# Integrated Silicon Photonic Two-Dimensional Grating Couplers

Galina Georgieva

► To cite this version:

Galina Georgieva. Integrated Silicon Photonic Two-Dimensional Grating Couplers. Engineering Sciences [physics]. Technische Universität Berlin, 2023. English. NNT: . hal-04272580

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Integrated Silicon Photonic
Two-Dimensional Grating Couplers

vorgelegt von

Galina Doneva Georgieva, M. Sc.
geboren in Yambol, Bulgarien

zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
- Dr.-Ing. -

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

<figure><img src="image_2.png" /></figure>

Technische Universität Berlin | Fakultät IV - Elektrotechnik und Informatik | Institut für Hochfrequenz- und Halbleiter-Systemtechnologien | Fachgebiet Hochfrequenztechnik-Photonik

Berlin 2023
<figure><img src="image_3.png" /></figure>
# Integrated Silicon Photonic
Two-Dimensional Grating Couplers

Galina Doneva Georgieva, M. Sc.

## Dissertation

to achieve the academic degree

## Doktor der Ingenieurwissenschaften (Dr.-Ing.)
<figure><img src="image_4.png" /></figure>
# Contents

<table><tr><td>Contents</td><td>i</td></tr><tr><td>Abstract</td><td>iv</td></tr><tr><td>Zusammenfassung</td><td>vi</td></tr><tr><td>List of Publications</td><td>viii</td></tr><tr><td>List of Figures</td><td>xi</td></tr><tr><td>List of Tables</td><td>xvi</td></tr><tr><td>List of Acronyms</td><td>xvii</td></tr><tr><td>List of Definitions</td><td>xix</td></tr><tr><td>1 Introduction</td><td>1</td></tr><tr><td>1.1 Optical Interfacing in Silicon Photonics</td><td>2</td></tr><tr><td>1.2 Fundamental Issues of Two-Dimensional Grating Couplers</td><td>7</td></tr><tr><td>1.3 Objectives of the Thesis</td><td>9</td></tr><tr><td>References</td><td>13</td></tr><tr><td>2 Methods for Analysis of Two-Dimensional Grating Couplers</td><td>23</td></tr><tr><td>2.1 Numerical Simulation and Evaluation</td><td>23</td></tr><tr><td>2.1.1 General Problem Description</td><td>23</td></tr><tr><td>2.1.2 Numerical Techniques</td><td>27</td></tr><tr><td>2.1.2.1 General Features of Numerical Approaches</td><td>27</td></tr><tr><td>2.1.2.2 Specific Numerical Methods for Photonic Applications</td><td>29</td></tr><tr><td>2.1.2.3 Generalized Numerical Methods in the Frequency Domain</td><td>30</td></tr><tr><td>2.1.2.4 Generalized Numerical Methods in the Time Domain</td><td>33</td></tr><tr><td>2.1.2.5 Open Boundary Conditions</td><td>39</td></tr><tr><td>2.1.2.6 Summary of Numerical Errors</td><td>40</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>2.1.3</td>
      <td>Simulation Procedure</td>
      <td>44</td>
    </tr>
    <tr>
      <td>2.1.3.1</td>
      <td>Modeling</td>
      <td>44</td>
    </tr>
    <tr>
      <td>2.1.3.2</td>
      <td>Definition of Excitation Sources</td>
      <td>46</td>
    </tr>
    <tr>
      <td>2.1.3.3</td>
      <td>Simulation Initialization</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.1.3.4</td>
      <td>Field Export</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.1.3.5</td>
      <td>Initial Geometry</td>
      <td>50</td>
    </tr>
    <tr>
      <td>2.1.4</td>
      <td>Methods for Parameter Evaluation</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.1.4.1</td>
      <td>Coupling Angles Determination from a 2D Radiation Pattern</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.1.4.2</td>
      <td>Coupling Efficiency</td>
      <td>53</td>
    </tr>
    <tr>
      <td>2.1.4.3</td>
      <td>Polarizations' Splitting and Orthogonality</td>
      <td>60</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>Measurement Methodology</td>
      <td>63</td>
    </tr>
    <tr>
      <td>2.2.1</td>
      <td>Basic Setup</td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.2.2</td>
      <td>Power Normalization in Loss Measurements</td>
      <td>65</td>
    </tr>
    <tr>
      <td>2.2.3</td>
      <td>A Measurement Technique for the Polarization Split Ratio</td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.2.4</td>
      <td>Wafer Statistics</td>
      <td>68</td>
    </tr>
    <tr>
      <td></td>
      <td>References</td>
      <td>71</td>
    </tr>
    <tr>
      <td>3</td>
      <td>Fabrication Platform</td>
      <td>76</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Bipolar CMOS</td>
      <td>76</td>
    </tr>
    <tr>
      <td>3.1.1</td>
      <td>CMOS vs. Bipolar Technology</td>
      <td>76</td>
    </tr>
    <tr>
      <td>3.1.2</td>
      <td>SiGe:C Heterojunction Bipolar Transistor</td>
      <td>78</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Monolithic Photonic BiCMOS</td>
      <td>78</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>BiCMOS With BEOL Photonic Layers</td>
      <td>81</td>
    </tr>
    <tr>
      <td></td>
      <td>References</td>
      <td>84</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Investigation of Fundamental Physical Effects in Two-Dimensional Grating Couplers</td>
      <td>88</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Diffraction</td>
      <td>88</td>
    </tr>
    <tr>
      <td>4.1.1</td>
      <td>Derivation of a 2D Diffraction Condition</td>
      <td>89</td>
    </tr>
    <tr>
      <td>4.1.1.1</td>
      <td>Phase Condition</td>
      <td>89</td>
    </tr>
    <tr>
      <td>4.1.1.2</td>
      <td>Sheared 2D GC Design Parameters</td>
      <td>92</td>
    </tr>
    <tr>
      <td>4.1.2</td>
      <td>Numerical Verification of the 2D Diffraction Condition</td>
      <td>92</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>In-Plane Scattering</td>
      <td>97</td>
    </tr>
    <tr>
      <td>4.2.1</td>
      <td>Analytical Investigation of In-Plane Scattering</td>
      <td>98</td>
    </tr>
    <tr>
      <td>4.2.2</td>
      <td>Cross-Polarization</td>
      <td>103</td>
    </tr>
    <tr>
      <td>4.2.2.1</td>
      <td>Split Ratio and Coupling Efficiency Limitations</td>
      <td>104</td>
    </tr>
    <tr>
      <td>4.2.2.2</td>
      <td>Polarizations' Non-Orthogonality</td>
      <td>112</td>
    </tr>
    <tr>
      <td>4.2.2.3</td>
      <td>Polarization-Dependent Loss</td>
      <td>115</td>
    </tr>
    <tr>
      <td>4.2.3</td>
      <td>Higher-Order Modes</td>
      <td>117</td>
    </tr>
    <tr>
      <td></td>
      <td>References</td>
      <td>124</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Design Optimization and Characterization</td>
      <td>127</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Improvement of the Out-Coupled Power Efficiency</td>
      <td>127</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>5.2 Optimized Polarization Handling</td><td>131</td></tr><tr><td>5.2.1 Segmented Two-Dimensional Grating Couplers</td><td>131</td></tr><tr><td>5.2.2 Two-Dimensional Grating Couplers with Elongated and Individually Oriented Perturbing Elements</td><td>139</td></tr><tr><td>5.2.2.1 Linear Two-Dimensional Grating Couplers</td><td>142</td></tr><tr><td>5.2.2.2 Focusing Two-Dimensional Grating Couplers</td><td>155</td></tr><tr><td>5.2.3 Experimental Validation</td><td>161</td></tr><tr><td>5.2.3.1 Results in C-band</td><td>161</td></tr><tr><td>5.2.3.2 Results in O-band</td><td>163</td></tr><tr><td>References</td><td>166</td></tr><tr><td>6 Conclusions and Outlook</td><td>170</td></tr><tr><td>Danksagung</td><td>174</td></tr></tbody></table>
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 µm² 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
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.
## Zusammenfassung

Die Integration von photonischen Komponenten auf Basis von vorhandenen CMOS Herstellungsprozessen hat in den letzten Jahren die Bedeutung des Siliziumphotonik-Forschungsbetriebes 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 Modenfehlanpassung zwischen dem kleinen, chip-integrierten Siliziumwellenleiter mit einer Querschnittsfläche von 0.1 µm² und der externen Einmodenfaser mit einem Kerndurchmesser von etwa 10 µm. Darüber hinaus gibt es eine bedeutende Diskrepanz bei der Polarisationsführung 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 Koppelstelle soll in der Lage sein, die Polarisation einer Eingangswelle zu manipulieren. Eine bevorzugte Ausführung solch einer Schnittstelle ist der Gitterkoppler. In seiner einfachsten, 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 ermöglichen, wird die Faser dieses Mal nicht nur unter einem kleinen Winkel geneigt, sondern auch entlang der Gittersymmetriechse ausgerichtet. Demnach soll das Koppelgitter nicht nur das Licht unter einem nahezu vertikalen Winkel beugen, sondern auch das Licht aus beiden Wellenleitern entlang der Symmetriechse orientieren.

Obwohl 2D Gitterkoppler längst bekannte Koppelschnittstellen sind, ist ihren effizienten und polarisationsunabhängigen Entwurf sehr anspruchsvoll. Um dieses Problem zu bewältigen, 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.
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 Multiplexing 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) Grating 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 Technology," 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 TM₀₀ 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
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.
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 Integration 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

<table><tr><td>1.1</td><td>A block diagram of a general optical communication system.</td><td>2</td></tr><tr><td>1.2</td><td>(a) A schematic comparison between a single-mode-fiber and a Si nanowire.<br/>(b) Horizontal coupling with a spot size converter. (c) Nearly vertical coupling<br/>with a diffraction grating coupler.</td><td>3</td></tr><tr><td>2.1</td><td>A photograph of a monolithically integrated dual-polarization coherent re-<br/>ceiver.</td><td>24</td></tr><tr><td>2.2</td><td>A description of a 2D grating coupler model.</td><td>25</td></tr><tr><td>2.3</td><td>An illustration of the difference between the numerical models of a 1D and<br/>2D grating coupler.</td><td>26</td></tr><tr><td>2.4</td><td>An exemplary representation of the Yee's cell, consisting of two staggered<br/>grids.</td><td>35</td></tr><tr><td>2.5</td><td>An exemplary space-time representation of a 1D wave propagation accord-<br/>ing to the leapfrog scheme.</td><td>36</td></tr><tr><td>2.6</td><td>An exemplary representation of the staggered grid used by the finite-integration-<br/>technique.</td><td>37</td></tr><tr><td>2.7</td><td>An exemplary model for a numerical convergence study.</td><td>40</td></tr><tr><td>2.8</td><td>The balance parameter of a simplified model for a different number of grid<br/>cells per minimal wavelength.</td><td>41</td></tr><tr><td>2.9</td><td>The balance parameter of a simplified model for different levels of the<br/>perfectly matched layer (PML) reflection.</td><td>43</td></tr><tr><td>2.10</td><td>An illustration of the components relevant for a complete 2D grating coupler<br/>(GC) model.</td><td>44</td></tr><tr><td>2.11</td><td>An illustration of the calculation of the diffracting elements' positions in a<br/>2D focusing array.</td><td>46</td></tr><tr><td>2.12</td><td>An exemplary Gaussian-sine excitation signal for simulations in O-band.</td><td>47</td></tr><tr><td>2.13</td><td>A decomposition of a full model into a source and imprint project (an example<br/>of a 2D focusing grating coupler).</td><td>48</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>2.14</td>
      <td>An illustration of the transformations, necessary to construct a tilted plane for results evaluation, representing the fixed position of a single-mode fiber (SMF).</td>
      <td>49</td>
    </tr>
    <tr>
      <td>2.15</td>
      <td>An illustration of the parameters of 2D grating couplers (2D GCs).</td>
      <td>51</td>
    </tr>
    <tr>
      <td>2.16</td>
      <td>An illustration of the adaption procedure for the determination of an appropriate 2D grating coupler geometry for desired coupling angles.</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.17</td>
      <td>A schematic of the typical power distribution in a 2D grating coupler (2D GC).</td>
      <td>54</td>
    </tr>
    <tr>
      <td>2.18</td>
      <td>A buried oxide (BOX) dependence of the out-coupled power of 1D grating couplers (1D GCs), designed for C- or O-band.</td>
      <td>55</td>
    </tr>
    <tr>
      <td>2.19</td>
      <td>A schematic with definitions used for the derivation of an overlap integral.</td>
      <td>56</td>
    </tr>
    <tr>
      <td>2.20</td>
      <td>A schematic for the illustration of two parameters, related to the polarization splitting in 2D grating couplers (2D GCs).</td>
      <td>59</td>
    </tr>
    <tr>
      <td>2.21</td>
      <td>An exemplary representation of two non-orthogonal polarization states on the Poincaré sphere (in a 2D cross-section).</td>
      <td>61</td>
    </tr>
    <tr>
      <td>2.22</td>
      <td>An exemplary setup for the characterization of passive on-chip components.</td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.23</td>
      <td>A schematic of a device for the measurement of a polarization split ratio.</td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.24</td>
      <td>An exemplary representation of 9 chips on a wafer.</td>
      <td>69</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>A schematic cross-section of a n-MOSFET and a npn-BJT.</td>
      <td>77</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>An illustration of the photonic modules' integration in a 0.25 µm electronic BiCMOS flow.</td>
      <td>79</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>A schematic cross-section of a photonic BiCMOS platform.</td>
      <td>80</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>A schematic representation of a 3D photonic BiCMOS platform.</td>
      <td>82</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>An illustration of a sheared 2D grating coupler (2D GC) with a rhombus-shaped grating area, placed in the plane (x,y,z = 0).</td>
      <td>89</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Two types of sheared 2D grating couplers (2D GCs) investigated in this work.</td>
      <td>93</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>A comparison of 2D grating couplers (2D GCs) without and with a shear angle.</td>
      <td>94</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>A comparison of 2D grating couplers (2D GCs) of Type I with different shear angles.</td>
      <td>95</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>A comparison of 2D grating couplers (2D GCs) of Type II with different shear angles.</td>
      <td>96</td>
    </tr>
    <tr>
      <td>4.6</td>
      <td>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°.</td>
      <td>97</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>A schematic representation of two scattering problems.</td>
      <td>99</td>
    </tr>
    <tr>
      <td>4.8</td>
      <td>An analytically calculated electric field distribution for the scattering problem of a dielectric cylinder array.</td>
      <td>101</td>
    </tr>
    <tr>
      <td>4.9</td>
      <td>A comparison of the target- and cross-polarization of different sheared 2D grating couplers (2D GCs).</td>
      <td>104</td>
    </tr>
    <tr>
      <td>4.10</td>
      <td>An electric field distribution and a power loss in percent caused by fields, propagating further in a 2D grating coupler.</td>
      <td>106</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>4.11</td>
      <td>Exemplary measured interferometric curves for the characterization of different sheared 2D grating couplers (2D GCs) in terms of split ratio.</td>
      <td>107</td>
    </tr>
    <tr>
      <td>4.12</td>
      <td>An exemplary simulation of the optical signal-to-noise ratio (OSNR) penalty for reaching a bit error ratio BER = 10<sup>-3</sup>, plotted over different 2D grating coupler (2D GC) split ratios.</td>
      <td>110</td>
    </tr>
    <tr>
      <td>4.13</td>
      <td>A test structure and a measurement setup used to investigate the polarization combining performance of a 2D grating coupler (2D GC).</td>
      <td>111</td>
    </tr>
    <tr>
      <td>4.14</td>
      <td>Bit error ratio vs. optical signal-to-noise ratio (BER vs. OSNR) for a transmission with a single active polarization and two active polarizations. A 2D grating coupler (2D GC) is included at the transmitter-side.</td>
      <td>112</td>
    </tr>
    <tr>
      <td>4.15</td>
      <td>A comparison of 2D grating couplers (2D GCs) for C- and O-band.</td>
      <td>114</td>
    </tr>
    <tr>
      <td>4.16</td>
      <td>An illustration of different incident polarizations at a receiver-side 2D grating couplers (2D GCs).</td>
      <td>115</td>
    </tr>
    <tr>
      <td>4.17</td>
      <td>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.</td>
      <td>116</td>
    </tr>
    <tr>
      <td>4.18</td>
      <td>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.</td>
      <td>118</td>
    </tr>
    <tr>
      <td>4.19</td>
      <td>A camera picture of an exemplary device used for the experimental determination of the polarization state and the modal composition of fields, excited by a 2D grating coupler (2D GC) in Si structures.</td>
      <td>119</td>
    </tr>
    <tr>
      <td>4.20</td>
      <td>Exemplary measured coupling spectra of the modes TE<sub>00,y</sub> (target-polarization), TE<sub>10,y</sub>, TE<sub>00,x</sub>, TE<sub>10,x</sub> for different 2D grating couplers (2D GCs).</td>
      <td>121</td>
    </tr>
    <tr>
      <td>4.21</td>
      <td>A coupling spectrum of the output 1D focusing grating coupler (1D FGC), measured at 10°.</td>
      <td>122</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>An a-Si:H 2D grating coupler (2D GC) design – a dependence of the normalized out-coupled power on the SiO<sub>2</sub> thickness separation from the Si substrate.</td>
      <td>129</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>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.</td>
      <td>130</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>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.</td>
      <td>131</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>A segmented a-Si:H 2D grating coupler (2D GC) for C-band, comprising 4 or 6 segments.</td>
      <td>132</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>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.</td>
      <td>134</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>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.</td>
      <td>135</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>5.7</td>
      <td>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.</td>
      <td>136</td>
    </tr>
    <tr>
      <td>5.8</td>
      <td>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.</td>
      <td>137</td>
    </tr>
    <tr>
      <td>5.9</td>
      <td>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.</td>
      <td>137</td>
    </tr>
    <tr>
      <td>5.10</td>
      <td>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.</td>
      <td>138</td>
    </tr>
    <tr>
      <td>5.11</td>
      <td>A schematic of an optimized 2D grating coupler (2D GC), comprising elon- gated perturbing elements with different orientations.</td>
      <td>140</td>
    </tr>
    <tr>
      <td>5.12</td>
      <td>A comparison between a reference and an optimized design of a C-band 2D grating coupler (2D GC).</td>
      <td>141</td>
    </tr>
    <tr>
      <td>5.13</td>
      <td>Even vs. odd polarizations of a reference sheared 2D grating coupler (2D GC) of Type II for C-Band.</td>
      <td>143</td>
    </tr>
    <tr>
      <td>5.14</td>
      <td>Even vs. odd polarizations of an optimized sheared 2D grating coupler (2D GC) of Type II for C-Band.</td>
      <td>144</td>
    </tr>
    <tr>
      <td>5.15</td>
      <td>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.</td>
      <td>145</td>
    </tr>
    <tr>
      <td>5.16</td>
      <td>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.</td>
      <td>146</td>
    </tr>
    <tr>
      <td>5.17</td>
      <td>A performance variation of the optimized 2D grating coupler (2D GC) in C-band, when the perturbing elements' major axis varies.</td>
      <td>146</td>
    </tr>
    <tr>
      <td>5.18</td>
      <td>A performance variation of the optimized 2D grating coupler (2D GC) in C-band, when the perturbing elements' minor axis varies.</td>
      <td>147</td>
    </tr>
    <tr>
      <td>5.19</td>
      <td>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.</td>
      <td>148</td>
    </tr>
    <tr>
      <td>5.20</td>
      <td>A schematic of an optimized 2D grating coupler (2D GC) design, comprising stretched perturbing elements, combined with circular perturbing elements.</td>
      <td>149</td>
    </tr>
    <tr>
      <td>5.21</td>
      <td>A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for C-band with a Metal3 back-reflector.</td>
      <td>150</td>
    </tr>
    <tr>
      <td>5.22</td>
      <td>A comparison between a reference and an optimized design of a O-band 2D grating coupler (2D GC).</td>
      <td>151</td>
    </tr>
    <tr>
      <td>5.23</td>
      <td>Even vs. odd polarizations of a reference sheared 2D grating coupler (2D GC) of Type II for O-Band.</td>
      <td>152</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>5.24 Even vs. odd polarizations of an optimized sheared 2D grating coupler (2D GC) of Type II for O-Band.</td>
      <td>152</td>
    </tr>
    <tr>
      <td>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.</td>
      <td>153</td>
    </tr>
    <tr>
      <td>5.26 A performance variation of the optimized 2D grating coupler (2D GC) in O-band, when the perturbing elements' major axis varies.</td>
      <td>153</td>
    </tr>
    <tr>
      <td>5.27 A performance variation of the optimized 2D grating coupler (2D GC) in O-band, when the perturbing elements' minor axis varies.</td>
      <td>154</td>
    </tr>
    <tr>
      <td>5.28 A comparison of optimized a-Si:H 2D grating couplers (2D GCs) for O-band with a Metal3 back-reflector.</td>
      <td>155</td>
    </tr>
    <tr>
      <td>5.29 A description of the modeling procedure of 2D focusing grating couplers (2D FGCs).</td>
      <td>156</td>
    </tr>
    <tr>
      <td>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.</td>
      <td>157</td>
    </tr>
    <tr>
      <td>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.</td>
      <td>158</td>
    </tr>
    <tr>
      <td>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.</td>
      <td>159</td>
    </tr>
    <tr>
      <td>5.33 An optimized 2D grating coupler (2D GC) for C-band comprising a zig-zag-tilted ovals array.</td>
      <td>161</td>
    </tr>
    <tr>
      <td>5.34 Experimental results for an optimized C-band 2D grating coupler (2D GC) comprising a zig-zag-tilted ovals array.</td>
      <td>163</td>
    </tr>
    <tr>
      <td>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-tilted ovals.</td>
      <td>164</td>
    </tr>
    <tr>
      <td>5.36 A comparison of SEM photographs of O-band zig-zag-tilted ovals 2D grating couplers (2D GCs).</td>
      <td>165</td>
    </tr>
  </tbody>
</table>
List of Tables

<table><tbody><tr><td>4.1</td><td>Numerically estimated coupling angles for 2D grating couplers (2D GCs), designed by different combinations of a shear angle and a grating period.</td><td>93</td></tr><tr><td>4.2</td><td>Numerically estimated split ratios different sheared 2D grating couplers (2D GCs).</td><td>105</td></tr><tr><td>4.3</td><td>Experimentally estimated mean split ratios for different sheared 2D grating couplers (2D GCs).</td><td>109</td></tr><tr><td>4.4</td><td>A summary of the notations, used to assign a desired target-polarization with a TE<sub>00</sub> modal component, resulting from diffraction and in-plane-scattering-related polarizations/modes.</td><td>119</td></tr><tr><td>4.5</td><td>2D grating coupler (2D GC) designs used for the investigation of their polarizations' and modal on-chip excitation.</td><td>120</td></tr><tr><td>4.6</td><td>Mean extinction ratios (ERs) ±σ between the target TE<sub>00,y</sub> and the other considered polarizations/modes averaged over 9 chips. Different 2D grating coupler (2D GC) designs are compared.</td><td>122</td></tr><tr><td>5.1</td><td>Sheared a-Si:H 2D grating couplers (2D GCs) of Type II, designed for C- and O-band.</td><td>128</td></tr><tr><td>5.2</td><td>Geometric parameters of sheared, segmented C-band a-Si:H 2D grating couplers (2D GCs) of Type II with a different number of segments.</td><td>133</td></tr><tr><td>5.3</td><td>Geometric parameters of a 2° sheared, segmented O-band a-Si:H 2D GCs of Type II with 4 sections.</td><td>136</td></tr><tr><td>5.4</td><td>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.</td><td>164</td></tr></tbody></table>
List of Acronyms

<table><tr><td>BEOL</td><td>backend of line</td><td>FEM</td><td>finite elements method</td></tr><tr><td>BER</td><td>bit error ratio</td><td>FEOL</td><td>frontend of line</td></tr><tr><td>BiCMOS</td><td>bipolar complementary metal oxide semiconductor</td><td>FGC</td><td>focusing grating coupler</td></tr><tr><td>BJT</td><td>bipolar junction transistor</td><td>FIT</td><td>finite integration technique</td></tr><tr><td>BOX</td><td>buried oxide</td><td>GC</td><td>grating coupler</td></tr><tr><td>CMOS</td><td>complementary metal oxide semiconductor</td><td>HBT</td><td>heterojunction bipolar transistor</td></tr><tr><td>DCI</td><td>data center interconnect</td><td>IC</td><td>integrated circuit</td></tr><tr><td>DBR</td><td>distributed Bragg reflector</td><td>IM-DD</td><td>intensity modulated direct detection</td></tr><tr><td>DoP</td><td>degree of polarization</td><td>LP</td><td>linearly polarized</td></tr><tr><td>DSP</td><td>digital signal processing</td><td>MFD</td><td>mode field diameter</td></tr><tr><td>DUV</td><td>deep ultra violet</td><td>MMI</td><td>multi-mode interferometer</td></tr><tr><td>EDFA</td><td>erbium doped fiber amplifier</td><td>MOS</td><td>metal oxide semiconductor</td></tr><tr><td>EIC</td><td>electronic integrated circuit</td><td>MOSFET</td><td>metal oxide semiconductor field effect transistor</td></tr><tr><td>EPIC</td><td>electronic-photonic integrated circuit</td><td>OSNR</td><td>optical signal-to-noise ratio</td></tr><tr><td>ER</td><td>extinction ratio</td><td>PBS/C</td><td>polarization beam splitter/combiner</td></tr><tr><td>FDM</td><td>finite differences method</td><td>PDL</td><td>polarization-dependent loss</td></tr><tr><td>FDTD</td><td>finite differences time domain</td><td>PIC</td><td>photonic integrated circuit</td></tr></table>
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
# 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 converted 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 positions, 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 the overlap integral
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 preserved 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 output 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 (ICs), allowing for an extremely efficient computational and signal processing performance with a remarkably low power consumption. Moreover, the phenomenal ICs' 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 development 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 1 Gbit/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-
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><img src="image_5.png" /><figcaption>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.</figcaption></figure>

With electronics and optics being inseparable in communication systems, the ICs' 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 (PICs) 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 PICs and electronic integrated circuits (EICs). 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
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><img src="image_6.png" /><figcaption>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 distribution, 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 mismatch 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.</figcaption></figure>

The most characteristic feature of Si photonics is the high index contrast Δn: Si photonic
waveguides are based on a silicon on insulator (SOI) technology with Si as a core material
and SiO2 as a cladding (Δn ≈ 2, e.g. [46]). This allows for the realization of waveguides
with a very small cross section: a typical waveguide height × width is 220 nm × 500 nm.
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
their price. The standard SMF, based on a low-index contrast technology ($\Delta n < 0.01$), has a typical mode field diameter (MFD) of around 10 µm. There is a significant obstacle for the simple interfacing between Si waveguides and SMFs, 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 (TE) and transverse magnetic (TM) polarizations $TE_{00}$ and $TM_{00}$ 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 $TE_{00}$ mode. On the other hand, the SMF supports two degenerated linearly polarized (LP) 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 $TE_{00}$. In summary, two crucial issues have to be overcome for an efficient fiber-to-chip interfacing:

1. Mode matching: the Si fundamental mode with a small MFD has to be adapted to the dimensions of the fiber mode and vice versa.

2. Polarization handling: both fiber polarizations $LP_{01,x}$ and $LP_{01,y}$ have to be transformed to the fundamental $TE_{00}$ 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 (SSCs). 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 $TE_{00}$ and $TM_{00}$ polarizations with a very low polarization-dependent loss (PDL). Additional on-chip polarization rotator splitter (PRS) is necessary to separate both polarizations and enable the conversion from $TM_{00}$ into $TE_{00}$.

Unfortunately, these advantages come in combination with significant physical, technological and economical constraints. To the most considerable disadvantages belong:
* 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 6.6 µm and a standard SMF with a MFD of 10.4 µm/ 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 manufacturing 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 PRSs 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 PICs. 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 technique, realized by the adoption of diffraction gratings. As a most basic form of such structures, the one-dimensional (1D) grating couplers (GCs) will be first presented (Fig.
1.2 (c)). A conclusive analysis of the 1D GCs' properties is provided in the doctoral thesis of Taillaert [68]. In 1D GCs, 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 $\theta$ with 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 consequence of the physical background of the diffraction gratings. Following fundamental limitations are present:

* Lower coupling efficiency: because a GC 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 enhanced 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 GCs. The maximal coupling is at the wavelength with the diffraction angle, matching the fiber tilt angle.
* Polarization dependence: 1D GCs are able to couple only the SMFs 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 (FGCs) has been proposed [77–79]. In parallel to the work on the present thesis, FGCs for the TM₀₀ 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 GCs are designed by combining some of the mentioned methods. This is the case for the record low-loss 1D GC from Ref. [83]. To improve the optical bandwidth, silicon nitride (Si₃N₄) or Si₃N₄-assisted GCs 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 GC needs to be modified. One possibility is the counter-directional coupling of TE and TM with a 1D GC designed in such a way that TE₀₀ has a coupling angle θ and TM₀₀ has a coupling angle -θ (see e.g. Ref. [88]). This method has the disadvantage that a PRS is still required to convert the TM₀₀ polarization into TE₀₀. This leads us to the main subject of the present thesis: the two-dimensional (2D) GC. Although 2D GCs are based on a natural idea - to simply combine orthogonally two 1D GCs 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 GC'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 GCs, can be naturally transferred to their 2D counterparts. Some 2D GC 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 GCs. There are several reasons for that. In early publications, no adaption
of the diffraction condition of 2D GCs 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 GC designs showed a very low coupling efficiency in the range from -8 dB to -6.5 dB [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 GCs has been considered. Current 2D GCs 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 -5 dB [98–100]. The firm Luxtera in collaboration with STMicroelectronics demonstrated 2D GCs with around -3 dB 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 GCs [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 (193 nm or 248 nm). 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 GCs, the spectrum shape is determined by the wavelength-dependence of both the nearly vertical angle $θ$ and the in-plane angle $φ$ (cf. the angles' definitions in Fig. 1.2 (c)). Potentially, the widening of the 2D GCs' bandwidth may be addressed in the same way as in 1D GCs, i.e. by a dual-layer principle. So far, no designs addressing this aspect have been reported, most possibly because the available 2D GCs' bandwidth is sufficient for the target applications. Another aspect is the spectrum wavelength adaption by the coupling angle $θ$. In 1D GCs, the maximum transmission wavelength can be adjusted by the choice of an appropriate coupling angle. In 2D GCs, there is another specific, which may be a problem for the optimal coupling. There is only one coupling angle $θ$ that is related to diffraction in the symmetry plane $φ = 45°$. The 2D GC's design must be sufficiently precise, so that the coupling angle $θ$ at the symmetry plane matches to the required wavelength. Spectrum shift via $θ$-adaption leads to a departure from the symmetry plane ($φ \neq 45°$), 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 GCs as well. However, the nature of these processes is quite different from the polarization-dependence of 1D GCs, 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
effects in 2D GCs 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 GCs. 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 GC’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 GC 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 GC to 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 GCs, available in the literature already more than 10 years ago, a large part of the research community considered the topic of 2D GCs 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 GCs remained to a large extent 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 GCs’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 GCs’ physical properties.
The second impulse for this work was given by the emerging topic of the coherent communications 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 GCs 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 GCs' suitability even for more challenging form factors is approved by Sicoya's product portfolio (e.g. a grating coupler based QSFP28 transceiver [122]). Naturally, having a polarization-diversity coupling interface, polarization-multiplexed modulation formats come into question. Therefore, the investigation of 2D GCs 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 GCs 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 GCs. 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 GCs, in-plane scattering has been identified as a dominant process that had been overlooked until that moment. With in-plane scattering designated as the most decisive issue in 2D GCs, design strategies for its minimization have been proposed and developed, using the systematically acquired basic knowledge about 2D GCs. 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 GCs, which has filled the gap of unexplained observations in older publications. A conclusive description of the 2D GCs' physical fundamentals and their practical manipulation 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 GCs' geometric parameters and the electromagnetic fields' deflection
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-orthogonality 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], scattering effects have been underestimated in 2D GCs. 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 understanding of the physical processes in 2D GCs. 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 GCs 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 GCs 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 (DCIs), the present work shows that O-band 2D GCs 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 description of the numerical and experimental methods for the analysis of 2D GC 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 calculations are described in more detail. In the second part of the chapter, the measurement
methodology for 2D GCs'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 GCs' 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 GCs 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 GCs 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 GCs 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 GCs 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 FGCs is considered. Subsequently, 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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# 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 GCs. The first section outlines the main challenges for the accurate numerical investigation of 2D GCs, 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 GC 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
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 (EPICs), 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><img src="image_7.png" /><figcaption>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].</figcaption></figure>

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 (MMIs) etc. can be characterized with comparatively small computational resources, monolithically
integrated 2D GCs 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 GCs 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 GC 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><img src="image_8.png" /><figcaption>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.</figcaption></figure>

Even with a strongly reduced computational volume, the simulation of a 2D GC remains challenging. As an illustration, we compare the numerical problems of a 1D and 2D GC shown in Fig. 2.3. For the simulation of the 1D GC, 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 GCs 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 GCs are defined on 220 nm 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 GC and only 5 µm above the 2D GC. For simulations covering C-band (1530–1565 nm), a typical minimal computation free-space wavelength is 1500 nm (the simulation window is usually
100 nm, i.e. in the range 1500-1600 nm). If we assume the simple case of a hexahedral grid and a resolution of 15 cells per wavelength, the computational domain of a 1D GC will consist of nearly 500 000 grid cells and the 2D GC'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><img src="image_9.png" /><figcaption>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.</figcaption></figure>

Due to the geometrical specifics of integrated 2D GCs, 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 GC simulation as well (e.g. for FGCs). 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.
### 2.1.2 Numerical Techniques

The development of numerical methods and their software implementation was an important 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]:

1) An unknown function $u(r)$, which is contained in an operator equation, needs to be expressed by the summation of a set of linear independent basic functions $u_n(r)$ weighted by the undetermined parameters $c_n$.

$$u(r) \approx \hat{u} = \sum_{n=1}^{N} c_n \cdot u_n(r) \quad (2.1)$$

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 $N \to \infty$.

The choice of basic functions $u_n(r)$ and the determination of their unknown parameters $c_n$ 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].

2) 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.

3) A principle for error minimization needs to be chosen, out of which the unknown parameters $c_n$ 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.

4) The solution of the matrix equation resulting in the previous step gives the approximate solution of the initial problem.
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:

$$ \mathcal{L}u = f \qquad (2.2) $$

The operator $\mathcal{L}$ 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 $x$ into the variable $y$. Thus, $x$ belonging to the interval $[-\infty, +\infty]$ is transformed to $y$ in the range from $[0, +\infty]$.

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 $u_0$ and a fixed normal derivative $g_0$. Due to the approximation in (2.1), residuals $R_{i=0,1,2}$ result in the equation domain $\Omega$ and at the boundaries $\Gamma_1, \Gamma_2$.

$$ \begin{array}{lll} \mathcal{L}u = f & \text{in the domain } \Omega & \underset{u \approx \hat{u}}{\stackrel{\text{(2.1)}}{\longrightarrow}} R_0 = \mathcal{L}\hat{u} - f \\ u|_{\Gamma_1} = u_0 & \text{at the boundary } \Gamma_1 & \underset{u \approx \hat{u}}{\stackrel{\text{(2.1)}}{\longrightarrow}} R_1 = \hat{u}|_{\Gamma_1} - u_0 \\ \frac{\partial u}{\partial n}|_{\Gamma_2} = g_0 & \text{at the boundary } \Gamma_2 & \underset{u \approx \hat{u}}{\stackrel{\text{(2.1)}}{\longrightarrow}} R_2 = \hat{g}|_{\Gamma_2} - g_0 \end{array} \qquad (2.3) $$

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

$$ \int_{\Gamma_1} R_1 \cdot W_1 \cdot d\Gamma_1 + \int_{\Gamma_2} R_2 \cdot W_2 \cdot d\Gamma_2 = 0 \qquad \int_{\Omega} R_0 \cdot W_0 \cdot d\Omega = 0 \qquad (2.4) $$

Boundary methods
* 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 approaches 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 $\exp(\mp j\beta z)$ ($\beta$ a propagation constant along the $\pm z$-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 electromagnetic 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 structures. 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**
**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 GC 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 GC 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 GC 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 to 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.
A typical discretized equation of BEM has the form:

$$[H]U = [G]Q \qquad (2.5)$$

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 **H** and **G** result 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 **H** and **G** are 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 commercialization 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 interpolation 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:
1) Discretization of the solution domain by elements. Within each element, the unknown function included in the governing equation $u$ is approximated by $\hat{u}_e$ (cf. (2.1)), where $e$ assigns the particular element. If we assume a triangular element with the vertices $i, j, m$, the unknown function $u$ will be approximated by the nodal values $u_i, u_j, u_m$ and the shape functions $N_i^e, N_j^e, N_m^e$:

$$u \approx \hat{u}_e = \sum_k N_k^e u_k \quad (k = i, j, m) \qquad (2.6)$$

2) The element matrix equation is derived by the weighted residuals principle or by the variational principle.

3) The matrix equation at every node of the domain is assembled to form a system matrix equation, which has the form:

$$K u = B \qquad (2.7)$$

with $u$ a column vector with the size $N$, where $N$ is the total number of nodes. The global matrix $K$ is called assembled matrix, system matrix or stiffness matrix. It is a sparse, symmetric and positive definite matrix with the order $N \times N$. The column matrix $B$ includes the known boundary conditions.

4) The solution of (2.7) delivers the values of the discretized function on every node.

Regarding the given 2D GC 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 $N$ could 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 GCs. For a given spectrum, the solution of the matrix equation must be carried out for a discrete number of wavelengths $N_\lambda$ until a certain error minimization criterion is fulfilled. This means that $N_\lambda$ 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.
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 electromagnetic 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 GCs 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 to 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].
**Discretization with FDTD** As the name suggests, the basic approximation step in FDM is the so called finite differences approximation, i.e. the definition of the first derivative of a function f at the point x = x₀ is substituted by a subtraction within a discrete element in the range x₀ ± Δ:

$$f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + \frac{h}{2}) - f(x_0 - \frac{h}{2})}{h} \approx \frac{f(x_0 + \Delta) - f(x_0 - \Delta)}{2\Delta} \quad (2.8)$$

Instead of using the limits of an infinitely small variation h around the variable x, we take a finite segment on the x-axis. Geometrically, we substitute the slope of the tangent at x₀ by the slope of a line passing through x₀ ± 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:

$$-\frac{\partial \vec{B}}{\partial t} = \nabla \times \vec{E} \quad (2.9)$$

In a Cartesian coordinate system, the equation can be decomposed as:

$$-\frac{\partial B_x}{\partial t} = \frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z} \quad (2.10)$$

$$-\frac{\partial B_y}{\partial t} = \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x} \quad (2.11)$$

$$\frac{\partial B_z}{\partial t} = \frac{\partial E_x}{\partial y} - \frac{\partial E_y}{\partial x} \quad (2.12)$$

With the finite differences approximation in (2.8), a set of scalar equations result. Define a spatial grid point as (i, j, k) = (iΔx, jΔy, kΔz) 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:

$$\frac{B_x^{n+\frac{1}{2}}(i,j+\frac{1}{2},k+\frac{1}{2}) - B_x^{n-\frac{1}{2}}(i,j+\frac{1}{2},k+\frac{1}{2})}{\Delta t} = \frac{E_y^n(i,j+\frac{1}{2},k+1) - E_y^n(i,j+\frac{1}{2},k)}{\Delta z} - \frac{E_z^n(i,j+1,k+\frac{1}{2}) - E_z^n(i,j+1,k+\frac{1}{2})}{\Delta y} \quad (2.13)$$

From (2.13), it is evident that the x-component of the magnetic flux is half-cell shifted from the position of E_y on the z-axis and from the position of E_z on the y-axis. The reason is that both the differences $\frac{\Delta E_y}{\Delta z}$ and $\frac{\Delta E_z}{\Delta y}$ should correspond simultaneously to a single clearly defined point B_x 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.
<figure><img src="image_10.png" /><figcaption>Figure 2.4: An exemplary representation of the Yee’s cell, consisting of two staggered grids. In the primary grid, electric field strength $\vec{E}$ and flux $\vec{D}$ 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 $\vec{H}$ and flux $\vec{B}$ 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]).</figcaption></figure>

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]

$$ \Delta t \le \left( c \sqrt{\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} + \frac{1}{\Delta z^2}} \right)^{-1} \qquad (2.14) $$

with $c = 1/\sqrt{\epsilon\mu}$ the light velocity in a material with a the permittivity $\epsilon$ and the permeability $\mu$. 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
time step $\Delta t \sim 10^{-17}$ 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><img src="image_11.png" /><figcaption>Figure 2.5: An exemplary space-time representation of a 1D wave propagation. A staggered grid for the electric and magnetic field strength $\vec{E}$ and $\vec{H}$ 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]).</figcaption></figure>

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 accumulated 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].
The FIT will be introduced, again using the Maxwell-Faraday's equation, but this time in integral form:

$$ -\iint_A \frac{\partial \vec{B}}{\partial t} d\vec{A} = \oint_{\partial A} \vec{E} d\vec{s} \qquad (2.15) $$

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 **e** and magnetic fluxes **b** are introduced (Fig. 2.6). Their bold notation assigns a full-grid vector. The Maxwell's equation in a single rectangular grid cell *n* with edges $L_i, i = 1, 2, 3, 4$ and a facet $A_n$ can be written as:

$$ e_1 + e_2 - e_3 - e_4 = -\dot{b}_n \\ \text{with } e_i := \int_{L_i} \vec{E}(\vec{r}, t) d\vec{s}, \quad i = 1, 2, 3, 4 \qquad b_n := \iint_{A_n} \vec{B}(\vec{r}, t) d\vec{A} \qquad (2.16) $$

where the dot denotes a time derivative. The grid voltages $e_i$ 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**:

$$ \begin{bmatrix} 1 & 1 & -1 & -1 & \cdots \\ \cdot & \cdot & \cdot & \cdot & \cdots \\ \cdot & \cdot & \cdot & \cdot & \cdots \\ \cdot & \cdot & \cdot & \cdot & \cdots \\ \cdot & \cdot & \cdot & \cdot & \cdots \end{bmatrix} \begin{bmatrix} e_1 \\ e_2 \\ e_3 \\ e_4 \\ \vdots \end{bmatrix} = - \begin{bmatrix} \dot{b}_n \\ \cdot \\ \cdot \\ \cdot \\ \vdots \end{bmatrix} \Rightarrow \mathbf{C}\mathbf{e} = -\dot{\mathbf{b}} \qquad (2.17) $$

Due to the similarity to the Maxwell's equation in differential form, the matrix **C** in (2.17) is called curl matrix. Similarly, all Maxwell's equations can be discretized and represented in a compact matrix form.

<figure><img src="image_12.png" /><figcaption>Figure 2.6: An exemplary representation of the staggered grid used by the finite-integration- technique (FIT). In the primary grid, electric grid voltages **e** and fluxes **d** are defined. A dual grid, shifted by a half grid cell in all directions, contains the magnetic voltages **h** and fluxes **b**. (Representation after Ref. [17].)</figcaption></figure>
In an analogous way, the magnetic grid voltage **h**, the electric grid flux **d**, the electric grid current **j** and the grid charge **q** in the dual grid cell $\tilde{n}$ with edges $\tilde{L}_i, i = 1, 2, 3, 4$ and a facet $\tilde{A}_n$ can be introduced:

$$h_i := \int_{\tilde{L}_i} \vec{H}(\vec{r}, t)d\vec{s}, \quad i = 1, 2, 3, 4 \qquad (2.18)$$

$$d_n := \iint_{\tilde{A}_n} \vec{D}(\vec{r}, t)d\vec{A} \qquad (2.19)$$

$$j_n := \iint_{\tilde{A}_n} \vec{j}(\vec{r}, t)d\vec{A} \qquad (2.20)$$

$$q_n := \iiint_{\tilde{V}_n} \rho(\vec{r}, t)dV \qquad (2.21)$$

(2.22)

The Maxwell's equations in matrix form can be summarized as:

$$\oint_{\partial A} \vec{E} d\vec{s} = - \iint_A \frac{\partial \vec{B}}{\partial t} d\vec{A} \quad \Rightarrow \quad \mathbf{C}\mathbf{e} = -\dot{\mathbf{b}} \qquad (2.23)$$

$$\oint_{\partial A} \vec{H} d\vec{s} = \iint_A \frac{\partial \vec{D}}{\partial t} + \vec{j} d\vec{A} \quad \Rightarrow \quad \tilde{\mathbf{C}}\mathbf{h} = \dot{\mathbf{d}} + \mathbf{j} \qquad (2.24)$$

$$\oint_{\partial V} \vec{B} d\vec{A} = 0 \quad \Rightarrow \quad \mathbf{S}\mathbf{b} = 0 \qquad (2.25)$$

$$\oint_{\partial V} \vec{D} d\vec{A} = Q \quad \Rightarrow \quad \tilde{\mathbf{S}}\mathbf{d} = \mathbf{q} \qquad (2.26)$$

Here, the tilde assigns a discretized equation on the dual grid, the matrices $\mathbf{C}, \tilde{\mathbf{C}}$ are called curl matrices and $\mathbf{S}, \tilde{\mathbf{S}}$ - source matrices. The matrices are sparse, which makes their storage less memory demanding. It is worth mentioning that the matrices fulfill basic identities:

$$\mathbf{S}\mathbf{C} = \tilde{\mathbf{S}}\tilde{\mathbf{C}} = 0 \quad \Leftrightarrow \quad \mathrm{div} \ \mathrm{rot} = 0 \qquad (2.27)$$

$$\mathbf{C}\tilde{\mathbf{S}}^T = \tilde{\mathbf{C}}\mathbf{S}^T = 0 \quad \Leftrightarrow \quad \mathrm{rot} \ \mathrm{grad} = 0 \qquad (2.28)$$

with $T$ assigning the transpose of a matrix.

The last aspect to discuss are the material relations. Since **e** and **d**, resp. **h** and **b** are 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 $f$ within the cell $n$ with the edges $L_i, i = 1, 2, 3, 4$ and the face $A_n$:

$$f(\vec{B}, \vec{H}) = \frac{\int_{A_n} \vec{B} d\vec{A}}{\oint_{\tilde{L}_i} \vec{H} d\vec{s}} \approx \mathbf{D}_{\mu,n}. \qquad (2.29)$$

As an approximation of $f$, the permeability matrix $\mathbf{D}_{\mu,n}$ can be defined. Analogously, we
obtain the permittivity and conductivity matrices:

$$d = D_{\varepsilon}e \qquad j = D_{\kappa}e \qquad (2.30)$$

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 $\sigma^E$ and magnetic conductivity $\sigma^H$. The PML is reflectionless, provided that the electric and magnetic conductivity are related by:

$$\frac{\sigma^E}{\varepsilon} = \frac{\sigma^H}{\mu} \qquad (2.31)$$

where $\varepsilon$ is the permittivity and $\mu$ 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 parameters 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
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 generalized 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 requirements 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><img src="image_13.png" /><figcaption>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. (c) An electric field distribution at a 1310 nm wavelength.</figcaption></figure>
**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 GC 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 GC. 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 220 nm on a BOX of 2 µm. In this work, 2D GCs 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 GCs, namely 400 nm diameter with a 620 nm 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 E<sub>tang</sub> = 0, reducing the total number of cells by a factor of two. Two ports are defined and the fundamental TE<sub>00</sub> 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 1310 nm is shown in Fig. 2.7 (c).

<figure><img src="image_14.png" /><figcaption>Figure 2.8: The balance parameter of a simplified model for a different number of grid cells per minimal wavelength N<sub>cells</sub>. (a) Results in C-band. (b) Results in O-band.</figcaption></figure>

To study the convergence the balance parameter *B* is used as a figure-of-merit. With two ports, *B* is calculated from the port S-parameters as:

$$B = \sqrt{|S_{1,1}|^2 + |S_{2,1}|^2} \qquad B[\text{dB}] = 20 \lg B. \qquad (2.32)$$
Because we have a radiating structure, $|B|^2$ 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 1500 nm in simulations covering C-band and 1260 nm 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 GC model, this resolution results in too large number of grid cells, for which no sufficient RAM resources were available. Since the difference between $N_{cells} = 15$ and $N_{cells} = 25$ at 1500 nm is about 0.1 dB (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 $N_{cells} = 15$ is chosen in C-band as well.

**Staggered grid errors** The staggered grid errors are related to the independent definitions 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 GC 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 wavelengths. Generally, this leads to an error of the 2D GC's phase condition for wavelengths, different from the central wavelength. Therefore, the simulation bandwidth window has always been centered at the 2D GC's design wavelength. A parameter that remains affected is the 2D GC'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 approximation 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 GC bandwidth. Nevertheless, the efficiency of the non-uniform
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 boundary 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 GC 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 GCs, 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 GC 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><img src="image_15.png" /><figcaption>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.</figcaption></figure>
### 2.1.3 Simulation Procedure

The complete procedure for the design of a 2D GC follows some basic steps, which will be summarized here.

<figure><img src="image_16.png" /><figcaption>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 $d$ as 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.</figcaption></figure>

#### 2.1.3.1 Modeling

Each simulation procedure starts with the modeling of a structure, which has to be optimized 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 220 nm, 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)).

1) Linear 2D GCs 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 GC itself starts at the taper's end. Additional simulations to obtain the proper taper length have been executed. The taper simulations have been kindly performed by the colleague Karsten Voigt using the
commercial software Fimmwave/PhotonDesign. The results show that a taper length of about 250 µm is sufficient for both C- and O-band devices.

2) Focusing 2D GCs comprise perturbing elements, which are placed along curved lines. In such a case, the feeding taper is significantly shorter (mostly between 20 µm and 40 µm), 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 GCs' 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 GC will be outlined.

**Linear 2D GCs** For linear 2D GCs, the modeling is straightforward - the 2D GC 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 FGCs) is more complex. The main challenge hides in the proper positioning of the diffracting
elements, which is not just given by the periodicity in a certain direction. In 2D FGCs, 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 FGCs, 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 FGCs were investigated at the last stage of this work, the modeling procedure is open for optimization in future works.

<figure><img src="image_17.png" /><figcaption>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).</figcaption></figure>

### 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
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 (1260-1360 nm). The stimulation is performed in $\Delta t$ time steps, with $\Delta t$ determined by the Courant condition (2.14) (typically $\Delta t \sim 10^{-17}$ s). The simulation is terminated after the signal's energy decays to a sufficiently low level (a default value is -40 dB). 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 GC, different types of field sources are applicable. Generally, a waveguide port is used for the excitation of the fundamental $TE_{00}$ 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><img src="image_18.png" /><figcaption>Figure 2.12: An exemplary Gaussian-sine excitation signal for simulations in O-band.</figcaption></figure>

Particularly challenging is the simulation of 2D GCs 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 GC'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 GCs, 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 $\le 4^\circ$) and a sufficiently fine mesh in the source and imprint projects, the error is small. The second violation of the equivalence principle is the
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 FGCs 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><img src="image_19.png" /><figcaption>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.</figcaption></figure>

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.
### 2.1.3.3 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 10 nm 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 10 nm 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><img src="image_20.png" /><figcaption>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 $(u', v', w')$ after a rotation of $45^\circ$ around the z-axis. (c) Coordinates $(u'', v'', w'')$ after a rotation of $\theta_F$ around the v'-axis. (d) The final coordinates $(u, v, w)$ after a rotation of $-45^\circ$ around the w''-axis.</figcaption></figure>

### 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 GC'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 $\phi_F$, $\theta_F$ (with a subscript F for fiber). The determination of the coordinates of a tilted evaluation plane (cf. Fig. 2.3 (b)) goes
through three steps: starting from the Cartesian coordinates as in Fig. 2.14 (a), the first step is a rotation of $\phi_1 = 45^\circ$ around the z-axis (Fig. 2.14 (b)). The coordinates after this first intermediate step are assigned as $(u', v', w')$. Next, a rotation of the angle $\theta_F$ around the v'-axis is carried out, reaching the plane in Fig. 2.14 (c) with the coordinates $(u'', v'', w'')$. Finally, a rotation $\phi_2 = -45^\circ$ 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 $\phi_F = \phi_1 = -\phi_2$, the final transform matrix is:

$$ \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \underbrace{ \begin{bmatrix} \cos^2 \phi_F \cos \theta_F + \sin^2 \phi_F & -\cos \phi_F \sin \phi_F (1 - \cos \theta_F) & -\cos \phi_F \sin \theta_F \\ -\cos \phi_F \sin \phi_F (1 - \cos \theta_F) & \sin^2 \phi_F \cos \theta_F + \cos^2 \phi_F & -\sin \phi_F \sin \theta_F \\ \cos \phi_F \sin \theta_F & \sin \phi_F \sin \theta_F & \cos \theta_F \end{bmatrix} }_{=A_T} \begin{bmatrix} x \\ y \\ z \end{bmatrix}. \quad (2.33) $$

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 $(u,v,w = \text{const.})$, we need to go through the opposite way of the transformation above. With the inverse transform matrix of $A_T$:

$$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \underbrace{ \begin{bmatrix} \sin^2 \phi_F + \cos^2 \phi_F \cos \theta_F & -\sin \phi_F \cos \phi_F (1 - \cos \theta_F) & \cos \phi_F \sin \theta_F \\ -\sin \phi_F \cos \phi_F (1 - \cos \theta_F) & \cos^2 \phi_F + \sin^2 \phi_F \cos \theta_F & \sin \phi_F \sin \theta_F \\ -\cos \phi_F \sin \theta_F & -\sin \phi_F \sin \theta_F & \cos \theta_F \end{bmatrix} }_{=A_T^{-1}} \begin{bmatrix} u \\ v \\ w \end{bmatrix}. \quad (2.34) $$

Once the results are exported and are available in the $(x,y,z)$, the transform matrix $A_T$ 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 $(u,v,w = \text{const.})$ 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 GC'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 2D GC'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 GC's arms. Otherwise, non-symmetry may result due to a 2D GC imbalanced position with respect to the waveguides. All 2D GCs in this work have the so called shear angle - a non-zero angle $\alpha$ 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
grating half-spaces, having different effective refractive indices. For the Type I 2D GC, the non-perpendicular propagation direction is caused by the angled interface between both half-spaces. For the Type II 2D GC, 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 $\vec{k}_{in}$ 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 $\varphi_{in}$ (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 $w_{\Lambda}$ and etch depth $d$ (for a given waveguide height $h_{WG}$). The propagation angles of the fields out-coupled by the grating $\varphi_{out}, \theta_{out}$ are related to the combination of a shear angle $\alpha$ and a grating period $\Lambda$ 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. $\varphi_{out} = \varphi_{F} = 45^{\circ}, \theta_{out} = \theta_{F}$.

<figure><img src="image_21.png" /><figcaption>Figure 2.15: An illustration of the parameters of 2D grating couplers (2D GCs). A shear angle $\alpha$ 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 $h_{WG}$, diffracting elements' diameter $w_{\Lambda}$ and etch depth $d$. The shear angle causes a non-zero angle of propagation $\varphi_{in}$ within the grating. The combination of $n_{eff,GC}$ and $\varphi_{in}$ determines the combination of $\alpha$ and $\Lambda$ for desired coupling angles $\varphi_{out} = 45^{\circ}, \theta_{out}$, which are equivalent to the fiber tilt angles, i.e. $\varphi_{out} = \varphi_{F}, \theta_{out} = \theta_{F}$.</figcaption></figure>

With other words, there is the dependence chain:

$$(w_{\Lambda}, d) \Rightarrow (n_{\text{eff,GC}}, \varphi_{\text{in}}) \Rightarrow (\alpha, \Lambda) \Rightarrow (\varphi_{\text{out}}, \theta_{\text{out}}). \qquad (2.35)$$
When a simulation has to be started for the first time, the question of the initial GC's geometry ($w_\Lambda, d$) occurs. The designer may choose values, which appear meaningful, e.g. $w_\Lambda$ such that no pre-defined design rules regarding permitted dimensions are violated, and etch depth $d$ around half of the total waveguide height (an empirical value). Next, desired coupling angles need to be specified, typically $\varphi_{out} = 45^\circ$ and $\theta_{out}$ in the range $4^\circ - 16^\circ$. 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 $\varphi_{in}$ 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$_2$ is considered as a cladding. For example, if we assume a waveguide height of $h_{WG} = 220$ nm, an etch depth of 120 nm (remaining Si thickness 100 nm) and a duty cycle of 0.5, the GC's effective refractive index $n_{eff,GC}$ is approximated as:

$$n_{eff,GC} \approx 0.5 \cdot [n_{eff,WG}(h_{WG} = 220 \text{ nm}) + n_{eff,WG}(h_{WG} = 100 \text{ nm})]. \quad (2.36)$$

The input angle $\varphi_{in}$ can be initially estimated using the Snell's law.

<figure><img src="image_22.png" /><figcaption>Figure 2.16: An illustration of the adaption procedure for the determination of an appropriate 2D grating coupler (2D GC) geometry for desired coupling angles ($\varphi_{out}, \theta_{out}$). First, initial diffracting elements' size and etch depth ($w_\Lambda, d$) are set. A corresponding grating effective refractive index and input angle ($n_{eff,GC}, \varphi_{in}$) are estimated. With the assumed ($n_{eff,GC}, \varphi_{in}$) and for desired coupling angles ($\varphi_{out}, \theta_{out}$), the required shear angle and grating period ($\alpha, \Lambda$) can be calculated. After a simulation with a duration in the range 5-15 hours, the actual ($\varphi_{out}, \theta_{out}$) and the corresponding ($n_{eff,GC}, \varphi_{in}$) can be determined by a spatial fast Fourier transform (FFT). If ($\varphi_{out}, \theta_{out}$) are not as desired, ($\alpha, \Lambda$) are adapted and the simulation is repeated. If ($w_\Lambda, d$) need to be changed for an efficiency optimization, the whole procedure starts all over again.</figcaption></figure>

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 ($\varphi_{out}, \theta_{out}$) can be determined. Out of the known relationship between ($\varphi_{out}, \theta_{out}$) and $n_{eff,GC}, \varphi_{in}$ (for a fixed grating shear angle and period), the latter two parameters can be calculated. If the angles ($\varphi_{out}, \theta_{out}$) are not as desired, the shear angle and period can be adapted, using
the just determined $n_{eff,GC}$, $\phi_{in}$. If ($w_A, d$) 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 $\phi_{out}, \theta_{out}$ via FFT from Fig. 2.16 will be described. Assume a general electric field, propagating in an arbitrary direction with a free-space wavelength $\lambda_0$ in a material with a refractive index $n_1$. Such a general propagation with a propagation vector $\vec{k}_{out}$ can be given by the angles in spherical coordinates $\phi_{out}, \theta_{out}$, defined as in Fig. 2.15. At a constant plane $z = \text{const.}$, the electric field has the form:

$$ \vec{E}(\vec{r}) = \vec{E}_0 \exp(\vec{k}_{out} \cdot \vec{r}) = \vec{E}_0 \exp(k_x x + k_y y) = \vec{E}_0 \exp(k_0 n_1 (\cos \phi_{out} \sin \theta_{out} x + \sin \phi_{out} \sin \theta_{out} y)), \quad (2.37) $$

with the free-space wave number $k_0 = \frac{2\pi}{\lambda_0}$. A FFT of a given electric field distribution at $z = \text{const.}$ delivers the vectors $k_x, k_y$. With the equivalence:

$$ \begin{aligned} k_x &= k_0 n_1 \cos \phi_{out} \sin \theta_{out} \\ k_y &= k_0 n_1 \sin \phi_{out} \sin \theta_{out} \\ k_x^2 + k_y^2 &= k_0^2 n_1^2 \sin^2 \theta_{out}, \end{aligned} \quad (2.38) $$

it is obvious that with known $k_x, k_y$, the propagation angles can be determined as:

$$ \begin{aligned} \varphi_{out} &= \arctan \frac{k_y}{k_x} \\ \theta_{out} &= \arcsin \frac{\sqrt{k_x^2 + k_y^2}}{k_0 n_1}. \end{aligned} \quad (2.39) $$

In our case, we assume a free-space coupling and $n_1 = 1$ (no index-matching gel considered).

#### 2.1.4.2 Coupling Efficiency

The 2D GC'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.
**Normalized Out-Coupled Power** There are several limitations for a 2D GC, 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 GC. The input power, which is used as a reference for the normalization of the out-coupled power, is assigned as $P_{WG}$. In general, the power is split into several components:

* $P_{up}$ - the part of the power, diffracted upwards. If a high diffraction efficiency is achieved, the whole power $P_{up}$ will be well directed under an angle $\theta_{out}$. The angle will be the same as the SMF's tilt angle $\theta_{F}$ for the desired central wavelength. Typically, $\theta_{out} \neq 0^{\circ}$ is chosen, otherwise the grating behaves as a second order Bragg grating with a strong back-reflection. In addition, with increasing diffraction angle $\theta_{out}$, 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 $P_{up}$ and the input power $P_{WG}$.

* $P_{down}$ - a part of the power, which is diffracted down towards the substrate.

* $P_{thru}$ - a part of the power, which propagates further within the grating and is not diffracted.

* $P_{refl}$ - 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><img src="image_23.png" /><figcaption>Figure 2.17: A schematic of the typical power distribution in a 2D grating coupler (2D GC): (a) $P_{up}$ - the upwards diffracted power - is the only contribution to the coupling efficiency. Power loss is caused by power diffracted towards the substrate $P_{down}$, 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 $P_{up}$, $P_{down}$ and $P_{thru}$. The reflection at the Si substrate and the BOX thickness have the most decisive impact.</figcaption></figure>
The ratio between $P_{up}$, $P_{down}$ and $P_{thru}$ is not only given by the grating parameters themselves, 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 GC, separating the waveguide from the Si substrate. The maximum value for $P_{up}$ 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 GC have also an impact. The latter is, however, smaller, because the reflection at the Si substrate is much stronger than the reflection at $SiO_2$, $Si_3N_4$ layers 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 GCs 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 GC designs for C- and O-band. The dependence is similar for 2D GCs. The figure shows the out-coupled power at 1550 nm and 1310 nm, depending on the BOX thickness. The refractive index of Si is 3.47 at 1550 nm 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 GC for C-Band with a period of 630 nm, a diffracting element's width of 315 nm and an etch depth of 70 nm; 2) 1D GC 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 400 nm in O-band.

<figure><img src="image_24.png" /><figcaption>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.</figcaption></figure>

**Overlap Integral** The second important factor for the coupling efficiency is the mode field overlap between the radiated 2D GC 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-