply looking at the picture. But to enable a

program to find the function of any topology is a difficult task. A

semantic analyzer, which searches the geometry for different

functional elements is required.

Therefore it is necessary to define a set of functional elements

with which any planar lightwave circuit can be constructed. In

Table 5.2 the different functional elements and their

corresponding descriptions are shown. Constraints of each

functional element define the limits of applicability. These

constraints are stored in a waveguide library which is linked to

the waveguide definition. The corresponding values can be

defined by the user according to experimental results or

experience.


Table 5.2. Definition of the functional elements used by the semantic

analyzer to extract the function of any filter topology. The

general variables X{, Yx, X-,, and F, define the start and end

points of the elements, ini and outl represent the interface

nodes. The description syntax is the one used by the system.

Element Description

W

outA

L

StraightGuide (/«,) -> {oiltx) (L, W)

StraightGuide (A7,, 7,, X2, Y2, W)

a

W \

VétvX

BentGuide (z;?,) -> (oü/j) (R,CX, W)

BentGuide (A" Y,, X„ 7„ R,a, W)

.mt (yivCT"

D [m w^

my j rOUt'y\.

StraightCoupler {inv in7) ->

(outv out,) (D, L, Wi,W2)

StraightCoupler (Xla, Yta, X2a, Y2a, Wv

^x\b> 1IA> 'v2/i' l2b> "2>

BendStraightCoupler (/«,, in-,) ~>

{outv out2) {Y R, Dmm, Dmax, Wv

IF,, a )

BendStraightCoup 1 er ( 1, Xla, Yla, X2a,

*Ice ''I'^li' '

W ^2A> *2b> **> "2>

D.

BendStraightCoupler (in{, in0) ~>

{out,, out,) (2, R,Dmm)Dm^ Wv

W2,a)

\ p) BendStraightCoupler (2, Xx , YXa,Xla,

Y2l,Wx.X]b,Yxl,XmY2b,R,W2)


Elementi

Description

BendBendCoupler [inx, in2) ->

{outv out2) ( R^, Rz ,Dmm, DmaK,Wl}

W2, a )

BendBendCoupler (XXa, YXa, X2a, Y2o, R

y¥x,AXB. lXy A-,b, J2b'

*2' "2'

BendlnBendCoupler ( /«,, m2) ->

(outv ouf) (1,7?,, R2, Dmm, Dm^, Wv

W2,a)

BendlnBendCoupler ( 1, XXa, YUl, X2a, Y2d)R W X Y X Y R W )

>' 1' MA' 1"' 2b> -/2A'iv2> rr2>

BendlnBendCoupler {mv m2) ->

{outv ouf) (2,RX,R2, Dmm, Dmax, Wv

W2,a)

BendlnBendCoupler (2, Xia, Y]a, X2a, Y2fi,R W Y Y X Y R W )

Table 5.2. Continued.

The functional description still contains all coordinates, but it

now contains information about related elements. The geometry

elements are cut accordingly to obtain the functional elements.

Almost every filter design can be constructed out of directional

couplers and connecting waveguides.

From the functional description created by the semantic

analyzer the generation of a netlist is relatively straightforward.It defines the connectivity between the functional elements. The

netlist definition contains a minimal set of parameters necessary

to completely describe each element. Additionally the input and

output ports are numbered to define how the elements are

connected. The netlist generator automatically detects any global

input and output ports. If the user does not define and number

the ports, the program numbers them.

A.

a

ID


The program can then transform the netlist description back

into a functional description and into a geometry. This will be

necessary for the operation of any optimizer. It must be

guaranteed that the transformation back and forth does not

modify the geometry of the structure.

If any illegal constellation of geometric elements is detected bythe semantic analyzer, an internal error code is generated. Thus,the optimizer can eliminate illegal structures.

5.3.4. The Scattering Matrix Compilation

The netlist represents a number of individual elements

connected to each other by ideal links. Every element can be

represented by an individual scattering matrix. Assuming that

the whole system is single mode, only one port is required for

each interface node. Figure 5.7 shows an example of connected

scattering matrices.

B

ial nodes (k)

al nodes (p)

C

Figure 5.7. Filter structures are composed of several elements. These can be

combined such that an overall scattering matrix can be definedto make the direct connection with the external ports.

Using the connection scattering matrix method [110, 111] it is

possible to combine the whole system into one scattering matrix

representing the characteristics of the overall system with

respect to the global system inputs and outputs. To do this, all

scattering matrices of the individual elements are first placed on

the diagonal of a matrix.


[4.

[4 [°] M [o][o] [4 [o] ... [0][o] [o] [4 [0] (5.1

I»! M [«I - [4,

where \S] is the scattering matrix of the element / and M is the

total number of elements. The resulting matrix is in general very

sparse.

The rows and columns of \S] are then rearranged to place theL hot ö ^

external nodes in the uppermost lines and in the leftmost

columns. With this operation \S] is divided into four distinct^ L lioi

VmYS[S]kk,[S]pp,[S}pk and [4,.

K

KFml 1

Mkk

mPk

5] kP

s\

a,

a,

ct..

Ü..

a.

a.

(5.2)

In (5.2) at and bt represent the incoming and outgoing waves

respectively of the port i. \S\k now depends on the external

ports only and [S]?depends on the internal ports only. The

other two ([S]A and [S]k7) refer to external and internal ports.

To make the connections between the internal ports a

connection matrix [C] is needed. The matrix [C] has the same

dimensions as [S]^. The value 1 is inserted wherever two ports

are connected.


M

C C

Mi L12

C C

r c

Ci\

G2\

\\

(5.3)

where

0, port k and port / not connected

1, port k and port / connected(5.4)

are the elements and X is the number of internal ports. Onlypairs of nodes can be connected together.

The external scattering matrix LSI can then be calculated as

follows:

(5.5)[^L-^L-^LfL-M lsL-

Since the individual scattering parameters are wavelengthdependent, it is more efficient to evaluate the expression

analytically before sweeping over the wavelength. In the system

implementation the matrix inversion is therefore solved

analytically with an LR-decomposition [50]. For realistic devices

consisting of 50 to 100 elements, this evaluation takes less than

one second of computation time, but requires a considerable peakamount of memory (up to several megabytes). It is then possibleto rapidly calculate the external parameters for different

wavelengths. Any calculated values that do not change for one

wavelength are cached. Therefore for one wavelength different

scattering matrix parameters can be calculated with virtually no

additional computation effort.

5.3.5. The Waveguide Library and the WaveguideDatabase

The semantic analyzer is coupled with a waveguide librarywhich includes information such as constraints (e.g., minimum

radius for bends or minimum and maximum waveguide widths),and other conditions used for the detection of couplers. This

information is mainly provided by the user and is based on

experience concerning the fabrication process, and on

measurements.


In order to accelerate the calculations the forward solver

makes use of a waveguide database. Eigenmodes, couplingcoefficients (as a function of the wavelength and waveguidewidths) and other information used by the forward solver are

stored in this database. The solver can then rapidly access this

information without having to recalculate it every time. As soon

as an information is missing in the waveguide database, the

corresponding solver (see the following sections) is launched and

the missing data is calculated and introduced into the waveguidedatabase. The data can then be accessed, and the forward solver

can continue its calculation.

5.3.6. The Eigenmode Solver

The eigenmodes and the effective indices of the individual

waveguides are computed using the imaginary-distance beam

propagation method, as described in chapter 2. The eigenmodesolver can either be directly called from a menu, or invoked

automatically when the forward solver detects a lack of

information in the waveguide database.

5.3.7. The Calculation of the Coupling Coefficients

The most important parameters to describe the physicalbehavior of directional couplers are the coupling length Lc and

the coupling coefficients kX2 and k2X ,which can be calculated

from the propagation constants/?, and ß0 of the even and the odd

mode of the coupled waveguide structure.

Lc represents the propagation distance that is required to

couple the maximum power from one into the adjacentwaveguide and can be determined as follows:

Therefore the difference between the even and the odd

propagation constants ßi — ßn is an essential quantity when

calculating waveguide couplers.

Exact coupling coefficients could be calculated if the even and

odd modes in a coupled waveguide structure are known


analytically or numerically with a high precision. But accuratelycalculating these modes in a general fashion is a difficult and

time consuming task. It would be much easier to operate with

the eigenmodes ElX and El2 of the individual waveguides only.

This is possible using the following approximation for ße—ß0[112]:

ße-ßo =

iklK]2K2X+(ßx- ß2y ß2(i~- N^y(5.7)

ß-(\-N-)

where k is the vacuum wave number, ßx and ß2 are the

propagation constants of the individual (uncoupled) waveguides,and ß represents the average propagation constant

ß = M+ßo) = HA+A)> (5-8)

and N is the overlap integral between the two separate

eigenmodes

N= IJEn-El2dxdy. (5.9)CO

Further the coefficients K2l smclKx2 are defined as follows:

K2X = \$[n;(x>y)-n:(x<yJ\(E,i EI2-NE„ • ElX)dxdy, (5.10)

and

Kn = jj nl(x,y) - n](x,y) (ElX • El2 - XEt2 El2)dxdy, (5.11)

where nx and n1 represent the index distributions with only the

core waveguide 1 or waveguide 2 respectively, and if is the index

distribution with no waveguide core at all. No waveguide core

means that the width of the corresponding waveguide is assumed

to be zero.

The coupling coefficients k]2 and k2] can then be determined as

follows:

^12 ~

T BN + yJl-B-(l-N2) (A ~ßo) (5.12)

and

'21BX + ^l-B"(i^X-) (£-#V (5.13)


with

B=ßylA_ (5.14)ßc-ßo

This theory gives very good results for symmetrical directional

couplers. A comparison of different methods is given in [113].

It can easily be observed from (5.12) and (5.13) that the two

coupling coefficients /c12 and /c21 are always different, unless the

two waveguides are identical (B = 0) or they are uncoupled

(A = 0).

Commonly two other quantities 5 and 5,called detuning and

effective detuning, are defined as follows

S = j(ß-ß2), (5.15)

where ßx and ß2 are the propagation constants of the two

individual waveguides, and

%« =Hßc-ßo) = ^M::W:^^' (5-16)

where Kn and k2] are the two coupling coefficients between the

two waveguides which are defined in equations (5.12) and (5.13).

The problem of asymmetrical directional couplers has been

extensively treated in literature [112, 114, 115, 116, 117], but no

satisfactory result has yet been reported. The simplificationsresult in a violation of the energy conservation law. The only wayto bypass the problem is to use rigorous methods without

simplifications which require precise knowledge of the even and

odd modes of the coupler. Then, the propagation constants of

these modes (ße and ßo) are exactly known and the coupling

coefficients result directly from equations (5.9) and (5.14), which

are introduced into equations (5.12) and (5.13).

5.3.8. Elementary Scattering Matrices

The most widely used scattering matrices are those of straightand bent waveguides, and directional couplers. In this work it is

assumed that the bending radii are large enough such that no

significant radiation losses occur. Therefore the scatteringmatrices of straight and bent waveguides have the same form.


V=

S=

^st/aiehl ^beiui

( 0 e

(iß a)Lq

(;/}-a)L\

(5.17)

where ß is the propagation constant, a represents the losses,

and L is the length of the waveguide. For bent waveguides, the

length L is approximated by their neutral path, which may be

adjusted, because the field propagates slightly on the outer side

of the bend.

For directional couplers, the scattering matrix depends on the

coupling coefficients /cp and /c7, as well as on the detuning factors

8 and 8cff .See Figure 5.8 for the definition of the nodes.

S.31

i3i n— cos(8cff L) - j

_

S$,m{8,, L)

o_

c_

°42~'

J24~~

V

f(

Vcos(8Lf,L)+ j

,<5sm(8 L)

8

532 = S2i - ~-[jKn8$ii\(8elj L))e

if

-ccL

J

\

J

-al

-aL

(5.18)

^41 =- S\4 = ~(/K12Ss'm(3tifL))e11=

k 12^

21~ *^22 = ^Vi = ^U

~

^4") "^ ^44 =

This scattering matrix is valid for directional couplers with

two parallel waveguides. If the coupler consists of two bent

waveguides or any other configuration, where the distance

Figure 5.8. Example of a directional coupler with varying gap between the

waveguides. To calculate the overall scattering matrix it may be

subdnided into several segments of length LI with constant

gaps d].


between the waveguides changes over the length of the coupler,the scattering matrix can be determined by cascading small

couplers with constant distances [118] (see Figure 5.8).

The overall scattering matrix can be calculated as the cascade

of the partial scattering matrices S^'Jsegment

V_

CO)A

C(-) A...AÇ(') A-.-A Ç(A-1) A <s(A') f5 1 Q">u

couplerusegment

' N usegment

' v ' x usegment

/x ' yusegment ' v usegment ' \u.a.*j;

evaluated from left to right, where the cascade operator a is

defined in the following manner

n7 çib , eu ob

Sa a Sb =

u o31o31 + o41o32 o3]o41-t-04,0,42

b . p« c*b eta c<b 1 pa oA0 0 s^ + s:;2s;2 $°2s°x + s:;2s.42

c7 c^ j„ cc/ c^1 c7 c^ -i_ 00 c^ a n^31 31 41^32 ^32^31 ' *-*42i->32

V03,o41-t-o41o42 o32o41-i-o42o42 U 0

(5.20)

After calculating the individual scattering matrices of all

separate elements, the overall scattering matrix of the lightwavecircuit can be determined using the connection scattering matrix

method that was described in the previous section.

5.4. Examples

5.4.1. Resonant Couplers

As primary examples a number of resonant couplers are

calculated. Such resonant couplers can be used as add-dropfilters in WDM systems. A number (1 to 9) of Mach-Zehnder

stages are cascaded to show the growing free spectral range (seeFigure 5.9 for one and two stages). In this example all stageshave the same parameters. The goal is to analyze the

performance of an add-drop filter with various stages and not to

construct an optimal filter.


Drop Add-

-Through In

Figure 5.9. One and two Mach-Zehnder interferometer stages. The outputsare interchanged for even and odd numbers of stages.

The first parameter to determine is the length difference of the

delay lines. To obtain this value, it is first necessary to calculate

the delay time difference Ar [107]

Ar = 4"-' (5-21)/o

where f0 is the desired spectral period, chosen to be 800 GHz for

this example. The corresponding wavelength period of 6.4 nm

gives Ar = 1.25-1 (T12 s and a delay line length difference of

253.4 urn for an effective index of 1.48. The coupling coefficients

and coupler lengths are chosen to couple 50% of the energy into

the adjacent waveguide at about 1.56 urn.

Table 5.3 shows the results of the simulation of the devices

from 1 to 9 stages. A strong wavelength dependency of the

devices can be observed mainly due to the wavelengthdependency of the directional couplers. Nevertheless this

dependency may be neglected over one period of 800 GHz.

The selectivity is increased as can be seen in the cases with

one, five and nine stages in Table 5.3. But without optimizationthe side lobe suppression is very poor, even for 9 stages. By justadding more identical stages this problem cannot be solved. It

can only be overcome with a non-uniform distribution of the

coupling coefficients and the delay line lengths [3, 69]. A flatter

passband and a smaller crosstalk can then be achieved. Usually,to meet these requirements, an optimization is necessary. Such

an optimization will be demonstrated in the next chapter.


Table 5.3. Filter characteristic of resonant couplers with 1 up to 9 Mach-

Zehnder interferometer stages. The drop output is shown on the

left, and the through, output on the right. The results were

obtained, considering a, waveguide loss of 0.1 d,B/cm.

Number

of

Stages

Add-

In-

Drop

• Drop/Through

"Through/Drop

Through

1.55 1.57 1 1.53 1.55 1.57 X

g0.5

H

1.57 X 1.55 1.57 X

30.5

H

&3 1.55 1.57 X

4

H

0 5|t

h y

( A

1.53 1.55

05

H

A A 1

W

odiLiiiiiïiiiii,

1.53 1.55 1.57 A 1.55 1.57 X


Number

of

Stages

Add-

In -

Drop

'Drop/Through

"Through/Drop

Through

6

57 X 1 57 X

57 X 1 53 1 55 1 57 X

8

.57 X 1 57 X

9

57 X f57 X

Table 5.3. Continued.

5.4.2. Add-Drop Filters Using Ring Resonator Devices

Many filter topologies using ring resonators and corresponding-

analysis methods were reported in literature [102, 103, 108, 109.

119, 120]. In this section two types of ring resonator devices are

calculated. Both types are compact add-drop devices that use

Vernier operation to obtain both a high wavelength selectivity as

well as a large free spectral range (FSR).


Add _

In —,

Figure 5.10. Compound triple coupler ring resonator (CTCRR) add-drop

filter ]109]'.

The first filter structure treated in this section is a compound

triple coupler ring resonator (CTCRR). The schematic diagram of

the CTCRR (see Figure 5.10) has two rings with radii r, and r2 as

well as three different types of directional couplers with

amplitude coupling ratios Kv K2 and Kv The two waveguides

external to the rings have a radius of curvature at least equal to

the smallest between the radii of curvature of the rings. The

same propagation constant ß is assumed for all waveguides of

the structure. More details about this configuration may be

obtained from [106].

To enable a direct comparison with literature, the following

parameters were chosen: i\ =5.7 mm, /*, = 6.5 mm, and a

waveguide loss of a = 0.5 dB/cm. The result of the calculation is

shown in Figure 5.11. Comparing this with the result in [106], a

slight difference is visible and comes from the wavelength

dependency of the directional couplers that is also taken into

account in the present calculation.


I I

ÖO

CO

0

-10

20

-30

-40

z

2 -io

TTYTlP/rHTH0: i

[1061

12 16 20 24

ßl/K

1.5495 1.55 1.5505

Figure 5.11. Transfer cliaracteristics (Out) of the compound triple couplerring resonator structure shown in Figure 5.10.

A second filter structure to be treated in this section is a

double-ring resonator (DRR). It consists of two ring resonators

with different radii r, and r2 located between the input and

output waveguides (see Figure 5.12). The waveguides are

coupled by three directional couplers with amplitude couplingratios of Kx, K2 and A"3. Each ring has a different resonance

wavelength. By correctly choosing the ratio between the two

radii r, and r2 it is possible to obtain a free spectral range which

is related to the least common multiple of the two resonance

wavelengths. This can be referred to as Vernier operation.

In

Add

^Out

Drop

Figure 5.12. Double-ring resonator (DRR) filter structure [121].

To enable a direct comparison with literature the structure has

been calculated with the parameters r, = 5.7 mm, ;\ = 6.5 mm, and


a waveguide loss of a - 0.2 dB/cm. The calculated transmission

characteristics (see Figure 5.13) is in very good agreement with

the measurements presented in [121].

CQ

«3

O

80

60

40

(YfWV\

'121]

37.2 GHz

0 10 11 3D »0 !0 $0

RELATIVE FREQUENCY (GHz)

201.549 1.55

X

1.551

Figure 5.13. Characteristics of the dropping output of the double-ringresonator structure, shown in Figure 5.12.

5.5. Discussion

The forward solver approach described in this chapter proves

to be very accurate and fast for practical filter structures.

Because the solver is based on a geometric description and does

not require the specification of any functionality by hand, it is

very suitable to be used by a general optimizer. It is possible to

freely switch between the three representation levels and

therefore to extract any information needed. The semantic

analysis is quite complex but still very fast.

Since each representation level is available as an ASCII text,the user can easily supervise the steps and also interact at

different levels. Editors and viewers are available at every stageof the forward solver.

The use of scattering matrices for the analysis of integratedwaveguide structures reduces the complexity of the calculations

while only degrading the obtained results very slightly. In this

implementation all waveguides are assumed to be single-mode.


For a multimode analysis (this is also possible for single mode

waveguides), all partial scattering matrices would have an order

that is proportional to the number of guided and radiation modes

considered. It would then imply that the coupling coefficients

between the individual modes have to be determined as well. The

overall complexity would grow to an extent that the analyticdetermination of the overall scattering matrix is no longer a

realistic approach. It would then be easier to use a numerical

analysis method for the calculation of the scattering matrix

elements. Then, they are combined into a global scatteringmatrix numerically. The result of such calculations may

potentially be more accurate than with a strict single-modeapproach, but the increase in calculation time would be too largefor the gain in precision.

6. An Optimization Concept for Planar

Integrated Optics

In an industrial context almost every design procedure results

in an optimization process. The optimization of integrated

lightwave circuits has not yet been completely solved. Onlyparameter optimizations have been presented so far. In this

chapter a new optimization strategy is developed based on the

forward solver described in Chapter 5. Since information is

available at different levels of abstraction, the optimizer is not

only able to optimize any filter characteristic, but it is also

possible to optimize structures with respect to other parameterssuch as chip size, aspect ratio, complexity, etc.

The resulting inverse problem solver is based on an

evolutionary optimization procedure. Modifications of the

structures are done by special mutation operators, acting directlyon the geometry definition. Because it would be very difficult to

define a crossover operator between two (possibly completelydifferent) lightwave circuits, a crossover operator has not yetbeen implemented.

The following implementation of the optimizer represents a

mandatory step in the direction towards a system capable of

"inventing" new lightwave circuits.

6.1. Introduction

To build a successful inverse problem solver with an

evolutionary optimization procedure, the following elements are

necessary:

1) A general format that can represent every generated structure

(this topic has been covered in Chapter 5).2) A robust forward solver that gives useful results for realistic

structures (this has also been described m Chapter 5).3) A fitness definition which allows a correct qualification of the

individuals with respect to the given specifications.4) Mutation operators to transform the structures.

81

82 Optimization concept for planar integrated optics

5) Crossover operators or other statistical procedures to

integrate information about several individuals into newly

generated ones.

6) A supervision method delivering an abortion criterion (such a

method has been proposed in Chapter 3).

First, the overall optimizing strategy is described, then the

points 3-5 are explained.

6.2. The Optimizer Strategy

The optimizer strategy is depicted in Figure 6.1. The forward

solver is not a black box for the optimizer, therefore it is possiblefor it to use all information about the different stages and levels

of representation. This enables a hybrid structural and

parameter optimization.

Inverse Problem Solver

Figure 6.1. General architecture of the inverse problem solver for planar

integrated lightwave circuits.

The optimization procedure itself resembles the breeder

genetic algorithm used in Chapters 3 and 4 to design the spot-

size converter. The only major difference is that no crossover

operator has been defined in this implementation.

It is possible to introduce several restrictions for the optimizer.Such restrictions are useful when the overall topoLogy should not

change, and optimization of the remaining parameters is desired.

For example, the user can forbid the separation of directional

Optimization concept for planar integrated optics 83

couplers, or let rings move in horizontal or vertical directions

only. Many restrictions act on the functional description. A

design rule check ensures the correct functionality of the

structures. If the optimizer produces a geometrical structure that

results from illegal operations, the structure is rejected, and a

new structure will be generated. The restrictions are mostly used

in cases where the designer knows that a structure is near the

topological optimum.

6.3. The Fitness Definition

The definition of the desired filter characteristics is a

piecewise linear function in the linear scale (see Figure 6.4 as an

example). Each part of the definition consists of either an upper

or a lower limit and a weighting factor that defines the

importance of that limit. For each segment a selection can be

made, which scattering parameter is concerned, and if the

corresponding power, group delay or the dispersion should be

considered for the limit.

The difference between the calculated filter response and the

constraint is integrated over the ranges where the constraints

are not met. This results in the following fitness definition:

F =

1 JX \[max(0Sl(X)^Eu(X))

y

7= 1 ?Af/l1î!ll

dX +

max(ÖJ'L(X)-~S'L(X)) dX

, (6.1)

-i'/ m,,/

where the symbols are defined as follows:

N0-, XL : number of upper and lower limits,

w'LJ, w'L : weighting factors for the limits,

^mm^l/nm'^mm^/max: wavelength ranges for the limits,

ELI (X) : definition of the upper limit /,

EL(X): definition of the lower limit /,

S'b(X): value of the transfer function, group delay or

dispersion that has to be compared with the

upper limit /,


S'L(X): value of the transfer function, group delay or

dispersion that has to be compared with the lower

limit /, and

p: defines the p-norm of the distance.

This fitness definition has a lower limit of zero and an upper

limit of one. The weighting factors Wv and w'L, can implement

any special behavior without modifying the fitness calculation

procedure.

Additionally the filter function may be shifted along the

wavelength axis. This makes perfect sense because a wavelengthshift of the final design can easily be achieved by scaling the

geometry. Tuning elements can also be implemented to achieve a

fine tuning of the filter response [3, 122].

6.4. Mutation Operators

The geometry is modified with different mutation operators.

Each mutation operator has a different probability that can be

set individually. Two of the most important are shown in Figure

6.2, shifting of any node and rotation of the node axis. In both

cases a number of statistically selected nodes is locked prior to

the mutation to limit the geometrical extent of the mutation.

H-..

ts<U

^ a\

a) b)

Figure 6.2. Two mutation operators, a) node shift, b) node rotation. The

black dots show fix-points that are not moved by the mutation

operator. Depending on these fix-points the operation may have

different results.


Other mutation operators are, e.g.. the addition and deletion of

rings as well as the scaling and displacement of ring elements.

For all mutation types the connectivity must be maintained and

constraints like the minimum bending radius or the minimum

distance between waveguides have always to be fulfilled.

Since the optimizer has access to functional information about

the current structure, constraints such as not to separate

couplers, not to modify certain types of elements, etc. can be used

to optimize the structure without modifying the network

topology.

6.5. Example: A Resonant Coupler Add-DropDevice

In this section an optimization is shown. It consists of a 5-

stage resonant coupler add-drop filter. As input, five identical

stages are introduced resulting in a poor dropping performanceand a large crosstalk. Figure 6.3 shows the device together with

its filter characteristic.

A number of constraint levels were defined for the

optimization. Figure 6.4 shows the constraints and the

corresponding weighting factors that were used (see the

definition of the fitness (6.1) for a explanation of the different

factors). A rectangular filter shape was given with several levels

of constraints. This prevents the optimizer from producingstructures that fulfill the constraints in a large wavelengthrange and have very poor performance in the remaining part.The constraints are much more demanding than the structure

can ever fulfill. The optimization will converge to a structure

that gives the best fitness. This structure best approximates the

given filter characteristics.

Optimization concept for planar integrated optics

-Drop

-Through

a)

1.56

1.56

3. a) Schematics of the 5-stage resonant coupler consisting of 6

directional couplers, b) and c) the initial filter characteristics

(drop-channel) with a linear and logarithmic scale respectively.This starting structure has large losses in the passband, and a

very poor performance in the stop band.


w[ =3000 ,--4W

w2L = 9000

>X4(^)

L\;(X)\

\\

4(X)\

= 600 \

4(x) 4(a.)

\ w* = 60° /\

i K = 200 l i w?,=200 ,

0.95

0.85

0.1

0.05

1.544

>A

un > 'W/r

1.55

1.5496 1.5504

1.556

Xl ^

!\\ ^3 "l

1,2 -Vt^t/nux ' ^C/max

AT/mm> 'Vmm'

T.1 ">2

Figure 6.4. The optimization, constraints are shown in this figure. 6

different constraints are introduced, each, of which, having a,

different weighting factor, shoivn beneath the lines.

For this optimization example, a population size of 20 was

chosen. Experience shows that for topological optimizations the

population size should be kept small. Figure 6.5 shows the

fitness evolution during the optimization.

0.7

0.6

0.5

0.4

^ 0.3

0.2

0.1

a

1

r-ffîy.^^'yfry^y-,

500 1000 1500

Optimization Steps2000

Figure 6.5. This graph, shows the evolution of the fitness during the

optimization. The top line represents the fitness of the best

individual, the line below the least fitness which is still included

in the population. Dots show the fitness of every evaluation.


The best performing structure was found after about 2000

optimization steps and has a much lower crosstalk than the

starting structure (see Figure 6.6). The losses in the passbandare also low. Depending on the fitness definition, some regions

may have a higher importance. In this example every part of the

filter characteristics had the same importance. The equiripple in

the stop band could be improved by introducing additional

constraints.

1.56

1.56

Figure 6.6. Filter characteristics of the optimized, structure, a) linear scale,

b) logarithmic scale. A large improvement is visible when

comparing this with, the characteristics shown in Figure 6.3.

For this particular example more efficient optimizationmethods exist. However, we show that this optimizer is able to

treat known problems. The advantage of the present optimizingapproach is that it is potentially able to treat problems where no

theory or design method exists. The user will not have to develop


such a theory, he will just have to enter the new topology and

optimize.

Figure 6.7 shows the coupling ratios and the difference of

delay line lengths for the optimized structure.

249 I —, ,

r——|

| 248 8 r—,

= 248 6-

S 248 4 •

g 248 2 -

123456 12345

Coupler Stage

a) b)

Figure 6.7. Distribution of a) the amplitude coupling ratios and b) the delaylines for the optimized filter structure. All values are valid at a

wavelength of 1.55 /urn.

6.6. Discussion

A procedure was implemented for the topological optimizationof planar lightwave circuits. It is very flexible and allows the

optimization of many aspects of the structure. Especially it is

possible to optimize the filter characteristics and the

corresponding dispersion at the same time.

Since the phase conditions are very critical to filter

characteristics, two almost identical structures may have

completely different performance. Therefore, finding a goodtopology by just randomly putting together basic waveguideelements is very unlikely. Given a topology that corresponds to

the desired structure, the optimizer is then able to find an

optimal configuration. A given topology defines the number and

types of directional couplers, as well as the feed-backs. If an

optimal structure cannot be found using the given topology the

optimizer is able to modify the functional structure to obtain a

new topology.

Additional work has to be done to enable the system to

autonomously discover new filter topologies. Such a functionality

30%

erf

U 20%

o

U

-i no/


could be implemented by acquiring statistical information duringthe optimization and using a database of well performingstructures. The statistical information could partially replace the

crossover operator. A structure database may be used to compose

filters out of more complex functional blocks. It is nevertheless

not yet clear, how the optimization would function in future. The

rapid development of the computer performance will certainlyallow "inventor" software to become a reality.

7. Conclusions and Outlook

In this thesis a number of topics were addressed in the domain

of optical lightwave circuit design. In a first part, the simulation

of waveguide structures with the beam propagation method was

investigated. In this context, an improved method for eigenmodeextraction has been developed. This technique is based on the

imaginary-distance beam propagation method. Using a

mathematical trick in the context of a finite difference solver it is

possible to directly extract higher order propagation modes of

arbitrary shaped waveguide structures. This method is very fast

and accurate, and the results are especially well adapted for

BPM simulations.

The beam propagation method, despite of its limitations, is

very suitable for complex calculations. This was shown by the

design of a very short spot-size converter of typically less than

140 pm length. The structure of the converter is a non-

periodically segmented waveguide with irregular tapering. Usingan evolutionary optimization procedure a solution that improvesthe coupling losses from an optical fiber to the chip from 3.5 dB

down to about 1.3 dB was found. Thus, evolutionary algorithmsare well suited for such optimizations. Using these algorithmswe have found that non conventional designs potentiallyoutperform straightforward implementations.

A thorough analysis of the genetic algorithm applied for the

optimization of the spot-size converter showed the internal

behavior of the evolution. The implicit parallelism was

demonstrated using the notation of traces of sub-populations.These sub-populations were also the basis for the definition of a

evolution quality figure. As a first application, this figure can be

used as an abortion criterion of the optimization. It is also

possible to qualify the efficiency of the evolution when analyzingthe temporal behavior of the evolution quality figure. This is a

promising method of qualifying an evolutionary algorithm.

In the following, a new optimizer architecture for planarintegrated lightwave circuits was designed and implemented.The forward solver part is based on three representationschemes (geometry, functional description and netlist). Each

level contains an increasing amount of information about the

91

92 Conclusions and outlook

functionality of the structure. A scattering matrix solver is then

used to calculate the output spectra of the structures. Because

information is available at every stage of the forward solver, an

optimizer can act on different levels of abstraction. In particularit can modify the geometry by using different mutation

operators, but it can also use the functional description to test

constraints on a higher level of abstraction. The actual

implementation of the optimizer is based on an evolutionaryprocedure. The presented implementation of the optimizer is a

very important step towards an autonomous program which is

able to construct new filter topologies based on different

previously designed structures.

During this work, a Microsoft Windows® based application was

written, combining all elements of this thesis into a user friendly

program. Editors and viewers are available for every step of the

design. An interpreted high level language (the syntax is similar

to that of C) allows a very flexible use of the BPM solver.

Further investigations are required to improve the "inventor"

part of the optimizer. Statistical information about the

optimization and the structures may be useful to guide the

optimizer towards the right topology that can implement a

desired filter characteristics. A database of designs could be used

as a template library to obtain different successful substructure.

These could then be combined into a new topology. Scientific

research should go on in this direction.

The program needs some further development if it should be

made commercially available. In particular, graphical editors

should be implemented for a simpler drawing and modification of

the optical structures. It would be very desirable to have such a

tool for BPM simulations as well as for the CAD of planarintegrated lightwave circuits.

References

[1] Denzel, W.E., and B. Meekers, "Photonics in the backbone

of corporate networks - the ACTS COBNET project" in

Photonic Networks, Optical Technology and Infrastructure,

NOC'97, D.W. Faulkner and A.L. Harmer, eds., IOS Press,

1997, pp. 25-32.

[2] Kawachi, M., "Silica wavegiddes on silicon and their

application to integrated-optic components" Opt. and

Quantum Electron., vol. 22, no. 5, September 1990,

pp. 391-416.

[3] Offrein, B.J., R. Germann, G.L. Bona, F. Horst, and

H.W.M. Salemink, "Tunable optical add/drop componentsin silicon-oxynitride waveguide structures," Proc. ECOC'98,

September 1998, pp. 325-326.

[4] Li, Y.P., and C.H. Henry, "Silica-based optical integratedcircuits" IEE Proc.-Optoelectron., vol. 143, no. 5, October

1996, pp. 263-280.

[5] He, J.-J., B. Lamontagne, A. Delâge, L. Erickson, M. Davis,and E.S. Koteles, "Monolithic integrated wavelengthdemulitplexer based on a waveguide rowland circle gratingin InGaAsP/InP" J. Lightwave Technol., vol. 16, no. 4,

April 1998, pp. 631-638.

[6] Gini, E., W. Hunziker, and H. Melchior, "Polarization

independent InP WDM multiplexer/demultiplexer module"

J. Lightwave Technol., vol. 16, no. 4, April 1998, pp. 625-

630.

[7] Feit, M.D., and J.A. Fleck, Jr., "Light propagation in

graded-index optical fibers," Appl. Opt., vol. 17, no. 24,December 1978, pp. 3990-3998.

[8] Jüngling, S., and J.C. Chen, "A study and optimization ofeigenmode calculations using the imaginary-distance beam-

propagation method," IEEE J. Quantum Electron., vol. 30,no. 9, September 1994, pp. 2098-2105.

93

94 References

[9] Spühler, M.M., D. Wiesmann, P. Freuler, and M. Diergardt,"Direct computation of higher-order propagation modes

using the imaginary-distance beam propagation method,"Opt. and Quantum Electron., vol. 31. no. 9, vol. 9/10,October 1999, pp. 751-761.

[10] Spuhler, M.M, B.J. Offrein, G.-L. Bona, R. Germann, 1.

Massarek, and D. Erni, "A very short planar silica spot-sizeconverter using a nonperiodic segmented waveguide," J.

Lightwave Technol., vol. 16, no. 9, September 1998,

pp.1680-1685.

[11] Mühlenbein, H., and D. Schlierkamp-Voosen, urpae science

of breeding and its application to the breeder genetic

algorithm (BGA)f' Evolutionary Computation, vol. 1, no. 4,Winter 1993. pp. 335-360.

[12] Erni, D., M. Spuhler, and J. Fröhlich, "A generalizedevolutionary optimization procedure applied to waveguidemode treatment in non-periodic optical structures" Proc.

ECIO'97, April 1997, pp. 218-221.

[13) Spuhler, M.M., D. Erni, and J. Fröhlich, "An evolutionaryoptimization procedure applied to the synthesis of integratedspot-size converters" Opt. and Quantum Electron., vol. 30,no. 5/6, May 1998, pp. 305-321.

[14] Erni, D., M.M. Spühler, and J. Fröhlich, "Evolutionaryoptimization of non-periodic coupled-cavity semiconductor

laser diodes," Opt. and Quantum Electron., vol. 30, no. 5/6,May 1998, pp. 287-303.

[15] Spuhler, M.M, B.J. Offrein, G.-L. Bona, R. Germann, and

D. Erni, "Compact spot-size converters using non-periodic

segments for high refractive index contrast planarwaveguides," Proc. CLEO/EUROPF/98, September 1998,

p. 234.

[16] Spuhler, M.M., B.J. Offrein. G.L. Bona, D. Erni, and I.

Massarek, "Design and implementation of short opticalspot-size converters" Bulletin SPG/SSP, vol. 15, no. 1,

February 1998, p. 94.

[17] Paul, H., and J. Tmdle, "Passive optical netivork planningin local access networks - an optimisation approachutilising genetic algoiithms" BT Technol. J., vol. 2, no. 2,

April 1996, pp. 110-115.

References 95

[18] Koh, S., and L. Ye, "Modeling, simulation, and verification

of photonic integrated circuits using VHDL." Proc. SPIE,vol. 2659, January 1996, pp. 234—211.

[19] Jakubczyk, Z., M. Florjanczyk, and V. Tzolov, "Modelingtools in waveguide optics," Proc. SPIE, vol. 2994, February1997, pp. 310-318.

[20] Glingener, C, J.-P. Elbers, and E. Voges, "CAE for photonic

waveguide devices," AEU Int. J. Electron. Commun.,vol. 51, no. 1, January 1997, pp. 20-28.

[21] Spofford, C.W., "A symposis of the AESD workshop on

acoustic propagation modelling by non-ray-tracing

techniques," Rpt. TN-73-05, Office of Naval Research,

Washington D.O., 1973.

[22] Tappert, F.D.. "The parabolic equation approximation," in

J.B. Keller and J.S. Papadakis, eds., Wave propagation and

underwater acoustics, Lecture Notes in Physics, vol. 70,

Springer, Berlin, 1977, pp. 224-287.

[23] Yevick D., and B. Hermansson, "Efficient beam propagationtechniques," IEEE J. Quantum Electron., vol. 26, no. 1,

January 1990, pp. 109-112.

[24] Tsuji Y., and M. Koshiba, "A finite element beam

propagation method for strongly guiding and longitudinally

varying optical waveguides," J. Lightwave Technol., vol. 14,no. 2, February 1996, pp. 217-222.

[25] Tsuji Y., M. Koshiba, and T. Shiraishi, "Finite element beam

propagation method for three-dimensional opticalwaveguide structures," J. Lightwave Technol., vol. 15, no. 9,

September 1997, pp. 1728-1734.

[26] Chung. Y., and N. Dagh, "An assessment of finite differencebeam propagation method" IEEE J. Quantum Electron.,vol. 26, no. 8, August 1990, pp. 1335-1339.

[27] Huang, W., C. Xu, S.-T. Chu, and S.K. Chaudhun, "The

finite-difference vector beam propagation method: analysisand assessment," J. Lightwave Technol., vol. 10, no. 3,March 1992, pp. 295-305.

[28] Hoekstra. H.J.W.M., G.J.M. Krijnen. and P.V. Lambeck,"New formulation of the beam propagation method based on

the slowly varying envelope approximation,'' Opt. Comm.,vol. 97, no. 5/6, April 1993, pp. 301-303.

96 References

[29] Lui, W.W., W.-P. Huang, and K. Yokoyama, "Precision

enhancement for optical waveguide analysis," J. LightwaveTechnol., vol. 15, no. 2, February 1997, pp. 391-396.

[30] Yamauchi J., J. Shibayama, O. Saito, 0. Uchiyama, and H.

Nakano, "Improved finite-difference beam-propagationmethod based on the generalized Douglas scheme and its

application to semivectorial analysis," J. LightwaveTechnol., vol. 14, no. 10, October 1996, pp. 2401-2406.

[31] Rao, H.. R. Scarmozzmo, and R.M. Osgood, "A bidirectional

beam propagation method for multiple dielectric interfaces,"IEEE Photon. Technol. Lett., vol. 11, no. 7, July 1999,

pp. 830-832.

[32] Hayashi, K., M. Koshiba, Y. Tsuji, S. Yoneta, and R. Kaji,"Combination of beam propagation method and mode

expansion propagation method for bidirectional opticalbeam propagation analysis," J. Lightwave Technol., vol. 16,

no. 11, November 1998, pp. 2040-2045.

[33] Mitomi, O., and K. Kasaya, "An improved semivectorial

beam propagation method using a finite-element scheme,"IEEE Photon. Technol. Lett., vol. 10. no. 12, December

1998, pp. 1754-1756.

[34] Xu, C.L., W.P. Huang, and S.K. Chaudhuri, "Efficient and

accurate vector mode calculations by beam propagationmethod" J. Lightwave Technol., vol. 11, no. 7, July 1993,

pp.1209-1215.

[35] Fogli, F., G. Bellanca, P. Bassi, I. Madden, and W.

Johnstone, "Highly efficient full-vectorial 3-D BPM

modeling of fiber to planar waveguide couplers," J.

Lightwave Technol., vol. 17, no. 1, January 1999, pp. 136-

143.

[36] Mansour I., A.-D. Capobianco. C. Rosa, "Noniterative

vectorial beam propagation method with a smoothingdigital filter" J. Lightwave Technol., vol. 14, no. 5, May1996, pp. 908-913.

[37] Xu, C.L., W.P. Huang, J. Chrostowski, and S.K. Chaudhuri,"A full-vectorial beam propagation method for anisotropicwaveguides," J. Lightwave Technol., vol. 12, no. 11,November 1994. pp. 1926-1931.

References 97

[38] Kriezis, E.E., and A.G. Papagiannakis, "A joint finite-difference and FFTfull vectorial beam propagation scheme"

J. Lightwave Technol., vol. 13, no. 4, April 1995, pp. 692-

700.

[39] Huang, W.P., C.L. Xu, S.T. Chu, and S.K. Chaudhuri, "A

vector beam propagation method for guided-wave optics,"IEEE Photon. Technol. Lett., vol. 3, no. 10, October 1991,

pp. 910-913.

[40] Heiblum, M., and J.H. Harris, "Analysis of curved optical

waveguides by conformai transformation," IEEE J.

Quantum Electron., vol. 11, no. 2, February 1975, pp. 75-

83.

[41] Lee C.-T., M.-L. Wu, L.-G. Sheu, P.-L. Fan, and J.-M. Hsu,

"Design and analysis of completely adiabatic taperedwaveguides by conformai mapping" J. Lightwave Technol.,vol. 15, no. 2,'February 1997, pp. 403-410.

[42] Wu, M.-L., P.-L. Fan. J.-M. Hsu, and C.-T. Lee, "Design ofideal structures for lossless bends in optical waveguides byconformai mapping" J. Lightwave Technol., vol. 14, no. 11,November 1996, pp. 2604-2614.

[43] Fan, P.-L., M.-L. Wu, and C.-T. Lee, "Analysis of abruptbent waveguides by the beam propagation method and the

conformai mapping method" J. Lightwave Technol., vol. 15,no. 6, June 1997, pp. 1026-1031.

[44] Rivera, M.. "A finite difference BPM analysis of bent

dielectric waveguides," J. Lightwave Technol., vol. 13, no. 2,

February 1995, pp. 233-238.

[45] Sewell, P., T.M. Benson, T. Anada, and P.C. Kendall, "Bi-

oblique propagation analysis of symmetric and asymmetricY-junctions" J. Lightwave Technol., vol. 15, no. 4, April1997, pp. 688-696.

[46] Lee, C.-T., M.-L. WYi, and J.-M. Hsu. "Beam propagationanalysis for tapered ivaveguides: taking account of the

curved phase-front effect in paraxial approximation," J.

Lightwave Technol., vol. 15, no. 11, November 1997,

pp.2183-2189.

[47] Haus, H.A., "Waves and fields in optoelectronics," Prentice-

Hall, Englewood Cliffs. NJ. 198 1, 402 p.

98 References

[48] Lee, P.-C, and E. Voges, "Three-dimensional semi-vectorial

wide-angle beam propagation method" J. LightwaveTechnol., vol. 12, no. 2, February 1994, pp. 215-225.

[49] Yamauchi, J., J. Shibayama, M. Sekiguchi, and H. Nakano,

"Improved multistep method for wide-angle beam

propagation"

IEEE Photon Technol. Lett., vol. 8, no. 10,October 1996, pp. 1361-1363.

[50] Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P.

Flannery, "Numerical recipes in C, the art of scientific

computing" 2nd ed., Cambridge University Press, 1992,994 p.

[51] Vinsome, P.K.W., "Orthomin, an iterative method forsolving sparse sets of simultaneous linear equations," Proc.

Fourth SPE Symp. on Numerical Simulation of Reservoir

Performance, Soc. Pet. Eng. of AIME, February 1976,

pp. 149-159.

[52] Hadley, G.R., "Transparent boundary condition for the

beam propagation method" IEEE J. Quantum Electron.,vol. 28, no. 1, January 1992, pp. 363-370.

[53] Vassallo, C. and J.M. van der Keur, "Highly efficienttransparent boundary conditions for finite difference beam

propagation method at order four," J. Lightwave Technol.,vol. 15, no. 10, October 1997, pp. 1958-1965.

[54] Vassallo, C, and F. Collino. "Highly efficient absorbingboundary conditions for the beam propagation method," J.

Lightwave Technol., vol. 14, no. 6, June 1996, pp. 1570-

1577.

[55] Hoekstra, H.J.W.M., G.J.M. Krijnen, and P.V. Lambeck,

"Efficient interface conditions for the finite difference beam

propagation method" J. Lightwave Technol., vol. 10, no. 10,October 1992, pp. 1352-1355.

[56] Yevick, D., and B. Hermansson, "New formulations of the

matrix beam propagation method: application to rib

waveguides," IEEE J. Quantum Electron., vol. 25, no. 2,

February 1989. pp. 221-229.

[57] Chen, J.C, and S. Jüngling, "Computation of higher-orderwaveguide modes by imaginary-distance beam propagationmethod," Opt. and Quantum Electron., vol. 26, no. 3. March

1994, pp. S199-S205.

References 99

[58] Yamauchi, J., M. Sekiguchi, 0. Uchiyama, J. Shibayama,and H. Nakano, "Modified, finite-difference formula for the

analysis of semivectorial modes in step-index opticalwaveguides," IEEE Photon. Technol. Lett., vol. 9, no. 7,

July 1997, pp. 961-963.

[59] Wijnands, F., H.J.W.M. Hoekstra, G.J.M. Krijnen, and

R.M. de Ridder, "Numerical solution method of nonlinear

guided modes with a finite difference complex axis beam

propagation method," IEEE J. Quantum Electron., vol. 31,

no. 5, May 1995, pp. 782-790.

[60] Lee, J-S., and S.-Y. Shin, "On the validity of the effective-index method for rectangular dielectric waveguides," J.

Lightwave Technol, vol. 11, no. 8, August 1993, pp. 1320-

1324.

[61] Yamauchi, J., G. Takahashi, and H. Nakano, "Modifiedfini te-difference formula for semivectorial H-field solutions

of optical waveguides," IEEE] Photon. Technol. Lett.,vol. 10, no. 8, August 1998, pp. 1127-1129.

[62] Wijnands, F., H.J.W.M. Hoekstra, G.J.M. Krijnen, and

R.M. de Ridder, "Modal fields calculation using the finitedifference beam propagation method," J. LightwaveTechnol, vol. 12, no. 12, December 1994, pp. 2066-2072.

[63] Huang, WT., and H.A. Haus, "A simple variational approachto optical rib ivaveguides," J. Lightwave Technol., vol. 9,no. 1, January 1991, pp. 56-61.

[64] Thylén, L., "The beam propagation method: an analysis ofits applicability" Opt. and Quantum Electron., vol. 15,no. 5, September 1983, pp. 433-439.

[65] Takagi. A.. K. Jinguji, and M. Kawachi, "Silica-based

waveguide-type wavelength -insensitive couplers (WINC's)with series-tapered coupling structure," L. LightwaveTechnol., vol. 10, no. 12, December 1992, pp. 1814-1824.

[66] Ramadan, T.A., R. Scarmozzmo, and R.M. Osgood, Jr.,

"Adiabatic couplers: design rules and optimization" J.

Lightwave Technol., vol. 16, no. 2, February 1998, pp. 277-

283.

100 References

[67] Veldhuis, G.J., J.H. Berends, and P.V. Lambeck, "Designand characterization of a mode-splitting lF-junction" J.

Lightwave Technol., vol. 14, no. 7, July 1996, pp. 1746-

1752.

[68] Klekamp, A., P. Kersten, and W. Rehm, "An improvedsingle-mode Y-branch design for cascaded 1:2 splitters," J.

Lightwave Technol., vol. 14, no. 12, December 1996,

pp. 2684-2686.

[69] Offrein, B.J., G.L. Bona, F. Horst. H.W.M. Salemink, R.

Beyeler, and R. Germann, "Wavelength tunable optical add-

after-drop filter with flat passband for WDM networks,"IEEE Photon. Technol. Lett., vol. 11, no. 2, February 1999,

pp. 239-241.

[70] Kuznetsov. M., "Cascaded coupler Mach-Zehnder channel

dropping filters for wavelength-division-multiplexed opticalsystems," J. Lightwave Technol., vol. 12, no. 2, February1994, pp. 226-230.

[71] Dragone, C, "Efficient N x N star couplers using fourieroptics" J. Lightwave Technol., vol. 7, no. 3, March 1989,

pp. 479-489.

[72] Fischer, U., T. Zmke, J.-R. Kropp, F. Arndt, and K.

Petermann, "0.1 dB/cm waveguide losses in single-modeSOI rib Waveguides," IEEE Photon. Technol. Lett., vol. 8,no. 5, May 1996, pp. 647-648.

[73] Soref, R.A., J. Schmidtchen, and K. Petermann, "Largesingle-mode rib waveguides in GeSi-Si and Si-on-Si02,"IEEE J. Quantum Electron., vol. 27. no. 8, August 1991,pp.1971-1974.

[74] Kohl, A.. R. Zengerle, H.J. Bruckner, H.J. Olzhausen, R.

Müller, and K. Heime, "Fabrication of buried InP/InGaAsP

waveguide-wires for use in optical beamwidth

transformers," Microelectronic Engineering, vol. 21, April1993, pp. 483-486.

[75] Kawano, K., M. Kohtoku, N. Yoshimoto, S. Sekine, and Y.

Noguchi, "2 x 2 InGaAlAs/InAlAs multiquantum well

(MQW) directional coupler waveguide switch modules

integrated with spotsize convertor^s," IEE Electron. Lett.,

vol. 30, no. 4, February 1994. pp. 353-355.

References 101

[76] Wenger, G., L. Stoll, B. Weiss, M. Schienle, R. Müller-

Nawrath, S. Eichinger, J. Müller, B. Acklin, and G. Müller,

"Design and fabrication of monolithic optical spot size

transformers (MOST's) for highly efficient fiber-chip

coupling," J. Lightwave Technol., vol. 12, no. 10, October

1994, pp. 1782-1790.

[77] Mitomi, 0., K. Kasaya, and H. Miyazawa, "Design of a

single-mode tapered waveguide for low-loss chip-to-fiber

coupling," [EEE J. Quantum Electron., vol. 30, no. 8,

AugusU994, pp. 1787-1793.

[78] de Ridder, R.M., R.A. Wijbrans, H. Albers, J.S. Aukema,P.V. Lambeck, H.J.W.M. Hoekstra, and A. Driessen, "A

spot-size transformer for fiber-chip coupling in sensor

applications at 633 nm in silicon oxynitride," Proc.

LEOS'95, vol. 2, October 1995, pp. 86-87.

[79] Zengerle, R., O. Leminger, W. Weiershausen, K. Faltin, and

B. Hubner, "Laterally tapered InP-InGaAsP waveguides forlow-loss chip-to-fiber butt coupling: a comparison ofdifferent configurations" IEEE Photon. Technol. Lett.,vol. 7, no. 5, May 1995, pp. 532-534.

[80] Zengerle, R., B. Jacobs, WT. Weiershausen, K. Faltin, and A.

Kunz, "Efficient spot-size transformation using spatiallyseparated tapered InP/InGaAsP twin waveguides," J.

Lightwave Technol., vol. 14, no. 3, March 1996, pp. 448-

453.

[81] Mitomi, 0.. K. Kasaya, Y. Tohmori, Y. Suzaki, H. Fukano,Y. Sakai, M. Okamoto, and S.-I. Matsumoto. "Optical spot-size converters for low-loss coupling between fibers and

optoelectronic semiconductor devices," J. LightwaveTechnol., vol. 14, no. 7, July 1996, pp. 1714-1720.

[82] Paatzsch, T., I. Smaglinski, M. Abraham, H.-D. Bauer, U.

Hempelmann, G. Neumann, G. Mrozynski, and W.

Kerndlmaier. "Very low-loss passive fiber-to-chip couplingwith tapered fibers," Appl. Opt., vol. 36, no. 21, July 1997,

pp.5129-5133.

[83] Kawano, K., M. Kohtoku, M. Wada. H. Okamoto, Y. Itaya,and M. Naganuma, "3-D semivectorial beam propagationanalysis of a spotsize-converter-integrated laser diode in the

1.3-pm-wavelength region," IEEE Photon. Technol. Lett.,vol. 9, no. 1, January 1997, pp. 19-21.

102 References

[84] Weissmann, Z., and I Hendel. "Analysis of periodically

segmented waveguide mode expanders" J. LightwaveTechnol., vol. 13, no. 10, October 1995, pp. 2053-2058.

[85] Chou, M.H., M.A. Arbore, and M.M. Fejer, "Adiabatically

tapered periodic segmentation of channel waveguides formode-size transformation and fundamental mode

excitation," Opt. Lett., vol. 21, no. 11, June 1996, pp. 794-

796.

[86] Weissmann, Z., and A. Hardy, "Modes of periodicallysegmented waveguides," J. Lightwave Technol., vol. 11,

no. 11, November 1993, pp. 1831—1838.

[87] Weissman, Z., and A. Hardy, "2-D mode tapering via

tapered channel waveguide segmentation," ÏEE Electron.

Lett., vol. 28, no. 16, July 1992, pp. 1514-1516.

[88] Liang, T., and R.W7. Ziolkowski, "Mode conversion of

ultrafast pulses by grating structures in layered dielectric

waveguides," J. Lightwave Technol., vol. 15, no. 10, October

1997^pp. 1966-1973.

[89] Offrem, B.J., G.L. Bona, R. Germann, I. Massarek, and

H.W.M. Salemmk, "High contrast and low loss SiON

optical waveguides by PECVD," Proc. IEEE/LEOS Symp.,Benelux Chapter, November 1996, pp. 290-293.

[90] Hoffmann, M., P. Kopka, and E. Voges, "Low-loss fiber-matched low-temperature PECVD waveguides with small-

core dimensions for optical communication systems," IEEE

Photon. Technol. Lett., vol. 9, no. 9, September 1997,

pp.1238-1240.

[91] Elyutin, P.V.. and F. Ladouceur, "Adiabaticity and random

wave propagation," Opt. Lett., vol. 22, no. 16, August 1997,

pp. 1241-1243.

[92] Ladouceur, F., "Roughness, inhomogeneity, and integratedoptics," J. Lightwave Technol., vol. 15, no. 6, June 1997,

pp.1020-1025.

[93] Goh, T., S. Suzuki, and A. Sugita, "Estimation of waveguidephase error in silica-based waveguides," J. LightwaveTechnol, vol. 15, no. 11, November 1997, pp. 2107-2113.

[94] Holland, J.H, "Adaptation m natural and artificialsystems" The University of Michigan Press, 1975, 183 p.

References 103

[95] Grefenstette, J.J., and Baker, J.E.. "How genetic algorithmswork: a critical look at implicit parallelism," Proc. Third

Intern. Conf. on Genetic Algorithms, June 1989, pp. 20-27.

[96] Winter, G., J. Périaux. M. Galân, and P. Cuesta, eds.,"Genetic algorithms in engineering and computer science,"

Wiley, New York, Chichester, 1995, 464 p.

[97] Fröhlich, J., "Evolutionary optimization for computational

electromagnetics" Diss. ETH No. 12232, 1997.

[98] Moosburger. R., C. Kostrzewa, G. Fischbeck, and K.

Petermann, "Shaping the digital optical switch usingevolution strategies and BPM," Proc. ECIO'97, April 1997,

pp. 22-25.

[99] Johnson, J.M., and Y. Rahmat-Samii, "Genetic algorithmsin engineering electromagnetics" IEEE Ant. Prop. Mag.vol. 39, no. 4, August 1997, pp. 7-25.

[100] Goldberg, D.E.. "Genetic algorithms in search, optimization,and machine learning," Addison-Wesley, Reading, MA,

1989, 412 p.

[101] Syswerda G., "A study of reproduction in generational and

steady-state genetic algorithms" m Foundations of Genetic

Algorithms, edited by G.J.E. Rawlins, Morgan Kaufmann

Publishers, San Mateo, CA, 1991, pp. 94—101.

[102] Orta, R., P. Savi, R. Tascone, and D. Trinchero, "Synthesis

of multiple-ring-resonator filters for optical systems," IEEE

Photon. Technol. Lett., vol. 7, no. 12, December 1995,

pp.1447-1449.

[103] Jinguji, K,, "Synthesis of coherent two-port optical delay-line circuit witJi ring waveguides" J. Lightwave Technol,vol. 14, no. 8, August 1996. pp. 1882-1898.

[104] Oppenheim. A.V.. and R.W. Schäfer. "Discrete-time signalprocessing," International Edition, Prentice-Hall,

Englewood Cliffs. NJ, 1989, 879 p.

[105] Dragone, C, C.H. Henry, LP. Kaminow, and R.C. Kistler,

"Efficient multichannel integrated optics star coupler on

silicon"

IEEE Photon. Technol Lett., vol. 1, no. 8, August1989, pp. 241-243.

[106] Dragone, C, "An N x N optical multiplexer using a planararrangement of two star couplers," IEEE Photon. Technol.

Lett., vol 3, no. 9, September 1991, pp. 812-815.

104 References

[107] Jinguji, K., and M. Kawachi, "Synthesis of coherent two-port

lattice-form optical delay-line circuit," J. Lightwave

Technol, vol. 13, no. 1, January 1995, pp. 73-82.

[108] Barbarossa, G., A.M. Matteo, and M.N. Armenise,

"Theoretical analysis of triple-coupler ring-based optical

guided-wave resonator," J. Lightwave Technol, vol. 13,

no. 2, February 1995, pp. 148-157.

[109] Barbarossa, G., A.M. Matteo, and M.N. Armenise,

"Compound triple-coupler ring resonators for selective

filtering applications in optical FDM transmission systems,"Proc. SPIE, vol. 2401, February 1995, pp. 86-94.

[110] Monaco, V.A., and P. Tiberio, "Automatic scattering matrix

computation of microwave circuits," Alta Frequenza,

vol. 39, no. 2, February 1970, pp. 165-170.

[Ill] Epprecht, G., "Die S-Matrix zusammengesetzter Systeme",

Skript, Kurs 35-136, Mikrowellen-Laboratorium ETHZ,

1974, pp. 1-9.

[112] Marcuse, D., "Tlieory of dielectric optical waveguides,"Academic Press, Inc., San Diego, CA, 1991, 380 p.

[113] Working Group I, COST 216, "Comparison of different

modelling techniques for longitudinally invariant integrated

optical waveguides," IEE Proc, Part J, Optoelectronics,vol. 136, no. 5, October 1989, pp. 237-280.

[114]Yariv, A., "Coupled-mode theory for guided-wave optics,"IEEE J. Quantum Electron., vol. 9, no. 9, September 1973,

pp. 919-933.

[115] Taylor, H.F., and A. Yariv, "Guided wave optics," Proc.

IEEE, vol. 62, no. 8, August 1974, pp. 1044-1060.

[116] Hardy, A., and W. Streifer. "Coupled mode theory of

parallel waveguides," J. Lightwave Technol, vol. 3, no. 5,

October 1985, pp. 1135—1146.

[117] März, R., "Integrated optics, design and modeling," Artech

House, Inc., Norwood, MA, 1995, 336 p.

[118] Little, B.E., "Filter syntliesis for coupled ivaveguides," J.

Lightwave Technol, vol. 15. no. 7, July 1997, pp. 1149-

1155.

References 105

[119]Madsen, CK., and G. Lenz, "Optical all-pass filters forphase response design with applications for dispersioncompensation," IEEE Photon. Technol. Lett., vol. 10, no. 7,

July 1998, pp. 994-996.

[120] Soref, R.A., and B.E. Little, "Proposed N-Wavelength M-

Fiber WDM CrossConnect switch using active microringresonators," IEEE Photon. Technol. Lett., vol. 10, no. 8,

August 1998, pp. 1121-1123.

[121] Oda, K., N. Takato, and H. Toba, "A wide-FSR double-ringresonator for optical FDM transmission systems," J.

Lightwave Technol, vol. 9, no. 6, June 1991, pp. 728-736.

[122] Chu, S.T., W. Pan, S. Sato, T. Kaneko. B.E. Little, and Y.

Kokubun, "Wavelength trimming of a, microring resonator

filter by means of a UV sensitive polymer overlay" IEEE

Photon. Technol. Lett., vol. 11, no. 6, June 1999, pp. 688-

690.

Acknowledgments

Many people were involved in this project. It is nearly

impossible to name all of them. Nevertheless I want to give some

special thanks.

First of all I want to thank my professor. Werner Bächtold for

his support and for giving me the possibility to make this work

at the IFH. I like also to thank René Dändliker and Gian-Luca

Bona for their interest in this work and for acting as co-

examiners. Special thanks go to Daniel Erni, who always had the

right answer and kept away from me a lot of administrative

stuff. He gave me a lot of support and 1 am very grateful to him.

I want to mention my colleagues and friends Nicolas Piller and

Eric Nicolet. Nicolas gave me the motivation to pursue the

studies at the ETH, without him I wouldn't stand here. I am also

very grateful to Eric. Not only was he a good friend during the

studies, our semester project was the basis for this Ph.D. thesis.

Many thanks go to my colleagues at the laboratory. EspeciallyI want to mention Martin Schmatz, with whom I had many

fruitful discussions, Hansruedi Benedickter. who was the movingspirit of the institute. Ray Ballisti was always present to helpwith the computer problems. Many thanks go to Jürg Fröhlich,

whose genetic algorithm was the bootstrap of my research

activities. I also thank the colleagues of the optics group,

especially Dorothea WTesmann, for the good ambiance and many

discussions. Finally I want to thank Charlotte Biber for the final

reading of this text.

From the IBM Research Laboratory I want to give specialthanks to Gian-Luca Bona. Bert Offrein, Roland Germann and

Folkert Horst. Their support and the many fruitful discussions

had a big influence for the successful outcome of this work.

I also want to thank my mother and my family for their

continuous support and love during my whole studies.

Finally, I want to thank my wife Eva for the infinite patience,help and love she gave me during this work.

107

Curriculum Vitae

Name: Michael—Martin SPÜHLER

Date of birth: April 30, 1969 in Fribourg, Switzerland

Nationality: Swiss, citizen of Zürich and Wasterkingen ZH

Married to Eva, one son Simon, born on

December 22nd 1999.

Education:

8.1976-7.1984: Primary and secondary school

8.1984-6.1988: Electromechanical apprenticeship10.1988-11.1991: Studies in electrical engineering at the Ecole

d'Ingénieurs de Fribourg.11.1991: Diploma (HTL) in electrical engineering;

diploma thesis: "Visualisation et commande

d'un processus chimique".11.1992—4.1996: Studies in electrical engineering at the Swiss

Federal Institute of Technology (ETH), Zurich.

4.1996: Diploma in electrical engineering; diplomathesis: "Charakterisierung von unscharfen

blobförmigen 3-D Strukturen.

10.1994-10.1997: Studies in teaching (Teaching Certificate in

Higher Education) at the ETH Zurich.

5.1996-1.2000: Research assistant and doctoral student at the

Swiss Federal Institute of Technology (ETH),

Laboratory for Electromagnetic Fields and

Microwave Electronics.

Professional Experience:

10.1989-12.1991: Electronic developments at Contrinex SA,

Fribourg, Switzerland.

1.1992—10.1992: Full-time software engineer at Sintro

Electronics AG, Interlaken, Switzerland.

1.1993-8.1995: Software developments at Sintro Electronics

AG, Interlaken. Switzerland.

since 10.1996: Partial-time lecturer at the University of

Applied Sciences (former HTL), Zurich,Switzerland.

109

Publications

Journal Papers and Chapters in Books:

[1] Spühler, M.M., D. Erni, and J. Fröhlich, "An evolutionaryoptimization procedure applied to the svnthesis of integrated

spot-size converters," Opt. and Quantum Electron., vol. 30,

no. 5/6, May 1998. pp. 305-321.

[2] Erni, D., M.M. Spuhler, and J. Fröhlich. "Evolutionaryoptimization of non-periodic coupled-cavity semiconductor

laser diodes," Opt. and Quantum Electron., vol. 30, no. 5/6,

May 1998, pp. 287-303.

[3] Spuhler, M.M., B.J. Öftrem, G.L. Bona, R. Germann, I.

Massarek, and D. Erni, "A very short planar silica spot-sizeconverter using a non-periodic segmented waveguide," J.

Lightwave Technol, vol. 16, no. 9, September 1998,

pp.1680-1685.

[4] Spuhler, M M., D. Wiesmann, P Freuler, and M. Diergardt,"Direct computation of higher-order propagation modes using

the imaginary-distance beam propagation method," Opt. and

Quantum Electron., vol. 31, no. 9/10, October 1999, pp. 751-

761.

[5] Erni, D., D. Wiesmann, M.M. Spuhler, S. Hunziker, B.

Oswald, J. Fröhlich, and C Hafner, "Evolutionaryoptimization algorithms in computational optics," Chapter mRecent Research Developments in Optical Engineering,Research Signpost. Trivandrum. India. 1999, pp. 19-36.

[6] Spuhler, MM., and D. Erni. "Towards structural

optimization of planar integrated lightwave circuits," Opt.and Quantum Electron

.in press

[7] Erni, Ü., D. Wiesmann, M.M. Spuhler, S. Hunziker, E.

Moreno, B. Oswald. J Fröhlich, and C. Hafner, "Applicationof evolutionary optimization algorithms in computationaloptics," ACES Journal submitted

111

112 Publications

Conferences and Workshops:

[8] Erni, D., M.M. Spühler, and J. Fröhlich, "A generalized

evolutionary optimization procedure applied to waveguidemode treatment in non-periodic optical structures," Proc.

Eurpean Conf. on Integrated Optics (ECIO), Stockholm,

Sweden, April 1997, pp. 218-221.

[9] Spühler, M.M., D. Erni, and J. Fröhlich, "Topological

investigations on evolutionary optimized non-periodic optical

structures," Int. WTorkshop on Optical Waveguide Theory and

Numerical Modeling, Twente, the Netherlands, September1997.

[10] Spühler, M.M., B.J. Offrein, G.L. Bona, D. Erni, and I.

Massarek, "Design and implementation of short optical spot-

size converters," SPG Jahrestagung, Bern, Bulletin

SPG/SSP, vol. 15, no. 1, February 1998, p. 94.

[11] Spuhler, M.M., B.J. Offrein, G.L. Bona, R. Germann, and D.

Erni, "Compact spot-size converters using non-periodic

segments for high refractive index contrast planar

ivaveguides," Conf. on Lasers and Electro-Optics (CLEO),

Glasgow, UK, Technical Digest, September 1998, p. 234.

[12] Spuhler, M.M., Ü. Wiesmann, P. Freuler, M. Diergardt, and

D. Erni, "Accelerated computation of higher-order

propagation modes using the imaginary-distance BPM" Int.

Workshop on Optical Waveguide Theory and Numerical

Modeling, Hagen, Germany, September 1998.

[13] Spuhler, M.M.. and D. Erni, "A design and optimization

platform for integrated optica devices" SPG Jahrestagung,Bern, Bulletin SPG/SSP, vol. 16, February 1999. p. 43.

[14] Spuhler, M.M., D. Wiesmann, and D. Erni, "Evolutionaryoptimization in computational optics," invited, Proc.

Progress in Electromagn. Research Symp. (PTERS), Taipei,Taiwan, March 1999, p. 765.

Publications 113

[15] Neuhold, S.M., M.L. Schmatz. M. Hässig, M.M. Spühler, and

G. Storf, "Combined broad and narrow band multichannel

PD measurement system with high, sensitivity for GIS" Proc.

Eleventh Int. Symp. on High-Voltage Engineering (ISH),

London, UK, vol. 5, August 1999, pp. 152-155.

[16] Spühler, M.M., and D. Erni, "Structural optimization in

planar integrated optics," Int. Workshop on Waveguide

Theory and Numerical Modeling. Saint-Etienne, France,

September 1999.

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