Document [original]

Integrated Silicon Photonic

Two-Dimensional Grating Couplers

vorgelegt von

M. Sc.

Galina Doneva Georgieva

an 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.-Ing. Ronald Freund

Technische Universität Berlin,

Fraunhofer Heinrich-Hertz-Institut

Gutachter: Prof. Dr.-Ing. Dr. h. c. Klaus Petermann

Technische Universität Berlin

Prof. Dr.-Ing. Manfred Berroth

Universität Stuttgart

Prof. Dr. Ir. Wim Bogaerts

Universiteit Gent - IMEC

Tag der wissenschaftlichen Aussprache: 15. Mai 2023

Berlin 2023

Fakultät IV - Elektrotechnik und Informatik

Institut für Hochfrequenz- und Halbleiter-Systemtechnologien, Fachgebiet Hochfrequenztechnik-Photonik

Integrated Silicon Photonic

Two-Dimensional Grating Couplers

Galina Doneva Georgieva, M. Sc.

Dissertation

to achieve the academic degree

Doktor der Ingenieurwissenschaften (Dr.-Ing.)

Submitted on: 6th December 2022

Defended on: 15th May 2023

Contents

Contents i

Abstract iv

Zusammenfassung vi

List of Publications viii

List of Figures xi

List of Tables xvi

List of Acronyms xvii

List of Deﬁnitions xix

1 Introduction 1

1.1 Optical Interfacing in Silicon Photonics . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Fundamental Issues of Two-Dimensional Grating Couplers . . . . . . . . . . 7

1.3 ObjectivesoftheThesis............................... 9

References.......................................... 13

2 Methods for Analysis of Two-Dimensional Grating Couplers 23

2.1 Numerical Simulation and Evaluation . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 General Problem Description . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.2.1 General Features of Numerical Approaches . . . . . . . . . 27

2.1.2.2 Speciﬁc Numerical Methods for Photonic Applications . . . 29

2.1.2.3 Generalized Numerical Methods in the Frequency Domain 30

2.1.2.4 Generalized Numerical Methods in the Time Domain . . . . 33

2.1.2.5 Open Boundary Conditions . . . . . . . . . . . . . . . . . . . 39

2.1.2.6 Summary of Numerical Errors . . . . . . . . . . . . . . . . . 40

CONTENTS

2.1.3 Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.1.3.1 Modeling.............................. 44

2.1.3.2 Deﬁnition of Excitation Sources . . . . . . . . . . . . . . . . . 46

2.1.3.3 Simulation Initialization . . . . . . . . . . . . . . . . . . . . . 49

2.1.3.4 FieldExport ............................ 49

2.1.3.5 Initial Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.1.4 Methods for Parameter Evaluation . . . . . . . . . . . . . . . . . . . . 53

2.1.4.1 Coupling Angles Determination from a 2D Radiation Pattern 53

2.1.4.2 Coupling Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.4.3 Polarizations’ Splitting and Orthogonality . . . . . . . . . . . 60

2.2 Measurement Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.2.1 BasicSetup.................................. 64

2.2.2 Power Normalization in Loss Measurements . . . . . . . . . . . . . . 65

2.2.3 A Measurement Technique for the Polarization Split Ratio . . . . . . 66

2.2.4 WaferStatistics................................ 68

References.......................................... 71

3 Fabrication Platform 76

3.1 BipolarCMOS..................................... 76

3.1.1 CMOS vs. Bipolar Technology . . . . . . . . . . . . . . . . . . . . . . . 76

3.1.2 SiGe:C Heterojunction Bipolar Transistor . . . . . . . . . . . . . . . . 78

3.2 Monolithic Photonic BiCMOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3 BiCMOS With BEOL Photonic Layers . . . . . . . . . . . . . . . . . . . . . . . . 81

References.......................................... 84

Investigation of Fundamental Physical Effects in Two-Dimensional Grating

Couplers 88

4.1 Diffraction....................................... 88

4.1.1 Derivation of a 2D Diffraction Condition . . . . . . . . . . . . . . . . . 89

4.1.1.1 PhaseCondition.......................... 89

4.1.1.2 Sheared 2D GC Design Parameters . . . . . . . . . . . . . . 92

4.1.2 Numerical Veriﬁcation of the 2D Diffraction Condition . . . . . . . . . 92

4.2 In-PlaneScattering.................................. 97

4.2.1 Analytical Investigation of In-Plane Scattering . . . . . . . . . . . . . . 98

4.2.2 Cross-Polarization..............................103

4.2.2.1 Split Ratio and Coupling Eﬃciency Limitations . . . . . . . . 104

4.2.2.2 Polarizations’ Non-Orthogonality . . . . . . . . . . . . . . . . 112

4.2.2.3 Polarization-Dependent Loss . . . . . . . . . . . . . . . . . . 115

4.2.3 Higher-OrderModes ............................117

References..........................................124

5 Design Optimization and Characterization 127

5.1 Improvement of the Out-Coupled Power Eﬃciency . . . . . . . . . . . . . . . 127

CONTENTS

5.2 Optimized Polarization Handling . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.2.1 Segmented Two-Dimensional Grating Couplers . . . . . . . . . . . . . 131

5.2.2

Two-Dimensional Grating Couplers with Elongated and Individually

Oriented Perturbing Elements . . . . . . . . . . . . . . . . . . . . . . . 139

5.2.2.1 Linear Two-Dimensional Grating Couplers . . . . . . . . . . 142

5.2.2.2 Focusing Two-Dimensional Grating Couplers . . . . . . . . . 155

5.2.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.2.3.1 Results in C-band . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.2.3.2 Results in O-band . . . . . . . . . . . . . . . . . . . . . . . . . 163

References..........................................166

6 Conclusions and Outlook 170

Danksagung 174

iii

Fakultät IV - Elektrotechnik und Informatik

Institut für Hochfrequenz- und Halbleiter-Systemtechnologien, Fachgebiet Hochfrequenztechnik-Photonik

Abstract

Silicon (Si) photonics has proven its importance in the recent years for enabling the integra-

tion of photonic components, using available complementary metal oxide semiconductor

(CMOS) fabrication ﬂows. The simultaneous realization of electronics and photonics in

the same manufacturing platform allows for the accomplishment of high-performance

electronic-photonic integrated circuits (EPICs). Their adoption in optical communication

systems meets one essential challenge - the optical interfacing to the transmission links,

based on single-mode ﬁbers (SMFs). On the one hand, there is a large mode-ﬁeld mismatch

between the tiny on-chip Si waveguide with an area of about 0.1 µm

and the external

SMF with a core diameter of around 10 µm. In addition, there is a severe discrepancy

between the polarization natures of both optical modes. While the external SMF supports

a mode with two orthogonal, degenerated polarization states, the integrated Si waveguides

and remaining components are mostly designed for the fundamental transverse electric

(TE) polarization. A Si photonic coupling interface is thus required to deliver polarization

handling capabilities as well. A favored component for a simple interfacing is the diffrac-

tion grating coupler (GC). In its most basic, one-dimensional implementation, a silicon

waveguide mode is laterally enlarged by a taper structure, feeding a periodically etched

grating. The latter deﬂects the light under a small angle with respect to the chip surface’s

normal. The external SMF is tilted under the same angle and placed at a small distance

above the grating. The coupling device in this form allows for mode matching, but not

for polarization manipulation. For that reason, its modiﬁed form - the two-dimensional

grating coupler (2D GC) - is necessary. The latter combines two one-dimensional GCs in

such a manner, that both orthogonal SMF polarizations are coupled into two separate Si

waveguides, supporting the fundamental TE polarization, and vice versa. This time, the

SMF is not only tilted with respect to the vertical, but also oriented towards the grating’s

symmetry axis between the feeding waveguides. Accordingly, the grating should be able

not only to diffract both Si modes in a nearly vertical direction, but also to direct them

along the symmetry plane.

Although 2D GCs have been known for many years, their eﬃcient and polarization in-

dependent design remained very challenging. To overcome this problem, a good basic

understanding of the fundamental physical effects in such structures is necessary. This

work is entirely dedicated to the investigation and systematization of the physical prop-

erties of 2D GCs. The gained theoretical base is used for the 2D GCs’ optimization in

different aspects. Starting with the theoretical description of the diffraction mechanisms

in two dimensions, the interplay between the grating’s geometry and coupling angles is

demonstrated. Furthermore, an in-depth characterization of the polarization behavior

Fakultät IV - Elektrotechnik und Informatik

Institut für Hochfrequenz- und Halbleiter-Systemtechnologien, Fachgebiet Hochfrequenztechnik-Photonik

of 2D GCs is presented. New aspects, such as the polarizations’ conversion, crosstalk

and non-orthogonality are analyzed, tracking their origins back to the in-plane scattering,

resulting from the ﬁnite size of the grating’s perturbing elements with respect to the Si

mode. The importance of this physical process in 2D GCs has been underrated until now,

and is revealed here as the most determining limitation for the optimal performance of 2D

GCs. After this conclusion, several methods for the in-plane scattering’s suppression are

considered. The feasibility of the proposed approaches is investigated by both numerical

simulations and wafer-level experiments. In the end, a novel optimization technique is

demonstrated, which allows for the design of eﬃcient 2D GCs with a low polarization

crosstalk, low non-orthogonality and low polarization-dependent loss.

Fakultät IV - Elektrotechnik und Informatik

Institut für Hochfrequenz- und Halbleiter-Systemtechnologien, Fachgebiet Hochfrequenztechnik-Photonik

Zusammenfassung

Die Integration von photonischen Komponenten auf Basis von vorhandenen CMOS Herstel-

lungsprozessen hat in den letzten Jahren die Bedeutung des Siliziumphotonik-Forschungs-

gebietes etabliert. Die Realisierung von photonischen und elektronischen Komponenten

auf einer gemeinsamen Plattform ermöglicht den Entwurf von leistungsstarken elektronisch-

photonischen Schaltkreisen. Ihre Anwendung in optischen Kommunikationssystemen trifft

eine essenzielle Herausforderung - die optische Ankopplung zu den Übertragungslinks

basierend auf Standard-Einmodenfaser. Auf einer Seite existiert eine große Modenfehl-

anpassung zwischen dem kleinen, chip-integrierten Siliziumwellenleiter mit einer Quer-

schnittsﬂäche von 0.1 µm

und der externen Einmodenfaser mit einem Kerndurchmesser

von etwa 10µm. Darüber hinaus gibt es eine bedeutende Diskrepanz bei der Polarisati-

onsführung in beiden Wellenleitern. Während die externe Einmodenfaser eine Mode mit

zwei orthogonalen, entarteten Polarisationszuständen unterstützt, sind die integrierten

Siliziumkomponenten stark polarisationsabhängig, sodass der Siliziumwellenleiter nur die

fundamentale transversal-elektrische (TE) Polarisation führt. Eine siliziumphotonische Kop-

pelstelle soll in der Lage sein, die Polarisation einer Eingangswelle zu manipulieren. Eine

bevorzugte Ausführung solch einer Schnittstelle ist der Gitterkoppler. In seiner einfachs-

ten, eindimensionalen (1D) Form wird zunächst der Siliziumwellenleiter lateral erweitert

und anschließend einer periodisch geätzten Gitterstruktur zugeführt. Letztere beugt das

einfallende Feld, sodass das Licht unter einem nahezu senkrechten Winkel aus dem Chip

ausgekoppelt wird. Dementsprechend wird die Faser unter demselben Winkel geneigt

und auf unmittelbarer Distanz zur Chipoberﬂäche angebracht. Diese Form von Kopplung

ermöglicht eine Größenanpassung der beiden Moden, aber nicht den Umgang mit deren

Polarisationen. Aus diesem Grund ist eine modiﬁzierte Form – der zweidimensionale (2D)

Gitterkoppler – notwendig. Diese Struktur kombiniert zwei eindimensionale Gitterkoppler

in solcher Art, dass beide orthogonale Polarisationen der externen Faser in zwei getrennte

Siliziumwellenleiter mit der Grund-TE-Polarisation angeregt werden. Um das zu ermög-

lichen, wird die Faser dieses Mal nicht nur unter einem kleinen Winkel geneigt, sondern

auch entlang der Gittersymmetrieachse ausgerichtet. Demnach soll das Koppelgitter nicht

nur das Licht unter einem nahezu vertikalen Winkel beugen, sondern auch das Licht aus

beiden Wellenleitern entlang der Symmetrieachse orientieren.

Obwohl 2D Gitterkoppler längst bekannte Koppelschnittstellen sind, ist ihren eﬃzienten

und polarisationsunabhängigen Entwurf sehr anspruchsvoll. Um dieses Problem zu be-

wältigen, ist ein gutes Verständnis der grundlegenden physikalischen Effekte in solchen

Strukturen notwendig. Die vorliegende Arbeit ist vollkommen gewidmet, die physikalischen

Eigenschaften von 2D Gitterkopplern gründlich zu untersuchen und klar zu systematisieren.

Fakultät IV - Elektrotechnik und Informatik

Institut für Hochfrequenz- und Halbleiter-Systemtechnologien, Fachgebiet Hochfrequenztechnik-Photonik

Das erworbene theoretische Grundverständnis ermöglicht die anschließende Optimie-

rung von Gitterkopplern in verschiedene Aspekte. Als Erstes wird die Beschreibung der

Beugungsmechanismen in zwei Dimensionen ausgeführt und das Zusammenspiel verschie-

dener Design-Parametren wird veranschaulicht. Weiterhin wird eine gründliche Charakte-

risierung des Polarisationsverhaltens in 2D Gitterkopplern dargelegt. Neue Aspekte wie

die Polarisationswandlung, -nebensprechen und -Nichtorthogonalität werden analysiert.

Als physikalische Ursprung dieser limitierenden Größen wird die Streuung in der Gitter-

ebene erkannt, die aufgrund der endlichen Größe der Gitterstörelemente im Vergleich

zur Siliziumwellenleitermode entsteht. Die Bedeutung der Streuung in der Gitterebene

als physikalischer Prozess in 2D Koppelgittern wurde bisher unterschätzt. In dieser Arbeit

wird die Streuung in der Gitterebene als die entscheidende Limitierung für das optimale

Strukturverhalten identiﬁziert. Nach dieser Erkenntnis werden verschiedene Methoden

zur Unterdrückung der ungewünschten Streuung untersucht. Die Umsetzbarkeit der vor-

geschlagenen Optimiertechniken wird durch numerische Simulationen und Experimente

auf Wafer-Ebene analysiert. Am Ende der Arbeit wird eine neuartige Optimierungsstrate-

gie vorgestellt, die es erlaubt, eﬃziente 2D Gitterkoppler mit niedrigen Polarisationsne-

bensprechen, geringe Nichtorthogonalität und kleine polarisationsabhängige Verluste zu

entwerfen.

List of Publications

Journal Articles

•G. Georgieva

, M. Sena, P. M. Seiler, K. Petermann, J. Fischer, L. Zimmermann, “Penalties

From 2D Grating Coupler Induced Polarization Crosstalk in Silicon Photonic Coherent

Transceivers,” in IEEE Photonics Journal, vol. 14, no. 5, pp. 1-11, Art no. 6653011, 2022.

doi: 10.1109/JPHOT.2022.3206619

•G. Georgieva

, C. Mai, P. M. Seiler, A. Peczek, L. Zimmermann, “Dual-Polarization Multi-

plexing Amorphous Si:H Grating Couplers for Silicon Photonic Transmitters in the

Photonic BiCMOS Backend of Line,” in Frontiers of Optoelectronics, vol. 15, Art. no. 13,

2022.

doi: 10.1007/s12200-022-00005-8

•G. Georgieva

, K. Voigt, P. M. Seiler, C. Mai, K. Petermann, L. Zimmermann, “A Physical

Origin of Cross-Polarization and Higher-Order Modes in Two-Dimensional (2D) Grat-

ing Couplers and the Related Device Performance Limitations,” in Journal of Physics:

Photonics, vol. 3, p. 035002, 2021.

doi: 10.1088/2515-7647/abf942

•G. Georgieva

, K. Voigt, C. Mai, P. M. Seiler, K. Petermann, L. Zimmermann, “Cross-

Polarization Effects in Sheared 2D Grating Couplers in a Photonic BiCMOS Technol-

ogy,” in Japanese Journal of Applied Physics, vol. 59, no. SO, p. SOOB03, 2020.

doi: 10.35848/1347-4065/ab8e21

•G. Georgieva

, K. Voigt, A. Peczek, C. Mai, L. Zimmermann, “Design and Performance

Analysis of Integrated Focusing Grating Couplers for the Transverse-Magnetic TM

Mode in a Photonic BiCMOS Technology,” in Journal of the European Optical Society-

Rapid Publications, vol. 16, Art. no. 7, 2020.

doi: 10.1186/s41476-020-00129-4

•

P. M. Seiler,

G. Georgieva

, A. Peczek, M. Oberon, C. Mai, S. Lischke, A. Malignaggi,

L. Zimmermann, “Monolithically Integrated O-Band Coherent ROSA Featuring 2D

viii

List of Publications

Grating Couplers for Self-Homodyne Intra Data Center Links,” in IEEE Photonics Journal,

vol. 15, no. 3, Art no. 6601306, 2023.

doi: 10.1109/JPHOT.2023.3272476

•

L. Zimmermann, P. M. Seiler, C. Mai,

G. Georgieva

, “Toward 2D Grating Coupler

Enabled O-Band Coherent Links Based on SiGe Photonic Electronic Technology”, in

Japanese Journal of Applied Physics vol. 62, no. SC, Art. no. SC0807, 2023.

doi: 10.35848/1347-4065/acb4fe

•

P. M. Seiler, K. Voigt, A. Peczek,

G. Georgieva

, S. Lischke, A. Malignaggi, L. Zimmermann,

“Multiband Silicon Photonic ePIC Coherent Receiver for 64 GBd QPSK,” in Journal of

Lightwave Technology, vol. 40, no. 10, pp. 3331-3337, 2022.

doi: 10.1109/JLT.2022.3158423

•

P. M. Seiler,

G. Georgieva

, G. Winzer, A. Peczek, K. Voigt, S. Lischke, A. Fatemi, L.

Zimmermann, “Toward Coherent O-Band Data Center Interconnects, ” in Frontiers of

Optoelectronics, vol. 14, no. 4, 2021.

doi: 10.1007/s12200-021-1242-0

•

S. Lischke, D. Knoll, C. Mai, K. Voigt, A. Hesse,

G. Georgieva

, A. Peczek, L. Zimmermann,

“Directly Silicon Nitride Waveguide Coupled Ge Photodiode for Non-SOI PIC and Epic

Platforms,” in ECS Transactions, vol 98, no. 5, 2020.

doi: 10.1149/09805.0315ecst

Conference Articles

•G. Georgieva

, P. M. Seiler, C. Mai, A. Peczek, K. Petermann, L. Zimmermann, “A

Polarization-Independent Zig-Zag-Tilted Ovals Grating Coupler in a 0.25 µm Photonic

BiCMOS Technology,” in The 2022 European Conference on Optical Communication

(ECOC), paper Mo3F.5, 2022.

•G. Georgieva

, P. M. Seiler, C. Mai, K. Petermann, L. Zimmermann, “2D Grating Coupler

Induced Polarization Crosstalk in Coherent Transceivers for Next Generation Data

Center Interconnects,” in Optical Fiber Communication Conference (OFC) 2021, paper

W1C.4, 2021.

doi: 10.1364/OFC.2021.W1C.4

•G. Georgieva

, K. Voigt, C. Mai, K. Petermann, L. Zimmermann, “Cross-Polarization

Effects in Sheared Two-Dimensional Grating Couplers for Silicon Photonics,” in 2019

24th Microoptics Conference (MOC), pp. 58-59, 2019.

doi: 10.23919/MOC46630.2019.8982796

•G. Georgieva

, K. Voigt, L. Zimmermann, “Focusing 1D Silicon Photonic Grating Coupler

in Photonic BiCMOS Technology for the Excitation of the Fundamental TM Mode,”

in 2019 PhotonIcs & Electromagnetics Research Symposium - Spring (PIERS-Spring), pp.

List of Publications

1667-1673, 2019.

doi: 10.1109/PIERS-Spring46901.2019.9017259

•G. Georgieva

, K. Petermann, “Analytical and Numerical Investigation of Silicon Pho-

tonic 2D Grating Couplers with a Waveguide-to-Grating Shear Angle,” in 2018 Progress

in Electromagnetics Research Symposium (PIERS-Toyama), 2018.

doi: 10.23919/PIERS.2018.8598162

•

F. Goetz, S. Lischke,

G. Georgieva

, A. Peczek, L. Zimmermann, “Toward FEOL Inte-

gration of SiN Waveguides into a Photonic BiCMOS Process,” in 2023 IEEE Silicon

Photonics Conference (SiPhotonics), 2023.

doi: 10.1109/SiPhotonics55903.2023.10141964

•

D. Steckler, S. Lischke, A. Kroh, A. Peczek,

G. Georgieva

, L. Zimmermann, “Germanium

Fin Photodiode with 3dB-Bandwidth >110 GHz and High L-Band Responsivity,” in

2023 IEEE Silicon Photonics Conference (SiPhotonics), 2023.

doi: 10.1109/SiPhotonics55903.2023.10141946

•

L. Zimmermann,

G. Georgieva

, P. M. Seiler, “Advances in Monolithic High-Speed SiGe

Photonic Electronic System Integration”, in 2022 International Conference on Solid State

Devices and Materials, 2022.

•

S. Lischke, D. Knoll, C. Mai, A. Hesse,

G. Georgieva

, A. Peczek, A. Kroh, M. Lisker, D.

Schmidt, M. Fraschke, H. Richter, A. Krüger, U. Saarow, P. Heinrich, G. Winzer, K.

Schulz, P. Kulse, A. Trusch, L. Zimmermann, “Silicon Nitride Waveguide Coupled 67+

GHz Ge Photodiode for Non-SOI PIC and ePIC Platforms,” in 2019 IEEE International

Electron Devices Meeting (IEDM), pp. 33.2.1-33.2.4, 2019.

doi: 10.1109/IEDM19573.2019.8993651

Patent Applications

G. Georgieva, “Optical Element”:

•

European Patent EP 4 145 200 A1, Application Nr. EP21194821, submitted: Sept. 3,

2021, published: March 08, 2023;

•

US Patent US2023/0085271 A1, Application Nr. 17/902,023, submitted: Sept. 2, 2022,

published: March 16, 2023.

List of Figures

1.1 A block diagram of a general optical communication system. . . . . . . . . . 2

1.2

(a) A schematic comparison between a single-mode-ﬁber and a Si nanowire.

(b) Horizontal coupling with a spot size converter. (c) Nearly vertical coupling

with a diffraction grating coupler. . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1

A photograph of a monolithically integrated dual-polarization coherent re-

ceiver. ......................................... 24

2.2 A description of a 2D grating coupler model. . . . . . . . . . . . . . . . . . . . 25

2.3

An illustration of the difference between the numerical models of a 1D and

2Dgratingcoupler................................... 26

2.4

An exemplary representation of the Yee’s cell, consisting of two staggered

grids........................................... 35

2.5

An exemplary space-time representation of a 1D wave propagation accord-

ingtotheleapfrogscheme.............................. 36

2.6

An exemplary representation of the staggered grid used by the ﬁnite-integration-

technique........................................ 37

2.7 An exemplary model for a numerical convergence study. . . . . . . . . . . . 40

2.8

The balance parameter of a simpliﬁed model for a different number of grid

cells per minimal wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9

The balance parameter of a simpliﬁed model for different levels of the

perfectly matched layer (PML) reﬂection. . . . . . . . . . . . . . . . . . . . . . 43

2.10

An illustration of the components relevant for a complete 2D grating coupler

(GC)model. ...................................... 44

2.11

An illustration of the calculation of the diffracting elements’ positions in a

2Dfocusingarray. .................................. 46

2.12 An exemplary Gaussian-sine excitation signal for simulations in O-band. . . 47

2.13

A decomposition of a full model into a source and imprint project (an example

of a 2D focusing grating coupler). . . . . . . . . . . . . . . . . . . . . . . . . . 48

LIST OF FIGURES

2.14

An illustration of the transformations, necessary to construct a tilted plane

for results evaluation, representing the ﬁxed position of a single-mode ﬁber

(SMF)........................................... 49

2.15 An illustration of the parameters of 2D grating couplers (2D GCs). . . . . . . 51

2.16

An illustration of the adaption procedure for the determination of an appro-

priate 2D grating coupler geometry for desired coupling angles. . . . . . . . 52

2.17

A schematic of the typical power distribution in a 2D grating coupler (2D GC).

2.18

A buried oxide (BOX) dependence of the out-coupled power of 1D grating

couplers (1D GCs), designed for C- or O-band. . . . . . . . . . . . . . . . . . . 55

2.19 A schematic with deﬁnitions used for the derivation of an overlap integral. . 56

2.20

A schematic for the illustration of two parameters, related to the polarization

splitting in 2D grating couplers (2D GCs). . . . . . . . . . . . . . . . . . . . . . 59

2.21

An exemplary representation of two non-orthogonal polarization states on

the Poincaré sphere (in a 2D cross-section). . . . . . . . . . . . . . . . . . . . 61

2.22 An exemplary setup for the characterization of passive on-chip components. 64

2.23 A schematic of a device for the measurement of a polarization split ratio. . 66

2.24 An exemplary representation of 9 chips on a wafer. . . . . . . . . . . . . . . 69

3.1 A schematic cross-section of a n-MOSFET and a npn-BJT. . . . . . . . . . . . 77

3.2

An illustration of the photonic modules’ integration in a 0.25 µm electronic

BiCMOSﬂow. ..................................... 79

3.3 A schematic cross-section of a photonic BiCMOS platform. . . . . . . . . . . 80

3.4 A schematic representation of a 3D photonic BiCMOS platform. . . . . . . . 82

4.1

An illustration of a sheared 2D grating coupler (2D GC) with a rhombus-

shaped grating area, placed in the plane (x,y,z=0). .............. 89

4.2 Two types of sheared 2D grating couplers (2D GCs) investigated in this work. 93

4.3

A comparison of 2D grating couplers (2D GCs) without and with a shear angle.

4.4

A comparison of 2D grating couplers (2D GCs) of Type I with different shear

angles. ......................................... 95

4.5

A comparison of 2D grating couplers (2D GCs) of Type II with different shear

angles. ......................................... 96

4.6

A comparison of the mode ﬁeld overlap of a sheared 2D grating coupler (2D

GC) when the ﬁber tilt angle corresponds to the design coupling angle or

deviates by ±3◦. ................................... 97

4.7 A schematic representation of two scattering problems. . . . . . . . . . . . . 99

4.8

An analytically calculated electric ﬁeld distribution for the scattering problem

of a dielectric cylinder array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.9

A comparison of the target- and cross-polarization of different sheared 2D

gratingcouplers(2DGCs)...............................104

4.10

An electric ﬁeld distribution and a power loss in percent caused by ﬁelds,

propagating further in a 2D grating coupler. . . . . . . . . . . . . . . . . . . . 106

xii

LIST OF FIGURES

4.11 Exemplary measured interferometric curves for the characterization of dif-

ferent sheared 2D grating couplers (2D GCs) in terms of split ratio. . . . . . 107

4.12

An exemplary simulation of the optical signal-to-noise ratio (OSNR) penalty

for reaching a bit error ratio BER = 10

–3

, plotted over different 2D grating

coupler (2D GC) split ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.13

A test structure and a measurement setup used to investigate the polariza-

tion combining performance of a 2D grating coupler (2D GC). . . . . . . . . 111

4.14

Bit error ratio vs. optical signal-to-noise ratio (BER vs. OSNR) for a trans-

mission with a single active polarization and two active polarizations. A 2D

grating coupler (2D GC) is included at the transmitter-side. . . . . . . . . . . 112

4.15 A comparison of 2D grating couplers (2D GCs) for C- and O-band. . . . . . . 114

4.16

An illustration of different incident polarizations at a receiver-side 2D grating

couplers(2DGCs). ..................................115

4.17

Coupling eﬃciencies of a target- and cross-polarization in parallel with an

even- and odd-polarization for 2D grating couplers (2D GCs) with the same

geometry and of different types. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.18

A schematic representation of a 2D grating coupler (2D GC) with a Gaussian

beam excitation from top. Several ﬁeld components may be excited in a Si

waveguide due to grating in-plane scattering. . . . . . . . . . . . . . . . . . . 118

4.19

A camera picture of an exemplary device used for the experimental determi-

nation of the polarization state and the modal composition of ﬁelds, excited

by a 2D grating coupler (2D GC) in Si structures. . . . . . . . . . . . . . . . . 119

4.20

Exemplary measured coupling spectra of the modes TE

00,y

(target-polarization),

TE10,y, TE00,x, TE10,xfor different 2D grating couplers (2D GCs). . . . . . . . . 121

4.21

A coupling spectrum of the output 1D focusing grating coupler (1D FGC),

measured at 10◦....................................122

5.1

An a-Si:H 2D grating coupler (2D GC) design – a dependence of the nor-

malized out-coupled power on the SiO

thickness separation from the Si

substrate. .......................................129

5.2

An a-Si:H 2D grating coupler (2D GC) for C-band at the TopMetal1 level. Bulk

Si and Metal3 are compared as grating back-reﬂectors. . . . . . . . . . . . . 130

5.3

An a-Si:H 2D grating coupler (2D GC) for O-band at the TopMetal1 level. Bulk

Si and Metal3 are compared as grating back-reﬂectors. . . . . . . . . . . . . 131

5.4

A segmented a-Si:H 2D grating coupler (2D GC) for C-band, comprising 4 or

6segments.......................................132

5.5

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for C-band in a single-port simulation. The relevant parameters

are the target- and cross-polarization per single waveguide port. . . . . . . 134

5.6

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for C-band in a single-port simulation. The relevant parameters are

the polarization split ratio per single waveguide port and the polarizations’

angular relation and crosstalk between the signals from both waveguides. . 135

xiii

LIST OF FIGURES

5.7

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for C-band in a dual-port simulation. The relevant parameters are

the even- and odd-polarization, resulting from in- or anti-phase superposi-

tion of the signals from both GC waveguides. . . . . . . . . . . . . . . . . . . 136

5.8

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for O-band in a single-port simulation. The relevant parameters

are the target- and cross-polarization per single waveguide port. . . . . . . 137

5.9

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for O-band in a single-port simulation. The relevant parameters are

the polarization split ratio per single waveguide port and the polarizations’

angular relation and crosstalk between the signals from both waveguides. . 137

5.10

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for O-band in a dual-port simulation. The relevant parameters are

the even- and odd-polarization, resulting from in- or anti-phase superposi-

tion of the signals from both GC waveguides. . . . . . . . . . . . . . . . . . . 138

5.11

A schematic of an optimized 2D grating coupler (2D GC), comprising elon-

gated perturbing elements with different orientations. . . . . . . . . . . . . . 140

5.12

A comparison between a reference and an optimized design of a C-band

2D grating coupler (2D GC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.13

Even vs. odd polarizations of a reference sheared 2D grating coupler (2D

GC)ofTypeIIforC-Band. ..............................143

5.14

Even vs. odd polarizations of an optimized sheared 2D grating coupler (2D

GC)ofTypeIIforC-Band. ..............................144

5.15

The performance of a sheared 2D grating coupler (2D GC) of Type I with the

same perturbing elements’ geometry and periodicity as the optimized 2D

GCofTypeII. .....................................145

5.16

A performance variation of the optimized 2D grating coupler (2D GC) in

C-band, when the perturbing elements’ size varies in direction of both axes. 146

5.17

A performance variation of the optimized 2D grating coupler (2D GC) in

C-band, when the perturbing elements’ major axis varies. . . . . . . . . . . . 146

5.18

A performance variation of the optimized 2D grating coupler (2D GC) in

C-band, when the perturbing elements’ minor axis varies. . . . . . . . . . . . 147

5.19

An a-Si:H 2D grating coupler (2D GC) for C-band with a Metal3 back-reﬂector -

an optimized design with individually oriented elongated perturbing elements.

148

5.20

A schematic of an optimized 2D grating coupler (2D GC) design, comprising

stretched perturbing elements, combined with circular perturbing elements.149

5.21

A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for C-band

with a Metal3 back-reﬂector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.22

A comparison between a reference and an optimized design of a O-band

2D grating coupler (2D GC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.23

Even vs. odd polarizations of a reference sheared 2D grating coupler (2D

GC)ofTypeIIforO-Band...............................152

xiv

LIST OF FIGURES

5.24

Even vs. odd polarizations of an optimized sheared 2D grating coupler (2D

GC)ofTypeIIforO-Band...............................152

5.25

A performance variation of the optimized 2D grating coupler (2D GC) in

O-band, when the perturbing elements’ size varies in direction of both axes. 153

5.26

A performance variation of the optimized 2D grating coupler (2D GC) in

O-band, when the perturbing elements’ major axis varies. . . . . . . . . . . . 153

5.27

A performance variation of the optimized 2D grating coupler (2D GC) in

O-band, when the perturbing elements’ minor axis varies. . . . . . . . . . . . 154

5.28

A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for O-band

with a Metal3 back-reﬂector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.29

A description of the modeling procedure of 2D focusing grating couplers

(2DFGCs). .......................................156

5.30

2D focusing grating couplers (2D FGCs) for C-band: a comparison between a

reference design with circular perturbing elements and an optimized design

with elongated perturbing elements. . . . . . . . . . . . . . . . . . . . . . . . 157

5.31

2D focusing grating couplers (2D FGCs) for C-band: a comparison between a

reference design with circular perturbing elements and an optimized design

with elongated perturbing elements. . . . . . . . . . . . . . . . . . . . . . . . 158

5.32

A comparison of 2D focusing grating couplers (2D FGCs) for C-band with

different waveguide heights. Both designs comprise elongated perturbing

elements with the same dimensions. . . . . . . . . . . . . . . . . . . . . . . . 159

5.33

An optimized 2D grating coupler (2D GC) for C-band comprising a zig-zag-

tiltedovalsarray....................................161

5.34

Experimental results for an optimized C-band 2D grating coupler (2D GC)

comprising a zig-zag-tilted ovals array. . . . . . . . . . . . . . . . . . . . . . . 163

5.35

An experimental comparison of a reference O-band 2D grating coupler (2D

GC) with circular perturbing elements and an optimized O-band 2D GC with

zig-zag-tiltedovals...................................164

5.36

A comparison of SEM photographs of O-band zig-zag-tilted ovals 2D grating

couplers(2DGCs). ..................................165

List of Tables

4.1

Numerically estimated coupling angles for 2D grating couplers (2D GCs),

4.2

Numerically estimated split ratios different sheared 2D grating couplers (2D

GCs). ..........................................105

4.3

Experimentally estimated mean split ratios for different sheared 2D grating

couplers(2DGCs). ..................................109

4.4

A summary of the notations, used to assign a desired target-polarization with

a TE

modal component, resulting from diffraction and in-plane-scattering-

related polarizations/modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.5

2D grating coupler (2D GC) designs used for the investigation of their polar-

izations’ and modal on-chip excitation. . . . . . . . . . . . . . . . . . . . . . . 120

4.6

Mean extinction ratios (ERs)

±σ

between the target TE

00,y

and the other

considered polarizations/modes averaged over 9 chips. Different 2D grating

coupler (2D GC) designs are compared. . . . . . . . . . . . . . . . . . . . . . 122

5.1

Sheared a-Si:H 2D grating couplers (2D GCs) of Type II, designed for C- and

O-band. ........................................128

5.2

Geometric parameters of sheared, segmented C-band a-Si:H 2D grating

couplers (2D GCs) of Type II with a different number of segments. . . . . . . 133

5.3

Geometric parameters of a 2

◦

sheared, segmented O-band a-Si:H 2D GCs

ofTypeIIwith4sections. ..............................136

5.4

An experimental comparison of a reference O-band 2D grating coupler (2D

GC) with circular perturbing elements and a proposed O-band 2D GC with

zig-zag-tiltedovals...................................164

xvi

List of Acronyms

BEOL backend of line

BER bit error ratio

BiCMOS bipolar complementary metal

oxide semiconductor

BJT bipolar junction transistor

BOX buried oxide

CMOS complementary metal oxide

semiconductor

DCI data center interconnect

DBR distributed Bragg reﬂector

DoP degree of polarization

DSP digital signal processing

DUV deep ultra violet

EDFA erbium doped ﬁber ampliﬁer

EIC electronic integrated circuit

EPIC electronic-photonic integrated

circuit

ER extinction ratio

FDM ﬁnite differences method

FDTD ﬁnite differences time domain

FEM ﬁnite elements method

FEOL frontend of line

FGC focusing grating coupler

FIT ﬁnite integration technique

GC grating coupler

HBT

heterojunction bipolar transistor

IC integrated circuit

IM-DD intensity modulated direct

detection

LP linearly polarized

MFD mode ﬁeld diameter

MMI multi-mode interferometer

MOS metal oxide semiconductor

MOSFET metal oxide semiconductor ﬁeld

effect transistor

OSNR optical signal-to-noise ratio

PBS/C polarization beam

splitter/combiner

PDL polarization-dependent loss

PIC photonic integrated circuit

xvii

List of Acronyms

PML perfectly matched layers

PRS polarization rotator splitter

QAM quadrature amplitude

modulation

QPSK quadrature phase shift keying

SMF single-mode ﬁber

SOI silicon on insulator

SSC spot size converter

TE transverse electric

TM transverse magnetic

WDM wavelength-division multiplexing

xviii

List of Deﬁnitions

Here, frequently used deﬁnitions within the thesis are listed.

•

Coupling eﬃciency: is deﬁned always as the product of two parameters - 1) the

power, which is diffracted by a grating coupler towards a coupling ﬁber and 2) the

mode ﬁeld overlap between the radiated grating mode and the ﬁber mode. (Details

in 2.1.4.2, p. 53.)

•

Cross-polarization: indicates the power of a given input polarization, which is con-

verted to another polarization state after passing through a grating coupler. Hereby,

the cross-polarization is orthogonal to the input polarization (and may have any

modal composition). For example, if we have a y-polarized input polarization and

an output polarization with both x- and y-polarized components, the x-polarized

component is referred as a cross-polarization. (Details in 2.1.4.3, p. 60.)

•

Even-polarization: a polarization, which is split by a two-dimensional grating coupler

with an even symmetry with respect to the symmetry axis between both feeding

arms. (Details in 4.2.2.3, p. 115.)

•

Focusing grating couplers: grating couplers with perturbing elements having posi-

tions, which are placed along curved lines. Focusing grating couplers have shorter

feeding tapers. (An illustration in Fig. 2.10 in 2.1.3.1, p. 44.)

•

Linear grating couplers: grating couplers with perturbing elements having positions,

which are placed along straight lines. Linear grating couplers require long adiabatic

feeding tapers. (An illustration in Fig. 2.10 in 2.1.3.1, p. 44.)

•

Normalized out-coupled power: is the power, which is diffracted by a grating coupler

towards a coupling ﬁber. The power is normalized to the input power of the Si

waveguide mode, which is fed into the grating. (Details in 2.1.4.2, p. 53.)

•

Mode ﬁeld overlap: describes the degree of similarity of two modal distributions.

Particularly for grating couplers, the mode ﬁeld overlap is given by and overlap integral

xix

List of Deﬁnitions

between the grating’ radiated ﬁeld distribution and the coupling ﬁber’s fundamental

mode. (Details in 2.1.4.2, p. 53.)

•

Odd-polarization: a polarization, which is split by a two-dimensional grating coupler

with an odd symmetry with respect to the symmetry axis between both feeding arms.

(Details in 4.2.2.3, p. 115.)

•

Polarization crosstalk: is the crosstalk between non-orthogonal polarization states

in terms of power. (Details in 2.1.4.3, p. 60.)

•

Polarization-dependent loss: is the power difference in dB between the polarizations,

coupled with a maximal and minimal coupling eﬃciency by the grating coupler.

(Details in 4.2.2.3, p. 115.)

•

(Polarizations’) non-orthogonality: assigns a relationship between two polarization

states on the Poincaré sphere, in which the two polarizations are not orthogonal to

each other. Orthogonal polarizations have zenith angles with the same magnitude

and a different sign. In addition, their azimuth angles differ by 180

◦

. Polarization

pairs, which do not fulﬁll this condition are non-orthogonal. For a 2D grating coupler,

the (non-)orthogonality is evaluated for the signals, originating from both 2D grating

coupler arms. (Details in 2.1.4.3, p. 60.)

•

(Polarization) split ratio: we assume an output 2D grating coupler and a waveguide

with a given input polarization. The grating coupler diffracts this polarization towards

a coupling ﬁber. If the input polarization is preserved, it is assigned as a target-

polarization. If it is converted to its orthogonal counterpart, it is assigned as a cross-

polarization. The ratio of target-vs. cross-polarization gives the polarization split

ratio. The deﬁnition considers a single-polarization, i.e. a single-waveguide excitation.

(Details in 2.1.4.3, p. 60.)

•

Segmented grating coupler: a grating coupler that comprises perturbing elements,

which have different sizes and periodicity. Elements with the same size and periodicity

are locally grouped, forming different segments. (Details in 5.2.1, p. 131.)

•

Target-polarization: assigns the power of a given input polarization, which is pre-

served after passing through the grating coupler and is not converted into another

polarization state. For example, if we have a y-polarized input polarization and an out-

put polarization with both x- and y-polarized components, the preserved y-polarized

component is referred as a target-polarization. (Details in 2.1.4.3, p. 60.)

•

Uniform grating coupler: a grating coupler that comprises perturbing elements,

which have the same size and periodicity. (A comparison between uniform and

segmented 2D grating couplers is given in 5.2.1, p. 131.)

1 Introduction

The rapid exchange of information between locations anywhere on the world is nowadays

taken for granted. In fact, it is not even a complete generation in humans’ life since the birth

of the world wide web. There are two crucial cohesive factors, which hide behind its fast

evolution. The ﬁrst of them is the great progress of the electronic circuits, which was driven

by the introduction of the complementary metal oxide semiconductor (

CMOS

) technology

in the 60s [1]. This enabled the realization of compact integrated circuits (

s), allowing for

an extremely eﬃcient computational and signal processing performance with a remarkably

low power consumption. Moreover, the phenomenal

s’ scalability, known as the Moore’s

law, enabled the attainment of high-complexity systems on only few square millimeters

area - a factor that made high-volume, low-cost production possible. With the develop-

ment of even more powerful, advanced computational devices, the quest to facilitate the

communication between these devices arose in parallel. With the strive to exchange more

information even faster, the communication channels had to support considerably higher

bandwidths, especially when data rates exceeded 1Gbit/s. This led to the development

of the second important element: the optical communication interconnects as the most

effective solution to realize a fast communication medium. Their success was predestined

by the invention of the low-loss optical single-mode ﬁber (

SMF

) [2], combined with the

decisive role of the erbium doped ﬁber ampliﬁer (

EDFA

) invention [3]. The latter allowed

for an optical signal transmission over thousands of kilometers. With increasing data rates

also in small-scale networks, optical links have been penetrating down to even shorter

distances, superseding the electric ones [4

–

9]. Valuable literature sources, covering various

theoretical and practical key aspects related to optical ﬁber communication components

and systems, are given in Refs. [4,14–24].

Apparently, the electrical domain offers the most eﬃcient resources for signal computation,

while the optical domain delivers the best means for a high-bandwidth signal transmission

over large distances. For that reason, present communication systems are a symbiosis of

high-performance microelectronics and high-speed, low-loss optics. The basic parts of such

systems are presented schematically in the block diagram in Fig. 1.1 [5], where different

color code is used to assign electrical, optical and electro-optical parts. In a typical system,

a light source such as a semiconductor laser is electrically pumped to generate a narrow-

1 Introduction

linewidth optical beam. In the next step, the beam is encoded by an optical modulator. The

latter may be controlled electrically by a digital-analog-converter (DAC), which transforms

a given bit code to an analog signal. Afterwards, the coded optical signal is transmitted

via a communication channel, comprising a single or multiple

SMF

spans. In each span,

loss or dispersion compensation may be applied. Finally, the optical signal is detected at

the receiver and decoded. A special part of this process is the opto-electrical conversion,

which allows for a more complex data handling, e.g. by a digital signal processing (DSP).

Figure 1.1:

A block diagram of a general optical communication system with electrical (red),

optical (blue) and electro-optical parts (mixed coloring). Representation after

Ref. [5], Chap. 1.

With electronics and optics being inseparable in communication systems, the

s’ paradigm

inspired many researchers to seek for mechanisms to apply the same principle for optical

components as well [25]. The ambition was to realize optical devices in a planar technology

and accomplish them in photonic integrated circuits (

PIC

s) with a greatly reduced footprint

and cost. In spite of the presence of different material platforms suitable for that purpose,

silicon (Si) was considered as an especially attractive one, being the fundamental of

CMOS

electronics as well. This gave rise to the research topic of silicon photonics - a technology

intended for the realization of optical components, by using standard

CMOS

or bipolar

complementary metal oxide semiconductor (

BiCMOS

) fabrication routines. Particularly

attractive was the idea of the potential monolithic co-integration of photonic and electronic

components on the same chip, which could allow for the shortest interconnects and lowest

parasitic effects between

PIC

s and electronic integrated circuits (

EIC

s). Presently, the Si

photonics fundamentals are summarized in many books, e.g. Refs. [5

–

13]. The following

sections within this chapter are mostly based on the books [9, 11, 13] and the review

papers [86,87].

1.1 Optical Interfacing in Silicon Photonics

Considering again the schematic in Fig. 1.1, in the ideal case all electrical and electro-optical

components would be co-integrated on a single transmitter, receiver or transceiver chip. At

this stage, we meet the ﬁrst constraint in Si photonics: since Si is an indirect semiconductor,

no laser sources can be realized monolithically on a purely Si-based platform. To address

this issue, hybrid III-V on Si integration keeps being a popular research topic over decades,

see Refs. [26

–

43]. At the present moment, a standard approach is still not fully established,

despite some promising developments of Intel [44] and the recent press release by Tower

Semiconductor, announcing a demonstration of a silicon photonics platform with InP-

based integrated lasers [45]. For now, the generation of an optical signal on a Si chip is

enabled by an external light coupling, mostly using a

SMF

. Even when the laser integration

1.1 Optical Interfacing in Silicon Photonics

would eliminate that necessity in the near future, the interfacing between a Si chip and

an external

SMF

remains still inevitable, because the integrated transmitter and receiver

need to be connected with the ﬁber optical transmission link. This leads us to one of the

essential challenges in Si photonics: the optical coupling problem.

Figure 1.2:

(Not to scale.) (a) A schematic comparison between a single-mode-ﬁber (SMF)

mode, comprising two degenerated polarizations with the same ﬁeld distri-

bution, and a Si nanowire mode, comprising two polarizations with different

spatial distributions and propagation constants. The plots show the electric ﬁeld

strength with an assigned polarization direction. A signiﬁcant mode size mis-

match is given, which can be addressed by two coupling schemes. (b) Horizontal

coupling with a spot size converter (SSC): amplitude and phase matching take

place in the propagation direction of the Si waveguide mode. (SSC schematic by

K. Voigt.) (c) Nearly vertical coupling with a diffraction grating coupler (GC): mode-

ﬁeld diameter (MFD) matching still takes place in horizontal direction, while

phase matching is realized in nearly vertical direction. One- or two-dimensional

(1D, 2D) implementation is possible. Representation after Ref. [9], Chap. 3.

The most characteristic feature of Si photonics is the high index contrast

n: Si photonic

waveguides are based on a silicon on insulator (

SOI

) technology with Si as a core material

and SiO

as a cladding (

≈

2, e.g. [46]). This allows for the realization of waveguides

with a very small cross section: a typical waveguide height

width is 220nm

500nm.

Moreover, the high index contrast is related to small waveguide bend radii, allowing for

a higher integration density. However, the advantages of the high index contrast have

1 Introduction

their price. The standard

SMF

, based on a low-index contrast technology (

n< 0.01), has

a typical mode ﬁeld diameter (

MFD

) of around 10 µm. There is a signiﬁcant obstacle for

the simple interfacing between Si waveguides and

SMF

s, caused by their large size- and

MFD

-mismatch. A schematic comparison between the sizes of a

SMF

and a Si nanowire

(slab waveguide) mode is shown in Fig. 1.2 (a). In addition to that, there is another signiﬁcant

characteristic of high-index contrast structures, which is the strong birefringence. The

fundamental transverse electric (

) and transverse magnetic (

) polarizations TE

and

have a quite different mode ﬁeld distribution, which is related to a large difference in

their effective refractive indices and propagation constants (see Fig. 1.2 (a)). Therefore, all

Si photonic components are strongly polarization dependent and mostly designed for the

fundamental TE

mode. On the other hand, the

SMF

supports two degenerated linearly

polarized (

) modes: LP

01,x

and LP

01,y

. This adds another diﬃculty to the optical coupling,

since both LP

01,x

and LP

01,y

should be converted to the Si fundamental TE

. In summary,

two crucial issues have to be overcome for an eﬃcient ﬁber-to-chip interfacing:

Mode matching: the Si fundamental mode with a small

MFD

has to be adapted to

the dimensions of the ﬁber mode and vice versa.

Polarization handling: both ﬁber polarizations LP

01,x

and LP

01,y

have to be trans-

formed to the fundamental TE00 on Si.

Different design strategies have been developed to reach these goals. We distinguish

between two basic paradigms, with the main properties summarized in the following.

Horizontal Coupling

Horizontal optical interfaces may be based on direct edge coupling or supported by optical

interposers, guiding evanescent ﬁelds. A lateral or edge coupling can be enabled by spot

size converters (

SSC

s). A

SSC

may be realized in different ways, using one or more taper

sections. An example with an inverted taper is shown schematically in Fig. 1.2 (b). The

signiﬁcant characteristic of such interfaces is that the amplitude and phase adaption of the

Si mode takes place in the mode’s propagation direction. The most important advantages

of the horizontal coupling method are:

•High coupling eﬃciency.

•Large optical bandwidth of the coupled signal.

•

Polarization independence: the ﬁber’s LP

01,x

and LP

01,y

polarizations are coupled to

the Si waveguide’s TE

and TM

polarizations with a very low polarization-dependent

loss (

PDL

). Additional on-chip polarization rotator splitter (

PRS

) is necessary to sepa-

rate both polarizations and enable the conversion from TM00 into TE00.

Unfortunately, these advantages come in combination with signiﬁcant physical, technologi-

cal and economical constraints. To the most considerable disadvantages belong:

1.1 Optical Interfacing in Silicon Photonics

•

Complex fabrication and high sensitivity to fabrication deviations: mostly, an

SSC

requires a tiny tip at the position, where the ﬁber coupling takes place. Moreover,

fabrication post-processing steps such as dicing and polishing are necessary to obtain

a ﬂat coupling surface. The higher fabrication effort inevitably increases the total

costs.

•

Small spot-size: typically, coupling to a ﬁber with a small

MFD

can be guaranteed.

Often, lensed ﬁbers are used for that purpose with a MFD of only few micrometers.

This requires a very precise alignment between ﬁber and chip, which is a critical

obstacle for the device packaging. To overcome this issue, different design strategies

have been considered, many of them related to an enhanced (post-)fabrication effort.

Moreover, index matching gel has been frequently applied. Lower complexity has

been often achievable at the price of excess loss (cf. Refs. [47

–

53]). The most notable

SSC

design has been patented by the ﬁrm Teraxion [54], which proposed the usage

of multiple optical strips in the backend of line (

BEOL

). Eﬃcient coupling to both

ﬁber with a

MFD

of 6.6µm and a standard

SMF

with a

MFD

of 10.4µm/ 9.2 µm in

C-/ O-band has been demonstrated with results reported in [55,56]. Nevertheless,

horizontal coupling to cleaved SMF with a large MFD is still not widely spread.

•

No automated wafer-scale measurement available: reliable automated wafer testing

capability is crucial during fabrication and sub-systems assembly in large scale man-

ufacturing processes [57,58]. Up to now, few manual wafer testing approaches have

been proposed. In Ref. [59] 3D printed elements have been used. Recently, edge

coupled wafer-scale measurements with reﬂecting optical ﬁber probes have been

reported [60].

•

Non-trivial

PRS

design: the most crucial problem for the horizontal coupling method

is the limited polarization separation of the

PRS

, leading to a deteriorated polarization

extinction ratio and large polarization crosstalk. For their minimization, the cascading

of multiple

PRS

s or the utilization of optical ﬁlters or other supporting devices is

necessary. Depending on the particular design, either increased conversion loss, or

reduced bandwidth or both may result. The overall device footprint (

SSC

PRS

) may

become very large as well (cf. Refs. [61–67]).

In spite of the convincing beneﬁts of horizontal coupling structures, their present drawbacks

are not negligible with regard to the practical implementation in large-scale

PIC

s. Further

development is necessary for a complete domination over other coupling alternatives,

such as the nearly vertical coupling by diffraction gratings, which will be introduced in the

following.

Nearly Vertical Coupling

The major competitor to the lateral coupling method is the nearly vertical coupling tech-

nique, realized by the adoption of diffraction gratings. As a most basic form of such

structures, the one-dimensional (1D) grating couplers (

s) will be ﬁrst presented (Fig.

1 Introduction

1.2 (c)). A conclusive analysis of the 1D

s’ properties is provided in the doctoral thesis

of Taillaert [68]. In 1D

s, the mode matching is achieved by the combination of two

measures: 1)

MFD

matching - the Si waveguide is gradually widened to match the size of the

coupling ﬁber. Until this point, the conversion still remains horizontal; 2) phase matching -

the phase of the waveguide mode is manipulated via diffraction grating, thus achieving a

mode deﬂection in an off-chip propagation direction. The phase matching mechanism is

the fundamental difference between both coupling strategies. Using diffraction gratings,

a perfectly vertical out-coupling is the most desired variant. However, it is related to a

second-order Bragg reﬂection, inevitably leading to a high power loss. For that reason, the

diffraction grating is designed in such a way that the waveguide mode is directed under a

small angle θwith respect to the vertical axis.

The nearly vertical coupling scheme comes with several advantages, such as:

•

Large spot size: the coupling to a cleaved

SMF

with an

MFD

of around 10 µm is

possible. The coupling tolerances are signiﬁcantly larger than for conventional

SSC

This promises for a more relaxed and inexpensive packaging.

•

Automated wafer-level testing: the nearly vertical optical coupling is very similar to

the electrical probing on a wafer. This allows for the simple automated wafer level

testing, which is mandatory for large-scale manufacturing platforms.

•

Easier fabrication without the necessity for post-processing steps. Also compared to

SSCs with PRS, smaller on-chip footprint is achievable.

These practical beneﬁts are combined with inherent drawbacks, which are a direct con-

sequence of the physical background of the diffraction gratings. Following fundamental

limitations are present:

•

Lower coupling eﬃciency: because a

diffracts light not only under a given angle

in upper direction, but also under the same angle downwards into the substrate, we

are generally not able to reach a maximal power directivity without the adoption of

a back-reﬂector or the modal manipulation by thickened (enhanced) gratings. The

fabrication of eﬃcient back-reﬂectors is mostly not

CMOS

compatible and requires

post-fabrication bonding steps. The reliable and repeatable Si growth to form en-

hanced gratings is also not trivial. In addition to the power directivity limitations, there

is a modal mismatch between the distributions of the grating ﬁeld and the coupling

ﬁber ﬁeld. This requires also more speciﬁc grating designs, which may not be easily

implementable under given technological constraints.

•

Optical bandwidth limitation: according to the diffraction condition of gratings, each

wavelength is diffracted under a different angle. On the other hand, the coupling

ﬁber is tilted under a ﬁxed angle. A Gaussian-like coupling spectrum is typical for

s. The maximal coupling is at the wavelength with the diffraction angle, matching

the ﬁber tilt angle.

1.2 Fundamental Issues of Two-Dimensional Grating Couplers

•

Polarization dependence: 1D

s are able to couple only the

SMF

s polarization,

which is aligned with the grating axis. Without an active polarization control, large

PDL results.

Many scientiﬁc works have been dedicated to address the ﬁrst two issues. To improve

the out-coupled power eﬃciency, metal [70

–

72] and distributed Bragg reﬂector (

DBR

)

[69] back-reﬂectors have been considered. Alternatively, gratings with an enhanced Si

thickness and multi-layer gratings have been examined [73

–

75]. Blazed gratings with a sub-

wavelength index-matching structure have been investigated as well [76] - the results have

been obtained at the cost of an increased fabrication complexity. Furthermore, footprint

optimization via the introduction of focusing grating couplers (

FGC

s) has been proposed

[77

–

79]. In parallel to the work on the present thesis,

FGC

s for the TM

polarization have

been investigated [80]. For the optimization of the mode ﬁeld overlap, apodized/chirped

gratings have been reported [79,81,82]. Often, the 1D

s are designed by combining

some of the mentioned methods. This is the case for the record low-loss 1D

from

Ref. [83]. To improve the optical bandwidth, silicon nitride (Si

) or Si

-assisted

have been reported as a successful option [75,82,84]. The present list of references is

by far not exhaustive due to the variety of designs and target application ﬁelds. For a

more comprehensive overview over coupling design strategies, the reader may refer to

the review papers [85–87].

To address the problem with the polarization dependence, the 1D

needs to be modiﬁed.

One possibility is the counter-directional coupling of

and

with a 1D

designed

in such a way that TE

has a coupling angle

and TM

has a coupling angle –

(see

e.g. Ref. [88]). This method has the disadvantage that a

PRS

is still required to convert

the TM

polarization into TE

. This leads us to the main subject of the present thesis:

the two-dimensional (2D)

. Although 2D

s are based on a natural idea - to simply

combine orthogonally two 1D

s and obtain thus polarization diversity (Fig. 1.2 (c)) - their

practical implementation presented unexpected issues.

1.2 Fundamental Issues of Two-Dimensional Grating Couplers

In this section, the main 2D

’s issues will be summarized. Each of these aspects has

been addressed in the scope of this work.

Inherent Issues

The limitations, known from the properties of 1D

s, can be naturally transferred to

their 2D counterparts. Some 2D

speciﬁc factors have an additional contribution to the

performance’s deterioration.

Insertion Loss The insertion loss, caused by a limited power directivity to a

SMF

and

by the non-optimal modal matching with the

SMF

’s fundamental mode, is even more

distinctive in 2D

s. There are several reasons for that. In early publications, no adaption

1 Introduction

of the diffraction condition of 2D

s was considered, which could ensure that an input

mode from any Si waveguide will be radiated at the grating’s symmetry plane. For that

reason, the reported 2D

designs showed a very low coupling eﬃciency in the range

from -8dB to -6.5dB [89

–

91]. It has been found that the adoption of a non-zero angle

between Si waveguide and grating (here called a shear angle) is necessary for a reduced

insertion loss [68,92

–

94]. Along with the shear angle, an improved etch depth larger than

the standard 70 nm for 1D

s has been considered. Current 2D

s on standard 220 nm

SOI

reach a coupling eﬃciency between -5 dB and -3.5 dB without the application of other

optimization methods [95

–

97]. Coupling structures to higher-order ﬁber modes show

similarly an eﬃciency of around -5dB [98

–

100]. The ﬁrm Luxtera in collaboration with

STMicroelectronics demonstrated 2D

s with around -3dB coupling eﬃciency, by using

a customized buried oxide (

BOX

) thickness below the grating [101]. For an enhanced

coupling eﬃciency, the adoption of backside reﬂectors has been reported, which is based

either on metal mirrors [102,103] or on Bragg mirrors on double-

SOI

substrate [104,105].

Furthermore, dual-etch [106], chirped/apodized 2D

s [107

–

112] have been investigated,

including sub-wavelength (metagrating) designs. The latter require mostly feature sizes,

which are challenging for the fabrication by a standard deep ultra violet (

DUV

) lithography

(193nm or 248nm). Because of the different technological background, sub-wavelength

or metamaterial structures are not directly compared to standard devices in this work.

Spectrum The grating’s spectrum shape indicates a bandwidth limitation due to the

different optimal coupling angles for different wavelengths. In 2D

s, the spectrum shape

is determined by the wavelength-dependence of both the nearly vertical angle

and

the in-plane angle

(cf. the angles’ deﬁnitions in Fig. 1.2 (c)). Potentially, the widening

of the 2D

s’ bandwidth may be addressed in the same way as in 1D

s, i.e. by a

dual-layer principle. So far, no designs addressing this aspect have been reported, most

possibly because the available 2D

s’ bandwidth is suﬃcient for the target applications.

Another aspect is the spectrum wavelength adaption by the coupling angle

. In 1D

the maximum transmission wavelength can be adjusted by the choice of an appropriate

coupling angle. In 2D

s, there is another speciﬁc, which may be a problem for the optimal

coupling. There is only one coupling angle

that is related to diffraction in the symmetry

plane

φ=

◦

. The 2D

’s design must be suﬃciently precise, so that the coupling angle

at the symmetry plane matches to the required wavelength. Spectrum shift via

-adaption

leads to a departure from the symmetry plane (

φ=

◦

), which could increase the coupling

loss to a ﬁber placed there. The deviation is tolerable to some extend, without a signiﬁcant

excess loss, but no large θ-range may be covered.

Polarization-Related Issues

Problems, related to the polarization of light, are present in 2D

s as well. However,

the nature of these processes is quite different from the polarization-dependence of 1D

s, which is simply caused by the absence of a diffraction mechanism to manipulate

the direction of one of the basic

SMF

polarizations. In fact, undesired polarization-related

1.3 Objectives of the Thesis

effects in 2D

s result from the desire for the simultaneous interaction with both

SMF

polarizations. The presence of these effects has consequences for all functional properties

of 2D GCs.

Polarization-Dependent Loss (PDL) The

PDL

is expressed in coupling spectrum shape or

coupling spectrum maximum, depending on the incident SMF polarization [93,114–116].

The issue became noticeable after the introduction of the non-zero waveguide-to-grating

angle and is among the most extensively investigated problems in 2D

s . As a possible

solution, the application of a phase shifter has been proposed [116]. A more popular

alternative was the adoption of a special shape of the 2D

’s perturbing elements,

proposed initially by the ﬁrm Luxtera [101,104,117]. Many research groups re-conﬁrmed

the success of this method [103,113,119

–

121]. As an alternative, stretched perturbing

elements have been reported, achieving, however, a lower coupling eﬃciency [118]. In

2021, an apodized 2D

was reported [109], relying on the variation of the perturbation

strength. Most recently, an apodized meta-grating, comprising perturbing elements with

different stretching direction, have been reported [112]. No experimental data on the

producibility of this method is available at the present moment.

Polarization Splitting The polarization splitting indicates the capability of the 2D

split or combine two polarization states properly. This aspect has not been addressed often

so far, partially due to its practical relevance for polarization-multiplexed systems, which

started gaining in importance only a few years ago. In early publications, the polarization

crosstalk or extinction ratio have been measured [89,92], but no subsequent analyses on

its physical background have been carried out. In Ref. [120], it has been shown that the

special perturbing elements’ shape may be advantageous for the improved polarization

splitting. In-depth investigations of the polarization related aspects have been performed

in the scope of this work, which will be summarized in the next section.

1.3 Objectives of the Thesis

With all designed 2D

s, available in the literature already more than 10 years ago, a large

part of the research community considered the topic of 2D

s as exhausted. However,

there were several aspects that motivated the present work in the beginning. On the one

hand, most of the known facts about 2D

s remained to a large extend unexplained in

the literature. First of all, no veriﬁcation of a diffraction mechanism in a two dimensional

direction was available. In spite of the introduction of a non-zero waveguide-to-grating

angle, its exact contribution to the 2D

s’s behavior was not evident. Furthermore, no

investigation has been carried out, which clearly shows, why a given perturbing element’s

shape should be more advantageous with regard to the

PDL

. In spite of the numerous

papers on that topic, the analysis on a physical level was missing. For these reasons, the

ﬁrst goal of this work was to prepare a conclusive systematic description of the 2D

s’

physical properties.

1 Introduction

The second impulse for this work was given by the emerging topic of the coherent commu-

nications for data center interconnects. In the recent years, the increasing data rates in

this domain established a competition between the classically used intensity modulated

direct detection (

IM-DD

) and the coherent technology. While it is predicted that

IM-DD

will predominate in the next few years, there is an extensive work ongoing towards the

optimization of the coherent technology in two directions - the minimization of the power

consumption and the reduction of the total transceiver cost. While Si photonics is the

best scalable technology to address the second point, it is almost evident that 2D

are the most mature option for optical interfacing in this practical context. Their decisive

properties are the automated wafer-level testing capability, which is essential for such

systems, and the relaxed packaging requirements, which are the most critical economic

factor. The 2D

s’ suitability even for more challenging form factors is approved by

Sicoya’s product portfolio (e.g. a grating coupler based QSFP28 transceiver [122]). Natu-

rally, having a polarization-diversity coupling interface, polarization-multiplexed modulation

formats come into question. Therefore, the investigation of 2D

s with respect to their

polarization handling characteristics became necessary. In this work, polarization-related

aspects, which have not been considered previously, have been studied. Particularly, the

polarization splitting/combining behavior of 2D

s has been analyzed for the ﬁrst time

in detail, exploring the signiﬁcant issue of a strong polarization conversion - the so called

cross-polarization. The latter assigns the power conversion of a given input polarization

state into its orthogonal counterpart. As a result, a limited polarization split ratio and

polarization crosstalk occur. Moreover, the cross-polarization is related to excess loss and

higher-order mode excitation in 2D

s. In this work, the impact of the design geometric

parameters has been ﬁrst investigated, showing that the cross-polarization scales with

the grating perturbation strength. Moreover, it has been shown that a deviation from the

initial polarization state can be related to the loss of signals’ orthogonality. Looking for

the physical origins of cross-polarization in 2D

s, in-plane scattering has been identi-

ﬁed as a dominant process that had been overlooked until that moment. With in-plane

scattering designated as the most decisive issue in 2D

s, design strategies for its min-

imization have been proposed and developed, using the systematically acquired basic

knowledge about 2D

s. In the end, it has been shown that the

PDL

is directly related to

the polarization-associated issues and it could be shown that reduced in-plane scattering

and cross-polarization are imperative to reach a low

PDL

. As a ﬁnal study, the potential

footprint reduction of 2D GCs by their modiﬁcation towards FGCs has been carried out.

In spite of the initially narrow target application domain, the extensive occupation with

polarization-related aspects has contributed to the development of a clearly structured

theory of 2D

s, which has ﬁlled the gap of unexplained observations in older publications.

A conclusive description of the 2D

s’ physical fundamentals and their practical manip-

ulation for different purposes has been obtained. In the following, the most signiﬁcant

contributions of the present thesis are summarized.

•

Explanation of the diffraction mechanism in two dimensions - the basic dependences

between the 2D

s’ geometric parameters and the electromagnetic ﬁelds’ deﬂection

1.3 Objectives of the Thesis

in two angular directions have been derived. The role of the non-zero waveguide-to-

grating angle, assigned here as a shear angle, has been outlined. The mathematical

formulation gives the possibility to develop designs for different coupling angles at

the grating’s symmetry plane, by properly combining the grating’s periodicity with a

given shear angle. Designs with a different practical realization of the shear angle

and for different target coupling angles have been compared.

•Investigation of issues, induced by non-ideal polarization handling:

–

Cross-polarization and the resulting limited polarizations’ split ratio, non-orthogo-

nality and crosstalk have been analyzed on device and system level. Several

predictions based on simulations have been conﬁrmed experimentally.

–

The importance of in-plane scattering processes has been explored. Although

scattering losses in photonic crystals are a well-known issue [123

–

125], scatter-

ing effects have been underestimated in 2D

s. This work shows that in-plane

scattering is the fundamental limitation of 2D GCs.

–

The dependence between in-plane scattering, cross-polarization and

PDL

has

been demonstrated. The latter is able to explain the secret behind the success

of the special perturbing elements’ shape.

•

Proposal of design modiﬁcations and optimization variants based on the understand-

ing of the physical processes in 2D

s. The developed designs ensure simultaneously

a large polarizations’ split ratio, a corresponding low polarizations’ crosstalk and a

broadband nearly perfect polarization’s orthogonality, a low

PDL

and an improved

coupling eﬃciency. Thus, their adoption in wavelength-division multiplexing (

WDM

)

systems becomes possible as well. Furthermore, the optimized designs distinguish

themselves by their simplicity, which allows for their fabrication with a low-resolution

lithography and without optical proximity corrections. Overall, the new 2D

s are

capable to reduce the total fabrication cost, when applied in Si photonic coherent

systems. It is the author’s wish that the acquired knowledge on the proper designing

of 2D

s will contribute to the efforts towards the cost-optimization of coherent

transceivers.

•

O-band vs. C-band: with regard to the discussion on the potential of O-band coherent

data center interconnects (

DCI

s), the present work shows that O-band 2D

s are

beneﬁcial, when applied in similar technologies. Along with the better polarization-

related parameters, a satisfying coupling eﬃciency is reachable with less effort.

The remaining chapters of the present thesis are organized as follows. In Chap. 2, a de-

scription of the numerical and experimental methods for the analysis of 2D

are given.

A signiﬁcant part of the chapter is dedicated to the proper numerical handling: starting

from the general description of the simulation problem, several numerical techniques

are evaluated as potential candidates for its proper solution. After the motivation for the

chosen method is given, the concrete simulation procedure and the post-processing calcu-

lations are described in more detail. In the second part of the chapter, the measurement

1 Introduction

methodology for 2D

s’s characterization is outlined. Along with different measurement

procedures for the determination of a given parameter, several sources of measurement

uncertainty are discussed. In Chap. 3, the speciﬁcs of the fabrication platform, used for

the 2D

s’ fabrication, are summarized. Particularly, the differences between

CMOS

and

BiCMOS

are brieﬂy reviewed and the basic characteristics of the monolithic photonic

BiCMOS

platform are given. Its possible extension in a 3D concept is also discussed due to

its potential for more optimization ﬂexibility of certain Si photonics devices, including 2D

GCs. Using the presented techniques for the 2D GCs’ examination, fundamental physical

effects related to their operational speciﬁcs and limitations are thoroughly studied in

Chap. 4. Diffraction and in-plane scattering are indicated as the most important processes.

Regarding the former, a suitable condition for 2D

s is formulated, which allows for the

designs with different target coupling angles. The derived formulas are veriﬁed numerically.

Second, the importance of in-plane scattering effects in 2D

s are shown. Starting with

a simple analytical proof for its existence, cross-polarization is investigated as the most

important consequence of scattering. Critical performance limitations are studied, such as

excess loss, polarization split ratio, polarizations’ non-orthogonality and

PDL

. In the end,

the excitation of higher-order modes due to scattering is shown as well. In Chap. 5, several

possibilities for the improvement of 2D

s are proposed. First, a 3D integration approach

is investigated for the enhancement of the grating’s out-coupled power. Furthermore, the

optimization of polarization-related issues is addressed, considering in-plane scattering

as their physical background. For scattering manipulation, segmented 2D

s are ﬁrst

analyzed, showing a satisfying performance in some aspects. Furthermore, an improved

technique to control in-plane scattering is proposed. The new grating design comprises

elongated perturbing elements with individual orientations, ensuring a local scattering

manipulation. The achieved overall in-plane scattering suppression results in the signiﬁcant

improvement of the polarizations’ split ratio, polarizations’ orthogonality and

PDL

, which is

shown numerically. The robustness of the proposed designs against fabrication deviations

is evaluated as well. The transformation towards low-footprint

FGC

s is considered. Subse-

quently, the feasibility of the proposed optimization techniques is shown in wafer-level

experiments, demonstrating a considerable

PDL

reduction. The observations and results in

this work are ﬁnally summarized in Chap. 6 and aspects of potential further investigations

are given.

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Sicoya, presented at: The 47th ECOC Exhibition, 2021, video available at:

https://www.

youtube.com/watch?v=HgAxj5op5Rs (online Oct. 20, 2023).

[123]

E. Yablonovitch, “Photonic Crystals: Semiconductors of Light,” in Scientiﬁc American,

vol. 285, no. 6, pp. 46-51, 54-55, 2001.

[124]

W. Bogaerts, Nanophotonic Waveguides and Photonic Crystals in Silicon-on-Insulator,

PhD Thesis, University of Ghent, 2004.

[125]

J. Joannopoulos, S. Johnson, J, Winn, R. Meade, Photonic Crystals: Molding the Flow

of Light, Princeton University Press, 2008.

2 Methods for Analysis of

Two-Dimensional Grating

Couplers

This chapter is dedicated to the description of all methods used for the analysis and

characterization of 2D

s. The ﬁrst section outlines the main challenges for the accurate

numerical investigation of 2D

s, which determine the choice of an appropriate simulation

approach. Numerical methods, which are currently established in the optical domain,

are reviewed and the most appropriate of them are explained in further detail. Special

attention is paid to the discussion on possible sources of numerical errors and their

minimization. Afterwards, a detailed description of all simulation steps is presented. Finally,

the procedures for the determination of coupling angles, eﬃciency and polarization state

in the post-processing steps are explained mathematically.

The second section within this chapter is focused on the measurement methodology, used

for the experimental determination of various 2D

parameters. Setup speciﬁcs, error

sources and the possibilities to achieve maximal accuracy of the measurement results are

discussed in detail.

2.1 Numerical Simulation and Evaluation

This section is focused on the speciﬁcs of the simulation and post-processing problems

and on the most important aspects for the achievement of accurate results.

2.1.1 General Problem Description

Structures operating in the optical (visible or infra-red) domain are well-known for their

challenging simulation. The reason is that in this spectral range we need to cope with

short wavelengths - typically between 0.5 µm and 2 µm. Thus, even small structures have

dimensions, extending over many wavelengths. In addition, for the appropriate wavelength

2 Methods for Analysis of Two-Dimensional Grating Couplers

resolution, a very ﬁne simulation grid is needed. These two factors lead to large models

requiring substantial computational resources and long simulation times to obtain accurate

numerical results. To illustrate the complexity of electronic-photonic integrated circuits

(

EPIC

s), a photograph of a fabricated dual-polarization coherent receiver is shown in Fig.

2.1. The realization of this receiver is a result of the collaboration between TU Berlin and

IHP Microelectronics, whereat more details can be found in the doctoral thesis of Pascal

M. Seiler [1].

Figure 2.1:

A photograph of a monolithically integrated dual-polarization coherent receiver

with a photonic integrated circuit (PIC) on the left hand side and an electronic

integrated circuit (EIC) on the right hand side (IQ: in-phase quadrature). 2D

grating couplers (GCs) are used as coupling interfaces, while 1D GCs are included

for testing purposes. The PIC comprises a variety of photonic components

with different requirements regarding the choice of a numerical method and

simulation procedure. Receiver design by Pascal M. Seiler [1].

As depicted, the receiver comprises photonic and electronic integrated circuits (PIC and

EIC), monolithically integrated in parallel (details in Chap. 3). The photonic segment includes

a large variety of components; the photonic circuit extends over several millimeter, making

its complete numerical simulation impossible without advanced hardware. Typically, each

single component is characterized in separate simulations, whereat different requirements

with regard to the choice of the most optimal numerical method are given.

While structures such as optical waveguides, tapers, multi-mode interferometers (

MMI

etc. can be characterized with comparatively small computational resources, monolithically

2.1 Numerical Simulation and Evaluation

integrated 2D

s are signiﬁcantly more challenging to simulate for three reasons. First,

since the structure has a 2D radiation pattern, full 3D simulation models are required (Fig.

2.2 (a)). This distinguishes 2D

s also from their 1D counterparts, which can be described

reasonably by 2D simulations. Second, in a monolithic conﬁguration, the coupling device is

covered by a

BEOL

stack, which is typically

∼

15 µm in height (Fig. 2.2 (b)). Its co-simulation

is necessary, in order to obtain correct coupling angles and insertion loss. Layers in the

vicinity of the grating have an impact on its effective refractive index and power directivity.

In addition, the total stack thickness determines the minimal achievable distance between

the 2D

and the ﬁber. The

BEOL

stack consideration makes the modeling procedure

more time-consuming and increases substantially the model’s volume, adding further to

the required number of grid cells. Third, to allow for the results’ evaluation at a tilted plane,

according to the design radiation angles, several micrometers in space above the

BEOL

stack is necessary. Because of the large space, which is required above the grating and

because of the typically long feeding tapers (several hundred micrometers), the simulation

domain need to be restricted (Fig. 2.2 (c)). Only small part of the tapers’ end and the grating

can be considered, including the BEOL stack and a free space for the ﬁelds’ evaluation.

Figure 2.2:

A description of the numerical model of a two-dimensional grating coupler (2D

GC). (a) A 2D GC with feeding linear tapers. (b) The complete structure is covered

by a backend of line (BEOL) stack. (c) Only a small volume is considered for

simulation. The tapers are typically simulated separately to avoid long simulation

times.

Even with a strongly reduced computational volume, the simulation of a 2D

remains

challenging. As an illustration, we compare the numerical problems of a 1D and 2D

shown in Fig. 2.3. For the simulation of the 1D

, a simpliﬁed 2D model is suﬃcient. The

exemplary models are framed by their computational boxes, which are equivalent to Fig.

2.2 (c). In both cases, the

s are deﬁned in the (x,y,z)-plane. The tilted plane (u,v,w) gives

the coordinates of the external

SMF

, which would be placed at w

const. In all simulations,

such a plane is used for the post-processing evaluation of the radiated ﬁelds.

In this example, the

s are deﬁned on 220nm

SOI

with a

BOX

thickness of 2 µm and

the same

BEOL

stack height. In addition, 15 µm empty space is left above the 1D

and

only 5µm above the 2D

. For simulations covering C-band (1530–1565nm), a typical

minimal computation free-space wavelength is 1500nm (the simulation window is usually

2 Methods for Analysis of Two-Dimensional Grating Couplers

100nm, i.e. in the range 1500-1600nm). If we assume the simple case of a hexahedral

grid and a resolution of 15 cells per wavelength, the computational domain of a 1D

will consist of nearly 500 000 grid cells and the 2D

’s model would result in almost

75 000 000 grid cells. The 2D GC’s model is 150 times larger than the 1D GC’s one in this

example. Naturally, the exact difference in the number of grid cells is dependent on the

particular speciﬁcations, i.e. which optical band, what grid size etc. In the most cases, 2D

and 3D models differ by at least two orders of magnitude in computational size. A careful

evaluation of the numerical methods, which are typically used in the optical domain, is

crucial to avoid time and resource extensive simulations of 2D GCs.

Figure 2.3:

An illustration of (a) 1D and (b) 2D grating coupler (GC) models with their

computational boxes. A 2D model can be used for the 1D GC, a 3D model is

required for the 2D GC. The GCs are deﬁned in the (x,y,z)-plane. The tilted

(u,v,w) coordinates give the position of the single-mode ﬁber, rotated according

the design coupling angles. Any ﬁeld evaluation takes place at w

const. in the

free space above a given structure. In this example, both 1D and 2D GCs have

the same minimal simulation wavelength, the same type of grid (hexahedral),

the same grid size and the same stack height; the 1D GC has 10 µm extra

free space above the stack. The 2D GC’s model consists of 150 times more

hexahedral grid cells than the 1D GC’s one.

Due to the geometrical speciﬁcs of integrated 2D

s, their numerical simulation combines

two problems: 1) the diffraction of a periodic 2D array, 2) the ﬁeld propagation of a Gaussian-

like beam along several tens of wavelengths distance from the grating. In some cases, the

light propagation in the complete feeding tapers needs to be simulated and combined with

the 2D

simulation as well (e.g. for

FGC

s). Thus, numerical methods, which are dedicated

to periodic structures alone, are not suﬃcient for the full structure analysis. Moreover,

because the coupling distance is not suﬃciently large, no far-ﬁeld approximations can

be assumed. Therefore, a suitable numerical technique should be able to address both

diffraction and propagation problems and to be able to cope with a large-volume problem

at the same time.

2.1 Numerical Simulation and Evaluation

2.1.2 Numerical Techniques

The development of numerical methods and their software implementation was an im-

portant step towards the solution of complex practical problems in many scientiﬁc ﬁelds,

including electrodynamics. Currently, there are numerous contributions in the literature,

covering a variety of numerical methods and electromagnetic applications, e.g. Refs. [2

–

4].

Here, several general characteristics of such methods will be outlined.

2.1.2.1 General Features of Numerical Approaches

Zhou [2] distinguishes in general between two basic approaches to solve a ﬁeld problem:

by using differential equations or integral equations. The ﬁrst possibility is assigned as a

ﬁeld approach, while the second one is given as a source distribution/boundary method.

All numerical algorithms can be divided into these two categories.

Independent of the particular implementation, each numerical technique follows the same

basic steps, summarized below from [2]:

An unknown function u(r), which is contained in an operator equation, needs to be ex-

pressed by the summation of a set of linear independent basic functions u

(r) weighted

by the undetermined parameters cn.

u(r)≈ˆ

n=1

cn·un(r) (2.1)

The latter expression is called a trial function and is equivalent to an approximate

solution.

This step can be compared e.g. to the Taylor polynomial approximation of a given

function. If the number of polynomials tends to inﬁnity, we obtain the exact function.

Similarly, the trial function tends to the real solution, when N→∞.

The choice of basic functions u

(r) and the determination of their unknown parameters

must be done in such a way that the operator equation together with all boundary

and initial value conditions are well satisﬁed. The particular numerical methods differ in

the way how they fulﬁll these requirements [5].

The solution domain needs to be discretized, resulting in a ﬁnite number of elements

and nodes. With this, the unknown continuous function u(r) with inﬁnite degrees of

freedom is approximated by a spatially discrete function with ﬁnite degrees of freedom.

Aprinciple for error minimization needs to be chosen, out of which the unknown param-

eters c

of the basic functions can be found. The initial operator equation results in a

matrix equation. Among the most commonly used principles for error minimization are

the variational principle and the principle of the weighted residuals.

The solution of the matrix equation resulting in the previous step gives the approximate

solution of the initial problem.

2 Methods for Analysis of Two-Dimensional Grating Couplers

In the following, the emphasized terms will be explained shortly, in accordance with Chap.

2 in Ref. [2]. The reader may refer to [2] for a more detailed explanation of the theoretical

concepts summarized here.

Operator Equation The general assignment of an operator equation has the form:

Lu=f(2.2)

The operator

maps a given element to another one, each of both belonging to a different

domain. For example, the exponential operator

exp

(.) in the equation y

=exp

(x) maps the

variable xinto the variable y. Thus, xbelonging to the interval [–

∞

] is transformed to y

in the range from [0,+∞].

An example for different operators are the derivation, integral, gradient, divergence, curl,

Laplacian and matrix transformations. One or more properties may be associated with a

given operator, e.g. whether it is linear, symmetric, positive-deﬁnite, self-adjoint, continuous,

etc. The properties of a given operator are determining for the choice of the approximation

approach for the formulation of a matrix equation (step 3)) and for the solution of this

equation (step 4)).

Principles for Error Minimization The most important principles for error minimization

used predominantly in numerical methods are the principle of the weighted residuals and

the variational principle. The former targets at the operator equation directly, while the

latter operates with the equivalent functional of the governing equation.

•

The principle of the weighted residuals - minimizes the approximation error in a

weighted average sense. An exemplary problem is given by the operator equation (2.2)

in a certain domain in combination with boundary conditions, e.g. a ﬁxed boundary

value u

and a ﬁxed normal derivative g

. Due to the approximation in (2.1), residuals

Ri=0,1,2 result in the equation domain Ωand at the boundaries Γ1,Γ2.

Lu=fin the domain Ωu≈ˆ

u(2.1)

−−−−−−−→ R0=Lˆ

u–f

u|Γ1=u0at the boundary Γ1u≈ˆ

u(2.1)

−−−−−−−→ R1=ˆ

u|Γ1–u0

∂u

∂n|Γ2=g0at the boundary Γ2u≈ˆ

u(2.1)

−−−−−−−→ R2=ˆ

g|Γ1–g0(2.3)

For the calculation of a weighted residual average, the residuals are multiplied by

weighting functions W

i=0,1,2

. The exact choice of these functions varies for the different

numerical methods. The average of the weighted residuals is forced to zero either

at the boundaries (for the domain methods) or in the equation domain (for the

boundary methods). A combination of both approaches is also possible.

Domain methods Boundary methods

ˆΓ1

R1·W1·dΓ1+ˆΓ2

R2·W2·dΓ2=0ˆΩ

R0·W0·dΩ=0 (2.4)

2.1 Numerical Simulation and Evaluation

•

The variational principle is equivalent to the principle of minimum energy. A certain

problem is expressed by an equivalent functional. A function, which minimizes this

functional, is the solution of the original equation. For example, a potential distribution

that satisﬁes the Laplace’s equation is in the same time a potential distribution that

minimizes the electric ﬁeld energy. The solution of a given problem is again expanded

in terms of trial functions as in (2.1). The unknown constants are determined this

time by forcing the equivalent functional to be stationary or minimized or maximized

with respect to these constants.

2.1.2.2 Speciﬁc Numerical Methods for Photonic Applications

After some basic concepts for the numerical approaches have been summarized, we

will move here towards the discussion of particular techniques. Several simulation ap-

proaches are used by current open-source and commercial software packages. Some of

them are more specialized and target at a limited number of applications. For example,

the beam propagation method (BPM) and the eigenmode expansion method (EME)

are particularly suitable for the eﬃcient simulation of long bounded structures instead of

radiating devices. BPM is an approximation method, which is based on the slowly varying

envelope approximation [10,11]. The method removes the fast varying term

exp

(

∓

z) (

propagation constant along the

z-direction) and delivers a solution for the slowly varying

ﬁelds. Due to this approximation, the method is not suitable for the solution of fast varying

or discrete structures, including periodic gratings. It works well for waveguides, tapers,

y-junctions under the condition that the refractive index contrast is not too large. With

this, the method is rather suitable for other material platforms than

SOI

[11]. By contrast,

EME is a rigorous solution of the Maxwell’s equations, which is one of its advantages over

BPM [12]. It implements the concept of eigenmodes, i.e. the representation of electro-

magnetic ﬁelds as a decomposition of modes, which have a certain distribution at the

waveguide cross-section’s plane (e.g. in the (x,y)-plane) and a harmonic dependence in

the propagation z-direction. The modes can be both guided and radiation modes, which

form together a basis set. The accuracy of EME is limited by the number of considered

modes. An advantage of the method compared to BPM is its bi-directionality, which makes

it possible to take all reﬂections into account [12]. Among the disadvantages of EME is its

complexity for structures with a large cross-section. This makes it applicable rather for

problems such as very long waveguides. Exemplary solutions of other practical problems

with EME can be found in Ref. [12].

Another family of methods is dedicated to the simulation of multi-layer or periodic struc-

tures. For a 1D periodicity, the transfer matrix method (TMM) can be applied. In a two-port

multi-layer system, the forwards- and backwards propagating waves of a given mode in

the beginning and the end are coupled by a 2

2 matrix. The elements of this matrix

result from the Fresnel coeﬃcients for reﬂection and transmission at each layer interface.

The method can be applied for the simulation of 1D photonic crystals, Bragg mirrors,

anti-reﬂection coatings, waveplates, and polarization converters [6, 7]. To account for

2D-periodic problems, the Fourier Modal Method (FMM), known also as the Rigorous

2 Methods for Analysis of Two-Dimensional Grating Couplers

Coupled Wave Analysis (RCWA) can be considered [7]. Here, the electromagnetic ﬁeld in

each layer is expanded into a number of eigenmodes with a characteristic propagation

constant. For each eigenmode, a forwards- and backwards propagating component can

be assigned. The relationship between eigenmodes in different layers is expressed by the

Floquet-Bloch theorem, stating that in a periodic array, the ﬁeld function in a given layer

differs from the function after one period by a ﬁxed phase shift. A complete ﬁeld function

can be thus given by the sum of the phase-shifted functions over all periods. Characteristic

for FMM is that the Floquet-expansion is truncated at the same number of elements as the

eigenmode expansion, determining with this the accuracy of the ﬁeld approximation [8].

Obviously, the classical method deﬁnition assumes an inﬁnite or suﬃciently large number

of periods. The particular problem must be such that it can be reasonably approximated

by these assumptions. In 2D

simulations, the number of periods is determined by the

mode spot size, to which the radiated grating ﬁeld must be matched. It is questionable,

whether the ﬁxed number of eigenmodes is suﬃcient to describe a high index contrast 2D

with a good accuracy. In Ref. [9], adapted FMMs are proposed, which have, however,

even in the best case a rather high computational cost.

Due to the limitations of the methods referred until here, the latter have not been consid-

ered for the solution of a full 2D

problem. For the simulation of radiating structures,

including their near-ﬁelds’ propagation, more generalized methods are necessary. In the

following, several of the most universal simulation techniques will be reviewed. We shall

focus particularly on the numerical method used within the scope of this thesis.

2.1.2.3 Generalized Numerical Methods in the Frequency Domain

We begin with the discussion of generalized approaches, which are more suitable for an

implementation in the frequency domain.

Boundary Element Method (BEM) As the name suggests, it belongs to the boundary type

of numerical methods and has its roots back to the 1960s [2]. It is an alternative to the well-

known domain methods - the ﬁnite elements method (

FEM

) and ﬁnite differences method

(

FDM

). Due to the similarities in terms of discretization, BEM has been often compared

FEM

[2,13]. In the following, a short description of the main method’s properties are

summarized from Ref. [2]. The reader may refer to Chap. 9 in Ref. [2] for an exhaustive

explanation of BEM.

Generally, BEM is based on the boundary integral equation and the fundamental solution

of the governing equation of the problem. The boundary integral equation is derived

from the governing equation e.g. through the principle of the weighted residuals. The

fundamental solution is chosen as a weighting function there to form a matrix equation.

The values of the function and its normal derivative along the boundary are unknown.

For the discretization of the equation, the boundary is divided into a number of elements

and nodes. The unknown function is interpolated by shape functions, which can be linear,

quadratic, cubic, etc.

2.1 Numerical Simulation and Evaluation

A typical discretized equation of BEM has the form:

[H]U=[G]Q(2.5)

with U- a vector with the nodal values of the unknown function on the boundary and Q- a

vector with the nodal values of the normal derivative of the unknown function. The square

matrices Hand Gresult from the fundamental solution of the governing solution and the

shape function of the discretization.

The main advantage of BEM is that it reduces the dimensions of a given problem by one.

For a 3D problem, only the domain’s surface needs to be discretized. This reduces the

matrices’ dimension and the corresponding number of equations to be solved. The method

is also useful for the solution of open boundary problems as well and requires no artiﬁcial

absorbing boundary conditions [13]. However, BEM has some signiﬁcant drawbacks. The

matrices Hand Gare fully populated and in general not symmetric [2]. Depending on

the particular problem, this can lead to a signiﬁcant computational effort and memory

requirements [13]. Another issue is the fundamental solution of the governing equation -

such a solution is not trivial for certain problems. In BEM, the calculation of the coeﬃcient

matrix in each element requires more time, e.g. compared to

FEM

[2]. The concerns

regarding computational complexity and time are the main reason to consider BEM not

suitable for the solution of a 2D GC problem.

Finite Elements Method (FEM)

FEM

is among the oldest numerical methods, which

is used nowadays for the simulation of problems in a large number of scientiﬁc ﬁelds.

The method dates back to the 1930s and was originally developed to solve problems

in structural mechanics. In the 1970s, its utilization for electromagnetic problems was

proposed. The progress towards computationally powerful hardware allowed for the com-

mercialization of many FEM-based tools, which were already well-known in the 1990s [2].

In the following, the basic features of

FEM

will be summarized, based on Chap. 4 in Ref. [2].

Interested readers may refer to Chaps. 4, 5 and 6 in Ref. [2], in which a comprehensive

revision of FEM is provided.

The workﬂow of

FEM

includes the subdivision of the computational domain into a number

of small interconnected subregions - “elements” - which may have generally an arbitrary

shape. Currently, triangles or tetrahedrons are most commonly used. Because of the

variable size and extension of the elements, a given geometry can be discretized very

accurately. Therefore, objects with complicated shapes and material composition can be

handled. The density of the elements may vary, depending on the particular problem.

Areas with strong ﬁeld gradients will be discretized more precisely, which can be applied

e.g. to calculate the ﬁeld strength at sharp metal edges with a good precision.

Similarly to BEM, the unknown function within the element is approximated by an in-

terpolation function. The weighted residuals or the variational principle can be used for

the composition of a matrix equation. In contrast to BEM, the resulting system matrix is

symmetric.The basic steps in FEM are:

2 Methods for Analysis of Two-Dimensional Grating Couplers

Discretization of the solution domain by elements. Within each element, the unknown

function included in the governing equation uis approximated by

(cf. (2.1)), where

eassigns the particular element. If we assume a triangular element with the vertices

i,j,m, the unknown function uwill be approximated by the nodal values u

and

the shape functions Ne

i,Ne

j,Ne

u≈ˆ

ue=X

kuk(k=i,j,m) (2.6)

The element matrix equation is derived by the weighted residuals principle or by the

variational principle.

The matrix equation at every node of the domain is assembled to form a system

matrix equation, which has the form:

Ku =B(2.7)

with ua column vector with the size N, where Nis the total number of nodes. The

global matrix Kis called assembled matrix, system matrix or stiffness matrix. It is

a sparse, symmetric and positive deﬁnite matrix with the order N

N. The column

matrix Bincludes the known boundary conditions.

4) The solution of (2.7) delivers the values of the discretized function on every node.

Regarding the given 2D

problem,

FEM

would be capable to model with excellent

accuracy the material composition given by the complex

BEOL

stack. However, due to

the simultaneous presence of very thin and very thick layers, the mesh quality may suffer.

Furthermore, due to the high index contrast and the geometrical extension of the problem,

a large number of mesh cells and nodes Ncould be expected. This leads to further practical

issues, including:

•time consuming meshing of the large computational volume,

•

large equation matrices, requiring large random access memory (RAM) and large

disk space for storing of the results.

In frequency-domain based methods, the latter issue becomes very critical when analyzing

large problems such as 2D

s. For a given spectrum, the solution of the matrix equation

must be carried out for a discrete number of wavelengths N

until a certain error mini-

mization criterion is fulﬁlled. This means that N

simulations will be necessary. By contrast,

time-domain methods solve the problem for the full spectrum range simultaneously. One

could argue that in time-domain methods we need many discrete time steps. However,

the solution of an equation within a given time-step is signiﬁcantly faster and less RAM

demanding compared to the solution within a ﬁxed frequency-domain step with FEM. To

reach the same speed of simulation, the FEM requires more powerful hardware than

its time-domain alternatives, which is a practical limitation, when such resources are not

given.

2.1 Numerical Simulation and Evaluation

Naturally, the implementation of

FEM

in the frequency-domain is not mandatory. However,

a time-domain implementation of

FEM

is very challenging. The reason is that it is diﬃcult

to realize an explicit time integration scheme, because the latter requires the inversion of

a non-diagonal material matrix. This process is computationally costly. There are strategies

to eliminate the off-diagonal elements, which, however, work well only in a hexahedral

(Cartesian) mesh. In tetrahedral meshes, indeﬁnite matrices may result, raising stability

concerns [14]. For these reasons, the most commercial

FEM

solvers for transient elec-

tromagnetic problems are implemented in the frequency domain. Due to the practical

limitations of

FEM

in the frequency domain, requiring large computational resources, this

method was not considered as suitable for the solution of the 2D GC problem.

In this work, the 2D

s are completely analyzed by a ﬁnite integration technique (

FIT

)

time-domain solver. In the following, the latter will be explained in more detail. Due to

its similarity to the well-known ﬁnite differences time domain (

FDTD

) technique, parallels

between both methods will be made.

2.1.2.4 Generalized Numerical Methods in the Time Domain

In the following paragraphs, the

FDM

and

FIT

methods in time domain will be discussed.

Both numerical approaches are closely related to each other and are available in many

commercial and open-source software packages for a large variety of applications. The

mathematical basis of

FDM

is among the oldest ones - it is suggested that even Gauss

and Boltzmann were familiar with it. Nevertheless, the practical application of

FDM

engineering problems started no earlier than in the 1940s. Its simple application made

the method a valuable means for the solution of a large variety of practical problems [2].

The

FDM

adapted for the solution of transient electromagnetic ﬁelds, known as the

FDTD

method, was ﬁrst proposed by Yee in 1966 [15]. Later on,

FIT

was developed by Weiland

in 1977 [16], which is considered as the more generalized version of

FDTD

, being able to

handle electrostatic, low- and high-frequency problems (cf. e.g. [17,18]). Some of the most

signiﬁcant common characteristics of FDTD and FIT in the time-domain are:

•

The usage of a dual grid, i.e a secondary computational grid, where the magnetic

ﬁeld components are half-step width shifted from the electric ﬁeld components.

•The usage of the leapfrog scheme as an update approach for the calculation of the

ﬁeld components at the next time step.

Both concepts will be explained in more detail in the following. The basic difference between

FDTD

and

FIT

is the starting point of these approaches. In

FDTD

, the Maxwell’s equations

in differential form are replaced by a set of ﬁnite difference equations, by using the

FDM

approximation of the ﬁrst derivative, which is contained in the operators for rotation or

divergence [15]. In

FIT

, the Maxwell’s equations in integral form are transformed to a set

of algebraic equations without an approximation at that stage. An approximation error

results from the material matrices [17].

2 Methods for Analysis of Two-Dimensional Grating Couplers

Discretization with FDTD As the name suggests, the basic approximation step in

FDM

the so called ﬁnite differences approximation, i.e. the deﬁnition of the ﬁrst derivative of a

function fat the point x

is substituted by a subtraction within a discrete element in

the range x0±Δ:

f′(x0)=lim

h→0

f(x0+h

2)–f(x0–h

h≈f(x0+Δ)–f(x0–Δ)

2Δ(2.8)

Instead of using the limes of an inﬁnitely small variation haround the variable x, we take a

ﬁnite segment on the x-axis. Geometrically, we substitute the slope of the tangent at x

the slope of a line passing through x

0±Δ

. It is intuitive that the approximation approaches

the deﬁnition for smaller segments Δ→0 [19].

In the following, an exemplary discretization of a Maxwell’s equation will be given from

Ref. [15]. Assume e.g. the Maxwell-Faraday’s equation:

–∂

∂t=∇×

E(2.9)

In a Cartesian coordinate system, the equation can be decomposed as:

–∂Bx

∂t=∂Ez

∂y–∂Ey

∂z(2.10)

–∂By

∂t=∂Ex

∂z–∂Ez

∂x(2.11)

∂Bz

∂t=∂Ex

∂y–∂Ey

∂x(2.12)

With the ﬁnite differences approximation in (2.8), a set of scalar equations result. Deﬁne

a spatial grid point as (i,j,k)

x,j

y,k

z) and a function of the space and the time with

index n(i.e. t=n·Δt): F(iΔx,jΔy,kΔz)=Fn(i,j,k). (2.10) can be then written as:

Bn+1

x(i,j+1

2,k+1

2)–Bn–1

x(i,j+1

2,k+1

Δt=

y(i,j+1

2,k+1)–En

y(i,j+1

2,k)

Δz

–En

z(i,j+1,k+1

2)–En

z(i,j,k+1

Δy

(2.13)

From (2.13), it is evident that the x-component of the magnetic ﬂux is half-cell shifted from

the position of E

on the z-axis and from the position of E

on the y-axis. The reason is that

both the differences

ΔEy

Δz

and

ΔEz

Δy

should correspond simultaneously to a single clearly

deﬁned point B

in the space coordinates [14,15]. This leads us to the concept of the Yee’s

staggered grid, which contains a primary grid for the electric ﬁeld components, which are

placed in the middle of the edges, and a half-cell shifted (in all directions) dual grid for the

magnetic ﬁeld components. With this, the magnetic ﬁeld components can be found at the

centers of the faces of the primary grid and in opposite - the electric ﬁeld components are

placed at the centers of the faces of the dual grid.

2.1 Numerical Simulation and Evaluation

Figure 2.4:

An exemplary representation of the Yee’s cell, consisting of two staggered grids.

In the primary grid, electric ﬁeld strength



and ﬂux



components are placed

in the middle of the edges. A dual grid, shifted by a half grid cell in all directions,

contains the magnetic ﬁeld strength



and ﬂux



components. With respect to

the primary grid, the magnetic ﬁeld strength and ﬂux are deﬁned in the center

of the faces. (Representation after Refs. [15,20]).

An illustration of the Yee’s grid can be seen in Fig. 2.4 [15,20]. Another novelty within the

FDTD

method was the introduction of a staggered grid also in time, i.e. the electric and

magnetic ﬁeld components are shifted by a half time step as well [20]. This arrangement

is nowadays known as the leapfrog scheme. A schematic representation of the leapfrog

principle is shown in Fig. 2.5. Basically, all electric components are computed in the space

domain and stored for a particular time point, by using previously stored magnetic ﬁeld

data. After this, all magnetic ﬁeld computations in the space are carried out by using the just

computed electric ﬁeld data. The process continues until the time stepping is completed.

The leapfrog method has the following properties [20]:

•

It is fully explicit, so that problems related to simultaneous equations and matrix

inversion are avoided.

•

The time derivatives are calculated with the same ﬁnite differences approach as the

spatial derivatives and have the same accuracy order (second order).

•

The algorithm is non-dissipative, i.e. the wave that propagates in the mesh does not

decay due to non-physical artifacts of the time-stepping algorithm.

Explicit time update schemes are conditionally stable and put a limitation on the maximal

time step, which can be chosen. As a stability criterion, the Courant condition is imperative:

[17]

Δt≤cs1

Δx2+1

Δy2+1

Δz2–1

(2.14)

with c

√εμ

the light velocity in a material with a the permittivity

and the permeability

Especially for simulations in the optical domain, this criterion puts serious constraints on

the time, which is necessary for a simulation. The short wavelengths in the optical range

require a very ﬁne spatial grid and a resolution in the nm-range is typical. This leads to a

2 Methods for Analysis of Two-Dimensional Grating Couplers

time step

∼

–17

s. During the past years, a huge effort was put to develop algorithms

that overcome this limitation and in Ref. [21] an explicit method using Faber polynomials

was proposed. Unfortunately, no commercial realization of timely-eﬃcient algorithms is

currently available, so that in the scope of this work the simulation’s duration was a limiting

factor for the structure’s development.

Figure 2.5:

An exemplary space-time representation of a 1D wave propagation. A staggered

grid for the electric and magnetic ﬁeld strength



and



is used both in space

and time. Central differences are used for the space derivatives and leapfrog

for the time derivatives. (Representation after Ref. [20]).

The last important characteristic of

FDTD

, which will be addressed in this paragraph is the

effect of the numerical (grid) dispersion. The latter results from the spatial discretization

of the

FDTD

grid, leading to a frequency-(resp. wavelength-) dependent phase velocity.

The numerical dispersion is especially critical for electrically large problems, where accu-

mulated delay or phase error can result. This may lead to non-physical effects such as

broadening of pulsed waveforms, anisotropy and pseudo-refraction. An extensive analysis

on the numerical dispersion is given in Chap. 4 in Ref. [20]. A number of contributions

in the literature have been dedicated to the analysis of grid dispersion in

FDTD

and its

minimization. Generally, this can be achieved either by choosing a ﬁner discretization

step or by employing a higher-order approximation of the derivatives [22]. In Ref. [22] the

derivative’s expansion in terms of Chebyshev and Butterworth polynomials is discussed.

The implementation of an artiﬁcial anisotropy has been proposed as well [23]. Modern

commercial

FDTD

software packages (e.g. Lumerical FDTD [24]) use a non-uniform, graded

mesh with a ﬁner discretization of regions with high refractive index, in order to reduce

the numerical dispersion.

Discretization with FIT The

FIT

shares the special features of the staggered spatial and

time grid, introduced in the

FDTD

method. In this paragraph, only the main differences in

terms of discretization will be outlined, based on Refs. [14,16,17].

2.1 Numerical Simulation and Evaluation

The

FIT

will be introduced, again using the Maxwell-Faraday’s equation, but this time

in integral form:

–¨A

∂

∂td

A=˛∂A



Ed

s(2.15)

Now, the Yee’s grid will be modiﬁed - instead of electric and magnetic components in the

middle of the primary or the dual grid, grid voltages eand magnetic ﬂuxes bare introduced

(Fig. 2.6 ). Their bold notation assigns a full-grid vector. The Maxwell’s equation in a single

rectangular grid cell nwith edges Li,i=1,2,3,4 and a facet Ancan be written as:

e1+e2–e3–e4=–˙

with ei:=ˆLi



E(

r,t)d

s,i=1,2,3,4 bn:=¨An



B(

r,t)d

A(2.16)

where the dot denotes a time derivative. The grid voltages e

are deﬁned in Fig. 2.6. It should

be noted that within the discrete cell, no approximation took place so far, in contrast to

FDTD

, where the derivative is approximated by a difference. Next, the coeﬃcients of the

algebraic equations of all grid cells can be grouped into a matrix C:







1 1 –1 –1 ···

· · · · ···



















=–













⇒Ce =–˙

b(2.17)

Due to the similarity to the Maxwell’s equation in differential form, the matrix Cin (2.17) is

called curl matrix. Similarly, all Maxwell’s equations can be discretized and represented in

a compact matrix form.

Figure 2.6:

An exemplary representation of the staggered grid used by the ﬁnite-integration-

technique (FIT). In the primary grid, electric grid voltages eand ﬂuxes dare

deﬁned. A dual grid, shifted by a half grid cell in all directions, contains the

magnetic voltages hand ﬂuxes b. (Representation after Ref. [17].)

2 Methods for Analysis of Two-Dimensional Grating Couplers

In an analogous way, the magnetic grid voltage h, the electric grid ﬂux d, the electric grid

current jand the grid charge qin the dual grid cell

with edges

1,2,3,4 and a facet

Ancan be introduced:

hi:=ˆ˜



H(

r,t)d

s,i=1,2,3,4 (2.18)

dn:=¨˜



D(

r,t)d

A(2.19)

jn:=¨˜



J(

r,t)d

A(2.20)

qn:=˚˜

ρ(

r,t)dV(2.21)

(2.22)

The Maxwell’s equations in matrix form can be summarized as:

˛∂A



Ed

s=–¨A

∂

∂td

A⇒Ce =–˙

b(2.23)

˛∂A



Hd

s=¨A

∂

∂t+

Jd

A⇒˜

Ch =˙

d+j(2.24)

‹∂V



Bd

A=0⇒Sb =0 (2.25)

‹∂V



Dd

A=Q⇒˜

Sd =q(2.26)

Here, the tilde assigns a discretized equation on the dual grid, the matrices C,

are called

curl matrices and S,

- source matrices. The matrices are sparse, which makes their storage

less memory demanding. It is worth mentioning that the matrices fulﬁll basic identities:

SC =˜

S˜

C=0⇔div rot =0 (2.27)

C˜

ST=˜

CST=0⇔rot grad =0 (2.28)

with Tassigning the transpose of a matrix.

The last aspect to discuss are the material relations. Since eand d, resp. hand bare

deﬁned independently, their quotient corresponding to the permittivity or the permeability

introduces a discretization error. The same applies to the conductivity as well. Assume the

relation fwithin the cell nwith the edges Li,i=1,2,3,4 and the face An:

f(

B,

H)=ˆAn



Bd

˛˜



Hd

s≈Dμ,n. (2.29)

As an approximation of f, the permeability matrix D

μ,n

can be deﬁned. Analogously, we

2.1 Numerical Simulation and Evaluation

obtain the permittivity and conductivity matrices:

d=Dεe j =Dκe(2.30)

The material matrices are diagonal in the case, when linear materials or diagonal tensor

materials are present.

2.1.2.5 Open Boundary Conditions

All discussed “element methods”, including

FEM

FDTD

and

FIT

require a computational

volume with a restricted size. This is particularly inconvenient, when radiating structures

need to be investigated. This motivated the research and development of an absorbing

boundary condition (ABC), which is able to truncate the computational space with an

absorbing material. Ideally, such a medium has a low thickness, no reﬂection for any type

of incident wave and is also able to handle the near-ﬁeld of a given source [20]. The

most successful work towards the realization of a good ABC was the development of the

novel technique of the perfectly matched layers (

PML

), which was able to substitute open

problems with boundless space by an artiﬁcial boundary condition. This approach was

ﬁrst proposed by Berenger [25]. In the following, some important speciﬁcs of the

PML

technique will be summarized from references [20,25].

The signiﬁcant contribution of

PML

is the fact that the technique allows for the absorption

of electromagnetic ﬁelds independent of the angle of incidence, the frequency or the

polarization. Moreover, ﬁeld distributions other than plane waves can be handled. This is

enabled by the introduction of a

PML

layer made of an artiﬁcial material with an electric

conductivity

σE

and magnetic conductivity

σH

. The

PML

is reﬂectionless, provided that the

electric and magnetic conductivity are related by:

σE

ε=σH

μ(2.31)

where

is the permittivity and

the permeability of a given medium [25]. Furthermore,

Berenger proposed a split-ﬁeld formulation of the Maxwell’s equations, in which each

vector ﬁeld component is split into a subset of two orthogonal components. With this, 12

components result, which are coupled by partial differential equations. The loss parame-

ters within these equations are chosen according to (2.31) to ensure a perfect matching

condition for any type of an incident ﬁeld. One of the weaknesses of the Berenger’s method

is the fact that the matching condition (2.31) is perfectly fulﬁlled only in the continuous

space. In the discrete

FDTD

(or

FIT

) lattice, the material properties are represented in a

discrete form as well, with spatially staggered electric and magnetic parameters. Berenger

investigated possibilities to reduce the numerical reﬂection. Aside from enhancing the

PML

thickness, another possibility is to gradually increase the

PML

loss in the direction

deﬁned by the interface’s normal [20,25].

The original Berenger’s method was limited to a free-space or homogeneous lossless

medium [26]. Moreover, it has been shown that the split-ﬁeld

PML

is not able to absorb

2 Methods for Analysis of Two-Dimensional Grating Couplers

evanescent ﬁelds [26,27]. Many of the subsequent works attempted to address these

issues. With a stretched-coordinate

PML

formulation from [28], in Ref. [29], a general-

ized

PML

was proposed. The latter accounts for the absorption of both propagating and

evanescent ﬁelds in lossless and lossy media. An alternative approach to handle lossy

and dispersive media was proposed by Gedney, referred as the uniaxial anisotropic PML

method [30]. Another ABC handling dispersive materials was given by Uno [26] with an

approach, basing on the original Berenger’s

PML

. Fully anisotropic [31] and nonlinear

PML [32] methods have been reported as well.

For the transfer of

PML

methods to commercial software, their computational require-

ments are decisive. The convolutional

PML

[33] and the complex-frequency shifted

PML

from [34] are implemented in some of the modern software packages. In this work, the

commercial software Simulia CST was used for all simulations. The latter uses the

FIT

in time domain with a convolutional

PML

. The method is reported to be able to handle

any type of materials, incl. lossy, inhomogeneous, dispersive or nonlinear materials in an

eﬃcient manner, without changing the original formulation and with a comparatively low

global error [33]. In this work, a

PML

reﬂection level of 0.1 ‰was found to be suﬃcient.

The consideration of a material dispersion, however, was found to be time consuming and

was omitted at a later stage. Potential errors with regard to this and other aspects will be

discussed in the next subsection.

2.1.2.6 Summary of Numerical Errors

In this section, the potential sources of numerical errors will be summarized, basing on

the software that has been used to obtain the numerical results in this work. As already

noted, all simulations have been performed with the commercial

FIT

time-domain solver

of Simulia CST Studio Suite (previously known as CST Microwave Studio). The list of errors

is derived from the documentation of CST [35] and of the very similar software Lumerical

FDTD [24]. The following basic sources of numerical errors can be indicated:

Figure 2.7:

(a) A generic schematic of a 2D grating coupler (2D GC). Only a single row

of perturbing elements, indicated by the red dashed line, is considered for

a convergence analysis. (b) A model in CST Studio Suite, the backend of line

(BEOL) stack is not shown. The mode excitation is in x-direction from Port 1 to

Port 2. (b) An electric ﬁeld distribution at a 1310nm wavelength.

2.1 Numerical Simulation and Evaluation

Spatial grid discretization The spatial grid discretization has a direct impact on the time

increment and the grid dispersion. Obviously, the spatial grid must be suﬃciently ﬁne

to guarantee for results with an acceptable accuracy. The recommended resolution by

CST is 15 grid cells per minimal wavelength. Whether this value delivers satisfying results

for a given problem can be proven by a convergence analysis. In the case of a full 2D

model, such an analysis becomes very diﬃcult. On the one hand, the simulations

become very time consuming with increasing spatial resolution. In addition, models with

a large number of grid cells require a hardware with a suﬃcient RAM. A convergence

analysis can be thus limited by the available computational resources. For these reasons, a

convergence analysis is performed in a simpliﬁed model, containing even in the worst-case

a reasonable number of grid cells . Figure 2.7 (a) shows a generic schematic of a 2D

. For

the simpliﬁed model, we consider one row of perturbing elements, indicated by the red

dashed line. Typically, circular perturbing elements are used. In Fig. 2.7 (b), the actual model

in CST can be seen. The waveguide width corresponds to the feeding taper’s end width.

The waveguide height is 220nm on a

BOX

of 2µm. In this work, 2D

s for both C-band

(1530–1565 nm) and O-band (1260–1360 nm) have been investigated. For that reason, the

simpliﬁed model comprises circular perturbing elements with typical dimensions for C-

or O-band 2D

s, namely 400nm diameter with a 620nm periodicity for C-band and

280 nm with a 480 nm periodicity in O-band. Their etch depth is intentionally chosen to be

large - 200 nm - to make sure that the model includes both radiated and propagating ﬁelds.

The analysis is carried out in both optical bands. A symmetry plane at y

0 is chosen with

tang =

0, reducing the total number of cells by a factor of two. Two ports are deﬁned and

the fundamental TE

is excited by Port 1 in x-direction, indicated by the black arrow. The

structure is 5 µm long to avoid potential interactions between both ports. An exemplary

electric ﬁeld distribution at 1310nm is shown in Fig. 2.7 (c).

Figure 2.8:

The balance parameter of a simpliﬁed model for a different number of grid cells

per minimal wavelength Ncells. (a) Results in C-band. (b) Results in O-band.

To study the convergence the balance parameter Bis used as a ﬁgure-of-merit. With two

ports, Bis calculated from the port S-parameters as:

B=q|S1,1|2+|S2,1|2B[dB] =20logB. (2.32)

2 Methods for Analysis of Two-Dimensional Grating Couplers

Because we have a radiating structure, |B|

will be < 1. The number of grid cells per minimal

wavelength N

cells

is varied as N

cells =

10,15,20,25,30. The minimal free-space wavelength is

1500nm in simulations covering C-band and 1260nm in O-band simulations.

Figure 2.8 shows the results of the convergence analysis in both optical bands. In O-band,

a slightly larger deviation can be observed for N

cells =

10. All other scenarios deliver almost

unchanged results, making the choice of 15 cells per minimal wavelength reasonable.

On the other hand, the simulation in C-band shows a larger difference, especially at the

minimal wavelength of 1500 nm. In this case, the choice of N

cells =

25 could be more reliable.

However, in a full 2D

model, this resolution results in too large number of grid cells, for

which no suﬃcient RAM resources were available. Since the difference between N

cells =

and N

cells =

25 at 1500nm is about 0.1dB (a relative error of < 2%) - a value, which is

below the accuracy limit of measurement results - the discretization error is considered as

acceptable. Thus, a resolution of Ncells =15 is chosen in C-band as well.

Staggered grid errors The staggered grid errors are related to the independent deﬁni-

tions of different ﬁeld or ﬂux parameters and to the half-space shift of the magnetic and

electric ﬁelds/ﬂuxes. An example for such a grid-related inaccuracy is the discretization

error of the material relation, shown in the previous subsection. Similar error will occur

also in the power ﬂux calculation, because electric and magnetic ﬁeld components are not

deﬁned at the same spatial point.

Geometry approximation error The circular diffracting elements of the 2D

cannot

be approximated perfectly by a hexahedral mesh. Another diﬃculty is the good

BEOL

stack

modeling, since the latter comprises of both very thick and very thin layers. Again, a ﬁne

enough spatial grid can minimize this kind of error.

Wavelength-dependent refractive index in the time domain Typically, ﬁtting models

are used for the transformation of a wavelength-dependent refractive index into the time-

domain, which inevitably cause a ﬁt error. In this work, the wavelength dependence is

omitted, especially in order to accelerate the simulation. For the same reason, no mode

tracking is used to calculate the waveguide’s effective refractive index for different wave-

lengths. Generally, this leads to an error of the 2D

’s phase condition for wavelengths,

different from the central wavelength. Therefore, the simulation bandwidth window has

always been centered at the 2D

’s design wavelength. A parameter that remains affected

is the 2D

’s bandwidth. Its exact simulation is not a trivial problem. The present work is

not focused on the bandwidth optimization - the indicated constraints should be taken

into account, in case that future works would address this aspect.

Non-uniform meshing Generally, this kind of mesh improves the geometric approxima-

tion and can be used to reduce the grid dispersion. However, it may cause some issues,

such as grid scattering or enhanced

PML

reﬂection. These effects could again have an

impact on the calculated 2D

bandwidth. Nevertheless, the eﬃciency of the non-uniform

2.1 Numerical Simulation and Evaluation

mesh is more decisive, so that its usage is now well-established in commercial software

packages.

Waveguide ports and external ﬁeld sources Since the ports have their own bound-

ary conditions, it must be ensured that they have no interaction with the excited ﬁelds.

The ports need to be large enough, in order to calculate a waveguide mode accurately,

considering properly its evanescent component. The port boundaries should not touch

PML

boundaries. Because the waveguide width at the 2D

is very large and supports

theoretically many modes, it should be ensured that higher-order modes will be absorbed

by the port. External ﬁeld sources should also be preserved from grating’s back-reﬂected

or scattered ﬁelds. Modeled structures with a large perturbation strength will be more

affected by such interactions.

Field monitors The electric, magnetic or power ﬂux distribution is spatially interpolated by

the monitors and thus contains an interpolation error. In the case of 2D

s, the monitor

ﬁeld results are projected on a tilted plane, which has also an impact on that kind of error.

Furthermore, ﬁeld sources from a source simulation may be used in a full 2D

simulation

to account for tilted waveguide excitation. A discretization mismatch may be introduced

in that case. Typically, the ﬁeld monitors’/sources’ interpolation is good enough to keep

errors from such deviations small.

PML

reﬂections The reﬂections at the PML boundaries are inevitable and depend on

the ﬁeld’s angle of incidence. The number of

PML

layers must be such that the reﬂection

is suﬃciently low. In CST, the parameter estimated reﬂection level can be chosen. The

recommended value of 0.1 ‰is suﬃcient for the purposes of this work. To illustrate this,

the simpliﬁed models for C- and O-band are simulated for different

PML

reﬂection levels

at a resolution of 15 cells per minimal wavelength. The results in Fig. 2.9 conﬁrm the low

impact of PML reﬂections on the simulation results.

Figure 2.9:

The balance parameter of a simpliﬁed model for different levels of the perfectly

matched layer (PML) reﬂection in (a) C-band, (b) O-band. The results do not vary

signiﬁcantly, showing that a reﬂection level of 10–4 is a reasonable choice.

2 Methods for Analysis of Two-Dimensional Grating Couplers

2.1.3 Simulation Procedure

The complete procedure for the design of a 2D

follows some basic steps, which will be

summarized here.

Figure 2.10:

An illustration of the components relevant for a complete 2D grating coupler

(GC) model (only one GC arm is depicted: waveguide (WG) 1). (a) 2D GCs are

designed for Si rib waveguides with a slab, deﬁned by the same etch depth

das the grating. (b) Depending on the 2D GC design - linear or focusing -

tapers with different properties are needed. The linear 2D GC requires a

long adiabatic taper, which can be simulated separately from the grating. The

focusing 2D GC requires short tapers, which have to be simulated together

with the grating.

2.1.3.1 Modeling

Each simulation procedure starts with the modeling of a structure, which has to be opti-

mized numerically. In a basic conﬁguration, a Si waveguide as in Fig. 2.10 (a) is widened by

a taper structure (Fig. 2.10 (b)) to obtain a suﬃciently large spot size. The enlarged Si mode

at the taper’s end is then fed into the grating to be phase-matched with the mode of a

SMF. In a 2D GC, we have two feeding sides respectively (only one depicted). The 2D GCs

in this work are mostly designed for Si rib waveguides with a Si thickness of 220nm, slabs,

deﬁned with the same etch depth as the grating’s etch depth, and core width between

400 nm and 500 nm, depending on the optical band (Fig. 2.10 (a)). The underlying

BOX

layer

is 2 µm thick, the cladding is formed by a complex

BEOL

stack (not shown). Depending on

the particular GC design, the tapers can be designed in two ways (cf. Fig. 2.10 (b)).

Linear 2D

s comprise perturbing elements, which are placed along straight lines.

In this case, an adiabatic linear taper is necessary. The latter is typically simulated

externally, so that the simulation of the 2D

itself starts at the taper’s end. Addi-

tional simulations to obtain the proper taper length have been executed. The taper

simulations have been kindly performed by the colleague Karsten Voigt using the

2.1 Numerical Simulation and Evaluation

commercial software Fimmwave/PhotonDesign. The results show that a taper length

of about 250 µm is suﬃcient for both C- and O-band devices.

Focusing 2D

s comprise perturbing elements, which are placed along curved lines.

In such a case, the feeding taper is signiﬁcantly shorter (mostly between 20 µm and

40µm), changing abruptly the mode size. The Si mode’ wave front becomes curved.

Because of the speciﬁc wave type, the taper has to be simulated together with the

2D GC. The exact modeling procedure will be given later in this section.

Regarding the waveguides’ and tapers’ geometry, it should be noted that in practice their

side-walls are angled and not perfectly vertical. The side-wall angles are not considered

here, because they have no direct impact on the 2D

s’ performance. In addition, the

waveguides have no sharp edges, in contrast to the numerical model. For that reason,

interested researchers, who would like to reproduce the simulations in CST, need to

disable the option “consider for snapping” within the mesh properties. The latter enforces

a strongly inhomogeneous mesh for the precise geometric representation of edges, which

is not necessary in our case. The omission of this option greatly reduces the total number

of mesh cells of a numerical model, leading to relaxed RAM and disk space requirements

and shorter simulation duration. Another important point is the geometric representation

of etched diffracting elements, which is also not exact. In the reality, the diffracting elements

have a conical shape, which depends on the etch depth. In CST, the diffracting elements are

generally modeled as cylinders and have therefore a larger volume as the fabricated ones.

This affects the exact representation of the grating’s perturbation strength. This fact must

be kept in mind, when numerical and experimental results are compared. No attempts

towards higher geometric accuracy have been undertaken, since it is very hard to determine

the exact shape after etching and to consider all factors that have a simultaneous impact

on it. Another relevant aspect are the proper material deﬁnitions. The material properties

are assumed constant within a given band (e.g. O- or C-band) - no wavelength dependence

of the refractive indices is considered. The constant refractive indices for O- and C-band

differ, however. For Si, the assumed refractive indices are 3.5 in O-band and 3.47 in C-band.

The refractive indices and reﬂective properties of the remaining materials are conﬁdential

information of IHP Microelectronics.

In the following, the modeling procedure of the linear and focusing 2D

will be outlined.

Linear 2D GCs For linear 2D

s, the modeling is straightforward - the 2D

with all

BEOL

layers and their speciﬁc pattern can be modeled with the basic shape objects and the

boolean operations provided in CST. The periodicity can be simply carried out by copying

the diffracting element’s object. CST allows for the deﬁnition of local coordinate systems,

which are angled with respect to the global Cartesian coordinates. With this, periodicity

under a certain angle with respect to the Cartesian axes is also possible. All geometric

dimensions are deﬁned parametrically and can be easily updated.

Focusing 2D GCs By contrast, the modeling of 2D focusing grating couplers (2D

FGC

is more complex. The main challenge hides in the proper positioning of the diffracting

2 Methods for Analysis of Two-Dimensional Grating Couplers

elements, which is not just given by the periodicity in a certain direction. In 2D

FGC

s, these

positions result from the crossing of two arc arrays, which are deﬁned by a curve equations.

The latter result from the focusing condition of 2D

FGC

s, which will be described in more

detail in Chap. 5. Figure 2.11 shows schematically the calculation of the diffracting elements’

positions from the crossing of two arrays of arcs. The mathematical determination of these

positions is not easy to implement in CST. For that reason, it is performed externally within

a MATLAB code. Since these positions are dependent on the periodicity, they must be

re-calculated every time, when the periodicity is updated. No simple update within CST is

possible. The second diﬃculty hides in the deﬁnition of the diffracting elements themselves.

Each calculated position is the center of an object, which describes such an element. In

CST, there is again the problem that there is no directly implemented command, which

can be used to assign an object to a certain central position. For that reason, the objects’

generation is also performed externally, by using a Python code. The objects’ array is thus

generated in Python, considering the pre-calculated objects’ positions. The objects are

then exported as a GDS-ﬁle (GDS: graphic design system) and imported into CST. The GDS

ﬁle should not contain too many points, otherwise the import procedure fails or takes too

long time. For that reason, curved or circular objects are constructed with a small number

of points, which makes their shape less accurate. After the import procedure is completed,

the 2D objects can be extended to the 3D diffracting elements with an extension given by

the grating’s etch depth. Thus, etch depth can be updated parametrically, but the size of

the objects - not. For their adaption, the Python code must be used each time to generate

the array of objects with changed sizes. Of course, the new array must be re-imported each

time into the CST model. This makes the whole modeling procedure rather cumbersome.

Since 2D

FGC

s were investigated at the last stage of this work, the modeling procedure is

open for optimization in future works.

Figure 2.11:

An illustration of the calculation of the diffracting elements’ positions in a

2D focusing array, resulting from the crossing of two 1D-periodic arc arrays

(depicted by continuous and dashed arcs).

2.1.3.2 Deﬁnition of Excitation Sources

Generally, a transient solver uses a signal in the time domain as stimulating excitation. The

signal refers to a globally deﬁned frequency (resp. wavelength) range. In CST, different kind

2.1 Numerical Simulation and Evaluation

of signals are available. For simulations, in which the user is not interested in the analysis

in the low-frequency range, it is recommended to use the Gaussian-sine pre-deﬁned signal.

The latter contains no DC component (DC: direct current). Figure 2.12 shows an exemplary

Gaussian-sine signal deﬁned in O-band (1260-1360nm). The stimulation is performed in

ttime steps, with

tdetermined by the Courant condition (2.14) (typically

∼

–17

s).

The simulation is terminated after the signal’s energy decays to a suﬃciently low level (a

default value is -40dB). Aside from a single-port excitation with a given signal, there is the

possibility to excite two ports simultaneously, whereat time- or phase-shift between the

excitation signals can be deﬁned [35].

Depending on the particular kind of 2D

, different types of ﬁeld sources are applicable.

Generally, a waveguide port is used for the excitation of the fundamental TE

mode directly

in the vicinity of the grating, i.e. at the taper’s wide end. The taper itself is not included. The

port is chosen wide enough to represent the waveguide mode accurately. It is ensured

that the port does not touch the open boundaries and that the higher-order modes are

absorbed. To account for the wavelength dependence of the waveguide refractive index,

the option broadband mode tracking may be activated.

Figure 2.12: An exemplary Gaussian-sine excitation signal for simulations in O-band.

Particularly challenging is the simulation of 2D

s with angled waveguides, where angled

excitation ports are necessary. The latter are not supported by the hexahedral grid in CST.

To overcome this problem, a separate simulation in a source project of a small waveguide

piece with a standard port excitation is carried out and a ﬁeld source monitor is deﬁned. The

waveguide segment corresponds to the feeding taper’s end width. The ﬁeld source bounds

the excitation port and the simulated waveguide. The ﬁeld source data is saved after the

simulation of the waveguide piece. In the next step, the ﬁeld source can be imported

into the 2D

’s imprint project and used as an excitation source, which can be tilted with

respect to the grating. The ﬁeld source replaces ﬁelds in its interior by equivalent sources,

radiating outside. The procedure is based on the Huygens’ equivalence principle and is

used typically for antenna simulations. The equivalence principle is exact only, when the

environment in the imprint project remains the same as in the source project [35]. In 2D

s, this is obviously not the case for two reasons. First, since the ﬁeld source is tilted

to account for the excitation from tilted waveguides, an additional interpolation error

occurs. With a small tilt angle (typically

≤

◦

) and a suﬃciently ﬁne mesh in the source and

imprint projects, the error is small. The second violation of the equivalence principle is the

2 Methods for Analysis of Two-Dimensional Grating Couplers

existence of the grating itself, because the latter can reﬂect or back-scatter light and thus

affect the ﬁeld source. Errors from such interactions may be responsible for differences

between numerical and experimental results. In spite of these drawbacks, presently, the

external ﬁeld source excitation is the only option to account for tilted waveguides in CST.

In 2D

FGC

simulations, the ﬁeld source must replace the complete focusing taper in front

of the grating. The reason is that otherwise the excitation port cannot be included and

consequently the ﬁeld source does not “radiate” ﬁelds. This makes the computational

domain of 2D

FGC

s signiﬁcantly larger. However, the space occupied by ﬁeld sources does

not contain any ﬁeld values in its inner domain and does not increase signiﬁcantly the RAM

and disk space requirements.

Figure 2.13:

A decomposition of a full model into a source and imprint project. A 2D

focusing grating coupler (2D FGC) is shown on example - the feeding taper

is simulated within the source project. The ﬁelds saved by the ﬁeld source

monitor are imported into the imprint project and used as a tilted excitation

source.

The decomposition of a full model into a source project and an imprint project is shown

schematically in Fig. 2.13 for the case of a 2D

FGC

. The feeding taper is simulated in the

source project and the equivalent source is imported into the imprint project, which takes

only the grating area into account.

2.1 Numerical Simulation and Evaluation

2.1.3.3 Simulation Initialization

The basic setup parameters are the same for any kind of simulation. After the appropriate

material constants are chosen and the simulation wavelength range and basic units are

deﬁned, the geometric model can be prepared. Afterwards, open boundary conditions

in all directions are chosen and a free space above the grating is deﬁned for further ﬁeld

evaluation. An excitation source and type of time domain signal are speciﬁed afterwards.

The pulse width of the time signal is directly related to the simulation wavelength range.

If a non-default signal excitation is used, one should be aware to activate the results’

normalization during the solver setup. Finally, ﬁeld monitors for the electric and magnetic

ﬁelds can be deﬁned at the wavelengths of interest - typically covering the complete

simulation bandwidth in 10nm increment. The ﬁelds will be determined after the time

domain excitation is ﬁnished by a Fourier transform [35]. The number of ﬁeld monitors

given by the 10nm increment is limited by the required hard disk space, as the ﬁeld

monitors cover the complete simulation domain and contain a large number of complex

electric and magnetic ﬁeld values.

Figure 2.14:

An illustration of the transformations, necessary to construct a tilted plane

for results evaluation. The tilted plane accounts for the ﬁxed position of a

single mode ﬁber (SMF). (a) A starting point. (b) Coordinates (u

′

) after a

rotation of 45

◦

around the z-axis. (c) Coordinates (u

′′

) after a rotation of

θF

around the v

′

-axis. (c) The ﬁnal coordinates (u,v,w) after a rotation of –45

◦

around the w′′-axis.

2.1.3.4 Field Export

Among the signiﬁcant strengths of CST is the possibility to export ﬁelds at arbitrarily tilted

planes. This is especially valuable for the determination of the 2D

’s eﬃciency at a plane,

corresponding to the ﬁxed angled position of a

SMF

. The advantage of this technique is

that at this plane, the

SMF

’s ﬁeld distribution can be directly deﬁned and no projection of

a Gaussian beam on the Cartesian coordinates is necessary. In the following, the ﬁxed tilt

angles of the SMF will be assigned as φF,θF(with a subscript Ffor ﬁber).

The determination of the coordinates of a tilted evaluation plane (cf. Fig. 2.3 (b)) goes

2 Methods for Analysis of Two-Dimensional Grating Couplers

through three steps: starting from the Cartesian coordinates as in Fig. 2.14 (a), the ﬁrst

step is a rotation of

φ1=

◦

around the z-axis (Fig. 2.14 (b)). The coordinates after this ﬁrst

intermediate step are assigned as (u

′

). Next, a rotation of the angle

θF

around the v

′

axis is carried out, reaching the plane in Fig. 2.14 (c) with the coordinates (u

′′

). Finally,

a rotation

φ2=

–45

◦

around the w

′′

-axis is done, obtaining the ﬁnal (u,v,w)-coordinates.

The last step is necessary, in order to adjust the same angle between the x- and u-axes

and the y- and v-axes. With φF=φ1=–φ2, the ﬁnal transform matrix is:











=





cos2φFcosθF+sin2φF–cosφFsinφF(1–cosθF) –cosφFsinθF

–cosφFsinφF(1–cosθF) sin2φFcosθF+cos2φF–sinφFsinθF

cosφFsinθFsinφFsinθFcosθF







|{z }

=:AT

·









. (2.33)

It should be noted that CST always expects points in the global (x,y,z) coordinates for

export. This means that for a desired

SMF

position (u,v,w

=const.

), we need to go through

the opposite way of the transformation above. With the inverse transform matrix of AT:











=





sin2φF+cos2φFcosθF–sinφFcosφF(1–cosθF) cosφFsinθF

–sinφFcosφF(1–cosθF) cos2φF+sin2φFcosθFsinφFsinθF

–cosφFsinθF–sinφFsinθFcosθF







|{z }

=A–1

·









. (2.34)

Once the results are exported and are available in the (x,y,z), the transform matrix A

is used to bring them in (u,v,w) coordinates for further post-processing in MATLAB. In

this coordinate system, all evaluation steps such as power normalization, mode overlap

and coupling eﬃciency calculation are carried out. Due to interpolation errors, the initially

deﬁned (u,v,w

=const.

) points are not exactly the same as the exported ones. The deviation

can be corrected by adding a constant offset prior to export.

2.1.3.5 Initial Geometry

Before explaining the basic simulation initialization and workﬂow, some of the basic 2D

’s

properties and dependences need to be depicted. For a full explanation of the physical

effects and relationships, the reader may refer to Chap. 4. All designs consider coupling to

or from a Si rib waveguide with a typical height of 220 nm and a rib etch depth equal to the

’s etch depth. With this, grating and waveguide are deﬁned at the same lithographic

etch step and the grating is placed at an equal distance from each of the 2D

’s arms.

Otherwise, non-symmetry may result due to a 2D

imbalanced position with respect

to the waveguides. All 2D

s in this work have the so called shear angle - a non-zero

angle

between the Si waveguide and the grating area (Fig. 2.15). A shear angle can be

deﬁned either by making the grating area rhombus-shaped or by keeping it rectangular

and tilt the Si waveguide instead. The two types of grating are assigned as Type I and Type II

respectively. A non-zero shear angle means that an incident wave will be deﬂected under a

certain angle into the grating area. The reason is the refraction between the waveguide and

2.1 Numerical Simulation and Evaluation

grating half-spaces, having different effective refractive indices. For the Type I 2D

, the

non-perpendicular propagation direction is caused by the angled interface between both

half-spaces. For the Type II 2D

, we have the classical oblique ﬁeld incidence between two

half-spaces, caused by the tilted waveguides. Thus, an electromagnetic wave propagates

within the grating with a propagation vector



kin

having two components k

GC,x

and k

GC,y

These two components can be related to the grating’s effective refractive index n

eff,GC

and an angle of propagation

φin

(details in Sect. 4.1, Chap. 4). The latter parameters are

dependent on the material distribution in the grating, and thus, on the diffracting elements’

diameter w

and etch depth d(for a given waveguide height h

). The propagation angles

of the ﬁelds out-coupled by the grating

φout

θout

are related to the combination of a a

shear angle

and a grating period

for a desired wavelength by a matching condition

(derivation in Sect. 4.1, Chap. 4). In all cases, we require that the grating radiation angles

are equivalent to the ﬁber tilt angles, i.e. φout =φF=45◦,θout =θF.

Figure 2.15:

An illustration of the parameters of 2D grating couplers (2D GCs). A shear

angle

can be deﬁned either by making the grating area rhombus shaped

(Type I) or by tilting the Si waveguides with respect to the grating (Type II).

The grating effective refractive index n

eff,GC

depends on parameters such as

waveguide height h

, diffracting elements’ diameter w

and etch depth d.

The shear angle causes a non-zero angle of propagation

φin

within the grating.

The combination of n

eff,GC

and

φin

determines the combination of

and

for

desired coupling angles

φout =

◦

θout

, which are equivalent to the ﬁber tilt

angles, i.e. φout =φF,θout =θF.

With other words, there is the dependence chain:

(wΛ,d)⇒(neff,GC,φin)⇒(α,Λ)⇒(φout,θout). (2.35)

2 Methods for Analysis of Two-Dimensional Grating Couplers

When a simulation has to be started for the ﬁrst time, the question of the initial

’s

geometry (w

,d) occurs. The designer may choose values, which appear meaningful, e.g. w

such that no pre-deﬁned design rules regarding permitted dimensions are violated, and

etch depth daround half of the total waveguide height (an empirical value). Next, desired

coupling angles need to be speciﬁed, typically

φout =

◦

and

θout

in the range 4

◦

–16

◦

. To

determine the shear angle and the period, which are necessary to obtain these angles, the

effective refractive index n

eff,GC

and the input angle

φin

need to be estimated. In the ﬁrst

iteration, this is a diﬃcult task, since no averaging techniques are applicable. The reason

are the complex

BEOL

ﬁlling layers and their speciﬁc patterns. Thus, in the ﬁrst step, a

simple approximation is used, which averages the waveguide effective refractive index

at full Si height and at Si height, reduced by the initial etch depth (e.g. Ref. [36]). Hereby,

only SiO

is considered as a cladding. For example, if we assume a waveguide height of

WG =

220

, an etch depth of 120nm (remaining Si thickness 100 nm) and a duty cycle

of 0.5, the GC’s effective refractive index neff,GC is approximated as:

neff,GC ≈0.5·[neff,WG(hWG =220nm)+neff,WG(hWG =100nm)]. (2.36)

The input angle φin can be initially estimated using the Snell’s law.

Figure 2.16:

An illustration of the adaption procedure for the determination of an ap-

propriate 2D grating coupler (2D GC) geometry for desired coupling angles

(

φout

θout

). First, initial diffracting elements’ size and etch depth (w

,d) are set.

A corresponding grating effective refractive index and input angle (n

eff,GC

φin

)

are estimated. With the assumed (n

eff,GC

φin

) and for desired coupling angles

(

φout

θout

), the required shear angle and grating period (

) can be calculated.

After a simulation with a duration in the range 5-15 hours, the actual (

φout

θout

)

and the corresponding (n

eff,GC

φin

) can be determined by a spatial fast Fourier

transform (FFT). If (

φout

θout

) are not as desired, (

) are adapted and the sim-

ulation is repeated. If (w

,d) need to be changed for an eﬃciency optimization,

the whole procedure starts all over again.

After the ﬁrst simulation and after any optimization of the etch depth or the diffracting

elements’ size, the actual effective refractive index of the grating and the input angle can be

determined with the help of a spatial fast Fourier transform (FFT). A spatial FFT transforms

a spatial ﬁeld distribution into the wave vector k-space (or the “spatial frequency” domain).

In the k-space, the propagation angles giving the propagation direction (

φout

θout

) can

be determined. Out of the known relationship between (

φout

θout

) and n

eff,GC

φin

(for a

ﬁxed grating shear angle and period), the latter two parameters can be calculated. If the

angles (

φout

θout

) are not as desired, the shear angle and period can be adapted, using

2.1 Numerical Simulation and Evaluation

the just determined n

eff,GC

φin

. If (w

,d) need to be changed for an eﬃciency optimization,

the whole procedure starts from the beginning. Since simulations require typically 5-

15 hours (depending on the material and geometric properties, the simulation band,

etc.), the complete optimization procedure may take up to several days. Note that the

given simulation duration does not consider acceleration techniques, e.g. using graphic

processing units (GPUs). Figure 2.16 shows schematically the adaption procedure. The

exact calculations behind the spatial FFT will be given in more detail in the next subsection.

2.1.4 Methods for Parameter Evaluation

In this subsection, methods for the numerical results’ post-processing are explained.

2.1.4.1 Coupling Angles Determination from a 2D Radiation Pattern

Here, the determination of the grating’s angles

φout

θout

via FFT from Fig. 2.16 will be

described. Assume a general electric ﬁeld, propagating in an arbitrary direction with a free-

space wavelength

λ0

in a material with a refractive index n

. Such a general propagation

with a propagation vector



kout

can be given by the angles in spherical coordinates

φout

θout

deﬁned as in Fig. 2.15. At a constant plane z=const., the electric ﬁeld has the form:



E(

r)=

E0exp(

kout ·

r)=

E0exp(kxx+kyy)=

E0expk0n1(cosφout sinθoutx+sinφout sinθouty),

(2.37)

with the free-space wave number k

0=2π

λ0

. A FFT of a given electric ﬁeld distribution at z

const. delivers the vectors kx,ky. With the equivalence:

kx=k0n1cosφout sinθout

ky=k0n1sinφout sinθout

x+k2

y=k2

0n2

1sin2θout, (2.38)

it is obvious that with known kx,ky, the propagation angles can be determined as:

φout =arctan ky

θout =arcsin qk2

x+k2

k0n1. (2.39)

In our case, we assume a free-space coupling and n

1 (no index-matching gel consid-

ered).

2.1.4.2 Coupling Eﬃciency

The 2D

’s eﬃciency results from two components: 1) the out-coupled power eﬃciency,

i.e. the part of the feed-in power, which is coupled towards the position of a

SMF

; 2) the

mode ﬁeld overlap between the grating’s radiated mode and the SMF’s mode.

2 Methods for Analysis of Two-Dimensional Grating Couplers

Normalized Out-Coupled Power There are several limitations for a 2D

, which do not

allow that the full power propagating in a given waveguide will be coupled out towards the

SMF

. Consider the illustration in Fig. 2.17 (a) assuming a transmitter-side 2D

. The input

power, which is used as a reference for the normalization of the out-coupled power, is

assigned as PWG. In general, the power is split into several components:

•

- the part of the power, diffracted upwards. If a high diffraction eﬃciency is

achieved, the whole power P

will be well directed under an angle

θout

. The angle will

be the same as the

SMF

’s tilt angle

θF

for the desired central wavelength. Typically,

θout =

◦

is chosen, otherwise the grating behaves as a second order Bragg grating

with a strong back-reﬂection. In addition, with increasing diffraction angle

θout

, the

diffraction eﬃciency decreases as well, i.e. other angular components result, which

have no contribution to the coupling eﬃciency. At a ﬁxed ﬁber facet position, the

out-coupled power eﬃciency is deﬁned by the quotient of P

and the input power

PWG.

•Pdown - a part of the power, which is diffracted down towards the substrate.

•

thru

- a part of the power, which propagates further within the grating and is not

diffracted.

•

reﬂ

- a reﬂected part of the power, which can be caused by the Fresnel reﬂection

at the waveguide-grating interface or by back-scattering at the grating diffracting

elements.

Figure 2.17:

A schematic of the typical power distribution in a 2D grating coupler (2D GC):

(a) P

- the upwards diffracted power - is the only contribution to the coupling

eﬃciency. Power loss is caused by power diffracted towards the substrate

down

, not diffracted power, further propagating within the structure P

thru

and

reﬂected or back-scattered power P

reﬂ

. (b) An illustration of the reﬂections at

layers with different refractive indices and thicknesses, including the backend

of line (BEOL) stack layers above the grating and the buried oxide (BOX) and

the Si substrate below the grating. These reﬂections are responsible for the

ﬁnal ratio of P

down

and P

thru

. The reﬂection at the Si substrate and the BOX

thickness have the most decisive impact.

2.1 Numerical Simulation and Evaluation

The ratio between P

down

and P

thru

is not only given by the grating parameters them-

selves, such as the perturbing elements’ size and etch depth (see Fig. 2.17 (b)). A signiﬁcant

factor is the thickness of the

BOX

layer below the 2D

, separating the waveguide from

the Si substrate. The maximum value for P

in a certain wavelength range can be reached,

depending on the combination of a

BOX

thickness, a waveguide thickness, refractive indices

of the waveguide and the

BOX

and the angle of diffraction [37]. Similarly, the thickness of

the layers above the 2D

have also an impact. The latter is, however, smaller, because

the reﬂection at the Si substrate is much stronger than the reﬂection at SiO

-, Si

- lay-

ers or air. Mostly, the layers’ thicknesses and material properties are pre-deﬁned by the

particular fabrication platform, which can limit the maximally reachable coupling eﬃciency

in a given wavelength range. It is also not guaranteed that 2D

s designed for different

bands reach the same performance, because the periodicity of their maxima is wavelength

dependent. Figure 2.18 shows as an example the

BOX

dependence of 1D

designs for

C- and O-band. The dependence is similar for 2D

s. The ﬁgure shows the out-coupled

power at 1550nm and 1310 nm, depending on the

BOX

thickness. The refractive index of

Si is 3.47 at 1550nm and 3.5 at 1310 nm. The refractive index of the

BOX

layer is around

1.45 in both bands. The exemplary devices are: 1) 1D

for C-Band with a period of

630nm, a diffracting element’s width of 315 nm and an etch depth of 70nm; 2) 1D

for

O-Band with a period of 500 nm, a diffracting element’s width of 250 nm and an etch depth

of 70 nm. It can be seen that the standard

BOX

thickness of 2 µm is similarly good for both

wavelengths, in spite of the different maxima periodicity, which is about 550 nm in C-band

and 400nm in O-band.

Figure 2.18:

A buried oxide (BOX) dependence of the out-coupled power of 1D grating

couplers (1D GCs), designed for C- or O-band. The dependence is similar for

2D GCs. Due to the different maxima periodicity, the performance is equally

good in both bands only at certain BOX thicknesses, e.g. 2 µm.

Overlap Integral The second important factor for the coupling eﬃciency is the mode ﬁeld

overlap between the radiated 2D

ﬁelds and the fundamental

SMF

mode. A theoretical

upper limit of around 80% for non-optimized gratings has been determined [37]. The

obtained value corresponds to the overlap between a Gaussian function with an expo-

2 Methods for Analysis of Two-Dimensional Grating Couplers

nentially decaying function, the later used as an approximation of the ﬁeld diffracted by a

uniform grating.

The mode ﬁeld overlap is characterized by an overlap integral. The latter will be derived

here, by considering the orthogonality relations for the SMF’s eigenmodes.

Figure 2.19:

A schematic with the deﬁnitions used for the derivation of an overlap integral

between the upwards diffracted grating ﬁelds



, related to the power

up ≡

and the single mode ﬁber (SMF) fundamental mode. The SMF is

placed at the local coordinates (u,v,w

=const.

), which are deﬁned by the angles

φF

θF

. The ﬁber’s effective area is assigned as A

. Arbitrary SMF eigenmodes



El,m



Hk,n

,(l,m,k,n

∈Z

), weighted by transmission a

l,m

k,n

and back-propagation

coeﬃcients bl,m,bk,nare shown.

Assume the 2D

’s radiated ﬁelds as incident ﬁelds at the ﬁber facet with an effective

area A

and the position (u,v,w

=const.

) (Fig. 2.19 ). In the following, all given ﬁelds are

complex.

Deﬁnitions

•Tangential incident ﬁelds: 

Ei,

•Fiber eigenmodes:



EF=X

l,m

(al,m+bl,m)

El,m,l,m∈Z



HF=X

k,n

(ak,n–bk,n)

Hk,n,k,n∈Z(2.40)

Here, the coeﬃcients arepresent forwards-propagating waves and determine the trans-

mission through the ﬁber facet with an effective area A

. The coeﬃcients baccount for

2.1 Numerical Simulation and Evaluation

backwards-propagating waves. Particularly, we are interested in the calculation of the

coeﬃcients a. An orthogonality relation between the ﬁber’s eigenmodes is given as:

ˆAF



El,m×

H∗

k,nd

A=–ˆAF



Hl,m×

E∗

k,nd

A=δlkδmn =





1, if l=kand m=n

0, otherwise (2.41)

where δrepresents the Kronecker’s delta.

Continuity Condition at w=const.

The unknown coeﬃcients aand bcan be determined by using the continuity of the tan-

gential ﬁelds at the ﬁber facet w=const.

(i.) 

Ei=



Ei=X

l,m

(al,m+bl,m)

El,m(2.42)

To make use of the orthogonality relations, we make on both sides of the equation the

cross product with 

H∗

k,nand integrate over the area AF.

ˆAF



Ei×

H∗

k,nd

A=X

l,mˆAF

(al,m+bl,m)

El,m×

H∗

k,nd

↓

l=k,m=nˆAF

(al,m+bl,m)

El,m×

H∗

l,md

⇒al,m+bl,m=ˆAF



Ei×

H∗

l,md

ˆAF



El,m×

H∗

l,md

(2.43)

Analogously,

(ii.) 

Hi=



Hi=X

k,n

(ak,n–bk,n)

Hk,n. (2.44)

This time, we make on both sides of the equation the cross product with –



E∗

l,m

and again

integrate over the area AF.

–ˆAF



Hi×

E∗

l,md

A=–X

k,nˆAF

(ak,n–bk,n)

Hk,n×

E∗

l,md

↓

l=k,m=n

–ˆAF

(al,m–bl,m)

Hl,m×

E∗

l,md

⇒al,m–bl,m=ˆAF



E∗

l,m×

Hid

ˆAF



E∗

l,m×

Hl,md

=ˆAF



E∗

l,m×

Hid

ˆAF



El,m×

H∗

l,md

(2.45)

2 Methods for Analysis of Two-Dimensional Grating Couplers

Finally, the unknown coeﬃcients acan be obtained.

(i.) + (ii.)

al,m=1

2ˆAF



Ei×

H∗

l,md

A+ˆAF



E∗

l,m×

Hid

A1

ˆAF



El,m×

H∗

l,md

(2.46)

The coeﬃcients bcan be calculated analogously ((i.) - (ii.)). Since we have no coupling to

back-propagating waves, we obtain in any case b

l,m=

0. The estimation of bcan be done

as a veriﬁcation of the intermediate steps and as a proof for the transparent boundaries.

Overlap

The overlap can be deﬁned as the quotient of the power transmitted into the ﬁber P

l,m

and the incident grating ﬁeld power. With Pup ≡Piat the ﬁber facet:

ηF=P+

l,m

ηF=ℜˆAF

al,m

El,m×a∗

l,m

H∗

l,md

ℜˆAF



Ei×

H∗

id

=|al,m|2ℜˆAF



El,m×

H∗

l,md

ℜˆAF



Ei×

H∗

id

(2.47)

Now, we substitute al,mfrom (2.46) and assume ˆAF



El,m×

H∗

l,md

Ato be real. We obtain:

ηF=1

4ˆAF



Ei×

H∗

l,md

A+ˆAF



E∗

l,m×

Hid

A

ℜˆAF



El,m×

H∗

l,md

A·ℜˆAF



Ei×

H∗

id

(2.48)

Coupling Eﬃciency The overlap integral (2.48) still does not take into account the limited

out-coupled power eﬃciency, deﬁned as the portion of the input waveguide power P

that

is directed by the grating into the proper direction P

up ≡

. We can assign this parameter

as an input-output grating eﬃciency:

ηio =Pi

PWG =ℜˆAF



Ei×

H∗

id

ℜˆAWG



EWG ×

H∗

WGd

(2.49)

with A

the ﬁber effective area, A

the waveguide effective ares,



EWG



HWG

the Si waveguide

ﬁelds.

The ﬁnal expression for the coupling eﬃciency is:

η=ηF·ηi,o

η=1

4ˆAF



Ei×

H∗

l,md

A+ˆAF



E∗

l,m×

Hid

A

ℜˆAWG



EWG ×

H∗

WGd

A·ℜˆAF



El,m×

H∗

l,md

. (2.50)

2.1 Numerical Simulation and Evaluation

SMF Reference Mode In (2.50), the

SMF

’s LP

01,u

-mode can be substituted as the mode

of interest deﬁned at (u,v,w

=const.

). The ﬁeld distribution is commonly approximated as

a Gaussian beam:

Eu(u,v)=E0exp(u–u0)2

0exp(v–v0)2

0

Hv(u,v)=E0

ZFexp(u–u0)2

0exp(v–v0)2

0ZF=Z0

nSMF =376.73

nSMF Ω

with n

SMF ≈

1.447 [39] and the mode ﬁeld diameter 2w

10.4µm at 1550 nm or 2w

9.2 µm at 1310nm, according to the Corning’s SMF-28 data sheet [40]. The central position

of the Gaussian beam is given by (u0,v0) and its ﬁeld amplitudes are normalized, so that:

¨AF

EuH∗

vdudv =1.

The LP

01,v

-mode can be deﬁned in an analogous way. In this work, the Gaussian beam’s

position is always set in such a way that the ﬁber-grating mode ﬁeld overlap is maximized.

The position is also balanced between both arms of the 2D GC.

Figure 2.20:

A schematic for the illustration of two parameters, related to the polarization

splitting in 2D grating couplers (2D GCs). (a) Waveguides (WG) 1 and 2 are

excited separately, delivering a certain ﬁeld distribution at a tilted plane with

coordinates (u,v,w

=const.

), related to the angles (

φF=

◦

θF

). Out of each

ﬁeld distribution, a set of Stokes parameters related to a polarization state

or P

results. (b) Exemplary polarization states P

and P

on the Poincaré

sphere. (c) Ideally, P

and P

are composed of one component (x- or y). The

polarization split ratio SR is a ratio between the coupling eﬃciency CE of the

desired component to its orthogonally converted counterpart. The split ratio

refers to a single waveguide. The angular relationship between P

and P

on the Poincaré sphere shows, whether the signals from WG 1 and 2 are

orthogonal to each other. Non-orthogonal relation results in polarization

crosstalk between both waveguides. (Adapted from Ref. [38] under a CC BY

4.0 license.)

2 Methods for Analysis of Two-Dimensional Grating Couplers

2.1.4.3 Polarizations’ Splitting and Orthogonality

Along with the coupling performance, another important task for a 2D

is to act as a

polarization splitter/combiner. Particularly, it is of large importance, whether the 2D

able to split/combine polarizations orthogonally. In the following, two parameters used

for the characterization of the polarization splitting/combining behavior of 2D

s will be

distinguished. The content in the following paragraphs has been partially published in our

previous work in Ref. [38].

Assume the conﬁguration in Fig. 2.20 (a), where the 2D

is placed in the Cartesian (x,y,z)-

plane. The input ﬁelds from either waveguide 1 or 2 are radiated towards a tilted plane,

which is rotated by (

φF=

◦

θF

) with respect to the grating’s coordinates and corresponds

to the position where a

SMF

would be placed (u,v,w

=const.

). First, waveguide 1 is excited

with the fundamental

mode of a silicon waveguide. The radiated ﬁeld evaluated at the

tilted reference plane can be evaluated in terms of coupling eﬃciency. In addition a set of

Stokes parameters (S

)

and angles (2

ψ1

χ1

) on the Poincaré sphere (Fig. 2.20

(b)) can be determined (polarization state

). The bold font used here denotes a Stokes

vector. Next, the procedure is repeated after the excitation of waveguide 2, resulting in a

polarization P2. Exemplary polarization states P1and P2are marked in Fig. 2.20 (b).

According to the waveguides’ deﬁnition, we could expect that excitation of waveguide 1 will

result in an x-polarized ﬁeld in the

SMF

and waveguide 2 will deliver a y-polarized ﬁeld. For

the corresponding waveguides, these polarizations are assigned as target-polarizations.

Any power conversion, which leads to the excitation of the orthogonal counterpart of the

target-polarization is assigned as a cross-polarization. With other words:

•waveguide 1 - x: target-polarization, y: cross-polarization;

•waveguide 2 - y: target-polarization, x: cross-polarization.

From the perspective of a single waveguide, we can deﬁne the polarization split ratio SR as

the ratio between the power coupling eﬃciencies CE of the target- and cross-polarization:

SR =CE target-polarization

CE cross-polarization ⇒SR1=CEx

CEySR2=CEy

CEx. (2.51)

Due to symmetry, the split ratios are equal SR

. This parameter gives us a good intu-

itive impression about the polarization splitting quality of a 2D

. In the ideal case, when

each waveguide excites only the target-polarization, the split ratio is inﬁnite and perfect

polarization separation is guaranteed. Otherwise, signals coded in different polarizations

will be erroneously superposed, e.g. the target x-polarization from waveguide 1 and the

cross-x-polarization originating from waveguide 2 (and in opposite). To estimate how critical

this superposition is, another parameter needs to be introduced. The latter accounts for

waveguides 1 and 2 not alone, as in the deﬁnition of split ratio, but characterizes the

interaction between the ﬁelds of both waveguides with their x- and y-components. The

important statement about this characterization is whether the signals originating from

waveguides 1 or 2 are orthogonal to each other or not. The investigation of this aspect

is important, because orthogonal signals can be separated from each other by applying

2.1 Numerical Simulation and Evaluation

orthogonal transformations - similarly to the polarization demultiplexing of a signal rotated

arbitrarily during its propagation in a

SMF

. By contrast, non-orthogonal signals cannot be

separated perfectly from each other by this means. A polarization crosstalk will be given in

such a case, which is the second important parameter of interest here. The essential differ-

ence between both used parameters is that the split ratio gives a polarization conversion

considering a single waveguide, while the polarization crosstalk refers to the relationship

between the signals from both waveguides (illustration in Fig. 2.20 (c)).

Figure 2.21:

An exemplary representation of two non-orthogonal polarization states on

the Poincaré sphere (in a 2D cross-section). (a) A rotation of the Stokes vector

by 180

◦

should result in

. In the non-orthogonal case, the polarization

states

differ by the angle 2

Δ˜

. (b) A Thales triangle for the determination

of a polarization crosstalk. (Adapted from Ref. [38] under a CC BY 4.0 license.)

In the following, the determination of the polarization crosstalk will be explained in more

detail. The orthogonality relation of two polarizations can be determined from the Stokes

vectors

and

with their corresponding angles on the Poincaré sphere (2

ψ1

χ1

) and

(2ψ2,2χ2) (Fig. 2.20 (a), (b)). The orthogonality condition is:

|ψ1–ψ2|=90◦

χ1=–χ2. (2.52)

First, the calculation of the Stokes parameters will be shown. The latter is based on the

procedure used for their practical measurement [41]. We use the deﬁnitions:

S0=I(0◦)+I(90◦)

S1=I(0◦)–I(90◦)

S2=I(45◦)–I(135◦)

S3=IRHC –ILHC, (2.53)

where Iis the ﬁeld intensity of the polarization states, given in the parentheses. The latter

are linear (0

◦

,90

◦

,45

◦

,135

◦

) and right-handed or left-handed circular (RHC, LHC). In our

case, the intensity is calculated as the square of the magnitude of a given electric ﬁeld

distribution, integrated over the evaluation plane. Since the ﬁelds are normalized, the

parameter S

, giving to the total ﬁeld intensity, is equal to 1. A normalized Stokes vector

can be represented in such a case as a set of (S

). The transformation through

2 Methods for Analysis of Two-Dimensional Grating Couplers

polarization plates is described mathematically with the help of their Jones matrices [42].

S0=I(0◦)+I(90◦)=¨|Eu|2+|Ev|2dudv=1 normalization (2.54)

S1=I(0◦)–I(90◦)=¨|Eu|2–|Ev|2dudv. (2.55)

Transformations representing polarizing plates:

linear polarization±45◦:"ˆ

Ev#=1

2"1±1

±1 1 #·"Eu

Ev#(2.56)

right/left hand circular polarization : 



Ev

=1

2"1±j

∓j1#·"ˆ

Ev#. (2.57)

Note the linear polarizations’ equivalence –45

◦≡

135

◦

. Passing the ﬁelds through the

polarizing plates yields:

S2=I(45◦)–I(135◦)=¨|ˆ

Eu,pol45|2+|ˆ

Ev,pol45|2dudv–¨|ˆ

Eu,pol135|2+|ˆ

Ev,pol135|2dudv

(2.58)

S3=IRHC –ILHC =¨|ˆ

Eu,pol45,RHC|2+|ˆ

Ev,pol45,RHC|2dudv–

–¨|ˆ

Eu,pol135,RHC|2+|ˆ

Ev,pol135,RHC|2dudv. (2.59)

The degree of polarization (

DoP

)pand angles on the Poincaré sphere can be calculated

from the Stokes parameters as given in Ref. [43], whereat the DoP here is p=1:

p=qS2

1+S2

2+S2

S0=1

S1=pcos2χcos2ψ=cos2χcos2ψ

S2=pcos2χsin2ψ=cos2χsin2ψ

S3=psin2χ=sin2χ

⇒2ψ=arctan S2

S1, 2χ=arcsinS3. (2.60)

After the determination of the angular relation between the signals from the two

arms, the corresponding polarization crosstalk can be calculated in the next step. For a

simplicity, we set 2

χ=

0. In Fig. 2.21 (a), a 2D cross-section of the Poincaré sphere is shown.

Here, the exemplary polarization vectors

and

are such, that the angular difference

is 2

Δψ

< 180

◦

. For orientation, the normalized Stokes vectors of a x-polarization (1, 0, 0)

and a y-polarization (-1, 0, 0) are given. For a system with two orthogonal Stokes vectors,

we should be able to cancel one of the two polarizations by rotating it by 180

◦

. If the

assumption of orthogonality is wrong, an error occurs. To determine this error, we rotate

e.g. the Stokes vector P1by 180◦, obtaining ˆ

P1.

2.2 Measurement Methodology

The polarization states ˆ

P1and P2are not identical and differ by the angle:

2Δ˜

ψ=180◦–(2ψ2–2ψ1)=180◦–2Δψ. (2.61)

Now, we consider the Thales right-angled triangle in gray, spanned by

, where

is the orthogonal counterpart of

(Fig. 2.21 (b)). In the case of orthogonality, the

polarizations

and

would be coincident and the depicted triangle would not exist. The

latter indicates otherwise different paths between the points

and

. To account

for this deviation, the line segment 2

mcan be used, where mis a fraction of the total

line length 2 (with p

1) between the points

and

. The parameter mis deﬁned in

the interval m

∈

[0,1] and its value gives the polarization crosstalk in terms of power. To

calculate mas a function of the polarization angles’ difference

Δψ

, we apply the geometric

mean theorem for the hypotenuse and the altitude h:

h2=h1+(1–2m)i·2m. (2.62)

Out of the smaller highlighted triangle, we obtain:

h=sin(2Δ˜

ψ)⇒h2=sin2(2Δ˜

ψ). (2.63)

By setting (2.62) and (2.63) to be equal, a quadratic equation for mresults in the end. With

the boundary condition that mshould vanish in the orthogonal case

Δψ =

◦

, i.e.

Δ˜

ψ=

◦

the ﬁnal solution is:

m=1

21–cos(2Δ˜

ψ)=sin2(Δ˜

ψ). (2.64)

The polarization crosstalk in dB can be calculated as m[dB]=10lgm.

2.2 Measurement Methodology

Although the experimental characterization of 2D

s requires a relatively simple means,

several important objectives should be considered to reach conﬁdence in the measurement

results. This includes:

•

Proper power normalization in loss measurements. Any loss contributions off- and on-

chip must be determined as precisely as possible. Any overestimation of the additional

loss terms may lead to rather optimistic coupling eﬃciency results. It is mandatory to

describe carefully the means for power normalization, when measurement results

are presented.

•

Reliable approach for relative measurements. Here, the focus is on the accurate

determination of the polarization split ratio. An appropriate test structure is required

for a coupling-eﬃciency-independent measurement. Robustness against mechanical

instability is desirable as well.

•

Wafer statistics. When we develop a component such as an integrated 2D

, which

is intended for a large-scale fabrication, the repeatability of its performance is crucial.

2 Methods for Analysis of Two-Dimensional Grating Couplers

It is not correct to give the best chip on a wafer as a benchmark. A given

’s

parameter should be characterized by a mean value and a standard deviation. A full,

automated wafer-scale measurement is best suited for that purpose. If this should

not be attainable, at least 9 chips at ﬁxed wafer positions should be measured. This

number of chips is typical for process control measurements.

In the following subsections, these points will be discussed in further detail.

Figure 2.22:

An exemplary setup for the characterization of passive on-chip components

(not to scale). A tunable laser source in the C- or O-band is used for a signal

generation. A manual or programmable polarization controller is used for

polarization management. Two single-mode ﬁbers (SMFs) are used for in- and

out-coupling. Their position is controlled by mechanical positioners. The signal

is received by a power meter and can be further processed on a computer.

A photonic chip contains a device under test (DUT), which comprises in the

simplest case back-to-back structures with 1D or 2D grating couplers (GCs) at

each end. The coupling is further supported by an external light source and a

microscope camera, which may introduce limitations in the SMF positioning

range. Such factors need to be considered, when laying out a test structure.

2.2.1 Basic Setup

In the beginning of our discussion, a basic measurement setup will be shown schematically

(Fig. 2.22). With small modiﬁcations, each passive device measurement involves these

components. A tunable laser source is used for a signal generation in the C- or O-band.

A polarization controller is placed prior to chip-coupling, either to ensure the proper

polarization state on chip, or to generate different polarization states or arbitrarily polarized

light via scrambling. In the ﬁrst case, a manual polarization controller is suﬃcient. In

the second one, programmable polarization controllers come into question, which have

typically a higher insertion loss. The in- and out-coupling is reached via cleaved standard

SMF

s, which are adjusted by mechanical positioners with three axes (six axes also possible).

The signal is ﬁnally received by a photodetector and programming tools can be used for

2.2 Measurement Methodology

the depiction of a measurement on a computer. The devices under test (DUT) include

in the simplest case 2D (or 1D)

s at both ends, connected by tapers and waveguides.

Depending on the investigated problem, the basic conﬁguration can be extended by other

components. Aside from mechanical positioners, an external light source and a microscope

camera are used as helping tools for coupling. Although these components have no direct

impact on the measurement, their placement may introduce restrictions regarding the

adjustable coupling angles and the positioning range of the

SMF

s. These factors need to

be taken into account, when laying out a test structure.

2.2.2 Power Normalization in Loss Measurements

The power normalization in coupling loss measurements needs to take into account off-

and on-chip losses. Off-chip losses are mostly caused by ﬁber attenuation, connector

losses and losses from other components, e.g. a programmable polarization controller.

Furthermore, the cleaved ﬁber facet may suffer from impurity or mechanical damages,

due to erroneous cleaving or undesired ﬁber contacts with the chip surface. The best way

to take all off-chip losses into account and to eliminate some of them is the calibration

with a photodetector directly in front of the facets of the in- and out-coupling ﬁbers. The

calibration takes place at both sides, so that for the receiver path, the ﬁber at the pho-

todetector must be switched over to the laser source. Apparently, it must be ensured that

all connectors and ﬁber facets are suﬃciently clean, so that no loss and a corresponding

error in the calibration occur, when the ﬁbers are switched back. In this work, a calibrated

photodiode Thorlabs SM05PD5A has been used to determine the off-chip loss in most of

the measurements. The measured photocurrent at the in- or the out-coupling ﬁber facet,

is transferred to an optical power, by using the provided responsivity data. Ideally, calibra-

tions at several wavelengths are made to cover a certain band, due to the responsivity’s

wavelength dependence. Alternatively, only the band central wavelength can be used. This

causes a wavelength-dependent error of the determined input power of maximally

0.1

according to the photodiode data sheet for C- and O-band. In some measurements, a

slim photodiode S132Ce is used for calibration purposes, combined with a power meter

PM100D by Thorlabs. In this case, the optical power instead of the photocurrent is directly

measured.

The second aspect is the determination of the on-chip loss. Typically, short deeply-etched

rib waveguides and bends with suﬃciently large radii are used to minimize the impact of

their loss on the measurement. For longer waveguides, the cut-back [44] or the optical

frequency domain reﬂectometry [45] methods are most commonly used for their loss

determination. These techniques distinguish themselves for their simplicity, reliability and

robustness, which make them suitable for automated wafer-scale measurements [46].

In the photonic

BiCMOS

platform relevant for this work, the loss for deeply-etched rib

waveguides is typically 3dB/cm. Another important component on the chip is the taper,

which scales a waveguide from the grating width of about 12 µm down to a single-mode

waveguide with a 400-500 nm width. Typically, simulations predict a nearly lossless behavior

for a suﬃciently long taper. However, the speciﬁc of our sheared 2D

s with a non-zero

2 Methods for Analysis of Two-Dimensional Grating Couplers

waveguide-to-grating angle, makes it possible that a mode, coupled by the grating, enters

the taper under an angle different from expected. The deviation can result from both

material/geometric variations or from ﬁber misalignment. The angled mode coupling could

generally cause higher-order modes. In addition, in rib waveguides, modes in the slab may

also occur, leading to spectrum oscillations. The latter could make the determination of

the maximum coupling value diﬃcult. The uncertainty due to such artifacts may reach

0.5

. Finally, Fabry-Perot oscillations in the measured spectrum should be mentioned.

The latter result in back-to-back conﬁgurations, due to back-reﬂections at the grating and

also due to reﬂections at the SMF’s facet. Typical spectrum variation of ±0.1dB is given.

In summary, there are different power loss and oscillation sources, which need to be quan-

tiﬁed for an exact determination of the 2D

’s coupling loss. The accumulated uncertainty

makes it diﬃcult to guarantee for a loss accuracy better than ±0.5dB.

Figure 2.23:

(a) A schematic of a device for the measurement of a polarization split ratio. (b)

An incident polarization for a split ratio SR

1, leading to an extinction ratio

(ER) of inﬁnity. (c) An incident polarization for a split ratio SR

→∞

, leading to

an ER of 1 and vanishing resonances. (d) A camera picture of an exemplary

fabricated device with assigned components. Other abbreviations: GC: grating

coupler, MMI: multi-mode interferometer. (Adapted from Ref. [47], under a

CC BY 4.0 license.)

2.2.3 A Measurement Technique for the Polarization Split Ratio

An approach for the reliable characterization of the 2D

’s polarization splitting per-

formance has been initially reported in our work, Ref. [47]. Here, the description of the

test structure design follows Ref. [47]. The characterization of the polarization split ratio

is enabled by the integrated device, schematically depicted in Fig. 2.23 (a). It comprises

a 2D

under study, connected with two Si waveguides via tapers. For linear 2D

adiabatic tapers with a length

≥

250µm can be used. The Si waveguides are typically rib

waveguides with the same etch depth as the 2D

(deeply etched rib waveguides). The

core width is 500nm and the slab width 2 µm in C-band. The tapers and the waveguides

2.2 Measurement Methodology

are not depicted in the schematic, but can be seen in the camera picture of an exemplary

fabricated device in Fig. 2.23 (d) . Furthermore, the test structure comprises a low-loss

delay line of 200 µm length on the lower arm, a

MMI

and two 1D

FGC

s as outputs. The

MMI’s split ratio is wavelength dependent, but the variation is small.

The device has the following working principle - the 2D

splits an incident wave of an

arbitrary polarization into the two waveguide arms. If we denote these amplitudes as

and

, the ratio of their squares corresponds to the 2D

’s split ratio SR, i.e. to the power

ratio between the polarizations in the two waveguides (Fig. 2.23 (a)). The amplitudes are

normalized to the 2D GC’s input power

SR =|a1|2

|b1|2,|a1|2+|b1|2=1. (2.65)

After the delay line and the

MMI

, the two signal paths interfere constructively or destructively

at the outputs, depending on the wavelength and the corresponding phase delay

Δφ

the delay line. The ratio between the maximal and minimal signal transmission is known as

an extinction ratio (ER) and can be measured after out-coupling with a 1D FGC.

From the

we are able to estimate the 2D

’s split ratio by using a simple matrix

model [48]. The relation between the input waves

and

and the output waves

and

can be mathematically written as a multiplication of a matrix

TMMI

describing the

MMI

transfer properties and a matrix TΔφ describing the delay line:

"a2

b2#=TMMI ·TΔφ "a1

b1#with (2.66)

TMMI =1

√2"1j

j1#,TΔφ ="ejΔφ 0

0 1#. (2.67)

If we take e.g. the ﬁrst output, the power measured there is proportional to the square of

the wave amplitude a2

Pout ∼|a2|2=1

2|a1|2+|b1|2+2|a1||b1|sinΔφ. (2.68)

The maximum and the minimum of P

out

can be obtained by maximizing or minimizing

sinΔφ

. The maximum-to-minimum ratio is the

. If we further replace

and

through

their split ratio SR, we obtain

ER =1+√SR

1–√SR2

. (2.69)

Measured

s can be easily translated into split ratios. The working principle is further

illustrated by two examples, which do not consider the

’s ﬁlter spectrum. In Fig. 2.23 (b),

the incident polarization is such that the signal is equally split between the two waveguides.

The split ratio is therefore 1 and from (2.69) we obtain an

of inﬁnity. If we consider

another case in Fig. 2.23 (c), where the incident polarization is such that all the power is

coupled in one of the waveguides , the split ratio is this time SR

→∞

and the

is 1, which

means that minimal and maximal levels are the same and no resonances result. In our

2 Methods for Analysis of Two-Dimensional Grating Couplers

experiment, the situation in Fig. 2.23 (c) is pursued: we look for a polarization, which will

be coupled into one of the 2D

arms. Ideally, smooth transmission without resonances

should result in this case. Out of the minimal measured

, we can estimate the best

achievable split ratio of the considered structure, according to (2.69).

To be able to measure the split ratio according to the given deﬁnition, we attempt to

reproduce the scenarios depicted in Fig. 2.23 (b), (c) during all measurements. In the ﬁrst

case, the incident polarization is split equally between both 2D

arms. Such a polarization

state can be found by using an auxiliary structure, comprising two 1D

FGC

s connected to

each other in a back-to-back conﬁguration. The 1D

FGC

s can be aligned on the chip in such

a way that maximal coupling eﬃciency results for a polarization state oriented as in Fig.

2.23 (b). After the desired polarization state is found, a measurement on the 2D

with a

delay line

MMI

can be done. Due to the equal splitting, very large

s will be observed. In

the next step, this polarization state can be rotated by 2

ψ=

◦

on the Poincarè sphere,

using a programmable polarization controller. This corresponds to the scenario in Fig. 2.23

(c). The corresponding

can be then measured as well and used for the determination

of the split ratio SR according to the deﬁnition used here. Although this method may have

a limited precision, it still gives us the opportunity to observe trends, which is the main

purpose of this investigation.

The principle of our measurement using a delay line interferometer may ﬁrst look rather

complicated, but in manual measurements, it has two obvious advantages. First, the signals

at the two

MMI

outputs are only phase shifted. Thus, it is suﬃcient to measure only one of

the outputs, which reduces the time for the measurement and/or the setup complexity.

Second, if we measure on two outputs, we need to guarantee that they are equally well

coupled, which is practically very diﬃcult in manual measurements. In this experiment, the

in- and out-coupling eﬃciency has no impact on the determined split ratios. The mechanical

coupling stability during the long wavelength sweep is also less crucial.

2.2.4 Wafer Statistics

Objectives Among the largest strengths of 2D

s compared to lateral spot-size con-

verters is the possibility for an automated wafer-scale characterization. This is especially

important for a high-yield production, which makes 2D

s currently a more mature

solution for

EPIC

s. Simultaneously, higher standards for the performance of 2D

s are

given: it is not important that a single device reaches a good behavior, even when it is

a record-high one. Rather, it is decisive that most of the devices show a similarly good

operation.

Several 2D

’s design parameters can vary on a wafer. Here, they will be categorized,

depending on their particular impact:

Parameters with an impact on the grating’s effective refractive index and the maximum

transmission wavelength. Such parameters include:

–The waveguide height;

–The size and the etch depth of the 2D GC’s diffracting elements;

2.2 Measurement Methodology

–The thickness of BEOL layers, covering the waveguide and the grating.

2. Parameters with an impact on the 2D GC’s out-coupled power such as:

–The BOX thickness;

–The thickness of BEOL layers, independent on their distance to the grating.

Typically, we target at a ﬁxed wavelength for a given application. The quality of any 2D

’s

parameter can be characterized in terms of a mean value and a 3

-interval (

the standard

deviation). Often, a standard Gaussian statistical wafer distribution can be assumed for the

majority of optical parameters. The 3

-interval deﬁnes a span, where 99% of the values of a

certain parameter are expected. The robustness of

s against variations is often given by

their high-index contrast nature and cannot be inﬂuenced signiﬁcantly. Mostly, the choice

of an appropriate oxide thickness below the grating or the choice of an appropriate target

wavelength can be helpful, provided that such ﬂexibility is given. Otherwise, an optimization

of the fabrication process itself is the only way towards improved performance stability. A

weighting between complexity and reachable best operation is necessary at this point.

In this work, all fabricated 2D

s have been characterized statistically. At TU Berlin, a

manual wafer-probe system has been mostly used for that purpose, which is the main

limiting factor for the number of chips, which could be measured on a wafer. The minimal

number of chips is 9 (typical for process control measurements), which can be placed as

illustrated in Fig. 2.24. The exact positions depend on the total number of chips on the

wafer.

Figure 2.24:

An exemplary representation of 9 chips on a wafer, at which 2D grating cou-

plers (2D GCs) can be characterized.

Fully-Automated Wafer-Scale Measurements Although automated wafer-scale mea-

surements are well-established in electronics, there are only very few vendors that are

specialized in automated optical measurement equipment. The reason is that optical

measurements are signiﬁcantly more sensitive to alignment, coupling distance adjustment,

environmental changes etc. Thus, the ﬁber alignment and positioning system is among

the most challenging components in a fully automated wafer test system. In contrast to

2 Methods for Analysis of Two-Dimensional Grating Couplers

electronic tests, where contact pads are available, the photonics coupling takes place

at a certain distance and under a non-zero coupling angle. The nearly vertical coupling

requires a precise control over the height and an appropriate calibration. In addition,

precise alignment and mechanical stability are required. Veriﬁcation and calibration are

not trivial and are connected with certain variations [49,50]. Exemplary automated wafer

probe systems are provided by Formfactor [49] and Keysight [50]. Both vendors invested

effort in high-precision software kits to fulﬁll the requirement of equal coupling conditions

over the whole wafer. Dedicated calibration tools are necessary to satisfy the stringent

requirements of optical wafer-scale characterization. In addition, it is of large importance

for volume production that high throughput is ensured. Thus, it is desirable that all electric,

optic and electro-optic test are performed in one step [49,50].

Furthermore, short- and long-term repeatability of the measured results is another issue. A

short-term interval covers several hours, while long-term periods may extend over several

months. It has been shown that coupling loss variation in short-term test remains below

0.1dB [51]. By contrast, long-term reproducibility is affected by more factors such as

contamination of the ﬁber facets, mechanical instability, temperature changes, polarization

shift due to ﬁber movement or other mechanical defects in connectors and other connect-

ing ﬁber patches. In the same work, the long term repeatability of measured coupling loss

of less than 0.8dB has been determined [51]. Automated wafer characterization at IHP

Microelectronics is enabled by a fully-automated system by Formfactor. Similar measure-

ment principles and performance in terms of repeatability apply for this test station. Some

of the experimental results in the present work are obtained on this setup.

Summary Every measurement, even performed with the best state-of the-art equipment,

contains inevitably a level of uncertainty. For any reported experimental results, a careful

description of all setup speciﬁcs is mandatory, including the aspects of calibration, stability

and repeatability. The range of possible deviations needs to be determined.

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3 Fabrication Platform

In early years, Si photonics was a niche research ﬁeld and the experimental work did not

consider constraints given by a certain foundry process. Presently, several Si photonics

platforms with a variable device portfolio are available, e.g. Refs. [1

–

4]. The 2D

s designed

and presented in this thesis consider the boundary conditions for an integration in a

0.25µm photonic

BiCMOS

technology. In this chapter, the basic differences between a

pure

CMOS

and a

BiCMOS

process will be explained. The advantages and speciﬁcs of

the SiGe:C heterojunction bipolar transistor (

HBT

) will be highlighted. Afterwards, the

combination of

SOI

photonics with the electronic process in bulk Si for the realization

of a photonic

BiCMOS

platform will be brieﬂy reviewed. An alternative approach for the

integration of photonics and electronics both on bulk Si wafers, in a 3D integration manner,

will be outlined in the end of this chapter.

3.1 Bipolar CMOS

This section gives a brief introduction to the idea of the bipolar

CMOS

technology and

discusses in further detail the realization of high-speed SiGe:C

HBT

s, which is the most

distinctive electronic component in the technology platform relevant for this work.

3.1.1 CMOS vs. Bipolar Technology

The following paragraphs are based on Refs. [5,6]. The information here is intended to

give a brief overview, but not to go into technological details on the

BiCMOS

integration.

For a more detailed information, the reader may refer to the publications given in the

bibliography.

The motivation for a bipolar

CMOS

technology is to combine the advantages of metal oxide

semiconductor (

MOS

) and bipolar transistors on a single platform. The

MOS

transistors are

known for their low power consumption and ease of design, however, they have limitations

in drive capability and speed. By contrast, bipolar transistors allow a larger current drive

and a better high-frequency performance [5].

3.1 Bipolar CMOS

Figure 3.1 gives a schematic comparison of the cross-sections of two classical components

- the n-type metal oxide semiconductor ﬁeld effect transistor (

MOSFET

) transistor and the

(planar) npn bipolar junction transistor (

BJT

). Apparently, in the

MOSFET

, the charge carrier

ﬂow and the electric current are horizontally directed. By contrast, the npn transition in

the

BJT

is vertical, which implies a vertical charge carrier ﬂow. This fundamental difference

is related to different design concerns, which are reﬂected in the fabrication technology

of both transistor types. The combination of these two technologies - the bipolar

CMOS

(BiCMOS) - is hence related to compromises in the performance of these devices.

Figure 3.1:

A schematic cross-section of (a) a n-MOSFET (adapted from Ref. [6], Chap. 2)

and (b) a npn-BJT (adapted from Ref. [5], Chap. 1). The components differ by

the direction of the charge carrier and electric current ﬂow. The co-fabrication

of both transistor types is related to performance trade-offs.

Among the most critical trade-offs is the n-well formation, which is necessary for both the

MOS

and the npn-

BJT

. For the p-

MOS

, the n-well optimization is performed with respect

to the threshold voltage, punchthrough voltage, source and drain capacitance and body

effect. On the other hand, for the

BJT

’s collector formation, the n-well doping and thickness

are related to a compromise between high transit frequency and large breakdown voltage.

The choice of an appropriate well proﬁle is also technology dependent (n-well, p-well or

twin-well) [6].

The efforts to combine properly

CMOS

with bipolar technology with a minimal number

of additional processing steps have been shifted with the time towards the pursuit to

reduce the total fabrication time. Current

BiCMOS

processes attempt rather to add bipolar

transistor modules with a minimal disturbance of the standard

CMOS

process. Additional

optimization steps can be added depending on the target application [5].

Often, a comparison between

CMOS

and

BiCMOS

has been made. In the ﬁrst place, it

should be noted that

CMOS

scales signiﬁcantly faster than

BiCMOS

. The integration of

bipolar modules gets even more challenging with decreasing

CMOS

nodes. However, there

are certain advantages of BiCMOS, which still makes the technology competitive [7]:

•

Reliable high-speed performance. The bipolar transistors come with a higher break-

3 Fabrication Platform

down voltage at the same speed than most of the CMOS transistors.

•Large transconductance and low frequency noise.

•

Better

BiCMOS

passive devices, due to the optimized

BEOL

, usually not available in

CMOS.

With these characteristics, the

BiCMOS

technologies target at applications for even increas-

ing frequencies and very demanding requirements with respect to power and reliability.

Exemplary application ﬁelds are the high-speed, high-data-rate communication systems or

smart mobility systems [7].

3.1.2 SiGe:C Heterojunction Bipolar Transistor

The IHP Microelectronics’ fabrication platform related to this work offers high-speed elec-

tronics enabled by SiGe:C

HBT

s based on a 200 mm technology. As a most decisive ﬁgure-

of-merit, the product of transit frequency and collector-emitter breakdown voltage is used:

T×

CEO

. Reaching simultaneously a high f

and a high BV

CEO

is a challenging task. The

HBT

s available in this platform are a result from a long development process and designs

for different target applications are available. The choice of a SiGe doped base instead of a

pure Si base has several advantages. First, the base bandgap in SiGe is reduced, compared

to Si. Thus, the barrier height for the emitter-to-base electron ﬂow is smaller, leading to a

larger collector current. With this, larger current gain can be reached with SiGe

HBT

s com-

pared to Si

BJT

s. Moreover, the SiGe base offers more design ﬂexibility. When a moderate

current gain is chosen, the base doping can be optimized for reaching higher f

and/or

higher maximal oscillation frequency f

max

. To maximize both parameters, the boron proﬁle

in the base must be as thin and highly doped as possible [5]. For the realization of such a

base, the boron diffusion during layer growth and high-temperature anneals must be mini-

mized. A possibility of the reduction of the boron diffusion is the additional carbon doping

of the base. This approach is used for the

HBT

s in this technology as well [8]. Another

important aspect is the collector design, responsible for the combination of f

T×

CEO

. The

IHP’s collectors are formed without subcollectors and deep trenches. The low collector

resistance is achieved by high-dose ion implantation after shallow trench formation [9,10].

The best in class f

max

performance has been consistently demonstrated in 0.13 µm

CMOS environment (fT/fmax of 505 GHz / 720 GHz) [11]. The latest generation of 0.25 µm

HBTs at IHP show fT/fmax values of up to 220 GHz / 290 GHz [12].

3.2 Monolithic Photonic BiCMOS

IHP’s SiGe:C

BiCMOS

is a prominent but not the only example of a high-performance

200mm technology. Most important characteristics of these technologies is the f

scaling

in vertical transistor structures, thus not necessarily advancing lithography beyond 248nm

DUV

capabilities, corresponding to a 0.13 µm

CMOS

. The photonic

BiCMOS

platform of

IHP has been developed from previous electronic

BiCMOS

technology families featuring a

3.2 Monolithic Photonic BiCMOS

0.25 µm

CMOS

and a 5-metal

BEOL

. The

BiCMOS

technologies based on a 0.13 µm

CMOS

have a 7-metal

BEOL

. Both technology families rely on a 248nm

DUV

lithography. For small

and intermediate fabrication volumes 200mm technologies have proven advantageous

in terms of cost. However, the 248nm

DUV

lithography is not an optimum for nano-

waveguide Si photonic structures. Due to the improved resolution at a 193 nm wavelength,

ﬁner features can be fabricated. So far, best performing 2D

s were fabricated using

193nm

DUV

lithography [13]. The question whether comparable 2D

s may be realized

with a 248nm DUV is addressed by this thesis.

As discussed in the previous section, the co-integration of

CMOS

and bipolar technology is

not a trivial task. Even more demanding is the ambition to go for monolithic

EPIC

s. Along

with the strict conditions for the realization of fast and reliable electronics, additional

requirements for the highly sensitive photonic circuits result.

Figure 3.2:

An illustration of the photonic modules’ integration in a 0.25 µm electronic

BiCMOS ﬂow. Abbreviations: (Bi)CMOS: (bipolar) complementary metal oxide

semiconductor, HBT: heterojunction bipolar transistor, BEOL: backend of line,

SOI: silicon on insulator, WG: waveguide. (Representation after Ref. [12].)

Currently, there are only few

EPIC

platforms worldwide. On the other hand, monolithic

EPIC

platforms have several clear advantages. Among the most important ones is the

possible realization of the shortest interconnects between photonic and electronic devices.

By contrast, hybrid platforms require wire-bonding or ﬂip chip techniques for interfacing

between

PIC

s and

EIC

s, which is inevitably related to additional parasitic resistances,

capacitances and inductances with an impact on the whole device performance [12,14

–

16].

In addition, a technology with a common

BEOL

stack for photonics and electronics is

3 Fabrication Platform

favorable compared to dual backend approaches with expensive photonic-electronic co-

packaging [14].

The main challenge for the monolithic photonic-electronic integration is the frontend of

line (

FEOL

) with different substrate requirements. On the one hand, high-performance

BiCMOS

devices are typically realized in bulk Si, while photonic devices require

SOI

wafers.

The translation of

BiCMOS

SOI

is limited by the deep collector and the high thermal

resistivity of the buried oxide, which is a signiﬁcant issue for the heat dissipation of high-

speed circuits [14]. To overcome these problems, a mixed-substrate technology - the local

SOI

approach [17] - has been introduced. First, an

SOI

substrate with a 2 µm thick buried

oxide and a 220nm

SOI

layer, intended for photonic applications, is taken. No

BiCMOS

requirements are considered at that point. Afterwards, the

SOI

layer and the buried oxide

are removed locally by a sequence of plasma and wet etch steps. These locally etched areas

are intended for the

BiCMOS

electronics and are re-ﬁlled in the next step by a selective Si

epitaxy. Chemical mechanical polishing (CMP) is used for the planarization. This approach

leads to the ﬁnal monolithic integration of photonics and

BiCMOS

electronics, achieving

an unaltered

BiCMOS

performance in comparison to the pure electronic technology [14].

Figure 3.2 illustrates the integration of photonic modules within the 0.25 µm

BiCMOS

according to Ref. [12].

Figure 3.3:

A schematic cross-section of a photonic BiCMOS platform. In the frontend

of line (FEOL) high-speed heterojunction bipolar transistors (HBTs) in bulk Si

are integrated in parallel with silicon on insulator (SOI) photonic devices, such

as waveguides, coupling interfaces, multimode interferometers (MMIs), Ge

photodetectors etc. Both electronics and photonics share the same backend

of line (BEOL) stack, consisting of multiple SiO

layers with different thicknesses

and material constants. The BEOL offers 5 metals for the realization of the

shortest interconnects between photonics and electronics. (Adapted from

Ref. [18] under a CC BY 4.0 license.)

3.3 BiCMOS With BEOL Photonic Layers

Figure 3.3 shows an exemplary cross-section of the IHP’s monolithic photonic-electronic

platform. In the

FEOL

SOI

photonics, including coupling devices, waveguides, tapers, direc-

tional couplers,

MMI

s, Mach-Zehnder modulators and Ge photodetectors are depicted on

the left-hand side. The photonic devices are realized in parallel with high-speed SiGe:C

HBT

s in bulk Si (depicted on the right-hand side). Both photonics and electronics share a

common

BEOL

layer stack, comprising multiple SiO

and Si

with different thicknesses

and refractive indices. The speciﬁc geometric and material properties of the covering

layers are taken into account in all 2D

designs. It should be noted that this has no

impact on the general observations and design rules, which have been derived in the

scope of this thesis. Generally, the 2D

designs would require only small modiﬁcations

to be adopted to other fabrication platforms. Finally, the platform offers three thin (Metal1,

Metal2, Metal3) and two thick metals (TopMetal1, TopMetal2) for the realization of the

shortest interconnects between photonics and electronics.

3.3 BiCMOS With BEOL Photonic Layers

The monolithic photonic

BiCMOS

platform presented in the previous section shows the

established fabrication routine for photonic-electronic integration. In the recent years, an

alternative approach has been investigated, which is based on the principles of the 3D

photonic integration. Some of our preliminary ideas and analyses on that concept have

been published in Ref. [18] and will be given in this section. 3D silicon photonics is based on

the idea of the extension of the number of photonic layers, thus improving the integration

density and the design ﬂexibility of photonic components [19]. Several materials come into

question for that purpose: hydrogenated amorphous silicon (a-Si:H) [20], silicon nitride

(Si

) [21,22] or aluminum nitride (AlN) [23]. From the perspective of

enabled devices,

a high-index contrast material is preferable. Among the listed materials, a-Si:H offers the

highest refractive index contrast, and has been proved suitable for inter- or intra-chip

vertical grating coupling [24]. The initial disadvantage of the high absorption loss of a-Si

at telecom wavelengths could be overcome by a hydrogen passivation [25]. For these

reasons, a-Si:H has been considered a promising material for 3D photonic transmitters or

transceivers as a part of the IHP’s photonic BiCMOS platform.

The basic motivation for a 3D photonic integration is the following. After years of research

in silicon photonics, it is now well-known that the silicon modulator and the modulator

drivers are the bottleneck for the realization of high-speed all-silicon transmitters [26].

Due to the fundamental limitations of all-silicon phase shifters, silicon Mach-Zehnder

modulators typically show opto-electrical bandwidth < 50

GHz

, large modulation loss and

and considerable power consumption [27

–

30]. For these reasons, many groups started

considering the accomplishment of silicon hybrid transmitters, in which the modulator is

made of material with distinct second-order nonlinearities (Pockels effect). Modulators

based on various materials have been reported, e.g. nonlinear polymers [31], lead zirconate

titanate [32], barium titanate [33] and lithium niobate [34

–

37]. Moreover, attempts towards

the co-integration on a full-ﬂow silicon platform have been undertaken [33,36]. On the

3 Fabrication Platform

one hand, polymers suffer from temperature and long-term stability issues. Polymers are

in general not

FEOL

compatible. On the other hand, all inorganic Pockels materials listed

here are not compatible with common

CMOS

processes and require a post fabrication

bonding step. From the perspective of a foundry process ﬂow, bonding of a Pockels

material in the

FEOL

would require signiﬁcant modiﬁcations of the

FEOL

fabrication routine,

adding processing time and complexity. In comparison, bonding after the

BEOL

has been

processed is less challenging and both IBM [33] and Sandia [36] demonstrated their

modulators in the

BEOL

. If we consider keeping the modulator there, an extension with

another photonic layer at that level is necessary.

Figure 3.4:

A schematic representation of a 3D photonic BiCMOS platform with electronic

and photonic components in the frontend of line (FEOL) extended by a photonic

amorphous Si:H (a-Si:H) layer at the TopMetal1 level. Grating couplers (GCs)

can be used as in- and out-coupling interfaces, which may or may not use

Metal3 as a back-reﬂector. A 2D GC is needed for a polarization-multiplexed

system. A Pockels material can be bonded for the realization of high-speed

modulators and transmitters in the backend of line (BEOL). The realization on

both silicon-on-insulator (SOI) and bulk Si wafers is possible. (Adapted from

Ref. [18] under a CC BY 4.0 license.)

Figure 3.4 shows an exemplary cross-section of the concept of a 3D photonic

BiCMOS

platform with an additional a-Si:H photonic layer in the

BEOL

at the TopMetal1 level. The

combination of the a-Si:H layer with a Pockels material of choice could enable the realization

of high-speed transmitters, which meet the requirements for future generation

DCI

s. The

separation of the metal levels, where Pockels materials and a-Si:H are integrated, requires

a thick enough interlayer stack. The interlayer dielectric stacks above the thin metal layers

(Metal1, Metal2, Metal3) have a thickness of around 900nm, which is not suﬃcient. For

that reason, the integration of a-Si:H after processing TopMetal1 is favorable, since the

3.3 BiCMOS With BEOL Photonic Layers

interlayer dielectric stack between TopMetal1 and TopMetal2 with a thickness of around

3µm fulﬁlls best the requirements.

In the photonic a-Si:H layer, typical components such as waveguides, directional couplers,

MMI

s etc. can be realized in the same way as their

SOI

counterparts. Grating couplers can

be used as in- and out-coupling interfaces, whereat the input interface can be substituted

in future by a hybrid integrated laser source, following the principles outlined e.g. in

Refs. [38

–

40]. For polarization-multiplexed systems, 2D

s are necessary as an output.

Regarding the eﬃciency of

s, a low thickness variation of silicon dioxide (SiO

) is very

decisive. There are several process points in the

BEOL

, which cause variation in the SiO

cladding. Before the fabrication of each metal layer, a CMP is done to create suitable

process conditions. This means that we have four CMP modules (PreMetal1, PreMetal2,

PreMetal3, PreTopMetal1), which will increase the thickness variation. If we consider the

whole

BEOL

with the planarized TopMetal1 topography, a typical thickness variation of the

SiO

cladding of around 330 nm results. In this context, the adoption of Metal3 as a mirror

below the

s is not only to increase the coupling eﬃciency, but – more important – to

reduce the inﬂuence of the SiO

thickness below the grating. For a SiO

thickness between

a-Si:H and Metal3 of around 3 µm, a thickness variation of around 210 nm can be expected.

A deviation in that range is acceptable for the 2D GCs, which will be shown in Chap. 5.

A typical transmitter conﬁguration requires integrated photodetectors for setting the

operation point of the modulator. In the case of a

BEOL

integrated modulator, an interlayer

coupling scheme such as in Ref. [23] will be needed to access a frontend photodiode. The

interlayer coupling is not shown in Fig. 3.4, being still under development. The possibility

for non-

SOI

photodetector has been investigated on our

BiCMOS

platform as well [41]. In

this concept, a Si

photonic layer has been used in the

FEOL

. This outlines the potential

possibility to realize complete transceivers on bulk Si wafers. As a very premature concept,

an in-depth analysis is still forthcoming. Nevertheless, the analysis of 2D

s in this context

is of interest for this thesis, since reachable maximal coupling eﬃciency limits can be

explored.

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4 Investigation of Fundamental

Physical Effects in

Two-Dimensional Grating

Couplers

Among the most signiﬁcant objectives of this work is the systematic explanation of the

physical properties of 2D GCs. This includes:

The description of the diffraction in two directions. This is possible by the derivation

of a 2D diffraction condition, in which all material and geometric 2D

’s parameters

are put into a clear relation with the two angles of diffraction

φout

θout

. The reliability

of the derived formulas can be shown in simulations and experiments.

The demonstration of the importance of in-plane scattering effects. The latter have

been underestimated in 2D

s until now. This work demonstrates the fundamental

limitations due to in-plane scattering and proposes new means to overcome in-plane-

scattering-related issues.

In this chapter, the basic description of these effects will be summarized. Methods for

the 2D

s optimization by an appropriate diffraction and scattering management will be

reviewed in Chap. 5.

4.1 Diffraction

This section is dedicated to the derivation of a 2D diffraction condition, which links the

geometric and material 2D

’s properties to two angles of diffraction - an azimuth angle

φout

and an off-plane angle

θout

(Fig. 4.1). Out of this condition, general design rules for our

sheared 2D GCs are derived. Their feasibility is veriﬁed in numerical simulations.

4.1 Diffraction

Figure 4.1:

An illustration of a sheared 2D grating coupler (2D GC) with a rhombus-shaped

grating area, placed in the plane (x,y,z

0). An incident wave vector



is trans-

formed to a vector



kin

due to the angled interface between waveguide and

grating. The vector



kin

is deﬁned by the effective refractive index of the 2D GC

eff,GC

and the angle of incidence

φin

. The 2D GC’s wave vector



is deﬁned by

the period

and the shear angle

. The grating’s wave vector



transforms

the input vector



kin

into the wave vector



kout

, which is related to the diffraction

angles (φout,θout).

4.1.1 Derivation of a 2D Diffraction Condition

A part of the following analyses has been previously published in Ref. [1]. The starting

point is the consideration of the wave vectors of the ﬁeld in the grating plane and the

ﬁeld radiated outside of the grating. The continuity of the tangential ﬁelds at the grating’s

interface leads to a phase condition, which gives an information about the grating geometry

that ensures light deﬂection under desired radiation angles. We focus here entirely on

the coupling to a given target

SMF

polarization. For that reason, only the target diffraction

order will be considered.

For all calculations, an output 2D GC is examined. The grating area is assumed rhombus-

shaped as illustrated in Fig. 4.1. A Si waveguide mode propagating in the grating is referred

as an input ﬁeld. A ﬁeld radiated by the grating is denoted as an output ﬁeld. All derivations

are made for the x-polarized waveguide mode. For a y-polarization all calculations may be

performed analogously. Note that the equations for the x-polarization can be converted to

equations for the y-polarization by taking into account, that the y-polarization is principally

the same ﬁeld, rotated by 90◦.

4.1.1.1 Phase Condition

We start with the formulation of the phase condition of a sheared 2D

with a rhombus-

shaped area and perpendicular feeding waveguides. A phase condition for a sheared 2D

GC with tilted waveguides can be found in Ref. [2].

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Input Field To implement the 2D nature of a 2D

with respect to a single input arm,

the grating and the input ﬁeld from this arm must be oblique to each other. A modiﬁed

input ﬁeld vector with two components and/or a sheared grating wave vector allows for a

2D diffraction pattern. We consider a waveguide with an effective refractive index n

eff,WG

When this waveguide is periodically perturbed to form a grating, a lower effective refractive

index n

eff,GC

within the grating results. Thus, for a wave propagating from the waveguide

into the grating (and vice versa), a Fresnel reﬂection and refraction can be expected (Fig.

4.1). In sheared 2D

s, the transition from the waveguide to the grating takes place at

an oblique angle. Depending on the 2D

’s type, we may have two scenarios. When the

grating area is rhombus shaped, an angled interface between waveguide and grating is

given. When the grating is square-shaped, but the feeding waveguide is angled, the incident

ﬁeld propagates in an oblique direction with respect to the interface. In any case, a purely

x-polarized waveguide mode that propagates in y-direction will change its propagation

direction after a transition into the grating. Thus, the following wave vector transformation

takes place:

x-pol. waveguide mode: 

kWG =k0neff,WG

grating mode: 

kGC =k0neff,GCcosφin

ex+sinφin

ey=:

kin (4.1)

with k

0=2π

λ0

the free-space wavelength, n

eff,WG

eff,GC

the effective refractive indices

of the waveguide and grating mode respectively,

φin

the angle of transmittance from the

waveguide into the grating, which can be initially estimated using the Snell’s law. The mode

that propagates within the grating is further referred as an input ﬁeld.

Grating Wave Vector The wave vector of a rhombus-shaped grating can be deﬁned as:



K=2π

Λsinα·

ex+2π

Λcosα·

ey(4.2)

with Λthe grating period and αthe shear angle.

If we consider the path difference between two diffracting elements

Δ

r=Λsinα·

ex+cosα·

ey(4.3)

we obtain



K·Δ

r=2π

Λ·Λsin2α+2π

Λ·Λcos2α=2π. (4.4)

This short calculation check conﬁrms that the condition for constructive interference

between two partial diffracted waves is fulﬁlled. Due to the non-zero shear angle, the

grating vector



can be used for the manipulation of the components of



kin

in two angular

directions.

Continuity Condition To obtain the phase condition for a sheared 2D

, the continuity

of the tangential ﬁelds at the interface between the grating and the outer space must be

4.1 Diffraction

considered. For the ﬁeld phases it means that the wave vectors must be continuous for all

x,y.

Input Wave Vector



kin =k0neff,GCcosφin

ex+sinφin

ey=k0˜

neff,GC

ex+˜

neff,GC

eywith (4.5)

neff,GC =neff,GC ·sinφin (4.6)

neff,GC =neff,GC ·cosφin. (4.7)

Grating Wave Vector



K=2π

Λsinα·

ex+2π

Λcosα·

ey. (4.8)

Output Wave Vector



kout =k0n1sinθoutcosφout

ex+sinφout

ey(4.9)

where

φout

is the angle in the horizontal plane z

const. and

θout

is the angle with respect

to the vertical. Normally, the propagation angle

θout

in the free space is of interest, the

outer space has a refractive index n

1 in this case. When we substitute the cladding’s

refractive index n

cladd

, we obtain the angle of propagation within the cladding

θout,cladd

. At

the cladding-air interface, the Snell’s law holds:

ncladd sinθout,cladd =n1sinθout (4.10)

Continuity at z=0



kin –

K=

kout (4.11)

x-component:

k0˜

neff,GC –2π

Λsinα=k0n1cosφout sinθout

neff,GC –λ

Λsinα=n1cosφout sinθout, (4.12)

y-component:

k0˜

neff,GC –2π

Λcosα=k0n1sinφout sinθout

neff,GC –λ

Λcosα=n1sinφout sinθout, (4.13)

with the deﬁnitions of

neff,GC

eff,GC

given in (4.6), (4.7). For the y-polarization the same

equation system results, if we let

φin →

◦

–

φin

φout →

◦

–

φout

α→

◦

–

under the

condition, that all named angles are mathematically positive.

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

4.1.1.2 Sheared 2D GC Design Parameters

For given 2D

geometric parameters - a period

and a shear angle

- the radiation

pattern may be determined from (4.12) and (4.13).

(4.13) : (4.12) ⇒φout =arctan˜

neff,GC –λ

Λcosα

neff,GC –λ

Λsinα(4.14)

From (4.13) ⇒θout =arcsin1

n1sinφout ˜

neff,GC –λ

Λcosα. (4.15)

Usually, we are interested in ﬁnding design parameters of a 2D

for a given desired

combination (

φout =

◦

θout

), which will be equivalent to the ﬁber position

φout =φF

θout =

θF

. To do so, we can use again (4.12) and (4.13), this time with the shear angle

and the

period Λas unknown parameters.

To linearize the equation system, following parameters may be introduced:

cosα

Λ=:A

sinα

Λ=:B. (4.16)

Thus, the linear system of equations that have to be solved with respect to Aand Bis:

"λ0

0λ#·"A

B#="˜

neff,GC –n1sinφout sinθ

neff,GC –n1cosφout sinθ#. (4.17)

Finally, the 2D GC’s design parameters αand Λare obtained:

α=arctan B

Λ=1

Bsinα=1

Acosα. (4.18)

It should be noted that the waveguide-to-grating shear angle, initially proposed by Taillaert

[2], has been reported often in the literature, e.g. Refs. [3

–

7], however, without a detailed

explanation, what the exact design procedure is and whether the latter describes well

strongly perturbed 2D

s with a shear angle. Strongly perturbed sheared gratings require

a full 3D numerical analysis [2]. Therefore, the derived relationships in this work (4.12),

(4.13), (4.18) will be veriﬁed numerically in the next subsection.

4.1.2 Numerical Veriﬁcation of the 2D Diffraction Condition

From the relations in (4.18), we can see that the combination of a shear angle

and a grating

period

results in a combination of the coupling angles (

φout

θout

). The dependence is

used to design different 2D

s for various coupling angles

θout

, combined with a constant

φout =

◦

. Numerical simulations are carried out for exemplary cases combining different

(α,Λ) and a non-sheared 2D GC is used as a reference.

4.1 Diffraction

As already mentioned in Chap. 2, there are basically two ways to realize a shear angle in

practice. A possibility is to etch a rhombus-shaped grating array and keep the waveguides

perpendicular - a design, which is assigned here as a sheared 2D

of Type I. The second

option is to keep the grating rectangular, but tilt instead the waveguides with respect to

the grating - a conﬁguration assigned as a 2D

of Type II. Figure 4.2 illustrates both types

of 2D

s, which have been investigated in this work. In the following, several exemplary

designs of sheared 2D GCs Type I and II for C-band will be analyzed.

Figure 4.2:

Two types of sheared 2D grating couplers (2D GCs) investigated in this work.

Type I - rhombus-shaped grating area and perpendicular waveguides. Type II -

square-shaped grating area an angled waveguides.

With regard to the results that will be shown in the remaining part of this work, the following

speciﬁcations need to be made. As noted in Chap. 2, all results obtained by simulations are

evaluated at the

SMF

-plane with coordinates (u,v,w

=const.

) calculated by (2.33), which are

tilted with respect to the Cartesian (x,y,z), according to the ﬁber angles

φF=φout

θF=θout

An on-chip input polarization E

is thus translated to the polarization E

, respectively E

. To avoid confusion, the following equivalence will be set: E

x≡

y≡

. Although the

polarizations Eu,Evare calculated, they will be further denoted as Exand Ey.

Table 4.1:

Numerically estimated coupling angles

φout

θout

at a 1550nm wavelength for

designs with varying shear angle

and grating period

. Both 2D grating coupler’s

(2D GC’s) polarizations are excited simultaneously. The designs are intended for

coupling to a Si rib waveguide with a 220nm height and slab etch depth equal

to the grating etch depth d. The buried oxide thickness is 2 µm.

Type and

shear angle αPeriod Λperturbing elements’

diameter wΛand etch depth d

x-pol.

φout,θout

y-pol.

φout,θout

0◦615nm 420nm and 120 nm 90◦,8◦0◦,8◦

Type I, 2◦620 nm 420 nm and 120 nm 45◦,8◦45◦,8◦

Type II, 2◦612 nm 420 nm and 120 nm 50◦,8◦40◦,8◦

Type I 3◦633nm 425nm and 120 nm 45◦,12◦45◦,12◦

Type II 3◦624nm 425nm and 120 nm 49◦,12◦41◦,12◦

Type I, 4◦643 nm 430 nm and 120 nm 45◦,15◦45◦,15◦

Type II, 4◦637 nm 430 nm and 120 nm 50◦,16◦40◦,16◦

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

In contrast to 1D

s, the perturbation strength of a single diffraction element in 2D

is lower. For that reason, 2D

s require a larger etch depth, compared to the “standard”

70nm. An etch depth of 120nm is found appropriate for 2D

s. The diameter of the

diffraction elements should be suﬃciently large, but its exact value is not determining.

It is important that the design rules for a 248nm

DUV

lithography are not violated. The

distance between two diffracting elements should not be less than 180nm, which deﬁnes

the limit for the maximal diffracting elements’ size. Table 4.1 summarizes the details of the

exemplary designs used here to demonstrate the principle of the 2D diffraction condition.

The angles are obtained via spatial FFT (cf. 2.1.4.1) after both input 2D

’s polarizations are

excited simultaneously. The evaluation wavelength is 1550nm.The geometric properties

of the perturbing elements are nearly the same - the small diameter variation ensures a

constant duty cycle of about 0.67-0.68 (depending on the type), where the duty cycle is

deﬁned as the ratio of the diffracting element’s diameter to the period. As stated in (4.12),

(4.13), (4.18), different combinations of shear angle and grating period result in different

combinations of the coupling angles (φout,θout).

Figure 4.3:

A comparison of (a) the normalized out-coupled power, (b) the mode ﬁeld

overlap and (c) the coupling eﬃciency of 2D grating couplers (2D GCs) without

and with a shear angle

(a grating of Type I). The ﬁber position is given by the

angles φF=45◦,θF=8◦.

In the case of zero shear angle, it is evident that no rotation in

-direction takes place.

The propagation angle

θout

is the same for both polarizations, but the angle

φout

differs by

◦

. It can be further observed that for a desired maximum transmission wavelength, the

coupling angle

θout

at a ﬁxed

φout =

◦

can be increased, by increasing simultaneously the

shear angle and the grating period. Although some of the designs have a slight deviation

Δφout =±

◦

, this is not critical and corresponds to a very small wavelength mismatch.

More signiﬁcant changes in terms of eﬃciency occur, when

Δφout

◦

, as will be shown

4.1 Diffraction

exemplary later in this section. The results in Tab. 4.1 conﬁrm that there is not a single

“proper” shear angle, but the latter can be adapted according to the desired coupling angle

θout

. The differences between gratings of Type I and II scale with an increasing shear angle

and can be explained by different effective refractive indices of both types. The latter are

caused by the different shape and material distribution of the grating’s area.

Next, the designs in Tab. 4.1 are examined with regard to their coupling eﬃciency. In Fig.

4.3 two designs without and with a shear angle (Type I) are compared ﬁrst. The evaluation

plane is oriented along the symmetry axis between the 2D

’s arms

φF=

◦

. The

coupling angle in both cases is

θout =θF=

◦

. Figure 4.3 (a) shows the out-coupled power

normalized with respect to the waveguide input power and Fig. 4.3 (b) the mode ﬁeld overlap

of both designs. The out-coupled power differs more profound for wavelengths larger

1560nm. More remarkably, the maximal mode ﬁeld overlap is signiﬁcantly deteriorated in

the non-sheared case, due to the missing rotation towards the symmetry plane and the

corresponding different optimal coupling positions for both polarizations. This results in

about 3dB lower coupling eﬃciency at 1550nm, as can be seen in Fig. 4.3 (c). The shear

angle is crucial for the improvement of the 2D

’s mode ﬁeld overlap and the overall

coupling eﬃciency.

Figure 4.4:

A comparison of (a) the normalized out-coupled power, (b) the mode ﬁeld

overlap and (c) the coupling eﬃciency of 2D grating couplers (2D GCs) with

different shear angles

α=

◦

, corresponding to different coupling angles

θout =

◦

,12

◦

,15

◦

φout =

◦

. The 2D GCs are of Type I - with a rhombus-

shaped grating area and perpendicular waveguides.

Figure 4.4 shows the normalized out-coupled power, the mode ﬁeld overlap and the

coupling eﬃciency of the considered 2D

designs of Type I. Analogously, Fig. 4.5 shows

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

the same parameters for 2D

s of Type II. In both cases, the mode ﬁeld overlap does

not depend signiﬁcantly on the chosen combination of a shear angle and a grating period.

Once again, it is shown that different shear angles

adjust different coupling angles

θout

the 2D

’s symmetry plane. It should be noted that the maxima of some of the overlap

curves are slightly larger than the theoretical limit of 80% [2]. The latter value results from

an estimation, assuming an exponential proﬁle of the grating ﬁeld distribution. Deviations

of the actual ﬁeld proﬁle from the assumed function may cause small deviations in the

range of ±5%.

Figure 4.5:

A comparison of (a) the normalized out-coupled power, (b) the mode ﬁeld

overlap and (c) the total coupling eﬃciency of a 2D GCs with different shear

angles

α=

◦

, corresponding to different coupling angles

θout =

◦

,12

◦

,16

◦

φout =

◦

. The 2D GCs are of Type II - with a square-shaped grating area

and tilted waveguides.

Regarding the out-coupled power eﬃciency, differences can be observed both between

structures of different types (because of the effective refractive indices n

eff,GC

variation)

and with different shear angles (because of the changed out-coupling angles

θout

). Both

parameters - n

eff,GC

and

θout

- have an impact on the

BOX

thickness dependence of the

out-coupled power at a given wavelength. The wavelength-dependence of the out-coupled

power may be responsible for an asymmetric spectrum of a 2D

, as e.g. in the case

of the structure with

α=

◦

, Type I in Fig. 4.4 (c). Another effect may be a (quasi-)shifted

maximum as in the case of the structures with

α=

◦

, Type II (Fig. 4.5 (c)). The spectrum

shift in this situation is not caused by a coupling angle mismatch that would be reﬂected in

the mode ﬁeld overlap (cf. Fig. 4.5 (b)). Appropriate designs have such n

eff,GC

and coupling

angle that the power distribution is nearly constant within the desired wavelength range.

With certain ﬁxed design speciﬁcations, this is not always achievable.

4.2 In-Plane Scattering

As could be shown, the role of the shear angle is to “shift” the desired coupling angle

θout

towards the symmetry plane

φout =

◦

for both polarizations. With this, coupling

angles

θout

for wavelengths different from the desired one are coupled out at

φout =

◦

An example for the coupling-angles-wavelength combinations obtained by a spatial FFT

is given for the structure with a shear angle

α=

◦

of Type I in Fig. 4.6. For this design, a

change of

Δθout =±

◦

is combined with

Δφout ≈±

◦

(for the x-polarization on example).

This means that unlike for 1D

s, there is only a limited possibility to shift the grating

spectrum’s maximum by adapting the

SMF

’s tilt angle

θF

. The reason is that a deviating

θF

would not be optimally coupled at

φF=

◦

, because the grating ﬁxes the required

combination of angles for each wavelength. The mismatch φout =φF=45◦is responsible

for a mode ﬁeld overlap deterioration. Figure Fig. 4.6 shows further the mode ﬁeld overlap

when the ﬁber is tilted by the default coupling angle

θF=

◦

and by two other angles with

ΔθF=±

◦

. The evaluation plane is always ﬁxed at

φF=

◦

. The overlap penalty due to the

symmetry plane deviation is about 8%.

Figure 4.6:

A mode ﬁeld overlap of a 2D grating coupler (2D GC) with a shear angle

α=

◦

of Type I and a design coupling angle

θout =

◦

φout =

◦

. A comparison

between the case of optimal ﬁber tilt angle

θF=θout

and two other coupling

cases with

θF=θout ±

◦

φF=

◦

is done. The table indicates the expected

(

φout

θout

) combinations for each wavelength, which are ﬁxed by the 2D GC.

The combined deviation from both (

φout

θout

) determines the wavelength shift.

The mismatch between

φout =φF=

◦

is responsible for a mode ﬁeld overlap

deterioration.

4.2 In-Plane Scattering

In this section, the importance of the grating in-plane scattering is demonstrated in theory

and experiment. As shown in the previous section, the out-coupled ﬁelds of 2D

s are

always associated with a constructive interference and can be described appropriately

by the formulation of a 2D diffraction condition using wave vectors. By contrast, the

ﬁelds deﬂected within the grating plane are not necessarily related to a coherent ﬁeld

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

superposition and their description using wave vectors may not be suﬃciently general. For

that reason, ﬁeld deﬂection within the grating plane, assigned here as in-plane scattering,

is mathematically handled in a more rigorous way, using analytical methods. The in-plane

scattering’s existence is proven mathematically in a simpliﬁed conﬁguration. Although only

qualitative, this analytical study is able to give us a good feeling about the origin of parasitic

in-plane ﬁeld components in 2D

s. Next, the effect of the cross-polarization, i.e. the

orthogonal conversion of a given polarization within the grating plane and its subsequent

ﬁber coupling, is thoroughly investigated as the most signiﬁcant consequence of in-plane

scattering. It is shown that the cross-polarization is related to coupling and polarization

splitting issues. For the experimental evidence of the in-plane scattering, higher-order

modes excitation in different 2D GC designs is ﬁnally studied.

4.2.1 Analytical Investigation of In-Plane Scattering

The following analytical study of the in-plane scattering in 2D

s has been initially pub-

lished in Ref. [8]. A simple analytical method is used to describe the scattering by an array

of dielectric cylinders with similar properties as the diffracting elements of a 2D

. Only a

limited number of cylinders can be taken into account. The ﬁeld problem is simpliﬁed in

several aspects, so that the results can be interpreted only qualitatively. The limitations of

the analytical description are summarized in the end of the subsection.

Si photonic

s are typically considered as diffractive structures, for which only the diffrac-

tion condition is of interest. In 1D

s, the grating perturbing element is deﬁned completely

along the waveguide’s cross-section and is continuous with respect to the wave front. For

this reason, diffraction is the dominant process in 1D

s. However, in 2D

s, the perturb-

ing elements are discrete and have a ﬁnite dimension with respect to the waveguide and

its wave front. For that reason, an incident wave will not only be diffracted according to the

grating’s diffraction condition, but also scattered in the grating plane. The scattering being

initially a random process [9] can receive a more systematic nature, when the scattering

elements are periodic along the propagation direction of the incident wave. Such situation

is given in 2D

s, therefore, scattering needs to receive as much attention as diffraction.

Here, it is shown that scattering is an issue, which can be responsible for many undesired

effects in 2D

s such as polarization conversion (i.e. cross-polarization), polarization

crosstalk,

PDL

and higher-order modes excitation in Si waveguides. These effects are not

predicted by the 2D

’s diffraction condition. The latter states that the polarization of a

wave will be preserved and only the propagation direction will be changed according to

the chosen grating period (no cross-polarization). The diffraction condition can be applied

to higher-order modes, however, their refractive index is signiﬁcantly smaller and requires

for the same design coupling angle a signiﬁcantly larger period than the 2D

s designed

for the fundamental TE

have. For that reason, lateral higher-order modes coupled to

2D-GC-interfaced Si components cannot result from diffraction.

Scattering of a plane wave by a dielectric or conductive cylinder has been a well-known

electromagnetic problem for many years. Currently, various solutions of special cases are

present in the literature. Good summaries of classical solutions were available already in

4.2 In-Plane Scattering

the 70s [10,11]. Simpliﬁed analyses consider the cylinder inﬁnite in its length. In addition,

the material properties include perfect metals or lossless dielectrics. In more advanced

calculations, ﬁnite conductivity, dielectric losses (see e.g. Ref. [12] and the references

therein), anisotropy [13] or ﬁnite length [14] are taken into account. For a certain group of

problems, e.g. in the optical domain, the plane wave as an incident wave is substituted

by a Gaussian beam [15]. Another interesting aspect is the scattering not only by a single

cylinder, but also by a periodic array [12,15].

The grating area of a silicon photonic 2D

combines many special conditions in its

scattering nature. It consists of a dielectric cylinder array, which is periodic in two direc-

tions. The cylinders have a ﬁnite length and a size comparable to the wavelength. The

material is dielectric with a wavelength-dependent refractive index. Moreover, in our case

the cylinders are radially stratiﬁed, due to the

BiCMOS BEOL

ﬁlling layers. Obviously, the

scattering taking place in 2D

s is far more complex to describe than is the diffraction. A

simple analytical formulation, which clearly states how a scattered ﬁeld is distributed, in

what direction it propagates, and what polarization components it has, is not available. The

aim in this work is not to derive an absolutely exact analytical description of the scattering

in 2D

s, but to deliver a mathematical proof, based on a simpliﬁed conﬁguration, that

scattering can be the reason for a polarization conversion in 2D GCs. The understanding

of this fact is almost intuitive, since a plane wave scattered by a circular cylinder results in a

cylindrical wave. A previous analysis of the scattering properties of 2D photonic crystals [16]

is used as a basis to illustrate that in 2D

s scattering takes place in parallel to diffraction.

For the next analysis, the following simpliﬁcations are assumed. The incident wave is a

plane wave instead of a Gaussian beam. The grating’s perturbing elements are considered

as inﬁnitely long dielectric cylinders. These assumptions would be eligible, if a 2D

fully etched, the waveguide mode is very well-conﬁned and the grating area is smaller

than the incident wave front. For the scattering’s description, the relations from Ref. [16]

are adapted to the particular problem. A scattered ﬁeld by a single cylinder and by a

cylinder array is represented. Since the derivations are well-explained in Ref. [16], only the

signiﬁcant basic steps are outlined.

Figure 4.7:

A schematic representation of two scattering problems: (a) scattering by a

dielectric cylinder, (b) scattering by a dielectric cylinder array. (Adapted from

Ref. [8] under a CC BY 4.0 license.)

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

For an incident P-polarized wave, the electric ﬁeld is parallel to the (x,y)-plane, the magnetic

ﬁeld is parallel to the cylinder axis (



z

) and is used for the determination of the

scattered ﬁeld. The assumed time-dependence in [16] is

exp

(–j

t). The latter is omitted in

the ﬁelds’ representation.

In general, a non-zero angle of incidence

φi

can be assumed (Fig. 4.7), which can be

related to a waveguide-grating shear angle. The propagation direction of the incident ﬁeld

is depicted by the vector



with the subscript ifor “incident”. Here, we are particularly

interested in the case

φi=

180

◦

(a propagation in x-direction, according to the deﬁnitions

in Fig. 4.7). Such a propagation is given in perfect vertically coupled gratings, for which no

polarization conversion and orthogonal polarization separation is expected [17]. Assuming

a propagation direction in



and a y-polarized incident ﬁeld, we expect no x-polarized

component of the total ﬁeld.

To obtain the magnetic ﬁeld in the exterior, the sum of the incident ﬁeld and the scattered

ﬁeld at the cylinder or the cylinder array is calculated. Once the superposition is calculated

for H

, the electric ﬁeld components result from the Ampere-Maxwell’s equation. In Fig. 4.7

the considered cases are illustrated - Fig. 4.7 (a) shows the simple case of a single cylinder,

while in Fig. 4.7 (b) we have the extended case of a cylinder array. We assign the refractive

indices n1,2 in the regions 1 and 2 , and a free-space wavelength λ0.

All calculations presented in the following are adapted from Ref. [16]. We begin with a

single cylinder (Fig. 4.7 (a)), which will be later extended to a cylinder array (Fig. 4.7 (b)). In

the ﬁrst half-space

, the incident wave and the scattered wave have a wave number k

and the wave impedance Z

, depending on the material refractive index n

. In our case, n

is the Si waveguide’s refractive index. In the cylinder with a radius a(half-space

), we have

the refractive index n

, for which we assume SiO

, with the corresponding wave number

k2and wave impedance Z2.The listed parameters are deﬁned as:

k1=2π

λ0n1Z1=sμ0

ε0n2

k2=2π

λ0n2Z2=sμ0

ε0n2

. (4.19)

The ﬁrst step is to expand the incident plane wave in a sum of cylindrical waves (Jacobi-Anger

identity). For both cases (single cylinder or cylinder array), the expansion in Bessel-functions

1ρ

) with the coeﬃcients p

is the same and can be written in matrix form, as a vector

product. Here, [.] represents a column vector with elements deﬁned by the index m.

Hz=H0e–jk1(cosφix+sinφiy)=H0e–jk1ρcos(φ–φi)=H0Ψi(ρ,φ), (4.20)

Ψi(ρ,φ)=ΦT

J·p0with ΦJ=[Jm(k1ρ)ejmφ], p0=[(–j)mejmφi], m=0,±1,±2,... (4.21)

The scattered ﬁeld of a single cylinder can be expressed as a sum of Hankel functions of

ﬁrst kind H(1)

m(k1ρ), weighted by unknown scattering coeﬃcients as

Ψs=ΦT

H·as

0with ΦH=[H(1)

m(k1ρ)ejmφ]. (4.22)

The scattering coeﬃcients are related to the incident wave coeﬃcients by the T-Matrix

T[16,18], which results from the continuity condition at the cylinder’s surface.

100

4.2 In-Plane Scattering

For a P-polarized wave the T-Matrix is given as:

T=[τmδmn], with τm=–Z2Jm(k1a)J′

m(k2a)–Z1Jm(k2a)J′

m(k1a)

Z2J′

m(k2a)H(1)

m(k1a)–Z1Jm(k2a)H(1)′

m(k1a)(4.23)

0=Tp0(4.24)

with

δmn

the Kronecker’s delta. The parameters J

′

(1)′

represent the ﬁrst spatial derivative

of the Bessel and Hankel functions. Now, we extend the solution for a single cylinder to

ah-periodic cylinder array along the y-axis (Fig. 4.7 (b)), using the Floquet principle. The

scattered ﬁeld is now modiﬁed as:

Ψs(x,y)=∞

l=–∞

ejk1cosφilhΦT

H,las

0with (4.25)

ΦH,l=[H(1)

m(k1ρl)ejmφl], ρl=qx2+(y–lh)2, sinφl=y–lh

ρl(4.26)

The T-Matrix Twill be substituted by the aggregate T-matrix Twith the following steps:

L=[Lmn], Lmn =∞

l=1

H(1)

n–m(k1lh)he–jk1cosφilh +(–1)n–mejk1cosφilhi(4.27)

T=(I–TL)–1T(4.28)

0=Tp0, (4.29)

where Iis the identity matrix.

Figure 4.8:

An analytically calculated electric ﬁeld distribution for the scattering problem

of a dielectric cylinder array, placed along the y-axis. The incident wave is a

y-polarized plane wave, propagating in x-direction. (Adapted from Ref. [8] under

a CC BY 4.0 license.)

101

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

The ﬁnal magnetic ﬁeld in the exterior is a superposition of the incident and the scattered

ﬁeld components. The electric ﬁeld can be obtained by the Ampere-Maxwell’s equation.

Ψ(x,y)=ΦT

Jp0+∞

l=–∞

ejk1cosφilhΦT

H,las

0(4.30)



H=H0Ψ(x,y)

ez(4.31)



E=1

jωε0n2

rot 

H(4.32)

In summary, the following scattered ﬁelds result for each of the two cases:

Hz=H0ΨP-polarized wave

Ψ=Ψi+Ψs i - incident, s–scattered

Ψi=ΦT

Jp0, with ΦJ=[Jm(k1ρ)ejmφ], p0=[(–j)mejmφi], m=0,±1,±2,...

Single cylinder Cylinder array

Ψs=ΦT

H·as

0Ψs=∞

l=–∞

ejk1cosφilhΦT

H,las

ΦH=[H(1)

m(k1ρ)ejmφ]ΦH,l=[H(1)

m(k1ρl)ejmφl],

ρl=qx2+(y–lh)2, sinφl=y–lh

ρl

0=Tp0as

0=Tp0(4.33)

Independent of the speciﬁc case, the ﬁnally resulting H

ﬁeld is a sum of cylindrical waves

with both x- and y-dependence. When calculating the curl of this ﬁeld, we will obtain not

only the initial polarization E

, but also an E

-polarized part, which is an indication for

the origin of cross-polarization in the 2D

s. Figure 4.8 shows the magnitude of the

calculated E

and E

ﬁeld components for a case, which considers the 2D

’s material

and geometric properties. The cylinders’ material is SiO

, while the exterior’s material

is Si. The matrix operations are carried out in MATLAB with a Bessel/Hankel functions

order m

1,...

19 and a wavelength of 1550nm. The discrete spatial locations have a

resolution of 15 points per wavelength. The calculated ﬁeld distribution in Fig. 4.8 results

from a y-polarized incident ﬁeld, which passes through the cylinder array in



-direction.

The plotted ﬁelds are scaled equally, so we can clearly see that the initial E

-polarization

predominates. The scattering at the cylinder array appears to spoil its modal purity, but the

effect is small for this short 1D array. The cross-polarization E

looks rather stochastically

distributed without a well-deﬁned propagation direction. This can be explained by the

small number of scattering elements and by the missing second periodicity in propagation

direction, which could enhance the Exstrength and directivity.

This example shows that an array of cylindrically shaped elements leads to the partial

conversion of a given incident polarization into its orthogonal counterpart. This happens

even when the incident wave propagates perpendicularly with respect to the given array.

102

4.2 In-Plane Scattering

Since 2D

s are basically an arrangement of such cylindrical elements, the orthogonal

polarization conversion there (i.e. the cross-polarization) is inevitable. The present analytical

model is limited in its capability to describe cross-polarization, as its complexity grows

signiﬁcantly, when we go for a 2D array of scattering objects. In this case, the scattered

ﬁelds of the ﬁrst row should be used as a new incident ﬁelds on the next row etc. For even

higher accuracy, the assumed plane wave must be substituted by a Gaussian beam. In such

a case, the Gaussian beam must be expanded in plane waves and the scattering problem

must be solved for each partial plane wave. Obviously, the complexity of the analytical

model becomes easily comparable with the one of a numerical model and looses the

manageability typical for analytical representations. Nevertheless, the simpliﬁed analytical

formulation is valuable as a qualitative mathematical proof of the origin of cross-polarized

ﬁelds in 2D

s. In the following subsections, the consequences of the in-plane scattering

will be presented. Their impact on chip or system level will be analyzed in simulations and

experiments.

4.2.2 Cross-Polarization

As already indicated, the conversion of a given target-polarization into its orthogonal

counterpart - an effect assigned as a cross-polarization - is a phenomenon in 2D

which can be explained by scattering processes in the grating plane. There are several

important consequences from the cross-polarization, particularly:

Limited coupling eﬃciency. The cross-polarization scales with the grating’s perturba-

tion strength, since larger objects enhance the scattering. This puts limitations to the

2D GC’s out-coupled power.

Limited split ratio. Especially in polarization multiplexed systems, the 2D

s’ pur-

pose is twofold. They act not only as a coupling interface, but also as a polarization

splitter/combiner. The presence of cross-polarization inevitably limits the splitting

performance of a 2D

. The latter two issues have been discussed in detail in the

scope of this work and have been initially published in Ref. [19].

Polarization-dependent loss (PDL). In receiver-side 2D

s random incident polar-

izations are given. Depending on their orientation, the grating splits them into x-

and y-polarized components with a different power ratio and/or phase relation. This

means that a given target-polarization will be superposed with the cross-polarization

from the other signal path with a different phase and amplitude relation. This leads to

a large PDL. The problem is relevant also for systems without polarization multiplexing.

An investigation of this issue has been published in Ref. [20].

Polarizations’ non-orthogonality and crosstalk. Consider two signals coded into two

different channels 1, 2, which are orthogonally polarized to each other. The presence

of cross-polarization rotates the polarization states 1, 2 on the Poincarè sphere, so

that the latter may become non-orthogonal to each other. A polarization crosstalk

results as a consequence.

103

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

System-level penalties. The presence of a cross-polarized part in a given signal

leads to an additional optical signal-to-noise ratio (

OSNR

) penalty. The latter may

be compensated in the case of orthogonal signals, but becomes critical for non-

orthogonal signals. The latter two issues were investigated in the previous works

[19,21,22] with a focus on dual-polarization quadrature amplitude modulation (

QAM

)

systems.

4.2.2.1 Split Ratio and Coupling Eﬃciency Limitations

The basic investigation of the cross-polarization and, particularly, its impact on the 2D

’s

split ratio and coupling eﬃciency are summarized here, following the previously published

work in Ref. [19]. The analysis is performed numerically and experimentally and considers

the two types of sheared 2D

s (cf. Fig. 4.2). The impact of different shear angles and

etch depths are studied in more detail.

Figure 4.9:

A comparison of the target- and cross-polarization of different sheared 2D

grating couplers (2D GCs). Two types of gratings are considered - Type I with a

rhombus-shaped grating area and Type II with angled waveguides. (a) A shear

angle

α=

◦

, an etch depth d

120

, (b) a shear angle

α=

◦

, an etch depth

, (c) a shear angle

α=

◦

, an etch depth d

120

. The evaluation

plane is tilted, according to the ﬁber angles

φF=

◦

θF=

◦

. (Adapted from

Ref. [19] under a CC BY 4.0 license.)

Numerical Analysis For the numerical analysis of cross-polarization related effects, out-

coupling 2D

s for the C-band are considered. Only one of the waveguides is excited, in

our case corresponding to the y-polarization. The radiated ﬁelds are evaluated at a tilted

plane (

φF=

◦

θF

), corresponding to the

SMF

position. We calculate the power, which is

104

4.2 In-Plane Scattering

properly coupled into the y-polarization in the

SMF

. The portion of power coupled into

the x-polarization is then referred as a cross-polarization. The difference between the

calculated y- and x-polarization powers in dB at a wavelength of interest corresponds to

the 2D

’s split ratio. A large split ratio indicates that the 2D

is a good polarization

beam splitter/combiner (

PBS/C

). Because in practical measurements it is diﬃcult to de-

termine exactly the maximal transmission wavelength, a mean split ratio in a wavelength

range of 15nm around the y-polarization’s maximum is evaluated. In this interval, the

coupling eﬃciency changes by no more than 0.3dB, which is typically the accuracy limit

in transmission measurements. For symmetry reasons, the obtained split ratios are the

same, also when a single-port simulation of the other 2D GC arm are carried out.

The designs for the investigation of cross-polarization effects are such that maximally large

diffracting elements are given. Their diameter and the grating period take into account

the design rules of the 248nm

DUV

lithography used for the fabrication. The designs

are chosen on the one hand to compare structures of Type I and II with different shear

angles and on the other hand - structures with different etch depths and ﬁxed remaining

geometric parameters. Following geometries and coupling angles are designed, assuming

a coupling to/from a Si rib waveguide with a 220nm height and slab etch depth equal to

the grating etch depth.

A shear angle

α=

◦

of Type I or II, a grating period

Λ=

622

, circular perturbing

elements with a diameter w

Λ=

440

, an etch depth d

120

, coupling angles

φout =45◦,θout =8◦at 1550nm.

A shear angle

α=

◦

of Type I or II, a grating period

Λ=

622

, circular perturbing

elements with a diameter w

Λ=

440

, an etch depth d

, coupling angles

φout =45◦,θout =8◦at 1590nm.

A shear angle

α=

◦

of Type I or II, a grating period

Λ=

636

, circular perturbing

elements with a diameter w

Λ=

450

, an etch depth d

120

, coupling angles

φout =45◦,θout =12◦at 1550nm.

Table 4.2:

Numerically estimated split ratios SR in a 15 nm wavelength range near the y-

polarization’s transmission maximum for different sheared 2D grating couplers

(2D GCs). The coupling angle is 8◦at the symmetry plane.

Shear angle αEtch depth d[nm] Wavelength [nm] 2D GC Type Split ratio

SR [dB]

2◦120 1530-1545 I 9.5

II 11.5

2◦90 1585-1600 I 14.2

II 16.5

3◦120 1570-1585 I 7.4

II 9.8

105

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

The ﬁrst two designs a) and b) are used to investigate the impact of the etch depth. Because

all other parameters are kept unaltered, the maximum transmission wavelength is shifted

to 1590nm for the decreased etch depth of 90 nm. The designs with equal etch depth a)

and c) can be compared regarding the impact of the shear angle. For these two designs,

different radiation angles

θout =

◦

,12

◦

result. To ensure the same coupling conditions e.g.

in terms of vertical distance to the grating, the evaluation angle is ﬁxed at

θF=

◦

. The ﬁxed

ﬁber coupling angle is more convenient for comparison of experimental results as well.

With θF=8◦the design with a shear angle α=3◦has a maximum shifted near 1580 nm.

Figure 4.9 shows the simulated out-coupling spectra for the two types of sheared gratings,

comparing the three different cases. In any of them, we see that the two types of gratings

Type I and Type II differ slightly from each other. Another common characteristic in all

cases is that the cross-polarization (x-polarized) has a central wavelength, which is shifted

to larger wavelengths compared to the y-polarization. The shift is larger in the case of

the gratings of Type II. Figure 4.9 (a) and (b) compares the geometries with different etch

depths, the remaining parameters ﬁxed. Figure 4.9 (a) and (c) compares models comprising

different shear angles. For each structure, a mean split ratio in a 15nm interval around

the maximal transmission wavelength is given in Tab. 4.2. Because of the larger cross-

polarization wavelength shift for the Type II gratings, their split ratios appear better than for

the Type I grating. Comparing the structures with shear angles 2

◦

and 3

◦

, we see that the

latter shows around 1-2dB worse split ratio. This shear angle is combined with a different

grating period, so that the worse behavior may be caused not by the shear angle alone,

but by the combination of shear angle and period.

Figure 4.10:

An electric ﬁeld distribution and a power loss in percent caused by ﬁelds,

propagating further in a 2D grating coupler (2D GC). The shear angle is

α=

◦

The input mode is purely y-polarized. After propagating through the 2D GC, a

portion of its power is radiated under

φout =

◦

θout =

◦

and coupled into a

SMF. Power losses caused by ﬁelds that propagate further within the grating

are shown. The propagating mode has two polarization states. For an etch

depth d

, the power remains mainly within the initial y-polarization

(24%) and 10% are converted into the cross-polarization. For an etch depth

120

the power loss is mainly caused by the cross-polarization conver-

sion (24%). (Adapted from Ref. [19] under a CC BY 4.0 license.)

106

4.2 In-Plane Scattering

A comparison between the structures with different etch depths in Tab. 4.2 shows clearly

that a larger etch depth, corresponding to a stronger grating perturbation strength, leads

to about 4-5dB worse split ratio. For a good splitting functionality, shallowly etched 2D

should be preferred. However, unlike the shear angle, the etch depth has a strong impact

on the grating out-coupled power and coupling eﬃciency. If the perturbation strength is too

weak, a large amount of power remains guided and is not radiated by the grating. With an

increasing etch depth, more cross-polarization occurs and again - further propagating (this

time) cross-polarized ﬁelds remain in the grating and limit the out-coupled power eﬃciency.

This trade-off is illustrated by the numerical example in Fig. 4.10 for the designs a) and b)

of Type I. On the left hand side, we have an input mode, which is purely y-polarized. During

the mode’s propagation through the 2D

, a portion of its power is coupled out towards a

SMF

. Power losses are caused by a radiation towards the substrate (not represented) and

by the power further guided in the structure. In Fig. 4.10, we clearly see that in both cases

the further propagating mode has an y-polarized part (in the waveguide on the right hand

side) and an x-polarized part (in the upper waveguide). For the etch depth d

the

power remains mainly within the initial y-polarization (24 %) and 10 % are converted into

the cross-polarization. On the other hand, for an etch depth d

120

the power loss is

mainly caused by the cross-polarization conversion (24%) and overall we do not increase

the amount of the radiated power. The cross-polarization conversion increases with an

increasing grating perturbation strength - compared to 1D

s, this is an additional loss

mechanism limiting the maximal achievable coupling eﬃciency.

Figure 4.11:

Exemplary measured interferometric curves for the characterization of differ-

ent sheared 2D grating couplers (2D GCs) in terms of split ratio. Two types of

gratings are considered - Type I with a rhombus-shaped grating area and Type

II with angled waveguides. (a) A shear angle

α=

◦

, an etch depth d

120

(b) a shear angle

α=

◦

, an etch depth d

, (c) a shear angle

α=

◦

, an

etch depth d=120nm. (Adapted from Ref. [19] under a CC BY 4.0 license.)

107

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Experimental Measurement of the Polarizations’ Split Ratio The experimental veriﬁca-

tion of the numerically predicted polarizations’ split ratios is carried out by manual wafer

measurements, using the test structure and method described in 2.2.3, Chap. 2.

We compare structures of Type I and II as well structures with different shear angles

α=

◦

and

α=

◦

on the same wafer. For the comparison of the structures with different

etch depths d

and d

120

, two separate wafers are available. To account for

fabrication variations on these wafers, we determine an averaged split ratio over 10 chips.

A laser source Agilent 81960 with a wavelength range from 1505 nm to 1625 nm is used,

followed by a programmable polarization controller Agilent 8169A and two standard

SMF

for the in- and out-coupling together with their alignment equipment. The signal is ﬁnally

detected by a power sensor Agilent 81634B. Back-to-back reference structures with 1D-1D

FGC

s and 2D-2D

s are used for the determination of the proper coupling angles at both

sides. They are chosen such, that the 2D

s at the input and the 1D

FGC

s at the output

have a maximal coupling centered at the same wavelength. The 1D-1D

FGC

back-to-back

structure is additionally used for polarization adjustment, as described in 2.2.3, Chap.

2. The 1D

FGC

s have a period of 610nm, a perturbing elements’ width of 315nm and

and etch depth of 70nm. The wavelength sweep is performed in 1pm increment for the

MMI

-based measurement, to make the

determination as accurate as possible. The

is ﬁrst averaged in the given 15nm bandwidth. Its value is then converted to a split

ratio SR according to (2.69) in 2.2.3, Chap. 2. The split ratios determined on different chips

are ﬁnally averaged to account for the wafer deviations (a mean value

and a standard

deviation σ).

Figure 4.11 shows exemplary interferometric curves for each of the considered 2D

designs. The polarization is such, that the incident ﬁeld is coupled into one of both 2D

arms. The depicted wavelength range covers the 15nm evaluation bandwidth around the

spectral maximum. The vertical axes are scaled equally. The transmission includes the

coupling losses at the input and the output. It has been observed that split ratios above

20dB, which correspond to

s of below 1.5dB become diﬃcult to measure, because

other effects like Fabry-Perot resonances and noise start to predominate. For that reason,

a larger distance between the

SMF

and the 2D

has been chosen for the measurements

of the structures with a 90

etch depth. Parasitic effects could be minimized at the cost

of a higher insertion loss (Fig. 4.11 (b)). Since a relative measurement is carried out, the

coupling loss is not critical for the ER estimation.

In Tab. 4.3 the bandwidth- and wafer-averaged split ratio

SR±σ

for each of the considered

cases are summarized. All structures, which are directly compared to each other, show a

similar standard deviation. The differences between devices of Type I and Type II vary from

one device geometry to another and we cannot generally state that one of the types has

an advantageous splitting behavior. The lack of systematic difference can be caused by

variations of the perturbing elements’ size. In addition,

SOI

height deviations have also an

impact, as they change the grating perturbation strength. These fabrication variations can

make it diﬃcult to observe differences between the gratings with shear angles

α=

◦

well. Being diﬃcult to distinguish from other fabrication variations, the contrast between

108

4.2 In-Plane Scattering

Table 4.3:

Experimentally estimated mean split ratios

in a 15 nm wavelength range near

the maximal transmission for different sheared 2D grating couplers (2D GCs).

The values are averaged over 10 chips with a standard deviation

. The coupling

angle is 8◦.

Shear angle αEtch depth d[nm] Wavelength [nm] 2D GC Type SR±σ[dB]

2◦120 1527-1542 I 16.4±1.4

II 17.7±1.6

2◦90 1579-1594 I 19.7±1.1

II 19.1±1.7

3◦120 1542-1557 I 15.3±1.0

II 17.7±0.9

the 2D

s of different types or with different shear angles appears to be of minor impor-

tance. Compared to simulations, the difference between the split ratios for d

120

and

is lower - between 1 dB and 3 dB and better pronounced for the 2D

s of Type

I. The deviations from the simulation can result for the following reason. In real structures,

the perturbing elements are not cylindrically, but conically etched. This results in a lower

perturbation strength than assumed in simulations, so that we generally obtain better split

ratios than expected. Moreover, deeply etched features are more affected by that problem

than shallowly etched ones. This means that the perturbation strength does not scale with

the etch depth in the same way, as assumed in the simulations. Because the perturbation

difference between 2D

s with d

and d

120

is lower than assumed, their

split ratios differ less from each other. In addition, effects related to non-orthogonality may

have also an impact on the maximally reachable split ratio. Nevertheless, the distinction

between structures with different etch depths is still observable, which shows that the

grating’s perturbation strength has the most pronounced impact on the 2D

’s splitting

performance.

System-Level Behavior The following paragraph is a kind contribution by Pascal M. Seiler,

who performed and evaluated system-level simulations and experiments.

In these analyses, the impact of the limited split ratio is examined in a coherent system

that does not take this issue into account (no dedicated

DSP

applied). As a criterion, the

OSNR

penalty at a bit error ratio (

BER

) of 10

–3

is used. The penalty is deﬁned as the

difference between the

OSNR

levels in dB, which are necessary to reach the target

BER

. In

simulations, penalties for different dual-polarization (DP)

QAM

formats are determined

(results originally published in Ref. [19]). Experimentally, 16-

QAM

single-polarization (SP)

and DP transmission are compared (results originally published in Ref. [21]).

Simulation Results for Different QAM Formats (by Pascal M. Seiler) In an integrated

DP coherent detection receiver, the 2D

’s limited polarization split ratio, can be expressed

as a superposition between two data streams with the same source. The system behavior

in such a case is comparable to the in-band crosstalk, discussed in long haul transmission

systems [23]. We perform an exemplary simulation, where the split ratio is varied and

109

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

the

OSNR

penalty for reaching the forward error correction (FEC) limit of

BER

= 10

–3

determined. The penalty results from the comparison of the same polarization in SP

and DP operation. A 28Gbaud transmission using a laser with a linewidth of 100kHz is

assumed (local oscillator effects are neglected). Polarization mode dispersion is not taken

into account.

Figure 4.12 shows the

OSNR

penalty over the 2D

split ratio for different

QAM

formats.

If we permit a 1 dB penalty (corresponding to the dashed line in Fig. 4.12), we can estimate

the minimal split ratio required for the desired

QAM

format. For quadrature phase shift

keying (

QPSK

), 8-PSK, 16-QAM, 32-QAM and 64-QAM, the required split ratios are 16.5 dB,

21.5dB, 23.5dB, 26.5dB and 29dB, respectively. The 2D

designs discussed in the

present section reach mostly split ratios better than 16.5dB, which make them suitable

for DP

QPSK

without the necessity of dedicated

DSP

compensation methods. However,

for higher order modulation formats, the limited split ratio becomes more critical. The

integration of 2D

s in such systems may be possible only with appropriate signal post-

processing. The latter may increase the receiver power consumption and complexity.

Therefore, the suppression of cross-polarization effects in 2D

s is desirable for the

realization of cost-eﬃcient integrated higher order DP QAM systems.

Figure 4.12:

An exemplary simulation of the optical signal-to-noise ratio (OSNR) penalty for

reaching the forward error correction (FEC) limit of a bit error ratio BER = 10

–3

plotted over different 2D grating coupler (2D GC) split ratios. The considered

DP QAM formats are QPSK, 8-PSK, 16-QAM, 32-QAM and 64-QAM. Further

abbreviations: DP: dual-polarization, QAM: quadrature amplitude modulation,

(Q)PSK: (quadrature) phase shift keying. (Adapted from Ref. [19], used under a

CC BY 4.0 license. Author: Pascal M. Seiler.)

A 16-QAM Experiment (by Pascal M. Seiler) To verify the predicted performance

deterioration, the polarization splitting/combining limitations of 2D

s are investigated in

a 16-

QAM

system experiment. The 2D

s are considered at the transmitter side, but due

to reciprocity, the same effects are present at the receiver as well. An exemplary 2D

design is chosen, which has been analyzed with respect to the split ratio in this section

(design c) of Type II on p. 105). To study the impact of the chosen 2D

, a simple on-chip

110

4.2 In-Plane Scattering

structure is designed, which comprises two 1D

FGC

s as inputs, combined at the two arms

of a 2D GC. The 1D FGCs have a period of 610nm, perturbing elements’ width of 315nm

and an etch depth of 70nm. An exemplary conﬁguration is shown in Fig. 4.13 (a), the shown

structures do not correspond to the measured ones. The test structure is fabricated in a

standard full photonic BiCMOS ﬂow.

Figure 4.13:

(a) A test structure (photonic integrated circuit, PIC) used to investigate the

polarization combining performance of a 2D grating coupler (2D GC). For a

single-polarization (SP) operation, a signal at only one of the inputs is coupled -

the 2D GC acts as a coupling interface. For a dual-polarization operation, both

inputs are active - the 2D GC acts as a coupler and polarization combiner. (b)

A schematic representation of the experimental setup used for a 16-QAM

transmission experiment. The inset shows 160k bits of a recovered 16-QAM

at 10GBd for a single polarization. Further abbreviations: QAM: quadrature

amplitude modulation, ECL: external cavity laser, DP: dual-polarization, IQM:

in-phase quadrature modulator, DAC: digital-to-analog converter, PBS: po-

larization beam splitter, VOA: variable optical attenuator, PC: polarization

controller, POW: power meter, OBPF: optical bandpass ﬁlter, OSA: optical

spectrum analyzer, Rx: receiver, RTO: real-time oscilloscope. (Author: Pascal

M. Seiler, re-print with permission. The ﬁgure is published in Ref. [21].)

An intradyne measurement setup is deployed to study the polarization combining per-

formance of a 2D

(Fig. 4.13 (b)). A commercial IQ transmitter (ID photonics OMFT,

1540.30nm) is used to generate a DP 16-

QAM

. The modulator is driven by a DAC (Fujitsu

OOLA DK 2A), which supplies four 10GBd random patterns with a peak-to-peak voltage

of approximately 290mV. The 10GBd is limited by the available equipment and has no

impact on the cross-polarization. The -0.5dBm (DP) output power are then ampliﬁed with

EDFA

(IPG EAD-1K-C) to +20dBm, before split into the respective orthogonal polariza-

tions (denoted as X and Y) using a PBS. The 1D

s, used as chip interfaces, also act as

polarization ﬁlters, removing the residual unwanted polarizations after the PBS. Remaining

power imbalances of the modulator and PBS are compensated using variable optical

111

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

attenuators (VOAs) and a calibrated photodiode at the ﬁber facet at the chip input. Using

the VOAs, one of the polarizations can be blocked of to have either a SP, or DP signal at the

. The coupling is closely monitored using a power meter and a 20dB splitter. The

SP/DP signal is then ﬁltered using a 1 nm ﬁlter to remove ampliﬁed spontaneous emission

(ASE) noise and subsequently noise loaded using an

EDFA

(Fiber Labs AMP-FL8011-CB).

The input power at the coherent receiver (Finisar CPRV1222A) for the SP and DP signal are

-3dBm and -0.2dBm, respectively. The local oscillator, LO (Agilent 81960A, 1540.05nm)

is set to +14dBm. Only one polarization is detected at the coherent receiver, since the

impact of cross-polarization on a single polarization is suﬃcient for this investigation. The

detected polarization state is not changed between the SP and DP measurements. The

polarization states within the setup are closely controlled using polarization controllers.

For this purpose, only the coherent receiver’s Y output is connected to the real-time os-

cilloscope (Tektronix DPO71304SX, 50GSa/s, 13GHz bandwidth). The Y output has per

speciﬁcation a polarization extinction ratio of 35.5dB at 1550nm.

An oﬄine processing is done with the Tektronix OM1106 optical modulation analyzer

software. The

DSP

includes a frequency offset compensation and a least-mean square

ﬁlter based on a radius-directed algorithm. The

OSNR

is varied between 21dB and 33 dB

per polarization and the respective

BER

are measured by switching between SP and DP

using the VOAs. The determined

BER

s for SP and DP signal are shown in Fig. 4.14. The DP

signal is overall showing a penalty in comparison to the SP signal. At a

BER

of 10

–3

, an

OSNR

penalty of approximately 2 dB can be found. According to Fig. 4.12, the corresponding split

ratio is about 16dB. This value is in the expected range for a structure with this geometry

(cf. Tab. 4.3 for a Type II design with a shear angle of 3

◦

and an etch depth of d

120

Figure 4.14:

Bit error ratio vs. optical signal-to-noise ratio (BER vs. OSNR) for a transmission

with a single active polarization (single-polarization, SP) and two active polariza-

tions (dual-polarization, DP). The 2D grating coupler (2D GC) is included at the

transmitter-side. The curves are evaluated for one of the receiver polarizations.

The OSNR penalty at BER = 10

–3

in the DP case is approximately 2dB. (Author

Pascal M. Seiler, re-print with permission. The ﬁgure is published in Ref. [21].

4.2.2.2 Polarizations’ Non-Orthogonality

Some of the results and discussions shown in the following paragraph have been published

in Ref. [22].The penalties presented in the previous paragraph are related to the splitting

112

4.2 In-Plane Scattering

performance limitations of 2D

s. It can be expected that these negative effects can be

compensated by appropriate

DSP

routines. Since a 2D

converts partially a x-polarization

into a y-polarization and vice versa, the ﬁnal effect can be compared to the polarization

rotation, which takes place in an optical ﬁber. In coherent receivers, the compensation

of this rotation is a standard procedure. However, the recovery of the original x- and

y-signals assumes that they are combined/rotated orthogonally. In systems, where the

orthogonality is not preserved (e.g. a ﬁber transmission with PDL), power and OSNR

imbalance between the two polarizations can occur [24]. This motivated the investigation

of the splitting/combining performance of 2D

s in terms of orthogonality. A numerical

analysis can be carried out, following the procedure in 2.1.4.3, Chap. 2.

In this paragraph, two exemplary 2D

s for C- and O-band are compared with regard to

the polarizations’ non-orthogonality. The design geometries for a coupling angle

θout =

◦

at the symmetry plane are:

•

C-band: a shear angle

α=

◦

, a grating period

Λ=

622

, circular perturbing ele-

ments with a diameter wΛ=440nm, an etch depth d=120nm.

•

O-band: a shear angle

α=

◦

, a grating period

Λ=

480

, circular perturbing ele-

ments with a diameter wΛ=280nm, etch depth d=120nm.

Both designs are of Type II and are intended for coupling to a Si rib waveguide with a 220 nm

height and slab etch depth equal to the grating etch depth. For the simulations, a single-

port excitation of a given target-polarization is done separately for both input waveguides

1 and 2. The evaluation plane is ﬁxed at the

SMF

coordinates with

φF=φout =

◦

and

θF=θout =

◦

. At this plane, we calculate ﬁrst the coupling eﬃciency of the target- and cross-

polarization from each waveguide. Due to symmetry, results for one of both waveguides

will be shown. In addition, the Stokes parameters of the ﬁelds from waveguide 1 and 2

are calculated as well. Out of them, the angles on the Poincaré sphere are determined,

whereat the azimuth angles’ difference

Δψ =ψ1

–

ψ2

is of interest (

χ1

χ2

negligible). Finally,

a polarization crosstalk is determined out of this difference, using (2.64) from 2.1.4.3 in

Chap. 2.

In the following we investigate the designs in a 40nm bandwidth around their maximum

transmission. The design for C-band is centered at 1540nm, the considered bandwidth is

1520-1560nm. The design for O-band is centered at 1310 nm, the considered bandwidth

is 1290-1330 nm. Figure 4.15 compares multiple parameters, relevant for the coupling and

splitting performance of both devices. The coupling eﬃciencies of both designs (target- and

cross-polarization) can be seen in Fig. 4.15 (a) for C-band and (b) for O-band. Apparently,

the O-band design outperforms its C-band counterpart in terms of eﬃciency, with a target-

polarization’s maximal coupling eﬃciency of -3.3dB at 1310nm vs. -5dB at 1550nm. In

parallel, the C-band 2D

shows higher levels of cross-polarization, which directly reﬂects

on the polarization split ratio (Fig. 4.15 (c) vs. (d)). The worse splitting behavior in C-band is

further reﬂected in the orthogonality relation

Δψ

between waveguides 1 and 2. Within the

considered bandwidth, the variation in C-band is in the range 77

◦

to 132

◦

(Fig. 4.15 (e)),

compared to 84◦to 102◦in O-band (Fig. 4.15 (f)).

113

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Figure 4.15:

A comparison of 2D grating couplers (2D GCs) for C- and O-band. (a) Target-

and cross-polarization spectra in C-band. (b) Target- and cross-polarization

spectra in O-band. (c) The polarization split ratio in C-band. (d) The polarization

split ratio in O-band. (e) The polarization angles’ difference in C-band. (f) The

polarization angles’ difference in O-band. (g) The polarization crosstalk in C-

band. (h) The polarization crosstalk in O-band. (Adapted from Ref. [22] used

under a CC BY 4.0 license.)

This is ﬁnally reﬂected into the polarization crosstalk, which reaches up to -3.5 dB at

1560nm, whereas the maximum in O-band is -13.5dB at 1330nm (Fig. 4.15 (g), (h)). It

should be noted that the overall better performance of the O-band design is rather related

to the boundaries of our fabrication platform, which are more advantageous for the O-band

114

4.2 In-Plane Scattering

s (also experimentally observed, cf. [25]). Interestingly, independent of the optical

band and the achieved split ratio, both 2D

s combine the signals orthogonally at their

central wavelengths. Accordingly, wavelengths other than the design wavelength show non-

orthogonality and crosstalk. Although this behavior looks like a good fortune, it becomes

obvious that non-optimized 2D

s can be used in a very narrow wavelength range. This

is particularly disadvantageous, when robustness e.g. against laser emission ﬂuctuations is

required. The latter is mandatory for systems using uncooled lasers, which are economically

reasonable for data centers. Since 2D

s are candidates for photonic sub-systems in

DCI

s, where stringent requirements in terms of power consumption and cost are given,

the improvement of the 2D

s’ polarization splitting/combining performance is essential.

In this work, substantial efforts were made to optimize 2D

s in this aspect, with results

following later in Chap. 5.

4.2.2.3 Polarization-Dependent Loss

With respect to undesired effects caused by scattering and cross-polarization, we focus

here particularly on receiver-side 2D

s, which are exposed to randomly polarized light.

Here, it will be illustrated that the cross-polarization and the related polarization-splitting

issues can be associated with a

PDL

at the receiver. This

PDL

is expressed in a different

coupling eﬃciency, depending on the incident polarization.

Figure 4.16:

An illustration of different incident polarizations at a receiver-side 2D grating

couplers (2D GCs). The different orientations are responsible for a different

signal splitting manner between both 2D GC arms. (a) and (b) An ideal coupling

of a purely y- or x-polarized incident wave. (c) An incident polarization for

an in-phase signal splitting, assigned as an even-polarization. (d) An incident

polarization for an anti-phase signal splitting, assigned as an odd-polarization.

For the subsequent analysis, we shall make parallels between single- and dual-port sim-

ulations. Out of the former, we can determine the coupling eﬃciency of a given target-

polarization, as well as its orthogonally converted component - the cross-polarization.

115

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Afterwards, we can continue with a dual-port excitation, i.e. the simultaneous activation

of Ports 1 and 2. There, the ﬁnal x-polarization, for instance, will be a superposition of

the target-polarization from Port 2 and the cross-polarization from Port 1. To account

for the polarization diversity at the receiver, different amplitude/phase relations between

the signals from the two ports have to be deﬁned. Particularly, we are interested in the

scenarios depicted in Fig. 4.16 (c) and (d). The polarization in Fig. 4.16 (c) is related to signals

in the two

arms, which are in-phase, e.g. +x- and +y-polarized. The incident polarization

in Fig. 4.16 (d) corresponds to port signals in anti-phase, e.g. –x- and +y-polarized. To

distinguish between these two cases, the polarization, leading to an in-phase splitting,

is called an even-polarization. Accordingly, the polarization for an anti-phase splitting is

assigned as the odd-polarization. It should be noted that these polarization states can

be met in the literature under different names, such as S- and P-polarization or 45

◦

- and

135

◦

-polarization [26

–

29]. In accordance to the assignment of the even- and odd- incident

polarization, the most important circumstance for any difference in their behavior is the

way, how the target-polarization and cross-polarization are superposed. In the even case,

the target- and cross-polarization are in-phase, while in the odd case, they are in anti-phase.

Respectively, any other polarization orientations result in different superpositions of the

target- and cross-polarization. The variable scenarios for the superposition of a target- and

cross-polarization may be expressed in a

PDL

. The largest

PDL

can be expected between

the even- and the odd-combination, because the cross-polarized signal is maximized in

these cases.

Figure 4.17:

Coupling eﬃciencies of a target- and cross-polarization in parallel with the

x-component of an even- and odd-polarization for 2D grating couplers (2D

GCs) with the same geometry and of different types. (a), (c) Type I; (b), (d)

Type II. Both 2D GCs reach a relatively high levels of cross-polarization. In the

same time, their even- and odd-polarization are strongly wavelength-shifted

from each other, leading to a large polarization-dependent loss (PDL). It is

hypothesized that the cross-polarization is responsible for the PDL.

116

4.2 In-Plane Scattering

To illustrate this behavior, we consider now two exemplary 2D

designs of different types.

A single-port simulation is used to show the coupling spectrum of the target- and cross-

polarization (scenarios in Fig. 4.16 (a) and (b) equivalent). Afterwards, the same structure is

simulated in a dual-port simulation (simultaneous excitation), whereat the according phase

conditions are set to reproduce the even or odd case (Fig. 4.16 (c) and (d)). The considered

structures operate in C-band and differ only by their type and have a waveguide-to-grating

shear angle of 2

◦

(Type I or II), a grating period of 622 nm and circular diffracting elements

with a diameter of 440nm and an etch depth of 120nm (cf. the structures in 4.2.2.1).

Compared to previous simulations, the full

BEOL

has been omitted here, in order to save

simulation time and hard disk space. Due to the evaluation at a lower distance to the

grating, the coupling eﬃciency is better than with a full

BEOL

stack. Since the coupling

eﬃciency is not a subject of investigation here, the deviation is acceptable.

Figure 4.17 (a) and (b) shows the single-port simulation of both designs, indicating the

coupling eﬃciency of the target- and cross-polarization. In parallel, Fig. 4.17 (c) and (d)

shows the x-component of the even- and odd-polarization, determined from the dual-port

simulation. Due to symmetry, the y-component behaves in the same way and is not shown.

Large cross-polarization and large

PDL

can be observed simultaneously. The

PDL

is a

result from the wavelength-shifted spectra of the even- and odd-polarizations. It is not

predeﬁned, whether the even or the odd polarization will be shifted in a given direction, as

can be seen from the comparison between 2D

s of Type I and Type II. The

PDL

is among

the most critical problems in 2D

s. Based on these observations, in this thesis, it is

hypothesized that the large difference between the even- and odd-polarizations is entirely

caused by the cross-polarization and the grating’s in-plane scattering as its physical origin.

All developed optimization methods (cf. Sect. 5.2, Chap. 5) are based on this assumption.

It is worth mention that 2D

s with a low coupling eﬃciency will inherently have also

a poor cross-polarization out-coupling. For that reason, low-eﬃciency 2D

s achieve

“automatically” a low

PDL

as well. The challenge for the optimization methods is thus not

only to reduce the scattering/cross-polarization and improve the

PDL

, but in the same

time to deliver a reasonable coupling eﬃciency.

4.2.3 Higher-Order Modes

To ﬁnd an experimental evidence for the grating’s in-plane scattering, the polarization

and mode coupling in 2D

interfaced Si waveguides is investigated here. With regard

to polarization splitting, it is expected that 2D

s with a vertical ﬁber coupling

θF=

◦

are able to separate x- from y-polarized signals perfectly [17]. If we consider Fig. 4.18, this

would mean that an incident y-polarization will propagate in waveguides (WG) 1 and 3,

and the x-polarization will be coupled to WG 2 and 4. However, this assumption is correct

only, when no in-plane scattering is present. In this subsection, it will be shown that even

zero-angle coupled 2D

s suffer from cross-polarization and limited polarization splitting

capabilities. Furthermore, it will be shown that 2D

s designed for different coupling

angles change only the modal composition of the undesired coupled ﬁelds. Even when

higher-order modes can be eﬃciently ﬁltered in single-mode waveguides, their excitation

117

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

may be a further limiting factor for the coupling eﬃciency.

For the subsequent analysis, the following notations will be made. Considering Fig. 4.18,

we assume an incident polarization, such that all power should be coupled into WG 1 (an

example with

θF=

◦

). Our target-polarization is thus the y-polarized TE

mode (short:

00,y

), which results from diffraction at the 2D

perturbing elements matching the

ﬁber angle

θout =θF

. In addition, we investigate, whether other higher-order y-polarized

modes are coupled, particularly TE

10,y

. The latter cannot result from out-of-plane diffraction,

because its effective refractive index does not fulﬁll the 2D

’s diffraction condition for

the given angle

θF

. The presence of TE

10,y

will be the ﬁrst indication of in-plane scattering

Figure 4.18:

A schematic representation of a 2D grating coupler (2D GC) with a Gaussian

beam excitation from top. Several ﬁeld components may be excited in a Si

waveguide due to grating in-plane scattering. The grating plane is deﬁned in

the Cartesian (x,y,z) coordinates, the Gaussian beam is tilted under the ﬁber

angles (

φF=

◦

θF

). The arrows indicate the polarization of the excitation

source (in yellow), which is chosen such that all the power should be completely

coupled into the TE

mode of the Si waveguide WG 1 by diffraction (target-

polarization: TE

00,y

). The example illustrates the case

θF=

◦

. The excitation of

higher-order modes, e.g. TE

10,y

can be attributed to in-plane scattering. The

power coupled in the other waveguide WG 2 (marked in blue) is assigned as a

cross-polarization, which can be also composed of TE

00,x

, TE

10,x

etc. For the

given incident ﬁeld, the presence of these components is an indication for

in-plane scattering as well. (Adapted from Ref. [8] under a CC BY 4.0 license.)

processes in 2D

s. Similarly, the cross-polarization (here the x-polarization), coupled

to waveguide WG 2 will be analyzed. Also in this case, two modes are considered: TE

00,x

and TE

10,x

. For the given incident ﬁeld, the presence of the TE

00,x

and TE

10,x

components in

the Si waveguides can be also attributed to grating in-plane scattering (cf. 4.2.1). Table 4.4

summarizes the notations used in this subsection. The excitation of undesired polarizations

and modes is investigated experimentally, considering 3 different 2D

designs in C-band.

118

4.2 In-Plane Scattering

Table 4.4:

A summary of the notations, used to assign a desired target-polarization with

a TE

modal component, resulting from diffraction and in-plane-scattering-

related polarizations/modes.

Full assignment Short assignment Desired? Physical process

target-polarization yin mode TE00 TE00,yyes diffraction

polarization yin mode TE10 TE10,yno in-plane scattering

cross-polarization xin mode TE00 TE00,xno in-plane scattering

cross-polarization xin mode TE10 TE10,xno in-plane scattering

As a ﬁgure-of-merit, the

between the target TE

00,y

and the remaining polarizations/-

modes is used, which is determined on different wafer sites. More detailed results and

discussion can be found in the original publication, Ref. [8].

Experimental Approach and Setup We begin with the description of the experimental

method, used for the higher-order mode’s characterization.

Figure 4.19:

(a) A camera picture of an exemplary device used for the experimental deter-

mination of the polarization state and the modal composition of ﬁelds, excited

by a 2D grating coupler (2D GC) in Si components. (b) A detailed picture of a

2D GC under test, the input 2D GC varies according to the three considered

designs. (c) A schematic of the components following the upper and the lower

arms of a 2D GC. For the experiments, only the upper part is relevant, which

contains a TE

10 →

mode converter and 1D GCs for out-coupling. Output

1 (O1) delivers the portion of power of the TE

mode. Output 2 (O2) delivers

the portion of power of the TE

mode. (d) A detailed picture of the output

sections. (Adapted from Ref. [8] used under a CC BY 4.0 license.)

119

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Test Structures All test structures are fabricated in a photonic

BiCMOS

short ﬂow,

in which the

BEOL

has not been completely processed. This is eligible for the current

investigation, since the

BEOL

has no impact on the higher-order mode excitation by the 2D

s and its omission reduces the fabrication time. The minimum feature size is deﬁned by

the 248nm

DUV

lithography we used. Figure 4.19 (a) shows an exemplary device picture,

taken with a microscope camera. The relevant device parts are indicated by the dashed

ellipse in yellow. Figure 4.19 (b) shows a more detailed picture of a 2D

under test.

Figure 4.19 (c) illustrates schematically the structures in the upper and the lower arms

of the 2D

. Only the upper arm is relevant for the experiments. There, we apply after

down-tapering a TE

10 →

mode converter, which can extract a TE

mode propagating

in the waveguide towards Output 1 (O1) and convert it to the fundamental TE

mode

propagating into the waveguide towards Output 2 (O2). Therefore, for the polarization in

question, O1 gives us information about the power coupled into the fundamental TE

mode, while O2 shows the share of TE

. At each output, identical 1D

FGC

s are used (a

period 610 nm, perturbing elements with a width of 315 nm, an etch depth 70 nm, a design

coupling angle 8

◦

at 1550nm). Figure 4.19 (d) shows a detailed camera picture of the

output sections.

Table 4.5:

2D grating coupler (2D GC) designs used for the investigation of their polariza-

tions’ and modal on-chip excitation.

Abbr. Coupling

angle

Shear

angle

Grating

period [nm]

Perturbing elements

diameter [nm]

Etch

depth [nm]

M1 0◦0◦585 360 120

M3 8◦2◦, Type I 622 440 120

M4 8◦2◦, Type II 622 440 120

Three different 2D

s are compared during the measurements. All of them are intended

for coupling to a Si strip waveguide with a 220nm height. The ﬁrst 2D

is designed

for a zero angle of incidence

θF

(cf. Fig. 4.18) and has no waveguide-to-grating shear

angle. The other two

s are designed for a 8

◦

coupling angle and differ by the realization

of their waveguide-to-grating shear angle (Type I vs. Type II). The geometrical details

are summarized in Tab. 4.5 listed with their abbreviations that will be used to refer to

the structures. The TE

10 →

mode converter consists of multiple adiabatic taper

sections, which are designed as in Ref. [30]. Its characterization is done by Karsten Voigt

in Fimmwave/PhotonDesign. The conversion eﬃciency of TE

to TE

(O2) is better than

98.6% (-0.06dB) for the whole C-band. The portion of TE

power further propagating

towards O1 is less than 1 ‰(-30 dB). The fundamental TE

arrives at O1 with an eﬃciency

better than 99.6% (-0.02 dB) and less than 4.5 ‰(-23.5 dB) can couple down and propagate

towards O2.

Measurement Setup and Procedure Manual wafer measurements are performed

on 9 chips. Our setup consists of a tunable laser source Agilent 81940A, followed by a

120

4.2 In-Plane Scattering

manual polarization controller. Standard

SMF

s are used for the in- and out-coupling. The

measured signal is detected by a power meter Agilent 81634B. The measurement steps

are as follows:

•Position and polarization at the 2D GC adjusted for a maximal out-coupling at O1:

→measure the signal at O1, corresponding to the target TE00,y,

→measure the signal at O2, corresponding to TE10,y.

•

Position kept constant and polarization at the 2D

adjusted for a minimal out-

coupling at O1:

→measure the signal at O1, corresponding to the cross-polarized TE00,x,

→measure the signal at O2, corresponding to the cross-polarized TE10,x.

By setting the polarization for a maximal or minimal transmission, we have to perform

the measurement only at the upper arm of the 2D

. For each structure, we compare

the

between the target TE

00,y

and the remaining polarizations/modes near the maxi-

mum transmission wavelength of the target TE

00,y

. For the statistical measurements, we

Figure 4.20:

Exemplary measured coupling spectra of the modes TE

00,y

(target-polarization),

10,y

, TE

00,x

, TE

10,x

for different 2D grating couplers (2D GCs). The considered

structures are (a) M1: 2D GC with a vertical coupling and without a shear angle;

(b) M3: 2D GC with a 10

◦

coupling angle and a 2

◦

shear angle (Type I); (c) M4:

2D GC with a 10

◦

coupling angle and a 2

◦

shear angle (Type II). (Adapted from

Ref. [8] under a CC BY 4.0 license.)

intentionally choose a larger coupling distance, in order to exclude the distance variation

as a factor and to move easily from one chip to another on the wafer. The insertion loss

121

4 Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers

Figure 4.21:

A coupling spectrum of the output 1D focusing grating coupler (1D FGC),

measured at 10◦. (Adapted from Ref. [8] used under a CC BY 4.0 license.)

(in- and out-coupling) is between 4dB and 5dB higher than for the case of an optimal

coupling height. Similarly to the measurements in 4.2.2.1 (see Tab. 4.3), we determine the

s by ﬁrst averaging in a 15nm range around the wavelength of maximal transmission.

The obtained value is then averaged over the 9 chips on the wafer.

Experimental Results and Discussion First experimental results indicated that the 1D

FGC

and the 2D

s M3 and M4 require a coupling angle of 10

◦

instead of 8

◦

(see e.g.

simulations in Fig. 4.9 (a)). The deviation is most likely caused by variations of the perturbing

elements’ size or etch depth. The coupling angle of M1 is 0

◦

. Figure 4.20 shows exemplary

measured coupling spectra of the considered polarizations/modes for the investigated

designs (for a maximized coupling). An exemplary spectrum of the output 1D

FGC

is shown

in Fig. 4.21. In all plots in Fig. 4.20 it is evident that 2D

s excite cross-polarized modes

00,x

, TE

10,x

, as well as higher-order TE

10,y

, independent of their particular geometry. The

statistical analysis helps us to examine the differences between the considered designs

more precisely.

Table 4.6:

Mean extinction ratios (ERs)

±σ

between the target TE

00,y

and the other consid-

ered polarizations/modes averaged over 9 chips. Different 2D grating coupler

(2D GC) designs are compared. The analyzed structures are M1: 2D GC with

a vertical coupling and without a shear angle; M3: 2D GC with a 10

◦

coupling

angle and a 2

◦

shear angle (Type I); M4: 2D GC with a 10

◦

coupling angle and a

2◦shear angle (Type II).

TE00,yvs. M1: mean ER ±σ[dB] M3: mean ER ±σ[dB] M4: mean ER ±σ[dB]

TE10,y10.6 ±1.7 12.7 ±2.1 11.4 ±2.2

TE00,x27 ±2.1 23.8 ±2.7 25.3 ±3.7

TE10,x16.8 ±0.9 16.8 ±1.1 17.0 ±1.4

The results for designs M1, M3, M4 are summarized in Tab. 4.6. The mean extinction ratio of

the target TE

00,y

vs. TE

10,y

, TE

00,x

and TE

10,x

is ﬁrst wavelength-averaged around the maximal

122

4.2 In-Plane Scattering

transmission wavelength of each structures. The wavelength ranges are 1545-1560nm

for M1 and 1552-1567nm for M3 and M4. Afterwards, the values are wafer averaged

with the corresponding standard deviation

. Comparing the results in Tab. 4.6, we can

basically observe that designs with a non-zero coupling angle reduce the parasitic coupling

to the TE

10,y

mode compared to the vertically coupled structure M1. In the same time,

the cross-polarized TE

00,x

becomes larger in M3 and M4. The different shear angle type is

responsible for the variation between both structures. With regard to TE

10,x

, all 3 designs

behave similarly in average. Since higher-order modes will be ﬁltered by single-mode

waveguides, the lower coupling to TE

10,y

means that structures M3 and M4 with a tilted

coupling angle improve their coupling eﬃciency, compared to the zero-angle coupled M1.

This happens at the cost of a deteriorated polarization splitting, as the TE

00,x

coupling

eﬃciency increases.

In the original publication Ref. [8] the role of the coupling position is discussed in addition.

Penalties with regard to imbalanced coupling are analyzed in terms of coupling eﬃciency,

bandwidth and

PDL

. As this investigation does not change the revealed trends and conclu-

sions here, interested reader may refer to Ref. [8] for further information on that matter.

As most important observation, and contrary to prior expectations, it becomes evident

that vertically coupled devices induce cross-polarization as well (TE

00,x

and TE

10,x

). Thus,

out-of-plane diffraction and in-plane scattering appear to be parallel existing physical

processes in all 2D GCs.

123

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126

5 Design Optimization and

Characterization

In this chapter, the 2D

’s design development with respect to coupling eﬃciency and

polarization handling is outlined. First, possibilities to enhance the out-coupled power

are investigated, regardless of polarization-related issues. Afterwards, methods to reduce

in-plane scattering and cross-polarization related problems are discussed. In the end, it

is shown that 2D

s with a satisfying coupling eﬃciency, a low

PDL

and an outstanding

polarization splitting performance can be realized without a complication of the standard

0.25 µm photonic BiCMOS process.

5.1 Improvement of the Out-Coupled Power Eﬃciency

There are two widely spread methods to enhance the out-coupled power eﬃciency of a

GC, which have been initially developed for 1D GCs [1]:

1) Via the adoption of an enhanced Si waveguide thickness [2–8].

2) Via the application of a back-reﬂector below the grating area [9–16].

The idea behind the ﬁrst concept is to improve the ratio between upwards and downwards

diffracted light, that is, to reduce the power of the downwards diffracted wave in favor

of the upwards diffracted one. This should be achieved by enhancing the thickness of

the Si waveguide in the vicinity of the grating. The abrupt height change, is related to an

increased power, diffracted in upwards direction. Typically, we start with a standard 220 nm

Si waveguide height, which is locally enhanced by an epitaxial growth in the area, where

the grating should be deﬁned. Although there has been a successful demonstration of

an enhanced 1D

on our platform [6], there are issues with regard to the integration

compatibility. For that reason, this optimization approach is not considered in this work.

The second way to increase the out-coupled power eﬃciency is to try to re-gain the

downwards diffracted light. To do so, a suﬃciently good metal or Bragg mirror is necessary.

However, this is not the only prerequisite. A proper phase match between the upwards

127

5 Design Optimization and Characterization

diffracted wave and the back-reﬂected wave must be ensured. The latter is typically fulﬁlled

for a proper oxide thickness between the grating and the mirror [1]. The main disadvantage

of this optimization method is the complicated fabrication procedure. For the formation of a

dielectric Bragg mirror, a dedicated wafer fabrication is required [10,11]. Alternatively, non-

standard double-

SOI

substrates [14] may be considered. On the other hand, the application

of a metal mirror often requires non-

CMOS

compatible materials and bonding techniques

[9,12,13,15,16]. It should be noted that in the recent years, an alternative approach for

reducing the downwards diffraction has been proposed - the dual-layer grating, designed

using the adjoint optimization method [17]. The proposed device combines two gratings,

the second one intended for the elimination of the downwards propagating light. By

additionally applying apodization as a method for an improved mode ﬁeld overlap, a nearly

unity coupling eﬃciency for 1D

s with perfectly vertical coupling angle is predicted.

However, in the case of 2D

s, the in-plane scattering, which takes place in multiple

directions, needs to be considered as a degree of freedom. This may result in an overall

very complex design with a non-trivial fabrication, even when the minimum feature size for

a 0.25µm photonic

BiCMOS

is guaranteed. Moreover, the elimination of the downwards

diffracted light is achievable in a very limited bandwidth, which could exclude e.g. the

possibility to use 2D

s in

WDM

systems. For that reason, such an optimization procedure

is not considered in this work.

Table 5.1:

Sheared a-Si:H 2D grating couplers (2D GCs) of Type II, designed for C- and

O-band. The designs are intended for an a-Si:H rib waveguide with a 220nm

height and a slab etch depth equal to the grating etch depth.

Band, refractive index shear angle &

grating period

perturbing elements’

diameter & etch depth

coupling angles

φout =45◦,θout

C-band, naSi = 3.57 2◦&593nm 400nm & 100nm 9◦

O-band, naSi = 3.66 2◦&458nm 280nm & 100 nm 9◦

Here, an alternative possibility to form a 2D

with a back-reﬂector has been investigated.

Some of the analyses below have been originally published in Ref. [18]. The concept, already

presented in Sect. 3.3 of Chap. 3, involves the realization of 2D GCs in an a-Si:H photonic

layer in the

BiCMOS BEOL

, whereat bulk Si substrate is considered (Fig. 3.4, p. 82). Doing

so, the metals in the

BEOL

can be used as back-reﬂectors to enhance the out-coupled

power of the 2D

s. The TopMetal1 level has been chosen as an appropriate position for

the deﬁnition of the a-Si:H layer. In contrast to

FEOL

s, there are no speciﬁc

BEOL

ﬁlling layers at TopMetal1 and only around 3 µm SiO

covers the 2D

s. One of the three

thin metal levels below TopMetal1 (Metal1, Metal2 or Metal3) is a candidate to form a

back-reﬂector. Among them, Metal3 allows for an appropriate SiO

thickness separation

of 3.06 µm between grating and mirror. The exact value is given by the

BEOL

speciﬁc layer

thicknesses in our

BiCMOS

platform. Moreover, the choice of Metal3 guarantees for the

lowest SiO

thickness variation, which is determining for the wafer-scale stability of the

’s performance. Table 5.1 shows sheared 2D

s of Type II, designed for C- and

128

5.1 Improvement of the Out-Coupled Power Eﬃciency

O-Band. A light coupling to an a-Si:H rib waveguide with a 220nm height and a slab etch

depth equal to the grating etch depth is considered.

To ﬁnd an appropriate SiO

thickness separation between the grating and an underlying

reﬂecting material, a parameter sweep is carried out. The refractive index of SiO

is about

1.45. Due to the long simulation times, the C-band design is considered only. The latter

requires less number of grid cells, typically

∼

70 million vs. > 100 million in O-band. In

addition, Si with a constant refractive index is assumed as a substrate material (n

Si =

3.47)

and only the upwards diffracted normalized out-coupled power is detected (no mode ﬁeld

overlap evaluation). Figure 5.1 shows the dependence of the out-coupled power on the

underlying SiO

thickness, out of which the separation to Metal3 of 3.06 µm has been

determined as appropriate.

Figure 5.1:

An a-Si:H 2D grating coupler (2D GC) design – dependence of the normalized

out-coupled power on the SiO

thickness separation from the Si substrate

(mode ﬁeld overlap not considered). The wavelength is 1550nm. (Adapted from

Ref. [18] used under a CC BY 4.0 license.)

Next, the C- and O-band 2D

s are simulated for two cases. 1) Bulk Si below the grating

- in the absence of Metal3, around 8.37 µm thick SiO

layer separates the grating from

the underlying bulk Si substrate. The thickness results as a sum of the layer thicknesses

deﬁned in the

BiCMOS BEOL

and its exact value may vary, depending on the exact position

of the a-Si:H layer. 2) Metal3 below the grating with a 3.06 µm thick SiO

separation. The

conductivity of Metal3 is calculated from the measured mean value of its sheet resistance,

which is 55mΩ. Figure 5.2 shows the out-coupled power, the mode ﬁeld overlap and the

coupling eﬃciency of the C-band 2D

. Figure 5.3 shows the corresponding parameters

of the O-band 2D

. For symmetry reasons, only one of both 2D

waveguides is excited

(single-port simulation), with a polarization assigned as the target-polarization (here, the

y-polarization). The corresponding cross-polarization (here, the x-polarization) is given as

well. For both C- and O-band, the mode ﬁeld overlap changes only little, depending on the

back-reﬂector below the grating. The small differences between the bulk Si and the Metal3

case are most possibly due to the different distances between the reﬂector and the grating.

As expected, the out-coupled power is increased by the metal mirror, however, not only

for the target-polarization, but also for the cross-polarization. Overall, the split ratio at the

129

5 Design Optimization and Characterization

maximum transmission wavelength remains unchanged in C-band (about 10dB) and is

minimally improved in the O-band (by about 1.5dB at 1300 nm: 14.5 dB vs. 16 dB). Finally,

the coupling eﬃciency in the C-band design with a metal mirror improves by about 1.6 dB -

from -5dB to -3.4 dB at 1550 nm. For the O-band design, an increase of 1.1 dB is achieved

- from -3.9dB to -2.8dB at 1300nm. Obviously, the coupling eﬃciency increase is not as

large as compared to reported 2D

s with gold back-reﬂectors [15,16]. This is due to the

lower reﬂectivity of Metal3 (AlCu alloy), compared to gold. Moreover, power losses, caused

by the conversion into cross-polarization cannot be compensated by a back-reﬂector.

A ﬁnal comment will be given regarding the O-band coupling eﬃciency, which is about

0.5-1dB better than in C-band - a phenomenon, observed in

SOI

s as well [21].

This may be given by the lower in-plane scattering strength of the perturbing elements in

O-band. A different a-Si:H waveguide thickness may be potentially more appropriate for

C-band 2D

. The current 220nm have been chosen to allow for a direct comparison

between a-Si:H and

SOI

waveguides and

s. The waveguide thickness may be adapted in

the future, depending on the desired application.

Figure 5.2:

An a-Si:H 2D grating coupler (2D GC) for C-band at the TopMetal1 level. Two

different back-reﬂectors are compared: bulk Si at a distance of 8.37 µm and

Metal3 at a distance of 3.06 µm. The curves represent (a) the normalized out-

coupled power, (b) the mode ﬁeld overlap and (c) the total coupling eﬃciency

of a 2D GC. (Adapted from Ref. [18] under a CC BY 4.0 license.)

A question that remains open for an experimental investigation is the impact of Metal3

imperfections on the wafer-scale performance of 2D

s. Particularly, metal thickness

variation and surface roughness need to be analyzed in more detail. It should be noted

that the performance improvement of 2D

s, achieved by an additional a-Si:H layer in

the

BEOL

and by the adoption of a

BEOL

metal as a back-reﬂector, is applicable only at

the transmitter side. The

BEOL

and a-Si:H temperature requirements are not compatible

with the fabrication of Ge photodetectors, so that a receiver-side 2D

still needs to be

realized on SOI.

130

5.2 Optimized Polarization Handling

Figure 5.3:

An a-Si:H 2D grating coupler (2D GC) for O-band at the TopMetal1 level. Two

different back-reﬂectors are compared: bulk Si at a distance of 8.37 µm and

Metal3 at a distance of 3.06 µm. The curves represent (a) the normalized out-

coupled power, (b) the mode ﬁeld overlap and (c) the coupling eﬃciency of a

2D GC. (Adapted from Ref. [18] under a CC BY 4.0 license.)

5.2 Optimized Polarization Handling

Apparently, the most signiﬁcant problems of 2D

s result from an undesired in-plane

scattering. This section is devoted to the analysis of approaches for its minimization, which

is related to an improved polarization splitting behavior and a reduced

PDL

. Along with

moderately successful methods, a new design concept is presented, which is a simple,

working solution for the elimination of scattering-related issues in 2D

s. In the ﬁrst two

subsections 5.2.1 and 5.2.2, the optimization methods are presented and numerically

supported. Experimental validation is given in the last subsection 5.2.3.

5.2.1 Segmented Two-Dimensional Grating Couplers

The strong in-plane scattering strength in 2D

s is most likely caused by the constructive

superposition of the local scattered ﬁelds of periodic objects with identical geometry. Since

it has been observed that the scattering strength is lower for shallowly etched objects, the

ﬁrst idea for in-plane scattering reduction was to combine shallowly and deeply etched

arrays, e.g. in a dual etch conﬁguration [19]. In gratings with such a geometry, the ﬁrst

several periods are shallowly etched for a lower in-plane scattering, followed by deeply

etched perturbing elements for an enhanced power out-coupling. However, such 2D

showed no improvement, because once the light reached the deeply etched region, strong

scattering and cross-polarization resulted rapidly after a small number of periods.

Therefore, going away from the unitary nature of the scattering array requires another

131

5 Design Optimization and Characterization

means. A possible alternative includes the design of gratings, which have objects with

different sizes and a varying distance between them. This leads us to the old concept of the

apodization, combined with period chirping. In apodized gratings, the diffracting elements

have gradually increasing sizes from the beginning of the array. To ensure a diffraction

in a single, clearly deﬁned direction, the periodicity is adapted, depending on the objects’

size and the corresponding local effective refractive index. Having a grating with objects

of different sizes and with different periodicity between them, a certain reduction of the

in-plane scattering could be expected. It should be noted that in this work apodization has

not been investigated with regard to the improvement of the mode ﬁeld overlap between

the grating and the ﬁber mode, which is the usual motivation to apply this optimization

method. In Ref. [20] an “inversely” apodized grating has been shown: the obtained results

reveal the potential of apodization for an improved polarization handling.

Figure 5.4:

A segmented a-Si:H 2D grating coupler (2D GC) for C-band, comprising (a) 4

or (b) 6 segments. The i-th segment has a local a local period

Λi

and a local

diffracting elements’ diameter wΛi.

The apodization of 2D

s can be realized in two ways: either by using diffracting elements

of different sizes in every single row or by deﬁning segments with a given number of rows,

which gather objects with the same geometry. The latter special type of apodization is

assigned here as a segmentation. With Nobjects in a given direction, we would need in

the ﬁrst case Ndifferent diameters for each diffracting element. The exact adjustment

of a very small difference between the diameters can be very challenging for lithography

techniques with a limited resolution. For example, if two neighboring diffracting elements

in a grating have a diameter difference of < 10 nm, this difference may not be transferred

to the fabricated structure at all. For that reason, the segmented design is the preferred

method in this work.

Due to the signiﬁcantly simpler modeling, the segmentation as a method to reduce in-

plane scattering and cross-polarization has been investigated for the case of a-Si:H

BEOL

integrated 2D

s (cf. Fig. 3.4, p. 82). Particularly, the modeling is signiﬁcantly easier in the

absence of the

BEOL

grating ﬁlling layers. Nevertheless, the same observations can be

translated to

SOI

integrated 2D

s. To ensure similar conditions as for

SOI

s, the

132

5.2 Optimized Polarization Handling

investigated designs have a Si substrate below the grating and no metal mirrors have been

modeled. Thus, the pure contribution of the segmentation can be observed. To make the

designs comparable with the a-Si:H 2D

s presented in the previous section, the SiO

thickness separation to the Si substrate is kept 8.37 µm. Results from simulations in the

C-band will be presented ﬁrst.

Table 5.2:

Geometric parameters of sheared, segmented C-band a-Si:H 2D grating couplers

(2D GCs) of Type II with a different number of segments. The designs are intended

for the coupling to an a-Si:H rib waveguide with a 220 nm height and a slab etch

depth equal to the grating etch depth. The i-th segment has a local period

Λi

and a

local diffracting elements’ diameter w

Λi

. For both given structures, the shear angle

is 2

◦

, the etch depth is 100 nm and the coupling angles are

φout =

◦

θout =

◦

Number of segments 4 6

Segment 1: Λ1,wΛ1573 nm, 300 nm 573nm, 300 nm

Segment 2: Λ2,wΛ2581 nm, 340 nm 577nm, 320 nm

Segment 3: Λ3,wΛ3589 nm, 380 nm 581nm, 340 nm

Segment 4: Λ4,wΛ4593 nm, 400 nm 585nm, 360 nm

Segment 5: Λ5,wΛ5— 589nm, 380 nm

Segment 6: Λ6,wΛ6— 593nm, 400 nm

In the following, three designs are compared - the uniform C-band a-Si:H 2D

from the

previous section and two segmented designs, comprising either 4 or 6 segments. Apart

from the modiﬁed gratings, all material and geometric details are as given in the previous

Sect. 5.1, cf. Tab. 5.1. Figure 5.4 illustrates the geometry of the segmented gratings, in which

a ﬁxed number of sections are deﬁned. In each of them, the diffracting elements’ diameter

and the periodicity

vary. As a general rule of thumb, a change of

Λ=±

or a

change of the etch depth

=±

, results in an effective refractive index change of

eff,GC =∓

0.02. Therefore, to keep the coupling angle constant and gradually increase the

diffracting elements’ diameter, e.g. with a

Λ=

increment, the local period must

be adapted by

ΔΛ =

. Table 5.2 summarizes the geometric speciﬁcs of the segmented

2D GCs.

We compare the structures regarding the following parameters:

•

Single-port simulations (Port 1 and 2 excited separately): the coupling eﬃciency of a

given target-polarization, the cross-polarization and the polarizations’ split ratio per

single waveguide; the polarizations’ angular relation

Δψ

(i.e. the non-orthogonality)

and the polarization crosstalk between waveguides with Ports 1 and 2. The latter two

parameters are evaluated in a 40nm bandwidth around the maximal transmission

wavelength.

•

Dual-port simulations (Port 1 and 2 excited simultaneously): the coupling eﬃciency of

the even- and odd polarizations, deﬁned as in Fig. 4.16 (c) and (d) on p. 115;

PDL

, i.e.

the absolute difference between the coupling eﬃciency of these two polarizations

(in dB).

133

5 Design Optimization and Characterization

We expect an improvement in any of these aspects, when the parasitic grating in-plane

scattering is reduced. The comparison with regard to in-plane scattering suppression can

be directly translated also to

SOI

s, since in the

SOI

designs only the local periods

need to be changed in accordance to the SOI refractive index.

Figure 5.5:

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for C-band in a single-port simulation. The relevant parameters are

the target- and cross-polarization per single waveguide port. Three designs

are compared - a uniform 2D GC as a reference, a segmented 2D GC with 4

sections and a segmented 2D GC with 6 sections. The curves represent (a) the

normalized out-coupled power, (b) the mode ﬁeld overlap and (c) the coupling

eﬃciency of a 2D GC.

We begin with results from the single-port simulations. Figure 5.5 shows a comparison

between the three designs in terms of target- and cross-polarization, whereat out-coupled

power, mode ﬁeld overlap and coupling eﬃciency are analyzed. It is evident that the seg-

mentation in any of the two variants reduces the cross-polarization’s power and improves

at the same time the target-polarization’s out-coupling eﬃciency. In segmented designs,

the mode ﬁeld overlap has only a minor improvement by about 5%, since it is not the target

parameter for optimization. The results show an unusual effect of the segmentation as a

type of an apodization technique - in 2D

s it can be used for enhancing the out-coupled

power by reducing the parasitic in-plane scattering and the related cross-polarization.

Accordingly the coupling eﬃciency is improved by 0.6-0.7 dB at 1550 nm, depending on the

particular design (-4.3/-4.4dB for the 6/4-segment design vs. -5 dB for the reference, see

134

5.2 Optimized Polarization Handling

Fig. 5.5 (c)). The consequences of the reduced cross-polarization can be further inspected

in Fig. 5.6, which shows the polarization split ratio (per single waveguide) and the polariza-

tions’ angular relation and crosstalk between the signals from waveguide Ports 1 and 2.

The zero-crossings of the crosstalk are indicated additionally, when they occur at points,

which are not included in the simulation. The cross-polarization reduction is expressed

in an increased polarization split ratio. The angular relation is corrected towards the or-

thogonal state

Δψ =

◦

particularly for wavelengths larger than the central wavelength of

1550nm. Accordingly, the polarization crosstalk becomes lower in this wavelength range.

The difference between the segmented structures with 4 and 6 sections is minimal, which

indicates that even more segments may be necessary. If we permit a minimal difference

between the local diameters of 20nm, the case with 6 segments represents the maximal

achievable improvement using this technique. Next, we continue with a simultaneous

Figure 5.6:

A comparison between uniform and segmented a-Si:H 2D grating couplers (2D

GCs) for C-band in a single-port simulation. The relevant parameters are the

polarization split ratio per single waveguide port and the polarizations’ angular

relation and crosstalk between signals from both waveguides. Three designs

are compared - a uniform 2D GC as a reference, a segmented 2D GC with 4

sections and a segmented 2D GC with 6 sections. The curves represent (a) the

polarization split ratio, (b) the polarizations’ angular relation, (c) the polarizations’

crosstalk.

dual-port excitation to investigate the coupling of the even- and odd-polarization, which

result from in-phase vs. anti-phase superposition of the target- and cross-polarization

originating from different waveguides. Figure 5.7 shows the x-component (yequivalent)

of the even- and odd-polarization in terms of normalized out-coupled power (Fig. 5.7 (a)),

mode ﬁeld overlap (Fig. 5.7 (b)) and coupling eﬃciency (Fig. 5.7 (c)). The

PDL

resulting from

the different coupling spectra of the even- and odd-polarization is shown in Fig. 5.7 (d).

Also in this case the zero-crossings are indicated additionally, when they occur at points,

which are not included in the simulation. The

PDL

results from the gap between the out-

coupled powers of both polarizations (Fig. 5.7 (a)) and from the wavelength-shifted mode

ﬁeld overlap maximum (Fig. 5.7 (b)). With an increasing number of segments, we achieve

especially a correction of the overlaps’ spectral shift. Thus, for wavelengths > 1550 nm, the

difference between the coupling eﬃciency of the even- and odd-polarization is decreased.

This corresponds to a

PDL

improvement in this wavelength range. If we permit a maximal

PDL of 0.5dB, the achieved correction is still insuﬃcient.

135

5 Design Optimization and Characterization

Figure 5.7:

A comparison between uniform and segmented a-Si:H 2D grating couplers (2D

GCs) for C-band in a dual-port simulation. The relevant parameters are the

even- and odd-polarization, resulting from in- or anti-phase superposition of

the signals from both GC waveguides. Three designs are compared - a uniform

2D GC as a reference, a segmented 2D GC with 4 sections and a segmented

2D GC with 6 sections. The curves represent (a) the normalized out-coupled

power, (b) the mode ﬁeld overlap and (c) the total coupling eﬃciency and (d)

the polarization-dependent loss (PDL) of a 2D GC.

Further suppression of the in-plane scattering is necessary.

For completeness, the same comparison between a reference, uniform 2D

and a

segmented 2D

with 4 sections will be made in the O-band as well. Compared to the

device with 4 segments, a design with 6 segments shows a minor improvement, which does

not justify the enhanced fabrication effort. For that reason, the results for the 6-segment

device will be omitted.

Table 5.3:

Geometric parameters of a 2

◦

sheared, segmented O-band a-Si:H 2D GCs of

Type II with 4 sections. The designs are intended for the coupling to an a-Si:H

rib waveguide with a 220nm height and a slab etch depth equal to the grating

etch depth. The etch depth is larger than that of the reference uniform 2D GC:

120nm vs. 100 nm. The i-th segment has a local period

Λi

and a local diffracting

elements’ diameter wΛi. The coupling angles are φout =45◦,θout =9◦.

Number of segments 4

Segment 1: Λ1,wΛ1453 nm, 180 nm

Segment 2: Λ2,wΛ2461 nm, 220 nm

Segment 3: Λ3,wΛ3469 nm, 260 nm

Segment 4: Λ4,wΛ4473 nm, 280 nm

136

5.2 Optimized Polarization Handling

The geometric parameters of the segmented O-band 2D GC are given in Tab. 5.3. A larger

etch depth of 120nm than the reference design’s etch depth of 100 nm is considered to

compensate for the small diffraction elements in the ﬁrst two segments. The etch depth of

120nm is applied for all segments.

Figure 5.8:

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for O-band in a single-port simulation. The relevant parameters are

the target- and cross-polarization per single waveguide port. Two designs are

compared - a uniform 2D GC as a reference and a segmented 2D GC with 4

sections. The curves represent (a) the normalized out-coupled power, (b) the

mode ﬁeld overlap and (c) the coupling eﬃciency of a 2D GC.

Figure 5.9:

A comparison between uniform and segmented a-Si:H 2D grating couplers

(2D GCs) for O-band in a single-port simulation. The relevant parameters are

the polarization split ratio per single waveguide port and the polarizations’

angular relation and crosstalk between signals from both waveguides. Two

designs are compared - a uniform 2D GC as a reference and a segmented 2D

GC with 4 sections. The curves represent (a) the polarization split ratio, (b) the

polarizations’ angular relation, (c) the polarizations’ crosstalk.

137

5 Design Optimization and Characterization

Figure 5.8 shows the target- and cross-polarization of the uniform and the segmented

design, in a similar way to C-band (single-port simulations). Accordingly, Fig. 5.9 shows the

polarization split ratio per single waveguide and the polarizations’ angular relation and

crosstalk for the signals from both

waveguides. The even- and odd-polarizations and

their PDL are presented in Fig. 5.10 (dual-port simultaneous excitation).

The segmented design has a slightly shifted spectrum from the reference one, but this

plays a minor role for their comparison. Also in the O-band, the same trends shown in

C-band can be observed. In segmented 2D

s, the maximal coupling eﬃciency is slightly

enhanced by about 0.7 dB (-3.2 dB at 1310 nm vs. -3.9 dB at 1300 nm for the reference). The

polarizations’ split ratio, the angular relation and the corresponding polarization crosstalk

are improved for wavelengths > 1310nm. Regarding the even- and odd-polarizations

and their

PDL

, better characteristics are achieved in the same wavelength range, while

performance at wavelengths < 1310 nm is sacriﬁced. Although O-band 2D

s outperform

their C-band counterparts, the achievement of suﬃciently low

PDL

levels is challenging for

these devices as well. In our recent publication [22], segmented 2D

s in C- and O-band

Figure 5.10:

A comparison between uniform and segmented a-Si:H 2D grating couplers (2D

GCs) for O-band in a dual-port simulation. The relevant parameters are the

even- and odd-polarization, resulting from in- or anti-phase superposition of

the signals from both GC waveguides. Two designs are compared - a uniform

2D GC as a reference and a segmented 2D GC with 4 sections. The curves

represent (a) the normalized out-coupled power, (b) the mode ﬁeld overlap

and (c) the coupling eﬃciency and (d) the polarization-dependent loss (PDL)

of a 2D GC.

based on

SOI

have been investigated as a coupling interface in coherent transmission

systems. The designs account for the reduction of

OSNR

penalties, caused by the limitations

of polarizations’ non-orthogonality and crosstalk. The combination by the segmented 2D

s (especially in O-band) with an appropriate

DSP

improves signiﬁcantly the system-level

138

5.2 Optimized Polarization Handling

performance. By contrast, conventionally designed 2D

s are related to large penalties

even when advanced

DSP

is used. A strategy for the further design improvement of 2D

s, which could erase the necessity of a dedicated

DSP

, will be presented in the following.

5.2.2 Two-Dimensional Grating Couplers with Elongated and Individually

Oriented Perturbing Elements

Undoubtedly, the out-standing work of Luxtera towards a 2D

optimization delivered

the most remarkable results. Excellent performance with a low insertion loss and a low

PDL

could be demonstrated, on both 200mm- and 300 mm platforms [23

–

30] . Luxtera’s

team was among the ﬁrst who implemented successfully a non-zero waveguide-to-grating

shear angle, improving with this the coupling eﬃciency. However, due to their commercial

development activities, no explanation of the design with such an angle has been provided.

In fact, an explicit information about a non-perpendicular angle between the 2D

’s

waveguides can be found in Luxtera’s patents only, e.g. Refs. [35

–

37]. In the present work,

the systematic design of our sheared 2D GCs have been described in Chap. 4.

Furthermore, to optimize the PDL Luxtera proposed the adoption of a special clover-like

scatterer’s shape for the 2D

s’ perturbing elements. The latter inspired many groups

to make investigations on the importance of the scatterer’s shape, see e.g. Refs. [31

–

33],

but also in this case, no explanation about the physical processes determining the choice

of such a shape has been given. It is possible that Luxtera pursued similar purposes as

we do in the present work, namely - to reduce the scattering in undesired directions and

preserve the diffraction eﬃciency towards the

SMF

(Luxtera assigns the latter often as

a scattering as well). Possibly, the choice of the name - scatterer’s shape - is a hint about

the actual effect that has been manipulated, which is the in-plane scattering and not the

out-of-plane diffraction.

Here, an approach for an in-plane scattering optimization is proposed, which can be

used as an alternative to the scatterer’s shape of Luxtera. Some of the descriptions and

analyses below have been previously published in Ref. [38]. The basic idea behind the

optimization method is the following. A starting point is the hypothesis from our previous

work [39] that the in-plane scattering’s strength is very large, when perturbing elements

with identical scattering pattern are periodically arranged. The possibility to use objects

with different sizes and periodicity helps reducing the in-plane scattering to some extend,

but not suﬃciently to reach a low

PDL

(cf. 5.2.1). Alternatively, we still may use perturbing

elements of identical size, shape and periodicity, but in the same time we have to ensure

that neighboring objects have abruptly different local scattering patterns. To reach that

goal, we need ﬁrst elongated elements such as ellipses or ovals, whereat the exact choice

of a shape is not determining. Second, two neighboring objects may be rotated by 90

◦

to each other. A grating comprising such objects has been assigned as a zig-zag tilted

ovals array [38]. The sudden change of the orientations of adjacent objects allows for

the cancellation of a forwards-scattered local wave by a backwards-scattered one. The

exact perturbing elements’ size is thus determining for the degree of cancellation of the

forwards- and backwards-scattered ﬁelds.

139

5 Design Optimization and Characterization

Figure 5.11:

A schematic of an optimized 2D grating coupler (2D GC), comprising elongated

perturbing elements with different orientations. The major axis of a perturbing

element is denoted as wl, the minor axis is denoted as ws.

Figure 5.11 illustrates an exemplary geometry of a 2D

, which may be optimized for

a low in-plane scattering in this manner. Its most distinctive property is the fact that the

scattering’s cancellation is not reached by elements of a complex shape, but by simple

objects, differing in their abruptly changing orientations and local scattering proﬁles. The

simple shape of the perturbing elements is the most signiﬁcant advantage of this optimiza-

tion approach, compared to the Luxtera’s one. Thus, the 2D

s can be realized with a

low-resolution

DUV

lithography such as the 248nm

DUV

used in our fabrication platform.

Furthermore, no optical proximity correction (OPC) is necessary, but only an appropriate

shape biasing. By contrast, Luxtera’s scatterers require OPC in combination with a 193 nm

DUV

lithography. In view of the expenses’ gap between 248nm and 193 nm

DUV

, the 2D

s proposed in this work can be realized with the lowest DUV fabrication cost. At this

point, other alternatives to Luxtera’s design should be acknowledged. 2D

s with a low

PDL

, comprising uni-directional elongated scatterers, have been reported in Ref. [40]. A

post-simulation of the given geometry (performed for comparison purposes in this work)

suggests that the low

PDL

results partially at the cost of a low out-coupled power. Fur-

thermore, although the feature sizes should be compatible with a 193 nm or 248 nm

DUV

lithography, the reported test device has been fabricated by an electron beam lithography.

It is diﬃcult to predict, whether the given

PDL

values will remain the same, when a

DUV

lithography is used. In another recent work Ref. [20], apodized 2D

with rectangular

scatterers has been proposed for a low

PDL

. Since the designed geometry differs strongly

from the fabricated one, it is hard to predict, whether the obtained results are in a close

relation to the design method. In a most recent work Ref. [41], an optimization approach,

including perturbing elements with different orientations has been proposed. However,

the basic principle of the reported device differs from those presented here. A gradual

orientations’ change is used to achieve and improved mode ﬁeld overlap - a technique,

which may be rather associated with the apodization/chirping method. Due to the larger

degrees of freedom used there, it can be concluded that the success of the reported

design is achieved by the suﬃcient number of apodized segments (cf. the discussion from

5.2.1 in this chapter). Lastly, the resulting feature sizes are signiﬁcantly smaller than the

proposed ones in this work and experimental validation of the suggested design is missing.

140

5.2 Optimized Polarization Handling

A ﬁnal remark will be made with regard to the experimentally reported

PDL

values. There

are only few works, in which the repeatability of the determined

PDL

has been investigated

statistically. Reference [31] provides a wafer map of the distribution of the PDL within the

1dB bandwidth for a variety of 2D

s. Unfortunately, the test structures were fabricated

by an electron beam lithography, which is not given in large-scale manufacturing platforms.

In Refs. [20,32,33,40] very low PDL values are reported, but again without any discussion

of the measurement uncertainty and the statistical range. In fact, Ref. [33] shows results

from 3 chips, which is, however, not suﬃcient for a statistical evidence. It is a well-known

problem that parameters taking very low values are challenging for a characterization with

a good conﬁdence. In this context, objective statistical studies, considering measurement

uncertainties, are still not widely available. In this work, statistical results will be provided to

evaluate the applicability of the proposed designs in the target 0.25 µm BiCMOS platform.

Figure 5.12:

A comparison between a reference and an optimized design of a C-band

2D grating coupler (2D GC). (a) The coupling eﬃciency of a target- and cross-

polarization per single waveguide, (b) the corresponding polarization split ratio,

Δψ

between signals from both GC arms,

(d) the corresponding polarization crosstalk. The optimized design shows a

signiﬁcant improvement in terms of polarization splitting.

In the following, the exact optimization procedure for the proposed grating is brieﬂy

outlined. As already indicated - the achievement of a suﬃcient suppression of in-plane

scattered ﬁelds requires an exact sizing of the perturbing elements. The optimization

procedure involves the simultaneous adaption of several parameters:

•

The perturbing elements’ minor and major axis, denoted as w

and w

. Typically, when

the minor axis is too short, the overall out-coupled power decreases. This leads to a

lower cross-polarization and thus a low

PDL

. However, in the same time the coupling

141

5 Design Optimization and Characterization

eﬃciency drops as well. Therefore, models with a low

PDL

in combination with a low

coupling eﬃciency are not of interest in this work. As a lower limit for the coupling

eﬃciency, at least -4.5dB for both polarizations has been set. The

PDL

should be

< 0.5dB within the 2D GC’s 1 dB-bandwidth.

•

The grating period. Each simulation run, in which new perturbing elements’ shape is

investigated, requires in the same time the adaption of the grating period, so that a

desired angle θout results at the 2D GC’s symmetry plane.

•

The etch depth. Generally, the elongated perturbing elements, designed with the

restrictions of a 248nm

DUV

lithography, have a lower perturbation strength than

their circular counterparts. For that reason, together with an appropriate perturbing

elements’ dimensions, we look for a proper etch depth that guarantees for the

suﬃcient out-coupled power.

5.2.2.1 Linear Two-Dimensional Grating Couplers

The optimization procedure is ﬁrst applied to linear 2D

s. In the following, the best

performing devices are presented. An analysis of tolerances regarding deviations from the

target perturbing elements’ shape is carried out as well.

Designs for C-Band The investigation of the optimization technique begins in C-band,

where both SOI and a-Si:H material platforms are considered.

SOI All models below are designed for the coupling angles

φout =

◦

θout =

◦

at a

wavelength of 1550nm. To demonstrate the success of the applied optimization approach,

a reference and an optimized

SOI

design are compared. The reference design is

already known from 4.2.2.3, Chap. 4. We begin with sheared 2D GCs of Type II. Later, it is

shown exemplary that 2D

s of Type I cannot be optimized, using an equivalent geometry.

Generally, the suﬃcient scattering suppression in Type I gratings appears very challenging.

The reference and the optimal design have the following geometries:

•

Reference: a shear angle

α=

◦

(Type II), a grating period

Λ=

622

, an etch depth

d=120nm, circular perturbing elements with a diameter wΛ=440nm.

•

Proposed optimized design: a shear angle

α=

◦

(Type II), a grating period

Λ=

594

an etch depth d

140

, elongated perturbing elements with a minor axis w

230nm and a major axis wl=320nm (cf. Fig. 5.11).

As in previous analyses, single-port simulations are used to determine the ﬁber coupling

eﬃciency of a target- and cross-polarization as well as their split ratio, when a given 2D

arm is used as a source. The polarizations’ angular relation and crosstalk, between the

signals from both 2D

arms are determined as well. A dual-port simultaneous simulation

is used to evaluate the ﬁber coupling eﬃciency of an even- and odd-polarization as well as

the PDL resulting from the difference between their coupling eﬃciencies.

142

5.2 Optimized Polarization Handling

Figure 5.13:

Even vs. odd polarizations (x-component) of a reference sheared 2D grating

coupler (2D GC) of Type II for C-Band. (a) The out-coupled power, (b) the mode

ﬁeld overlap, (c) the coupling eﬃciency and (d) the polarization-dependent

loss (PDL).

Figure 5.12 shows a comparison between the reference and the optimized design in terms

of cross-polarization suppression (Fig. 5.12 (a)), split ratio (Fig. 5.12 (b)), polarizations’ non-

orthogonality (Fig. 5.12 (c)) and polarization crosstalk (Fig. 5.12 (d)). In the optimized design,

the cross-polarization’s maximum is reduced by 16dB, compared to the reference. This

is directly reﬂected in the polarizations’ split ratio - while in the reference design the split

ratio does not exceed 20dB, the optimized design has a split ratio > 23

in the range

1500nm-1580nm. Finally, the polarizations’ angular relation

Δψ

in a wavelength range

covering the 1dB-bandwidth of the optimized design shows a remarkable improvement

in terms of orthogonality with a maximally 3.5

◦

deviation from the ideal case of 90

◦

. The

orthogonality uniformity is signiﬁcantly better. The corresponding polarization crosstalk

remains below -24dB within the considered bandwidth.

It can be expected that the improved polarization splitting performance will be directly

translated to the

PDL

levels (even- vs. odd-polarization) in receiver-side 2D

s. Figure 5.13

shows (a) the out-coupled power, (b) the mode ﬁeld overlap and (c) the coupling eﬃciency

of the even- and odd-polarization (x-component) and (d) the

PDL

of the reference design.

For comparison, Fig. 5.14 (a)-(d) shows the same parameters for the optimized design.

Thanks to the cross-polarization suppression in the optimized design, the gap between

the out-coupled power of the even- and odd-polarization is signiﬁcantly decreased, as

can be seen in Fig. 5.13 (a) and Fig. 5.14 (a). The mode ﬁeld overlap spectra approach

the same central wavelength (cf. Fig. 5.13 (b) and Fig. 5.14 (b)). Overall, this leads to a

remarkable improvement of the coupling eﬃciency spectra of both polarizations, which are

now centered at the same wavelength with only a minor spectrum difference (Fig. 5.14 (c)).

The

PDL

becomes thus signiﬁcantly lower. The maximal coupling eﬃciency of the optimized

143

5 Design Optimization and Characterization

design is -4.1 dB at 1550 nm with a 34 nm 1 dB-bandwidth - from 1534 nm-1568 nm. In this

range, the

PDL

reaches maximally 0.55 dB, which is slightly higher than the target value. In

structures with conically etched perturbing elements, the

PDL

might be lower due to the

decreased perturbation strength.

Figure 5.14:

Even vs. odd polarizations (x-component) of an optimized sheared 2D grating

coupler (2D GC) of Type II for C-Band. (a) The out-coupled power, (b) the mode

ﬁeld overlap, (c) the coupling eﬃciency and (d) the polarization-dependent

loss (PDL).

It becomes evident that there is an optimization chain: waveguide split ratio

→

orthogonality

and polarization crosstalk between signals from both

arms

→PDL

between signals from

both

arms combined with different phase relations. This can be well explained: with an

improving split ratio, the signal of one of the waveguides coupled to the ﬁber approaches

the x-polarization state and the signal from the other one - the y-polarization state. Ac-

cordingly, the polarizations’ orthogonality comes even closer to 90

◦

and the polarization

crosstalk becomes lower. Thus, an arbitrary combination of orthogonal signals without

crosstalk causes no

PDL

. From Figs. 5.12 (b)-(d) and 5.14 (d), we can estimate a required

value for each of the examined parameter to reach a

PDL

of 0.5dB (via interpolation and

evaluation at 1565nm, where the PDL is exactly 0.5dB).

We obtain the following limits:

Minimal split ratio 25 dB

→

maximal deviation from the orthogonality state 3

◦

, i.e.

maximal polarization crosstalk of -26 dB →maximal PDL of 0.5 dB.

Because the parameters are closely connected with each other, the analyses within the

next paragraphs will be mostly focused on the

PDL

. Out of its behavior, we can directly

144

5.2 Optimized Polarization Handling

conclude in what value range the other parameters are. In some cases, where the simula-

tion duration is too long, the single-port simulations will be considered instead.

Next, it will be illustrated that sheared 2D

s of Type I are more challenging with respect to

in-plane scattering suppression. We use the design parameters of the previously designed

of Type II and change only the way of deﬁning the shear angle, i.e. the grating

area is now rhombus-shaped. Figure 5.15 shows (a) the coupling eﬃciency of the target-

and cross-polarization, (b) the coupling eﬃciency of the even- and odd-polarization. It is

evident that for the same perturbing elements’ size and periodicity, the cross-polarization

is signiﬁcantly higher. For that reason, the even- and odd- polarization are still shifted to

each other. Within the permitted geometric range for a compatibility with a 248nm

DUV

lithography, no appropriate perturbing elements’ geometry could be found for 2D

s of

Type I, which fulﬁlls the criteria in terms of coupling eﬃciency and PDL.

Figure 5.15:

The performance of a sheared 2D grating coupler (2D GC) of Type I with the

same perturbing elements’ geometry and periodicity as the optimized 2D GC of

Type II. (a) The coupling eﬃciency of the target- and cross-polarization (single-

port simulation), (b) the coupling eﬃciency of the even- and odd-polarization

(dual-port simulation).

Lastly, the optimal 2D

design is investigated, regarding its robustness against deviations

of the perturbing elements’ shape. Only the coupling eﬃciency of the even- and odd-

polarization and the PDL are presented. We consider the following cases:

•Δws=Δwl=+10nm, i.e. both axes are 10 nm larger →ws=240nm,wl=330nm

•Δws=Δwl=–10nm, i.e. both axes are 10 nm smaller →ws=220nm,wl=310nm

•Δ

s=const.

+10

, i.e. the minor axis is unchanged, the major axis is 10nm

larger →ws=230nm,wl=330nm

•Δ

s=const.

–10

, i.e. the minor axis is unchanged, the major axis is 10nm

smaller →ws=230nm,wl=310nm

•Δ

+10

l=const.

, i.e. the minor axis is 10nm larger, the major axis is un-

changed →ws=240nm,wl=320nm

•Δ

–10

l=const.

, i.e. the minor axis is 10nm smaller, the major axis is

unchanged →ws=220nm,wl=320nm

145

5 Design Optimization and Characterization

Figure 5.16:

A performance variation of the optimized 2D grating coupler (2D GC) in C-

band, when the perturbing elements’ size varies:

s=Δ

+10

nm →

240

330

s=Δ

–10

nm →

220

310

. The

compared parameters are (a)-(b) the coupling eﬃciency of the even- and odd-

polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).

Figure 5.17:

A performance variation of the optimized 2D grating coupler (2D GC) in C-band,

when the perturbing elements’ size varies:

s=const.

+10

nm →

230

330

s=const.

–10

nm →

230

310

The compared parameters are (a)-(b) the coupling eﬃciency of the even- and

odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).

Figure 5.16 shows the trade-offs, which occur in the ﬁrst two cases:

s=Δ

l=±

When the perturbing elements’ size is larger, the

PDL

is affected, becoming maximally

0.65 dB within the 1 dB-bandwidth. The coupling eﬃciency remains unchanged, the spectral

146

5.2 Optimized Polarization Handling

shift is small. When the perturbing elements’ shape is smaller than expected, the

PDL

remains below 0.5 dB within the wavelength range of interest. In fact, there is even a small

improvement at 1568nm of around 0.15dB. This is at the cost of a coupling eﬃciency

decrease of about 0.3dB. The wavelength shift is again not signiﬁcant.

As can be seen in Figs. 5.17, 5.18, similar results can be observed for the both cases

s=±

l=±

. A larger axis leads to a

PDL

deterioration with maximal

values of about 0.6-0.65dB and is thus more critical than a deviation towards smaller

dimensions, for which the

PDL

remains < 0.5

. The larger

PDL

is related to a slightly better

maximal coupling eﬃciency of -4 dB, while in the opposite case, the eﬃciency decreases to

about -4.3 dB. In summary, deviations of the perturbing elements’ size within a 20 nm range

are well tolerable within the 2D

’s 1dB-bandwidth in C-band. No signiﬁcant wavelength

shift occurs. Basically, an increase of a given dimension leads to a

PDL

deterioration and

a coupling eﬃciency improvement and vice versa. The worst-case

PDL

becomes 0.65dB.

Conically etched perturbing elements in real structures may have better tolerances, due

to the lower perturbation strength.

Figure 5.18:

A performance variation of the optimized 2D grating coupler (2D GC) in C-band,

when the perturbing elements’ size varies:

+10

l=const. →

240

320

–10

l=const. →

220

320

The compared parameters are (a)-(b) the coupling eﬃciency of the even- and

odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).

a-Si:H The advances, which are achievable in

SOI

structures, can be directly translated

also to the a-Si:H based structures, discussed in Sect. 5.1. Here, an improved a-Si:H design

with a Metal3 back-reﬂector is brieﬂy presented. Since the devices are applicable at the

transmitter-side, only the relevant aspects there are discussed, that is, the coupling eﬃ-

ciency, the split ratio and the polarizations’ angular relationship

Δψ

with the corresponding

polarization crosstalk. An optimized design may have the following geometry: a shear angle

α=

◦

(Type II), a grating period

Λ=

582

, stretched perturbing elements with a minor

147

5 Design Optimization and Characterization

axis w

210

and a major axis w

330

, an etch depth d

140

. The design

angles are

φout =

◦

θout =

◦

at 1550nm. Figure 5.19 shows the coupling eﬃciency of

the target- and cross-polarization and the corresponding split ratio (per single waveguide).

The polarizations’ angular difference and the related polarization crosstalk between signals

from both

arms is given as well. Compared to the design with a back-reﬂector from

Sect. 5.1 we observe a 1dB improvement of the maximal coupling eﬃciency, which is

here -2.4 dB at 1550 nm. A split ratio better than 23 dB is reachable in a 55 nm range from

1500nm-1555 nm. The orthogonality deviation within the bandwidth 1530 nm-1570 nm is

maximally 4

◦

, corresponding to a polarization crosstalk below -23.5dB. The parameters’

improvement is comparable to the

SOI

case. The example shows that the optimization

method works well also for 2D GC with a comparatively high coupling eﬃciency.

Figure 5.19:

An a-Si:H 2D grating coupler (2D GC) for C-band with a Metal3 back-reﬂector -

an optimized design with individually oriented elongated perturbing elements.

(a) The coupling eﬃciency of the target- and cross-polarizations. (b) The cor-

responding split ratio (per single waveguide). (c) The polarization’ angular

difference

Δψ

. (d) The corresponding polarization crosstalk (between signals

from both GC arms).

A Brief Outlook - Possible Modiﬁcations The proposed way to deﬁne an array of

stretched perturbing elements with alternating orientations is not the only one think-

able possibility to design 2D

s with a low in-plane scattering. More degrees of freedom

may be included to achieve an even better performance. A possible extension will be

shown here, using the a-Si:H design from the previous paragraph as a starting point.

148

5.2 Optimized Polarization Handling

Although the example here considers a-Si:H as a material, the same conﬁguration may

be used in

SOI

as well. Because the potential modiﬁcations have been investigated at a

later point, no experimental structures were available at the moment of the dissertation’s

preparation. The example given in this paragraph may be considered as an outlook for

the further design optimization. The design modiﬁcation is illustrated in Fig. 5.20. Along

Figure 5.20:

A schematic of an optimized 2D grating coupler (2D GC) design, comprising

stretched perturbing elements, combined with circular perturbing elements.

with the stretched objects, circular perturbing elements may be included as well. This

adds an additional local scattering pattern and increases the degrees of freedom for an

in-plane scattering cancellation. The exemplary design has the following dimensions: a

shear angle

α=

◦

, a grating period

Λ=

582

, stretched perturbing elements with a

minor axis w

210

and a major axis w

330

, circular perturbing elements with a

diameter w

circ =

265

and an etch depth d

140

. The only modiﬁcation, compared to

the previously developed a-Si:H design is the inclusion of the circular perturbing elements.

Their diameter is calculated such, that the circle has the same area as the ellipse/oval. If

we assume an ellipse:

wcirc =√ws·wl(5.1)

If we consider an oval, the surface is calculated from the two half-circles with a diameter

wsplus a small connecting rectangle with sidewalls lengths wsand wl–ws, that is:

wcirc =rw2

s+4

πws·(wl–ws) (5.2)

The calculated w

circ

differs little in dependence on the chosen formula. Figure 5.21 shows

the target- and cross-polarization’s coupling spectra, the corresponding split ratio together

with the polarizations’ angular difference and crosstalk. The ﬁgure compares the previously

shown optimized design as a reference and the modiﬁed optimized design. The latter is

slightly wavelength-shifted, the maximal coupling eﬃciency remains unaltered. The cross-

polarization is further reduced (by 2-3dB), compared to the initially optimized design. For

wavelengths > 1540

a constant split ratio improvement of about 2.5 dB can be achieved.

The polarization’ orthogonality is further improved with a maximal deviation of only 2.5

◦

The polarization crosstalk is below -27.5dB.

149

5 Design Optimization and Characterization

Figure 5.21:

A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for C-band

with a Metal3 back-reﬂector - an optimized design with stretched perturb-

ing elements (PE) vs. an optimized design combining stretched and circu-

lar perturbing elements (PE). (a) The coupling eﬃciency of the target- and

cross-polarization. (b) The corresponding split ratio (per single waveguide). (c)

The polarization’ angular difference

Δψ

. (d) The corresponding polarization

crosstalk (between signals from both GC arms).

Designs for O-Band Here, the same analyses are repeated in O-band.

SOI We begin with the comparison between a reference and an optimized sheared 2D

of Type II in

SOI

. The structures are developed for the coupling angles

φout =

◦

θout =

8◦at a wavelength of 1310nm and have the following geometries:

•

Reference: a shear angle

α=

◦

, a grating period

Λ=

480

, an etch depth d

120nm, circular perturbing elements with a diameter wΛ=280nm.

•

Proposed optimized design: a shear angle

α=

◦

, a grating period

Λ=

480

, an

etch depth d

140

, elongated perturbing elements with a minor axis w

180

and a major axis wl=260nm.

Figure 5.22 compares the target- and cross-polarization coupling eﬃciency and the cor-

responding split ratio of both designs (single-ports simulation). The reference O-band

structure has a better split ratio than its C-band counterpart. Further improvement can be

achieved by the proposed new design. For wavelengths around the maximal transmission,

the split ratio’s enhancement is more than 10dB. The split ratio is > 23

in the range

1275nm - 1330nm. The optimized design is also responsible for the improvement of

the uniformity of

Δψ

. In a 40nm bandwidth around the maximum, the deviation from the

orthogonal state is maximally 3.2

◦

, corresponding to a maximal polarization crosstalk of

-25dB.

150

5.2 Optimized Polarization Handling

Figure 5.22:

A comparison between a reference and an optimized design of a O-band

2D grating coupler (2D GC). (a) The coupling eﬃciency of a target- and cross-

polarization per single waveguide, (b) the corresponding polarization split ratio,

Δψ

between signals from both GC arms,

(d) the corresponding polarization crosstalk. The optimized design shows a

signiﬁcant improvement in terms of polarization splitting.

In the next step, the even and odd polarization’s coupling eﬃciency and

PDL

are determined

from dual-port simultaneous simulations. Figures 5.23 and 5.24 can be used for the

comparison between the reference and the proposed 2D

. Again, the proposed design

with a lower cross-polarization makes it possible that the even- and odd-polarization’s

spectra come closer to each other: the out-coupled power’s gap is closing and the overlap’s

wavelength shift vanishes. The optimized design reaches a coupling eﬃciency of -3.3dB

at 1310nm, the 1dB-bandwidth is 21nm from 1303nm to 1324 nm. Regarding the

PDL

values lower than 0.5dB can be obtained for wavelengths > 1280nm.

For completeness, the 2D

’s robustness against perturbing elements’ size deviations is

analyzed for the O-band design as well. The considered cases cover the same range as in

C-band with the following length combinations:

•Δws=Δwl=+10nm, i.e. both axes are 10 nm larger →ws=190nm,wl=270nm

•Δws=Δwl=–10nm, i.e. both axes are 10 nm smaller →ws=170nm,wl=250nm

•Δ

s=const.

+10

, i.e. the minor axis is unchanged, the major axis is 10nm

larger →ws=180nm,wl=270nm

•Δ

s=const.

–10

, i.e. the minor axis is unchanged, the major axis is 10nm

smaller →ws=180nm,wl=250nm

151

5 Design Optimization and Characterization

•Δ

+10

l=const.

, i.e. the minor axis is 10nm larger, the major axis is un-

changed →ws=190nm,wl=260nm

•Δ

–10

l=const.

, i.e. the minor axis is 10nm smaller, the major axis is

unchanged →ws=170nm,wl=260nm

Figure 5.23:

Even vs. odd polarizations (x-component) of a reference sheared 2D grating

coupler (2D GC) of Type II for O-Band. (a) The out-coupled power, (b) the mode

ﬁeld overlap, (c) the coupling eﬃciency and (d) the polarization-dependent

loss (PDL).

Figure 5.24:

Even vs. odd polarizations (x-component) of an optimized sheared 2D grating

coupler (2D GC) of Type II for O-Band. (a) The out-coupled power, (b) the mode

ﬁeld overlap, (c) the coupling eﬃciency and (d) the polarization-dependent

loss (PDL).

Figures 5.25, 5.26 and 5.27 show the simulation results for the considered cases. The

results conﬁrm the observations in C-band. Again, the spectrum shift is small. A shrunk

152

5.2 Optimized Polarization Handling

perturbing element in either direction is not an issue, with the

PDL

remaining < 0.5

and

a maximal coupling eﬃciency of –3.4dB.

Figure 5.25:

A performance variation of the optimized 2D grating coupler (2D GC) in O-

band, when the perturbing elements’ size varies:

s=Δ

+10

nm →

190

270

s=Δ

–10

nm →

170

250

. The

compared parameters are (a)-(b) the coupling eﬃciency of the even- and odd-

polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).

Figure 5.26:

A performance variation of the optimized 2D grating coupler (2D GC) in O-band,

when the perturbing elements’ size varies:

s=const.

+10

nm →

180

270

s=const.

–10

nm →

180

250

The compared parameters are (a)-(b) the coupling eﬃciency of the even- and

odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).

153

5 Design Optimization and Characterization

As previously observed, the larger perturbing elements are responsible for a

PDL

enhance-

ment with a maximum of 0.7 dB when both

s=Δ

+10

, and an upper limit of 0.6 dB

in the other cases. The

PDL

deterioration comes with an improved coupling eﬃciency of

–3.1

. Generally, independent of the given band, during fabrication it should be ensured

that the perturbing elements’ size does not get too large or too circular. An appropriate

shape biasing may be advantageous.

Figure 5.27:

A performance variation of the optimized 2D grating coupler (2D GC) in O-band,

when the perturbing elements’ size varies:

+10

l=const. →

190

260

–10

l=const. →

170

260

The compared parameters are (a)-(b) the coupling eﬃciency of the even- and

odd-polarization, (c)-(d) the corresponding polarization-dependent loss (PDL).

a-Si:H For completeness, the proposed design will be translated to an a-Si:H 2D

with a Metal3 back-reﬂector. Again, the transmitter-side relevant parameters will be shown,

i.e. the coupling eﬃciency, the split ratio, the polarizations’ angular relationship

Δψ

and

the polarization crosstalk. For a-Si:H, the only parameter that is changed is the grating

period, which is now 470nm. The complete design geometry includes: a shear angle

α=

◦

, a grating period

Λ=

470

, stretched perturbing elements with a minor axis

180

and a major axis w

260

, an etch depth d

140

. The design angles

are

φout =

◦

θout =

◦

at 1320nm, with a small deviation from the target wavelength of

1310nm. An extended, optimized design with a combination of stretched and circular

perturbing elements has the same geometric parameters, further including the circular

elements’ diameter wcirc =216nm.

Figure 5.28 shows the coupling eﬃciency of the target- and cross-polarization, the cor-

responding split ratio (per single-waveguide), the polarization angles’ difference and the

polarization crosstalk (between both

waveguides) for both optimized cases. Compared

to the reference design with Metal3 from Sect. 5.1, a small coupling eﬃciency improvement

is achieved - from -2.8 dB to -2.5 dB. As expected, the most signiﬁcant advance is in terms

154

5.2 Optimized Polarization Handling

of polarization splitting: the improved design with elongated elements has a split ratio

> 23

from 1290nm to 1330 nm. The extended optimized design, combining elongated

and circular elements, has an even better split ratio with values > 23dB from 1285 nm to

1353nm. In a 40nm bandwidth around the central wavelength, the design with elliptical

elements has a variation from

Δψ =

◦

Δψ =

◦

and the combined design both elon-

gated and circular elements - from

Δψ =

◦

Δψ =

◦

. In both cases the polarization

crosstalk lower than -27.5dB. Although no further improvement could be achieved by

the design with combined elements’ shape in this aspect, both structures have a decent

polarizations’ orthogonality and a low polarization crosstalk.

Figure 5.28:

A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for O-band

with a Metal3 back-reﬂector - an optimized design with stretched perturb-

ing elements (PE) vs. an optimized design combining stretched and circu-

lar perturbing elements (PE). (a) The coupling eﬃciency of the target- and

cross-polarization. (b) The corresponding split ratio (per single waveguide). (c)

The polarization’ angular difference

Δψ

. (d) The corresponding polarization

crosstalk (between signals from both GC arms).

5.2.2.2 Focusing Two-Dimensional Grating Couplers

After an approach for an improved polarization handling has been introduced and ap-

propriate designs for linear 2D

s in C- and O-band have been found, the last aspect

that remains open for an investigation is the potential size reduction of the proposed 2D

s. Here, 2D

FGC

s are analyzed as a low-footprint alternative to linear 2D

s. Particu-

larly, we look for polarization-optimized designs, using the same approach with elongated

perturbing elements with an individual orientation (zig-zag tilted ovals). The simulation

procedure is as given in 2.1.3.1, Chap. 2. Because the present analyses have been carried

out at the ﬁnal stage of this work, the following paragraphs will be used as an outlook and

will indicate some points that need to be addressed in future works.

155

5 Design Optimization and Characterization

Before looking for an appropriate diffracting elements’ design, the determination of their

locations will be brieﬂy explained. As a construction method, design1 in combination

with tilted waveguides from Ref. [34] was chosen. To obtain the required positions, the

intersections of two arc arrays I and II have to be found. First, each arc array I or II can

be deﬁned by the parametrized positions (x

n,I

) and (x

n,II

), with n

1,2

···

N,Nthe

number of arcs within an array. The positions of the grating perturbing elements result

from the intersections of these two arc arrays.

Figure 5.29:

A description of the modeling procedure of 2D focusing grating couplers (2D

FGCs). The grating perturbing elements’ positions are found by the intersec-

tion of two arc arrays I, II. The arc array I consists of the parametrized arcs

1,I

),(x

2,I

···

n,I

···

,(x

N,I

). The arc array II can be deﬁned analo-

gously.

The following parametrization can be used for the arc arrays I and II, deﬁned in the angular

range [–γ,γ] (Fig. 5.29):

xn,I=xoffset,I+(x0,I+(n–1)·Λ)·cost1,tI∈[–γ,γ]

yn,I=yoffset,I+(y0,I+(n–1)·Λ)·sint1,tI∈[–γ,γ]

xn,II =xoffset,II +(x0,II +(n–1)·Λ)·cost2,tII =tI+90◦(5.3)

yn,II =yoffset,II +(y0,II +(n–1)·Λ)·sint2,tII =tI+90◦

n=1,2,···,N,

where

is the period in an arc array. The offset points (x

offset,I

),(x

offset,II

) are

156

5.2 Optimized Polarization Handling

chosen in such a way that the arc arrays cross each other at their central region. The

points (x

0,I

), (x

0,II

) give the position, where the ﬁrst arc of a given array I and II is

deﬁned. This determines the length of the focusing taper, which transforms a waveguide

with an initial width w

start

to the ﬁnal w

end

directly at the grating (Fig. 5.29). Exemplary arcs

1,I

), (x

1,II

) and (x

2,I

), (x

2,II

) and their intersections can be seen in Fig. 5.29.

The parameters depicted for the arc array I can be deﬁned analogously for the arc array II.

For the following analyses, we choose the values:

x0,I=y0,II =20μm, γ=24◦

xoffset,I=x0,I+Λ·N

yoffset,I=0

xoffset,II =0

yoffset,II =y0,II +Λ·N

2(5.4)

After the discrete mathematical deﬁnition of the arc arrays, their intersections have to be

found. This is enabled by the MATLAB function intersections.m, which is an open-source

code, available from Ref. [42].

Figure 5.30:

2D focusing grating couplers (2D FGCs) for C-band: a comparison between a

reference design with circular perturbing elements and a proposed optimized

design with elongated perturbing elements. The plots show a target- vs. cross-

polarization (single-port simulation). (a) The normalized out-coupled power,

(b) the mode ﬁeld overlap, (c) the coupling eﬃciency, (d) the corresponding

polarizations’ split ratio.

In the following, reference and proposed optimized designs for C- and O-band will be

presented. Because of the large simulation models and computational time, only the

157

5 Design Optimization and Characterization

excitation from one of both 2D

arms is considered. The evaluated parameters are:

target- and cross-polarization and the polarizations’ split ratio (per single waveguide). If

a large split ratio is reached, a good performance in terms of orthogonality between the

signals of both

arms and a low

PDL

between arbitrary incident polarizations can be

expected. We begin with SOI designs for C-band with the following geometries:

•

Reference: a shear angle

α=

◦

, a grating period

Λ=

588

, an etch depth d

140nm, circular perturbing elements with a diameter wΛ=330nm.

•

Proposed optimized design: a shear angle

α=

◦

, a grating period

Λ=

594

, an

etch depth d

140

, elongated perturbing elements with a minor axis w

210

and a major axis wl=510nm.

The coupling angle is

θout =

◦

at the symmetry plane. Similarly, a reference and an opti-

mized design for O-band can be deﬁned:

•

Reference: a shear angle

α=

◦

, a grating period

Λ=

487

, an etch depth d

140

, circular perturbing elements with a diameter w

Λ=

280

, a coupling angle

θout =8◦at the symmetry plane.

•

Proposed optimized design: a shear angle

α=

◦

, a grating period

Λ=

490

, an

etch depth d

140

, elongated perturbing elements with a minor axis w

180

and a major axis wl=440nm, a coupling angle θout =10◦at the symmetry plane.

Figure 5.31:

2D focusing grating couplers (2D FGCs) for O-band: a comparison between a

reference design with circular perturbing elements and a proposed optimized

design with elongated perturbing elements. The plots show a target- vs. cross-

polarization (single-port simulation). (a) The normalized out-coupled power,

(b) the mode ﬁeld overlap, (c) the coupling eﬃciency, (d) the corresponding

polarizations’ split ratio.

158

5.2 Optimized Polarization Handling

Figures 5.30 and 5.31 compare the reference and the optimized design for C-band and

O-band with respect to the parameters: (a) normalized out-coupled power, (b) mode ﬁeld

overlap, (c) coupling eﬃciency and (d) polarizations’ split ratio in a single-port simulation.

In both cases, a clear advantage of the optimized design can be observed - both the

out-coupled power and the mode ﬁeld overlap of the cross-polarization are signiﬁcantly

suppressed compared to the reference design. In C-band, this results in a split ratio

better than 23 dB within the bandwidth 1535 nm - 1580 nm. In O-band, the corresponding

wavelength range is from 1305 nm - 1345 nm. Further optimization would be advantageous

for O-band, however, the current dimensions are at the limit of the design rules for the

target 0.25 µm

BiCMOS

technology. The adoption of a modiﬁed grating, comprising both

stretched and circular elements (cf. 5.2.2.1), may be a better alternative for O-band 2D

FGCs.

Figure 5.32:

A comparison of 2D focusing grating couplers (2D FGCs) for C-band with

different waveguide heights h

Si =

220

vs. h

Si =

300

. Both designs

comprise elongated perturbing elements with the same dimensions. The plots

show a target- vs. cross-polarization (single-port simulation). (a) The normalized

out-coupled power, (b) the mode ﬁeld overlap, (c) the coupling eﬃciency, (d)

the corresponding polarizations’ split ratio.

There are some basic differences in comparison to the linear 2D

s. First, the perturbing

elements’ shape for 2D

FGC

s must be further stretched. While the linear 2D

s show a

good suppression of the cross-polarization’s out-coupled power in combination with a

moderate mode ﬁeld overlap eﬃciency, the 2D

FGC

s reach an improvement mainly by a

strong deterioration of the mode ﬁeld overlap. This has the disadvantage of excess loss,

due to the persisting power out-coupling into the undesired polarization. Looking at the

maximally reachable coupling eﬃciency in C-band, we observe about 1.5 dB deterioration

compared to linear 2D

s: the maximum coupling eﬃciency of the optimized 2D

FGC

159

5 Design Optimization and Characterization

design is -5.7dB (cf. -4.1dB for optimized linear 2D

s). Similarly, in O-band we obtain

-5.5dB, which is more than 2dB worse compared to its linear 2D

alternative with a

-3.3 dB eﬃciency. Apparently, the large gap in terms of eﬃciency cannot be caused only by

the cross-polarized power. An inspection of both models showed that the low coupling

eﬃciency is related to a large power in the target-polarization, which is not coupled out

by the grating and keeps propagating within the structure. Typically, this issue is caused

either by a too shallow etch depth or by an inappropriate BOX thickness.

Particularly, the computationally smaller C-band model is investigated regarding these

two parameters. A variation of the etch depth and the

BOX

values shows that the present

coupling eﬃciency is the maximally reachable one. Regarding this matter, the following

explanation can be given: for a given design wavelength, there is an optimal combination

of a waveguide height, an etch depth (both related to the effective refractive index), a

BOX

thickness and a coupling angle. Reaching a local maximum for a given set of ﬁxed

parameters does not necessarily mean that the global maximum is found. The behavior of

our 2D

FGC

s suggests that the waveguide height is potentially not optimal for the coupling

eﬃciency. To verify this hypothesis, a modiﬁed C-band 2D

FGC

is simulated with the geo-

metric parameters: a waveguide height h

Si =

300

, a shear angle

α=

◦

, a grating period

Λ=

575

, an etch depth d

200

, elongated perturbing elements with a minor axis

210

and a major axis w

510

. While the perturbing elements’ size is kept

constant, the remaining parameters are adapted for a new, exemplary waveguide height

and a coupling angle of θout =10◦at the symmetry plane.

Figure 5.32 shows a comparison of the C-band designs for both waveguide heights. In-

deed, a better power out-coupling is reached by the 300 nm 2D

FGC

, leading to a coupling

eﬃciency improvement of about 0.7dB. Apparently, a slightly modiﬁed design of the per-

turbing elements’ shape is necessary to reduce further the cross-polarization and reach

an additional coupling eﬃciency improvement. In O-band, an enhancement of the power

out-coupling in a dedicated design could be observed as well, however, at the cost of a

mode ﬁeld overlap deterioration. The latter may be related to a non-optimal perturbing

elements’ shape and/or periodicity.

The present analysis showed that the zig-zag-tilted ovals technique can be applied success-

fully also to low-footprint 2D

FGC

s. The lower coupling eﬃciency of the initially developed

designs is not related to the perturbing elements’ optimization, since reference 2D

FGC

with circular diffracting elements suffer from the same problem. The coupling eﬃciency

limitation is rather a combination of the impact of several factors, including the waveguide

geometry. The concerns on the 2D

FGC

s’ optimization in terms of eﬃciency remain open

for a potential investigation in future works. The present work suggested the transition

to other waveguide thicknesses, making the 2D

FGC

s potentially more suitable for the

integration in the

BEOL

of this platform, in which the used a-Si:H waveguides offer more

design ﬂexibility.

160

5.2 Optimized Polarization Handling

5.2.3 Experimental Validation

In this subsection, experimental results for

SOI

designs, comprising elongated zig-zag

tilted perturbing elements, are presented. A focus is set on the measurement of the

PDL

, which is expressed as the difference between the coupling eﬃciencies (in dB) of the

polarization with a maximal transmission and the polarization with a minimal transmission.

This corresponds to the

PDL

deﬁnition used in simulations. Low values for this parameter

are an indication that the polarizations’ split ratio is suﬃciently large and the polarizations’

non-orthogonality and crosstalk are low.

5.2.3.1 Results in C-band

In the ﬁrst fabrication iteration, the producibility of the proposed elongated perturbing

elements has been investigated, starting with the linear 2D

s for C-band. The target

geometry includes a waveguide-grating shear angle of

α=

◦

, a grating period

Λ=

594

an etch depth d

140

, elongated perturbing elements with a minor axis w

230

and a major axis w

320

(as proposed in 5.2.2.1). It has been found that an appropriate

shape biasing is necessary, i.e. the diffracting elements must be laid out with larger and/or

stretcher dimensions. Both elliptical/oval and rectangular layout shapes may be used

to obtain a desired geometry. Figure 5.33 shows a scanning electron microscope (SEM)

picture after the reactive ion etch step. The determination of the exact dimensions of

the gratings’ elements out of the SEM photograph is not achievable with a large accuracy,

because of the limited resolution. Deviations of at least

are possible. A roughly

estimated range of the dimensions is: 210-230nm for the minor axis and 300-320nm for

the major axis. Thus, it is possible that the fabricated features are slightly smaller than

desired. This can be expressed in a lower coupling eﬃciency compared to simulations (cf.

the analysis of tolerances in 5.2.2.1).

Figure 5.33:

An optimized 2D grating coupler (2D GC) for C-band comprising a zig-zag-tilted

ovals array: (a) a schematic representation, (b) an exemplary array after the

reactive ion etch step. (The ﬁgure is published in Ref. [38]).

161

5 Design Optimization and Characterization

The 2D

shown in Fig. 5.33 is the ﬁrst design that has been investigated thoroughly

in experiments. Because of the back-to-back test structure in the layout, the parameter

that can be best examined in such a measurement is the

PDL

, out of which we can get

an idea of the polarization handling properties of a 2D

. The results below have been

initially reported in our work, Ref. [38]. To characterize the proposed design, back-to-back

test structures, comprising two 2D

s, connected by linear tapers and waveguides, are

fabricated in our 0.25 µm photonic

BiCMOS

technology on 200mm wafers with a partially

processed

BEOL

stack. The stack height has no importance for the

PDL

investigation. A

setup, including a tunable laser Agilent 81960A (1505nm - 1625nm), followed by a pro-

grammable polarization controller Agilent 8169A, is employed. Cleaved

SMF

s support the

light in- and out-coupling on a chip. The signal is detected by a power meter Agilent 81634B.

For the accurate power normalization, a slim photodiode S132Ce is placed in front of both

SMF

facets to measure the optical power loss in both off-chip paths, using a power meter

PM100D by Thorlabs. Due to the short Si waveguides, no waveguide loss is considered. To

prepare a full wafer statistic a semi-automated 300mm wafer probe system by Formfactor

is used. The measurements are performed at a ﬁxed height. Different polarization states

are scanned: for a ﬁxed polarization state, a wavelength sweep is carried out. The ﬁrst and

the last wavelength sweeps during the polarization scan are done for the same polarization

state, and are used to control the coupling stability in the end of the polarization sweep.

The

PDL

is determined in the following way. First, the wafer mean maximum transmission

wavelength is found and a 20nm bandwidth around this wavelength is chosen. In this

wavelength range, the maximal

PDL

is determined using 9 polarizations. The number of

polarizations is suﬃcient to ﬁnd the maximal coupling eﬃciency difference within the

given bandwidth and has been determined in preliminary experiments. Finally, the

PDL

averaged over the 61 chips on the wafer.

Figure 5.34 (a) shows exemplary measured coupling spectra of the 9 different polariza-

tions with a zoom in the considered bandwidth: the wafer-averaged central wavelength is

1555nm with a mean coupling eﬃciency of -4.7dB (the standard deviation

σCE

is 0.2dB).

The lower mean coupling eﬃciency may be related to smaller perturbing elements. The

20nm bandwidth considered for the PDL evaluation is from 1545 nm to 1565 nm. Figure

5.34 (b), (c), (d) shows a wafer map, a histogram and a cumulative distribution function

(CDF) of the maximal

PDL

. Because no index-matching gel can be used on this setup, there

is an uncertainty in the

PDL

determination, caused by Fabry-Perot ripples. To reduce their

impact, the measured curves are smoothed by a local regression ﬁtting algorithm. The

wafer maximal root-mean-square error, representing the measurement uncertainty, is

0.1

. The wafer-averaged maximal

PDL

in the investigated wavelength range is 0.5dB

with a standard deviation

σPDL

of 0.18dB. Furthermore, 67% of the wafer chips have a

maximal

PDL

< 0.55

(Fig. 5.34 (d)). Although further optimization of the shape biasing

is desirable to obtain an even better wafer uniformity, the present results conﬁrm the

suitability of the proposed C-band design for a fabrication in a 0.25 µm photonic

BiCMOS

technology. Furthermore, the present work shows for the ﬁrst time statistical results for

the PDL of optimized 2D GCs, fabricated with a 248nm DUV lithography.

162

5.2 Optimized Polarization Handling

Figure 5.34:

Experimental results for an optimized C-band 2D grating coupler (2D GC) com-

prising a zig-zag-tilted ovals array. (a) Exemplary measured coupling spectra

of different polarizations on a single chip. A zoom of the evaluation bandwidth

from 1545 nm to 1565 nm is shown. (b) A wafer map, (c) a histogram and (d) a

cumulative distribution function (CDF) of the maximal polarization-dependent

loss (PDL) within the considered bandwidth. (A similar ﬁgure is published in

Ref. [38]).

5.2.3.2 Results in O-band

Next, results from ﬁrst experiments for linear O-band 2D

s will be presented. Due to the

limited availability of the automated wafer probe system, the present analyses have been

performed on dice measurements. The chosen 9 chips originate from wafer locations,

which are used for process control measurements. For that reason, statistical range can be

evaluated also in this case. Another missing component at the time of the measurements

was a programmable polarization controller for O-band. No polarization sweep could be

performed. Thus, the

PDL

determination is based on the minimization or maximization of

the optical power transmission.

In the following, we compare designs with circular and elongated perturbing elements.

The former design is used as a reference and has the parameters: a shear angle

α=

◦

, a

grating period

Λ=

480

, an etch depth d

140

, circular perturbing elements with a

diameter w

Λ=

280

. The target design with elongated diffracting elements comprises

a shear angle

α=

◦

, a grating period

Λ=

480

, an etch depth d

140

, elongated

163

5 Design Optimization and Characterization

perturbing elements with a minor axis w

180

and a major axis w

260

(as

proposed in 5.2.2.1). Several biased shapes have been tested in the layout. Here, the best

performing variant will be presented.

The measurement setup includes a lightwave measurement system by Agilent (8164B) with

a tunable laser Agilent 81600b (sweep range 1280-1340nm) and an optical power meter

Agilent 81634B. The laser is followed by a manual polarization controller. Cleaved single

mode ﬁbers are used for in- and out-coupling. The loss in the off-chip paths is determined

using the calibrated photodiode Thorlabs SM05PD5A, which is placed in front of the facet

of each ﬁber.

Table 5.4:

An experimental comparison of a reference O-band 2D grating coupler (2D GC)

with circular perturbing elements and a proposed O-band 2D GC with zig-zag-

tilted ovals. The considered parameters are the average coupling eﬃciency CE

with a standard deviation

σCE

and the average maximal PDL in 20 nm bandwidth

with a standard deviation σPDL.

Design Bandwidth CE ±σCE PDL ±σPDL

Reference 1305-1325nm –4.5dB±0.4dB @1315 nm 2dB±1.2dB

Proposed 1300-1320 nm –4.8dB±0.5dB @ 1310nm 1.1dB±0.5dB

The back-to-back test structures are fabricated analogously to their C-band equivalents

and comprise two 2D

s, connected by linear tapers and waveguides. Also in this case,

the devices are fabricated in our 0.25 µm photonic

BiCMOS

technology on 200 mm wafers

with a partially processed

BEOL

stack. It should be noted that the 2D

s’ etch depth is

this time 130nm, instead of the target 140 nm. For that reason, a higher coupling loss is

expected.

Figure 5.35:

An experimental comparison of (a) a reference O-band 2D grating coupler

(2D GC) with circular perturbing elements and (b) an optimized O-band 2D GC

with zig-zag-tilted ovals.

Because of the lack of a programmable polarization controller, we use a 1D-1D

back-to-

back test structure to ﬁnd the polarization state, which corresponds to the Si TE

mode

on the chip. Due to the test structures’ orientation, this polarization state will be equally

split into both 2D

arms. The choice of this polarization state is necessary to ensure a

164

5.2 Optimized Polarization Handling

balanced coupling to the 2D

. With the adjusted polarization, the coupling to the 2D-2D

conﬁguration is optimized. After the best coupling position is found, the polarization is

changed so that a maximal or minimal power transmission results. The signal is controlled

at different wavelengths to make sure that the largest difference is found. The procedure

is repeated for all considered test structures on 9 chips.

Figure 5.35 shows exemplary coupling spectra of the reference design with circular diffract-

ing elements and the investigated design with elongated diffracting elements. The curves

are measured on the same chip. In addition, Tab. 5.4 compares both structures in terms

of mean coupling eﬃciency and

PDL

, including their standard deviations. The

PDL

is again

evaluated in a 20nm bandwidth around the central wavelength of each structure. For the

reference design, the bandwidth is 1305-1325nm. For the zig-zag tilted ovals design, the

bandwidth is 1300-1320nm. While the device with elongated perturbing elements shows

a slight deterioration in terms of eﬃciency, the average

PDL

and its variation are fairly

improved. In spite of the achieved progress, the investigated zig-zag-tilted ovals design

does not reach the performance benchmarks of its C-band equivalent.

The inspection of SEM photographs shows that the desired aspect ratio is not achieved

for the optimized design, i.e. the ovals are still too circular (Fig. 5.36 (a)). Alternative test

structures have also the problem that their ovals are either too circular or with too large

dimensions. For that reason, further shape biases are investigated next. Our observations

show that the smaller the desired minor axis w

, the stretcher the major axis w

must be

laid out. Thus, the ﬁrst-generation test ovals were laid out with a too low bias. To optimize

the layout design procedure, new test structures for O-band 2D

s are presently under

development, which examine variants in a more suitable range. An SEM of an improved

design is shown in Fig. 5.36 (b).

Figure 5.36:

A comparison of SEM photographs of O-band zig-zag-tilted ovals 2D grating

couplers (2D GCs). (a) Shape bias of the presently tested device. (b) An im-

proved shape biasing for stretcher perturbing elements.

165

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169

6 Conclusions and Outlook

High-index contrast 2D

s are fascinating devices: although simple in their composition,

they tend to interact with light in an unusual way. Understanding the behavior of 2D

s is

not as trivial as it initially seems. The present thesis has the ambition to provide the reader

with a systematic and comprehensive description of all important physical properties of

s. The author hopes that the summary in this work may be a helpful guideline for

interested researchers, who would like to develop own designs for a particular fabrication

platform.

The present dissertation was based on 3 milestones, namely: 1) The development of reli-

able methods for the analysis of 2D

s. 2) The thorough investigation of the interplay

between different physical effects in 2D

s. 3) The optimization of 2D

s considering

the requirements of a 0.25 µm photonic

BiCMOS

technology. The progress within this work

went from the bottom to the top through these three steps.

The ﬁrst aspect of the methods’ advancement was an essential basis to make proper con-

clusions in the subsequent analyses. Hereby, it was particularly important to get acquainted

with the trade-offs between numerical accuracy and computational cost. Moreover, dedi-

cated evaluation techniques and mathematical formulations were necessary to guarantee

for the appropriate post-processing of simulation results. The establishment of a reliable

chain of methods for the proper modeling, simulation and post-simulation analysis was

the key to determine 2D

s’ design metrics reliably. On the other hand, the accurate

experimental handling of the sensitive coupling devices was not less critical. The interplay

between theoretical and experimental analysis was the best means to understand the

complex behavior of 2D GCs as a next step.

In this second part of the work, the central objective of all analyses became the polarization.

As a ﬁrst important advance, a diffraction condition for two basic orthogonal polarizations

was formulated. For a given wavelength and effective refractive index, the combination of

a non-zero waveguide-to-grating shear angle and a grating period were responsible for

the variation of the resulting coupling angle at the symmetry plane. It was shown that a

larger coupling angle at the symmetry plane requires an increasing of both the shear angle

and the grating period. The mathematical formulation of these dependences made the

development of coupling structures for different optical bands, material compositions, and

170

coupling angles possible. Keeping the polarization as a central objective, the next exam-

ined aspect was the polarization splitting capability of 2D

s. Three performance-related

parameters were introduced. First, it was discovered that a 2D

converts partially a given

target-polarization state into its orthogonal counterpart, assigned as a cross-polarization.

The ratio between the target- and cross-polarization - the split ratio - was the ﬁrst investi-

gated parameter. The results showed that the cross-polarization scales with the grating

perturbation strength and contributes not only to the split ratio’s deterioration, but is also

a limiting factor for the coupling eﬃciency. In addition, the excitation of a cross-polarization

means that signals originating from both

arms are no more ﬁber coupled to purely

x- and y-polarized ﬁelds respectively, but have a mixed polarization composition. It was

shown that these modiﬁed polarizations are generally not orthogonal and can be associ-

ated with a polarization crosstalk. This was the second analyzed parameter, for which a

large wavelength dependence was found. Although orthogonal signals could be observed

at the 2D

’s central wavelength, all other wavelengths were affected by the polarizations’

non-orthogonality and crosstalk, deﬁning strict limitations of the usable 2D

bandwidth.

The last contribution of the cross-polarization could be observed in receiver-side 2D

where a given target signal (channel 1) can be superposed with the cross-polarized compo-

nent from the other communication channel 2 with a different phase relation. Depending

on the particular superposition, a large

PDL

results, which was the third important pa-

rameter for this analysis. In the end, a dependence chain could be deﬁned between split

ratio, polarizations’ non-orthogonality and crosstalk, and

PDL

. A good split ratio means

that polarizations from both

arms approach the desired x- and y-polarization states

after ﬁber coupling. Accordingly, the signals from the two

arms become orthogonal to

each other and the polarization crosstalk vanishes. Thus, any combination of orthogonal

polarizations without crosstalk is coupled with the same eﬃciency, leading thus to a low

PDL

. Therefore, a 2D

has ideally a high split ratio, a low polarization crosstalk and a low

PDL.

Apparently, to obtain these parameters as desired, the cross-polarization needs to be

eliminated. It became crucial to discover the physical origin of the cross-polarization. The

most important contribution of this work was the identiﬁcation of the grating in-plane

scattering as the physical effect behind the cross-polarization and the consequential issues.

The in-plane scattering results from the ﬁnite dimensions of the perturbing elements with

respect to the optical waveguide mode. In summary, the analysis of physical effects in

s showed that diffraction and in-plane scattering take place in parallel and are not

necessarily interrelated. The former physical effect is the desired one, the latter has to be

engineered in such a way that the 2D

handles properly any polarizations’ combination.

Only the proper in-plane scattering control is a guarantee for an eﬃcient 2D

with good

polarization splitting capabilities in terms of a low polarizations’ non-orthogonality and

crosstalk, and a low PDL.

After the systematization of these fundamental dependences, the ﬁnal objective in this

work became the exploration of a method to reach the goals given above, namely, to realize

an eﬃcient 2D

with excellent polarization handling properties. While it is not possible

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6 Conclusions and Outlook

to eliminate the in-plane scattering completely, an appropriate local scattering design can

be applied to reduce the total scattering strength and reach thus a low cross-polarization.

Two different approaches were proposed and analyzed, considering two optical bands -

C- and O-band - and two material platforms -

SOI

and a-Si:H in the platform’s

BEOL

. Both

optimization approaches have the basic idea that large total in-plane scattering may be

avoided, when the grating’s perturbing elements have different local scattering proﬁles.

As a ﬁrst technique, the segmented 2D

s were developed. Different local scattering

patterns were achieved by using circular perturbing elements with different sizes and peri-

odicity, grouped into several segments. The numerical analyses showed an improvement

in terms of enhanced polarizations’ split ratio, reduced polarizations’ non-orthogonality

and crosstalk, and improved coupling eﬃciency. However, the achieved advancement was

not suﬃcient to obtain a low

PDL

. For that reason, a second optimization approach had to

be developed, which can be considered as the highlight of the present work. The method

relies on a grating, comprising zig-zag tilted elongated perturbing elements. In such an

array, adjacent perturbing elements have abruptly different local scattering pattern, due

to their varying orientations. An appropriate design of the minor and major axis of the

diffracting elements ensures the local cancellation of forwards- and backwards scattered

waves of adjacent objects. In an extended variant of this method, both elongated and

circular elements can be used, achieving even better scattering suppression. The most

important properties of the proposed design is its simplicity - basic geometric shapes

can be used with dimensions appropriate for a 248nm

DUV

lithography. In addition, no

optical proximity correction, but only a suitable shape biasing is necessary. The feasibility

of the optimization technique was demonstrated by wafer-level statistics, investigating the

PDL

. In fact, this work provided for the ﬁrst time statistical results for the

PDL

of optimized

s fabricated with a 248nm

DUV

lithography. Satisfactory results were obtained for

C-band designs, without further shape biasing optimization. The measured wafer-averaged

maximal

PDL

in a 20nm was 0.5 dB. The results in O-band showed that further optimiza-

tion of the shape biasing is necessary. Possibilities to improve the O-band layout design

procedure were analyzed. Presently, optimized O-band devices are being prepared for

fabrication. Finally, numerical analyses of 2D

s with optimized zig-zag tilted elongated

perturbing elements and a metallic back-reﬂector showed that such devices may reach

a coupling eﬃciency between -2.5dB and -2 dB, while keeping the excellent polarization

handling properties unaltered.

As an outlook, the on-chip footprint reduction via 2D

FGC

s was brieﬂy investigated. Al-

though the polarization splitting performance could be improved in a similar way with the

zig-zag tilted ovals technique, the focusing designs required an additional optimization

in terms of coupling eﬃciency. Further work is necessary in the future to investigate this

matter. The transition to other waveguide geometries may be necessary. In addition, the

extended optimization technique, involving stretched and circular perturbing elements,

may be more advantageous for 2D FGCs.

The most of the analyses showed that O-band 2D

s developed under the boundaries

of the given 0.25 µm

BiCMOS

technology have better characteristics both in terms of

172

coupling eﬃciency and polarization handling, compared to their C-band equivalents. The

consideration of a metallic back-reﬂector promises for the achievement of a performance,

comparable to the best reported devices so far, with the decisive advantage of a reduced

complexity. This observation is particularly interesting with regard to the potential deploy-

ment of O-band coherent

DCI

s, willing to combine the best of the

IM-DD

and the coherent

detection worlds. Because the proposed 2D

s can be realized with a relatively low cost

and effort, the optimization technique may contribute to the cost reduction of coherent

transceivers, which is necessary for them to penetrate in the data center domain. The

proposed design technique is presently undergoing a patent registration procedure, which

could make it commercially available.

As a ﬁnal remark, the present work demonstrates that the good understanding of the

physical background of any optical component is crucial for its appropriate optimization.

Although the behavior of high-index contrast photonic devices is often not trivial to com-

prehend, it is worth investing efforts in ﬁnding an appropriate model of their physical

properties, instead of relying on blind searching methods. The devoted time to understand

single devices guarantees in return a more relaxed and reliable design of full systems.

173

Danksagung

An dieser Stelle möchte ich meinen ganz herzlichen Dank an alle richten, die den Erfolg

dieser Doktorarbeit möglich gemacht haben.

Ich danke Herrn Prof. Klaus Petermann dafür, dass er mich als letzte Doktorandin vor

seiner Pensionierung aufgenommen hat. Es war für mich eine besondere Ehre, von einer

der lebenden Legenden der optischen Kommunikationstechnik betreut werden zu dürfen.

Vor allem bin ich Prof. Petermann sehr dankbar, immer die richtigen Fragen gestellt zu

haben, sodass ich Wichtiges und Unwichtiges voneinander trennen konnte. Die Heraus-

forderungen von Herrn Petermann haben mich dazu motiviert, nach Antworten zu suchen,

die nicht sofort und nicht leicht zu ﬁnden waren. Ohne seine Motivation wäre ich nicht

zu den Einsichten gekommen, die letztlich den eigentlichen wissenschaftlichen Kern der

Doktorarbeit darstellen.

Ein ganz spezieller Dank geht an Herrn Prof. Lars Zimmermann für seine ausgeprägte

Unterstützung in diesen Jahren. Dabei meine ich nicht nur fachlichen Beratungen und die

zahlreichen wissenschaftlichen Diskussionen. Prof. Zimmermann hat mich motiviert, nicht

allein im bequemen Bereich der Feldtheorie zu verbleiben, sondern mich ebenso intensiv

mit experimentellen Untersuchungen auseinander zu setzen. Während der Laborarbeit

konnte ich die echten Probleme viel besser verstehen. Es ist für mich daher auch sehr

wichtig geworden, eine praktisch relevante Lösung zu liefern. Dank Prof. Zimmermann

hat sich die Balance zwischen Theorie und Praxis in dieser Arbeit deutlich verbessert. Des

Weiteren bin ich Prof. Zimmermann für seinen moralischen Beistand dankbar, den ich in

manch kritischer Situation besonders geschätzt habe.

Einen weiteren Dank geht an Herrn Prof. Heino Henke, der sich auch nach seiner Pen-

sionierung bemüht, eine interessante Vorlesung für seine Studenten anzubieten. Die dort

behandelten elektromagnetischen Fragestellungen haben mich zum ersten Mal auf die

Idee gebracht, was der mögliche Kern aller Problemen im Rahmen meiner Arbeit sein

könnte.

174

Danksagung

Bei Herrn Prof. Rolf Schuhmann möchte ich mich für die fachmännische Hilfe bei der

Einarbeitung in die Simulationssoftware CST bedanken. So konnte ich eine gute Überein-

stimmung zwischen theoretische und praktische Ergebnisse erreichen.

Einen ganz herzlichen Dank möchte ich an alle Kollegen von der TU Berlin und vom

IHP richten. Mit Pascal Seiler hatte ich eine angenehme und fruchtbare Zusammenarbeit,

da sich unsere Dissertationen erfreulich gegenseitig ergänzten. Unsere gemeinsamen

Diskussionen waren in vielen Situationen hilfreich, um voran zu kommen. Besonders

dankbar bin ich für die enorme Hilfe bei Laborarbeiten, die Spitze davon das nicht-triviale

kohärente Übertragungsexperiment in der TU-Berlin-Umgebung. Weiterhin waren die

System-Simulationen von Pascal Seiler eine wertvolle Ergänzung meiner Analysen. Ein Dank

geht an Karsten Voigt, der mir sowohl durch Simulationen verschiedener Hilfsstrukturen

für meine Layouts als auch durch seine praktische Erfahrung im Labor viel geholfen hat.

Ebenso danke ich Georg Winzer, der eine zuverlässige Hilfestellung bei jedem Layout-

Problem liefern konnte und viele Layout-Zellen für mich elegant programmiert hat. Ich

bedanke mich auch für die Unterstützung des IHP Technologie-Teams, insbesondere bei

Christian Mai, der bei der technologischen Realisierung vieler spezieller Designwünsche

sein Bestes gab und damit die Erzielung vieler Messergebnisse erst ermöglichte. Ich bin

weiterhin dankbar für die wirklich angenehme Zusammenarbeit mit Jochen Kreissl, von

dem ich auch etwas über andere Gitter gelernt habe. Vielen Dank an Anna Peczek und

Alexandra Kroh, die mich mit verschiedenen Messungen oder Hilfe im Labor am IHP unter-

stützt haben. Für die schönen Bilder während und nach meiner Verteidigung danke ich

Andrzej Gajda.

Abschließend möchte ich mich bei den restlichen Mitgliedern des Promotionsausschusses

– Herrn Prof. Manfred Berroth, Herrn Prof. Wim Bogaerts und Herrn Prof. Ronald Freund –

für Ihre Zeit bedanken. Während dieses Prüfungsverfahrens habe ich viel gelernt.

Meine tiefste Dankbarkeit und Liebe gehört aber meiner Familie in Bulgarien, ohne die

nichts von dem, was ich erreicht habe, möglich gewesen wäre. Eine solche Familie ist eine

Gottesgabe und ein großes Glück, das ich jedem wünsche.

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