Document [original]

GaAs-based components for photonic

integrated circuits

vorgelegt von

Master of Science

Bassem Arar

geb. in Daraa

von der Fakultät IV −Elektrotechnik und Informatik

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

−Dr.-Ing. −

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Bernd Tillack

Gutachter: Prof. Dr. Günther Tränkle

Gutachter: Prof. Dr.-Ing. Klaus Petermann

Gutachter: Prof. Dr.-Ing. Rainer Michalzik (Universität Ulm)

Tag der wissenschaftlichen Aussprache: 25. Januar 2019

Berlin, 2019

Abstract

Semiconductor lasers based on GaAs/AlGaAs double heterostructures have recently re-

ceived increasing interest for atomic spectroscopy applications. For example GaAs/Al-

GaAs double heterostructure laser radiation at the wavelengths of 780 nm and 1064 nm

is used for rubidium spectroscopy, and for molecular iodine spectroscopy (at 532 nm

using frequency-doubling techniques), respectively. Optical systems used for these

applications should feature compact, mechanically stable, and highly efficient compo-

nents in order to be suitable for operation in harsh environments (e.g. quantum optic

precision experiments in drop towers or in space). State-of-the-art optical systems use

micro-integrated laser modules with a small footprint. However, passive components

that are necessary for the manipulation of the laser light such as phase and amplitude

modulators, fiber couplers, and beam splitters are only commercially available on

macro-scales. These components are then integrated into the laser systems platforms

where additional opto-mechanical components are indispensable. Because of their ex-

cessive request for space, commercially available laser systems and electro-optic setups

are not suitable for deployment in the field or in space. The miniaturization of passive

components for reduction of mass and form factor of electro-optical systems as well as

for improving the overall robustness and reliability of the latter is a prerequisite for

field- and space deployment of quantum sensors.

This work presents GaAs/AlGaAs double heterostructure electro-optic phase mod-

ulators for the first time at the wavelength of 780 nm and at the wavelength of 1064 nm.

The specifications of the phase modulators meet the micro-integration requirements

into hybrid laser modules and spectroscopy modules with respect to optical aperture

and electrical interface. Details related to design and fabrication of the phase modu-

lators are included. The characterization of the electric and electro-optic properties of

the phase modulators is then presented. The GaAs/AlGaAs double heterostructure

of the phase modulators is further used to develop, design, and realize waveguide cou-

plers. Multi-mode interference (MMI) couplers for applications at the wavelength of

780 nm are presented for the first time. MMI couplers can replace fiber-based couplers

in complex optical systems and are core building blocks for photonic integrated circuits

(PICs). The compatibility of the double heterostructure phase modulators and MMI

couplers for implementation of PICs is then demonstrated through a monolithically-

integrated Mach-Zehnder-intensity modulator at 780 nm for the first time.

Further, a method based on heterodyne interferometry is developed for the in-

depth investigation of phase modulators including phase and amplitude modulation

response, including modulation efficiency, residual amplitude modulation, and signal

distortion in GaAs-based phase modulators. During hybrid integration of a phase

modulator chip into an electro-optical hybrid system, this method can be applied

to optimize the coupling efficiency in real time while at the same time reducing the

residual amplitude modulation. The method further provides the means to separate

linear and the non-linear effects in the phase modulation signal. It is shown how

this provides the means to determine separately linear and quadratic electro-optics in

GaAs/AlGaAs double heterostructures. For the first time, the quadratic electro-optic

coefficient is determined without having to calculate the contribution of free carriers

effects to phase modulation.

This thesis work is organized as follows:

First, the fundamentals of guided wave optics are presented (chapter 2). Electro-

magnetic wave propagation in planar waveguides is described, conditions for guided op-

tical modes are provided, and concepts for waveguide couplers are discussed. Then the

electro-optic properties of GaAs are presented and phase modulation in GaAs/AlGaAs

double heterostructures due to electro-optic effects is introduced (chapter 3).

Next, the design of GaAs/AlGaAs double heterostructure phase modulators (chap-

ter 4) and waveguide couplers (chapter 5) is presented. For the design of the phase

modulators we study the state-of-the-art GaAs-based electro-optic phase modulators

at the wavelength of 1.31 nm and transfer these concepts to the wavelengths of 780 nm

and 1064 nm. Efficient GaAs/AlGaAs double heterostructures with phase modulation

efficiencies larger than 15 deg /(V ·mm) are designed. The design of low loss and po-

larization maintaining waveguide couplers and phase modulators then allows for the

development of a Mach-Zehnder intensity modulator (chapter 6) at the wavelengths

of 780 nm.

In the experimental part of this work, after fabrication (chapter 7), the electro-

optic performance of these devices is characterized experimentally. The performance of

phase modulators, waveguide couplers, and MZI modulators is determined (chapter 8).

The modulation efficiency of 16 deg /(V ·mm) is demonstrated for phase modulators

at 780 nm and at 1064 nm. The propagation losses amount to 1.2 dB/cm at 780 nm and

to 4.3 dB/cm at 1064 nm (which corresponds to an improvement beyond state-of-the-

art for GaAs-based phase modulators at 1064 nm). For the Mach-Zehnder-intensity

modulator, the extinction ratio of more than 10 dB and excess loss of less than 3 dB

are demonstrated. Further, the novel heterodyne analysis method for in-depth char-

acterization of the electro-optic performance of phase modulators is developed and

implemented (chapter 9). As an application of the heterodyne analysis method, the

linear and the quadratic electro-optic coefficients of GaAs/AlGaAs double heterostruc-

tures are determined. For the linear electro-optic coefficient we find results that are

in agreement with literature. However, the results for the quadratic electro-optic co-

efficients differ from values given in the literature. The discrepancy is discussed and

suggestions to solve it are provided.

Kurzfassung

Halbleiterlaser auf Basis von GaAs/AlGaAs Doppelheterostrukturen erhalten in let-

zter Zeit zunehmend Interesse für Anwendungen in quantenoptischen Experimenten,

z.B. in der Atomspektroskopie. Zum Beispiel wird mit GaAs/AlGaAs Diodenlasern

erzeugte, kohärente Strahlung bei den Wellenlängen von 780 nm und 1064 nm für

die Rubidium- und Iodspektroskopie eingesetzt. Elektro-optische Systeme für diese

Anwendungen müssen aus kompakten, mechanisch stabilen, und hocheffiziente Kom-

ponenten aufgebaut sein, um den Einsatz in rauen Umgebungen (z.B. in Falltürmen

oder im Weltraum) zu erlauben. In aktuellen optischen Lasersystemplatformen werden

mikrointegrierte Lasermodule mit geringem Platzbedarf eingesetzt. Jedoch sind pas-

sive Komponenten, die für die Manipulation des Laserlichts notwendig sind, wie z.B.

Phasen- und Amplitudenmodulatoren, Faserkoppler und Strahlteiler, kommerziell nur

auf Makroskalen verfügbar. Diese Komponenten werden dann in die Lasersystemplat-

tformen integriert, was zusätzliche opto-mechanische Komponenten erforderlich macht.

Aufgrund ihres erheblichen Platzbedarfs sind State-of-the-Art Lasersysteme für den

Einsatz im Feld oder im Weltraum nicht geeignet. Die Miniaturisierung von passiven

Bauelementen zur Reduzierung von Masse und Formfaktor von elektrooptischen Sys-

temen sowie zur Verbesserung der Robustheit und Zuverlässigkeit der Systeme ist eine

Voraussetzung für den Einsatz von Quantensensoren im Feld oder im Weltraum.

Im Rahmen dieser Arbeit werden GaAs/AlGaAs Doppelheterostruktur elektro-

optische Phasenmodulatoren zum ersten Mal für die Wellenlänge von 780 nm und

für die Wellenlänge von 1064 nm realisiert. Die Entwicklung, Herstellung, und an-

schließend die Charakterisierung der elektrischen und elektrooptischen Eigenschaften

der Phasenmodulatoren wird in Details präsentiert. Die GaAs/AlGaAs Doppelhetero-

struktur der Phasenmodulatoren wird außerdem verwendet, um Wellenleiterkoppler zu

entwickeln und zu realisieren. Multi-Mode-Interferenz (MMI) Koppler und Richtkop-

pler für Anwendungen bei der Wellenlänge von 780 nm werden zum ersten Mal vorgest-

ellt. MMI Koppler können Faser-basierte Koppler in komplexen, Optischesystemen

ersetzen und sind Kernbausteine für photonische integrierte Schaltungen. Die Kom-

patibilität der Doppelheterostruktur Phasenmodulatoren und MMI-Koppler für die

Realisierung photonischer integrierten Schaltungen wird außerdem durch die Real-

isierung eines Mach-Zehnder-Intensitätsmodulators bei der Wellenlängen von 780 nm

zum ersten Mal demonstriert.

Darüber hinaus wird ein auf heterodyner Interferometrie basierendes Messverfahren

zur Untersuchung der Modulationseffizienz, der Restamplitudenmodulation, und der

nichtlineare Signalverzerrung in Phasenmodulatoren entwickelt. Während der hy-

briden Integration eines Phasenmodulatorchips in ein elektrooptisches Hybridsystem

kann dieses Verfahren angewendet werden, um in Echtzeit die Kopplungseffizienz zu

optimieren und gleichzeitig die Restamplitudenmodulation zu reduzieren. Außerdem

können mit diesem Verfahren linearer und nicht-linearer Response im Phasenmod-

ulationssignal getrennt werden. Es wird gezeigt, wie dies die Möglichkeit bereit-

stellt, lineare und quadratische elektro-optische Koeffizienten in GaAs/AlGaAs Dop-

pelheterostrukturen unabhängig voneinander zu bestimmen: zum ersten Mal kann der

quadratische elektrooptische Koeffizient bestimmt werden, ohne dass der Beitrag der

freier Ladungsträger zur Phasenmodulation ab initio berechnet werden muss.

Diese Arbeit ist wie folgt strukturiert:

Zunächst wird die Physik der geführten Wellenoptik vorgestellt (Kapitel 2). Die

elektromagnetische Wellenausbreitung in planaren Wellenleitern wird beschrieben, Be-

dingungen für geführte optische Moden werden bereitgestellt, und Konzepte für Wellen-

leiterkopplern werden diskutiert. Schließlich werden die elektrooptischen Eigenschaften

von GaAs vorgestellt und die elektrooptischen Effekte in GaAs/AlGaAs Doppelheteros-

trukturen beschrieben (Kapitel3).

Im Anschluss werden Phasenmodulatoren (Kapitel 4) und Wellenleiterkoppler (Kapi-

tel 5) basierend auf GaAs/AlGaAs Doppelheterostruktur entworfen. Für das Design

der Phasenmodulatoren wird der Stand der Technik bei GaAs-basierten elektroop-

tische Phasenmodulatoren analysiert und ausgehend davon die Übertragung der für

1.31 µmbestehenden Konzepte auf die Wellenlängen 780 nm und 1064 nm vorgenom-

men. Das Design von Phasenmodulatoren basierend auf einer GaAs/AlGaAs Dop-

pelheterostruktur wird erarbeitet. Die Simulationen versprechen eine Modulation-

seffizienz von mehr als 15 deg /(V ·mm). Schließlich wird das Design der Doppel-

heterostruktur genutzt, um das Design von Wellenleiterkoppler zu erstellen, so dass

diese Komponenten zusammen mit dem Phasenmodulator für das Design eines Mach-

Zehnder Intensitätsmodulator (MZI) genutzt werden können. Das Design des MZIs

folgt in Kapitel 6.

Im experimentellen Teil werden die Strukturen nach der Herstellung (Kapitel 7)

charakterisiert. Die Phasenmodulatoren, die Wellenleiterkoppler, und die MZI Mdu-

latoren werden experimentell untersucht (Kapitel 8). Die für 780 nm und 1064 nm real-

isierten Phasenmodulatoren zeigen eine Phasenmodulationseffizient von 16 deg /(V ·mm).

Die Ausbreitungsverluste betragen 1.2 dB/cm bei 780 nm und 4.3 dB/cm bei 1064 nm

und stellen daher eine Verbesserung gegenüber dem Stand der Technik (12 dB/cm

für GaAs/AlGaAs Phasenmodulatoren bei 1.06 µm.) dar. Mit dem Mach-Zehnder-

Intensitätsmodulator werden ein Extinktionsverhältnis von mehr als 10 dB und zusät-

zliche Verluste (excess loss) von weniger als 3 dB demonstriert. Ferner werden die

Phasenmodulatoren mit dem Heterodynanalyseverfahren untersucht (Kapitel 9). Mit-

hilfe des neuartigen Messverfahrens werden lineare und nichtlineare Effekte in dem

Phasenmodulationssignal getrennt. Daraus können der lineare elektro-optischen (LEO)

und der quadratische elektro-optische (QEO) Koeffizienten der GaAs/AlGaAs Doppel-

heterostrukturen bestimmt werden. Für den LEO Koeffizienten finden wir Ergebnisse,

die in guter Übereinstimmung mit der Literatur stehen. Die Ergebnisse für den QEO

Koeffizienten unterscheiden sich jedoch von den in der Literatur beschriebenen Werten.

Die Diskrepanz wird diskutiert und Vorschläge zu ihrer Lösung werden skizziert.

Acknowledgments

First and foremost I want to express my sincere gratitude to Professor Günther

Tränkle who gave me the opportunity to accomplish this thesis at the

Ferdinand-Braun-Institut, Leibniz Institut für Höchstfrequenztechnik (FBH) which

has been of a great value for me, both, academically and personally. I also thank him

for his generous professional supervising and constant support.

It has been a great honor for me to work with Dr. Andreas Wicht, an exceptional

scientist whose commitment to research have inspired me every day. I greatly

appreciate his personal qualities and his openness and I thank him for his constant

support, his patience, and his contribution to this work. I also thank his invaluable

help proofreading my thesis.

I am grateful to Dr. Hans Wenzel for sharing his robust experience in the theory of

semiconductor lasers and his valuable contribution to this work.

I thank Dr. Reiner Güther for the valuable discussions and for his contribution to

understanding the electro-optic properties of GaAs.

I would like to thank Dr. Harendra H. J. Fernando from Leibniz Institute for

Astrophysics Potsdam, for introducing me to the simulation software RSoft CAD.

I deeply thank Dr. Olaf Brox for the wafer layout design and processing, Dr. Andre

Maaßdorf for growing the structures, Dr. Peter Ressel for the AR-coating, Arnim

Ginolas and Sabrina Kreutzmann for mounting of the passive waveguide chips.

My warm gratitude goes to all members of the group "Lasermetrologie" for the

motivating, inspiring, and warm working atmosphere. I thank my colleagues

Christian Kürbis and Christoph Pyrlik for the valuable discussions at different stages

in this work. I also thank Heike Christopher, Max Schiemangk, Dr. Ahmad

Bawamia, and Robert Smol for their support and continuous willingness to help

whenever required.

Thank you Rashed AlToma, my friend, for your support and encouragement.

I gratefully acknowledge the funding sources that made my Ph.D. work possible. I

would like to thank the DLR for the funding under the grant number 50WM1141.

My deepest thanks to my family who have always encouraged me and believed in me.

Words can not express how grateful I am to my father Mohammad Arar and my

mother Fatima Kewan. My two brothers and best friends, Moammer and Thabet

whose unconditional support have always accompanied me. My sisters Samah, Hiba,

Wafa, and Rawda. And finally, thank you my dear wife Doaa, with love.

Contents

Motivation 1

Preamble ..................................... 1

Micro-integrated laser systems, state of the art . . . . . . . . . . . . . . . . . 3

1 Introduction 6

1.1 Developments in III-V semiconductor phase modulators . . . . . . . . . 6

1.2 Thesis objectives: GaAs-based passive photonic components . . . . . . 7

1.2.1 Objective: GaAs-based phase modulators . . . . . . . . . . . . . 8

1.2.2 Objective: GaAs-based couplers . . . . . . . . . . . . . . . . . . 9

1.2.3 Application: GaAs-based amplitude (intensity) modulator . . . 10

1.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Fundamentals of guided-wave optics 13

2.1 Electro-magnetic wave propagation in planar waveguides . . . . . . . . 14

2.1.1 Planar electro-magnetic waves . . . . . . . . . . . . . . . . . . . 14

2.1.2 Reflection of planar waves . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Planar waveguides . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.4 Ridgewaveguides.......................... 20

2.1.5 Bentwaveguides .......................... 21

2.2 Planar waveguide couplers . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Concepts for ridge waveguide couplers . . . . . . . . . . . . . . 22

2.2.2 Self-imaging in multi-mode interference couplers . . . . . . . . . 24

2.2.3 Interference mechanisms of self-imaging . . . . . . . . . . . . . . 27

2.2.4 Characteristics of waveguide couplers . . . . . . . . . . . . . . . 28

3 Theory of GaAs-based electro-optic phase modulators 29

3.1 Electro-optic effects in GaAs . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 Linear and quadratic electro-optic effects . . . . . . . . . . . . . 30

3.1.2 The index ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Free carrier effects (carrier density-related effects) . . . . . . . . . . . . 34

3.3 GaAs-based double heterostructures . . . . . . . . . . . . . . . . . . . . 34

3.3.1 The p-n junction and the heterojunctions . . . . . . . . . . . . . 35

3.3.2 GaAs/AlGaAs double heterostructures phase modulators . . . . 35

3.3.3 Phase modulation efficiency in double heterostructures . . . . . 36

vii

4 Design of GaAs-based phase modulators 38

4.1 Material profile for GaAs-based phase modulators . . . . . . . . . . . . 38

4.2 Design of phase modulators for laser radiation at 780 nm . . . . . . . . 39

4.2.1 Electro-optic coefficients of Al0.35Ga0.65As ............ 39

4.2.2 Vertical waveguide at 780 nm . . . . . . . . . . . . . . . . . . . 40

4.2.3 Lateral waveguide at 780 nm . . . . . . . . . . . . . . . . . . . . 46

4.3 Design of phase modulators for laser radiation at 1064 nm . . . . . . . 47

4.3.1 Electro-optic coefficients of GaAs at 1064 nm . . . . . . . . . . 47

4.3.2 Vertical waveguide at 1064 nm . . . . . . . . . . . . . . . . . . . 48

4.3.3 Lateral waveguide at 1064 nm . . . . . . . . . . . . . . . . . . . 50

5 Design of GaAs-based waveguide couplers 51

5.1 General remarks on multi-mode interference couplers . . . . . . . . . . 51

5.2 Design rules of the multi-mode interference couplers . . . . . . . . . . . 52

5.3 Numerical methods for simulation of waveguide couplers . . . . . . . . 53

5.4 Design of MMI couplers for applications at 780 nm . . . . . . . . . . . 54

5.4.1 Access waveguides . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4.2 Geometry of the multi-mode waveguide . . . . . . . . . . . . . . 54

5.4.3 Interfaces of the multi-mode waveguide to access waveguides . . 59

5.4.4 S-bends ............................... 60

5.5 Design of directional couplers . . . . . . . . . . . . . . . . . . . . . . . 60

5.6 Conclusions ................................. 61

6 Application: Mach-Zehnder intensity modulator 62

7 Fabrication of GaAs-based passive components 64

7.1 Materialgrowth............................... 64

7.2 Lithography................................. 65

8 Electro-optic Performance 67

8.1 Electro-optic Performance: GaAs-based phase modulators . . . . . . . 67

8.1.1 Electrical properties of phase modulators . . . . . . . . . . . . 67

8.1.2 Phase modulation efficiency . . . . . . . . . . . . . . . . . . . . 69

8.1.3 Propagation losses . . . . . . . . . . . . . . . . . . . . . . . . . 70

8.1.4 Conclusions ............................. 73

8.2 Electro-optic Performance: GaAs-based couplers . . . . . . . . . . . . . 74

8.2.1 Bentwaveguides .......................... 74

8.2.2 Directional couplers . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.2.3 Multi-mode interference couplers . . . . . . . . . . . . . . . . . 78

8.2.4 Conclusions ............................. 81

8.3 Electro-optic performance: Application . . . . . . . . . . . . . . . . . . 82

8.3.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3.2 Mach-Zehnder intensity modulator with 1x2 couplers . . . . . . 82

8.3.3 Mach-Zehnder intensity modulator with 2x2 couplers . . . . . . 84

8.3.4 Conclusions ............................. 85

viii

9 Heterodyne analysis of GaAs-based phase modulators 87

9.1 Novel method for electro-optic characterization of phase modulators . . 88

9.1.1 Method description . . . . . . . . . . . . . . . . . . . . . . . . 88

9.1.2 In-depth analysis of the modulation signal . . . . . . . . . . . . 89

9.1.3 Sensitivity of novel method . . . . . . . . . . . . . . . . . . . . 91

9.2 Implementation of the novel method . . . . . . . . . . . . . . . . . . . 91

9.2.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . 91

9.2.2 Measurement accuracy . . . . . . . . . . . . . . . . . . . . . . . 93

9.2.3 Preparation of modulator chips for characterization . . . . . . . 93

9.3 Measurement of the phase modulation efficiency . . . . . . . . . . . . . 95

9.3.1 Phase modulator chip C3059-3 010118 at 1064 nm . . . . . . . . 96

9.3.2 Phase modulator chip D2043-3 080511 at 780 nm . . . . . . . . 97

9.4 Residual amplitude modulation . . . . . . . . . . . . . . . . . . . . . . 98

9.5 Modulation bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.6 Determination of the electro optic coefficients . . . . . . . . . . . . . . 101

9.6.1 Electro-optic coefficients of GaAs at 1064 nm . . . . . . . . . . 101

9.6.2 Electro-optic coefficients of Al0.35Ga0.65As at 780 nm . . . . . . 104

9.7 Conclusions .................................105

10 Conclusions and Outlook 107

10.1 GaAs-based phase modulators and waveguide couplers . . . . . . . . . 107

10.2 In-depth characterization of phase modulators . . . . . . . . . . . . . . 108

List of Abbreviations and Symbols 111

Appendix A: GaAs (AlxGa1−xAs) compound semiconductors 113

.1 Zinkblendestructures............................113

.2 GaAswafers.................................114

.3 Optical properties of GaAs . . . . . . . . . . . . . . . . . . . . . . . . . 115

.3.1 Energy band gap and absorption in GaAs . . . . . . . . . . . . 115

.3.2 Refractive index of AlxGa1−xAs ..................115

Appendix B: List of measured chips using the FP method 117

List of Publications 119

Bibliography 121

Motivation

Preamble

The way air traffic has revolutionized our life is indispensable. We check into the board-

ing machines to travel thousands of miles taking time-saving and safety for granted. In

their cockpits, pilots relay on navigation systems, e.g. the global positioning system

(GPS), to determine their position and navigate safely into the crowded air corri-

dors. However, the GPS navigation does not provide an error-free transmission of

information. The main source of error in GPS navigation systems is the inaccurate

time keeping by the clock of the receiver1. The key solution for a new generation of

navigation systems are quantum sensors. Atom interferometry-based quantum sensors

can be used for precise measurement of acceleration and rotation providing the basic

platform for ultra-precise navigation systems [1]. They can also be employed for navi-

gation in harsh environments such as in deep-space or under water where the GPS may

be extremely disturbed or even unavailable [2]. Further, atom interferometry-based

quantum sensors are used for fundamental physics experiments under microgravity (in

drop towers or in space) [2], [3]. In order for quantum sensors to be ready to leave the

laboratory, appropriate space-qualified and portable laser systems should be available.

This explains the increasing interest over the last decade in building compact and

robust micro-integrated laser systems for quantum sensors [3].

As an example for applications of quantum sensors, figure 1 shows the laser plat-

form for the QUANTUS-2 laser system2for testing the equivalence principle by using

ultra-cold rubidium and potassium atoms [4]. In principle, the setup is divided into

two parts. The first part uses laser light at the wavelength of 780 nm for the manipu-

lation of rubidium and the other part uses a laser light at 767 nm for the manipulation

of potassium. Both parts fill a platform with a total diameter of 65 cm which is in-

tegrated in a "capsule". The corresponding quantum sensor experiment is carried out

inside the "drop tower" (Fallturm) in Bremen3. During the experiment, the capsule is

dropped inside the evacuated chamber (micro-gravity during free fall on the order of

1×10−6g) of the drop tower. This requires the laser platform to be robust enough

to withstand the mechanical stress it is subjected to during the catapult launch and

deceleration when caught in polystyrene pellets. For this purpose, the laser system

of the QUANTUS-2 experiment uses miniaturized Master-Oscillator-Power-Amplifier

(MOPA) laser modules (figure 1). Micro-integrated laser modules are compact devices

1GPS accuracy and error sources: see for example http://www.mio.com/technology-gps-accuracy.htm

2High resolution interferometry with ultra-cold mixtures in microgravity https://www.iqo.

uni-hannover.de/iqo_quantus2.html?&L=1

3https://www.zarm.uni-bremen.de/de/fallturm.html

Figure 1: CAD model of the catapult-capable laser system of the QUANTUS-2 experiment

for testing the equivalence principle by means of atom interferometry with ultra-cold rubid-

ium (Rb) and potassium (K) atoms.

with a very small footprint. Typically, a micro-integrated laser module consists of a

semiconductor laser chip, micro optics, and the corresponding electronic interface. All

these elements are micro-integrated on a suitable platform, e.g. aluminum nitride

(AlN) ceramic. Hybrid integration techniques (adhesive bonding, soldering) are used

to integrate these components on the smallest footprint possible as well as with the

highest degree of mechanical stability that can be achieved. The latter calls for omit-

ting any adjustment possibilities after bonding, i.e. any movable parts are omitted.

Mechanical stability is achieved through micro-integration simply for geometrical rea-

sons: as the dimension dof an object shrinks down, its mounting surface scales like d2

while its mass scales like d3. Further, misalignment through bending scales like dor d2.

Even with the successful micro-integration of laser modules, the total size of the

laser system platform is actually defined by the passive components for signal pro-

cessing such as beam splitters, fiber couplers, phase and amplitude modulators. In

the actual quantum sensor experiments, these devices are only commercially avail-

able. The resulting complexity and volume requirement for their integration into laser

system platforms are very huge. For example, for the micro-integration of a laser

module, typically a footprint of 60 ×50 mm2is required (e.g. the MOPA module in

the QUANTUS-2 laser system). The footprint of a (macroscopic) commercial phase

modulator for applications at the wavelength of 780 nm is typically about 25×50 mm2

(see photo on the upper left side of figure 1), where also additional space is required for

the deflecting mirrors and lenses for beam collimation. Besides, the macroscopic mod-

ulators require high-power drivers (a few Watt RF-power) which increases the space

requirement of about 350 ×250 ×100 mm3. Further, phase modulators are usually

combined with couplers, splitters, and other optical and mechanical components for

manipulation of the laser light to generate the necessary signals for the experiments.

These "distribution modules" are extremely complex and volume-consuming which is

not appropriate for applications in space. Hence, the miniaturization of couplers,

phase and amplitude modulators, e.g. by realizing them on the basis of semiconduc-

tors (GaAs-based) should reduce the place and power requirements, as well as the

complexity of the total optical system which is a prerequisite for field-capable and

space-qualified quantum sensors.

For the realization of the GaAs-based components one may benefit from the rich ex-

perience that has been made in the field of optical telecommunications (InP-based

photonic integrated circuits (PICs) for applications in the range of 1.3 - 1.6 µmwave-

length) in which a high level of complexity has been achieved [5]. This requires to

adjust the semiconductor technology for the modulators and coupler devices from the

telecommunication field into new wavelength ranges, for example, for quantum sensors

applications at the wavelengths of 780 nm (for rubidium spectroscopy) and at 1064 nm

(hyperfine transitions in molecular iodine at 532 nm 4).

In this work, GaAs-based phase modulators and couplers should be developed. The

aperture of the components should be compatible with the state of the art GaAs-based

edge-emitting lasers. This should make it feasible to realize micro-integrated laser and

spectroscopy modules that combine passive photonic components (e.g. phase modula-

tors) with active components (e.g. edge emitting lasers). The successful demonstration

of these two basic photonic components (GaAs-based phase modulators and couplers)

should in the future allow to integrate them on the chip (monolithic integration) which

should further decrease the complexity of the system and drastically improve its ro-

bustness and reliability.

Micro-integrated laser systems, state of the art

When the passive optical components from this thesis are demonstrated, the next

milestone for the future work is to employ these components into the state of the art

micro-integrated laser modules.

Micro-integrated laser modules for applications in field and in space have already

been demonstrated [3]. For example, an extended cavity diode laser (ECDL) module

for potassium spectroscopy is shown in figure 2. The diode laser chip, micro optics,

electronic interface, and a micro-thermoelectric cooler (µ-TEC) that carries a volume

holographic Bragg grating (VHBG) are all micro-integrated on an AlN ceramic micro-

optical bench. The footprint of the device corresponds to 25 ×80 mm2[3].

The deployment of micro-integrated laser modules in fundamental physics experi-

ments in space has been successful. For example, the MOPA platform have been em-

ployed in the first optical atomic frequency reference in space, on a sounding rocket [6]

(the FOKUS experiment 5). Shortly thereafter extended cavity diode lasers have

been demonstrated in space on-board a sounding rocket and demonstrated reliable

frequency stabilization to the potassium D2 line as well as to each other through a

4Optical frequency standard based on molecular iodine for sounding rockets https://www.physics.

hu-berlin.de/en/qom/research/jokarus

5REXUS 9 and 10, see for example: http://www.dlr.de/desktopdefault.aspx/tabid-6840/86_

read-29274/

Figure 2: Extended cavity laser module for Potassium spectroscopy. VHBG: volume holo-

graphic Bragg grating, µ-TEC: micro-thermoelectric cooler. (taken from [3]).

Figure 3: Laser modules based on the Milas technology. The platform allows the integration

of two arbitrary SC chips, all together with micro-optics, electronic interfaces, and integrated

fiber couplers.

frequency- offset stabilization [7] , and KALEXUS experiment 6). Both the FOKUS

and KALEXUS experiment were sponsored by the German Space Agency (DLR).

However, the platform of the laser module in figure 2 allows for micro-integration

of a single chip. To bring the micro-integration technology to the next level, the new

platform for micro-integrated optical systems in figure 3 has been developed by FBH

within the Milas7project. The new platform allows for micro-integration of arbitrary

combinations of two semiconductor chips. These combinations may include active

components (e.g. diode laser or amplifier chips), as well as passive components (like

GaAs chip-based phase modulators). Further, the platform has two fiber coupling

ports for input into and output from the laser module using polarization maintaining

single mode optical fibers. The Milas technology offers a suitable platform for the

micro-integration of the GaAs-based phase modulators that are developed within this

work.

6TEXUS 53, see for example: http://www.dlr.de/dlr/desktopdefault.aspx/tabid-10081/151_

read-16493/#/gallery/21758

7Mikrointegrierte Diodenlasersysteme (Milas): supported by the German Space Agency DLR with funds

provided by the Federal Ministry of Economics and Technology (BMWi) under grant number 50WM1141.

Chapter 1

Introduction

1.1 Developments in III-V semiconductor phase modulators

Milestones in the development of III-V semiconductor-based phase modulators are

summarized in chronological order in table 1.1. The last three results are essential

results of this work.

blank line

Table 1.1: The chronological development of GaAs-based and InP-based electro-optic phase

modulators. The last 3 papers are based on this this thesis work.

author year waveguide substrate junction λMod. Eff.∗∗ losses

[µm] [1/(V·mm)] dB/cm

[8] 1964 GaP InP p-n 0.550 75◦-

[9] 1983 InGaAsP/InP InP p-n 1.32 - >10

[10] 1986 GaAs/AlGaAs GaAs P-n-N 1.06 56◦-

[11] 1987 GaAs/AlGaAs GaAs P-i-N 1.09/1.15 38◦/36◦>13

[12] 1987 InGaAs/InP InP P-i-N∗1.52 12◦9.8

[13] 1988 GaAs/AlGaAs GaAs P-p-n-N 1.06 96◦>12

[14] 1988 GaAs/AlGaAs GaAs P-p-i-n-N 1.55 2.9◦1.2

[15] 1989 GaAs/AlGaAs GaAs P-I-i-I-N 1.09 28◦20

[16] 1992 InP/GaInAsP InP P-I-n-N 1.55 11◦1.0

[17] 1997 GaAs/AlGaAs GaAs P-p-i-n-N 1.31 35◦0.6

[18] 2003 InGaAs/InP InP P-p-n-N 1.55 34◦<4.5

[19] 2013 GaAs/AlGaAs GaAs P-p-i-n-N 0.780 11◦<1.4

[20] 2014 GaAs/AlGaAs GaAs P-p-n-N 0.780 23◦<1.4

[21] 2017 GaAs/AlGaAs GaAs P-p-n-N 1.064 16◦<2.7

∗Multi-Quantum-Wells (MQWs) inside the guiding region. blank ank bla blank blank blank la nkb

∗∗ Phase modulation efficiency (phase shift in deg/(V·mm))

The first III-V semiconductor electro-optic phase modulators were demonstrated a few

decades ago [8]. In 1964, D. F. Nilson and F. K. Reinhart observed for the first time

phase modulation of the light signal guided in a reversed biased gallium phosphide

p-n junction on InP substrate and related the modulation to the linear electro-optic

(LEO) effect. Twenty years later in 1983, H. J. Bach et al. demonstrated phase

7 1.2. Thesis objectives: GaAs-based passive photonic components

modulation at the wavelength of 1.32 µmin double heterostructures using InGaAsP

p-n heterojunctions for the first time [9]. The authors were also the first to describe

the quadratic electro-optic (QEO) effect in p-n diodes. Later in 1986, the first electro-

optic GaAs/AlGaAs double heterostructure phase modulator waveguide for integrated

optics was demonstrated [10] (operation at the wavelength of 1.06 µm). Since then,

GaAs-based phase modulators have received increasing interest for optical intercon-

nects and fiber coupling [22]. The year after, in 1987, J. Faist and F. K. Reinhart

reported the orientation dependence of the phase modulation in the GaAs/AlGaAs

double heterostructures for laser radiation at the wavelength of 1.09 µm[11]. They

measured phase modulation for both, the TE and TM modes for light propagating in

the [110] and [1¯

10] crystallographic directions in a P-i-N (with an intrinsic (i) core and

P (p-doped) and N (n-doped) cladding layers) phase modulator. No difference was

found between the measurements for the TM modes in both directions. However, for

the TE mode, the LEO effect was found to add in the [1¯

10] direction and subtract in

the [110] direction. Later in 1988, J. G. Mendoza-Alvarez et al. quantified the contri-

bution of carrier density-related effects to the modification of the refractive index in

highly-doped GaAs/AlGaAs double heterostructures. They presented phase modula-

tors at the wavelength of 1.06 µmwith a high modulation efficiency (phase shift per

volt per unit length) [13] due to contribution from the LEO effect, the QEO effect,

and the carrier density-related effect.

As shown by table 1.1 the performance of different phase modulators is characterized

by two parameters: the phase modulation efficiency and the propagation losses. For

some of these devices, the high modulation efficiency is accompanied with extremely

large propagation losses which are caused by electro-optic and free carriers absorp-

tion (see for example [13] and [11] in table 1.1). An efficient phase modulator was

demonstrated in 1997 by Y. T. Byun et al. [17]. The authors used a so-called W-shape

P-P-p-i-n-N-N double heterostructure ridge-waveguide phase modulator and demon-

strated a phase modulation efficiency of 34 ◦/(V ·mm) and very low propagation losses

(0.6 dB ·cm−1). This means that for example for a 2 mm long phase modulator (a typi-

cal length of a GaAs-based chip), a phase shift of 180◦can be achieved by only applying

1.32 V. The corresponding propagation losses of about 0.24 dB are negligible compared

to the coupling losses (typically 2.2 dB for 60% coupling efficiency) which makes this

structure very suitable for micro-integration applications. We carefully studied this

particular double heterostructure and applied the elements of the W-shaped concept

to develop the phase modulators in this work.

1.2 Thesis objectives: GaAs-based passive photonic compo-

nents

GaAs-based phase modulators require a careful design of the GaAs/AlGaAs double

heterostructure in order to efficiently use the electro-optic effects and the free carriers

effects in GaAs. For compatibility reasons, GaAs-based couplers in this work are

realized based on the GaAs/AlGaAs double heterostructures of the phase modulators.

With the successful demonstration of phase modulators and waveguide couplers, as

an application of monolithic integration of passive photonic components for future

works, an integrated intensity modulator can be realized in the simplest layout of a

Mach-Zehnder Intensity (MZI) modulator.

8 1.2. Thesis objectives: GaAs-based passive photonic components

As a medium term target, the passive components that are developed in this work

are intended to meet the micro-integration requirements on the hybrid laser modules.

These requirements and the specifications of individual components are discussed in

details in the following parts of this section.

1.2.1 Objective: GaAs-based phase modulators

The first objective of this work is to realize GaAs/AlGaAs double heterostructure

phase modulators.

Principles of phase modulation in GaAs-based waveguides should be investigated. The

GaAs/AlGaAs double heterostructures from the literature should be studied to acquire

the knowledge to design phase modulators at two different wavelengths (780 nm and

1064 nm). An efficient design requires to model the electro-optic response of the

modulator double heterostructure. The design criteria are the phase modulation ef-

ficiency (the amount of phase shift per volt per mm), the propagation losses (free

carrier-absorption losses, modal losses, and losses arising from the lateral waveguide

structure), the polarization maintenance, and the spectral dispersion of the phase

modulator.

The waveguide should be achieved based on a ridge waveguide concept. The design

should be optimized so that coupling a coherent light signal (laser beam) into and out

of the modulator can be achieved using the state of the art micro-optics and micro-

integration approaches.

The next step is to realize the phase modulators in the GaAs technology. The real-

ized devices should be characterized. For the characterization, a coherent light signal

from a diode laser should be coupled into the modulator and the transmitted power,

the polarization, the phase modulation, the residual amplitude modulation, and the

modulation bandwidth should be measured. We emphasize here that some of the char-

acteristic parameters of GaAs-based phase modulators such as phase and amplitude

non-linear distortion has not previously been investigated in the literature. This is

why efforts have to be made within this work to develop new methods for in-depth

characterization of GaAs-based phase modulators. These methods should be experi-

mentally implemented.

The required performance of the phase modulator at 780 nm and the phase mod-

ulator at 1064 nm is specified as the following:

•single mode waveguides: the ridge parameters (ridge width and etching depth)

should be selected for the waveguide to support only the fundamental guided

optical mode. This is a requirement for a well-defined phase modulation [19], [23].

•the far field of the modulator’s output signal must satisfy the requirements for

hybrid integration with active and passive optical elements. Typical divergence

angles (95%) are for example 15◦in the lateral and 25◦in the vertical direction.

At these values, the beam can be collimated using commercial lenses to provide

a beam diameter that is as close as possible to the optimal value of 0.6 mm [24].

•The 3-dB modulation bandwidth (3 dB/45◦phase delay) should be at least 8 MHz

(for example for the generation of the sidebands for Rb spectroscopy at 780 nm).

9 1.2. Thesis objectives: GaAs-based passive photonic components

The design of the electrical connection to the modulator should allow for modu-

lation frequencies of at least 8 MHz.

•the half-wave voltage (with direct driving) shall be smaller than 5 V to dispense

the demand for diving electronics (a driving voltage of 5 V can be achieved even

at bandwidths up to 10 MHz).

•polarization maintaining waveguide: the polarization extinction ratio (PER) is

defined as the ratio of optical powers of perpendicular polarizations. The PER

should exceed 60 dB provided that the injected beam features a PER of 60 dB or

better.

•very low propagation losses: typical values from the literature for low propaga-

tion losses in GaAs/AlGaAs double heterostructure phase modulator waveguides

are between 0.6 dB cm−1at 1.31 µm[17] and 1.2 dB cm−1at 1.2µm[14]. The ob-

jective is to reduce the propagation losses of the phase modulators in this work

beyond the state of the art.

1.2.2 Objective: GaAs-based couplers

The second objective is to realize waveguide couplers. Ridge waveguide couplers should

be realized based on the layer structure of the phase modulators. This facilitates the

monolithic integration of phase modulators together with couplers for the realization of

complex devices such as intensity modulators. A waveguide coupler generally consists

of three sections: The input waveguide/waveguides (usually referred to as the access

waveguides), the coupling section in which the mechanism for transmission of the

field into the output channels is determined, and the output waveguide/waveguides.

An M to N (or M×N) coupler describes a device with Minput waveguides and N

output waveguides. In their simplest layout, 2 to 1 (2×1) couplers can be used to

combine the optical field from two different input devices into one output path, or

as a 1×2 coupler also to split the optical field into two different paths (splitters).

Another common coupler concept is the 2×2 3dB coupler with two input waveguides

and two output waveguide. The optical field from either of the input waveguides

is divided equally (nominally) between the two output waveguides. For the optimum

design of ridge waveguide couplers, different coupling concepts from the literature (e.g.

evanescent coupling or self-imaging in multi-mode waveguides) should be compared.

Wave propagation in 1×2 couplers and 2×2 3dB couplers should be modeled. The

design criteria are the excess losses and the accuracy of the splitting ratio of the input

field into the output waveguides (imbalance).

Following to the design, the waveguide couplers should be realized and characterized.

A coherent light signal from a diode laser should be coupled into the couplers and the

transmitted power (the excess loss as a measure), the polarization, and the imbalance

between output ports should be measured.

The specifications for the couplers are given as the following:

•operation at the wavelength of 780 nm.

•single mode access waveguides for which the far field of the output signal must

satisfy the requirements for hybrid integration with active and passive optical

10 1.2. Thesis objectives: GaAs-based passive photonic components

elements (divergence angles (95%) are 15◦in the lateral and 25◦in the vertical

direction in similar lines to the waveguide of the phase modulators).

•excess loss: for the couplers in this work, the excess losses should be small enough

so that when they are used to realize an integrated MZI modulator, the total loss

of the modulator is comparable to the state-of-the-art. The excess loss of state-

of-the-art GaAs/AlGaAs double heterostructure MZI modulator corresponds to

about 8 dB [25]. The MZI is realized using 2 couplers and 2 phase modulators at

the actives arms. The length of the phase modulators is typically 2 mm to 4 mm.

Assuming that the propagation losses in one phase modulator waveguide are less

than 0.5 dB (this follows from the assumption that the propagation losses in the

phase modulator waveguide are about 1.2 dB cm−1[14]). If we further assume

that the coupling losses are 1 dB or better, the excess loss of the each of the two

waveguide couplers should not exceed 3.0 dB.

•imbalance: the imbalance of the input couplers for an MZI modulator translates

directly into cross-talk and extinction ratio. For example, the power imbalance

of 0.2 dB limits the extinction ratio to 33 dB [26]. Therefore, the imbalance

(the splitting ratio) for 1×2 splitter (typically the input coupler of the MZI

modulator) should not exceed 0.2 dB in order not to limit the performance of

the Mach-Zehnder intensity modulator. A typical value for the imbalance of 2×2

3dB couplers is 0.2 dB to 0.6 dB [26].

1.2.3 Application: GaAs-based amplitude (intensity) modulator

As a proof of concept, the monolithic integration of two phase modulators and two

waveguide couplers is demonstrated in the application of a MZI modulator. The opti-

cal field fed into the first (input) coupler is divided between the two phase modulators

(arms of the MZI modulator) and then the two arms are recombined using the second

(output) coupler. The connection between the couplers and the phase modulators

(active arms of the MZI modulator) is typically realized using bent waveguides such

as S-bends [26]. S-bends are required to guarantee a sufficient lateral spacing between

the two active arms so that the modulating electric field can be applied separately

on each arm. The design criteria for the MZI modulator are the extinction ratio and

the excess losses. In the S-bends, losses may arise due to radiation losses at the bend

structure or reflections due to the optical mode mismatch between the straight and

the bend waveguides. The S-bends should be modeled and a suitable structure with

minimal losses should be found. A suitable concept should be chosen for the input

and output couplers of the MZI modulator.

Next, the MZIs should be fabricated and characterized. A coherent light signal from a

diode laser should be coupled into the MZI modulator input coupler and the transmit-

ted power (the excess loss as a measure) and the extinction ratio when a modulating

electric field is applied should be measured.

We specify the MZI modulator as the following:

•operation at the wavelength 780 nm.

•single mode access waveguides. The far field of the output signal must satisfy

the requirements for hybrid integration with active and passive optical elements

(as in the phase modulators and couplers).

11 1.3. Structure of the thesis

•the excess loss must be comparable to or less than 8 dB (excess loss of state-of-

the-art GaAs/AlGaAs double heterostructure MZI modulators [25]).

•extinction ratio: extinction ratios of GaAs/AlGaAs MZI modulators in the liter-

ature range from 3.0 dB in [25] to 24 dB in [27]. Future applications of the MZI

modulators, for example in high-speed IQ modulators, require to reach extinc-

tion ratios beyond 20 dB to achieve a sufficient signal-to-noise ratio at high data

rates, see [28]. As a proof of concept, the MZI modulators in this work should

provide an extinction ratio well beyond 3 dB. The improvement of the extinction

beyond 20 dB should be feasible.

1.3 Structure of the thesis

Based on the discussed objectives for this work, the rest of the thesis is structured as

follows:

•in chapter 2 we briefly present the fundamentals of guided wave optics. The

physics of wave propagation in planar waveguides and the cut-off conditions for

guided modes in single mode and multi-mode ridge waveguides are introduced.

Concepts of waveguide couplers are discussed. These are then used later in

chapter 4 and in chapter 5 for the design procedure of the waveguides for the

phase modulators and coupler devices.

•the electro-optic properties of GaAs are discussed in chapter 3. These properties

are used to optimize the design of the GaAs/AlGaAs double heterostructure of

the phase modulators in chapter 4. The design requires to model the distribution

of both electrical and optical fields inside the double heterostructure in order to

calculate the contribution of different effects to phase modulation.

•in chapter 4 details to the design of the waveguides for electro-optic phase mod-

ulators at 780 nm and 1064 nm including the calculation of the electro-optic

coefficients of the double heterostructures, developing the multi-layer structure

for the vertical design, and the lateral design of single mode ridge waveguides are

presented.

•the developed multi-layer structure for the phase modulator at 780 nm in chap-

ter 4 is considered for the design of waveguide couplers. Details related to the

design of two different couplers concepts (directional couplers and MMI couplers)

are presented in chapter 5.

•as an example of the monolithic-integration of GaAs-based phase modulators and

GaAs-based couplers, the applications of a Mach-Zehnder intensity modulator in

chapter 6.

•chapter 7 is dedicated to the fabrication details of the designed GaAs-based

passive components.

•in chapter 8 the standard characterization of the GaAs-based phase modulators

is presented. The electrical properties, the phase modulation efficiency, and the

12 1.3. Structure of the thesis

propagation losses of phase modulators at 780 nm and at 1064 nm are experimen-

tally investigated using established methods. Then, the performance of GaAs-

based waveguide couplers is characterized. Finally, as an application the main

performance factors of Mach-Zehnder intensity modulators are characterized.

•a novel method for the in-depth characterization of GaAs-based phase modu-

lators is presented in chapter 9. The mathematical model of the method and

its experimental implementation are described. The experimental results of a

phase modulator at 780 nm and a phase modulator at 1064 nm are presented

and followed by a discussion of the results.

•finally, the conclusion and outlook are given in chapter 10.

Chapter 2

Fundamentals of guided-wave optics

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxx xxxxxxxxxx

J. Tyndall (1820-1893) was the first to demonstrate

guiding of a light beam within a falling water stream

(http://i-fiberoptics.com/pdf/12_0120-if_514.pdf)

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

An optical waveguide is a medium in which the optical signal remains confined due

to total internal reflection. To meet the requirements for total internal reflection, the

light guiding material (core) is imbedded in the (cladding) material with a refractive

index that is smaller than the refractive index of the core. Examples are optical fibers

for long distance propagation and planar (dielectric) waveguides. Planar waveguides

are of particular importance for integrated optics. The possibility to integrate planar

active devices such as diode lasers and ridge waveguide amplifiers (RWAs) with passive

components (phase modulators, couplers,...) make planar waveguides very attractive

for the realization of compact devices and complex PICs.

14 2.1. Electro-magnetic wave propagation in planar waveguides

In this chapter, the basic principles of wave propagation in planar waveguides are

introduced. These principles provide a guidance to rely on for the design procedure

which follows later in chapter 4 and in chapter 5. The derivation of guided modes of

a simple planar waveguide from the wave equation and the corresponding boundary

conditions are briefly presented. Then, the requirements for single mode and multi-

mode waveguides are discussed. Further, the properties of multi-mode waveguide

couplers are concluded.

2.1 Electro-magnetic wave propagation in planar waveguides

Maxwell’s equations describe the propagation of electro-magnetic waves in vacuum as

well as in any kind of matter. In the following, wave propagation in homogeneous

non-magnetic plane (lossless, infinitely extended) is considered. The light source (e.g.

laser beam) is assumed to have a harmonic time dependence with angular frequency

ω.

2.1.1 Planar electro-magnetic waves

In a homogeneous, linear, and lossless dielectric medium, the wave equation for the

electric wave vector ˜

E(x, y, z, t)as derived from the Maxwell’s equations follows the

following form [29, p. 22–24]:

∇2˜

E(r, t) = µµ00

∂2˜

E(r, t)

∂t2(2.1)

where r= (x, y, z)the position vector, µand µ0are the relative permeability and the

vacuum permeability, respectively, the relative dielectric constant, 0the vacuum

permittivity, and the operator ∇2=∂2/∂x2+∂2/∂y2+∂2/∂z2.

By considering a non-magnetic material with µ= 1, the wave equation is then

given by:

∇2˜

E(r, t) = ¯n2

∂2˜

E(r, t)

∂t2(2.2)

with c= 1/√µ00is the vacuum speed of light and ¯n=√is the refractive index.

A simple solution of the wave equation is a plane wave with a harmonic time

dependence of ˜

Eaccording to exp (iωt)that is given by:

E(r, t) = E(r) exp (iωt) + E∗(r) exp (−iωt)(2.3)

with E(r) = (Ex, Ey, Ez),r= (x, y, z)is a complex function that depends only on

the position. E(r)is called the phasor. The wave equation of the plane wave in

equation 2.3 can then be described by the Helmholtz equation [30, p. 36]:

∇2E(r) + ¯n2ω2

c2E(r)=0 (2.4)

Let us now consider a monochromatic, plane wave with a constant amplitude vector

E0propagating in the direction of the wave vector k= (kx, kykz). The corresponding

phasor of this wave is written as:

E(r) = E0exp (−ikr)(2.5)

15 2.1. Electro-magnetic wave propagation in planar waveguides

By inserting equation 2.5 in 2.4, the so-called separation condition [30, p. 37] is

achieved:

x+k2

y+k2

z= ¯n2k2(2.6)

with k=ω/c is called the vacuum wave number.

2.1.2 Reflection of planar waves

Let us consider two homogeneous, lossless media with a planar boundary at x=

0. A plane wave Ei, with a linearly polarized monochromatic electric field E=

(Ex= 0, Ey6= 0, Ez= 0) is propagating in the first medium with the refractive index

¯n1and is obliquely incident with the propagation vector kiat the planar boundary as

shown by figure 2.1.

Figure 2.1: Reflection of a linear monochromatic plane wave obliquely incident on a planar

boundary between two homogeneous, lossless media.

Ei=E0iexp (−ikir)(2.7)

The refractive index of the second medium ¯n2is assumed to satisfy ¯n2<¯n1. Thus,

the incoming wave is reflected at the interface between the two media. The reflected

wave is then given by:

Er=E0rexp (−ikrr)(2.8)

The wave vectors of the incident and reflected waves can be extracted from figure 2.1:

ki= (−kx,i,0, kz,i)=(−¯n1kcos Θi,0,¯n1ksin Θi)

kr= (kx,r,0, kz,r) = (¯n1kcos Θr,0,¯n1ksin Θr)(2.9)

In the second medium with the refractive index ¯n2<¯n1, the refractive wave propagates

with the wave vector ktat an angle Θtthat is different from the incidence angle Θi:

Et=E0texp (−iktr)(2.10)

with ktgiven by:

kt= (−kx,t,0, kz,t) = (−¯n2kcos Θt,0,¯n2ksin Θt)(2.11)

16 2.1. Electro-magnetic wave propagation in planar waveguides

Note that the wave vectors ki,kr,ktin equations 2.9 and 2.11 necessarily satisfy the

corresponding separation conditions (equation 2.6).

By taking into consideration the continuity of the tangential component of the electric

field at the planar boundary, equations 2.7 to 2.11 yield at x= 0:

E0iexp (−i¯n1kz sin Θi) + E0rexp (−i¯n1kz sin Θr) = E0texp (−i¯n2kz sin Θt)(2.12)

Equation 2.12 is valid for all values of z. This requires the phase components of the

quantities in equation 2.12 to be equal:

n1sin Θr=n1sin Θi=n2sin Θt(2.13)

The left-hand equality gives the reflection law (Θi= Θr) whereas the right-hand

equality corresponds to the refraction law, namely (n1sin Θr=n2sin Θt). By inserting

these two conditions in equations 2.9 and in 2.11, we find:

kz,i =kz,r =kz,t =k¯n1sin Θi(2.14)

Further, from the reflection law and the refraction law, the relation between the am-

plitudes from equation 2.12 is donated by:

E0i+E0r=E0t(2.15)

It can be further shown (see [29, p. 35]) that the continuity of the tangential component

of the magnetic field at the boundary between the two media delivers the following

equation:

E0i¯n1cos Θi−E0rn1cos Θr=E0tn2cos Θt(2.16)

With the conditions provided in equations 2.13 to 2.16, a common solution delivers

the relations between the magnitudes of the indecent and reflected waves:

E0r=¯n1cos Θi−¯n2cos Θt

¯n1cos Θi+ ¯n2cos Θt

Ei0(2.17)

with Θtrelated to Θiby:

cos Θt=∓v

t1−¯n2

¯n2

sin2Θi(2.18)

Since ¯n2<¯n1, i.e. Θi>ΘCwith sin ΘC= ¯n2/¯n1and ΘCdenoting the critical angle

for internal reflection, then the total reflection condition is achieved. At this condition,

the refraction angle in equation 2.18 becomes pure imaginary:

cos Θt=−iv

¯n2

sin2Θi−1(2.19)

The corresponding wave vector is therefore rewritten as kt= (−ikx,t,0, kz,t). This

means physically that the wave continues to propagate in the second medium along

the z-direction with an exponentially decaying component in the x-direction according

to exp (x/xp)where xp= 1/kx,t is called the penetration depth since the electric field

17 2.1. Electro-magnetic wave propagation in planar waveguides

decreases to 1/e of its value at the interface between the two media. By inserting

equation 2.19 into 2.17 we find:

E0r=¯n1cos Θi+iq¯n2

1sin2Θi−¯n2

¯n1cos Θi−iq¯n2

1sin2Θi−¯n2

Ei0(2.20)

It is clear from equation 2.20 that the intensities of the incident and reflected waves are

equal |E0r|2=|E0i|2. This means that the entire incident power is reflected. Further,

their amplitudes are equal |E0r|=|E0i|and their phases differ which corresponds to

a standing wave in the positive x-direction. It is typical to write equation 2.20 in the

form:

E0r=E0iexp (2iΦr)(2.21)

where the phase shift Φris given by:

tan Φr=+q¯n2

1sin2Θi−¯n2

¯n1cos Θi

(2.22)

The evaluation of equations 2.7, 2.8 and 2.10 under the condition of total reflection

corresponds to a wave that propagates in the z-direction (see equation 2.14) with a

period that is given by λz=λ/(¯n1sin Θi)where λ= 2πc/ω. In the positive x-direction

a stand wave (equations 2.21 and 2.22) with the period λx=λ/(¯n1cos Θi)is formed

whereas it decays exponentially in the negative x-direction (equation 2.19).

The previous analysis can be applied to understand wave propagation in planar

waveguides. If the reflected wave from figure 2.1 faces a second planar boundary to

another lossless homogeneous medium that has ¯n3<¯n1, the wave shall be reflected

again and thus continue to propagate in a zig-zag pattern along the first medium

(with the refractive index ¯n1). This is the case of wave guiding which is interesting for

real applications. In this case, solutions of the wave equation necessary satisfy both

boundary conditions and are referred to as guided modes.

The analysis of the wave guiding in planar waveguides leads to guided modes that

are classified into Transverse Electric (TE) and Transverse Magnetic (TM) waves. For

the TE waves only the y-component of the electric field is non-zero (E= (0, Ey,0)).

For the TM modes, only the y-component of the magnetic field is non-zero, or (H=(0,

Hy,0)). In the following, we explicitly consider the solution of the TE plane wave.

The solution for the TM waves follows along similar lines.

2.1.3 Planar waveguides

The simplest form of a planar waveguide is a 3-layer slab waveguide. The guiding

core is sandwiched between a cladding and a substrate. Such a planar waveguide is

shown in the GaAs/AlGaAs material system in figure 2.2. The GaAs layer forms

the waveguide core and has a refractive index ¯nfthat is larger than the refractive

indices of the cover layers: the substrate at x= 0 with the refractive index ¯nsand

the cladding at x=dwith the refractive index ¯nc. All the three layers are assumed

to extend infinitely in the y-direction and in the z-direction and form a so-called slab

18 2.1. Electro-magnetic wave propagation in planar waveguides

Figure 2.2: Cross-section of a GaAs/AlGaAs planar waveguide and the phase constant k¯nf

of a guided mode propagating in the core layer in the z-direction. The orientation of the

wave vector is indicated by the arrow direction.

waveguide. We further consider a TE plane wave that propagates along the z-direction

with the phasor Ey(x, z) = E0(x) exp (−iβz), with β=k¯nfsin Θ = kzand ky= 0

(no dependency on the y-coordinate). Optical wave propagation in the cover layers is

for practical reasons not much of interest and it is expected to be absorbed or radiated

out of the waveguide. Therefore, we restrict our analysis to cases where the field

propagates along the core region (guided modes).

The Helmholtz equation 2.4 for the considered TE plane wave can be written as:

∂2

∂x2+∂2

∂z2!Ey(x, z) + ¯n2k2Ey(x, z) = 0 (2.23)

Let us consider the wave propagation as depicted in figure 2.2. For a guided wave

in the film region, the propagation angle to the x-axis Θsatisfies the requirement for

total internal reflection, i.e. Θ>ΘC. Thus, the wave continues to propagate along

the z-direction. Due to total internal reflections at the interfaces between the GaAs

core and the AlGaAs cover layers, a standing wave is created along the x-direction. In

this case the field distribution inside the core varies sinusoidally along the x-axis with

βf=k¯nfcos Θ. In the cladding (or substrate) layers on the other hand, the signal is

reflected at the cladding-core (or substrate-core) interface and it decays exponentially

with αc= 1/xp,c and αs= 1/xp,s, where xp,c and xp,s are the penetration depths of the

fields in the cladding and in the substrate, respectively. This condition is valid for all

values of z, which means the confined power flow of the optical signal (in x-direction)

follows the propagation direction zand corresponds to a guided mode.

The optical field described here corresponds to a plane wave which is a solution of

the Helmholtz equation. Therefore, the resulting separation conditions according to

equation 2.6 can be written for both, the cover and core layers as the following:

for the cover layers (cladding and substrate)

x,c +k2

z,c = ¯n2

ck2

x,s +k2

z,s = ¯n2

sk2(2.24)

19 2.1. Electro-magnetic wave propagation in planar waveguides

and for the core region

x,f +k2

z,f = ¯n2

fk2(2.25)

with kz,c =kz,s =βm=β(propagation along the z-direction), kx,f =βf(propa-

gation inside the film region), and k2

x,c =−α2

c,k2

x,s =−α2

s(exponentially decaying

field in the x-direction into the cladding and into the substrate layers, respectively),

equations 2.24, 2.25 can be rewritten as:

for the cover layers

¯n2

ck2−β2=−α2

¯n2

sk2−β2=−α2

(2.26)

and for the core region

¯n2

fk2−β2=β2

f(2.27)

The solution of the Helmholtz equation 2.23 that satisfies the separation conditions

in equations 2.26 and 2.27 is then a guided TE wave that decays exponentially in the

cladding and the substrate. Such a solution can be written as:

Ey(x, z) = 









Ecexp (−αc(x−d)) exp (−iβz)x>d

Efcos (βfx−ΦS) exp (−iβz) 0 ≤x≤d

Esexp (αsx) exp (−iβz)x < 0

(2.28)

with Efis being the real field amplitude in the core, Ecand Es=Ecare being the real

field amplitudes in the cladding and in the substrate, respectively. ΦSis the half-phase

shift at the interface between the core layer and the substrate (see equation 2.22).

The terms in the first and third lines in equation 2.28 correspond to the decaying

optical field in the cover materials, whereas the term in the second line corresponds to

the standing wave in the core material. In order to fully describe the guided TE modes,

the boundary conditions at the interfaces between the core and the cover have to be

solved. We enforce the continuity of the tangential components of Eyand ∂Ey/∂x at

the interfaces (at x= 0 and x=d). The following equations is then achieved:

tan (βfd−ΦS) = tan (ΦC)(2.29)

with tan (ΦC) = αc/βfis the half-phase shift at the boundary between the core and

the cladding at x=d. Both of ΦCand ΦSare calculated according to equation 2.22

for TE modes.

By considering the periodicity of the tangent function, the so-called characteristic

equation of the guided modes [30] can be found:

βf·d−ΦC−ΦS=mπ (2.30)

The characteristic equation reveals the discrete nature of guided modes. Please refer

to [29, p. 56-60] for more details to the derivation of equation 2.30.

The electric field distribution of some guided TE modes is shown in figure 2.3. Note

that with βf= ¯nfkcos Θ, according to the characteristic equation 2.30 the number

of guided modes is determined by three parameters which are the waveguide width

d, the refractive index step between the guide and the cover material, and by the

frequency ω=kc. Each of the guided modes inside the core can be attributed to

20 2.1. Electro-magnetic wave propagation in planar waveguides

Figure 2.3: Schematic electric field distribution of some guided TE modes. mis the order of

the guided mode. Original image [30, p. 44].

a propagation constant βm=k¯nfsin Θmwith the incidence angle Θm>ΘC. The

mode with the lowest index mis usually referred to as the fundamental mode. As

a rule of thumb Θm>Θm+1 >ΘCwhich means that the oscillation period in the

x-direction (see section 2.1.2) λx,m =λ/(¯nfcos Θm)for the mode mis larger than the

oscillation period of the mode m+ 1. We define the effective refractive index ¯neff as

a characteristic parameter of the guided mode ¯neff,m =βm/k for all values of m(all

the guided modes). Please not the ¯neff for any guided mode necessarily satisfies:

¯ns,c <¯neff <¯nf(2.31)

Further, since the guided optical modes have a finite penetration depth into the

cladding and into the substrate, we define the effective waveguide width deff as:

deff =d+xp,c +xp,s (2.32)

For a waveguide with a sufficiently large refractive index step (¯nf−¯ns,c), the effective

width deff can replace din the characteristic equation 2.30.

2.1.4 Ridge waveguides

The planar waveguide depicted in figure 2.2 is assumed to be infinitely extended along

the y-direction. The electrical field is confined along the x-direction. However, for

the design of photonic integrated circuits (PICs) the guided optical signal has to be

well-defined in two transverse directions (in the xy-plane) which corresponds to a two-

dimensional (2D) waveguide geometry. Some 2D waveguides geometries are the strip

waveguide, embedded strip, and the ridge waveguide (see figure 2.4). Lateral wave

guiding (lateral mode confinement) in these structures is achieved due to the effec-

tive refractive index step between the central region (the strip, embedded strip, or

the ridge) and the region on the sides which is usually referred to as the slab region.

The ridge waveguide structure is easy to fabricate and thanks to the advances in the

III-V material technology it is easy to control the lateral and vertical index profile

for optimum design [31]. This is why ridge waveguides are the most interesting struc-

tures for the design of PICs. On the other hand, the modeling and analysis of ridge

waveguides becomes complicated due to the vector nature of the electromagnetic field

and the simultaneous presence of various boundary conditions that arise at the various

interfaces. In the literature, ridge waveguides were investigated either using approxi-

mate analytical approaches, e.g. the effective index methods [32] or using numerical

21 2.2. Planar waveguide couplers

Figure 2.4: Some types of 2D waveguide geometries: (a) strip, (b) embedded strip, (c) ridge

waveguide.

methods such as the finite difference (FD) and finite element (FE) methods [31]. The

later methods are computationally intensive but they have been widely used due to

their accuracy in comparison to the effective index methods.

Many efforts have been made in the literature to describe the design rules of single

mode ridge waveguides [33], [34]. The term single mode means here that only a single

lateral guided TE and (or) TM mode is supported. This is a very important require-

ment for the design of single optical components such as chip-based phase modulators

or for the design of PICs in order to control the propagation of the optical field through

the PIC and for further coupling of the light into single mode fibers. Multi-mode ridge

waveguides that support a finite number of guided TE or TM modes are also used

for certain applications, e.g. for the realization of multi-mode interference (MMI)

couplers [26].

2.1.5 Bent waveguides

Bent waveguides are indispensable for the design of PICs. They are used to adjust

the lateral position or the direction of ridge waveguides to achieve complex devices at

a small foot-print (optical systems on the chip [35]). A widely used bent structure is

the S-bend (see figure 2.5). For example in ridge-waveguide based MZI modulators,

S-bends are used at the output of the couplers to increase the spatial distance between

the active arms of the MZI modulator.

The S-bend, despite its simply appearing structure, raises a more challenging modeling

problem than the straight ridge waveguide [36]. The propagation vector kof the S-

bend has a non-vanishing lateral component (ky6= 0). Thus, the analysis of S-bends

should account for the transition losses between a straight and a bent waveguide and

the phase constants of the field upon propagation in the bend.

2.2 Planar waveguide couplers

Couplers are used in PICs to combine, i.e. spatially overlap optical fields or divide

(splitters) the field from one waveguide into two or more paths. Based on ridge waveg-

uides, waveguide couplers can easily be realized.

22 2.2. Planar waveguide couplers

Figure 2.5: Ridge waveguide couplers. (a) 1×2 Y-coupler with S-bends, (b) 2×2 directional

coupler with S-bends, and (c) the schematic of the lateral layout of a general M×N MMI

coupler.

2.2.1 Concepts for ridge waveguide couplers

The Y-coupler

For the realization of ridge waveguide couplers, different coupling concepts can be

applied. The simplest concept is the Y-junction (or Y-coupler) which is shown in fig-

ure 2.5 (a). This device can for example be used as a power splitter or combiner. In the

splitter regime, the input waveguide supports a single guided mode. The optical power

is entirely stored in this eigenmode. At the Y-junction , the waveguide expands and

begins to support a second guided mode. At the end of the Y-junction a compound

waveguide structure is created (see figure 2.5(a)). The optical power is adiabatically

tapered into the fundamental mode of this compound waveguide and is then divided

into the output waveguides with nominally equal power and equal phases [37]. Bent

waveguides are then used to introduce a sufficient lateral separation of the output

waveguides.

If the device is used as a power combiner, the optical power is adiabatically tapered

from the fundamental mode of the compound waveguide into the eigenmode of the

single mode waveguide. The Functionality of the power combiner is directly affected

by the differences between the phases of the incoming optical fields at the Y-junction

(assuming equal amplitudes). Phase-matched optical fields maximize the power stored

in the eigenmode of the single mode waveguide at the end of the junction whereas fields

with a πphase difference cancel each other [38, p. 524].

The performance of the the Y-junction depends on the quality of the compound waveg-

uide which is defined by means of the lithography process. Due to fabrication artifacts,

23 2.2. Planar waveguide couplers

reflection and radiation losses may arise which make this kind of couplers less desirable

for PICs.

The directional coupler

This concept is shown in (figure 2.5(b)). The directional coupler uses two identical

Figure 2.6: Principle of the overlap of the input optical field from the input waveguide with

the coupling mode of the coupling waveguide of a directional coupler.

parallel waveguides. Bent waveguides are used to bring the cross-sections of the two

waveguides at a distance dCthat is small enough such that the evanescent optical field

from one waveguide can couple into the guided mode of the neighboring waveguide [37,

p. 264]. This type of couplers is called a directional coupler since only modes which are

traveling in the same direction and with about the same velocity can couple [39, p. 342].

The waveguides spacing dC(see figure 2.5(b)) is maintained for a certain longitudinal

length LCthat is called the coupling length. The bent waveguides at the output side

increase the waveguides spacing to terminate the coupling effect.

Directional couplers have a characteristic parameter that is referred to as the transfer

distance L0[37, p. 267] which is defined as the shortest coupling length at which

the injected input power is fully coupled from the input waveguide to the coupling

waveguide. Different functionalities of the couplers can be realized at fractional lengths

of L0. For example, at a coupling length LC=L0a cross-coupler is achieved, whereas

at LC=L0/2the function of a 3dB coupler is realized. In fact, 1/L0is proportional

to the overlap of the input field EIand coupled field ECunder the ridge region of the

coupling waveguide. Both fields with their corresponding lateral expansion under the

ridge (in y-direction) are shown in figure 2.6. The overlap of both fields is calculated

using the coupling coefficient ΓC[37, p. 265] which is given by:

ΓC=1

2¯n2

eff −¯n2

c·2π¯neff

λZWr

EI(y)EC(y)dy (2.33)

where ¯neff is the effective index of the guided TE fundamental mode (equal for both

input and couple fields) and ¯ncis the effective refractive index of the slab region at

the sides of the ridge (see section 2.2.2 for the determination of the effective refractive

index of a slab region). The integral in equation 2.33 is carried out along the width of

the coupling waveguide Wr. The corresponding transfer distance is given by:

L0=π/ (2ΓC)(2.34)

24 2.2. Planar waveguide couplers

The larger ΓCis, the shorter the transfer distance L0, which then enables to realize

short couplers. The experiment shows that the performance of directional couplers is

very sensitive to the fabrication variations.

The multi-mode interference coupler

A widely used coupler device is the multi-mode interference (MMI) coupler [26] which

is shown in figure 2.5 (c). The so-called self-imaging (see the next section) in multi-

mode waveguides is used to realize MMI couplers with multiple input/output channels.

MMI couplers are known to be more tolerant to fabrication tolerances/variations in

comparison to couplers such as directional couplers or Y-junctions [26]. The MMI

couplers are the most interesting devices to realize MZI modulators [40]. The self-

imaging principle of MMI couplers is discussed in the following section.

2.2.2 Self-imaging in multi-mode interference couplers

In an MMI coupler, the input signal from one or more single mode input (access)

waveguides (waveguides 1 to M in figure 2.5 (c)) is launched into the multi-mode

waveguide of a width Wmand a length Lmthat supports a number of mguided lateral

modes (m≥N). At the length Lm, a number Nof images of the input fields is

generated by means of the self-imaging mechanism.

To simplify the description of self-imaging in multi-mode ridge waveguides, the 2D

geometry (in the xy-plane) of the multi-mode waveguide is reduced to one dimension

(1D). Such a 1D structure can be achieved by applying the effective refractive index

method [32]. In this method the cross-section of a ridge waveguide is divided into three

regions (see figure 2.7), each region is attributed to a characteristic parameter that

is the effective refractive index of the fundamental guided mode of an infinite planar

waveguide. The planar waveguide is assumed to have the same vertical structure of the

considered region. The lateral discontinuities of the original 2D structure (which cor-

Figure 2.7: Effective refractive index method. (a) ridge cross section, (b) two slab waveg-

uides, (c) corresponding effective refractive indices of the slab waveguides.

respond to the ridge boundaries) define the 1D region with different effective refractive

indicies as can be seen by figure 2.7(a) and (b). This new structure can be regarded

as a planar waveguide along the y-direction that has a core with the refractive index

25 2.2. Planar waveguide couplers

Figure 2.8: Two dimensional (2D) representation of a multi-mode ridge waveguide. (a)

the (effective) refractive index profile (¯nr= ¯neff2,¯ns= ¯neff1), (b) top view of the ridge

boundaries with examples of modal field distributions Enof the guided modes 0, 1, ··· 8.

(Reproduced photo, original taken from [26]).

¯neff2and cover layers with the refractive index ¯neff1. Thus, the analysis of the guided

optical modes follows in similar lines to the guided waves in a planar waveguide that

was presented in section 2.1.

The multi-mode ridge waveguide of a width Wmis represented by the effective

refractive indices of the ridge ¯nr= ¯neff2and the slab regions ¯ns= ¯neff1as shown by

figure 2.8(a) and supports mguided lateral modes. Each of these modes should satisfy

the separation condition for the guiding region:

y,n +β2

n= ¯n2

rk2(2.35)

with βnthe propagation constant of the guided lateral mode with the index nand

n= 0 ···m−1. The corresponding characteristic equation is given by:

ky,n =(n+ 1)π

Weff

(2.36)

Equation 2.36 results from the characteristic equation 2.30 by using the following:

•ΦC= ΦSfor a symmetric waveguide.

•the effective refractive index step is large enough to assume that the optical

guided mode is well confined under the ridge. Hence, Wmis well approximated

by Weff .

By considering that the optical field is TE polarized, equation 2.22 applies to

calculate the phase shift ΦCat the interfaces. The second condition above implies

that the incidence angle at the interfaces between the ridge region and the side regions

satisfies Θ→0and thus 2ΦC→π. This implies that the amplitude of the electric

field at the interfaces is too small so that the evanescent field from the ridge region

into the side regions Et=Ei−Er→0which corresponds to Et=Ei−Er→0(see

figure 2.1).

26 2.2. Planar waveguide couplers

At a sufficiently large multi-mode waveguide width Wmso that Wmλwith

λ= 2π/k a binomial expansion with k2

y,n ¯n2

rk2can be applied. In this case, by

inserting 2.35 in 2.36, the propagation constants of the guides modes are given by:

βn∼

=¯nrk−π(n+ 1)2λ

4¯nrW2

eff

(2.37)

Now let us consider an optical field that is imposed at z= 0 in figure 2.8(b). Suppose

En(y), n = 0,1,2,···m−1are the modal field distributions of the guided modes in

the multi-mode waveguide. The imposed optical field at lateral position (y, z=0) can

then be decomposed into the En(y)space [26]:

E(y, 0) =

m−1

n=0

anEn(y)(2.38)

The estimation of the coefficients anfollows using the overlap integrals:

an=RE(y, 0)En(y)dy

RE2

n(y)dy (2.39)

based on the field orthogonality relations [26].

Equation 2.38 is valid if the imposed field Eis totally contained within the multi-

mode waveguide effective width Weff which can be approximated by Wmfor waveg-

uides with a sufficiently large effective refractive index step (∆¯n= ¯nr−¯ns). In other

words, the field Eshould have a spatial spectrum that is narrow enough such as that

only the guided modes are excited which is a condition that is satisfied for all prac-

tical applications [26], since the power of the radiation modes leaks out pf the guide

rapidly [51].

Each of the guided (lateral) modes propagate in the z-direction according to βn. At

a position z > 0, the optical field becomes a superposition of the propagating guided

modes. Thus, by assuming an implicit harmonic dependence according to exp(iωt),

the field at position z > 0is written as:

E(y, z) =

m−1

n=0

anEn(y) exp(−iβnz)(2.40)

Now, by taking the propagation constant of the fundamental mode β0as a common

factor and then dropping it (without loss of generality), equation 2.40 can be written

as:

E(y, z) =

m−1

n=0

anEn(y)ei(β0−βn)z(2.41)

We define the beat length Lπof the fundamental and first order guided modes:

Lπ=π

β0−β1(2.42)

where ¯

β0, β1are the propagation constants of the fundamental mode and first order

guided mode, respectively. By inserting equations 2.37 and 2.42 in 2.41, we find:

E(y, z) =

m−1

n=0

anEn(y)ein(n+2)z

3Lπ(2.43)

27 2.2. Planar waveguide couplers

Equation 2.43 means that at a certain distance z=Lthe filed E(y, z =L)becomes

a reproduction (self-image) of E(y, 0) if:

ein(n+2)L

3Lπ= 1 or (−1)n(2.44)

This property of multi-mode waveguides is called self-imaging. If for example the first

Figure 2.9: Self-imaging in multi-mode waveguides. Interference pattern of the injected

optical field into a multi-mode waveguide (Simulation using a 2D beam propagation method).

Input field launched at the center of the multi-mode waveguide.

condition in 2.44 was fulfilled, i.e. (exp(in(n+2)L

3Lπ)=1), this means that all guides modes

overlap at the position z=Lwith phase differences of multiples of 2π. The resulting

field is then called a direct image of the input field. The second condition in 2.44

correspond to phase differences alternatively changing between odd or even multiples

of π. As a result, the self-image is a mirrored-image -with respect to the center of

the multi-mode waveguide- of the input field E(y, 0) (or a direct image of E(−y, 0)).

The conditions in equation 2.44 are satisfied at lengths Ll=l·3Lπ, l = 0,1,2, . . . (a

direct image at l even, mirrored image at l odd). It can also be shown (see [26]) that

Nimages of the input field E(y, 0) occur at Ll=l·3Lπ/N with l= 1,3,5, . . .. For

the special case of N= 2, two images with the equal amplitudes 1/√2(3dB power

divider) are created at Ll=l·3Lπ/2, l = 1,3,5, . . ..

2.2.3 Interference mechanisms of self-imaging

Generally, if the y-position of the input field is set arbitrarily, the first N-images of

the input occur at lengths of LN= 3Lπ/N. This interference mechanism is called a

general interference mechanism [26]. However, by carefully selecting the y-position of

the input field E, it may be possible to excite only a certain number of the guided

modes (selective excitation). This mechanism is referred to as a restricted interference

mechanism. It enables to reduce the periodicity of the phase factors in equation 2.43

which makes it feasible to create single or multiple images at fractional lengths of the

beat length [26]. The most commonly applied restricted interference mechanisms in

the literature are the symmetric interference mechanism and the paired interference

mechanism:

28 2.2. Planar waveguide couplers

•The symmetric interference mechanism: this is the simplest restricted in-

terference mechanism. It is usually applied to realize 1×NMMI couplers. Here

the field is launched at the center of the multi-mode waveguide which then ex-

cites only the even symmetric modes [26]. This means that the coefficients an= 0

for n= 1,3,5, . . . in equation 2.43. By noting that mod4(n·(n+ 2) = 0) for

n even, the length LNfor the first N-fold images is given by:

LN=3Lπ

4N(2.45)

which is 4-times shorter than the multi-mode waveguide length achieved by using

the general interference mechanism.

•The paired interference mechanism: this mechanism is usually applied to

realize 2×2MMI couplers. It is achieved when the input optical field is launched

at a position y=∓Wm/6. It can be shown that at this y-position the modes with

the indices n= 2,5,8, ... don’t contribute to the self-imaging (see figure 2.8). In

this mechanism, each excited even mode leads to its odd partner. Mode pairs are

for example {0, 1}, {3, 4}. At the propagation length L3dB =Lπ/2, the phase

differences between the paired modes are π/2which leads to the 3dB coupler.

Cross couplers are realized at Lπwhere the phase differences between the paired

modes are equal to π. Generally, the length of the multi-mode waveguide that

produces N-fold images using the paired interference is given by:

LN=Lπ

N(2.46)

The symmetric and paired interference mechanism will be applied in order to realize

1×2MMI splitters and the 2×2MMI couplers as will be later shown in chapter 5.

2.2.4 Characteristics of waveguide couplers

Generally, the performance of different types of waveguide couplers is characterized

by two parameters: the access loss, and the imbalance. The excess loss describes the

amount of optical power that does not emerge from the output ports of the coupler

due to propagation losses and radiation losses (coupling losses are not included). In

a coupler with a number Nof output ports, if the optical power Pin at z= 0 is

injected into the coupler, and the power that is transmitted to each output port is

P1, P2,···PNthe excess loss (in dB) is given by [40]:

exccess loss [dB] = 10 ×log Pin

P1+P2+···+PN

(2.47)

The imbalance is the splitting ratio of the output power into the output ports. It

can be measured directly at the output ports of a coupler device. By assuming equal

coupling losses at all the output ports, the imbalance (in dB) is given by:

imbalance [dB] = 10 ×log Poutmax

Poutmin

(2.48)

where Poutmax =max (P1, P2,···PN), and Poutmin =min (P1, P2,···PN).

Chapter 3

Theory of GaAs-based electro-optic

phase modulators

Applying an external electric field across certain materials (electro-optic (EO) materi-

als) can distort the orientation, position, or the shapes of the molecules building these

materials [37, p. 697]. As a result, their optical properties are modified. For example,

the refractive index of an electro-optic material can be perturbed with an electric field.

The resulting effect is called the electro-optic effect. This effect is used in electro-optic

phase modulators [23]. Electro-optic phase modulators can be for example realized

using single electro-optic crystals such lithium niobate (LiNbO3) or using compound

semiconductor material systems such as GaAs-based double heterostructures [23].

Chip-based electro-optic phase modulators based on GaAs/AlGaAs double heterostruc-

tures are of particular interest due to the unique integration properties of GaAs with

electronic devices. Besides, they reduce space and power requirements in comparison

to LiNbO3phase modulators which is very important for the realization of micro-

integrated optical devices and PICs.

In the GaAs/AlGaAs double heterostructure phase modulators, a waveguide in a multi-

layer epitaxial structure (double heterostructure) is realized and the optical field is

confined inside the guiding region of this waveguide. For the different layers a suitable

doping profile is selected such that an external electric field is applied on the guiding

region to modify the refractive index in this region. Further, in the GaAs-material

the spatial distribution of the free carrier density inside the GaAs-based double het-

erostructure can be modified by applying an electric field. The presence of the free

carriers in the GaAs (or AlGaAs) material also modifies the refractive through the

Kramers-Krönig relations [23]. This effect is called the free carrier effect (also the

carrier density-related effects).

In this chapter, we give a brief introduction to the electro-optic properties in GaAs

which include both the electro-optic effects and the free carriers effects. Further, we

discuss the properties of GaAs/AlGaAs double heterostructures and their deployment

for the realization of electro-optic phase modulators.

30 3.1. Electro-optic effects in GaAs

3.1 Electro-optic effects in GaAs

3.1.1 Linear and quadratic electro-optic effects

In electro-optic materials, the change of refractive index ¯nwith the electric field Eis

small enough such as that it can be expanded into the Taylor series [37, p. 698]:

¯n(E) = ¯n+ (d¯n/dE)|E=0E+1

2d2¯n/dE2|E=0E2+··· ,(3.1)

where the linear electro-optic (LEO) and the quadratic electro-optic (QEO) effects

arise in the second and third terms in equation 3.1, respectively. Higher order terms

in equation 3.1 can be safely ignored. For the derivation of the electro-optic coefficients

it is useful to derive from equation 3.1 an expression:

∆η= ∆ 1

¯n2=−2

¯n3(d¯n/dE)|E=0E−1

¯n3d2¯n/dE2|E=0E2(3.2)

where (η=0/ = 1/¯n2)is the electrical impermeability of the electro-optic material.

The LEO coefficient or Pockels (Friedrich Pockels (1865-1913)) coefficient is given by

r=−2

¯n3(d¯n/dE)|E=0 and the QEO coefficient or Kerr (John Kerr (1824-1907)) coef-

ficient ¯

R=−1

¯n3(d2¯n/dE2)|E=0 [37, p. 696–700]. Equation 3.2 can then be written as:

∆η=¯

rE+¯

RE2(3.3)

3.1.2 The index ellipsoid

The index ellipsoid is a geometric construction that can be used to describe the optical

properties of anisotropic materials [37, p. 712]. For example, the description of the

electro-optical properties of GaAs using the index ellipsoid is given by [37, p. 712–719]:

ηijxixj= 1, i, j = 1,2,3(3.4)

Using the ellipsoid, the refractive indices of two normal modes ¯naand ¯nbof an optical

wave propagating in a given direction kcan be derived (see figure 3.1). ηij are elements

of the impermeability tensor η=0/, and is the permittivity tensor. blanck

If an electric field E= (E1, E2, E3)is applied, the elements of the tensor ηbecome

functions of the electric field, i.e., ηij =ηij (E1, E2, E3). Thus, each of these elements

can be expanded in a Taylor’s series about E=0:

ηij (E) = ηij (0) + X

¯rij,kEk+X

Rij,klEkEl, i, j, k, l = 1,2,3(3.5)

with ¯rij,k =∂ηij

∂Ek|E=0,¯

Rij,kl =∂2ηij

∂Ek∂El|E=0 are the components of the LEO and QEO

tensors [¯

r]and [¯

R], respectively, which can be regarded as generalization of the electro-

optic coefficients in equation 3.3.

To describe the electro-optic coefficients of GaAs using the index ellipsoid, let’s

consider the unit cell of the GaAs crystal which is shown on the right side of figure 3.1.

The principal axes of the GaAs crystal correspond to the ellipsoid’s principal axes

31 3.1. Electro-optic effects in GaAs

Figure 3.1: (left) Index ellipsoid. The principal axes match the coordinates (x, y, z). ¯n1,¯n2,

¯n3are the principal refractive indices. ¯naand ¯nbare the refractive indices of two normal

modes of the wave traveling in direction k. See [37, p. 713]. (right) GaAs crystal with [iii]

being the crystallographic directions and (iii) the principal planes and i∈ {0,1,¯

1}.

(x=x1,y=x2,z=x3). The principal refractive indices are ¯n1,¯n2, and ¯n3.

Being a cubic crystal of the group ¯

43m, the GaAs crystal possesses only a single

Pockels component ¯r41 (or ¯r32,1) [41] and three Kerr components, namely, ¯

R11,¯

R12,

and ¯

R44 [42]. They can be written as the complex third-rank tensor [¯

r]and the fourth-

rank tensor [¯

R]as follows:

[¯

r] =







000

¯r41 0 0

0 ¯r41 0

0 0 ¯r41







,[¯

R] =







R11 R12 R12 000

R12 R11 R12 000

R12 R12 R11 000

000R44 0 0

0000R44 0

00000R44







, R44 =R11 −R12

(3.6)

Let us assume ,without loss of generality, that the electric field E= (E1, E2, E3)is

applied along the z-direction (E1= 0, E2= 0). In order to determine the electro-optic

effects, the ellipsoid can be rewritten using the modified impermeability (ηij (E) =

ηij (0)+∆ηij), with ηij (0) being a diagonal matrix whose elements are 1/¯n2

1,1/¯n2

2,1/¯n2

Due to the fact that the GaAs crystal is isotropic, the principal refractive indices are

equal: ¯n1= ¯n2= ¯n3= ¯n.

Pockels effect (LEO effect):

In order to extract the expression for the Pockels effect we consider only the first order

32 3.1. Electro-optic effects in GaAs

of the Taylor series in equation 3.5 with the coefficients (¯rij,k =∂ηij

∂Ek|E=0):

ηij (E) = ηij (0) + X

¯rij,kEk, i, j, k = 1,2,3(3.7)

By inserting equations 3.6 and 3.7 in 3.4, the modified ellipsoid for the Pockels effect

(LEO effect) can be written as [37]:

1+x2

2+x2

¯n2+ 2¯r41 ·E3·x1x2= 1 (3.8)

Let us now consider the modified coordinates v1=x1−x2

√2,v2=x1+x2

√2, and v1=x3. By

taking into account that 2·x1x2=v2

2−v2

1, equation 3.8 yields:

¯n2

1(E)·v2

1+1

¯n2

2(E)·v2

2+1

¯n2

3(E)·v2

3= 1 (3.9)

where: 1

¯n2

1(E)=1

¯n2+ ¯r41 ·E

¯n2

2(E)=1

¯n2−¯r41 ·E

¯n2

3(E)=1

¯n2

(3.10)

Equations 3.9 and 3.10 shows that the modified principal axes correspond to v1,v2,

v3which are rotated by 45◦with respect to the original coordinates x1,x2,x3. The

modified ellipsoid is now given in the new coordinate system (v1,v2,v3) and the final

step is to extract the associated refractive index change due the LEO effect. This

follows from equation 3.10 by using the approximation 1/√1 + δ≈1−1

2δwhich leads

to:

¯n1(E)≈¯n−1

2¯n3¯r41 ·E3

¯n2(E)≈¯n+1

2¯n3¯r41 ·E3

¯n3(E) = ¯n

(3.11)

The previous discussion yields the following: for the GaAs crystal with the principal

axes x=x1and y=x2matched to the [100] and [010] crystallographic directions,

the modified ellipsoid due LEO effect is rotated with respect to the principal axes

by 45◦. Which means that the corresponding modified v1, v2coordinates v1, v2are

then oriented to the [110] and [1¯

10] crystallographic directions, respectively. A plane

wave that is propagating in the xy-plane (parallel to the principal plane (001), see

figure 3.1 (right)) can observe the modification of the refractive index by the LEO

effect as long as the oscillation of its electric field in also parallel to the xy-plane,

which is the case of a TE polarized wave. If the propagation vector kof the TE wave

is chosen to be parallel to the [110] crystallographic direction (waveguide along the

v1access), the modification of the refractive index by the LEO effect corresponds to

¯n1in equation 3.11. For propagation in the [1¯

10] direction, the modification of the

refractive index by the LEO effect corresponds then to ¯n2in equation 3.11. If the

plane wave is on the other hand TM polarized, with the oscillation of the electric field

33 3.1. Electro-optic effects in GaAs

perpendicular to the xy-plane, no contribution of the LEO effect is observed according

to ¯n3in equation 3.11.

Kerr effect (QEO effect):

Following a discussion similar to what was described for the LEO effect, we drop this

time the first derivative (linear coefficient) which reduces equation 3.5 to:

ηij (E) = ηij (0) + X

Rij,klEkEl, i, j, k, l = 1,2,3(3.12)

By inserting equations 3.6 and 3.12 in 3.4, the modified ellipsoid for the Kerr effect

(QEO effect) can be written as [37]:

1

¯n2+¯

R12 ·E32(x2

1+x2

2) + 1

¯n2+¯

R11 ·E32x2

3= 1 (3.13)

Equation 3.13 shows that the Kerr effect corresponds to an equal modification of the

ellipsoid in the principal xy-plane and another, different modification in the principal

z-direction.

The resulting refractive index change due to the QEO effect (using the approximation

1/√1 + δ≈1−1

2δ) is:

¯n´a= ¯n1,2(E)≈¯n−1

2¯n3¯

R12 ·E2

¯n´

b= ¯n3(E)≈¯n−1

2¯n3¯

R11 ·E2

(3.14)

here ¯n´aand ¯n´

b, are the modified refractive indices of the normal modes in the xy-

principal plane and in the z-axis, respectively.

Hence, for a plane wave that is propagating in the xy-plane (parallel to the princi-

pal plane (001)), the QEO effect has no orientation dependence on the crystallographic

direction of the waveguide. The first and second terms in equation 3.14 correspond to

the contribution to phase modulation of the TE mode (with the QEO coefficient ¯

R12),

and the TM mode (with the QEO coefficient ¯

R11), respectively.

LEO and QEO effects in GaAs-based waveguides

The previous discussion of the LEO and QEO effects yields the following: in a GaAs-

based phase modulator waveguide that is realized on a (001) GaAs substrate with the

electric field perpendicular to the multi-layer, the waveguide (the propagation direc-

tion) should be realized parallel to the [110] or [1¯

10] directions to use the LEO effect.

The following two equations describe the refractive index modification by the LEO

and QEO effects depending on the modal excitation (TE or TM) [23], [19]:

blFor the TE mode:

∆¯n=±¯n3

2¯r41 ·E−¯n3

2¯

R12 ·E2(3.15)

34 3.2. Free carrier effects (carrier density-related effects)

bl and for the TM mode:

∆¯n=−¯n3

2¯

R11 ·E2(3.16)

The positive and negative signs in the first term of 3.15 correspond to light propagation

in the [1¯

10] and [110] directions, respectively.

3.2 Free carrier effects (carrier density-related effects)

Free carriers in GaAs which are introduced by doping or injection reduce the refractive

index [23], [30]. This change of the refractive index is attributed to a modification of

the absorption in two regions of the spectrum, namely, in the mid-infrared and at

the absorption edge. In the mid-infrared region, the refractive index changes due to

the plasma and intervalence-band absorption. At the absorption edge the band-filling

effect and the many-body effect modify the refractive index. In [23], J. Faist and F.-K.

Reinhart found that the overall contribution of the free carrier effects is proportional

to the free carrier concentration and independent of the effective mass of the free car-

riers. They were able to give an analytic formula for the free carrier effects in GaAs.

Accordingly, the modification of the refractive index by free carriers is given by [23]:

for holes:

∆¯n=−



8×10−22

+1.46 ×10−21

(Eg+kT)2−E2

ph 

·nh(3.17)

and for electrons:

∆¯n=−



2.8×10−21

+1.99 ×10−21

(Eg+kT)2−E2

ph 

·ne(3.18)

where nhand neare the holes and electrons densities in [cm−3] unit, respectively, Eg

is the bandgap energy (in [eV]), and Eph is the photon energy (in [eV]). kT is the

thermal energy (also in [eV] with kbeing Boltzmann constant and Tthe temperature.

GaAs-based electro-based phase modulators use both, the electro-optic effects and

the free carriers effects. The phase modulator waveguides are realized in double het-

erostructures for optical field confinement with intentionally doped (or undoped) layers

for electric field and carrier confinement as discussed in the following sections.

3.3 GaAs-based double heterostructures

GaAs is one of a few binary semiconductors that can exist as a high-quality sub-

strate. It can also be mixed with atoms from the same groups of the periodic system

to form ternary or quaternary compound semiconductors such as AlxGa1−xAs and

(AlxGa1−x)yIn1−yAs, with 0< x < 1and 0< y < 1. Mixed compound semiconduc-

tors from the III-V groups are very attractive for the realization of photonic integrated

35 3.3. GaAs-based double heterostructures

circuits. They can be grown in thin layers which are called epitaxial layers. As a rule

of thumb, within a system of multiple epitaxial layers with different compound semi-

conductors, the material with the lower bandgap has a larger refractive index. This

is a key for defining the functionality of photonic devices such as laser diodes and

waveguides. For example, in the planar waveguide in figure 2.2, the AlxGa1−xAs with

0<x<1has a larger bandgap than GaAs. Thus for the purpose of wave guiding, the

GaAs is used as a core material and the cover is made of AlxGa1−xAs with 0<x<1.

The waveguide is realized using GaAs/AlxGa1−xAs multi-layers. The entire waveg-

uide is grown on a GaAs substrate. Further, in GaAs-based electro-optic devices, the

multi-layers are doped with negative (n-type) free electrons or positive (p-type) free

holes in order to form heterojunctions for electric field confinement.

3.3.1 The p-n junction and the heterojunctions

The p-n junction occurs between p-type and n-type regions of the same semiconductor

material. When the p- and n-type regions are brought in contact the electrons diffuse

from the n-region into the p-region where they then recombine with excess holes. In

a similar way, the holes diffuse from the p-region into the n-region and recombine

with excess electrons. The diffusing electrons leave positively charged ionized doner

atoms whereas the diffusing holes leave negatively charged ionized acceptor atoms.

This diffusion on both sides of the junction leaves a narrow area at both sides of the

junction which is totally empty of free carriers and called the depletion region [24].

The tension that is created due to this situation creates a so-called built-in electric

field. Applying a reverse biased voltage to the p-n junction will drive the holes in the

p-side of the junction and the electrons in the n-side further away from the junction

which increases the depletion width.

In double heterostructures, the junction can be created between different semicon-

ductor materials (for example between GaAs and AlxGa1−xAs, or between Alx1Ga1−x1As

and Alx2Ga1−x2As with x16=x2) with different doping types, and is therefore called a

heterojunction. The energy band-diagram of the heterojunction suffers a local discon-

tinuity at the junction. This creates a barrier that can be used for example to create

prevented zones for selected carriers (carrier confinement in desired regions of space).

This property is used in GaAs-based double heterostructures for the design of diode

lasers, photodiodes, and for phase modulators as we will show in the following section.

3.3.2 GaAs/AlGaAs double heterostructures phase modulators

GaAs-based phase modulators use GaAs/AlGaAs double heterostructures to enforce

index guiding of the optical field confinement. Appropriate free carrier injection into

the different multi-layers is applied to create a p-n heterojunction in the waveguide

with a depletion area for electric field confinement.

Let us consider for example the multilayer in figure 3.2. The GaAs core with

the lower band-gap is sandwiched between the AlxGa1−xAs cladding layers to form a

double heterostructure. The layers are doped as follows: upper cladding p-doped(P-

AlxGa1−xAs), core p-doped (p-GaAs), lower cladding n-doped (N-AlxGa1−xAs). Such

a doping profile is called: P-p-N. The device is assumed to be grown on a n+-GaAs

(n-type highly doped) substrate that forms the n-contact layer and an additional

p+-GaAs (p-type highly doped) upper layer forms the p-contact layer. We further

36 3.3. GaAs-based double heterostructures

Figure 3.2: Concept of a double heterostructure phase modulator. Light beam is focused

into the guiding layer. The application of reverse-biased voltage changes the electric field

and carrier density in the junction which modifies the refractive index. ∆Φ(V)is the phase

shift due to applied reversed bias voltage V.

assume that the concentration of the free carriers in the core layer is much lower

than their concentration in the cover layers. In the absence of external electric field

(corresponding to a reverse biase voltage V=V0= 0) a depletion region is formed due

to the built-in voltage (VD). This region has one side in the p-GaAs core layer and

one side in the N-AlxGa1−xAs cladding layer. Free carriers are pushed on the sides of

this region. The depletion widths into the p-side (xp) and into the N-side (xN) are

given from the Poisson’s equations by [13]:

xp=s2pN

q·NN

np·VD−V

N·NN−p·np

xN=s2pN

q·np

NN·VD−V

p·np−N·NN

(3.19)

where pand Nare the dielectric constants of GaAs and AlxGa1−xAs, respectively,

qis the elementary charge, npthe doping in the p-GaAs core, NNthe doping in the

N-AlxGa1−xAs cladding. Please note that equation 3.19 is only valid for the case when

xpdoes not exceed the total thickness of the p-GaAs core d. For the cases when xpex-

tends beyond d, a different equation (without loss of generality) arises from solving the

Poisson’s equations in the three regions (upper-cladding, core, lower-cladding), see [13].

Since npNNin equation 3.19, the depletion region extends more into the p-side

of the junction, hence xp> xN. The built-in electric field covers the entire depletion

region with the width xp+xN. If a reverse bias voltage V=V1<0is applied to the

heterojunction, the width of the depletion region expands and the free carrier density

is pushed to its edges. As a result the local refractive index is changed due to both,

the electro-optic effects, and the free carriers effects as we previously explained.

3.3.3 Phase modulation efficiency in double heterostructures

Phase modulation is attributed to the overlap of the optical field with the change of

the refractive index due to electro-optic effects (LEO and QEO effects) and due to the

depletion of the carrier density by electric field (carrier density-related effects). The

electric field is present inside the depletion region of the modulator which is assumed

to overlap with the guiding region for the optical field whereas the free carriers density

37 3.3. GaAs-based double heterostructures

is pushed to the sides of the depletion region. Therefore, the resulting phase modu-

lation due to electro-optic effects and carrier effects can be separately handled. This

is the key for designing GaAs/AlGaAs double heterostructure phase modulators. By

carefully choosing the doping profiles of the different layers, the contribution of the

free carriers effects to phase modulation can be controlled in order to optimize the

performance of the phase modulators.

Let I(x)be the intensity distribution (assumed to be uniform in y-direction) of

the wave propagating along the modulator in figure 3.2 in the guiding core with the

refractive index profile ¯n(x). Then the modal refractive index change ∆¯neff (not to

be confused with the refractive index change ∆¯n) that contributes to the phase shift

is given by [19]:

∆¯neff =1

¯neff ·R+∞

−∞ ¯n(x)∆¯n(x)I(x)dx

R+∞

−∞ I(x)dx (3.20)

where ¯neff is the effective refractive index (the modal refractive index). The overlap

integral in equation 3.20 can be used to determine the contribution of each of the three

effects individually (LEO, QEO, and carrier effects) to ∆¯neff , and hence the relevant

phase shift in a modulator of length Lis given by:

∆Φ = 2π

λ·L·∆¯neff .(3.21)

Chapter 4

Design of GaAs-based phase

modulators

Laser emission at the wavelengths of 780 nm and 1064 nm is realized in the GaAs-

based technology using GaAs/AlGaAs double heterostructures [43], [44]. The layer

structure of lasers includes a region to provide gain that is usually referred to as ac-

tive region. This region typically consists of a p-n junction where injected carriers

permanently recombine [24]. GaAs/AlGaAs double heterostructures can also be used

to realize passive components i.e. components that do not generate (coherent) light

such as electro-optic phase modulators [23]. Unlike for active components, for passive

components the layer structure does not contain an active region.

This chapter is dedicated for the design of GaAs-based phase modulators. The pa-

rameters for the phase modulator double heterostructure multi-layer (material, thick-

nesses, doping profiles) are calculated and the corresponding electro-optic coefficients

for the guiding core at the operation wavelength are estimated. Next, the design of

the lateral waveguides (ridge width and etching depth) of the phase modulators is

presented.

4.1 Material profile for GaAs-based phase modulators

In the literature, GaAs/AlGaAs double heterostructure phase modulators are typically

realized at the wavelength range of 1.05 µmto 1.32 µm, (corresponding to the photon

energy range of 1.18 eV to 0.94 eV). The GaAs material is suitable for guiding of optical

signals with a photon energy below the bandgap of GaAs (1.43 eV) [29, p. 42-44] which

corresponds to wavelengths larger than λ=872 nm (see figure 4.1). This is why GaAs

is used for the guiding region of the waveguide of phase modulators at the wavelength

range of 1.05 µmto 1.32 µm. The natural choice for the cladding layers is in this case

AlxGa1−xAs (0<x<1) with a refractive index lower than that of GaAs. However, for

applications at photon energies larger than the bandgap of GaAs, i.g. at the wavelength

of 780 nm with the photon energy of 1.59 eV, GaAs suffers strong intraband absorption

[19]. This is why we chose AlGaAs with the Aluminum concentration x= 0.35. The

bandgap energy of Al0.35Ga0.65As corresponds to 1.9 eV which is sufficiently larger than

the photon energy at the wavelength of 780 nm (1.59 eV) [19]. This is the first time

a core material different from GaAs is used in GaAs/AlGaAs double heterostructure

39 4.2. Design of phase modulators for laser radiation at 780 nm

Figure 4.1: Fundamental absorption in GaAs and AlGaAs. Figure taken from [30].

phase modulators.

4.2 Design of phase modulators for laser radiation at 780 nm

In this section we present a detailed description of the design of the phase modulator

for operation at the wavelength of 780 nm. We calculate the electro-optic coefficients

of Al0.35Ga0.65As at 780 nm to determine the contribution of the LEO and QEO effects

to phase modulation. Then we present the multi-layer epitaxial structure for optimum

wave guiding. Further, the doping profile is carefully selected to use the contribution

of the carrier density-related effects to phase modulation in addition to the electro-

optic effects (LEO and QEO). Finally, the lateral design of the cross-section of the

modulator waveguide that supports single lateral mode operation (single mode ridge

waveguide) is presented.

4.2.1 Electro-optic coefficients of Al0.35Ga0.65As

Electro-optic effects in GaAs-based double heterostructures are the LEO effect, QEO

effect, and the carrier density-related effects (see chapter 3). In the following analysis

we consider the case where the phase modulator waveguide is grown on a (001) GaAs

substrate and the heterojunction electric field is applied along the [001] principal axis.

The modification of the refractive index by the LEO ad QEO effects follows equa-

tions 3.15 and 3.16 in chapter 3.

The electro-optic coefficients for Al0.35Ga0.65As are unknown and have to be esti-

mated for the design of the phase modulator. In the following, we suggest a method to

40 4.2. Design of phase modulators for laser radiation at 780 nm

estimate the electro-optic coefficients of Al0.35Ga0.65As at the wavelength of 780 nm us-

ing the reported values of these coefficients for GaAs. This method has been published

in [19]. In the previous works [45], [15], the LEO coefficients for GaAs with the bandgap

wavelength λ1g= 0.872 µmwere reported within the wavelength range of 1.05 µmto

1.31 µm. We assume that for AlxGa1−xAs material with 0≤x≤1, the electro-

optic coefficients are determined solely by means of the spacing between the desired

wavelength and the bandgap wavelength (independently from the Al-content) [19].

For the Al0.35Ga0.65As device, the design wavelength λ2= 0.780 µmist 0.129 µm

below the bandgap wavelength λ2g= 0.651 µm. We hence assume, that the LEO coeffi-

cient of Al0.35Ga0.65As at 0.780 µmcorrespond to the coefficients of a GaAs device oper-

ated 0.129 µmabove its bandgap, i.e. at λ1= 1.001 µm. The final step is to interpolate

the value of the LEO coefficient for GaAs at the wavelength λ1from the values which

are given in [45]. This leads to a value of ¯r41(GaAs, λ1= 1.001 µm) = 1.84 ×10−12 m/V

which is then assumed to correspond to the requested value of the LEO coefficients:

¯r41(Al0.35Ga0.65As, λ2= 0.780 µm) = 1.84 ×10−12 m/V.

Next, we determine the QEO coefficient ¯

R12 for Al0.35Ga0.65As. In [23], ¯

R12 for GaAs

was calculated using the expression:

R12 =C·Eph2

¯n4(Eph)E2

g−E2

ph2(4.1)

with Egthe gap energy, Eph the photon energy, and Cis a constant. Equation 4.1

is valid for AlxGa1−xAs, 0≤x≤1. Thus, by relating ¯

R12 for Al0.35Ga0.65As at the

requested wavelength λ2= 0.780 µmto ¯

R12 of GaAs at a given wavelength λ1:

R12(λ2) = ¯

R12(λ1)·



¯n2(λ1)·λ2·λ2

2g·λ2

1−λ2

1g

¯n2(λ2)·λ1·λ2

1g·λ2

2−λ2

2g



(4.2)

The experimental value of ¯

R12 =−1.7×10−20 m2/V2was given for GaAs at the wave-

length λ1= 1.15 µm[23]. The corresponding bandgap wavelength is λ1g= 0.872 µm.

With ¯n(λ1)=3.4496 for GaAs and ¯n(λ2)=3.4265 for Al0.35Ga0.65As. The resulting

QEO coefficient is:

R12(Al0.35Ga0.65As, λ2= 0.780 µm) = −2.3×10−20 m2/V2.

The phase modulator double heterostructure can be modeled using the estimated

values of the LEO and QEO coefficients and the contribution from the carrier density-

related effects that were presented earlier in chapter 3.

4.2.2 Vertical waveguide at 780 nm

Multi-layer structure for operation at the wavelength 780 nm

For a well-defined phase modulation, the vertical waveguide has to be single mode [23].

Phase modulation in the waveguide double heterostructure is achieved by the electro-

optic effects and contributions from the free carriers effect.

The first design criteria of the vertical multi-layer structure of the phase modulator is

41 4.2. Design of phase modulators for laser radiation at 780 nm

the phase modulation efficiency. To increase the phase shift due to the electro-optic

effects, an optimal overlap between the optical field and the applied external electric

field should be achieved [19]. Further, the contribution of the free carriers effect to

the phase shift can be increased by increasing the overlap of the optical field with the

change of the carrier density. The free carriers effect can be controlled by carefully

selecting the doping profile of the different layers in the double heterostructure.

The second design criterium of the vertical waveguide is the propagation loss. In ad-

dition to fundamental absorption, losses arise in the vertical waveguide due to free

carrier absorption, and by leaky modes in the cladding layers [46]. This has to be

taken into consideration when choosing the material, the doping profile, and thickness

of the different layers [46].

Let us first consider the most simple vertical planar waveguide which is a 3-layer

(cladding-core-cladding) waveguide. In order for the waveguide with core thickness d

to support only one guided mode, the so-called low-index guiding concept has to be

applied. This concept is based on that the difference between the refractive indices of

the core and the cladding materials are chosen small enough such that only one mode,

the fundamental mode, is guided [37, p. 252]. For example for the Al0.35Ga0.65As-based

waveguide with the core thickness d= 2 µmthe refractive index of the core corresponds

to ¯ncore = 3.426. To support single mode operation at 780 nm, the refractive index

¯ncladd has to be ¯ncladd >3.421. The corresponding refractive index difference is ∆¯n <

0.01 which means that a significant fraction of the guided mode will be propagating

outside the core, i.e. inside the cladding [29, p. 19-47]. If, for example, the modulating

electric field was selected to be well-confined within the guiding core, inefficient phase

modulation can be expected due to the small overlap between optical and electrical

fields.

Thus, to fulfill first design criterium, the vertical waveguide has to support a single

guided mode who’s optical field is well confined within the waveguide core. To enforce

this, we add a thin intermediate confinement layer between the core and each of the

cladding layers to create a 5-layer vertical structure. In order for this multi-layer to

support only the guided fundamental mode, the refractive index difference between the

core and cladding has to be chosen small enough (on the order of 10−2) in order to fulfill

the low-index guiding requirement. Further, the additional thin confinement layer

should have a much lower refractive index than the cladding layer. This confinement

layer serves as a mode filter [46]. This is because the fundamental mode is confined

inside the core region whereas the higher order modes leak through this layer into the

substrate [46]. The resulting refractive index profile is also referred to as a W-shape

multi-layer.

Further, suppose the refractive indices of the core, confinement, and cladding layers

are ¯ncore,¯nconf , and ¯ncladd, respectively. Then the following two conditions must be

met [46]:

¯neff1<¯ncladd <¯neff0,¯nconf <¯ncladd <¯ncore (4.3)

with ¯neff0and ¯neff1in equation 4.3 are the effective refractive indices of the funda-

mental mode and the first higher order mode, respectively.

The first (left) condition in equation 4.3 describes the single mode requirement in

weakly (low-index) guiding structures and the second (right) condition in equation 4.3

implies mode confinement in the W-shape multi-layer.

42 4.2. Design of phase modulators for laser radiation at 780 nm

The waveguide multi-layer for operation at 780 nm is described in table 4.1. The Al-

Table 4.1: Multi-layer structure of the phase modulator waveguide at the wavelength of

780 nm.

index function Material Thickness [nm]

1 upper cladding Al0.37Ga0.63As 1000

2 confinement Al0.7Ga0.3As 150

3 core Al0.35Ga0.65As 1900

4 confinement Al0.7Ga0.3As 150

5 lower cladding Al0.37Ga0.63As 1500

contents in the materials for the core, confinement, and cladding layers are (x= 0.35),

(x= 0.7), and (x= 0.37), respectively. This corresponds to the refractive indices

¯ncore = 3.426,¯nconf = 3.193, and ¯ncladd = 3.413, respectively, which satisfies the second

condition in equation 4.3. The thickness of the 5 layers has been selected such that

Figure 4.2: Refractive index profile (W-shape) of the vertical multilayer structure described

in table 4.2 and the intensity profile of the fundamental TE mode. Adapted/Reprinted with

permission from Ref [19], [Springer Nature].

the effective refractive indices of the TE (vertical) modes are ¯neff0= 3.421 for the

fundamental TE mode, and ¯neff1= 3.407 for the first higher order (leaky) TE mode.

This satisfies the first condition in equation 4.3. In figure 4.2 (which corresponds to

the vertical structure in table 4.2), the W-shape refractive index profile can be clearly

seen. Please note that the minor steps in the index profile correspond to additional

thin AlGaAs layers with a suitable Al-content to introduce an intermediate refractive

index step between the main layers. The optical field profile of the fundamental TE

mode in figure 4.2 and the effective refractive indices ¯neff0and ¯neff1were calculated

using a commercial software tool that utilizes a finite element method (FEM) [47].

It can be seen in figure 4.2 that the optical field intensity profile is well confined

inside the core region. The full width at half maximum (FWHM) of the intensity

profile of the far field is 22.5◦which satisfies well the requirements for integration

43 4.2. Design of phase modulators for laser radiation at 780 nm

of the phase modulator into hybrid laser modules with other passive/active optical

elements.

Table 4.2: Double heterostructure of a phase modulator at 780 nm with a p-i-n doping profile

of the waveguide core. Adapted/Reprinted with permission from Ref [19], [Springer Nature].

Layer Material Thickness [nm] Doping [cm−3]

contact p-GaAs 270 2×1019

cladding p-Al0.37Ga0.63As 1000 2×1018

confinement p-Al0.7Ga0.3As 150 7×1017

p-core p-Al0.35Ga0.65As 200 3×1017

intrinsic core p-Al0.35Ga0.65As 1500 2×1015

n-core n-Al0.35Ga0.65As 200 3×1017

confinement n-Al0.7Ga0.3As 150 7×1017

cladding n-Al0.37Ga0.63As 1500 2×1018

buffer n-GaAs 300 2×1019

substrate n-GaAs 130000 2×1019

Modal losses

In order to create the modulator heterojunction for electric field confinement, free

carriers (holes/electrons) are introduced by doping. The doping profile has to consider

the free carriers absorption in order to realize a low-loss waveguide. In [48, p. 175],

the free carrier absorption αfc in GaAs near the absorption edge has been given by:

αfc([cm−1]) ≈3×10−18ne+ 7 ×10−18nh(4.4)

where neand nhare being the free carrier density for electrons and holes, respectively

(both in cm−3]). The constants in equation 4.4 correspond to the electron cross-section

(3×10−18cm2) and to the hole cross-section (7×10−18cm2).

Based on the waveguide given in table 4.1, the complete vertical multi-layer struc-

ture of a phase modulator at the wavelength of 780 nm is given in table 4.2. The

central part of the waveguide core is intentionally not doped (intrinsic core), whereas

a200 nm thick region on either side of the core are moderately doped with ne=nh=

3×1017 cm−3. The low index (confinement) and cladding layers are strongly doped in

order to enforce a large overlap between the electric field and the optical field inside

the core which accounts to modal losses amount of only αmod = 0.1 cm−1.

Optimization of the phase modulation efficiency

In order to determine the phase modulation efficiency of the double heterostructure

one has to calculate the electric field and free carriers density for different modulat-

ing voltage signals. Let us compare the electric field and free carrier density for the

phase modulator of table 4.2 for two different settings of the modulating reverse-bias

voltage, for example at V0= 5 V and at V1= 15 V. We determine the dependence of

the distribution of the electric field on the voltage by using the drift-diffusion equa-

tions. The equations are solved using an in-house software that takes into account

44 4.2. Design of phase modulators for laser radiation at 780 nm

Figure 4.3: Simulation of the modification of the electric field and carrier density

(holes+electrons) inside the modulator heterojunction of table 4.2 when the reverse bias

voltages V0= 5 V and V1= 15 V are applied.

the Fermi statistics and the hetero-boundaries of the double heterostructure. As we

explained earlier in the previous chapter, in the absence of an external electric field

(at V=V0), the free carriers deplete and a built-in electric field is confined inside

the depletion region. When the reverse-bias voltage is applied (V=V1), the width of

the depletion region increases and the free carriers are pushed to it’s boundaries. The

resulting electric field is larger than the built-in electric field as shown in figure 4.3.

The dependence of the distribution of the electric fields on the voltage was determined

by solving the drift-diffusion equations by taking into account Fermi statistics and the

hetero-boundaries. The phase shift can be determined from the modification of the

modal refractive index ∆¯neff and the overlap integral that was previously given in

equation 3.20.

The doping profile of the double heterostructure multi-layer can be further op-

timized in order to increase phase modulation efficiency. To better understand the

effect of the doping profile on the phase modulation efficiency, we compare the P-p-

i-n-N double heterostructure of table 4.2 with another double heterostructure with

identical multi-layer and a slightly different doping profile.

Instead of the p-i-n doping profile, the core in the second double heterostructure has a

symmetric p-n (p-n core) doping profile. The doping profiles of the cladding layers and

the substrate remain unchanged. This new multi-layer structure is given in table 4.3.

We refer to the new doping profile as P-p-n-N.

blank

The comparison between the two double heterostructures is shown in figure 4.4. The

modification of the electric field and carrier density is plotted together with the cor-

responding intensity profile of the TE mode in order to demonstrate the difference of

the overlap between optical and electrical fields for both double heterostructures. The

blue curve corresponds to the change of the electric field inside the heterojunction and

the red curve is related to the modification of the electrons and holes (carrier) density.

The evaluation of figure 4.4 leads to the following conclusions:

45 4.2. Design of phase modulators for laser radiation at 780 nm

Table 4.3: Vertical multi-layer structure of a phase modulator at 780 nm with a p-n doping

profile of the waveguide core.

Layer Material Thickness [nm] Doping [cm−3]

contact p-GaAs 270 2×1019

cladding p-Al0.37Ga0.63As 1000 2×1018

low index p-Al0.7Ga0.3As 150 7×1017

p-core p-Al0.35Ga0.65As 450 3×1017

p-core p-Al0.35Ga0.65As 500 3×1016

n-core n-Al0.35Ga0.65As 500 3×1016

n-core n-Al0.35Ga0.65As 450 3×1017

low index n-Al0.7Ga0.3As 150 7×1017

cladding n-Al0.37Ga0.63As 1500 2×1018

buffer n-GaAs 300 2×1019

substrate n-GaAs 130000 2×1019

Figure 4.4: Modification of the electric field and carrier density of two modulators hetero-

junctions when the reversed bias voltage changes from 5 V to 15 V. (top) the phase modulator

with p-i-n core from table 4.2. (bottom) the phase modulator wih p-n core (table 4.3).

46 4.2. Design of phase modulators for laser radiation at 780 nm

p-i-n double heterostructure (table 4.2): the core is almost fully depleted. The

modification of the free carriers density occurs at the limits of the guiding core

whereas the optical field is well-confined inside the core. As a result of the minor

overlap between the optical field and carrier density, phase modulation due the

free carrier effects is expected to show a very low efficiency. Nevertheless, the

modification of the electric field, on the other hand, overlaps very well with the

optical field. The EO effects (LEO and QEO) are therefore expected to be the

dominant effects to phase modulation for the p-i-n double heterostructure.

p-n double heterostructure (table 4.3): the contribution of the free carriers ef-

fect to phase modulation is expected to increase due to the larger overlap of

carrier density with the optical field.

The individual contribution of the LEO and QEO effects to phase modulation can

be calculated from equations 3.15, 3.16 and the overlap integral in equation 3.20.

For the carrier effects, the local refractive index change can first be calculated using

equations 3.17 and 3.18, then the overlap integral in (3.20) can be used. The resulting

values of the phase modulation efficiency for each of the three effects is given in table 4.4

for both double heterostructures. The enhancement in the carrier effect in the p-n

Table 4.4: Modulation efficiency of phase modulators at 780 nm with different doping profiles

from tables 4.2, 4.3.

λstructure linear effects quadratic effects

[nm] [deg/(V·mm)] [deg/(V2·mm)]

Pockels carrier total Kerr

780 table 4.2 8.41 0.17 8.58 0.96

780 table 4.3 10.73 0.63 11.36 1.13

double heterostructure increases the modulation efficiency in comparison to the p-i-n

double heterostructure. The p-i-n structure, on the other hand, has a smaller junction

capacitance, which is more suitable for high speed operation. In both structures the

QEO effect is very small due to the large difference between the photon energy and the

band gap energy of the core layer [19]. If one decreases the spacing between the band

gap energy and photon energy (by reducing the Al-content of the guiding layer), the

QEO effect can be increased [19]. Another way to increase the QEO effect requires the

employment of quantum wells [22] in the guiding region. However, the use of quantum

well structures is beyond the scope of this work but is certainly interesting for the

future.

4.2.3 Lateral waveguide at 780 nm

The lateral confinement can be provided by means of the ridge waveguide (RW). The

ridge waveguide shall only support the fundamental lateral guided mode in order to

ensure a well-defined phase shift [19]. We use the finite element method [47] to pose

the conditions for lateral single modes in the RW. The cutoff condition between the

multi-mode and single mode regimes as a function of the ridge width and the etching

depth is given in figure 4.5. For example, if the ridge width is set to 3µm, the etching

depth should be less than 1.6µmin order for the ridge to guide only the fundamental

47 4.3. Design of phase modulators for laser radiation at 1064 nm

Figure 4.5: (left): Simulated single mode propagation condition at the wavelength 780 nm

for the vertical structure in table 4.2. The region over the solid line corresponds to the

multi-mode regime, and the region under the ridge represents the single mode regime. (right)

fundamental TE mode with the ridge width 2µmand etch depth 1.9µm. Adapted/Reprinted

with permission from Ref [19], [Springer Nature].

TE mode. To ease coupling of a laser beam into the phase modulator, a wider RW

may however be recommended. This implies a smaller etching depth which reduces the

effective index step of the lateral waveguide. The small effective refractive index step

should not be problematic for phase modulators with straight or tilted waveguides.

However, waveguides that feature a small step of the effective refractive index may be

problematic for the design of more complex photonic circuits. The smaller the step,

the less the wave is confined so that in more complex PICs, where bent waveguides

have to be implemented, significant radiation loss may occur. This requirement has

been taken into consideration for the design of the bent waveguides for the couplers

in the next chapter.

4.3 Design of phase modulators for laser radiation at 1064 nm

4.3.1 Electro-optic coefficients of GaAs at 1064 nm

As earlier discussed in section 4.1, the GaAs is a suitable material for the waveguide

core of the phase modulator at 1064 nm. The corresponding LEO coefficient for GaAs

at 1064 nm follows directly from the linear interpolation in [45]:

¯r41(GaAs, λ= 1.064 µm) = 1.77 ×10−12 m/V.

To determine the QEO coefficient we use equation 4.1 for GaAs at 1064 nm with the

photon energy of 1.156 eV, bandgap energy 1.424 eV, the refractive index ¯n= 3.4805,

and C= 8.5×10−15 (eV)2cm2/V2[23]:

R12(GaAs, λ= 1.064 µm) = −1.95 ×10−20 m2/V2.

The design of the phase modulator at 1064 nm follows in similar lines to the procedure

that was earlier presented for phase modulators at 780 nm. First, the vertical multi-

layer structure is optimized for low propagation losses and efficient phase modulation,

then a single mode lateral waveguide is designed.

48 4.3. Design of phase modulators for laser radiation at 1064 nm

4.3.2 Vertical waveguide at 1064 nm

Two vertical layer structures of phase modulators at the wavelength of 1064 nm with

GaAs core and AlGaAs cladding lasers is given in table 4.5 and in table 4.6. The

Table 4.5: Vertical layer structure of a phase modulator for operation at the wavelength of

1064 nm with a p-p-n-n guiding core.

Layer Material Thickness [nm] Doping [cm−3]

contact p-GaAs 270 2×1019

cladding p-Al0.05Ga0.95As 1300 2×1018

low index p-Al0.4Ga0.6As 180 7×1017

p-core p-GaAs 600 3×1017

p-core p-GaAs 600 3×1016

n-core n-GaAs 600 3×1016

n-core n-GaAs 600 3×1017

low index n-Al0.4Ga0.6As 180 7×1017

cladding n-Al0.05Ga0.95As 1800 2×1018

buffer n-GaAs 300 2×1019

substrate n-GaAs 130000 2×1019

Table 4.6: Vertical layer structure of a phase modulator for operation at the wavelength of

1064 nm with a p-n guiding core.

Layer Material Thickness [nm] Doping [cm−3]

contact p-GaAs 270 2×1019

cladding p-Al0.05Ga0.95As 1300 2×1018

low index p-Al0.4Ga0.6As 180 7×1017

p-core p-GaAs 1200 5×1016

n-core n-GaAs 1200 5×1016

low index n-Al0.4Ga0.6As 180 7×1017

cladding n-Al0.05Ga0.95As 1800 2×1018

buffer n-GaAs 300 2×1019

substrate n-GaAs 130000 2×1019

layers thicknesses and doping profile were optimized in similar lines to the modulators

at 780 nm. The modal losses of the structure in table 4.5 amount to αmod = 0.59 cm−1.

The structure in table 4.6 has lower modal losses αmod = 0.28 cm−1. Figure 4.6 shows

for the modulator structure in table 4.5 the simulated modification of the electric field

and carrier density when the applied modulation reverse-bias voltage is changed from

5 V to 15 V.

The phase modulation efficiency is calculated from the overlap integrals of the optical

field with the electric field and the free carrier density. The calculated phase modula-

tion due to the LEO effect, the QEO effect, and the carrier density-related effects are

given in table 4.7.

49 4.3. Design of phase modulators for laser radiation at 1064 nm

Figure 4.6: Modification of the electric field and carrier density of the modulator heterojunc-

tionS at the wavelength of 1046 nm in table 4.5 (top), and in table 4.6 (buttom) when the

reversed bias voltage changes from 5 V to 15 V.

Table 4.7: Modulation efficiency of phase modulator at the wavelength of 1046 nm with the

layer structure in table 4.5.

λstructure linear effects quadratic effects

[nm] [deg/(V·mm)] [deg/(V2·mm2)]

Pockels carrier total Kerr

1064 table 4.5 8.46 1.41 10.27 0.88

1064 table 4.6 8.88 2.03 10.91 1.18

50 4.3. Design of phase modulators for laser radiation at 1064 nm

4.3.3 Lateral waveguide at 1064 nm

The lateral waveguide has been designed to support only the fundamental guided TE

mode. The ridge width was set to 3µm. The corresponding etching depth was set to

2µm.

The simulated intensity profile (using the Fimmwave mode solver) of the single guided

TE mode with the corresponding vertical and horizontal cross-sections are shown in

figure 4.7. The FWHM lateral and vertical far field angles corresponds to 16.5 deg and

26.5 deg, respectively. blank

Figure 4.7: Simulated intensity profile (using a commercial software tool that uses a

film mode matching method (https://www.photond.com/products/fimmwave/fimmwave_

features_21.htm) of the fundamental guided TE mode in the modulator waveguide at the

wavelength of 1.064 nm. Ridge width is 3µm. Etching depth is 2µm.

Later on, in chapter 7, phase modulators based on the two multi-layers structures in

table 4.2, table 4.3 at 780 nm, and on table 4.5, and table 4.6 at 1064 nm are realized

in the GaAs technology.

Chapter 5

Design of GaAs-based waveguide

couplers

GaAs-based couplers are realized in this work based on the GaAs/AlGaAs double

heterostructure of the phase modulators at the wavelength of 780 nm. For this purpose,

two different coupling concepts are realized. The MMI coupler and the directional

coupler. MMI couplers are based on the self-imaging principle. Directional couplers

are based on evanescent mode coupling between two ridge waveguides. Please refer to

chapter 2 for the description of these two couplers.

5.1 General remarks on multi-mode interference couplers

The first step towards the design of an MMI coupler is to select the lateral parameters

of the input and output waveguides of the MMI coupler which we refer to as access

waveguides. The fundamental mode of the access waveguides is considered to be the

input mode of the MMI coupler and is launched into the multi-mode waveguide (multi-

mode waveguide) to excite higher order modes. The interference mechanism of these

modes generates the self-imaging effect. For an MMI coupler with Noutput ports,

the multi-mode waveguide has to support at least m≥N+ 1 guided modes which

can be set by the lateral parameters of the waveguide (width and etching depth of

the multi-mode waveguide). The longitudinal parameter of the multi-mode waveguide

(length LN) corresponds to the length at which Nfold-images are created [26]. The

length depends directly on the lateral position yof the exciting field at the input side

of the multi-mode waveguide as earlier described in section 2.2.2 in chapter 2. The

design of MMI couplers should consider the following requirements:

•vertical structure: the same vertical structure for phase modulators in order

to enable integration of the couplers and phase modulators to realize a MZI

modulator.

•access waveguides: the lateral parameters of the access waveguide (ridge width

and etching depth) can be set using the single-mode condition that was given in

figure 4.5.

•etching depth: in a multi-mode waveguide, the beat length is given by Lπ=

λ/ (2 ·∆¯neff )(equation 2.42 in chapter 2) with ∆¯neff = ¯neff0−¯neff1denoting

52 5.2. Design rules of the multi-mode interference couplers

the difference between the effective refractive indices of the guided fundamental

and first higher order modes in the multi-mode waveguide, respectively. For

shorter multi-mode waveguides, the etching depth should be increased to increase

∆¯neff . However, the etching depth must not exceed the value for the single-mode

cut-off condition of the access waveguide (see figure 4.5).

•width of the multi-mode waveguide: the design of some types of MMI couplers

requires to increase the number of guided modes inside the multi-mode waveguide

as we will explain in the following sections. This increases the width of the multi-

mode waveguide. As a rule of thumb, Lπincreases quadratically with Wm(see

equation 2.37 and 2.42 in chapter 2), which then accounts for longer MMI coupler

devices.

In figure 5.1, two types of MMI couplers are simulated as the etching depth is varied. A

1×2MMI coupler with Wm= 15 µm, and a 2×2MMI 3dB coupler with Wm= 20 µm.

The lengths of the muli-mode waveguides are calculated from equations 2.45 and 2.46

for N= 2. The simulation shows that the increment of the etching depth by 0.5µm

from 1.7µmto 2.2µmreduces the length of the multi-mode waveguide of the 1×2

devices from 624 µmto about 524 µm. Further, the increment of the etching depth of

the 2×2MMI 3-dB coupler by 0.5µmfrom 1.8µmto 2.3µmreduces the length of

the multi-mode waveguide from 1370 µmto about 1230 µm.

Figure 5.1: The calculated length of the multi-mode waveguide from the beat length of 1×2

MMI coupler (Wm= 15 µm) and 2×2MMI 3-dB coupler (Wm= 20 µm) as a function of

the etching depth.

5.2 Design rules of the multi-mode interference couplers

As a result of the foregoing discussion in the previous section, the design of the MMI

coupler has to go through five steps that are ordered as follows:

1. set the lateral parameters (ridge width and etching depth) of access waveguides

to define the input mode into the multi-mode waveguide.

2. select a width of the multi-mode waveguide that allows a sufficient number of

guided modes and calculate the guided modes inside the multi-mode waveguide.

53 5.3. Numerical methods for simulation of waveguide couplers

3. the beat length can be calculated from the propagation constants of the fun-

damental mode and the first higher order mode, β0, and β1, respectively, using

Lπ=π/(β0−β1).

4. depending on the number of input and output ports, the device length can be

calculated from the beat length as we explained above.

5. run numerical simulation to find the optimum length of the multi-mode waveg-

uide.

5.3 Numerical methods for simulation of waveguide couplers

The beam propagation method is recommended for simulation of light propagation

along couplers that are based on ridge waveguides [50]. We used a commercial software

tool (RsoftCAD) which allows to model the full 3D structure of the coupler device (2D

cross-section of the ridge waveguide, and one longitudinal dimension). The simulation

of the 3D structure is accurate but computationally intensive. An effective way to

reduce the computing time is to downsize the 3D structure into a two-dimensional (2D)

structure. This is achieved by reducing the 2D cross section (the ridge waveguide) into

one dimension using the effective refractive index method [32]. As earlier described

in chapter 2 in this method (see figure 5.2), every multilayer section of the lateral

waveguide structure is attributed to a characteristic parameter, the effective index of

the fundamental guided mode of a corresponding slab waveguide. This slab waveguide

Figure 5.2: Effective refractive index method. (a) 5-layers ridge cross section, (b) two slab

multi-layers waveguides, (c) corresponding effective refractive indices ¯neff0,1,¯neff0,2, of the

regions 1 and 2, respectively.

is considered to feature the same vertical multilayer structure and is assumed to extend

infinitely in the lateral direction. The reduction of the structure by means of the

effective index method significantly reduces the computing time.

54 5.4. Design of MMI couplers for applications at 780 nm

5.4 Design of MMI couplers for applications at 780 nm

5.4.1 Access waveguides

We consider the multi-layer structure in table 4.2. We set the ridge width to Wr=

2.2µm, the etching depth of de= 2.15 µm. The resulting access waveguide is single-

mode (see figure 4.5). It also ensures a sufficiently large effective refractive index step

for a good image quality [26]. Further, the etching depth is large enough so that

s-bends without significant radiation losses can be realized as will be shown in this

section and demonstrated later in the experiment. The FWHM vertical and lateral

far field angles of 23.5◦and 13.2◦, respectively, satisfy the requirements defined by

hybrid integration technologies for micro-integration of laser chips with optics and

PICs [19]. In fact, the simulation of the beam collimation using WinABCD-3D1shows

that even with one round aspheric lens (f= 1.45 mm) the horizontal and vertical

beam diameters correspond to 623 µmand 718 µm, respectively, which is close to the

optimal value of 600 µm[24] and should account for a high coupling efficiency into a

single mode optical fiber.

5.4.2 Geometry of the multi-mode waveguide

Now that the input mode of the MMI coupler is defined, the next step is to set the

width of the multi-mode waveguide. The number of guided modes min a multi-mode

waveguide of a width Wmis calculated using numerical simulations (a finite-element

method [47]).

Figure 5.3 shows the intensity profiles of the highest order guided modes for three

different values of the ridge width Wmof the multi-mode waveguide. At a ridge width

of Wm= 5 µm(Figure 5.3 (a)) the waveguide supports 3 guided modes. 8 modes are

supported at Wm= 15 µm(figure 5.3 (b)). As a rule of thumb, the more the mode

is confined under the ridge, the more efficient it may contribute to the self-imaging

mechanism and hence, to the quality of the images. For example, one may expect

that in a multi-mode waveguide with Wm= 20 µmthat supports 11 guided modes

(figure 5.3 (c)), the 11th (or the 10th highest order) guided mode may contribute less

efficiently to self-imaging in comparison to the 10th (or the 9thhigher order) guided

mode (figure 5.3 (d)) that has a better confinement under the ridge.

In table 5.1, the number of guided modes mis given with the confinement factors

(under the ridge) of the two highest order guided modes Γn, n =m−1m−2under

the ridge. Starting with the ridge width of the access waveguide ( Wr= 2.2µm), only

the fundamental mode is guided with a confinement factor of Γ0= 0.93 under the

ridge. At a ridge width of 3µm, the first higher order TE mode is supported and has

a confinement factor of Γ1= 0.57. Three modes are supported at a ridge width of

5µm. The confinement factor of the first two higher order modes are Γ2= 0.68, and

Γ2= 0.94.

1×2 MMI coupler: For the design of an MMI coupler with the number of output

ports N= 2, at least m=N+ 1 modes should contribute to the self-imaging in

1software product from the FBH, see for example: https://application.wiley-vch.de/berlin/

journals/op/08-02/OP0802_S48-S51.pdf

55 5.4. Design of MMI couplers for applications at 780 nm

Figure 5.3: TE guided modes in a mutli-mode ridge waveguide from the structure in table 4.2

at the etching depth de= 2.15 µm. (a), (b), and (c) the intensity profiles of the higher order

guided modes for the ridge width of the multi-mode waveguide Wm= 5 µm,Wm= 15 µm,

and Wm= 20 µm, respectively. (d) the 10th (or the 9th higher order) guided mode at a ridge

width of Wm= 20 µm.

Table 5.1: Number of the guided modes mwith respect to the ridge width with the con-

finement factor of the two higher order guided modes. Ridge etching depth is de= 2.15 µm.

Vertical waveguide structure in table 4.2. Simulated using a commercial software tool that

uses a film mode matching method.

ridge width [µm]mΓm−1Γm−2

2.2 1 0.93 −

3 2 0.57 0.96

5 3 0.68 0.94

7 4 0.75 0.94

9 5 0.78 0.93

11 6 0.81 0.93

13 7 0.84 0.94

15 8 0.86 0.94

17 9 0.87 0.94

19 10 0.88 0.94

20 11 0.75 0.93

56 5.4. Design of MMI couplers for applications at 780 nm

order to achieve a good image quality [26]. For example, for a 1×2 splitter, at

least 3 guided modes are required to contribute to the images. In this case, if the

coupler is realized using the symmetric interference mechanism, the multi-mode

waveguide must support at least 5 guided modes (modes 0, 1, 2, 3, 4) since the

modes 1 and 3 are not excited. In order for the contributing guided modes to

self-imaging to be well-confined under the ridge, the width of the multi-mode

waveguide has to be Wm≥10 µmas can be seen in table 5.1. If the width is

further increased to support more guided modes, the image resolution increases

which is preferred for beam splitting applications [51]. This is why for the design

of the 1×2 MMI splitter, the width of the multi-mode waveguide has been set to

15 µm, for which 4 guided modes are excited.

Figure 5.4: 2-D simulation of optical signal propagation along a 1×2MMI coupler with

the corresponding electric field at the end of the multi-mode waveguide section. LMMW =

520 µm, Wm= 15 µm.

The effective refractive indices of the first two guided modes are ¯neff0= 3.42103,

¯neff1= 3.42077. The corresponding beat length is Lπ= 1500 µm. Due to

the symmetric interference mechanism, the expected length of the multi-mode

waveguide for TE operation should be L= 3Lπ/8 = 562 µm.

Figure 5.4 shows for example the intensity distribution along the different sections

of a 1×2MMI coupler using a 2D beam propagation 2simulation. It can be

clearly seen in figure 5.4 that the positions of the two images at the output are

well-matched to the output ports at the end of the multi-mode waveguide. The

distance between the center of the output ports is Wm/2=7.5µm. For practical

reasons, for example, later for the realization of MZI modulators, a sufficient

lateral distance is required between the two outputs of the MMI coupler in order

to incorporate contacts for the active arms of the MZI modulator. This is why

S-bends are required to be added. The complete 1×2MMI coupler with S-bends

is shown by figure 5.5.

2×2 MMI coupler: For the 2×2MMI couplers, the first excited three modes are

mode 0, 1, 3 (due to paired interference mechanism). Thus, the minimum width

2https://www.synopsys.com/optical-solutions/rsoft/passive-device-beamprop.html

57 5.4. Design of MMI couplers for applications at 780 nm

of the multi-mode waveguide should be Wm>6µm. In this work, the width of

the multi-mode waveguide was set to Wm= 20 µmwhich supports a total number

of 11 of guided modes as given by table 5.1. Only 8 of these modes shall then

contribute to self-imaging imaging due to the paired interference mechanism. In

the simulation, the TE input field was injected at lateral position y= +Wm/6.

The corresponding beat length amounts to Lπ= 2480 µmwhich is the length of

the multi-mode waveguide of a cross coupler. The corresponding length of the

multi-mode waveguide of a 3dB coupler is therefore Lπ/2 = 1240 µm.

Figure 5.5: 2D simulation of beam propagation in complete 1×2MMI coupler and 2×2

MMI 3dB coupler, respectively, with S-bends to separate the output waveguides. (a.1), (b.1)

TE electric field at the outputs of the 1×2MMI coupler and the 2×2MMI 3dB coupler,

respectively. Lis the total length of the device.

Any modification in the geometry of the multi-mode waveguide should modify the

performance of the MMI coupler. The performance is characterized by the splitting

ratio of the input power into the output ports (imbalance, see equation 2.48 in chap-

ter 2), and by the excess loss (equation 2.47 in chapter 2). In figure 5.6, the length

of the multi-mode waveguide of a 1×2MMI coupler and a 2×23dB MMI coupler

was first optimized (using a 2D simulation) for the lowest excess loss. Then, the effect

of the deviation of the width of the multi-mode waveguide on the device performance

was simulated. Here we assumed a symmetric modification of the width of the multi-

mode waveguide. As a result, the excess loss of the 1×2MMI coupler increases at a

slower rate (about factor 2) than in the case of the 2×23dB MMI coupler when the

multi-mode waveguide deviates from the nominal value by ∆Wm. The imbalance of

the 1×2MMI coupler does not depend on ∆Wmdue to the symmetric interference

58 5.4. Design of MMI couplers for applications at 780 nm

mechanism [26]. As for the 2×23dB MMI coupler, the imbalance changes with ∆Wm

which was predicted in [26] for paired interference devices.

The simulation also shows that for the 2×2MMI 3dB coupler, the minimum of the

Figure 5.6: Simulated (2D structure) excess loss and imbalance vs. deviation of the width of

the multi-mode waveguide ∆Wmfrom its optimal value Wmusing a 2D beam propagation

method. Nominal multi-mode waveguide width is Wm= 15 µm(optimum length 546 µm)

for the 1×2MMI coupler and Wm= 20 µmfor the 2×2MMI 3dB coupler.

excess loss and the minimum imbalance don’t occur simultaneously. In fact, only one

of these two parameters can be optimized if the width of the multi-mode waveguide

is fixed. If the 3dB coupler is used in a switch or a Mach-Zehnder modulator, the

imbalance directly translates into extinction ratio [26]. Please note that the minimum

excess loss does not equal 0 dB due to radiation losses at the interfaces between the

multi-mode waveguide and the access waveguides.

The numerical simulation (beam propagation in a 2D model then a 3D model) of the

MMI coupler deliver values of the multi-mode waveguide length which slightly differ

from the that predicted by the simple approach (using the beat length) as shown by

table 5.2. Table 5.2 also shows that the error of the optimum length between the re-

Table 5.2: Simulated optimum multi-mode waveguide length of a 2×23dB MMI coupler

using 2D and 3D beam propagation methods with the corresponding length calculated from

the beat length (Wm=20 µm).

etching depth [µm] multi-mode waveguide length [µm]

Lπ/22D BPM 3D BPM

2.0 1273 1320 1290

2.15 1240 1280 1265

2.3 1210 1240 1235

sults of the 2D and the 3D simulations decreases with increasing etching depth. This

is attributed to the fact that, with increasing etching depth, the guided modes of the

multi-mode waveguide become more confined under the ridge region, which then in-

59 5.4. Design of MMI couplers for applications at 780 nm

creases the accuracy of the refractive index approximation in the 2D method [32].

However, despite being efficient, the 2D model may not fully reflect the real 3D

structure due to the effective index approximation. Please note that the 2D simulation

considers the guided mode of a slab waveguide that does not take the modal losses of

the actual input mode into consideration. Thus, the actual values of the excess loss are

expected to be larger than the values depicted in figure 5.6. A 3D model that considers

the boundary conditions of the ridge waveguide and the lateral guided modes instead

of the slab modes of the effective waveguide should deliver an estimation of the actual

excess loss and imbalance in the MMI couplers. These are given in table 5.3 for one

value of the etching depth 2.15 µm.

Table 5.3: Simulated excess loss and imbalance of a 2×23dB MMI coupler using 2D and

3D beam propagation methods (Wm=20 µm,de=2.15 µm).

model excess loss [dB] imbalance [dB]

2D-BPM 0.17 0.3

3D-BPM 0.74 0.3

5.4.3 Interfaces of the multi-mode waveguide to access waveguides

The output ports of the MMI coupler are realized at the end of the multi-mode waveg-

uide where the images of the input field are formed. In order to avoid reflections at

Figure 5.7: Simulated excess loss vs. normalized access waveguide width Wa/Wmof a 2×2

MMI 3dB coupler. Wm= 20 µm,de= 2 µm.

the interface between the access waveguides and the multi-mode waveguide and to

efficiently collect all the light at the end of the multi-mode waveguide [40], the width

of the access waveguides is linearly tapered from Wrto Wa> Wrat the input ports

and from Wato Wrat the output ports. Such a procedure, modifies the excess loss

of the device. Thus in order to find an optimum value of Wawe calculated the excess

loss as a function of the normalized width Wa/Wmas shown by figure 5.7. Minimum

loss is attained with Wa/Wm≥0.25 with Wais the taper width of the (adiabatic)

60 5.5. Design of directional couplers

access waveguide. With the width of the multi-mode waveguide Wm= 20 µm, which

corresponds to minimum taper width of Wa= 5 µm.

5.4.4 S-bends

S-bends are used at the output of the MMI coupler (see figure 5.4 (right)) to increase

the spacial distance between the output ports.

In comparison to straight waveguides, radiative losses in S-bends may not be ne-

glected in comparison to absorption or scattering losses. These additional losses oc-

cur due to optical mode radiation at the bends and to transition losses at the in-

terface between the straight and the curved sections of the S-bend [52, p. 328], [36].

For the MMI couplers with the ridge waveguide of the access waveguides selected

above (Wr= 2.2µm,de= 2.15 µm), the lateral effective refractive index contrast is

∆¯neff = 6.6×10−3for which the confinement factor of the guided optical TE mode

under the ridge is 0.93 (see figure 5.8 in the following section). Thus, the amount of

radiative losses depends on the curvature of the bent waveguide [53] (the bend radius

and the bend angle).

A widely used S-bend profile for low lateral confinement structures is the cosine

profile. The equation for the cosine S-bend with a propagation length lalong the

z-direction and a lateral offset WSis given by:

y(z) = WS

π·sin 2πz

l+WS·1−2·z

l+WS(5.1)

Using this equation, we designed S-bends with l= 700 µm,WS= 40 µm, and WS=

20 µm, respectively. This corresponds to a large bend curvature in order to limit the

effect of radiative losses at the bend.

5.5 Design of directional couplers

In this section only one configuration of directional couplers is considered. Operation

wavelength is 780 nm and the input field is assumed to be TE polarized. The ridge

waveguide of the access waveguides (Wr= 2.2µm,de= 2.15 µm) is selected for both,

the input waveguide, and the coupling waveguide of the directional coupler. The

distance between the ridges is set to dC= 500 nm. Due to the well-defined ridge

etching depth, only a small fraction of the input field expands towards the coupling

waveguide. A small overlap of both fields is expected.

The simulated optical fields for this configuration are shown in figure 5.8. The

values of the effective refractive indices for the calculation of the coupling coefficient

ΓC(equation 2.33 in chapter 2) correspond to ¯neff = 3.4182 (the effective index of the

guided TE fundamental mode which is equal for both input and coupled fields), and

¯nc= 3.4116 (the effective refractive index of the region at the sides of the ridges). The

resulting transfer distance (using a 3D model) is given by L0=π/ (2ΓC) = 508 µm

which corresponds to a cross coupler. A 3-dB coupler is then expected at LC= 254 µm.

61 5.6. Conclusions

Figure 5.8: Overlap of the input optical field with the coupling waveguide in a directional

coupler. (Vertical structure from table 4.2 in chapter 4).

5.6 Conclusions

In this chapter the multi-layer structure of the phase modulators at the wavelength of

780 nm has been used to design waveguide couplers. The lateral design of 1×2MMI

couplers and 2×2MMI couplers was first estimated using the self-imaging properties

of a multi-mode waveguide. Then the design was optimized by using a numerical

2D model that uses a finite difference beam propagation method. The simulation has

shown that the 1×2MMI couplers can provide a 0 dB imbalance due to the symmetric

interference. For the 2×2MMI couplers using the paired interference mechanism, it

seems possible to optimize the lateral geometry of the multi-mode waveguide for low

imbalance and reasonably low excess losses. The actual excess loss and imbalance

were then estimated using a 3D model of the MMI couplers. Further, the lateral

structure of a directional coupler of the same length scales of the MMI couplers has

been considered. The ridge parameters of the directional coupler are identical to the

ridge parameters of the access waveguides of the MMI couplers. Hence, the directional

coupler serves as a reference to evaluate the performance of the 2×2MMI couplers.

Chapter 6

Application: Mach-Zehnder

intensity modulator

The integration properties of the designed phase modulators, bend waveguides, and

MMI couplers can be demonstrated through a MZI modulator which is one of the

simplest PICs for applications at the wavelength of 780 nm.

MZI modulators in III-V compound semiconductors (InP-based) that use MMI cou-

plers have already been reported in the literature, see [28]. In [25] and in [27] integrated

MZI modulators based on GaAs/AlGaAs double heterostructures and Y-couplers were

reported. The MZI modulators in this work are designed using MMI couplers. The

Figure 6.1: Schematic of MZI modulators. (left) MZI with 1x2 input and output MMI

couplers. (right) MZI with 2×2MMI 3dB coupler at the output (with input ports A and

B and output ports C and D). The p-contact of the sections with the phase modulators are

marked with the golden color.

schematic description of both devices is shown in figure 6.1. The first MZI modulator

uses two identical 1×2MMI couplers, one 1×2MMI (splitter) at the input and one

1×2MMI (combiner) at the output. The input field is (equally) split between the

two phase modulators (arms of the MZI). A modulation voltage signal can be applied

on one or both arms to modify the phase difference between the optical signals propa-

gating in the two arms. The optical signals are then fed into the output MMI coupler.

Signals which are in-phase fulfill the conditions for a single image with a maximum

intensity at the output waveguide of the MZI. If the signals (nominally with identical

amplitudes) were out of phase (phase difference is πrad), then the interfere destruc-

tively at the output waveguide and the transmitted power is minimum. Thus, phase

difference is converted into intensity modulation.

The second MZI modulator uses a 1×2MMI splitter at the input and a 2×2MMI

3dB coupler (combiner) at the output. Each of the inputs ports of the output MMI

coupler generates two images at the two output ports with a relevant phase shift of

π/2[26] between the two images. Here we assume a perfect 3dB coupler (imbalance

of 0 dB) and equal amplitudes of the signals at the input ports of the 2×2MMI

3dB coupler. Let the initial phases of the signals at the input ports A and B of the

2×2 MMI coupler be ΦA0and ΦB0. The relevant phases (in radian) of the images at

the output ports C and D resulting from the input at port A can be -without loss

of generality- written as: ΦAC = ΦA0and ΦAD = ΦA0+π/2, respectively. And for

the input from port B, the relevant phases of the images at the output ports C and

D can be -without loss of generality- written as: ΦBC = ΦB0+π/2and ΦAD = ΦB0,

respectively. At each output port, the resulting image is then a super-position of two

images, one from each input port. The phase difference between these two images

corresponds to ΦC= Φ0−π/2and ΦD= Φ0+π/2, where Φ0= ΦA0−ΦB0is the

initial phase difference. Hence, if Φ0=p·2π+π/2 rad (pinteger), then the two

images at port C are in-phase while they are out of phase at port D. As a result the

transmitted power is maximum at port C and minimum at port D. If on the other

hand Φ0=p·2π+ 3π/2 rad, the situation is reversed. Thus, for this type of MZI

modulator, if a modulating signal is applied on one arm, the voltage required between

a maximum and a minimum of the transmitted power corresponds to a phase shift of

πrad at the phase modulator arm.

The performance of an MZI modulator in this work is characterized by its excess loss

and its extinction ratio. Both device types should feature low excess loss due to the

low losses of the phase modulators, the S-bends, and the MMI couplers.

According to [25], the transmission of the MZI modulator with a low extinction ratio

follows the expression:

P/Pmax = (1 −Pmin/Pmax) cos2πV

2Vπ

+ Φ0+Pmin/Pmax (6.1)

where Pmax and Pmin are the maximum and the minimum transmitted powers, respec-

tively, Vis the applied voltage, and Vπis the half-wave voltage.

The extinction ratio at an output port of the MZI modulator is then given by:

extinction ratio [dB] = 10 ×log Pmax

Pmin

(6.2)

Chapter 7

Fabrication of GaAs-based passive

components

In this chapter an overview of the semiconductor-technology related aspects of the

GaAs-based passive components is presented. The devices were exclusively fabricated

in-house. The fabrication was carried out by colleges of the material technology de-

partment and of the optoelectronic department.

During the different stages of this work, four technology process runs of GaAs-based

passive components have been carried out. Three of these processes runs were to man-

ufacture the basic photonic components (phase modulators, S-bends, couplers, and

MZI modulators) for operation at 780 nm. The fourth process was dedicated only to

phase modulators operating at a the wavelength of λ= 1064 nm.

Following the material growth that defines the vertical layout of the devices, lithog-

raphy was used to implement lateral structuring that defines the lateral waveguide

sections for the phase modulators, couplers, and MZI modulators. The realization of

the couplers and MZI modulators was restricted to the wavelength of 780 nm.

7.1 Material growth

The epitaxial layers of the four different processes were grown on 3-inch (001) n-GaAs

substrates (wafers). Table 7.1 gives an overview of the different process runs with the

corresponding wafers, application wavelength, and the types of devices that have been

realized. Each process is designated by a unique number (Z1 xxxx). As shown in

table 7.1, a unique identifier (I through IV) is assigned to each of the four processes

and will be used to refer to the corresponding process in the following parts of the

thesis.

In the process run I (or Z1 6105) the multi-layer structure was fabricated that

delivered the P-p-i-n-N double heterostructure phase modulator from table 4.2 (see

chapter 4). The structure was grown on the GaAs substrate using metal-organic vapor

phase epitaxy (MOVPE). Si and C were used as n- and p-type dopants, respectively. In

process run II (Z1 6520), the multi-layer structure of a P-p-n-N double heterostructure

phase modulator from table 4.3 has been realized. The doping profile of the P-p-

n-N structure in process II is intended to increase the phase modulation efficiency

when compared to the structure of process I. The comparison of the two structures

from process runs I and II have already been presented in chapter 4 (see figure 4.4).

65 7.2. Lithography

The multi-layer structure realized in process run III (or Z1 7471) is a repetition of

the layer structure of the process I. Here the lateral layout has been modified as

explained in the next section. In process run IV (Z1 6894), the multi-layer structures

of phase modulators for operation at the wavelength of 1064 nm (tables 4.5 and 4.6)

were realized.

Table 7.1: Fabrication plan of GaAs-based photonic components. PMod: phase modulator,

D. coupl.: directional coupler, MZI (a): layout with only 1×2MMI couplers, MZI (b):

layout with input 1×2MMI couplers and output 2×2MMI couplers.

ID Process # Wafer λ[nm] PMod bend D. coupl. MMI MZI

I Z1 6105 D2043-2 780 table 4.2 - - - -

D2043-3 780 table 4.2 - - - -

II Z1 6520 D2231-3 780 table 4.3 X- 1x2 (a)

D2231-4 780 table 4.3 X- 1x2 (a)

III Z17471 D2230-3 780 table 4.2 X X 1x2, 2x2 (b)

D2904-4 780 table 4.2 X X 1x2, 2x2 (b)

IV Z1 6894 C3059-3 1064 table 4.5 - - - -

C3062-3 1064 table 4.6 - - - -

7.2 Lithography

For the phase modulators on all wafers, a standard ridge waveguide process was applied

with the orientation of the waveguides parallel to the [1¯

10] crystallographic direction in

order to provide access to the linear electro-optic effect (see section 3.1.2 in chapter 3).

The ridge waveguides were defined by reactive ion etching (RIE) and encapsulated with

a100 nm thick SiNx layer. The isolation layer was opened on top of the waveguides for

lateral carrier confinement. Ti/Pt/Au layers were deposited for p-side contacts and

before evaporation of the n-metalization the wafer was thinned to 150 µm.

Coupler devices and MZI modulators were realized in process runs II and III. For

wafers of these two process runs, I-line stepper lithography followed by reactive ion

etching was applied for the lateral definition of the MMI couplers and the 2.2µmwide

ridges for linear and bent waveguides.

The MZI modulators of process run II were created on a total chip length of 10 mm.

The layout has 1×2MMI couplers at the input and at the output sides of the MZI

modulators. Tapered waveguides (at the interface between the multi-mode waveguides

and the access waveguides of the MMI couplers) were introduced. The layout of process

III includes 1×2MMI couplers, 2×2MMI 3dB couplers, 2×2MMI cross couplers,

directional couplers, and MZI modulators. The MZI modulator were realized in two

different layouts. Namely, MZIs with 1×2MMI couplers at the input and at the

output, and MZIs with 1×2MMI couplers at the input and with 2×2at the output

(see figure 6.1) were realized. Devices of process run III were realized on a total

length of 10 mm. The lithography steps of process III followed in similar lines to

that in process II. Individual details to different devices will be later included in the

experiment. Figures 7.1 and 7.2 shows the cross-section scanning electron micrograph

(SEM) of some of the fabricated devices.

66 7.2. Lithography

Figure 7.1: SEM image of: (left) fabricated phase modulator at the wavelength 780 nm from

process I. (right) output ports of a 1×2MMI coupler at the wavelength of 780 nm from

process II. (down center) part of the MZI modulator from process III showing the phase

modulators and S-bends.

Figure 7.2: SEM image of a directional coupler from process run III.

Chapter 8

Electro-optic Performance

Experimental results of the main performance factors of GaAs/AlGaAs double het-

erostructure phase modulators, directional couplers, MMI couplers, and MZI modula-

tors are presented in this chapter.

8.1 Electro-optic Performance: GaAs-based phase modula-

tors

We apply the well-known Fabry-Pérot (FP) method to determine the phase modulation

efficiency of the phase modulators fabricated in process runs I and II (at the wavelength

of 780 nm), and IV (at the wavelength of 1064 nm). We also use the FP method to

determine the propagation losses in the phase modulator waveguides.

8.1.1 Electrical properties of phase modulators

The phase modulators are operated in the reverse-bias direction in order to apply

the electric field necessary to use the electro-optic effects. The total reverse bias

Figure 8.1: Static I-V characteristics of (left) phase modulators from the process runs I and

II ( Wafer D2043-3:multi-layer in table 4.2, Wafer D2231-3: multi-layer in table 4.3), and

(right) phase modulator from the process IV (wafer C3059-3: multi-layer in table 4.5).

voltage must not exceed the break-down voltage of the modulator heterojunction.

68 8.1. Electro-optic Performance: GaAs-based phase modulators

To determine the break down voltage, the current through the phase modulator was

measured as a function of the applied reverse voltage. As a voltage source we used a

power supply (hp 6632b) which provides output ratings of 0-20 V. Figure 8.1 shows

for example the result of such a measurement for three phase modulators chips from

wafers from process runs I, II and IV. No break through is observed up to reverse

voltage of 20 V. i.e. the break-down voltage is larger than 20 V. This result is valid

for all the processed wafers in process runs I, II and IV.

The small-signal equivalent circuit of the reverse-biased modulator junction can be

well described by the circuit in figure 8.2 , where Rsis the series resistance and Cjthe

junction depletion capacity [54, p. 98]. An LCR-bridge (hp 4274 LCR meter) was used

Figure 8.2: Equivalent electrical circuit of a reverse-biased phase modulator junction at low

frequencies. Cjis the junction capacity, Rsis the conductance.

to determine the series resistance Rsand the capacity Cj1. Phase modulator chips

from process run I were mounted on a C-mount. The LCR bridge applies a sinusoidal

Figure 8.3: Series resistance and capacitance of phase modulator chips on C-mount from

process I (multi-layer in table 4.2) measured at different reversed-bias voltages.

modulation signal with a frequency fmand allows to add a DC voltage (bias voltage).

It determines the response (amplitude and phase components of the transfer function)

of the load (the phase modulator chips on C-mount), from which Rsand Cjcan be

determined. A C-mount without a modulator chip was used for calibration. The result

of measurements of 1 mm and 2 mm chips are shown in figure 8.3.

The capacitance of a 2 mm long modulator at 0 V is about 250 pF. This limits

the 3-dB modulation bandwidth for direct modulation with a 50 Ω source to about

1The measurements of the Series resistance and capacitance of phase modulator chips on C-mount were

carried out by Armin Liero from the Microwave department at FBH.

69 8.1. Electro-optic Performance: GaAs-based phase modulators

12.75 MHz (with Rs= 50 Ω). This value of the modulation bandwidth is sufficient for

the generation of the modulation side bands at 8 MHz for Rb spectroscopy as a direct

application for the phase modulators from this work.

8.1.2 Phase modulation efficiency

A simple method to estimate the modulation efficiency is the Fabry-Perot (FP) inter-

ference method [15], [19]. In this method, the FP cavity is formed by the uncoated

facets of the modulator’s waveguide. For the generation of the FP fringes a slowly

varying sawtooth voltage signal is applied to modify the optical path length of the

modulator due the modification of the effective refractive index. The transmitted

power is then recorded. The half-wave voltage is then required between two maxima

or two minima of the transmitted power. The experimental realization of this method

Figure 8.4: Fabry-Perot experimental setup for measurement of phase modulation efficiency

and propagation losses of GaAs-based phase modulators.

is shown in figure 8.4. The optical field of a DFB laser at the wavelength of 780 nm (or

at 1064 nm) is coupled into the waveguide by means of a polarization maintaining, sin-

gle mode lensed-fiber. A polarization controller that consists of a zero-order half-wave

plate and a polarizing beam splitter selects the TE polarization. The lensed fiber is

fixed on a rotation mount (Thorlabs HF R007). The output of the lensed fiber is fed

into the waveguide. This setup provides a polarization extinction ratio (PER) of 20 dB

and allows for setting the input polarization parallel to TE-polarization to better than

2 deg, corresponding to 29 dB of suppression of the TM-polarization. The PER has

been verified experimentally as the output of the lensed fiber was collimated using a

round aspheric lens then directed onto a polarizing beam splitter (Linos G335725000)

to select the required polarization. To prepare chips for characterization, 1 mm,2 mm

and 4 mm chips were cleaved and mounted on C-mounts. The output signal from the

modulator chip was collimated using a coated round aspheric lens and was detected

using a photoreceiver. Please refer to Appendix B for the list of measured chips with

different lengths and ridge parameters. A 1 kHz sawtooth on top of 10 V DC reverse

bias voltage signal is swept from 0 V to −20 V. The voltage Vπcorresponding to a

phase shift of πalong the modulator length Lis determined from the reverse-biased

voltage required between two maxima (or two minima) of the transmitted power [19].

As an example, figure 8.5 shows the FP fringes of a 2 mm phase modulator chip de-

70 8.1. Electro-optic Performance: GaAs-based phase modulators

Figure 8.5: The oscilloscope photo of the resulting FP fringes of a 2 mm phase modulator from

process run I. The upper curve shows the applied voltage, the lower curve is the corresponding

transmitted intensity [19].

veloped for application at 780 nm, see table 4.2 (process run I).

The results of the FP measurement are given in table 8.1 and are compared with

the theoretical values. The experimental results are to a certain extent consistent

with the values expected from the simulation which demonstrates the efficiency of the

FP method. The difference between the experimental and the theoretically expected

values the calculation may be attributed to deviation of the actual doping profile of

the modulator heterojunction from the nominal.

Table 8.1: Phase modulation efficiency of phase modulators from process runs I, II, IV.

Experimental values (FP method) vs. theoretical values.

process Wafer structure λmeasured (FP) theory

[nm] [deg/(V·mm)] [deg/(V·mm)]

I D2043-2 table 4.2 780 10.8 9.54

I D2043-3 table 4.2 780 8.2 9.54

II D2231-3 table 4.3 780 13.1 12.49

II D2231-4 table 4.3 780 15.6 12.49

IV C3059-3 table 4.5 1064 11.2 10.75

IV C3062-3 table 4.6 1064 13.2 12.09

8.1.3 Propagation losses

The FP technique is also suitable to perform a loss measurement. For this purpose,

the transmission is recorded for different chip lengths. In this way, the propagation

loss can be determined from transmission loss measurements carried out on waveguides

with different lengths according to [55]:

−ln 1−√1−K2/K=α·L−ln R, (8.1)

where Lis the length of the FP cavity (corresponding to the chip length), Ris the

reflectivity of the uncoated facets of the modulators chips, and Kis the contrast

of the Fabry-Perot resonances (K= (Pmax −Pmin)/(Pmax +Pmin)[55], [56], where

Pmax, Pmin are the maximum and minimum transmitted powers, respectively.

71 8.1. Electro-optic Performance: GaAs-based phase modulators

Propagation losses of phase modulators at 780 nm

In the absence of a modulating electric field, the free carriers density is expected to

be largest. Thus, the propagation losses are expected to be largest, when no electric

field is applied. In the FP setup in figure 8.4, the application of electric field modifies

the carrier density and the electro-absorption associated with the Franz-Keldysh effect

(QEO effect) [23]. We therefore modify the FP setup in figure 8.4 in order to optically

Figure 8.6: Transmission losses: left hand side of equation (8.1) as a function of the length

of the modulator cavity. The slope of the linear fit corresponds to the propagation loss

coefficient in [1/mm] units [19]. Chips from wafer D2043-3, process run I.

tune the phase as the following: we use a New Focus (Velocity 6312) tunable external

cavity diode laser (ECDL) instead of the DFB laser. Its wavelength can be tuned

within the range 765 nm-781 nm. We tune the wavelength of ECDL by 0.1 nm around

the wavelength 780 nm. This fine tuning is sufficient to observe the FP fringes of

the intensity profile at the output of the uncoated modulator chip with the smallest

cavity length of 1 mm. The measurement is carried out on devices with 3 different

lengths: 1 mm,2 mm, and 4 mm. For each of these lengths three nominally identical

chips (process run I) are selected and their transmission profile and Imax and Imin are

measured to calculate the FP contrast K. In figure 8.6 the term on the left hand

side of equation (8.1) is plotted as a function of the chip length. The loss coefficient

α= 0.28 cm−1(which corresponds to 1.2 dB/cm) can be extracted from the slope

of the fitted straight line (0.028 mm−1). This value is consistent with the value of

0.1 cm−1calculated in 4.2.2, which only accounts for the free-carrier absorption. The

deviation could be attributed to additional losses introduced by the etched planes of

the ridge waveguide. The reflectivity of the facets extracted from the interception

with the Y-axis at is R=0.275 which is slightly smaller than the theoretical value of

0.305 [19].

Propagation losses of phase modulators at 1064 nm

For the phase modulators at the wavelength of 1064 nm a tunable laser was not avail-

able. Therefore, the FP fringes for the loss measurements were generated by electrically

72 8.1. Electro-optic Performance: GaAs-based phase modulators

tuning the phase of the optical signal. This assumes the electro-absorption losses as-

sociated with the Franz-Keldysh effect to be negligible. In fact this assumption can

be verified by investigating the peak levels of the transmission of the phase modu-

lator chips as the modulating voltage is increased. Figure 8.7 shows that the peak

levels are almost independent of the modulation voltage which accounts to negligible

electro-absorption losses. This agrees very well with the fact that the QEO effect has a

Figure 8.7: The measured power transmission of two uncoated, 4 mm long phase modulators

from process run IV as a function of the modulation voltage.

minimum contribution to phase modulation in all the investigated phase modulators.

Chips from wafers C3059-3 and C3062-3 of process run IV with lengths of 2 mm and

4 mm were characterized. For each of these lengths 6 nominally identical chips (pro-

cess run IV) were selected and their transmission profile and Pmax and Pmin were

measured to calculate the FP contrast K. The resulting values for interception of the

Figure 8.8: Transmission losses: left hand side of equation (8.1) as a function of the length

of the modulator cavity of phase modulator chips from wafers from process run IV (C3062-3

from table 4.6 and C3059-3 from table 4.5). The slope of the linear fit corresponds to the

propagation loss coefficient αin [1/mm] units.

linear fits with the y-axes which correspond to the facets reflectivity R≈0.32 are very

convenient despite that only 2 different chip lengths were used for the measurements.

73 8.1. Electro-optic Performance: GaAs-based phase modulators

The propagation losses amount to 4.34 dB/cm and 2.63 dB/cm for chips from wafers

C3059-3, and C3062-3 respectively. The measured propagation losses for the phase

modulators at 1064 nm are consistent with the theoretically calculated values, which

only accounts for the free-carrier absorption. Any deviation could be attributed to

additional losses introduced by the etched planes of the ridge waveguides.

8.1.4 Conclusions

In this section, the main performance factors of phase modulators at 780 nm and

1064 nm were investigated. The break-down voltage for all the devices was found to

be larger than -20 V. For a 2 mm phase modulator chip at 780 nm from process I, the

total phase modulator capacitance at 0 V was estimated to be 250 pF, limiting direct

modulation with a 50 Ωsource to 12.75 MHz which is sufficient for the generation of

modulation sidebands for rubidium spectroscopy. In the future, the reduction in the

modulator capacitance by one to two orders of magnitude seems feasible by adding a

benzocyclobutene polymer (BCB) passivation layer between the p-metallization and

the upper cladding instead of the thin SiNx layer. This should allow GaAs/AlGaAs-

based modulators to provide access to modulation frequencies beyond 1 GHz with

direct driving.

The well-known FP method was applied to measure the phase modulation as a

function of the reverse-biased voltage for modulator chips from processes I, II, and IV.

Phase modulation efficiencies beyond 10 deg/(V·mm) were demonstrated for all the

measured modulators. The measured values are consistent with the values expected

from the simulation which demonstrates the efficiency of the FP method. Further,

using the FP method, the propagation losses of modulators chips from process I and

IV were estimated from the transmission loss measurements carried out on waveguides

with different lengths. Propagation losses of the waveguides of the phase modulators

at 780 nm were found to amount to only 1.2 dB/cm. The propagation losses of

the waveguides of modulators at 1064 nm were found to be less than 4.7 dB/cm

which is beyond the state-of-the-art for GaAs/AlGaAs double heterostructure phase

modulators with a GaAs guiding core.

74 8.2. Electro-optic Performance: GaAs-based couplers

8.2 Electro-optic Performance: GaAs-based couplers

In this section the electro-optical performance of GaAs-based bent waveguides and

waveguide couplers is presented.

8.2.1 Bent waveguides

Relevant parameters for the application of bent waveguides (S-bends) are the radiative

loss of the bent waveguides as well as the loss (reflection and scattering) at the inter-

face between bent and straight waveguide section due to different mode sizes (mode

mismatch).

In order to determine radiative losses of the S-bends, we included appropriate test

structures in the layout of process III. Each of these test structures contains two

waveguides: a straight reference waveguide and total of 14 repetitions of a basic bent

waveguide structure, which is an S-bend (see equation 5.1). Both waveguides of the test

structure are 10 mm long. For the characterization, the test structures were cleaved

Figure 8.9: (left) Schematic of the layout of the test structure with detailed layout of a

section with only one S-bend. (right) Light coupling into the bent waveguide by means of a

lensed-fiber.

and the facets were AR-coated to eliminate the FP reflections. The measurement

setup is shown in figure 8.10. As a light source we used a DFB laser at the wavelength

Figure 8.10: Measurement setup for the characterization of S-bends (WS= 20 µm). DUT:

device under test, PBS: polarizing beam splitter, T: transmission beam of the PBS, R:

reflection beam of the PBS.

75 8.2. Electro-optic Performance: GaAs-based couplers

of 780 nm. We used a polarization maintaining lensed-fiber to couple the light into the

waveguides. To excite one operation mode (TE or TM), the lensed-fiber was fixed on

a rotation mount (Thorlabs HF R007). The rotation mount can be manually adjusted

to achieve an experimental polarization extinction ratio of 20 dB (see section 8.1.2) at

the input of the tested waveguides for both, TE and TM mode operation. The out-

put beam was collimated using a mounted aspheric round lens (f=1.46 mm), then a

polarizing beam splitter (PBS) selected the polarization of interest. The transmission

signal of the PBS was detected using a powermeter. A second powermeter detected

the reflected beam (with the undesired polarization). The second PBS in the way of

the reflected beam purifies the TM reflected beam for an accurate measurement of

the polarization extinction ratio. Piezo-actuated XYZ-translation stages with 1 nm

resolution (Smaract SLC-series) were used to set the position of the lensed-fiber for

optimum coupling.

By comparing the power at the transmission (T) (at power meter 1) and reflection

(R) (at power meter 2), the polarization extinction ratio of the output signal can be

determined. The comparison for both, TE and TM-mode operation has revealed that

experimental input PER of 20 dB is maintained by both bent and straight waveguides.

In the following, we consider only operation in the TE mode.

The next step is to determine the insertion loss of the bent waveguide which includes

the propagation losses and coupling losses. We apply the following procedure: for

the input power P0delivered by the lensed-fiber, the output power of the reference

waveguide Pref , and of the bent waveguide Pbare given by:

Pref =ηref P0·e−(αc·Lref )

Pb=ηbP0·(1 −γ)·e−(αc·Lb)(8.2)

with ηref ,ηbthe coupling efficiencies into the reference and into the bent waveguides,

respectively, Lref the reference waveguide length (10 mm), Lbthe bent waveguide

length, αcthe propagation loss, and γthe radiation losses of the bent waveguide.

Actually, the reference waveguide suffers also radiative losses at the sides of the ridge.

Therefore, γhere is attributed solely to the radiative losses at the bends. Furthermore,

the optical path along the waveguide material in the bent waveguide is longer than

the length of the straight waveguide Lref . However, the path difference is negligible

due to the large curvature radius of the bent waveguide. Therefore we use Lb≈Lref .

We assume that ηref =ηbfor the following reasons: both the straight waveguide

and the bent waveguide have the same cross-section (at both waveguides facets). In

addition, for measurement repetition, we perform the measurement on 4 different chips

taken from the same bar. As a result, the radiation losses of the bent waveguide can

be directly calculated from the output powers of the bent and the straight waveguides:

γ= 1 −Pb/Pref

The resulting radiation loss of the bent waveguide is γ= 0.04. The corresponding sin-

gle S-bend loss is then γS=γ/14 = 3×10−3which is given in dBs by γS(dB) = 0.01 dB.

This concludes that the radiation losses of the S-bends can be safely discarded in com-

parison to the propagation losses (1.2 dB/cm).

76 8.2. Electro-optic Performance: GaAs-based couplers

In fact, the assumption that ηb=ηref can be verified by investigating the beam

profiles of the bent and straight waveguides. For this purpose, we use a CCD camera

instead of powermeter 1 in figure 8.10. The camera detects the collimated beam from

each or the reference and bent waveguides. The far field (collimated) profiles for both,

the reference waveguide (straight waveguide) and the bent waveguide for operation

in the TE mode as imprinted at the CCD camera are shown in figure 8.11. The

intensity profiles along the horizontal direction are fitted using a Gaussian function

y=y0+A/wqπ/2exp (−2(x−xc)2/w2). The comparison of the Gaussian fits for

Figure 8.11: (a), (b) Far field profiles of a straight waveguide and a bent waveguide, re-

spectively. (c) The intensity profiles along the horizontal direction. FWHM: full width at

half-maximum. Collimation with an aspheric round lens (f= 1.46 mm).

the horizontal profiles in figure 8.11 shows that the output profiles of the reference and

bent waveguides are well matched to each other. In the vertical direction, the intensity

profiles are expected to be identical since the vertical structures for both, straight and

bend waveguides are identical.

Equation 8.2 shows that in order to determine the coupling losses of the reference

waveguide ηref , it is sufficient to determine the input power P0since αcis known

(1.2 dB/cm). To determine P0the beam of the lensed fiber was directly collimated us-

ing the aspheric round lens and then detected using the powermeter. The correspond-

ing coupling losses were found to be less than 0.8 dB. The well-matched beam spot

of the lensed fiber to the guided mode of the reference waveguide and the AR-coating

make it feasible to achieve such a high coupling efficiency using the piezo-actuated

XYZ-translation stages with 1nm resolution.

8.2.2 Directional couplers

Directional couplers were realized in process (III) together with MZI devices (see

figure 7.2 in chapter 7). The waveguide spacing dCwas fixed to 500 nm to allow

for spatial overlap of the guided mode of the input waveguide with the coupling mode

of the coupling waveguide as previously discussed in section 5.5 in chapter 5.

77 8.2. Electro-optic Performance: GaAs-based couplers

Figure 8.12: Setup for the characterization of directional couplers and MMI couplers. DUT:

device under test, PBS: polarizing beam splitter, T: transmission beam of the PBS.

Setup for the characterization of directional couplers

The measurement setup for the characterization of the couplers is shown in figure 8.12.

At the input of the DUT, the setup is identical to the setup for the characterization of

S-bends in figure 8.10. At the output, the two beams (from the two output ports) are

collimated using one aspheric round lens, then a polarizing beam splitter (PBS) allows

for the transmission of the TE polarization (the interesting polarization) and reflects

the TM polarization. At a sufficient distance from the PBS, the transmitted two beams

are separated and detected using power meters 1 and 2 which are equidistant from the

chips facets to allow for equal illumination of the sensor of the photodetectors.

Coupling ratio and excess loss of directional couplers

Results of measurements of the coupling ratio and excess loss of coupler devices are

given in table 8.2. The values for the coupling length LCin the table correspond to

the nominal length of the straight coupling waveguides. Please note that the layout of

directional couplers includes S-bends at the end of the straight coupling waveguides to

introduce a smooth lateral separation. The coupling effect is expected to continue in

the S-bends. The S-bends were introduced in the couplers layout according to equa-

tion 5.1 at a vertical spacing of 40 µm. Hence, the actual coupling length is expected

to extend to a few tens of micrometers beyond the straight waveguides.

The experimental results in table 8.2 do not really reflect the expected performance.

For example samples 3 and 4 share an exactly identical layout but perform totally

different from each other. Sample 3 performs as a 3dB coupler whereas sample 4 has a

large imbalance. The same applies to samples 5 and 6. This discrepancy can only be

attributed to the fabrication tolerances. During the fabrication process, a well-defined

waveguide spacing dCcan be maintained which is not necessarily the case for the etch-

ing depth. The actual etching depth may vary from the theoretical value along the

coupling length LC, see figure 8.15. This causes a different lateral confinement of the

optical field which changes the coupling efficiency from one waveguide to the other,

and hence, modifies the transfer length L0. In order to compensate for fabrication

artifacts, one may vary the coupling length LCaround the theoretical value. However,

a reproduction process may not be feasible and the performance is expected to vary

from one process to another or even within the same process from one device to the

other as the measurement reveals. Additional excess losses in comparison to the the-

78 8.2. Electro-optic Performance: GaAs-based couplers

oretical values may be attributed to light scattering at the etched planes of the ridge

waveguides.

In conclusion, the performance of GaAs-based directional couplers is limited by the

quality of the fabrication process. A different concept that tolerates the fabrication

errors, e.g. MMI couplers, is recommended for the realization of ridge waveguide

couplers in GaAs-based PICs.

Table 8.2: Performance of directional couplers fabricated in process run III, wafer D2904-4.

LCgives the nominal (design) length of the coupling section according to figure 2.5(b). Total

devices length is 10 mm, operation at λ= 780 nm.

ID chip LC(nom.) port coupling ratio excess loss

TFRRDD µm % dB

theor. exp. theor. exp.

1 060211 180 A-C 65 95 0.1 0.7

A-D 35 05

2 060212 180 A-C 65 88 0.1 0.6

A-D 35 12

3 060213 280 A-C 45 54 0.1 0.7

A-D 55 46

4 060313 280 A-C 45 83 0.1 0.5

A-D 55 17

5 060214 380 A-C 25 52 0.1 0.6

A-D 75 48

6 060215 380 A-C 25 85 0.1 0.7

A-D 75 15

7 060217 580 A-C 12 09 0.1 0.7

A-D 88 91

8 060218 580 A-C 12 15 0.1 0.7

A-D 88 85

8.2.3 Multi-mode interference couplers

MMI couplers from process runs II and III were fabricated on 3-inch wafers, see fig-

ure 8.13. Each wafer consists of a number of test fields. A test field includes a stack

of bars, each with a total of 20 chips at a total length of 10 mm. Within the same

bar, two of the 20 chips are straight waveguides that can be used as reference devices

for insertion loss measurements. The remaining chips include one MZI (10 mm long)

based on a 1×2 MMI input and a 1×2 MMI output splitter as well as one MZI based

on the same 1×2 MMI coupler and a 2×2 MMI output coupler. The 1×2 MMI cou-

pler is intended to perform as a 50/50 power splitter. The splitting ratio (power at

output 1 to power at output 2) is insensitive to process tolerances (length and width of

79 8.2. Electro-optic Performance: GaAs-based couplers

WG, etch depth). The imbalance of the 2×2 MMI coupler depends on the multi-mode

waveguide lateral geometry (width and length of the MMI).

Figure 8.13: Scan electron micrograph (SEM) photo of a 1×2 MMI coupler (left), and a 2×2

MMI coupler (right).

Setup for the measurement of MMI couplers

For the characterization of the MMI couplers, the 10 mm long bars with MZI layout

were to be split into two parts such that the input and output waveguides of each

input and output coupler of the MZI could be accessed optically (see figure 8.14). The

Figure 8.14: The layout of three different chips from the wafers in process run III. (a) an MZI

modulator chip with a 1×2 MMI splitter and a 2×2 MMI coupler, (b) an MZI modulator

chip with only 1×2 MMI couplers, and (c) chip with a reference straight waveguide and a

bent waveguide.

resulting bars at a length of 5 mm were cleaved and were AR-coated to reduce the

facets reflectivity. We used a vacuum holder to hold the bars on the test fixture. This

kind of measurements of complete bars instead of individual chips has many advan-

tages. For example, it allows to test similar or different devices from the same bar

under (almost) identical conditions in order to improve the repetition rate of the mea-

surements. Thanks to the vacuum holder, bars can be easily demounted and replaced

by new bars for new experiments. Further, it reduces the costs of the characterization

in comparison to the mounting costs of individual chips. However, one disadvantage

80 8.2. Electro-optic Performance: GaAs-based couplers

of this procedure, is that the bars are very fragile and may easily be destroyed if not

carefully handled.

The measurement setup for the characterization of the MMI couplers is identical

to the setup in figure 8.12 in section 8.2.2. The light signal is fed into one input port

of the MMI couplers. The coupling is optimized for maximum power at the output.

Excess loss and imbalance of MMI couplers

The measurement results are summarized in table 8.3. According to equation 2.48, the

imbalance is determined from the ratio between the detected powers at powermeters

1 and 2. The excess loss is determined according to equation 2.47

Table 8.3: Performance of the MMI couplers from process runs II and III. Operation in the

TE mode. Input port A. X-coupler: cross-coupler.

Process wafer MMI chip WmLmexcess loss imbalance

ID TFRRDD [µm] [µm] [dB] [dB]

theor. exp. theor. exp.

III D2904-4 1×2080217 15 560 0.3 1.7 0.0 0.2

080218 15 560 0.3 1.4 0.0 0.2

II D2231-4 1×2010604 15 560 0.3 1.5 0.0 0.2

010605 15 560 0.3 1.4 0.0 0.2

III D2904-4

2×2

080201 19 1250 1.5 2.7 1.6 1.2

080301 19 1250 1.5 2.6 1.6 1.2

080208 20 1250 0.7 1.7 0.1 0.2

3dB 080308 20 1250 0.7 1.5 0.1 0.3

070208 20 1250 0.7 1.7 0.1 0.6

080215 21 1250 1.7 2.7 0.5 1.2

080315 21 1250 1.7 2.1 0.5 1.0

III D2904-4 2×2 060203 20 2500 0.8 2.0 30.0 10.8

X-coupler 060303 20 2500 0.8 1.8 30.0 11.0

Table 8.3 shows that -unlike to directional couplers- the performance of the three

types of MMI couplers (splitter, 3dB, and cross-coupler) does not depend on the

fabrication tolerances, see figure 8.15.

The 1×2 MMI coupler has the lowest imbalance (0.2 dB). Thus it can be used as the

input splitter in an MZI modulator to allow for an experimental extinction ratio of at

least 33 dB [26]. The imbalance of the 2×2 MMI 3dB coupler at the optimum width of

the multi-mode waveguide (Wm= 20 µm) is found to range between 0.2 dB and 0.6 dB.

The cross-coupler suffers a low extinction ratio ( 10 - 11 dB) when compared to the

expected value from the theory (30 dB). It is assumed that with increasing length of

the multi-mode waveguide, the contribution of leaky modes (only guided modes are

considered for the design) may no longer be discarded which may then contribute to

the reduction of the extinction ratio of the cross-coupler.

The comparison between table 8.3 and table 8.2 shows that the access loss of the MMI

couplers is larger than that of the directional couplers. These additional losses in

the MMI devices are attributed to the radiation losses in the multi-mode waveguide.

81 8.2. Electro-optic Performance: GaAs-based couplers

Nevertheless, the access loss of MMI couplers remains low enough so that MMI couplers

remain a more suitable option than directional couplers for the realization of MZI

modulators.

Figure 8.15: Two SEM images of the same 1x2 MMI coupler at different lateral longitudinal

positions showing different values of the etching depth.

8.2.4 Conclusions

In this section, the performance of GaAs/AlGaAs MMI couplers and directional cou-

plers for applications at the wavelength of 780 nm has been experimentally investi-

gated.

Directional couplers have demonstrated a very low insertion loss (0.7 dB) . However,

the investigation of nominally identical couplers has revealed different coupling ratios.

This demonstrates the dependence of the performance of the GaAs-based directional

couplers on the fabrication tolerances.

MMI couplers have been demonstrated to be an efficient alternative to directional

couplers. Unlike to directional couplers, the imbalance and excess loss of a 1×2 MMI

coupler and a 2×2 MMI 3dB coupler were found to be almost independent of the

transverse structures. The imbalance and excess loss of a 1×2 MMI coupler was found

to correspond to 0.2 dB and to 1.7 dB, respectively. The maximum imbalance of an

optimum 2×2 MMI 3dB coupler was determined to less than 0.6 dB. The excess loss

was estimated to 1.7 dB. Both types of MMI couplers offer an efficient solution for

integration with phase modulators to realize an MZI modulator.

82 8.3. Electro-optic performance: Application

8.3 Electro-optic performance: Application

Figure 8.16: SEM photos of parts of an MZI from process III.

In this section, results of MZI modulators as an application of monolithic integra-

tion of electro-optic phase modulators and MMI couplers are presented. We investigate

the insertion loss measurements and the modulation performance of two different MZI

devices. The first MZI modulator from process run II uses 1×2 MMI couplers at both,

the input and output sides (see figure 6.1 (left)). The active arm length (length of the

sections with electrodes for phase modulation) is 2 mm. The second MZI modulator

from process run III has 1×2 MMI coupler at the input and features a 2×2 MMI

coupler at the output side (see figure 6.1 (right)). The active arms length is 3.4 mm.

8.3.1 Measurement setup

After fabrication (see chapter 7) the MZI devices were cleaved and AR-coated. The

coating process is necessary to reduce the Fabry-Perot resonances which may arise

due to self-imaging during the OFF-state of MZIs with two 1×2 MMI couplers [26].

The measurement setup for the characterization of the MZI modulators (figure 8.17)

is similar to the setup for the characterization of couplers in figure 8.12. Please note

that powermeter 2 is only required in the case of a 2×2 MMI coupler at the output of

the MZI modulator. The modulation signal to modify the phase at one of the arms of

the MZI modulator is applied using an RF probe needle that can be directly put on

the electrical pads on the chip (figure 8.17).

8.3.2 Mach-Zehnder intensity modulator with 1x2 couplers

The schematic of the MZI under test is shown in figure 8.18. In order to determine

the excess loss, the transmission of the MZI (without modulation) was compared to

that of a reference single mode ridge waveguide. The estimated excess loss is less than

2.5 dB. The insertion loss is about 3.3 dB and includes both, the excess loss and the

coupling losses which were found to be about 0.8 dB (see section 8.2.1). This value of

the insertion loss is well beyond the excess less of the state-of-the-art GaAs/AlGaAs

double heterostructure MZI modulator in [25] which is larger than 8 dB.

83 8.3. Electro-optic performance: Application

Figure 8.17: (left) Setup for the characterization of MZI modulators. DUT: device under

test, Mod: modulating electric signal, PBS: polarizing beam splitter, T: transmission beam

of the PBS. (right) part of the CAD layout of an MZI modulator chip showing the electric

pads for chip soldering or connection using the RF probe needle.

For the determination of the extinction ratio, a slowly varying sawtooth signal

(frequency 1 kHz) is applied to modulate the phase of the optical field at one arm of

the MZI. The phase difference is imprinted at the output 1×2 MMI coupler (or the

2×1combiner).

Figure 8.18: (left) Layout of MZI with 1×2 MMI couplers in single-arm operation

regime.(right) electro-optic performance of an MZI modulator (process run II, wafer D2231-3,

chip 010514), [20] c

2014 IEEE.

The measured transmission of the MZI normalized to the maximum power is shown

on the left side of figure 8.18 as a function of the modulating voltage. The transmis-

sion is fitted using equation 6.1 and a good agreement is found. For the device the

measurement yields a maximum and minimum power of 1.99 mW and 0.93 mW, re-

spectively, which corresponds to the extinction ratio of 3.3 dB which also agrees with

the value from the fit. This value is marginally larger than the value of 3.0 dB which

was reported in [25].

The half-wave voltage that is calculated from the intensity profile in figure 8.18 is

84 8.3. Electro-optic performance: Application

Vπ= 3.9 V which corresponds to a modulation efficiency of 23 ◦/V·mm of the phase

modulator at the arm of the MZI. This is so far the largest modulation efficiency

reported for a GaAs/AlGaAs electro-optic phase modulator operating at 780 nm.

8.3.3 Mach-Zehnder intensity modulator with 2x2 couplers

The MZI modulators from process run III use 2×2 MMI couplers at the output. This

should improve the extinction ratio of the MZI in comparison to MZI modulators with

only 1×2 MMI couplers where internal resonances during the OFF-state of the MZI

which may not be fully eliminated [26]. The MZI modulators have an active arm

Figure 8.19: (a) Layout of an MZI modulator with 1×2 and 2×2 MMI couplers, S-bends,

and phase modulators as active arms. (b) Electro-optic performance of the MZI modulator

(process run III, wafer D2904-3, chip 080310) in single-arm operation regime with 2×2 MMI

coupler at the output. Modulating signal 1 kH sawtooth.

length of 3.4 mm. The layout of the MZI modulator is shown in figure 8.19 (a). the

two beams at the output of the 2×2 MMI coupler are separated using a dielectric

mirror (see figure 8.17). The powermeters 1 and 2 detect the transmission at both

output ports simultaneously.

The excess loss of less than 3 dB of the MZIs has been determined by comparing the

output power (sum of the powers at both ports C and D) of the MZI modulator to the

output power of reference ridge waveguide that has the same length as the MZI. To the

best of our knowledge, the reported values of the excess loss for the MZI modulators in

this work are well beyond the state of the art for GaAs-based MZI modulators. In [27],

the excess loss of GaAs/AlGaAs double heterostructures modulators was found to be

11.9 dB. Very large losses (8 dB) were also found in [25]. Both modulators in [27]

and in [25] use Y-couplers which reflects the advantage for using MMI couplers in this

work.

A slowly varying sawtooth signal (frequency 1 kHz) is used to modulate the opti-

cal field at one arm of the MZI. The result of such a measurement is shown on the

left-hand side of figure 8.19.

85 8.3. Electro-optic performance: Application

In order to estimate the extinction ratio, the measurement is fitted using equa-

tion 6.1. The normalized transmission (P/Pmax) at both output ports as a function

of the modulating voltage is shown in figure 8.20 for both output ports C and D with

the corresponding fit curves. The values of the extinction ratios resulting from the fit

correspond to 10.2 dB and 10.8 dB for the output ports C and D, respectively. The cor-

responding values of the relative initial phase shift Φ0of the fit curves in equation 6.1

are −2.43 rad for port C and to −0.79 rad for port D. The difference between the initial

phases is ∆Φ = 1.64 rad ≈π/2 rad. Please note that the output MMI coupler is a

3dB coupler based on the paired interference mechanism in multi-mode waveguides

which explains why the maximum output power at port C occurs alternatively with

the minimum output power at port D (see chapter 6).

Figure 8.20: The normalized transmission at the two output ports of the MZI modulator in

figure 8.19 and its fit using equation 6.1.

8.3.4 Conclusions

In comparison to GaAs-based MZI modulators in the literature, the achieved value of

the extinction ratio of 10.8 dB for the MZI modulator in this work is beyond the value

of the extinction ratio in [25] which amounts only to 3.0 dB (at 1.550 nm) but less than

23.5 dB that was demonstrated in [27] (at 878 nm). Please notice that the modulator

in [27] suffers very large excess losses that were attributed to the losses in the Y-

couplers which are undesirable for PICs. The GaAs-based MZI modulators in this

work were realized by combining phase modulators and MMI couplers without further

adjustment of the waveguides between the phase modulators and MMI couplers or of

the semiconductor technology which is beyond the scope of this work. Possible factors

that may contribute to the reduction of the extinction ratio in the current design of the

MZI modulators are for example, the presence of unmodulated scattered light that is

further fed into the output MMI couplers or non-optimal splitting ratios of the input

and output MMI couplers. The use of mode transformers was suggested in [25] to

filter the unmodulated scattered light and hence reduce its possible contributions to

86 8.3. Electro-optic performance: Application

the reduction of the extinction ratio. When applied between the phase modulators

and the MMI couplers, mode transformers allow to separately optimize the multi-layers

of the phase modulators and the MMI couplers. In this way, the phase modulators

may even include multiple quantum wells to deliver high modulation efficiencies. The

quantum wells are then etched away in the mode convertor and the MMI couplers

can be optimized to improve their extinction ratio which should further improve the

performance of the MZI modulator. MZI modulators with mode convertors were

successfully demonstrated in the III-V compound semiconductors technology. In [28]

an InP-based MZI modulator that incorporates a mode convertor between the phase

modulators (with quantum wells) and the MMI couplers features low excess losses

(about 3 dB) and the extinction ratio is 24 dB. The adjustment of the technology into

the GaAs material system seems to be feasible, so that efficient complex GaAs/AlGaAs

double heterostructure PICs can be achieved.

Chapter 9

Heterodyne analysis of GaAs-based

phase modulators

Phase modulation in electro-optic phase modulators is accompanied by residual am-

plitude modulation (RAM) [21], [57]. It is typical for the RAM to cause systematic

errors in applications of phase modulators. For example, RAM produces a drifting and

systematic frequency offset. This causes a degradation of the frequency stabilization

and is therefore a limiting factor, for example, for laser interferometers [57], where

the accuracy of the measurements is directly affected by the stability of the frequency

locking. In a conventional crystal-based EO phase modulator, the RAM mainly oc-

curs due to birefringence of the crystal [57]. In GaAs-based modulators, the RAM

may occur due to spatial inhomogeneities of the electric field of the modulating signal,

due to modulated free carriers absorption, by reflections at the modulators facets, by

inefficient coupling (excitation of higher order modes), and polarization impurity.

In optical terms, GaAs chip-based phase modulators are designed to operate as a single

transverse mode waveguide. Unlike active structures (i.g. laser diodes), the waveguide

material in phase modulators is transparent, so that light scattered out of the mode

through the chip upon propagation or upon coupling into the modulator chip contin-

ues to travel through the transparent material and interferes at the output with the

fundamental mode [21]. That is why phase modulation efficiency, non-linearity (phase

distortion), and RAM are expected to depend on the design of the waveguide and the

mode matching of the injected beam to the wave guide. In electrical terms, GaAs

chip-based modulators behave as reversed biased p-n diodes, and hence the modula-

tion efficiency, signal distortion (non-linearities), and RAM are expected to depend

on the operating parameters, which are offset bias voltage, TE/TM mode excitation,

modulation voltage amplitude, and modulation frequency. Although the first dou-

ble heterostructure (DH) GaAs-based electro-opitc phase modulators were presented

many years ago [23], works since then (to the best of our knowledge) included neither

measurements on RAM nor methods to quantify RAM in GaAs-based phase modula-

tors. However, the analysis of non-linearities and RAM dependence on the operation

parameters is very essential for the application of GaAs-based phase modulators.

88 9.1. Novel method for electro-optic characterization of phase modulators

9.1 Novel method for electro-optic characterization of phase

modulators

9.1.1 Method description

In [21], we proposed a method that allows to accurately determine phase shifts in

GaAs/AlGaAs double heterostructure phase modulators that are less than a degree

so that, for example, the modulation efficiency can be determined as a function of

the bias voltage or of the modulation voltage. Further, this method allows for an

analysis of non-linearities of the modulator device as will be explained in the following

sections. Another advantage of this method is that measurement data analysis is

carried out in real-time which makes it possible to optimize coupling of the light signal

into the modulator waveguide with respect to modulation efficiency and minimum

RAM. It should be noted here that this methods can be applied to characterize the

electro-optic performance of any kind of phase modulators independent of its physical

implementation, i.e. for GaAs-based and for crystal-based phase modulators.

The method is based on the heterodyne interferometer principle as shown in figure 9.1.

A single frequency local oscillator (LO) provides the optical field that is injected into

the interferometer. The first beam (the reference beam) is frequency shifted (typically

by several 10 MHz) by means of an acusto-optical modulator (AOM). The second beam

is coupled into the phase modulator waveguide (device under test, DUT) where phase

modulation and RAM are imprinted. The output of the DUT then interferes with

the reference beam on a fast photoreceiver that generates an RF beat note signal.

The I&Q quadrature components of the beat note signal are analyzed to extract the

information on the modulation response of the DUT.

Figure 9.1: schematic description of measurement method. DUT: device under test; S:

frequency shifter, UM: modulation signal applied to the DUT, PR: photoreceiver; I&Q:

signal sampling and I&Q demodulation. τXdenote the group delay for propagation of the

optical field from the input beam splitter to the frequency shifter (τS1)and to the DUT

(τD1), from the frequency shifter to the output beam combiner (τS2), and from the DUT to

the output beam combiner (τD2).

89 9.1. Novel method for electro-optic characterization of phase modulators

9.1.2 In-depth analysis of the modulation signal

The LO coherent optical field ELO(t)·eiϕLO(t)+c.c. with real-valued amplitude ELO(t)

and phase ϕLO(t)is injected into a Mach-Zehnder-interferometer. To account for

amplitude and phase noise of the LO during the measurement time Twe write:

ELO(t) = hELOiT(t)·[1 + δLO(t)] (9.1)

with hδLO(t)iT= 0 and ϕLO(t) = ωLO ·t+δϕLO(t)with hδϕLO(t)iT= 0, i.e. δLO(t)and

δϕLO(t)being the relative amplitude noise and phase noise of the LO, respectively. In

the special case of small amplitude noise, |δLO(t)|  1, the relative power noise of the

DUT equals twice the relative amplitude noise:

δPLO(t)/hPLO(t)iT≈2·δLO(t)(9.2)

In the reference arm of the interferometer, the frequency of the optical field is shifted

by ωS/(2π)(a frequency in the RF domain). The contributing electric field of this

arm at the input of the photoreceiver is described by:

ES(t)∝Ein(t−τS1−τS2)·eiωS(t)+c.c.

The other arm contains the DUT (the phase modulator). We restrict the discussion

to the quasi-static limit. A description that includes the dynamic response of the

modulator (and the driving network) is beyond the scope of this work. Also, it is not

necessary for the investigation on the LEO, QEO effects addressed by this work. In

the quasi-static limit, the response of the DUT to a modulation voltage UM(t)can be

described by:

E0(t) = EI(t)·[1 + δM(UM)]·exp (iφM(UM))

where EI+c.c. is the optical field injected into the DUT, EO+c.c. the optical field

retrieved from the DUT, δMthe relative amplitude modulation (corresponding to the

RAM), and φMthe phase modulation imprinted onto the optical field by the DUT. At

the output of the Mach-Zehnder interferometer, the fields of both arms are overlapped

and interfere on a fast photoreceiver. The corresponding RF (voltage) signal at the

photoreceiver output is proportional to EO(t)·E∗

S(t) + c.c. and satisfies:

UPR(t)∝[1 + δUPR(t)]·eiδφP R(t)eiωSt+c.c. (9.3)

where: δUP R(t) =δM(UM(t−τD2)) + δLO (t−τD1−τD2)

+δLO (t−τD1−τD2+ ∆τ)

+OδM·δLO, δLO2

(9.4)

and: δφP R(t) =φM(UM(t−τD2))

+δφLO (t−τD1−τD2)

−δφLO (t−τD1−τD2+ ∆τ)

(9.5)

Here τXwith X⊆ {D1, D2, S1, S2}are the corresponding group delays for prop-

agation of the optical signal through the interferometer (see figure 9.1), and ∆τ=

τD1+τD2−τS1−τS2denotes the group delay difference between the two arms of the

90 9.1. Novel method for electro-optic characterization of phase modulators

interferometer. An electrical spectrum analyzer (ESA) carries out I&Q demodulation

at IF frequency ωS/2πof the signal provided by the photoreceiver. This makes it

possible to reconstruct the modulation information, i.e. the phase modulation and

noise δφP R(t), as well as relative amplitude modulation and noise, δUP R(t), of the RF

beat note signal as a function of time.

The relative amplitude noise can be well approximated by δLO(t−∆τ)≈δLO(t)for

typical time scales of ∆τ <0.17 ns (corresponding to arm length difference <5 cm), so

that according to equations 9.2 and 9.4:

δUP R(t)≈δM(UM(t−τD2)) + δPLO(t−τD1−τD2)

hPLO(t−τD1−τD2)iT

(9.6)

Equation 9.6 shows that the relative amplitude modulation caused by the DUT is

given by the relative amplitude modulation of the photoreceiver RF signal. Further,

the relative power noise of the LO within the resolution bandwidth of the measurement

limits the sensitivity of the measurement method to amplitude modulation imprinted

by the DUT. Please note that for sufficiently small relative amplitude modulation,

|δM|  1, the relative power modulation caused by the DUT equals twice the relative

amplitude modulation.

Next, we consider the special case of a sinusoidal modulation. The modulation

frequency is assumed to be small enough to regard the response of the DUT as quasi-

static:

UM(t) = ˆ

UM+ ∆UM·sin (ΩMt)

The phase modulation response of the DUT can then be written as (Taylor’s expansion

about ˆ

UM):

φM(t) =φM(ˆ

UM) + 1

2∂2

UMδφM(ˆ

UM)

+∂UMδφM(ˆ

UM)·∆UM·sin (ΩMt)

−1

2∂2

UMδφM(ˆ

UM)·UM2·cos (2ΩMt) + O∆UM3

(9.7)

where the second and fourth terms in equation 9.7 are derived by using (2 cos2x=

cos 2x+ 1). It can be directly seen from equation 9.7 that fast Fourier transformation

(FFT) of the time-domain phase modulation signal - or more precisely the Fourier com-

ponent at the modulation frequency - delivers the linear phase modulation efficiency

of the DUT at the working point ˆ

UM:

∂UMδφM(ˆ

UM)(9.8)

The determination of the linear phase modulation efficiency at various working points

UMallows for the reconstruction of φM(UM). Further, determining the FFT compo-

nents at multiples of the modulation frequency delivers the non-linear response of the

device at the working point ˆ

UM.

The relative amplitude modulation can be determined in a analogous way. Lin-

ear relative amplitude modulation can be determined by FFT of the time-domain

relative amplitude signal in equation 9.4 and subsequent identification of the level

91 9.2. Implementation of the novel method

∂UMδM(ˆ

UM)·∆UMas the spectral component at the modulation angular frequency

ΩM. Higher-order residual amplitude modulation is determined accordingly. Determi-

nation of higher order RAM is relevant for applications, where signal demodulation is

carried out at harmonics of the modulation frequency in order to suppress the influ-

ence of RAM (present at the modulation frequency).

We emphasize that the I&Q demodulation is essential to this method. If the pho-

toreceiver signal is analyzed with a scalar electrical spectrum analyzer (ESA), it would

not in general allow for unambiguous separation of phase modulation from amplitude

modulation. This is because the side bands for phase and amplitude modulation are

coherent to each other, but amplitudes and relative phase of the side bands of phase

and amplitude modulation cannot be reconstructed simultaneously from the RF power

spectrum. Relative phases may be known a priori only in specific cases, e.g. in the

quasi-static limit.

9.1.3 Sensitivity of novel method

From equation 9.5 we find that the phase modulation of the RF signal of the photore-

ceiver corresponds to the phase modulation caused by the DUT. Its sensitivity (within

the resolution bandwidth of the measurement δfRBW ) is limited by the phase noise of

the LO. However, the suppression of the LO phase noise at an arm length difference

∆Lfollows from the transfer function of the heterodyne interferometer that can be

described by an effective first-order high pass filter for noise frequencies significantly

smaller than fHP = 1/(2π∆τ), where ∆τ= ∆L/c, with cbeing the speed of light in

vacuum. The expected phase noise for the measurement for modulation frequencies

well below fHP , i.e. fMfHP is then given by:

δφRMS =qSf(fM)/f2

M·δfRBW ·(fM/fHP )(9.9)

Here, Sf(fM)denotes the frequency noise power spectral density of the laser. For

example, if we assume that a sinusoidal modulation signal at fM=500 kHz is applied,

the corresponding frequency noise power spectral density for a DFB laser (typically

used for experimental implementation of the new method) Sf(fM)=4×105Hz2/Hz

[58]. As a result, the LO phase noise should be suppressed by 66 dB for the arm length

difference of 5 cm, and by and by 46 dB for an arm length difference of 0.5 m. Hence,

very small phase modulation amplitudes can be detected even without the need for

narrow line-width lasers simply by matching the interferometer arm length sufficiently

well.

9.2 Implementation of the novel method

The new method is experimentally implemented and applied to characterize a phase

modulator chip from wafer C3059-3 from the process IV (operation wavelength is

1064 nm) and a phase modulator chip from wafer D2043-3 from process I (operation

wavelength is 780 nm). Parts of the following results were published in [21].

9.2.1 Measurement setup

The setup is shown in figure 9.2. A DFB laser diode emitting at 780 nm or at 1064

92 9.2. Implementation of the novel method

Figure 9.2: Experimental setup for in-depth characterization of GaAs-based phase modula-

tors. (AOM: acousto-optic modulator, HWP: half wave plate, T: 2-1 telescope (f2= 40 mm,

f1= 20 mm), BS: 50/50 non-polarizing beam splitter, PBS: polarizing beam splitter).

Adapted/Reprinted with permission from Ref [21], [OSA].

nm is used as the local oscillator. An acousto-optic modulator (AOM, IntraAction

ATM-200C1) implements a 200 MHz frequency shift as well as the input beam splitter

of the heterodyne interferometer. The driving signal for the AOM is provided by a

signal generator (HP 8657-B). It is then amplified by means of an RF amplifier (Mini-

Circuits ZHL-2-12). The 0th order output of the AOM is fed into the phase modulator

(DUT). The polarization optics (two polarizing beam splitters (PBS) and a half wave

plate, see figure 9.2) are used to set the required polarization at the input of the

DUT. A 2:1 telescope increases the mode overlap between the fundamental mode of

the waveguide and the incoming optical field by about 25%. Further, aspheric round

lenses (f= 2.0 mm) are used for the coupling into the phase modulator. Another

(f= 2.0 mm) lens is used at the output of the phase modulator to collimate the

output beam. The beam is sent through a PBS to provide separate analysis for TM-

mode and TE-mode performance. A fraction of the beam is then picked of using a non

polarizing beam splitter (BS) in order to monitor the coupling efficiency by means of a

powermeter. A fiber coupler is used to combine the beam transmitted through the BS

with the 1st order output of the AOM (frequency shifted by fS=200 MHz). In order to

generate the RF beat note signal, the fiber coupler is connected to a fast photoreceiver

(New Focus 1544-A). Finally, the output of the photoreceiver is coupled into the

electrical signal analyzer (Rohde &Schwarz FSW26). The sinusoidal modulation

signal for the phase modulator is generated by means of a function generator (Agilent

33250A). Both, the internal clock of the modulation signal generator, and the internal

clock of the signal generator operating the AOM, are synchronized to the internal

clock of the signal analyzer. To run the measurement we employ the operation mode

analog demodulator of the FSW. In this operation mode the instantaneous phase and

amplitude modulation signals are derived and displayed by the signal analyzer in real

93 9.2. Implementation of the novel method

time. The signal analyzer then carries out an FFT of the time domain data to reveal

the phase and amplitude modulation spectra, again in real time. The spectra are then

saved an analyzed.

9.2.2 Measurement accuracy

Systematic errors can only arise in the measurement and data analysis through the

dynamic response of the photoreceiver and systematic errors of the I&Q demodulator

that is implemented by the RF signal analyzer.

Errors introduced by the photodetector can be minimized by using devices that fea-

ture a signal bandwidth significantly larger than the modulation frequency. For our

measurements a New Focus 1544-A photoreceiver featuring a bandwidth of 12 GHz is

used, and we measure at a beat note frequency of 200 MHz, with 0.5 MHz modulation

sidebands. No estimate is available on how accurately the photoreceiver tracks the

phase modulation of the beat note signal.

The I&Q demodulator tool of the signal analyzer Rohde &Schwarz FSW-26, Option

B-126, has an I&Q demodulation bandwidth of 160 MHz. For the PM demodulation

at modulation frequencies (PM rate) ≤1 MHz, the phase deviation uncertainty as per

specification sheet is ±(0.002 rad + 0.002×measured value). We have measured phase

shifts of up to 100 deg (1.74 rad), so that the inaccuracy we expect to be introduced

by the signal analyzer is on the order of 0.002 rad ··· 0.005 rad (0.1 deg ··· 0.3 deg).

The amplitude demodulation uncertainty at an AM rate ≤1 MHz as per specification

sheet is ±(0.2% + 0.001×measured value).

Further, the spurious harmonics of the signal generator were measured with a spec-

trum analyzer and found to be 60 dB below the carrier. Therefore, their contribution

to the phase shift signal (mainly at the quadratic component in the Fourier spectrum)

can be safely ignored.

9.2.3 Preparation of modulator chips for characterization

The waveguide of the investigated chips was tilted with respect to the cleaved facets

by 3◦. This has been suggested by [59] in order to reduce the effective reflectivity

by a factor of 103to104Chips with 4 mm long modulators were cleaved and their

facets were additionally AR-coated and then mounted on AlN-submounts. In order

to electrically match the device to the modulation signal generator, a resistive load of

50 Ω was added in parallel to the phase modulator chip (see figure 9.3).

A sinusoidal modulation signal with a frequency of 500 kHz is used to drive the

phase modulator. A sufficient reverse-bias voltage offset (VDC =−1.5 V) is applied to

operate the phase modulator.

The following two chips were investigated:

λprocess run wafer structure chip chip length

780 nm I D2043-3 table 4.2 080511 4 mm

1064 nm IV D3059-3 table 4.5 010118 4 mm

Table 9.1: List of characterized phase modulator chips using the spectral analysis.

94 9.2. Implementation of the novel method

(a) (b)

(c)

Figure 9.3: (a) Submount assimply: modulator chip on AlN submount, (b) fixture for sub-

mount assembly, (c) part of measurement setup showing the submount assembly, the fixture

for submount assembly and the motorized lens holders with lenses and electrical interfaces.

95 9.3. Measurement of the phase modulation efficiency

9.3 Measurement of the phase modulation efficiency

The spectrum of the phase modulation (PM) signals of a TE-mode for the two chips

in table 9.1 are shown in figure 9.4.

Let us for example consider the spectrum of the phase modulation signal at 1064 nm in

Figure 9.4: Spectrum of the phase modulation for TE-mode excitation of (left) phase mod-

ulator chip at 1064 nm, (right) phase modulator chip at 780 nm. The modulating signal

amplitude corresponds to 1.4 V, offset voltage is VDC =−1.5 V, and δfRBW = 3.8 kHz.

figure 9.4(left). The noise floor of about 0.02 degrees is due to the non-zero optical arm

length difference between the arms of the interferometer and from the phase noise of the

DFB laser. By taking into account the arm length difference of the interferometer in

Figure 9.5: Spectrum of the phase modulation for TE-mode excitation at two different values

of the arm length difference of the interferometer (at 50 cm and 100 cm). The amplitude of the

modulating signal corresponds to 0.95 V, offset voltage is −1.5 V DC, and δfRBW = 3.8 kHz.

the experiment setup which is about 50 cm, and the resolution bandwidth of δfRBW =

3.8 kHz, which corresponds to fHP = 95.49 MHz, using equation 9.9 we find that

δφRMS = 0.023 deg is consistent with our experimental findings, see figure 9.4.

The effect of the arm length difference on the noise floor can be clearly seen in figure 9.5.

Initially, measurements were taken with an arm length difference of about 100 cm, then

96 9.3. Measurement of the phase modulation efficiency

reduced it to about 50 cm which reduced the noise floor by a factor of 2 as expected.

If the arm length difference were reduced to 5 cm, the noise floor would be reduced to

2×10−3deg. However, at an arm length difference of 50 cm the noise floor is already

small enough to provide a sensitivity significantly better than 0.1 deg which is sufficient

for our application.

9.3.1 Phase modulator chip C3059-3 010118 at 1064 nm

In figure 9.4, the Fourier components at the modulation frequency fM(linear compo-

nent) describe the total phase shift due to linear effects. The Fourier component at

the 2nd harmonic of the modulation signal can be solely attributed to the quadratic

effects in the phase modulator chip which are the QEO effect and (possibly) quadratic

contributions from carrier density-related effects. However, according to [23], carrier

density-related effects contribute linearly to phase modulation in GaAs-based electro-

optic phase modulators. Thus, the quadratic phase modulation can be attributed

solely to the QEO effect and quadratic effects. The linear and quadratic components

are attributed to the phase shift due to the linear effects and to the quadratic effects,

respectively. The linear effects (not to confuse with LEO effect) include both, the LEO

effect and linear contributions from the carrier density-related effects [23].

Figure 9.6: Fourier components of the phase modulation at the fundamental and at the 2nd

harmonic of the modulation frequency as a function of the modulation voltage amplitude

of the phase modulator chip C3059-3 010118 at 1064 nm. (A), (C): for TE-mode operation

at fundamental and 2nd harmonic frequency, respectively. (B), (D): TM-mode operation

at fundamental and 2nd harmonic frequency, respectively. The modulator offset is set to

VDC =−1.5 V. Adapted/Reprinted with permission from Ref [21], [OSA].

The Fourier components of the phase modulation at the modulation frequency and

its 2nd harmonic as a function of the amplitude of the modulating signal are shown

for both, the TE and TM modes, in figures 9.6. The contribution of the quadratic

effects to phase modulation is about two orders of magnitude less than that of the

linear effects and the total phase shift can be well approximated by only the linear

97 9.3. Measurement of the phase modulation efficiency

component of the Fourier spectrum of the PM signal. In order to determine the linear

phase modulation efficiency, the amplitudes of the phase modulation at the modulation

frequency fMas a function of the amplitude of the driving signal (curves A and B

in figure 9.6) are fitted using a linear function (enforcing φM(UM= 0) = 0). The

resulting slope of the linear fit corresponds to 25.25 ◦/V(fit standard error 0.01 ◦/V)

for the TM-mode and 60.33 ◦/V(fit standard error 0.03 ◦/V) for the TE-mode, which

is the phase modulation efficiency of a 4 mm long modulator. Therefore, the (linear)

modulation efficiency (per mm) for TM and TE-modes correspond to 6.31 ◦/(V ·mm)

and 15.08 ◦/(V ·mm), respectively.

To determine the quadratic phase modulation efficiency, the experimental data

for the Fourier component at the 2nd harmonic of the modulation frequency (curves

C and D in figure 9.6) are fitted using the function y=√A2+C2·x4with A

being the noise floor and Cthe corresponding (quadratic) phase modulation effi-

ciency. The resulting fit values are (A=0.02◦,C=0.59 ◦/V2) for the TM-mode and

(A=0.02◦,C=0.69 ◦/V2) for the TE-mode.

9.3.2 Phase modulator chip D2043-3 080511 at 780 nm

Figure 9.7: Fourier component of PM signal at the fundamental modulation frequency (A

and B) and at its second harmonic (C and D) versus modulation voltage amplitude of TE

and TM-modes. The phase modulator is biased at an offset VDC =−1.5 V. Operation at

the wavelength 780 nm.

The Fourier components of the phase modulation at the modulation frequency and

its 2nd harmonic as a function of the amplitude of the modulating signal are shown

for both, the TE and TM modes, in figure 9.7.

The determination of the phase modulation efficiency follows in a similar way to

the steps taken for the modulator chip at 1064 nm. The amplitudes of the phase mod-

ulation at the modulation frequency fMas a function of the amplitude of the driving

signal (curves A and B in figure 9.7) are fitted using a linear function y=B·x(en-

forcing φM(UM= 0) = 0). The resulting slope corresponds to 27.84 ◦/V(fit standard

98 9.4. Residual amplitude modulation

error 0.03 ◦/V) for the TE-mode and 10.08 ◦/V(fit standard error 0.02 ◦/V) for the

TM-mode. Since the total length of the modulator is 4 mm, the (linear) modulation

efficiencies for TE and TM-modes correspond to 6.96 ◦/(V ·mm) and 2.7◦/(V ·mm),

respectively.

The analysis of the Fourier component at the 2nd harmonic of the modulation frequency

follows in similar lines. The measured values in figure 9.7 (curve C) is fitted using the

function y=√A2+C2·x4with Ais being the noise floor and Cthe corresponding

(quadratic) phase modulation efficiency. The resulting fit values are (A= 0.12◦,C=

0.85 ◦/V2) for the TE-mode. The PM Fourier components at the 2nd harmonic of the

modulation frequency for the TM mode excitation lay beneath the the noise floor.

Therefore, the corresponding quadratic phase shift could not be determined.

9.4 Residual amplitude modulation

For this measurement, we consider the chip C3059-3 010118 at 1064 nm. The relative

amplitude of the RAM at the modulation frequency and its harmonics is directly read

of the relative amplitude modulation spectrum, figure 9.8. According to [60], RAM

Figure 9.8: Spectrum of the RAM for TE-mode excitation of the phase modulator chip at

1064 nm. The modulating signal amplitude corresponds to 1.4 V, offset voltage is VDC =

−1.5 V, and δfRBW = 3.8 kHz.

is proportional to the phase modulation efficiency. Hence, since the linear effects are

dominant in phase modulation the same is expected for residual amplitude modula-

tion, so that only the RAM signal at the modulation frequency has to be analyzed.

Figure 9.9 shows the Fourier component at the fundamental frequency of the RAM

as a function of the amplitude of the modulation voltage. The amplitude of RAM

increases with increasing phase modulation, but it remains smaller than 5×10−3for

modulation amplitudes smaller than 1.4 V.

For the measurement of the RAM amplitudes in figure 9.9, the coupling efficiency

of the incoming light beam (both TE and TM) into the modulator chip was optimized

for minimum RAM at a modulation voltage amplitude of 0.1 V, then the RAM level as

a function of the modulation voltage amplitude was recorded. The resulting data for

99 9.5. Modulation bandwidth

Figure 9.9: Fourier component of the residual amplitude modulation at the fundamental

modulation frequency as a function of the modulation voltage amplitude for both, TE and

TM-modes. Offset VDC =−1.5 V. Adapted/Reprinted with permission from Ref [21], [OSA].

both TE and TM modes as shown by figure 9.9 are quite scattered with a general trend

of increasing RAM level with increasing modulation voltage. The scattering behavior

may be attributed to the alignment (coupling efficiency) of the optical mode into the

phase modulator waveguide. To investigate the effect of coupling misalignment on

RAM, we move the lens away from the optimum position, first in the lateral direction

with respect to the modulator waveguide (parallel to the guiding layers), and then in

the vertical direction (parallel to the pn-junction).

For a lateral misalignment of 500 nm, the RAM relative amplitude exceeds 10−2and

the output power of the phase modulator (detected by the powermeter, see figure 9.2)

decreases by about 15%. This value apply to an operation at a bias voltage of -1.5

V with a modulation amplitude of 1.4 V. The strong dependence of RAM on the

coupling could be attributed to the excitation of higher order lateral modes which

interfere at the output with the fundamental mode. For coupling misalignment in

the vertical direction we observed that the RAM level is significantly more sensitive

to the position of the coupling lens. A vertical misalignment of the coupling lens of

less than 100 nm increases the relative amplitude to 10−2. The corresponding output

power decreases by about 30%. We assume, that in addition to the possible excitation

of higher order modes, the free carriers absorption of the optical field in the doped

cladding layers may be responsible for this decrement in the output power by vertical

misalignment. It is suggested that the effect of free carrier’s absorption on the RAM is

determined by further investigation of phase modulators with different doping profiles.

9.5 Modulation bandwidth

For this measurement, we consider the chip C3059-3 010118 at 1064 nm. Figure 9.10

shows the Fourier components of the PM spectrum at the fundamental modulation fre-

quencies and at the 2nd harmonic of the modulation frequency for various modulation

frequencies. It can be seen that the amplitude of the PM signals remains independent

100 9.5. Modulation bandwidth

Figure 9.10: Amplitude of the phase modulation as a function of the modulation frequency.

(A) and (B) are the Fourier components of the PM spectrum at the modulation frequencies

and their corresponding 2nd harmonics, respectively. Input mode TE.

of the modulation frequency up to 10 MHz(within the resolution of the 50 Ω driving

source). The dependence of the modulation amplitude on the modulation frequency

can be described by a simple first order RC low pass filter, see figure 8.2. Hence, only

the Fourier component at fMcan be considered.

Figure 9.11: Normalized (to the PM amplitude at 1 V) amplitude of the Fourier components

of the PM spectrum at the fundamental modulation frequencies and a fit to a simple RC

low pass filter.

The transfer function of the low pass filter is given by H(jω) = A/ (1 + jωRsC).

The amplitude can hence be described by f(ω) = A/q1 + ω2(RsC)2. The amplitude

of the Fourier components of the PM spectrum at the fundamental modulation fre-

quencies fMcan then be fitted to the function f(x) = A/√1 + B2·x2as shown by

figure 9.11. The fit parameter Bcorresponds to 2πRsC, with Rsthe series resistance,

and Cthe modulator capacity. Ais an amplitude scaling factor. Values of the series

101 9.6. Determination of the electro optic coefficients

resistance at -1.5 V of 2 mm long phase modulators with ridge widths of 2µmand 4µm

were found (see figure 8.3) to correspond to 125 Ω and to 65 Ω, respectively. Thus,

for the 4 mm modulator chip with a ridge width of 3µm,Rsis expected to be about

48 Ω. Hence, using the value of the fit parameter B= 1.10 ×10−7sec, the modulator

capacity is found to correspond to C= 360 pF. Please note that the junction capacity

at VDC =−1.5 V corresponds to 2.7 pF (the ridge width is 3µm, the modulator length

is 4 mm, and the depletion width is 0.5µm). The capacitance of the modulator could

be reduced by using PCB passivation between the upper cladding and the p-contact

layer instead of the SiNx layer [19].

9.6 Determination of the electro optic coefficients

In this section, the heterodyne analysis of the phase modulation signal is used to de-

termine the electro-optic coefficients of the investigated double heterostructures. The

effects responsible for phase modulation in the GaAs/AlGaAs double heterostructures

are the LEO effect, the QEO effect, and the carrier density-related effects. The layout

of the tested modulators (see section 3.1.2 in chapter 3) is chosen so that the LEO

effect contribute to phase modulation only in TE-mode operation. Hence, the con-

tribution of the LEO effect to phase modulation can be extracted from the difference

between the linear phase shifts of the TE and TM-modes [15]. For operation in TM

mode, the linear phase shift is then solely attributed to linear contributions from the

carrier density-related effects.

Results of the spectral analysis of the phase modulators are used to experimentally

determine the electro-optic coefficients for GaAs at the wavelength of 1064 nm, and

for Al0.35Ga0.65As at the wavelength of 780 nm.

9.6.1 Electro-optic coefficients of GaAs at 1064 nm

Linear electro-optic coefficient

In figure 9.6, the linear effects are separated from the quadratic effects by means of

spectral decomposition of the phase shift signal.

Using equations 3.15, 3.20, and 3.21, the LEO coefficient ¯r41 is given by:

¯r41 =−∆φT E (V)−∆φT M (V)·λ

πL¯n3

·R+∞

−∞ I(x)dx

R+∞

−∞ (E(V+VDC, x)−E(VDC , x)) I(x)dx

(9.10)

where ∆φTE (V),∆φT M (V)are the resulting phase modulation amplitudes of the TE

mode and TM mode, respectively, , ¯nhere is the refractive index of the waveguide

core, Vis the amplitude of the modulating voltage signal, Lis the modulator length,

λis the vacuum wavelength, I(x)is the intensity distribution of the optical field which

was found to be almost identical for the TE and TM modes, see figure 9.12). Further,

E(V+VDC, x)and E(VDC , x)denote the electric fields for the respective modulation

voltage signals. The values of ∆φT E (V)and ∆φT M (V)are taken from figure 9.6

(graphs A and B for TE and TM mode, respectively).

According to equation 9.10, and from the linear fit in figure 9.6, the resulting LEO

102 9.6. Determination of the electro optic coefficients

Figure 9.12: Calculated (vertical) field of the TE and TM modes of the modulator waveguide

(at 1064 nm) and their overlap with the linear (∆E) and quadratic (∆E2) electric field varia-

tions where ∆E=E(V+VDC, x)−E(VDC, x), and ∆E2=E2(V+VDC, x)−E2(VDC, x).

Adapted/Reprinted with permission from Ref [21], [OSA].

coefficient ¯r41 for the P-p-n-N GaAs/AlGaAs double heterostructure at λ= 1.064 µm

is:

¯r41 =−1.70 ×10−10 cm/V

In [61], the value of ¯r41 =−1.80 ×10−10 cm/Vwas given for a phase modulator at

λ= 1.06 µmwith a p-i-n GaAs guiding layer. Besides, in [15] the values of ¯r41 for

GaAs-based phase modulators at λ= 1.09 µmand λ= 1.15 µmwith P-i-N, and

P-n-N doping profiles were found to correspond to ¯r41 =−1.68 ×10−10 cm/Vand

¯r41 =−1.72 ×10−10 cm/V, respectively. Hence, the value of ¯r41 =−1.70 ×10−10 cm/V

measured here is consistent with the literature (values reported in [61] and [15]).

Furthermore, in figure 9.13, the values of the LEO coefficient ¯r41 are derived (using

equation 9.10) from the individual data points in figure 9.6 at different modulation

voltages. It can be clearly seen that the variation of the value of ¯r41 is smaller than

0.5% within the modulation voltage amplitude range of 0.2 V to 1.4 V.

Quadratic electro optic coefficient

We follow the assumption made in [23], where quadratic phase modulation is assumed

to be solely due to the QEO effect. Hence, the QEO coefficients for the TE and TM-

modes can be calculated from the Fourier components of the phase modulation at the

2nd harmonic of the modulation frequency.

Following the lines of the discussion of the LEO coefficients, we use equation 3.15,

103 9.6. Determination of the electro optic coefficients

Figure 9.13: Values of the electro-optic coefficients ¯r41 ,¯

R11 , and ¯

R12 for GaAs at 1064 nm

determined using equation 9.10 from the individual data points in figure 9.6.

equation 3.16, equation 3.20, and equation 3.21. The QEO coefficients are given by:

R12,11 =−φTE,TM (V)·λ

πLn3

·R+∞

−∞ I(x)dx

R+∞

−∞ (E2(V+VDC, x)−E2(VDC , x)) I(x)dx

(9.11)

Where ¯

R11,¯

R12 are the QEO coefficients and φT M (V), φTE (V)are the quadratic

phase modulation amplitudes for the TM and TE-modes, respectively. The values

of φTM (V), φT E (V)are calculated from the fit of the data points (C) and (D) in

figure 9.6. The resulting values of the QEO coefficients for the GaAs/AlGaAs P-p-n-

N double heterostructure are:

R12 =−2.85 ×10−17 cm2/V2,

R11 =−2.44 ×10−17 cm2/V2.

As can be seen by figure 9.13, the variation of the values of ¯

R12,¯

R11 is smaller than

8% within the modulation voltage amplitude range of 0.2Vto 1.4V.

Unlike to our findings regarding the LEO coefficient, the measured values here of the

QEO coefficients are one order of magnitude smaller than the values that were reported

in [61] and [15] for GaAs at the wavelengths of 1.06 µmand 1.09 µm. Please note, that

the value for LEO coefficient was derived from the phase modulation spectra at the

fundamental modulation frequency and the measured value of the LEO coefficient

agrees very well with the literature. If we further assume with [23] that the quadratic

phase modulation is solely due to the QEO effect, then the QEO coefficients are

expected to also be accurately determined from the phase modulation spectra, i.e. from

the phase modulation amplitude at the 2nd harmonic of the modulation frequency.

In [15] and in [61] the authors used planar waveguide phase modulators. In their

experiment in [15], the light signal was coupled onto both the [110] and h1¯

10icleaved

104 9.6. Determination of the electro optic coefficients

facets of the planar waveguide such that the phase shift due to LEO effect could be

subtracted from the total phase shift using symmetry considerations. Then, the carrier

density-related phase shift was calculated and subtracted from the remaining phase

shift (see equations (3) and (4) in [15]). The authors emphasize that this approach

relies on the carrier effect to be calculated accurately.

Hence, the discrepancy between our results and the results presented in [15] and [61]

suggests, that in contrast to [23] other effects than the QEO effect may contribute to

the quadratic phase modulation and/or the models used in [15] and [61] to calculate

the carrier density-related effects may be inaccurate. We believe that further measure-

ments on GaAs/AlGaAs double heterostructure phase modulators, e.g. for different

waveguide orientations, are required to resolve this discrepancy.

9.6.2 Electro-optic coefficients of Al0.35Ga0.65As at 780 nm

The determination of the LEO and QEO coefficients of Al0.35Ga0.65As at the wave-

length 780 nm follows from the measurements of the phase shift signal for both, the TE

and TM-modes (figure 9.7). The procedure that was already applied to analyze the

Figure 9.14: Calculated (vertical) field of the TE and TM modes of the modulator waveguide

(at 780 nm) and their overlap with the linear (∆E) and quadratic (∆E2) electric field varia-

tions where ∆E=E(V+VDC, x)−E(VDC, x), and ∆E2=E2(V+VDC, x)−E2(VDC, x).

electro-optic coefficients of GaAs at 1064 nm can be applied here. Next, equation 9.10

is used to determine the values for the LEO coefficient for a P-p-i-n-N GaAs/AlGaAs

double heterostructure at the wavelength of 780 nm. These values correspond to:

¯r41 =−1.01 ×10−10 cm/V.

This values agrees well with the value of ¯r41 =−1.84 ×10−10 cm/Vthat we estimated

in section 4.2 for the LEO coefficient which is quite satisfying. Further comparison

with values from the literature is not available. Please note that this is the first time

a double heterostructure with Al0.35Ga0.65As guiding core at the wavelength 780 nm is

used to realize electro-optic phase modulators.

105 9.7. Conclusions

The value for the QEO coefficient ¯

R12 as determined from the evaluation of curve

C in figure 9.7 using equation 9.11 is:

R12 =−0.76 ×10−16 cm2/V2.

In section 4.2 we estimated the value of ¯

R12 =−2.3×10−16 cm2/V2which is about 3

times larger than the experimental value. However, the estimation of the QEO coeffi-

cient in section 4.2 was based on experimental values provided for the QEO coefficient

for GaAs. We believe that the previous discussion regarding the experimental value of

QEO for GaAs at the wavelength 1064 nm is also valid here. Namely, the discrepancy

suggests that in the in contrast to [23], the quadratic phase shift may not solely be

due to the QEO effects or/and the models used for the calculation of the free carriers

effects may be inaccurate.

Furthermore, the Fourier components for the TM mode at the 2nd harmonic of the

modulation frequency are below the noise floor (see curve D in figure 9.7). Hence,

no experimental value of ¯

R11could be determined for Al0.35Ga0.65As at the wavelength

780 nm. One possible explanation would be that for the considered structure, ¯

R11 is

significantly smaller than ¯

R12.

Please notice that in the literature, the values of the QEO coefficients ¯

R11 and ¯

R12

for GaAs at different values of the photon energy were reported to be very close to

each other. This also agrees with our experimental findings for the QEO coefficients

for GaAs at the wavelength 1064 nm. If our experimental finding for the QEO co-

efficients for Al0.35Ga0.65As are corrects, then this may suggest that unlike to GaAs,

for the AlxGa1−xAs material, the difference between the QEO coefficients ¯

R11 and

R12 increases with increasing Al-mole fraction. As far as we know, no experimental

values are available for the QEO coefficients for AlxGa1−xAs, with 0<x<1. Further

investigations are required in the future to confirm or disagree with our suggestions.

9.7 Conclusions

The heterodyne analysis method described here is based on a heterodyne interferom-

eter. The I&Q spectral analysis of the beat note signal delivers the amplitude and

phase modulation in the time domain. The method can be applied to measure the

RAM in real time and thus can be used to optimize the coupling efficiency for lowest

RAM level.

The I&Q spectral analysis of the phase modulation signal delivers an accurate

measurement of the phase modulation of less than 1 deg. It overcomes the limitation

of the FP method that requires a phase modulation amplitude of at least 90 deg. The

spectral analysis of phase modulation for both, TE mode and TM mode excitation has

been further used to determine the individual contribution of the LEO effect, the QEO

effect, and the carrier density-related effects to phase modulation in GaAs/AlGaAs

double heterostructure phase modulators. The contribution of the free carriers effect

to phase modulation can be calculated by subtracting the LEO contribution to the

linear phase shifts of the TE and TM mode. Table 9.2 summarizes the measured (using

the spectral analysis) phase modulation efficiencies of the two phase modulator chips

at 780 nm and at 1064 nm and compares them to the theoretically expected values. A

106 9.7. Conclusions

Table 9.2: Experimental and theoretical modulation efficiencies of GaAs-based phase mod-

ulators.

λstructure LEO carrier QEO

[nm] [deg/(V·mm)] [deg/(V·mm)] [deg/(V2·mm)]

theory experiment theory experiment theory experiment

780 table 4.2 8.41 4.44 0.17 2.52 0.96 0.21

1064 table 4.5 8.46 8.77 1.41 6.31 0.88 0.17

good agreement between theory and experiment is found for the modulator at 1064 nm

with respect to phase shift due to the LEO effect. For both, 780 nm and 1064 nm,

the carrier density related effects turn out to be more pronounced than predicted by

simulation. This suggests that the doping of the guiding layers are larger than the

nominal values in table 4.2 and in table 4.5. Finally, the experiment shows that (as

expected from the simulations) the contribution of the QEO effect to phase modulation

is very small in comparison to the linear effects (LEO effect and free carrier effects).

Furthermore, as a direct application, we were able to experimentally determine the

electro-optic coefficients for GaAs at the wavelength of 1064 nm and Al0.35Ga0.65As at

the wavelength of 780 nm.

Chapter 10

Conclusions and Outlook

In order to be ready for leaving the labs, electro-optical systems for quantum precision

experiments such as atom interferometers should be robust and compact to provide re-

liable mechanical and thermal stability. Laser radiation in the state-of-the-art electro-

optical systems is achieved using compact and robust micro-integrated laser modules.

However, passive components such as phase modulators, splitters, and fiber couplers

that are required for the manipulation of the light signal (for example at 780 nm for

rubidium spectroscopy and at 1064 nm for molecular iodine spectroscopy) are only

commercially available on macro-scales. They are implemented in the electro-optical

systems with a huge demand on space and from factor which correspond to a reduced

robustness and mechanical stability. The miniaturization of the passive components

to realize photonic integrated circuits or to micro-integrate them into laser and spec-

troscopy modules is a prerequisite for mature quantum sensors for applications in the

field and in space.

In this work, GaAs-based electro-optic phase modulators for operation at the wave-

lengths of 780 nm and at 1064 nm and waveguide couplers at the wavelength of 780 nm

have been developed, and experimentally investigated. Chip-based phase modulators

provide the means to substitute crystal-based electro-optic modulators in the state-

of-the-art optical systems. Waveguide couplers may replace the fiber couplers and are

required in photonic integrated circuits (PICs). The monolithic integration of GaAs-

based couplers with phase modulators is demonstrated by the implementation of a

Mach-Zehnder intensity modulator at 780 nm.

10.1 GaAs-based phase modulators and waveguide couplers

The phase modulators that were designed and fabricated are based on a GaAs/AlGaAs

double heterostructure, which allows for optical and electrical fields confinement for

efficient modulation and to account for low free carrier absorption. The devices were

realized based on the ridge waveguide optical design. The waveguide parameters were

optimized to meet the micro-integration requirements of phase modulators with ac-

tive devices such as edge-emitting GaAs-based lasers. Phase modulators with phase

modulation efficiencies larger than 15 deg/(V.mm) have been designed.

We applied established techniques in order to determine the electro-optic performance

of the phase modulators. The well-known Fabry-Perot (FP) method was used to de-

termine the phase modulation efficiency and the propagation losses. The largest phase

107

108 10.2. In-depth characterization of phase modulators

modulation efficiency achieved using the FP method with a phase modulator designed

for operation at 780 nm (epitaxial design according to table 4.3) was found to corre-

spond to 16 ◦/(V ·mm) with propagation losses smaller than 1.2 dB/cm. This allows

for phase modulators that are 4 mm long, so that the half wave voltage can be as small

as 2.8 V. The capacity of the phase modulator (epitaxial design according to table 4.2)

at the wavelength of 780 nm was found to be 250 pF which allows for a modulation

bandwidth of 12.8 MHz when the phase modulator is driven directly with a 50 Ω signal

source, which is sufficient for example for generation of modulation sidebands for Rb

spectroscopy applications. In the future, the SiNx isolation layer could be replaced

by BCB passivation to reduce the capacity by one to two orders of magnitude so that

GaAs-based phase modulators can provide access to modulation frequencies beyond

1 GHz with direct driving from a 50 Ω signal source .

For phase modulators at the wavelength of 1064 nm, propagation losses of 2.7 dB/cm

and 4.3 dB/cm were determined experimentally (epitaxial design according to ta-

ble 4.6 and table 4.5, respectively). The propagation losses measured for the devices

clearly demonstrate progress beyond state-of-the-art of GaAs/AlGaAs phase modula-

tors (12 dB/cm for modulators at 1.06 µm[13]).

Waveguide couplers for operation at the wavelength of 780 nm have been demon-

strated for the first time. The design of the couplers relies on the double heterostruc-

ture of the phase modulators at 780 nm (epitaxial design according to table 4.2 and

table 4.3). This is meant to ease the integration of phase modulators and couplers.

For the implementation of waveguide couplers two concepts have been analyzed, both

theoretically and experimentally. Multi-mode interference (MMI) couplers based on

the self-imaging principle in multi-mode waveguides were used to implement 1×2

and 2×2couplers. Directional couplers based on evanescent mode coupling were

used to implement 2×2couplers. The comparison between the MMI couplers and

the directional couplers showed the MMI couplers are less dependent on fabrication

tolerances related to the transverse geometry than the directional couplers. The MMI

couplers feature low excess losses (1.4 dB for 1×2 MMI splitter and 1.6 dB for a 2×2

MMI 3dB coupler) and low imbalance (0.2 dB for 1×2 MMI splitter and 0.2-0.6 dB

for a 2×2 MMI 3dB coupler) which allows for efficient integration of the couplers with

GaAs/AlGaAs double heterostructure phase modulators. The Mach-Zehnder inten-

sity modulator realized using phase modulators and MMI couplers features a very low

excess loss (less than 3 dB), and the extinction ratio is larger than 10 dB.

10.2 In-depth characterization of phase modulators

The waveguide material of the GaAs-based phase modulator is transparent. Upon

propagation or upon coupling into the modulator chip, light scattered out of the mode

(into higher order modes) is expected to continue to travel through the transparent

waveguide material and thus to interfere at the output with the guided fundamen-

tal mode. This scattered light is a main source for residual amplitude modulation

(RAM). RAM is a main source of systematic errors, and hence, RAM is an impor-

tant performance factor in the applications of phase modulators. The analysis of the

modulation efficiency, RAM, and the non-linearities in dependence of the operation

parameters (modulation voltage, coupling efficiency) is essential for the applications of

chip-based phase modulators. Even though the first GaAs-based electro-optic phase

109 10.2. In-depth characterization of phase modulators

modulators were presented many years ago, to the best of our knowledge, none of

the previous works on GaAs-based phase modulators have included measurements on

RAM or methods to quantify RAM.

In this work, a new method was developed to investigate linear and non-linear

phase and amplitude modulation of electro-optic phase modulators. Unlike to the

FP method that requires to determine the half-wave voltage, this method allows to

determine phase shifts less than a degree so that, the modulation efficiency can be

determined as a function of the bias voltage or of the modulation voltage. The method

is based on a heterodyne interferometer (see figure 9.1). The optical field provided by

a local oscillator is divided into two paths. The beam from the first path is coupled

into the modulator chip where both, phase and amplitude modulation are imprinted.

The second beam (reference beam) is frequency shifted, typically by several 10 MHz

using an acousto-optic modulator (AOM). The reference beam interferes then with the

output of the modulator on a fast photoreceiver. The in-phase and quadrature (I&Q)

components of the resulting beat note signal are then analyzed following lines that were

explained in section 9.1.2 in chapter 9. This novel method has been experimentally

implemented (see section 9.2 in chapter 9) and applied to phase modulator chips

designed for and operated at 780 nm and at 1064 nm.

The analysis of the I&Q components of beat note allows to separately analyze the

information of amplitude modulation. The RAM is measured from the Fourier spec-

trum of the amplitude modulation (see for example figure 9.9). The measurement is

carried out in real time. Thus, it can be applied for optimizing the coupling efficiency

into the modulator waveguide for lowest RAM level. The measurement has shown that

the RAM can be reduced to the 10−3level by means of optimum coupling of the light

signal into the guided fundamental mode of the phase modulator waveguide. It can be

very advantageous to apply this method for RAM measurement, for example, during

the micro-integration process of phase modulators into hybrid laser modules. In the

future, the phase modulator waveguide may use a mode filter, e.g. a bent geometry,

to reduce the overlap of undesired light with the modulated optical field which should

account to further reduction of the RAM.

By applying the I&Q analysis to the phase modulation signal, the instantaneous

phase shift could be determined. Using a Fourier transformation approach linear and

quadratic response were determined. From the linear response for TE and TM mode

operation, the linear electro-optic (LEO) coefficient as well as the phase modulation

coefficient describing the effect of carrier density modulation could be determined.

The quadratic response is solely due to the quadratic electro-optic effect and hence

allows for determination of the quadratic electro-optic (QEO) coefficient. As a direct

application of the separation of the linear and quadratic effects, we were able to exper-

imentally determine for the first time the LEO and QEO coefficients of Al0.35Ga0.65As

at the wavelength of 780 nm and for GaAs at the wavelength of 1064 nm. This is also

the first time the QEO coefficients were directly determined by an experiment. The

values of the LEO coefficients derived from the experimental data agree very well with

the values from the literature. However, the estimated QEO coefficients for GaAs at

the wavelength of 1064 nm are one order of magnitude smaller than the values that

were reported in the literature. Please note that the determination of the QEO coeffi-

cients was based on the assumption that the quadratic phase shift is solely due to the

110 10.2. In-depth characterization of phase modulators

QEO effects which is supported by literature. The discrepancy may be resolved by

investigating modulator chips with different doping profiles in order to challenge this

assumption and to investigate whether carrier density related effects also contribute

to the quadratic response.

List of Abbreviations and Symbols

AlGaAs Aluminum gallium arsenide

AlN Aluminum nitride

AOM acousto-optic modulator

AR Anti-reflection

BCB benzocyclobutene polymer

BPM Beam Propagation Method

CCD charge-coupled device

DFB Distributed feedback

DH Double Heterostructure

DLR German Space Agency (Deutsche Zentrum füer Luft- und Raumfahrt)

DUT Device under test

ECDL External.cavity Diode Laser

EO Electro-Optic

EZA Electrical Spectrum Analyzer

FE Finite Element

FEM Finite Element Method

FD Finite Difference

FOKUS First Orbital Curing Experiment of University Students

FP Fabry Perot

FWHM Full width at half maximum

GaAs Gallium arsenide

GPS Global Positioning System

I&Q In-phase and quadrature

KALEXUS Kalium Laser-Experimente unter Schwerelosigkeit

LEO Linear Electro-optic

LO Local Oscillator

Milas Mikro-Integrierte Diodenlasersysteme

MMI Multi-Mode Interference

MOVPE Metalorganic vapour phase epitaxy

MOPA Master-Oscillator-Power-Amplifier

MZI Mach-Zehnder-Interferometer

PBS Polarizing beam splitter

PER Polarization Extinction Ratio

PICs Photonic Integrated Circuits

PM phase modulation

QEO Quadratic Electro-optic

RAM Residual Amplitude Modulation

RBW Resolution bandwidth

111

112 10.2. In-depth characterization of phase modulators

RF Radio Frequency

RIE Reactive ion etching

RW Ridge Waveguide

RWA Ridge Waveguide Amplifier

TE Traverse Electric

TEC Thermoelectric cooler

TM Traverse Magnetic

VHBG Volume Holographic Bragg Grating

A: GaAs (AlxGa1−xAs) compound

semiconductors

.1 Zinkblende structures

Examples of III-V compounds semiconductors that crystallize in the Zinkblende struc-

ture are InSb, InAs, InP, GaSb, GaAs, and GaP. The unit cell of GaAs is shown in

figure 1. The crystallographic directions are defined by the components of a vector

Figure 1: Unit cell of GaAs as example of the Zinkblende structure. ˆais the lattice parameter.

Original picture in [30].

that is oriented in a given direction. For the cubic crystal (zinkblende), any direction

can be fully described using the orthogonal basic vectors ˆa, ˆ

b, and ˆcwith the same

length as can be shown in figure 2. A unit of three numbers from the group {0,1,¯

Figure 2: Crystallographic directions and planes in a cubic crystal. Picture taken from [30].

in square bracket describes then a given direction. Examples for these directions are

113

114 .2. GaAs wafers

given in figure 2. Where 1and ¯

1numerical codes refer to the opposite directions. The

crystallographic planes are given by the Miller indices that determine the Intersection

points of the plane with the axes ˆa, ˆ

b, and ˆc. The corrsponing planes are given in

a units of three numbers within round brackets. For example, the crystallographic

planes (110) and (100) are give in figure 2.

Figure 3: Photo of a model of GaAs crystal. The corresponding crystallographic directions

are the vertical directions. Original picture taken from [30].

.2 GaAs wafers

In the zinkblende strcutures different atoms can occupy neighboring position in the

crystal pattern. For example in GaAs, each Ga-atom is surrounded by 4 As atoms

which are positioned at the edges of a regular tetrahedral. Figure 3 shows photos

of a model of the GaAs pattern. The distribution of the Ga and As atoms in these

models revels very important properties of the crystal. Suppose that the crystal is

cleaved in the [111] direction. The resulting (111) layers would have Ga-atoms on the

surface whereas the (¯

1¯

1) have As-atoms on the surface. The chemical properties of

these two layers would then differ and are expected to respond in different manners

to processing chemicals. The same argument is valid for layer planes in the [100]

Figure 4: Crystallographic directions in a GaAs wafer. OF: orientation flat, IF: identification

flat. Picture taken from [30].

direction where the Ga-atoms and As-atoms are positioned alternatively over each

other in equidistant distance. The distance between atomic-layers are largest in the

[111] direction. These layers are hold together with only one binding per atom. The

115 .3. Optical properties of GaAs

same applies to atoms in the perpendicular direction to the [110] direction. However,

the binding in the [110] direction is weaker as in the [111] direction. This is why

the (110) crystallographic plane is considered as a natural cleaving plane [30]. As a

direct advantage of this property, a GaAs wafer can be cleaved along this plane and

the cleaved facets form perfect mirrors for edge-emitting semiconductor laser diodes.

GaAs wafers are available in diameters of 2, 3, 4, and even 6-inches. A typical wafer

surface is the (101) layer. The crystallographic direction in a GaAs wafer are typically

(in Europe and Japan) given in the form in figure 4.

.3 Optical properties of GaAs

.3.1 Energy band gap and absorption in GaAs

The latice parameters of AlAs and GaAs are very close to each other as shown by

figure 5. This makes AlxGa1−xAs at any value of x(0 <x<1) lattice-matched to

GaAs.

Figure 5: Energy bandgap and lattice constants of III-V semiconductors at 300 K.

.3.2 Refractive index of AlxGa1−xAs

According to [62], below the direct band gap edge the real part of the refractive index

¯nof a zinkblende material (such as AlxGa1−xAs) can be expressed due to a simplified

interband-transition model as:

¯n(λ) = rA0f(χ)+1/2 [E0/(E0+ ∆0)]3/2f(χSO)+B0

with:

f(χ) = χ−22−(1 + χ)1/2−(1 −χ)1/2

χ=~ω/E0

χSO =~ω/ (E0+ ∆0)A0and B0are constants. They are experimentally found to be

written as A0= 6.3−19.0x, and B0= 9.4−10.2x.

E0and E0+ ∆0are critical point energies (see section VI in [62]).

116 .3. Optical properties of GaAs

The refractive indices of GaAs and AlxGa1−xAs are given in figure 7.

Figure 6: Fundamental absorption coefficient and absorption coefficients in AlGaAs (left),

and in n-doped GaAs (right). Original picture taken from [30].

Figure 7: Refractive index of GaAs and AlGaAs a function of the wavelength. Picture taken

from [30].

B: List of measured chips using the

FP method

Table 1: List of chips measured to determine the phase modulation efficiency and propagation

losses of phase modulators using the Fabry-Perot method. ∗chip number is determined by

TFRRDD (TF: test field, RR: bar, DD: diode). Examples: (single chip 010212: TF=1,

RR=2, DD=12), (012(a,b)02: two different chips at fractional lengths of one bar: RR=2,

fractions a and b), (01(03,04)02): two different chips with diode number 2 at different bars 3

and 4), (012(a+b)02 chip on more than one section in one bar), (0102(01+02) chip extends

at two diodes from the same bar).

Process Wafer ∗chip λridge width length Mount coating

ID TTRRDD µmµm mm type

Z1 6105

D2043-2 016a05 780 2 1 C-mount no

D2043-2 016d05 780 2 1C-mount no

D2043-3 016d15 780 4 1 C-mount no

D2043-3 015(b+c)04 780 2 2 C-mount no

D2043-2 017(b+c)05 780 2 2 C-mount no

D2043-3 017(a+d)15 780 4 2 C-mount no

D2043-2 016(a+b+c+d)03 780 2 4 C-mount no

D2043-3 016(a+b+c+d)04 780 2 4 C-mount no

D2043-3 016(a+b+c+d)15 780 4 4 C-mount no

Z1 6520 D2231-3 0001(01,02,03,04,05) 780 2.2 2 C-mount no

D2231-4 0001(01,02,03,04,05) 780 2.2 2 C-mount no

Z1 6894

C3059-3 0201(17,18,19,20) 1064 3 2 C-mount no

C3059-3 0202(17,18) 1064 3 2 C-mount no

C3059-3 0203(17,18,19,20) 1064 3 4 C-mount no

C3059-3 0204(17,18) 1064 3 4 C-mount no

C3062-3 0201(17,18,19,20) 1064 3 4 C-mount no

C3062-3 0202(17,18) 1064 3 2 C-mount no

C3062-3 0205(17,18,19,20) 1064 3 4 C-mount no

C3062-3 0206(17,18) 1064 3 4 C-mount no

117

List of Publications

The following scientific publications have been prepared in connection with this thesis:

Articles

Bassem Arar, Hans Wenzel, Reiner Güther, Olaf Brox, Harendra J. Fernando, Andre

Maaßdorf, Andreas Wicht, Achim Peters, Markus Weyers, Götz Erbert, and Gün-

ther Tränkle,"Double heterostructure ridge-waveguide GaAs/AlGaAs phase modulator

for 780 nm lasers", Appl. Phys. B, vol. 116, pp. 175-181 (2014).

Bassem Arar, Max Schiemangk, Hans Wenzel, Olaf Brox, Andreas Wicht, Achim Pe-

ters, and Günther Tränkle, "Method for in-depth characterization of electro-optic phase

modulators", Appl. Opt. vol. 56, pp. 1246-1252 (2017).

Conference Contributions

Bassem Arar, Harendra Fernando, Olaf Brox, Andre Maaßdorf, Andreas Wicht, Achim

Peters, Markus Weyers, Götz Erbert, and Günther Tränkle, "Double Heterostructure

AlGaAs/ GaAs W shaped Waveguide MZI Modulator for 780 nm Lasers", Confer-

ence on Lasers and Electro-Optics (CLEO), OSA, ISBN: 978-1-55752-999-2, paper

JW2A.51, San Jose (2014).

Bassem Arar , Hans Wenzel, Olaf Brox, Andre Maaßdorf, Andreas Wicht, Markus Wey-

ers, Götz Erbert, and Günther Tränkle, "GaAs-based Phase Modulator for Laser Radi-

ation at 1070 nm", Conference on Lasers and Electro-Optics/Europe (CLEO/Europe-

EQEC 2015), ISBN: 978-1-4673-7475-0, paper CD-P-40, Munich (2015).

Bassem Arar , Hans Wenzel, Olaf Brox, Andre Maaßdorf, Andreas Wicht, Markus

Weyers, Achim Peters, Götz Erbert, and Günther Tränkle, "GaAs-based Phase Modu-

lators for Frequency Stabilization of Diode Lasers at 780 nm and 1070 nm", Semicon-

ductor and Integrated Opto-Electronics (SIOE) conference, Cardiff (2015).

119

120

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