Document [original]

Berlin 2018

Polarization Multiplexed Photonic Integrated

Circuits for 100 Gbit/s and Beyond

vorgelegt von

M.Sc. Moritz Friedrich Baier

geboren in Mannheim, Deutschland

Von der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

- Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzende: Prof. Dr. Kathy Lüdge

1. Gutachter: Prof. Dr. Martin Schell

2. Gutachter: Prof. Dr. Kevin Williams

Tag der wissenschaftlichen Aussprache: 23. Juli 2018

“Science is imagination in a straitjacket." - Richard P. Feynman

“Der Mensch spielt nur, wo er in voller Bedeutung des Wortes Mensch ist, und er ist nur da

ganz Mensch, wo er spielt." - Friedrich Schiller

“It never gets easier, you just go faster.” Greg LeMond

Abstract

In this thesis, a fully integrated dual-polarization (DP) optical transmitter is proposed,

developed and demonstrated. The transmitter proposed in this work consists of two key

building blocks: a polarization rotator (PR) and two electro-absorption modulators (EAM),

laid out in a serial configuration. Furthermore, a laser source is monolithically integrated. All

devices are realized in the semiconducting material system of indium phosphide (InP) and its

related quaternary alloys (InGaAsP). The technology used for fabrication is referred to as

generic, meaning that it is not intended for fabrication of a specific device but rather to

enable as many different applications as possible, much like modern silicon processes in

electronics.

The thesis begins with the theoretical implications of optical polarization in integrated

waveguides. Known principles on how to make integrated polarization rotating devices are

discussed and it is shown that they prove to be too sensitive on typical fabrication

tolerances. It is shown that the Jones formalism, originally intended for free space optics,

can be used to describe integrated waveguides on the micro-scale. Based on this formalism,

an optimization technique is derived that lowers the sensitivity to fabrication errors by 25%

and brings down theoretical optical losses from 0.8 dB to 0.1 dB. The same approach is

used to design devices that rotate polarization not by 90°, but by 45°. Next, new device

structures are proposed for polarization resolved reception and transmission of optical

signals. The iSTOMP (integrated STOkes MaPper) uses an interferometric structure loaded

with several PRs and allows the complete characterization of the polarization ellipse of

incoming signals. The mathematical groundwork is given as well as a geometric way of

understanding the device. For data transmission, a serial DP EAM design is proposed. It

consists of two EAMs interconnected by a single 90° PR. The DP EAM makes use of the fact

that EAMs in InP typically only modulate one polarization, so that the cascade of devices

enables DP modulation. The EAMs make use of multi-quantum well (MQW) layers, thin

sheets of semiconductor that break the symmetry of the InP crystal. The electronic and

optoelectronic effects in MQW-based EAMs are studied to verify that the proposed DP EAM

can indeed be demonstrated with good performance. In particular, a polarization resolved

model of the dominating light-matter interaction, the quantum-confined Stark effect

(QCSE), is given. Finally, a design of an EAM is derived. System-level simulations of the DP

EAM are carried out to conclude the theoretical part of the thesis. It is shown that for an

implementation penalty below 1 dB, the PR has to have an extinction ratio above 16 dB.

To carry out detailed analysis of the fabricated devices, a new approach for polarization

resolved measurements is given. It makes use of the Müller/Stokes formalism and is

implemented in an experimental setup. This setup uses only optical fibers and no free space

optics, thus it is suitable for automated measurements of many devices. The polarization in

this setup is shown to be accurate within 2° across the entire Poincaré sphere and C-band.

This setup allows a fast and stable characterization of the fabricated PRs, EAMs and the DP

EAM. It is shown how the polarimetric measurement equation can be solved easily in the

fiber-based setup and how decomposition of Müller matrices can give insights into the

various devices. It is shown further that for devices with strong polarization dependence,

phase effects like the chirp parameter can be deduced from Müller measurements.

Fabricated PRs with polarization extinction ratios of up two 25 dB (rotation within ±4°

around 90°) and losses below 1 dB are achieved. The fabricated EAMs show a very strong

polarization dependence of over 20 dB, just like theory suggests. In the first fabricated

generation, electro-optic bandwidths of up to 17 GHz are measured. These devices are

capable of transmitting 39 Gbit/s. The new DP EAM is characterized and it is shown that it

can indeed modulate two distinct states of polarization. In a system experiment using

28 GBaud PAM-4 signaling, error-free transmission of 100 Gbit/s is shown over 80 km of

fiber. Finally, a fully integrated transmitter is demonstrated, comprising a distributed

feedback (DFB) laser, a 45° PR and the aforementioned DP EAM. This is the first

demonstration of a monolithically integrated transmitter PIC capable of polarization

multiplexing in InP.

Zusammenfassung

In der vorliegenden Arbeit wird ein voll integrierter, optisch doppelpolarisierter Transmitter

vorgeschlagen, entwickelt und demonstriert. Der vorgeschlagene Transmitter besteht aus

zwei grundlegenden Bausteinen: einem Polarisationsdreher (PR) und zwei

Elektroabsorptionsmodulatoren (EAM), die in einer seriellen Konfiguration

aneinandergereiht sind. Weiterhin ist eine Laserquelle monolithisch integriert. Alle

Komponenten sind im Halbleitermaterialsystem Indiumphosphid (InP) und seinen

verwandten quaternären Verbundmaterialien (InGaAsP) verwirklicht. Die zur Umsetzung

genutzte Technologie wird als generisch bezeichnet, da sie nicht für ein spezielles

Bauelement, sondern vielmehr für die Verwirklichung einer größtmöglichen Anzahl an

verschiedenen Bauelementen vorgesehen ist. Sie ist damit der modernen

Siliziumtechnologie für elektronische Komponenten wesensverwandt.

Die Arbeit beginnt mit der Diskussion der theoretischen Aspekte der optischen Polarisation

in integrierten Wellenleitern. Bekannte Prinzipien zur Realisierung integrierter

polarisationsdrehender Komponenten werden ausgeführt um zu zeigen, dass sie für

typische Fertigungstoleranzen zu kritisch sind. Es wird gezeigt dass der Jones-Formalismus,

der ursprünglich für die Freistrahloptik konzipiert war, auch für die Beschreibung

photonisch integrierter Komponenten dienlich sein kann. Mithilfe dieses Formalismus wird

eine Optimierungsstrategie abgeleitet, deren Ergebnisse die Empfindlichkeit gegenüber

Fertigungstoleranzen um 25% entspannt und die theoretischen optischen Verluste von 0.8

auf 0.1 dB senkt. Dieselbe Strategie wird genutzt um Komponenten zu berechnen, die die

optische Polarisation nicht nur um 90°, sondern auch um 45° drehen. Im Weiteren werden

photonisch integrierte Schaltkreise (PICs) vorgeschlagen, die den polarisationsaufgelösten

Empfang und die Übertragung von optischen Signalen ermöglichen. Der iSTOMP nutzt ein

Interferometer, welches mehrere Arme besitzt, die alle mit Polarisationsdrehern bestückt

sind. Die aus dem Interferometer austretenden Signale erlauben die vollständige

Bestimmung der Polarisationsellipse des eintretenden Signals. Es wird die mathematische

Beschreibung hergeleitet und außerdem eine geometrische Veranschaulichung gegeben.

Zur Übertragung polarisationsmultiplexter Signale wird schließlich ein neuer

doppelpolarisierter EAM (DP EAM) vorgeschlagen. Der DP EAM besteht aus zwei EAMs, die

mit einem 90° PR verbunden sind. Das Funktionsprinzip beruht auf der Tatsache, dass EAMs

auf InP-Basis typischerweise nur die TE Polarisation modulieren, so dass die EAM-PR-EAM

Kaskade einen doppelpolarisierten Modulator bildet. Die EAMs basieren auf einer aktiven

Schicht aus mehreren Quantentöpfen (MQW), also einer Abfolge aus dünnen

Halbleiterschichten die die Symmetrie des InP Kristalls brechen. Die elektronischen und

opto-elektronischen Eigenschaften von MQW-basierten EAMs werden untersucht um die

Realisierbarkeit des angestrebten DP EAM Konzepts zu bestätigen. Insbesondere wird ein

polarisationsaufgelöstes Modell der wichtigen Wechselwirkung zwischen Licht und Materie

in den EAMs gegeben, dem quantum-confined Stark Effekt (QCSE). Schließlich wird der

EAM selbst entworfen. Zum Abschluss des theoretischen Teils der Arbeit werden

Systemsimulationen des DP EAMs durchgeführt. Es wird gezeigt, dass für eine

Implementierungseinbuße des DP EAM von weniger als 1 dB der PR ein Auslöschverhältnis

von über 16 dB haben muss.

Zur detaillierten Analyse der hergestellten Bauelemente wird ein neuer Ansatz für

polarisationsaufgelöste Messungen aufgezeigt. Der Ansatz beruht auf dem Müller/Stokes

Formalismus und wird in einem experimentellen Messaufbau verwirklicht. Der Messaufbau

basiert dabei vollständig auf einmodigen Glasfasern und verzichtet für die

Bauteilcharakterisierung vollständig auf Freistrahloptik. Die optische Polarisation kann in

diesem Aufbau mit einer Genauigkeit von 2° über die gesamte Poincaré Sphäre und das

gesamte C-band eingestellt und ausgelesen werden. Es wird eine schnelle und stabile

Vermessung der hergestellten PRs, EAMs und des DP EAM ermöglicht. Dafür wird gezeigt,

wie die polarimetrische Messgleichung gelöst werden kann und wie die Zerlegung der

gewonnen Müller Matrizen auf die verschiedenen Bauelementeigenschaften schließen lässt.

So kann im Grenzfall der starken Polarsationsabhängigkeit zum Beispiel sogar der Chirp

Parameter durch Müller Messungen gewonnen werden. Die hergestellten

Polarisationsdreher weißen Auslöschverhältnisse von bis zu 25 dB (äquivalent zu ±4°

Genauigkeit um 90°) und Einfügeverluste von unter 1 dB auf. Die hergestellten EAMs

zeigen in Übereinstimmung mit theoretischen Überlegungen eine starke

Polarsationsabhängigkeit von über 20 dB. Die erste Generation von EAMs weißt

elektrooptische Bandbreiten von bis zu 17 GHz auf. Mit ihnen wird Datenübertragung von

39 Gbit/s gezeigt. Schließlich wird anhand des hergestellten DP EAM experimentell gezeigt,

dass er die unabhängige Modulation von zwei getrennten Polarisationszuständen erlaubt.

In einem Systemexperiment wird er verwendet um mit 28 GBaud PAM-4 Signalen eine

fehlerfreie Datenübertragung von 100 Gbit/s über 80 km Glasfaser zu zeigen. Schließlich

wird ein vollständig monolithisch integrierter Transmitter gezeigt, der aus einem DFB Laser,

einem 45° PR und dem bereits erwähnten DP EAM besteht. Damit liegt das erste

vollintegrierte Transmitter PIC für Polarsationsmultiplexing in InP vor.

Table of Contents

1 Introduction ............................................................................................................. 10

1.1 Motivation ......................................................................................................... 10

1.2 Objectives .......................................................................................................... 11

1.3 Structure of the Thesis ....................................................................................... 11

2 Polarization in Photonic Integrated Circuits ............................................................... 12

2.1 Optical Anisotropy and Birefringence ................................................................. 12

2.2 The Jones Formalism .......................................................................................... 13

2.2.1 Jones Vector and Polarization Ellipse ............................................................ 13

2.2.2 Jones Matrix ................................................................................................ 16

2.3 The Müller/Stokes Formalism .............................................................................. 16

2.3.1 Stokes Vectors and the Poincaré Sphere ...................................................... 16

2.3.2 Distance on a Sphere ................................................................................... 18

2.3.3 Müller Matrices ........................................................................................... 19

2.4 Optical Waveguides ........................................................................................... 22

2.4.1 Analogy between plane waves and guided waves ........................................ 22

2.4.2 Symmetric Waveguides ............................................................................... 22

2.4.3 Asymmetric Waveguides.............................................................................. 23

2.5 Integrated Polarization Converters ...................................................................... 26

2.5.1 Basic Principle and Single Section Device ...................................................... 26

2.5.2 Two Section Device ..................................................................................... 27

2.5.3 Transitions between Waveguides ................................................................. 28

2.5.4 Design Optimization .................................................................................... 32

2.5.5 Discussion of Final Design ............................................................................ 38

2.5.6 45° Polarization Rotators ............................................................................. 39

3 Transmitters and Receivers for PDM ......................................................................... 41

3.1 iSTOMP: Stokes Space Receiver .......................................................................... 41

3.1.1 Mathematical Description ............................................................................ 41

3.1.2 Numerical Analysis ...................................................................................... 43

3.1.3 Geometrical Analysis ................................................................................... 44

3.2 Dual Polarization Electro-Absorption Modulators ................................................ 47

3.2.1 A New DP Modulation Scheme .................................................................... 47


 
2.2 Theoretical Background of EAMs ................................................................. 48 
2.3 EAM Design ................................................................................................ 58 
2.4 DP EAM Simulations .................................................................................... 64 
Experiments ............................................................................................................. 68 
1 Experimental Methodology – Fiber Based Stokes Measurements ......................... 68 
1.1 Experimental Setup ..................................................................................... 68 
1.2 Solving the Polarimetric Measurement Equation ........................................... 76 
1.3 Müller Matrix Decomposition ....................................................................... 78 
1.4 Software Implementation ............................................................................ 79 
2 Polarization Rotator Results ................................................................................ 80 
2.1 90° Polarization Rotators ............................................................................. 81 
2.2 45° Polarization Rotators ............................................................................. 84 
3 EAM Results ...................................................................................................... 85 
3.1 DC Measurements ....................................................................................... 85 
3.2 Small Signal Measurements ......................................................................... 89 
3.3 Large Signal Measurements ......................................................................... 91 
4 DP EAM Results ................................................................................................. 92 
4.1 DC Measurements ....................................................................................... 93 
4.2 Transmission Experiments ............................................................................ 94 
5 DPEML Results ................................................................................................... 97 
5.1 DC Measurements ....................................................................................... 98 
5.2 Large Signal Measurements ....................................................................... 101 
Conclusion & Outlook ............................................................................................ 103 
1 Conclusion ...................................................................................................... 103 
2 Outlook ........................................................................................................... 103 
3 Acknowledgements ......................................................................................... 104 
Appendices ............................................................................................................ 106 

1 Introduction

1.1 Motivation

The tremendous technological leaps made by mankind in the 20th century can be traced

back to two main innovations: the transistor in 1948 and the laser in 1960 [1], [2]. Within a

few decades, the interplay of both lead to the information society as we know it today.

Sophisticated lithographic techniques enable mass production of transistor-based devices

on a scale that seemed unthinkable before. On the other hand, these devices are capable of

generating, processing and storing the gargantuan amount of data that is transmitted

through optical fiber links that span the globe. Up to 21 billion transistors are integrated on

an area of less than 10 cm² [3] in today’s commercially available devices. Integration density

still grows exponentially, as was already observed by Moore in 1965 [4]. Global internet

traffic in 2016 reached 1.2 ZB (1021 bytes) over the whole year, or an average 3.7 TB per

second (1012), with a projected annual growth of 22% until 2021 [5].

For the exponential growth of optical communications to continue, the components also

have to obey a Moore-like law. If the integration density of communication capacity does

not keep up with the total capacity, the systems will become impractical. Not only the size,

also power dissipation prohibits an approach of “just adding more boxes”. Therefore,

photonic integration needs to scale. Smit et al. established the photonic Moore’s law in [6],

observing around 20% annual growth in integration density.

Modern optical communication systems typically make use of all five physical dimensions

that photons exhibit: time, space, frequency, quadrature and polarization [7], [8]. Usually,

the latter four parameters are thought of as varying with time. Space division multiplexing

(SDM) is usually achieved via arrays of components [9], but mode multiplexing is also

getting explored [10]. Wavelength division multiplexing (WDM) makes use of the frequency

parameter. The oldest and simplest modulation scheme just switches the carrier intensity in

an on-off keying (OOK) scheme. To make use of both quadratures components, IQ

modulation can be employed. Finally, polarization division multiplexing (PDM) makes use of

the two orthogonal directions that are perpendicular to the carrier propagation direction.

At least in indium phosphide (InP) platforms, PDM schemes have evaded full integration so

far [11]–[13]. Silicon photonics (SiP) devices can make use of 2D grating couplers [14], [15]

or non-standard platforms [16] to achieve PDM, but most demonstrated SiP transceivers

also lack PDM functionality.

Classically, PDM schemes in optical communications were attributed to coherent detection

schemes [11], [13], [17]–[19]. In the last decade, however, PDM was also explored in direct

detection (DD) systems [20]–[25]. The PDM devices in this thesis all fall into the DD

category. It has been shown [20] that DD PDM schemes can operate using four-dimensional

modulation, just like coherent PDM schemes. The transmitter and receiver structures in DD

schemes tend to be much simpler, however. This makes them an attractive alternative for

fully integrated solutions, offering a potentially smaller footprint and cost.

1.2 Objectives

The first goal of this work is bringing PDM functionality to generic photonic integrated

circuits (PICs) in InP. This is enabled by bringing manufacturable polarization rotators (PR)

into the platform. As will be seen in chapter 3, a PR building block is all that is needed to

enable both Tx- and Rx-type PDM functionality.

Second, to demonstrate the capabilities that a PR brings to integrated devices, a fully

integrated PDM transmitter shall be designed, implemented and tested.

1.3 Structure of the Thesis

After this introduction, chapter 2 discusses optical polarization in PICs in general and

particularly important aspects of PRs. Furthermore, a new design methodology for

designing PRs is presented. Chapter 3 introduces new PDM-capable receiver- and

transmitter designs. They make direct use of the PRs of the chapter before. Finally, chapter

4 presents experimental results of fabricated PRs, EAMs, and PDM transmitters. For this, a

new measurement methodology is presented that characterizes devices in terms of their full

Müller matrix. Further, high-frequency experiments and full system measurements are

carried out.

2 Polarization in Photonic Integrated Circuits

The purpose of this chapter is to design integrated PRs in HHI’s generic photonic integration

technology. The theory used is the Jones formalism. The Jones formalism in turn requires

understanding of the principles of optical anisotropy and birefringence. The Müller/Stokes

formalism is also discussed in this chapter since it follows naturally from the Jones calculus

and it will be needed for experiments.

2.1 Optical Anisotropy and Birefringence

The refractive index experienced by an electromagnetic wave is an effect of the wave

interaction with the medium it propagates in ([26], chapter 5). When a crystalline medium

is anisotropic, the refractive index will therefore also be anisotropic. In the most general

case of biaxial crystals, the refractive index is different along all three spatial dimensions.

Uniaxial crystals can be described by two different indices, and isotropic materials by just

one. We will focus on the uniaxial case in the following. This is because in waveguide optics

the longitudinal field components are often negligible, effectively rendering the refractive

index along this direction negligible.

In uniaxial systems with two different indices 𝑛𝑥 and 𝑛𝑦 along the x- and y-axis,

birefringence Δ𝑛 is defined as: 𝛥𝑛=𝑛𝑥−𝑛𝑦(2.1)

It can be shown that in an anisotropic crystal, only waves oscillating along the crystal axes

can propagate ([27], p. 8f). Polarizations entering the crystal which are not aligned with its

principal axes should therefore be decomposed into the two polarizations which can

propagate. This is useful to understand the working principle of wave plates, which are

used to retard one polarization with respect to another. For a wave plate of thickness 𝑡, the

retardation 𝛿 can be expressed as:

𝛿=𝛥𝑛 𝑡

𝜆 (2.2)

where 𝜆 is the wavelength of the incident light. Wave plates with 𝛿=0.5 are commonly

referred to as half-wave plates, whereas 𝛿=0.25 signifies a quarter-wave plate.

The practical aspects of this thesis revolve around the InP/InGaAsP material system, which

forms a Zinc blende crystal. Due to the cubic symmetry of this crystal, the material system is

optically isotropic. Any optical anisotropy that is present is due to a symmetry breaking of

either the waveguide geometry (this chapter) or the electronic structure of the material (e.g.

in hetero structures, see chapter 3).

There are other material systems, however, in which the definition for birefringence as in

eq. (2.1) does not work. Most famously in sugar solutions, but also many other liquids and

organic solids, the index of refraction depends on the helicity of a light wave, rather than

the direction along a linear axis. In those systems, the birefringence becomes:

𝛥𝑛=𝑛𝑅𝐻𝐶−𝑛𝐿𝐻𝐶 (2.3)

with RHC and LHC denoting right-handed and left-handed circularly polarized light. The

effect is sometimes referred to as optical activity or circular anisotropy in literature [28],

[29]. As will be seen later in chapter 4, circular anisotropy can also be observed as an

emergent phenomenon when arranging linearly anisotropic materials at different angles.

2.2 The Jones Formalism

2.2.1 Jones Vector and Polarization Ellipse

A common way of describing the polarization of light is the Jones Formalism [29]–[33]. Let

polarized light propagate along the z-direction, with transverse components as follows:

𝐸𝑥=𝐸𝑥𝑒𝑘𝑥𝑧−𝜔𝑡; 𝐸𝑦=𝐸𝑦,0𝑒𝑘𝑦𝑧−𝜔𝑡 (2.4)

In this notation, the two amplitudes 𝐸𝑥,0 and 𝐸𝑦,0 are complex, i.e. they carry a phase and

an amplitude. Combining the two in a 2-vector yields the Jones vector of the wave:

𝑗=(𝐸𝑥

𝐸𝑦)=(|𝐸𝑥|𝑒𝑖𝜙𝑥

|𝐸𝑦|𝑒𝑖𝜙𝑦)(2.5)

In literature, the Jones vector can also be referred to as Maxwell column [31]. The electric

field vector oscillates on a trajectory in the xy-plane depending on the Jones vector. In

general, it follows an elliptic trajectory. Hence, the trajectory covered after one full optical

oscillation 𝑇=𝜆𝑐 is called the polarization ellipse. Its shape only depends on the lengths of

the two components and the relative phase in between – not on the absolute phase. The

oscillation of the electric field in the xy-plane for different Jones vectors is shown in figure

2-1.

Figure 2-1. Examples for different Jones vectors and the corresponding trajectories (blue) of the electric field in

the xy-plane. The field vector for different times within one optical period 𝑇=1𝑓=𝜆𝑐 is shown. In the case of

the first four Jones vectors, the electric field vector oscillates along a straight line, hence the polarization is

called to be linear. Circular polarization can oscillate left handedly or right handedly (fifth and sixth Jones

vector). The most general case is elliptical polarization (bottom), of which the others can be regarded as

special case.

The general polarization ellipse is shown in figure 2-2. It is constituted by two angles: 𝜓 and

𝜒. The former describes the rotation of the principal axes with respect to the x-axis. The

latter describes the ellipticity. They can both be calculated directly from the Jones vector via

(see p.8 in [29]):

𝑡𝑎𝑛2𝜓= 2|𝐸𝑥||𝐸𝑦|

|𝐸𝑥|2−|𝐸𝑦|2𝑐𝑜𝑠(𝜙𝑥−𝜙𝑦) (2.6)

The two radii of the ellipse are

𝑎2=|𝐸𝑥|2𝑐𝑜𝑠𝜓+ |𝐸𝑦|2𝑠𝑖𝑛𝜓+2|𝐸𝑥|2|𝐸𝑦|2𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜓𝑐𝑜𝑠(𝜙𝑥−𝜙𝑦)

𝑏2=|𝐸𝑥|2𝑠𝑖𝑛𝜓+ |𝐸𝑦|2𝑐𝑜𝑠𝜓−2|𝐸𝑥|2|𝐸𝑦|2𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜓𝑐𝑜𝑠(𝜙𝑥−𝜙𝑦) (2.7)

From this, the ellipticity becomes:

𝑡𝑎𝑛(𝜒)=±𝑏𝑎 (2.8)

With the sign in (2.8) being the sign of 𝜙𝑥−𝜙𝑦. The sign of 𝜒 gives the helicity or

handedness of the ellipse. For positive 𝜒, the electric field vector rotates in the xy-plane in

the mathematically positive sense (right handed).

Figure 2-2. The general polarization ellipse and its parameters. The semi-major and -minor axes are a and b,

respectively. The axes are rotated by 𝜓. The ellipticity is characterized by 𝜒. The helicity (i.e. the sense of

rotation of the electric field vector) is given by the sign of 𝜒. If one of the axes has vanishing length, the

polarization is linear. If their lengths are equal, the polarization is circular.

Since Jones vectors are coherent representations of light, they can describe interference.

The Jones vector of two light beams being superimposed can be obtained by calculating

the superposition of the two initial Jones vectors.

2.2.2 Jones Matrix

Light propagating through a medium will in general change its Jones vector. In the regime

of linear optics, this change is described by a matrix multiplication. The matrix operator

mapping one jones vector onto another usually is referred to as Jones matrix [29]–[31].

Since Jones vectors are complex 2-vectors, Jones matrices are complex 2x2 matrices. With

the thoughts and notations from 2.1, the Jones matrix of a birefringent crystal becomes:

𝐽𝛿=(𝑒𝑖𝜋𝛿 0

0 𝑒−𝑖𝜋𝛿) (2.9)

Rotating the crystal around the optical axis by an angle 𝜃 amounts to the transformation

𝐽𝛿,𝜃=𝑅(𝜃)∙𝐽𝛿∙𝑅(−𝜃) (2.10)

With 𝑅(𝜃) being the rotation matrix:

𝑅(𝜃)=(𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃

𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃) (2.11)

In this framework, matrices of the following wave plates can readily be understood:

𝐽𝜆2, 0°=(−𝑖 0

0 𝑖); 𝐽𝜆2, 45°=(0 −𝑖

−𝑖 0); 𝐽𝜆4, 0°=1

√2(1−𝑖 0

0 1+𝑖); 𝐽𝜆4, 45°=1

√2(1 −𝑖

−𝑖 1) (2.12)

Hence, a half-wave plate rotated by 45° can be regarded as a polarization converter, as it

exchanges the elements of the Jones vector (apart from adding a 90° phase term to both).

2.3 The Müller/Stokes Formalism

2.3.1 Stokes Vectors and the Poincaré Sphere

The Stokes formalism is based on two mathematical objects: Stokes vectors and Müller

matrices, which act on Stokes vectors [26], [29], [31], [34], [35]. The former describe the

state of polarization (SOP) of a given signal, the latter a medium and how it acts on the

SOP. The Stokes vector of a given optical signal can be measured with a photodetector, a

linear polarizer and a circular polarizer. Six measurements are carried out with the polarizers

in front of the detector: four with the linear polarizer and two with the circular polarizer.

The first four measurements are done with different rotations of the linear polarizer around

the optical axes: 0°, 90°, 45° and 135°. The last two measurements are done with the

circular polarizer transmitting left-handed and then right-handed polarized light. Switching

from one to the other can be achieved by flipping the circular polarizer. The six measured

powers then comprise the Stokes vector as follows:

𝑠=(𝑠0

𝑠1

𝑠2

𝑠3)=(𝑃0°+𝑃90°

𝑃0°−𝑃90°

𝑃45°−𝑃135°

𝑃𝑅𝐻𝐶−𝑃𝐿𝐻𝐶) (2.13)

Because no phase measurements need to be done, the Stokes vector is much more

convenient experimentally. Definitions with different signs for 𝑠3 also exist. We use the

present definition because it orients the polarization ellipse in the mathematically positive

sense for positive 𝑠3. It can easily be shown (p.23 in [29]) that with the definition for the

Jones vector (2.5) and 𝜙=𝜙𝑥−𝜙𝑦, s and j are related by:

𝑠=(𝑠0

𝑠1

𝑠2

𝑠3)=

(

|𝐸𝑥|2+|𝐸𝑦|2

|𝐸𝑥|2−|𝐸𝑦|2

2|𝐸𝑥||𝐸𝑦|𝑐𝑜𝑠𝜙

2|𝐸𝑥||𝐸𝑦|𝑠𝑖𝑛𝜙

)

(2.14)

Since 𝑠0 only gives the total power of a signal, it makes sense to normalize the Stokes

vector via 𝑠/𝑠0. Then, it can be shown that:

𝑠12+𝑠22+𝑠32≤1 (2.15)

In the limit of fully polarized light, the inequality becomes an equality. In this case, the triple

(𝑠1,𝑠2,𝑠3) exists on a sphere with unity radius, the Poincaré sphere [36]. Each point on the

Poincaré sphere uniquely determines the polarization ellipse and vice versa. With (2.14) and

the equations for the polarization ellipse (2.6)-(2.8), it can easily be shown that the angles

defining the polarization ellipse, 𝜓 and 𝜒, are twice the spherical coordinates of the Stokes

vector on the Poincaré sphere:

𝑠=(𝑠0

𝑠1

𝑠2

𝑠3)=( 𝑠0

𝑐𝑜𝑠2𝜓𝑐𝑜𝑠2𝜒

𝑠𝑖𝑛2𝜓𝑐𝑜𝑠2𝜒

𝑠𝑖𝑛2𝜒 ) (2.16)

Figure 2-3. Correspondence between polarization ellipse (left) and Poincaré sphere (right). The Stokes vector is

a point on the Poincaré sphere, with the spherical coordinates (2𝜓,2𝜒), so twice the angles that define the

polarization ellipse. The numerical values used for the plots are 𝜓=𝜋7 and 𝜒=𝜋6, giving 𝑠≈

(1,0.31,0.39,0.87) or 𝑗≈(0.81,0.24+0.53𝑖).

To complement the SOPs visualized in figure 2-1, the Stokes vectors of linear polarized light

along 0°, 90°, 45° and 135°, as well as left-handed and right-handed circular polarized

light are

𝑠0°=(1100); 𝑠90°=(1

−1

00); 𝑠45°=(1010); 𝑠135°=(10

−1

0); 𝑠𝑅𝐻𝐶=(1001); 𝑠𝐿𝐻𝐶=(100

−1) (2.17)

The correspondence between Poincaré sphere, polarization ellipse and Stokes vector is

illustrated in figure 2-3. As opposed to Jones vectors, superimposing Stokes vectors does

not represent the polarization that emerges by superimposing two polarizations. The Stokes

Vector of x-polarized light is (1 1 0 0), that of y-polarized light is (1 −1 0 0). The

superposition yields (2 0 0 0), which corresponds to unpolarized light of twice the

power. This can be intuitively understood since the Stokes vector does not contain

information on the absolute phase and therefore cannot describe interference. The

superposition of Stokes vectors does hold, however, if the two beams are mutually

incoherent [31].

As has been pointed out by Feynman and Vernon in 1956 [37], a geometrical

representation like the Stokes vector on the Poincaré sphere is useful not just in optics but

also throughout quantum mechanics. It can be used to describe any two level system with

two complex amplitudes (like the two elements of the Jones vector). In this case, the

geometrical representation works the same but is commonly referred to as Bloch sphere (p.

15 in [38]). A Bloch sphere then represents a so-called qubit.

2.3.2 Distance on a Sphere

In both theory and experiment, it is useful to have a metric that quantifies the difference

between two SOPs – we need such a metric in chapter 4. One can think of various ways of

quantifying the difference between two SOPs. The Euclidean metric between two Strokes

vectors does not reflect the spherical nature of the problem. It would represent “short

cuts” through the sphere, where only curved lines along the sphere exist. Another

possibility might be a metric using the differences between the two spherical angles, e.g.

√Δ𝜓2+Δ𝜒2. But it is not a physically useful measure, as can be understood at the poles of

the Poincaré sphere, i.e. at 𝜒=±45°. At this latitude, all longitudes 𝜓 give the same point,

namely the upper (lower) pole of the sphere. Then, √Δ𝜓2+Δ𝜒2 might be large even when

the SOPs to be compared almost coincide. Therefore, the metric used throughout this thesis

is the central angle.

It can be shown (see appendix D1)) that the central angle Δ𝜉 between two points on a

sphere with spherical coordinates (2𝜓1,2𝜒1) and (2𝜓2,2𝜒2) is given by:

𝛥𝜉=𝑎𝑟𝑐𝑐𝑜𝑠(𝑠𝑖𝑛2𝜒1𝑠𝑖𝑛2𝜒2+𝑐𝑜𝑠2𝜒1𝑐𝑜𝑠2𝜒2𝑐𝑜𝑠(2𝜓1−2𝜓2)) (2.18)

The total power 𝑠0 is normalized to unity

The factor 2 in front of the angles is used here so two points represent a polarization ellipse

with angles 𝜓𝑖 and 𝜒𝑖 as in chapter 2.3.1. 𝛥𝜉 can be thought as the geodesic between two

points on a sphere of unity radius. The geodesic generally is an arc segment of a great circle

around that sphere. The relationship is illustrated in figure 2-4.

Figure 2-4. Illustration of the central angle 𝛥𝜉 between two polarization states 𝑠1 and 𝑠2 on the Poincaré

sphere. The angle can be calculated with (2.18). The measure is also referred to as great circle distance or

geodesic. This is because the central angle 𝛥𝜉 is the arc connecting the two vectors along their common great

circle on the sphere. The great circle (dashed black circle) is defined as the circle around the axis 𝑠1×𝑠2. The

central angle is a mathematically well-behaved measure of the difference between two polarization states.

2.3.3 Müller Matrices

The Müller matrix acts on 𝑠 as follows:

𝑀𝑠=(𝑚00 𝑚01 𝑚02 𝑚03

𝑚10 𝑚11 𝑚12 𝑚13

𝑚20 𝑚21 𝑚22 𝑚23

𝑚30 𝑚31 𝑚32 𝑚33)(𝑠0

𝑠1

𝑠2

𝑠3) (2.19)

Just like Stokes vectors, Müller matrices are real-valued. R. C. Jones himself called the

calculus using these matrices “more powerful” than Jones matrices [35]. Even though they

are generally attributed to Hans Müller for his contributions in the 1940s [39], they were

already considered by Paul Soleillet in 1929 [40]. A Müller matrix does not necessarily have

to be unitary, as in the case of (di-) attenuation or depolarization. Diattenuation describes a

process of polarization dependent attenuation. The name stems from the fact that such an

element generally has two eigenvectors of physical significance

. These eigenpolarizations

Only eigenpolarizations with positive real eigenvalues have physical significance in the Stokes formalism.

Negative or complex eigenvalues would correspond to negative or complex power and energy.

remain unchanged by the element except for an attenuating factor, the corresponding

eigenvalue. For instance, a linear diattenuator with attenuations 𝑞 and 𝑟 acts on linear

polarized light as:

𝑀𝐷=12

(

𝑞+𝑟 𝑞−𝑟 0 0

𝑞−𝑟 𝑞+𝑟 0 0

0 0 2√𝑞𝑟𝑐𝑜𝑠(𝛿)2√𝑞𝑟𝑠𝑖𝑛(𝛿)

0 0 −2√𝑞𝑟𝑠𝑖𝑛(𝛿)2√𝑞𝑟𝑐𝑜𝑠(𝛿)

)

; 𝑀𝐷∙(1100)=(𝑞𝑞00); 𝑀𝐷∙(1

−1

00)=(𝑟

−𝑟

00) (2.20)

Retarding elements are represented by unitary matrices. The eigenvectors of retarders are

generally elliptical states of polarization. If these eigenpolarizations are linear or circular, the

corresponding element is referred to as a linear or circular retarder, respectively. Linear

retarders are by far the most common retarders, as birefringent crystals with their

birefringent axes introduce phase retardation to linear states of polarization. Elliptical

retardation can, however, emerge from the combination of several linear retarders which

are rotated with respect to one another. The matrix of a linear retarder aligned with the xy

coordinate system is (chapter 22, [26]):

𝑀𝛿=(1 0 0 0

0 1 0 0

0 0 𝑐𝑜𝑠 (𝛿) 𝑠𝑖𝑛 (𝛿)

0 0 −𝑠𝑖𝑛 (𝛿) 𝑐𝑜𝑠 (𝛿)) (2.21)

Rotation of an element around the optical axis is represented by

𝑀𝛿,𝜃=𝑅(𝜃)∙𝑀𝛿∙𝑅(−𝜃) (2.22)

The rotational matrix for Müller matrices is

𝑅(𝜃)=(1 0 0 0

0𝑐𝑜𝑠 (2𝜃) 𝑠𝑖𝑛 (2𝜃) 0

0 −𝑠𝑖𝑛 (2𝜃) 𝑐𝑜𝑠 (2𝜃) 0

0 0 0 1) (2.23)

Just like for Jones matrices, this framework of linear retardation and rotation is enough to

write down the matrices of waveplates, e.g.:

𝑀𝜆2,0°=(1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 1); 𝑀𝜆2,45°=(1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 −1); 𝑀𝜆4,45°=(1 0 0 0

0 0 0 −1

0 0 1 0

0 1 0 0) (2.24)

Since waveplates are unitary operators, they can be thought of as rotational operators on

the Poincaré sphere. A comparison of the lower right 3x3 kernel of 𝑀δ,θ with the matrix

rotating a 3D vector around an axis 𝑅

󰇍



by 𝛼 yields:

𝑅

󰇍



=(𝑐𝑜𝑠(2𝜃)

𝑠𝑖𝑛(2𝜃)

0) (2.25)

𝛼=−𝛿 (2.26)

These relations are useful to understand a linear retarder on the Poincaré sphere. They are

visually summarized in figure 2-5. The axis around which the element rotates the state of

polarization is in the 𝑠1𝑠2-plane. It is impossible to rotate the state of polarization around

the 𝑠3-axis. The axis’ angle is given directly by the rotation of the retarder around the

optical axis. The angle of rotation on the Poincaré sphere is given by the element’s phase

retardation.

𝑚𝑅=𝑚 s𝛿

s 𝛿

0= s𝛿

s 𝛿

0=𝑅

Figure 2-5. Operation principle of a linear retarder on the Poincaré sphere. A general linear retarder Müller

matrix is given at the top. The retardation is 𝛿 and the element is rotated around the optical axis by 𝜃. The

only real-valued eigenvector of the retarder is 𝑅

󰇍



. It is a linear polarization state and hence lies on the 𝑠1𝑠2-

circle. The polarization 𝑅

󰇍



is not affected by the element represented by 𝑀. Rather, 𝑅

󰇍



is the axis around which

the retarder rotates. It encloses an angle of 2𝜃 with the 𝑠1-axis. The angle by polarization is rotated is 𝛿.

Another way of looking at the rotation on the Poincaré sphere is as follows: since the

lower-right 3x3 kernel 𝑚δ,θ of 𝑀δ,θ rotates Stokes vectors around an axis 𝑅

󰇍



, applying 𝑚δ,θ

to 𝑅

󰇍



itself must have no effect: 𝑚𝛿,𝜃𝑅

󰇍



= 𝑅

󰇍



(2.27)

In other words, 𝑅

󰇍



is an eigenvector to 𝑚δ,θ with eigenvalue 1. But since the eigenvectors (or

eigenpolarizations) of linear retarders are linear polarizations, they cannot have a 𝑠3-

component. So generally, rotation around the 𝑠3-axis requires circular or elliptical retarders.

In this general case, 𝑅

󰇍



can have any direction in Stokes space. Therefore, the

eigenpolarizations of the corresponding retarders are elliptical. Their eigenpolarizations

consequently have a latitude on the Poincaré sphere 2𝜒≠0. Together with elliptical

diattenuators, their mathematics are summarized in appendix A).

The connection between Jones and Müller matrices has been derived by Gerrard and Burch

in appendix F of [31]. The derivation of the Müller matrix from a Jones matrix is exact,

whereas the opposite only exists for non-depolarizing elements and has arbitrary phase.

The analogous statement is true for Jones and Stokes vectors.

2.4 Optical Waveguides

This section outlines how optical waveguides and their polarization properties can be

understood with matrix methods. Since waveguide design and simulation usually makes use

of complex amplitudes for each mode involved, the Jones formalism is a natural choice. It

will be used for the remainder of the entire chapter.

2.4.1 Analogy between plane waves and guided waves

Although the Jones formalism is intended for plane waves, it can also be used as an

approximation for guided waves. The quality of this approximation will be discussed in

chapter 2.5. There are clear analogies between plane waves in anisotropic crystals and

guided modes in optical waveguides: the principal axes in the crystal correspond to the

polarizations of the guided modes. Birefringence in the crystal corresponds to the

difference of effective indices of the guided modes.

The polarization of a guided mode can be expressed in terms of its relative energy polarized

along the x- and y-axes:

𝑃𝑜𝑙𝑋=∫|𝐸𝑥|2𝑑𝑥𝑑𝑦

∫(|𝐸𝑥|2+|𝐸𝑦|2)𝑑𝑥𝑑𝑦; 𝑃𝑜𝑙𝑌=∫|𝐸𝑦|2𝑑𝑥𝑑𝑦

∫(|𝐸𝑥|2+|𝐸𝑦|2)𝑑𝑥𝑑𝑦 (2.28)

The angle 𝜃 relative to the x-axis of the guided mode then writes as:

𝜃=𝑎𝑟𝑐𝑐𝑜𝑠(√𝑃𝑜𝑙𝑋) (2.29)

With this definition of a mode angle, the orthogonal modes of a waveguide (like TE- and

TM-modes) also have orthogonal angles. So, just like in an anisotropic crystal, incoming

waves should be thought of as getting decomposed into the two modes, which then

propagate. Jones matrices as in 2.2.2 can be used as propagators. All simulations of mode

profiles are performed by the commercial software package MODE that is supplied by

Lumerical Solutions Inc. [41]. It uses an FDE solver. The detailed solver parameters are given

in Table 3.

2.4.2 Symmetric Waveguides

Waveguides with a symmetric cross section generally support modes that are polarized

parallel or perpendicular to the plane of symmetry. For typical integrated planar

waveguides, this means that they are polarized parallel and perpendicular to the substrate.

In literature, the mode parallel to the substrate is typically called the (quasi-) TE mode, while

the perpendicular mode is referred to as (quasi-) TM mode

[42]–[44]. So, if we define the

𝑥-axis to be parallel to the substrate, the TE mode has 𝜃=0°, while the TM mode has 𝜃=

90°.

Figure 2-6 shows the index distribution and the mode profile of the deeply etched

waveguide used in this work. Its guiding core consists of an iron doped quaternary InGaAsP

layer with a photoluminescence (PL) peak at 1.06 µm. We will refer to the PL peak

wavelength of a material simply as PL in the following. Since the core is fully etched, it is

referred to as a deeply etched waveguide. The details of the material models used in this

chapter are in appendix B). It can be seen that although the material system InP is optically

isotropic, guided modes exhibit birefringence. This birefringence is only due to the

geometry of the waveguide and is therefore called geometric or waveguide birefringence,

as opposed to material birefringence. The birefringence in this example amounts to Δ𝑛=

1.7×10−3.

2.0 µm

Figure 2-6. Left: Refractive index profile of a deeply etched waveguide on InP substrate.The guiding core is

InGaAsP with a PL of 1.06 µm. The waveguide is 2 µm wide and supports two modes. Middle: profile of the

total electric field of the guided mode with 𝜃=0°, i.e. the TE mode. Right: same profile of the TM-mode (𝜃=

90°). The TE mode has an effective index of 3.2006, while the TM mode has an effective index of 3.1989.

2.4.3 Asymmetric Waveguides

If a waveguide has no distinct symmetry, it may still support two orthogonal modes, i.e.

their angle relative to each other is still 90°. Their absolute angles, however, can take on

any value and depend on the waveguide geometry. To exploit this behavior, a waveguide

etch in the generic platform that is based on HBr wet chemistry is used. Figure 2-7 shows

the cross section of this waveguide. In the case of a waveguide width of 1.4 µm as shown,

the modal coordinate system is tilted by 38° with respect to the substrate coordinate

system. The birefringence is Δ𝑛=1.8×10−3.

An asymmetric waveguide with a modal basis tilted by an angle 𝜃 can be thought of as a

symmetric waveguide rotated around the optical axis by this angle. This enables a clear

In real dielectric waveguides, there is always some non-vanishing longitudinal component of both the electric

and the magnetic fields. The convention comes from the solutions of infinite slab waveguides, which really

have vanishing longitudinal E-field for one mode, and vanishing longitudinal H-field for the other.

path of engineering any desired Jones matrix in an integrated planar device: the retardation

is determined by the waveguide geometry (giving the birefringence) and length. Inducing

an asymmetry then enables rotation of the retarder by in principle any arbitrary angle. One

way of altering both birefringence and rotation is to change the width of the waveguide.

Since this parameter is technologically easily accessible (via the mask), all devices will be

designed using the waveguide width as the design parameter. The dependence of

birefringence and rotation on waveguide width are shown in figure 2-8.

1.4 µm

Figure 2-7. Left: Refractive index profile of an asymmetric waveguide on InP substrate. The waveguide has a

core material with a peak photoluminescence of 1.06 µm (green) and is surrounded by air (purple). The top

width is 1.4 µm. Middle: electric field profile of the “slow mode”, with an effective index of 3.1847 and 𝜃=

38°. Right: electric field profile of the “fast mode”, with an effective index of 3.1829 and 𝜃=−52°.

width

Figure 2-8. Simulations of the properties of an asymmetric waveguide (left) as a function of its width. Both the

birefringence (middle) and mode rotation (right) are shown. The rotation is shown for the mode that

approaches the TE-regime (𝜃=0) for 𝑤𝑖𝑑𝑡ℎ→∞. The other guided mode always stays orthogonal to this

mode.

To facilitate the design of the asymmetric waveguide devices, fitting formulas are used. For

the birefringence, the fit as a function of the waveguide width 𝑤 writes as:

𝛥𝑛=𝑎𝑤3+𝑏𝑤2+𝑐𝑤+𝑑

𝑤+𝑒

𝑤2+𝑓

𝑤3+𝑔 (2.30)

For the mode rotation 𝜃, the fit is: 𝜃=𝛼𝑎𝑟𝑐𝑡𝑎𝑛(𝛽𝑤+𝛾)−𝜖 (2.31)

The quality of these fits and the values of the parameters are shown in figure 2-9. With

these fits, writing down the Jones Matrix of a given piece of waveguide of length 𝐿 and

width 𝑤 becomes just a matter of inserting equations (2.30) and (2.31) into (2.10), using

𝛿=Δ𝑛L

𝜆. The result is of the form:

𝐽𝑊𝐺(𝑤)=(𝑒𝑖𝜋𝛿(𝑤)𝑠𝑖𝑛2(𝜃(𝑤))+𝑒−𝑖𝜋𝛿(𝑤)𝑐𝑜𝑠²(𝜃(𝑤))12(1−𝑒2𝑖𝜋𝛿(𝑤))𝑒−𝑖𝜋𝛿(𝑤)𝑠𝑖𝑛(2𝜃(𝑤))

12(1−𝑒2𝑖𝜋𝛿(𝑤))𝑒−𝑖𝜋𝛿𝑠𝑖𝑛(2𝜃(𝑤)) 𝑒−𝑖𝜋𝛿(𝑤)𝑠𝑖𝑛2(𝜃(𝑤))+𝑒𝑖𝜋𝛿(𝑤)𝑐𝑜𝑠²(𝜃(𝑤))) (2.32)

So, the Jones Matrix of any waveguide can be written down in closed form, at least in

principle. If the waveguide is mirrored along the y-axis, 𝜃 changes sign. In the following, we

write the Jones matrix of a waveguide as in figure 2-7 as JWG, while we write the matrix of

the waveguide mirrored along the y-axis as J′WG(i.e. JWG with 𝜃 of opposite sign). The

complexity of (2.32) makes it impractical to be handled by hand. A modern computer,

however, can compute the numerical value of the matrix for any given waveguide width

with relative ease. In particular, this calculation is in general much faster than a full

simulation in a commercial beam propagation tool like MODE, as will be shown in the next

chapter.

-7e-3 µm-3 𝛼=26.1°

b=9.23e

-2 µm-2

𝛽

=-4.48 µm-1

-4.92e-1 µm-1 𝛾=6.24

-2.01 µm 𝜖=-39.3°

-1.51 µm2

-4.46e-1 µm3

1.36

Figure 2-9. Top: Tabulated values for the fitting parameters of the asymmetric waveguide from Figure 2-7.

The analytic fits for birefringence (left) and mode rotation (right) are in good agreement for waveguide widths

from 1 to 3.5 µm. The average errors of the fits for birefringence and modal tilt are 6×10−6 and 0.14°,

respectively.

2.5 Integrated Polarization Converters

2.5.1 Basic Principle and Single Section Device

To recall the result from chapter 2.2.2, polarization conversion is achieved with a device

with δ=0.5 and 𝜃=45°. With the results from figure 2-9, it is found that 𝜃(1.34 µm)≈

45°. This in turn means Δ𝑛=1.914×10−3. With 𝛿=Δ𝑛L

𝜆=!0.5, this forces 𝐿=!𝐿𝜋=

409.2 µm at a wavelength of 1.55 µm. To verify this design, simulations are carried out

using the eigenmode expansion (EME) method in Lumerical’s MODE. Since the modes have

to be propagated, the EME package is used. The figure of merit is the polarization

extinction ratio (PER) at the output:

𝑃𝐸𝑅=𝑃𝑜𝑙𝑌

𝑃𝑜𝑙𝑋=∫|𝐸𝑦|2𝑑𝑥𝑑𝑦

∫|𝐸𝑥|2𝑑𝑥𝑑𝑦 (2.33)

The desired PER is ∞ when the input is purely x-polarized and 0 when the input is purely y-

polarized. In MODE simulations, the PER can easily be calculated by calculating the field

profile at the output for the corresponding input field. In Jones calculus, we can probe the

device matrix 𝐽𝑊𝐺 with an x-polarized input vector and calculate the output:

𝑗𝑜𝑢𝑡=𝐽𝑊𝐺∙𝑗𝑋=𝐽𝑊𝐺∙(10) (2.34)

The PER then writes as:

𝑃𝐸𝑅=|𝑗𝑜𝑢𝑡,1|2

|𝑗𝑜𝑢𝑡,0|2(2.35)

The comparison of both calculations for the 1.34 µm wide and 405 µm long device is

shown in figure 2-10, together with a cartoon of the device. The Jones calculations match

the EME simulations excellently. The results makes it evident, however, how sensitive this

device would be to fabrication errors. Even if the compositions and thicknesses of the

epitaxial layers are matched perfectly, 20 dB of PER are hard to achieve. The waveguide

width has to be exact to ~50 nm. For a PER above 10 dB, the accuracy still has to better

than ~150 nm. For the processes used in HHI’s generic InP technology, this accuracy is

impractical. The following sections will therefore explore options to relax the requirements

on the process technology.

Figure 2-10. Left: Rendered 3D cartoon of the polarization converter based on the asymmetric waveguide as

discussed in chapter 2.4.3. Right: dependence of the PER on the waveguide width deviation from the ideal

width of 1.34 µm. The dependence is calculated using Lumerical’s EME solver as well as the Jones model. For

20 dB of PER, the width deviation has to be below 56 nm.

2.5.2 Two Section Device

One way of relaxing the fabrication accuracy has been proposed by Dzibrou et al. at the

Technical University of Eindhoven [45], [46]. The idea is also filed as an US patent [47].The

core idea is to use two asymmetric waveguides instead of one. The geometry of both

should be the same, but the cross sections are mirror images of one another. As a result,

the mode rotations of the two sections become 𝜃1=−𝜃2=45°. Further, it is proposed to

use 𝛿1=0.25 and 𝛿2=0.75. Since the total retardation then amounts to 1, the proposed

device is twice as long as the device in chapter 2.5.1. The advantage of this two section

design is its decreased sensitivity to fabrication errors, as shown in figure 2-11. Again, the

device is simulated in MODE as well as using the Jones model. In the Jones model, the total

device matrix writes as:

𝐽𝑃𝐶=𝐽𝑊𝐺(𝑤,𝐿𝜋

2)∙𝐽′𝑊𝐺(𝑤,3𝐿𝜋

2) (2.36)

with 𝑤=1.34 µm and 𝐿𝜋=409.2 µ𝑚 as before. The two section design can sustain

around 130 nm of waveguide width deviation to operate above 20 dB PER. This a more

than twofold improvement over the one section design (56 nm), but still impractical for

contact lithography, which can deviate from the design by up to ±100 nm (including etch

variations). Another issue that has not been tackled yet is the insertion loss of the device.

Figure 2-11. Left: Rendered 3D cartoon of the two-section polarization converter based on the asymmetric

waveguide as discussed in chapter 2.4.3. Right: dependence of the PER on the waveguide width deviation

from the ideal width of 1.34 µm. The dependence is calculated using Lumerical’s EME solver as well as the

Jones model. For 20 dB of PER, the width deviation has to be below 131 nm. The average deviation between

the two curves is 1.7 dB. The insertion loss in Lumerical is 0.8 dB.

2.5.3 Transitions between Waveguides

2.5.3.1 Interface between Asymmetric and Symmetric Waveguides

For application of the device in a PIC, it has to be connected to regular (symmetric)

waveguides, in particular the one from chapter 2.4.2. Due to the different mode profiles

and the resulting non-ideal mode overlaps, this interface is associated with optical loss. A

correct lateral offset has to be used at the interface in order to optimize mode overlap. In

addition, the modes in the two mirrored sections have mirrored profiles. Since the mode

profiles are asymmetric (even if just slightly, see figure 2-7), they will also not perfectly

overlap. Hence, the interface between the two asymmetric sections will exhibit loss too.

Again, the correct lateral offset is necessary for optimum coupling between the two

asymmetric sections.

First, the interface between symmetric and asymmetric waveguide is considered. The key

parameter is the mode overlap, since it determines the transmission from one mode into

the other. The transmissions 𝑆21 from the symmetric waveguide modes into the asymmetric

waveguide modes are defined such that they reflect the transmission for the TE and TM

polarizations:

𝑆21,𝑇𝐸=𝑜𝑣𝑒𝑟𝑙𝑎𝑝(𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝑇𝐸,𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐0)+ 𝑜𝑣𝑒𝑟𝑙𝑎𝑝(𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝑇𝐸,𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐1)

𝑆21,𝑇𝑀=𝑜𝑣𝑒𝑟𝑙𝑎𝑝(𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝑇𝑀,𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐0)+ 𝑜𝑣𝑒𝑟𝑙𝑎𝑝(𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝑇𝑀,𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐1) (2.37)

This quantity can be computed in MODE using pre-built functions that write as:

𝑜𝑣𝑒𝑟𝑙𝑎𝑝(𝑊𝐺1𝑖−𝑡ℎ 𝑚𝑜𝑑𝑒,𝑊𝐺2𝑗−𝑡ℎ 𝑚𝑜𝑑𝑒)=

|𝑅𝑒[(∫𝐸𝑊𝐺1,𝑖−𝑡ℎ 𝑚𝑜𝑑𝑒×𝐻𝑊𝐺2,𝑗−𝑡ℎ 𝑚𝑜𝑑𝑒

∗)(∫𝐸𝑊𝐺2,𝑗−𝑡ℎ 𝑚𝑜𝑑𝑒×𝐻𝑊𝐺1,𝑖−𝑡ℎ 𝑚𝑜𝑑𝑒

∗)

∫𝐸𝑊𝐺1,𝑖−𝑡ℎ 𝑚𝑜𝑑𝑒×𝐻𝑊𝐺1,𝑖−𝑡ℎ 𝑚𝑜𝑑𝑒

∗]1

𝑅𝑒[∫𝐸𝑊𝐺2,𝑗𝑖−𝑡ℎ 𝑚𝑜𝑑𝑒×𝐻𝑊𝐺2,𝑗−𝑡ℎ 𝑚𝑜𝑑𝑒

∗]| (2.38)

The results of the mode overlap calculations are summarized in figure 2-12. We find that

optimum coupling is given when the widths at the interface obey 𝑤𝑖𝑑𝑡ℎ𝑎𝑠𝑦𝑚𝑚=

0.96𝑤𝑖𝑑𝑡ℎ𝑠𝑦𝑚𝑚−20 m with a lateral offset of -100 nm. We choose widths of the

symmetric and asymmetric waveguides of 3.5 µm and 3.34 µm, respectively. This then gives

𝑆21=0.015 dB, with negligible polarization dependence.

With the results from figure 2-12, it becomes clear that the simple designs from chapters

2.5.1 and 2.5.2 will be non-ideal with respect to loss. Wider asymmetric waveguides,

however, lose their modal tilt 𝜃. To make use of the more advantageous wider waveguide

interfaces, while still exploiting the asymmetric waveguide in a strongly tilted regime, tapers

become necessary. On the other hand, tapers break the design flow for the one- and two-

section devices that were discussed so far. Since tapers introduce a modal tilt 𝜃(𝑧) that is

not constant and certainly not 45°, the operation principle becomes problematic. We

discuss how to design working devices with 𝜃≠45° and even 𝜃(𝑧)≠𝑐𝑜𝑛𝑠𝑡 in chapter

2.5.4. First, the actual tapers have to be simulated and designed.

neg.

offset

width symm.

width asymm.

a) b) c)

d) e) f)

Figure 2-12. Investigation of the coupling efficiency from the asymmetric (a) to the symmetric (b) waveguide.

𝑆21 is calculated for widths ranging from 1.5 µm to 4.0 µm in 20 nm increments for both waveguides. At

each combination of the two widths, lateral offsets along the x-axis ranging from -300 nm to 100 nm are

computed in 25 nm increments (c) for a particular combination of widths). An exaggerated illustration of a

negative offset is shown in d). 𝑆21 is maximum for an offset of -100 nm for all width combinations that are

along the pink line in the two bottom colormaps. e) and f) show the dependence of 𝑆21 versus the two

waveguides’ width. The inset pink line is a linear fit through the width combinations that yield the highest 𝑆21.

It is the same for TE and TM and follows 𝑤𝑖𝑑𝑡ℎ𝑎𝑠𝑦𝑚𝑚=0.96𝑤𝑖𝑑𝑡ℎ𝑠𝑦𝑚𝑚−20 𝑛𝑚.

Since simulating the tapers requires accurate modelling of the light propagation,

Lumerical’s FDTD [41] package is used. The results are shown in figure 2-13. With this, a

choice for the taper parameters is made: 100 µm long tapers on both sides of the interface

leading up to it the transition. On the symmetric side, the interface width is 3.34 µm, on

the asymmetric side it is 3.5 µm.

Figure 2-13. FDTD simulations of the tapered transition between symmetric and asymmetric waveguide (see

cartoon on the left). The transmission is plotted (right) for a wavelength of 1.55 µm. The widths at the

interface are 3.5 µm for the symmetric and 3.34 µm for the asymmetric waveguide, respectively. The starting

widths are 1.3 µm for the asymmetric and 2.0 µm for the symmetric waveguide, respectively. The dashed line

at 0.015 dB indicates the pure mode overlap at the interface, which is the upper limit for the taper

transmission. Making either of the sections longer than 100 µm has negligible impact on the overall

transmission.

2.5.3.2 Interface Between two Mirrored Asymmetric Sections

For a two-section design, the interface between the two asymmetric waveguides is also

important. Like in the previous chapter, tapered transitions reduce the loss significantly. This

adds more flexibility to the polarization converter design: when the two sections are joined

by tapered transitions, they do not necessarily have to be of the same width. Without

tapers, unequal widths would increase the loss due to mode mismatch. Using the same

methodology as previously in 2.5.3.1, a summary of the simulations of the asymmetric-

asymmetric interface is presented in figure 2-14. The optimal lateral offset is exactly twice

the offset of the symmetric-asymmetric case, i.e. 200 nm.

It can be seen that in the absence of a tapered transition and an asymmetric waveguide

width of 1.3 µm, the interface gives around 0.3 dB loss even at optimum offset. When

moving to tapered transitions, however, the choice of parameters is constrained by the

overall performance of the polarization converter. As will be seen in the next chapter, a

taper that is too long degrades the performance of the overall design. Therefore, the length

of the transitions is limited to two tapers with 25 µm each. With figure 2-15, the width at

the interface is then chosen to be 2.5 µm, giving around 0.05 dB total loss for this

transition.

offset

width asymm.

Figure 2-14. Investigation of the coupling efficiency between two mirrored asymmetric waveguides (as shown

on the left). The colormap (right) shows 𝑆21 for different waveguide widths and lateral offsets. The two

mirrored waveguides always have the same width.

width asymm.

Figure 2-15. FDTD Simulations of the transmission from one tapered asymmetric waveguide into its mirrored

counterpart. For different lengths, there are different optimum interface widths. Lowest loss is given by long

tapers with wide interfaces. If one wishes to constrain the taper length, however, the interface width should

not be too wide.

2.5.4 Design Optimization

2.5.4.1 Two Section Design without Tapers

With the results of the chapters 2.5.3.1 and 2.5.3.2, the goal is to design a polarization

converter that has minimal excess loss. This requires waveguides much wider than what

gives 𝜃=45°. At the same time, even the two-section design from 2.5.2 is very hard to

fabricate with the process variations that occur in HHI’s photonic integration technology. A

new concept is therefore needed, to achieve polarization conversion with 𝜃(𝑧)≠𝑐𝑜𝑛𝑠𝑡 and

in particular 𝜃(𝑧)≠45°. A first step is to consider a two-section device with different

waveguide widths 𝑤1≠𝑤2 in the two sections (each of lengths 𝐿1≠𝐿2). If we further

neglect any tapered transitions and loss, the Jones Matrix of the complete device writes as:

𝐽𝑃𝐶=𝐽𝑊𝐺(𝑤1,𝐿1)∙𝐽′𝑊𝐺(𝑤2,𝐿2)=!(0 1

1 0) (2.39)

Figure 2-16. Flow diagram of the numerical design optimization. The fit formulas (2.30) and (2.31) for 𝛥𝑛

and 𝜃 versus width together with 𝛿=𝛥𝑛𝐿

𝜆 define the behavior of a device in terms of its section widths and

lengths. A python script runs an optimization of this dependency to find the device with the highest PER

across a given width deviation 2𝛥𝑤. The output are the numerical values of the two widths and two lengths

of the respective sections.

To find an (approximate) solution to (2.39), it is numerically implemented in python [48].

Python is a scripting language that comes with many extensions that are useful for scientific

computing [49]. A flow diagram of the functionality of the python model is in figure 2-16.

In this model, all the relevant dependencies like (2.30) and (2.31) are included. The result is

a function that computes JPC(w1,L1,w2,L2). To perform optimization of the process

tolerance, a target function is defined that should be minimized. The target function 𝑓 of a

given device JPC(w1,L1,w2,L2) is defined as:

𝑓(𝑤1,𝐿1,𝑤2,𝐿2)= ∑ 𝐸𝑟𝑟(𝑤1+𝛥𝑤,𝐿1,𝑤2+𝛥𝑤,𝐿2)

𝛥𝑤𝑚𝑎𝑥

𝛥𝑤=−𝛥𝑤𝑚𝑎𝑥 (2.40)

The error function Err gives a quantitative measure of how much polarization is converted

by the device: 𝐸𝑟𝑟(𝑤1+𝛥𝑤,𝐿1,𝑤2+𝛥𝑤,𝐿2)=|𝑗𝑜𝑢𝑡,2| (2.41)

with 𝑗𝑜𝑢𝑡=𝐽𝑃𝐶∙𝑗𝑋=𝐽𝑃𝐶∙(10) (2.42)

An ideal polarization converter will transfer all energy from the first Jones’ element to the

second. Then, if the converter does not exhibit any fabrication error, 𝛥𝑤=0 and 𝐸rr=0.

For 𝛥𝑤≠0, 𝐸rr will generally grow. The faster it grows, the less fabrication errors can be

sustained before the PER suffers. This is expressed in large values for 𝑓. So, the goal

becomes to find the minimum value for 𝑓 in the (w1,L1,w2,L2) space.

Python’s SciPy package offers a whole library to tackle minimization problems of scalar

functions like 𝑓. For the following calculations, the Nelder-Mead method is used [50]. The

result of the optimization when optimizing across 2Δ𝑤=250 m is given in figure 2-17.

The entire optimization takes less than seven seconds. During the optimization process, 801

different device designs are calculated. For each design, 20 different width deviations are

evaluated to calculate the tolerance. With the EME solver on the other hand, just

calculating the tolerance of the single optimal design takes over two hours with a

granularity of 140 different width deviations. Hence, the Jones model is almost 10,000

times faster. figure 2-18 shows how well the design matches EME simulations.

Figure 2-17. Left: Result of the optimization implemented using scipy. The target function (2.40) is evaluated

up to 𝛥𝑤𝑚𝑎𝑥=125 𝑛𝑚. The optimized parameters are in the yellow inset. 20 dB PER are maintained across

width deviations of up to 177 nm. The widths of the short and long sections correspond to 𝜃1=38° and 𝜃2=

40°, respectively. On the right is the two-section design from 2.5.2 that can sustain 135 nm of width

deviation. For the optimized design, the two sections are not of equal width and also some 50 nm wider than

in the design that uses 𝜃1=𝜃2=45°. It has 0.7 dB loss (the un-optimized design has 0.8 dB).

A direct comparison between the two designs (𝜃≠45° and 𝜃=45°) shows clear

advantages for the optimized design. Across ±100 nm of width deviation, the optimized

design has 4 dB higher average PER (25.2 versus 21.4 dB). Because the waveguides are

wider and the mode overlaps are higher for wider waveguides, the loss is also slightly lower

(0.7 versus 0.8 dB). Most importantly, the optimized design is the proof that designs with

𝜃≠45° exist at all. It should therefore be possible to achieve similar PER values with

tapered designs. As was shown in section 2.5.3, this can severely reduce the insertion loss.

Figure 2-18. Comparison of the predictions of the Jones Model and the EME solver. The standard deviation of

around 1.4 dB in PER indicates that the Jones Model also works for devices operating with 𝜃≠45°. The

calculation of the 300 points in the Jones curve takes 200 msec on a personal computer. The 140 EME

simulations for the other curve take over around 140 minutes on the same machine.

The deviations between Jones model and eigenmode expansion in figure 2-18 amount to

an average of around 1.5 dB PER. This can be explained by the fact that the x- and y-

components of the mode fields are not homogeneously distributed. In particular, the

quantities 𝐸𝑥

𝐸𝑥+𝐸𝑦 and 𝐸𝑦

𝐸𝑥+𝐸𝑦 are not constant across the mode. This will generally lead to

coupling between the modes at waveguide interfaces that differs from the case of

completely homogenous mode distributions. But since Jones matrices rely on plane waves,

they can not reflect this inhomogeneity.

2.5.4.2 Two Section Design with Tapers

To take advantage of the considerations in 2.5.3, the Jones model needs adaptation. It is

clear that the overall loss can be significantly reduced by using tapers. The impact on the

actual polarization conversion is unclear, though. Since both 𝛿 and 𝜃 change along z, they

cannot be described by a simple matrix as (2.32). To analyze the achievable conversion

with tapered sections, they are modelled as a series of discrete elements as (2.32), and

then the matrix product of all elements gives the taper matrix. Figure 2-19 shows the

convergence of the effective taper matrix versus the number of discretization steps. We will

use five discrete sections in the following to model the tapers.

win

wout

Figure 2-19. Convergence of polarization evolution versus taper discretization. The input polarization is linearly

polarized at 45°, i.e. 𝜓=45° and 𝜒=0°. The two taper variants are the ones that are relevant for the

tapered polarization converter design. The predicted output polarization converges for both variants.

With the results from chapter 2.5.3, a set of taper parameters at the in- and outputs of the

PR is chosen: widths of 3.34 µm and 3.5 µm for the asymmetric and symmetric waveguide,

respectively. The length for each tapered waveguide is 100 µm. With these parameters

fixed, the length of the middle interface (asymmetric-asymmetric) is investigated. For low

loss, a longer taper is desirable. As can be seen in figure 2-20, however, the length of this

taper impacts the PER performance of the optimum design. Therefore, the middle taper

length is restricted to 25 µm in length. For this length, the lowest loss is achieved with an

interface width of 2.5 µm. This choice acts as a boundary condition for the optimization of

the actual design and the values are summarized and illustrated in figure 2-20. These

parameters will give 2x 0.015 dB loss for the asymmetric-symmetric interfaces and 0.07 dB

for the symmetric-symmetric interface, so 0.1 dB total loss due to the transitions. This is a

eight-fold reduction compared to the 2x 0.25 dB plus 0.3 dB in the absence of tapers.

𝑤𝑡𝑎𝑝𝑒𝑟, ,s mm

3.5 µm

𝑤𝑡𝑎𝑝𝑒𝑟, , s mm

3.34 µm

𝑤𝑡𝑎𝑝𝑒𝑟,𝑚𝑖𝑑

2.5 µm

𝐿𝑡𝑎𝑝𝑒𝑟, ,s mm

100 µm

𝐿𝑡𝑎𝑝𝑒𝑟, , s mm

100 µm

𝐿𝑡𝑎𝑝𝑒𝑟,𝑚𝑖𝑑

25 µm

Figure 2-20. Left: dependence of the PER of the optimal PR versus the middle taper length. Beyond 25 µm

taper length, the PR performance becomes worse than a non-tapered design. The tapered PR is rendered in

the center (not to scale). The parameters for the tapered PR design are summarized in the right table.

With the choice of taper parameters, an optimal rotator design is derived. The optimization

takes some 10 seconds on a personal computer, i.e. around one order of magnitude longer

than the non-tapered optimizations. This is a natural consequence from the additional

Jones matrices that are computed for each discretized taper element. The final design and

its tolerance are given in figure 2-21.

Figure 2-21. Comparison of the predictions of the Jones Model and the EME solver. The design is numerically

optimized and comprises tapered transitions at all junctions. The average deviation between Jones Model and

Lumerical is less than 1 dB. The calculation of the 300 points in the Jones curve takes 200 msec on a personal

computer. The 140 EME simulations for the other curve take over around 140 minutes on the same machine.

The device exhibits 0.1 dB loss.

2.5.5 Discussion of Final Design

The performance of the final design from section 2.5.4.2 is summarized in Table 1. To easily

compare it with the other designs, a figure of merit (FOM) is defined. For each design, the

average 𝑃𝐸𝑅𝑎𝑣 across a width deviation of ±100 nm is calculated. Together with the

insertion loss, we then define:

𝐹 𝑀=𝑃𝐸𝑅𝑎𝑣

𝑙𝑜𝑠𝑠 (2.43)

Using this measure, the untapered design operating at 𝜃≠45° gives a 30% improvement

over the 𝜃=45° design from literature. The final tapered design gives a 800%

improvement, mostly due to the achieved loss reduction.

Table 1. Performance comparison of different PR designs. The figure of merit is the average PER across

±100 nm of width deviation over insertion loss. The Jones-based optimization gives a 30% improvement for a

2-section design without tapered transitions. With tapered transitions, another 590% are achieved, or 800%

in total. The tapered design is the final PR design.

To further estimate the manufacturability of the final design, its dependence on parameters

other than waveguide width is analyzed: waveguide core thickness, waveguide core

composition and photon wavelength.

HHI’s epitaxial layers are specified with ±10 nm error for the photoluminescence, which is

an effective measure for the accuracy of the layers’ composition. The thickness variation is

around ±5%. As can be seen in figure 2-22, the PER stays above 15 dB within the epitaxial

accuracy in the case of no width deviation.

Figure 2-22. PER dependence on epitaxial errors of the waveguide core. Left: dependence on

photoluminescence (PL). The core PL is specified by 1.06 µm ± 10 nm. The specified deviation is indicated as

dashed lines. Right: dependence on the core thickness, which is specified by 1.045 µm ± 5%, again indicated

by the dashed lines. Both plots show the behavior for width deviations of -100, 0 and 100 nm.

All previous calculations are done for a wavelength of 1.55 µm. The mode solver model,

however, includes proper material dispersion. Also, even in the absence of material

dispersion, the modes’ shape and therefore tilt and birefringence depend on wavelength.

The impact of those circumstances is summarized in figure 2-23.

Figure 2-23. Wavelength dependence of the final design PER. The dependence is shown for width deviations

of -100, 0 and 100 nm.

2.5.6 45° Polarization Rotators

So far, only devices that fully convert between TE and TM are considered. Because they

rotate the respective polarizations by 90°, they are referred to as 90° PRs. For some

applications however, it may be desirable to rotate polarization by only 45°. Effectively, this

converts half the power from TE to TM and vice versa. Just as a 90° rotator can be thought

of as a half waveplate under a 45° angle, a 45° rotator can be though as a quarter

waveplate under the same angle. We use the same design methodology as for the 90°

rotators, but aim for 0 dB PER instead of maximum PER. Using the same taper configuration

as in figure 2-20, a design is derived and shown in figure 2-24.

Figure 2-24. PER of a 45° polarization rotator according to the Jones model (purple) and Lumerical’s MODE

(green). It is a tapered design with the same taper parameters as the 90° rotator. The widths and lengths are

given in the yellow inset. The average deviation between Jones and EME is 0.22 dB, the device operates

within ±0.5 dB PER across width deviations of 330 nm. The simulated insertion loss in MODE is 0.1 dB.

3 Transmitters and Receivers for PDM

3.1 iSTOMP: Stokes Space Receiver

With the previous chapter, a complete design methodology for arbitrary polarization

rotators in planar integrated circuits is given. In this chapter, an application example that

makes use of the rotators is discussed: iSTOMP (integrated STOkes MaPper). It is a PIC

designed for the measurement of Stokes vectors.

As pointed out by Augustin in [51], polarization rotators can be placed in Mach-Zehnder

interferometers (MZI) to yield polarization beam splitters (PBS). The basic concept is

illustrated in figure 3-1. In a DD scheme, however, a PBS does not enable the retrieval of

the input state of polarization. The information on the relative phase between X and Y

polarization is lost. To overcome this limitation, the classical 2-arm MZI configuration is

expanded to an N-arm configuration in this chapter. The device also has N outputs, each of

which the power is to be measured. The following will show how the Stokes vector of

incident light can be deduced from the powers exiting the device and what N needs to be.

2x2

MMI

2x2

MMI

Figure 3-1. 2-way Mach-Zehnder configuration for polarization beam splitting as proposed by Augustin [51].

The blue waveguides in the MZI are birefringent, so that the relative position of the two PRs changes the

overall interference condition for TE and TM light. By choosing the right position, the two output powers are

proportional to the powers in the two input polarizations, respectively.

3.1.1 Mathematical Description

Since the device to be discussed is based on interference, it cannot be understood in the

Stokes formalism but only in Jones terminology. To describe the interference of polarized

waves, one may simply add the two corresponding Jones vectors. Consider a configuration

as in figure 3-2, with light incident to the top waveguide on the left.

Figure 3-2. Schematic of the proposed receiver structure. An N-arm MZI is loaded with polarization rotators

(PR) at different relative positions in each arm. Each arm has the same total length and each PR is equal. The N

output powers can be described as the norms of the respective Jones vectors. The powers depend on the

polarization at the input, i.e. 𝑗𝑖𝑛.

The Jones vector at the 𝑖-th output is then given by:

𝑗𝑜𝑢𝑡,𝑖=(𝐽𝑀𝑀𝐼,1→𝑖∙𝐽1,2∙𝐽𝑃𝑅∙𝐽1,1∙𝐽𝑀𝑀𝐼,1→1+⋯+𝐽𝑀𝑀𝐼,𝑁→𝑖∙𝐽𝑁,2∙𝐽𝑃𝑅∙𝐽𝑁,1∙𝐽𝑀𝑀𝐼,1→𝑁)∙𝑗𝑖𝑛 (3.1)

where 𝐽𝑀𝑀𝐼,𝑖→𝑗 denotes the Jones matrix describing the transfer from the 𝑖-th input to the 𝑗-

th output of an MMI. The general phase relations 𝜙𝑖𝑗 of a NxN MMI between 𝑖 and 𝑗 are

given by Paiam et al. in [52]:

𝜙𝑖𝑗=𝜙1−𝜋2(−1)𝑖+𝑗+𝑁+𝜋

4𝑁(𝑖+𝑗−𝑖2−𝑗2+(−1)𝑖+𝑗+𝑁(2𝑖𝑗−𝑖−𝑗+12)) (3.2)

with some offset phase 𝜙1 which is given in [52]. Then, one can write:

𝐽𝑀𝑀𝐼,𝑖→𝑗=𝑒𝑖𝜙𝑖𝑗

√𝑁(1 0

0 1) (3.3)

The 𝐽𝑘,1 and 𝐽𝑘,2 in the 𝑘-th arm are simple birefringent elements and therefore write as:

𝐽𝑘,𝑙=(𝑒𝑖𝜋𝛿𝑘,𝑙 0

0 𝑒−𝑖𝜋𝛿𝑘,𝑙) (3.4)

We note that δk,1+δk,2=𝑐𝑜𝑛𝑠𝑡 for all 𝑘. This is true because the total length of all arms is

equal. Finally, the polarization rotator is

𝐽𝑃𝑅=𝑅(𝜃𝑃𝑅)∙𝐽𝛿𝑃𝑅∙𝑅(−𝜃𝑃𝑅) (3.5)

with the definition for rotation as in section 2.2.2. We may write the total Jones matrix

from the input to the 𝑖-th output as

𝑗𝑜𝑢𝑡,𝑖=𝐽𝑖∙𝑗𝑖𝑛; 𝐽𝑖=∑𝐽𝑀𝑀𝐼,𝑘→𝑖∙𝐽𝑘,2∙𝐽𝑃𝑅∙𝐽𝑘,1∙𝐽𝑀𝑀𝐼,1→𝑘

𝑁

𝑘=1 (3.6)

So, we can write the powers exiting the device as

𝑃𝑖=|𝐽𝑖∙𝑗𝑖𝑛|2 (3.7)

To continue, we encapsulate the elements of Ji in parameters defined as

(𝑎𝑖𝑏𝑖

𝑐𝑖𝑑𝑖)∶=𝐽𝑖 3.8

We can now analyze the powers further:

𝑃𝑖=|𝐽𝑖∙𝑗𝑖𝑛|2=|𝑎𝑖𝐸𝑥+𝑏𝑖𝐸𝑦|2+|𝑐𝑖𝐸𝑥+𝑑𝑖𝐸𝑦|2=|𝑎𝑖𝐸𝑥|2+|𝑏𝑖𝐸𝑦|2+𝑎𝑖𝐸𝑥𝑏𝑖𝐸𝑦+𝑎𝑖𝐸𝑥𝑏𝑖𝐸𝑦+⋯

=|𝑎𝑖𝐸𝑥|2+|𝑏𝑖𝐸𝑦|2+2𝑅𝑒{𝑎𝑖𝐸𝑥𝑏𝑖𝐸𝑦}+|𝑐𝑖𝐸𝑥|2+|𝑑𝑖 𝐸𝑦|2+2𝑅𝑒{𝑐𝑖𝐸𝑥𝑑𝑖𝐸𝑦} (3.9)

We define 𝜖𝑖,1∶=𝑎𝑟𝑔{𝑎𝑖𝑏𝑖} and 𝜖𝑖,2∶=𝑎𝑟𝑔{𝑐𝑖𝑑𝑖} and recall Re{𝑧}=|𝑧| s (𝑎𝑟𝑔{𝑧}) as well

as 𝑎𝑟𝑔{𝑧+𝑤}=𝑎𝑟𝑔{𝑧}+𝑎𝑟𝑔{𝑤}. The relative phase of the input polarization 𝑎𝑟𝑔{𝐸𝑦

𝐸𝑥}=

𝜙 is defined as before. Then:

𝑃𝑖=|𝑎𝑖𝐸𝑥|2+|𝑏𝑖𝐸𝑦|2+2|𝐸𝑥𝐸𝑦||𝑎𝑖𝑏𝑖|𝑐𝑜𝑠(𝜖1−𝜙)+|𝑐𝑖𝐸𝑥|2+|𝑑𝑖𝐸𝑦|2+2|𝐸𝑥𝐸𝑦||𝑐𝑖𝑑𝑖|𝑐𝑜𝑠(𝜖2−𝜙)(3.10)

We use s(𝛼−𝛽)= s(𝛼) s(𝛽)+s (𝛼)s (𝛽) and start regrouping:

𝑃𝑖=|𝑎𝑖𝐸𝑥|2+|𝑏𝑖𝐸y|2+|𝑐𝑖𝐸𝑥|2+|𝑑𝑖𝐸y|2+

2|𝑎𝑖𝑏𝑖||𝐸𝑥𝐸𝑦|( s(𝜖i,1) s(𝜙)+s (𝜖i,1)s (𝜙))+2|𝑐𝑖𝑑𝑖||𝐸𝑥𝐸𝑦|( s(𝜖i,2) s(𝜙)+s (𝜖i,2)s (𝜙))

=(|𝐸𝑥|2+|𝐸y|2)(|𝑎𝑖|2+|𝑏𝑖|2+|𝑐𝑖|2+|𝑑𝑖|2)

2+(|𝐸𝑥|2−|𝐸y|2)(|𝑎𝑖|2−|𝑏𝑖|2+|𝑐𝑖|2−|𝑑𝑖|2)

+2|𝐸𝑥𝐸𝑦| s(𝜙)(|𝑎𝑖𝑏𝑖| s(𝜖i,1)+|𝑐𝑖𝑑𝑖| s(𝜖i,2))+2|𝐸𝑥𝐸𝑦|s (𝜙)(|𝑎𝑖𝑏𝑖|s (𝜖i,1)+|𝑐𝑖𝑑𝑖|s (𝜖i,2)) (3.11)

This is the form needed to substitute the Stokes elements using (2.14):

𝑃𝑖=𝑠0(|𝑎𝑖|2+|𝑏𝑖|2+|𝑐𝑖|2+|𝑑𝑖|2)

2+𝑠1(|𝑎𝑖|2−|𝑏𝑖|2+|𝑐𝑖|2−|𝑑𝑖|2)

So, we find that the output powers are linear in the incident Stokes vector. We may write

the output powers in an N-vector that results from some linear operation 𝑍 on the Stokes

vector. The operation can be represented by an Nx4 matrix:

𝑧𝑖0=(|𝑎𝑖|2+|𝑏𝑖|2+|𝑐𝑖|2+|𝑑𝑖|2)

2 ; 𝑧𝑖1=(|𝑎𝑖|2−|𝑏𝑖|2+|𝑐𝑖|2−|𝑑𝑖|2)

𝑍=[𝑧𝑖𝑗] (3.13)

Then we may simply write:

(𝑃1

…

𝑃𝑁)=𝑃

󰇍



=𝑍𝑠 (3.14)

Finally, we take the (pseudo-) inverse 𝑍−1:

𝑠=𝑍−1𝑃

󰇍



(3.15)

If 𝑍 is of rank 4, this equation opens the path to using the device as a polarimeter and/or

Stokes space receiver. It enables the determination of any arbitrary input polarization by

measuring the output powers. By measuring all four Stokes parameters, the device also

measures the total power of the incident signal.

3.1.2 Numerical Analysis

A closed analytical description of the equations involved in the proposed polarimeter

scheme is hardly possible. Numerically, however, 𝑍 can be calculated. This is useful to study

the conditions under which 𝑍 is of rank 4 and the device is therefore fully functional and

linear.

First, N has to be determined. Since 𝑍 needs to be of rank 4, we immediately note that N

should be at least 4. The equations of the previous chapter are implemented in python and

parametrized as follows: the polarization rotator is characterized by its retardance 𝛿𝑃𝑅 and

rotation 𝜃𝑃𝑅. The total retardance in each arm of the MZI is denoted 𝛿𝑀𝑍. The retardances

before and after the PR in the 𝑖-th arm are 𝛿𝑖,1 and 𝛿𝑖,2, respectively. They are written as

𝛿𝑖,1=𝑅𝑖𝛿𝑀𝑍 and 𝛿𝑖,2=(1−𝑅𝑖)𝛿𝑀𝑍, so that each arm is fully characterized by only the ratio

𝑅𝑖 of the lengths.

No set of parameters is found for 𝑁<5 that yields a 𝑍 of rank 4. For 𝑁≥5, however,

many solutions exist. It is found that the rank of 𝑍 only depends on the ratios 𝑅𝑖, not on the

total birefringence 𝛿𝑀𝑍 or the polarization rotator parameters 𝛿𝑃𝑅 and 𝜃𝑃𝑅. This is a rather

striking result because the ratio of the length of two waveguides can be controlled with

high accuracy. We find that for 𝑁≥5, the set of 𝑅𝑖 needs to contain at least three different

values, so that 𝑍 is of rank 4. 𝜃𝑃𝑅 must not be an integer multiple of 𝜋2, i.e. the PR axes

should not be aligned with the device substrate.

5x5

MMI

5x5

MMI

Figure 3-3. Working principle of the proposed Stokes receiver in the case N=5. The top left shows a cartoon

of the receiver arrangement. The five plots show the dependence of the five output powers on the input

polarization.

3.1.3 Geometrical Analysis

3.1.3.1 N=3

Even though for N=3 𝑍 is not of rank 4, there is still an interesting mapping between 𝑠 and

𝑃

󰇍



that can be understood geometrically. Because 𝑃1+𝑃2+𝑃3=𝑐𝑜𝑛𝑠𝑡 and 𝑃𝑖>0, 𝑃

󰇍



exists

on an eighth of the surface of an octahedron, i.e. a triangle. Curiously, we find that the set

of all 𝑃

󰇍



is a smaller set than the triangle spanned by the octahedron. Rather, it exists on an

ellipse within that triangle. The mapping for a particular MZI configuration is illustrated in

figure 3-3. The mapping can be thought of as a projection of a 3D sphere onto a 2D plane.

Since the depth information is lost this way, one cannot tell if the measured point

corresponds to the front of the sphere or the back. Hence, each measurable point

corresponds to two distinct polarizations. This is why 𝑍 is not rank 4.

 =  ,

3x3

MMI

3x3

MMI

=  ,

, , ,

Figure 3-4. Geometrical interpretation of the working principle of a 3-arm iSTOMP. The PRs in each arm have

𝛿𝑃𝑅=𝜃𝑃𝑅=𝜋/8, the ratios are 0.25, 0.5, 0.75. The surface of the Poincaré sphere gets mapped to an ellipse

in the space spanned by the three output powers. The ellipse itself exists on the triangular plane defined by

𝑃1+𝑃2+𝑃3=1, i.e. the energy conservation condition (the input power is 1). The ellipse corresponds to the

projection of the Poincaré sphere onto the triangle, viewed under some angle. Therefore, the 3-arm iSTOMP

does not yield a bijection.

3.1.3.2 N>3

To be able to visualize iSTOMP of full rank as the one in figure 3-3, the space of 𝑃

󰇍



inconvenient since it has five dimensions. To get around this, we take inspiration from the

original definition for the Stokes vector. We recall that six powers define a Stokes vector 𝑠,

but the SOP can still be understood as a three dimensional point. We can define an

analogous vector 𝑠, whose elements are differences between the N powers 𝑃𝑖. A

completely analogous example would be an iSTOMP with N=6 and the definition:

𝑠=(𝑠0

𝑠1

𝑠2

𝑠3)=(𝑃1+𝑃2+𝑃3+𝑃4+𝑃5+𝑃6

𝑃1−𝑃2

𝑃3−𝑃4

𝑃5−𝑃6) (3.16)

We can generalize this expression using some 𝑞𝑘,𝑖∈[−1,1]

𝑠=

(

∑𝑃𝑖

𝑁

𝑖=1

∑𝑞1,𝑖𝑃𝑖

𝑁

𝑖=1

∑𝑞2,𝑖𝑃𝑖

𝑁

𝑖=1

∑𝑞3,𝑖𝑃𝑖

𝑁

𝑖=1

)

(3.17)

So, we can write this in matrix form using 𝑄=[𝑞𝑘,𝑖], and we get:

𝑠=𝑄∙𝑍 𝑠=𝑍𝑠 (3.18)

Note that the first row of 𝑄 only consists of ones. Further note that 𝑄 is a 4xN matrix, while

𝑍 is Nx4, so eq (3.18) can be written with a simple 4x4 matrix 𝑍=𝑄∙𝑍. The first row of 𝑍 is

always [1 0 0 0]𝑇. Because 𝑠0=𝑠0 is independent of the input SOP, the vector

[𝑠1𝑠2𝑠3]𝑇 completely characterizes the incident SOP. In fact, 𝑍 is an affine map (p.38 in

[53]). So, 𝑠 can be graphically understood in a similar way that SOPs can be understood as a

point on a sphere. There are two major differences, however: the mapped affine surface is

not a sphere but the surface of an ellipsoid. Second, the ellipsoid is generally not centered

on the origin. 𝑠 is illustrated with the example of a 5-way iSTOMP in figure 3-5.

Figure 3-5. Illustration of the mapping between the Poincaré sphere (left) to the affine 𝑠-space (right). The

conjugate ellipsoid is centered around [0.4 0.6 0.57] and has a volume that is 0.5% of that of the unity

sphere (i.e. its axes are on average around 6.5 times shorter).

The 𝑄 used for figure 3-5 is:

𝑄=[1 1 1 1 1

1 −1 −1 1 1

1 1 −1 −1 1

1 1 1 −1 −1] (3.19)

To summarize, iSTOMPs with 𝑁≥5 enable the measurement of a Stokes vector. To

maximize the signal-to-noise ratio (SNR), the volume of the mapping 𝑠=𝑍𝑠 should be as

big as possible. The variable parameters to achieve this are: the total retardance in the MZI

𝛿𝑀𝑍, the retardance and angle of the PR 𝛿𝑃𝑅 and 𝜃𝑃𝑅 and the ratios 𝑅𝑖. To implement

iSTOMP designs in HHI’s generic platform, MMI building blocks are still missing, as only 1x2

and 2x2 MMIs are available right now. Due to time constraints this has not been done.

3.2 Dual Polarization Electro-Absorption Modulators

3.2.1 A New DP Modulation Scheme

Since many practical electro-optical modulators have a strong polarization dependence,

they are only operated with one particular polarization. For most devices in InP, TE

polarization is used [9], [13], [54]. Almost all dual-polarization modulators found in

literature use a configuration as in figure 3-6 ([13], [16], [19]). Incident light is split into two

branches with a modulator in each. One of the modulated signals then is rotated in

polarization, and the two paths are merged by a polarization beam combiner. Since in most

cases only the two modulators are integrated and the other elements are implemented in

bulk optics, the chips are often referred to twin-modulators ([19]). A fully integrated device

has not been demonstrated in InP yet. Silicon-on-insulator (SOI) demonstrations exist,

however ([16]).

Figure 3-6. Classical scheme for dual-polarization modulation: the polarization diversity approach. Normally, it

is implemented for coherent applications with two nested IQ modulators.

Even though normally implemented with IQ modulators, the polarization diversity scheme

can also be used in direct detection systems. In those systems, however, modulation is

often done by the laser directly (DML), or by electro-absorption modulators (EAM). These

devices typically have a footprint one order of magnitude smaller than IQ modulators in the

same technology (compare e.g. [9] and [12]). This size advantage would at least partially be

counteracted by the parallel nature of the polarization diversity scheme. Therefore, a new

PDM scheme is proposed in figure 3-7. It makes use of the typically strong polarization

dependence of electro-absorption modulators (EAM). Given that both EAMs only modulate

TE and the PR rotates polarization by 90°, the working principle is as follows: light with

both TE- and TM-components is launched into the device. The first EAM modulates only the

TE component. The PR rotates the modulated signal into the TM state, while the

unmodulated signal is rotated into the TE state. The second EAM will not affect the

previously modulated signal, as it cannot interact with TM. Rather, its modulation is

imprinted onto what is now TE. As a result, the optical signal leaving the second EAM is

intensity modulated in both polarizations.

Figure 3-7. Alternative scheme for dual-polarization modulation.

The proposed scheme can in principle also be used with MZMs or even IQ modulators, as

they normally also have strong polarization dependence. In this thesis, however, we will

focus on EAMs for short reach applications. The following sections discuss the theoretical

details and implementation of an EAM Building Block (BB) in HHI’s generic photonic

integration technology. Together with the PR from the previous chapter, the EAM will

enable the implementation of the entire dual-polarization modulator.

3.2.2 Theoretical Background of EAMs

To implement these modulators, multi-quantum well- (MQW) based EAMs shall be used.

MQW-based EAMs are typically compressively strained. As will be discussed in this chapter,

this makes them strongly polarization dependent, which is exactly what is required for a

device as in figure 3-7.

3.2.2.1 Electronic Properties of Multi-Quantum Wells

The valence states in covalently bound crystals exist in three distinct subbands: heavy hole,

light hole and spin orbit split-off bands. Unlike unbound states with their s-like orbitals,

bound states are p-like, i.e. they have three distinct symmetry planes.

Barrier

Well

Barrier

Figure 3-8. Schematic of the band diagram across four quantum wells. We use y as the normal coordinate to

the MQW layers, to be consistent with previous definitions.

In unstrained bulk InGaAsP, the heavy hole (HH) and light hole (LH) bands are degenerate,

i.e. at the Γ point their energies are equal. When electrons and holes are confined in

quantum wells, however, the degeneracy no longer holds. To see this, the electronic states

in a finite quantum well are considered. Following the derivation in [42], chapter 3, we first

define 𝑚𝑤

∗ and 𝑚𝑏∗ the effective electron masses in the well and barrier, respectively. Δ𝐸 is

the potential depth. Further:

𝑘=√2𝑚𝑤

∗𝐸

ℏ (3.20)

and

𝛼=√2𝑚𝑏∗(𝛥𝐸−𝐸)

ℏ (3.21)

The well is of thickness 𝑡𝑤. Then, the states can be found by finding the eigenenergies 𝐸 of

𝛼=𝑚𝑏∗𝑘

𝑚𝑤

∗𝑡𝑎𝑛(𝑘𝑡𝑤

2) (3.22)

for even wave functions and

𝛼=−𝑚𝑏∗𝑘

𝑚𝑤

∗𝑐𝑜𝑡(𝑘𝑡𝑤

2) (3.23)

for uneven wave functions.

Due to their different effective masses, the energies of heavy and light holes generally are

different. Hence, the band gap energy in a quantum well is not the same for heavy and

light holes, even in the absence of strain. This fact is illustrated in figure 3-9.

In the presence of strain, the degeneracy of HH and LH bands is lifted even in bulk InGaAsP

[55], [56]. In an epitaxial layer with lattice constant 𝑎𝑒 grown on a substrate with lattice

constant 𝑎𝑠, we call the epitaxial layer strain 𝜖=(𝑎𝑒−𝑎𝑠)/𝑎𝑠. Two contributions lead to an

energy shift of the two bands: hydrostatic strain Δ𝐸𝐻𝑦 and uniaxial strain Δ𝐸𝑆ℎ [42], [57].

𝛥𝐸𝐻𝑦=−2𝑎𝐶11−𝐶12

𝐶11 𝜖 (3.24)

𝛥𝐸𝑆ℎ=−2𝑏𝐶11+2𝐶12

𝐶11 𝜖 (3.25)

Consequentially, 𝑎 is referred to as hydrostatic deformation potential and 𝑏 as shear

deformation potential. 𝐶11 and 𝐶12 are the elements of the material elastic constant tensor.

All four parameters generally depend on material composition and can be interpolated

between the respective binary values as explained in appendix B) or chapter 2 in [42]. Using

these values, figure 3-10 shows how compressive strain separates the HH band PLs and the

LH band PL. Tensile strain can lead to a crossing. The unstrained case reveals the pure

splitting induced by the different effective masses.

Figure 3-9. Splitting of heavy and light hole bands in the absence of strain (left) and 0.5% compressive strain

(right). The splitting generally increases with decreasing well thickness.

Figure 3-10. Difference of the HH and LH PLs versus strain for a MQW stack with 10 nm thick wells. The HH

PL is kept constant to 1.55 µm for all strain values by adjusting the material composition. For tensile strain

around -0.4%, the bands cross and the resulting PLs become equal.

3.2.2.2 Optical Absorption in Multi-Quantum Wells

Using Fermi’s golden rule (p.351, [55]), the absorption coefficient of bulk semiconductor is

found to be:

𝛼(𝐸𝑝ℎ)=2𝐶0

𝑉∑∑|𝑒∙𝑝𝑐𝑣|2𝛿(𝐸𝑐−𝐸𝑣−𝐸𝑝ℎ)(𝑓𝑣−𝑓𝑐)

𝑏𝑎 (3.26)

where 𝐶0 contains natural constants and the optical angular frequency 𝜔:

𝐶0=𝜋𝑒2

𝑛𝑐𝜖0𝑚02𝜔 (3.27)

|e∙p

󰇍



cv|2 is the momentum matrix element. e is the unit vector along the optical field and

󰇍



cv is the linear momentum associated with an electronic transition from the conduction to

the valence band. The momentum matrix element can be derived in closed form for

quantum wells [55]:

|𝑥∙𝑝𝑐𝑣|2={32 ℎ𝑒𝑎𝑣𝑦 ℎ𝑜𝑙𝑒𝑠

12 𝑙𝑖𝑔ℎ𝑡 ℎ𝑜𝑙𝑒𝑠 (3.28)

|𝑦∙𝑝𝑐𝑣|2={0 ℎ𝑒𝑎𝑣𝑦 ℎ𝑜𝑙𝑒𝑠

2 𝑙𝑖𝑔ℎ𝑡 ℎ𝑜𝑙𝑒𝑠 (3.29)

These matrix elements determine the polarization dependence of electro-absorption in

MQWs. Even in the case of energetically degenerate heavy- and light hole bands (e.g. in

bulk material), heavy hole transitions contribute three times as much to optical absorption

of TE polarized light (x). But because the two subbands are typically split in MQWs (see

previous section), light hole transitions can be neglected all together (their band gap energy

is much bigger). For TM-polarized ( ) light, however, heavy hole transitions are completely

forbidden. This polarization can only interact with the light hole band. As a result, in MQWs

with any compressive strain in the wells, the edge of the absorption spectrum of TE light is

red-shifted with respect to TM light. This is because the heavy hole band gap is smaller than

the light hole band gap.

It should be stressed that (3.28) and (3.29) only hold for quantum wells. For quantum

dots, the polarization dependence disappears in principle due to their isotropic nature. Real

quantum dots are always strained due to the involved wetting layers, however, leading to a

polarization dependence. Quantum wires are interesting because the polarization selectivity

depends on which of the wire sidewalls is shorter [58]. Hence, precise control of the wire

dimensions enables engineering of the polarization dependent gain.

3.2.2.3 Refractive Index of Multi-Quantum Wells

While most models for the refractive index of InGaAsP work reliably far away from the

band edge, they normally diverge close to it. EAMs naturally are operated at wavelengths

close to the active layer’s band edge. A different model than in chapter 2 is therefore

needed. Theoretically, the dispersion of the refractive index’ real part can be calculated by

the well know Kramers Kronig relation for linear, causal systems [59]–[62]. In general, it

gives a closed form relationship of the real and imaginary part of the system response in the

frequency domain. With it, the refractive index can be written as a function of the

absorption spectrum 𝛼(𝜔):

𝑛(𝜔)−1=𝑐0

𝜋𝑃∫𝛼(𝜔)

𝜔′2−𝜔2

∞

0𝑑𝜔′(3.30)

𝑃 denotes that the Cauchy principal value must be taken at the integrand singularity, i.e.

𝜔′=𝜔. A similar relation can be deduced to calculate the absorption coefficient from the

refractive index.

Besides computational complexity, (3.30) requires knowledge of 𝛼 over an infinite spectral

range, or at least a very wide range. Tanguy proposed a model in [63] that approximates

(3.30) in closed form using a decomposition of 𝛼 into various physical effects. It writes as:

𝑛2(𝜔)−1= 𝑎

𝑏−𝐸+𝐴√𝑅

𝐸2(𝑙𝑛 𝐸𝑔2

𝐸𝑔2−𝐸2+𝜋(2𝑐𝑜𝑡(𝜋√𝑅

𝐸𝑔)−𝑐𝑜𝑡(𝜋√𝑅

𝐸𝑔−𝐸)−𝑐𝑜𝑡(𝜋√𝑅

𝐸𝑔+𝐸))) (3.31)

, where 𝐸=ℏ𝜔+𝑖Γ, i.e. the photon energy with an imaginary broadening factor. 𝑎,𝑏,𝐴

and 𝑅 are fitting parameters. It has been shown experimentally by Seifert et al. in [64] that

for InGaAsP, all open parameters can be fitted linearly versus the band gap. The resulting

dispersion curves for the real and imaginary parts of bulk unstrained InGaAsP with a PL of

1.55 µm are shown in figure 3-11.

Figure 3-11. Dispersion of the real (purple) and imaginary (green) parts of the refractive index of bulk InGaAsP

with no strain and a PL of 1.55 µm. Since bulk material is isotropic, the indices are equal for TE and TM.

As pointed out by v.d. Ziel et al. in [65], MQW layer stacks can be described by an effective

index that is composed of the indices of the wells and barriers. The effective description

depends on polarization, as follows:

𝑛𝑇𝐸,𝑒𝑓𝑓

2=𝑛𝑇𝐸,𝑤

2∙𝑡𝑤+𝑛𝑇𝐸,𝑏

2∙𝑡𝑏

𝑡𝑤+𝑡𝑏 (3.32)

𝑛𝑇𝑀,𝑒𝑓𝑓

2=𝑡𝑤+𝑡𝑏

𝑡𝑤

𝑛𝑇𝑀,𝑤

2+𝑡𝑏

𝑛𝑇𝑀,𝑏

2 (3.33)

The indices of the wells and barriers for the two polarizations can be calculated from the

indices resulting from the different band edges for heavy- and light holes, taking into

account the momentum matrix elements from the previous chapter. As a result, the

anisotropic dispersion curves for an entire MQW stack can be calculated. An example is

shown in figure 3-12. This index model can then be used in a mode solver to calculate the

indices of the modes propagating in an MQW-loaded waveguide.

Figure 3-12. Anisotropic index dispersion of an MQW stack after eq. (3.32) and (3.33). The individual indices

of wells and barriers are calculated according to the selection rules in section 3.2.2.2. The MQW stack shown

here as 10 nm thick wells that are compressively strained by 0.5%.

3.2.2.4 The Quantum Confined Stark Effect

The dominating effect leading to electro-absorption in MQWs is the quantum-confined

Stark effect (QCSE). The QCSE was first experimentally reported 1984 by Wood et al. in [66]

and was then theoretically explained by the same group in the same year (Miller et al.,

[67]). It is a field effect that decreases the band gap energy in the wells. This leads to a shift

of the entire absorption spectrum towards lower photon energies or larger photon

wavelengths. The effect is illustrated in figure 3-13. The applied electric fields tilts the band

structure, which in turn changes the shape of the wave functions for electrons and holes.

This leads to a preference of electrons for positions inside the well that have lower energies,

while holes will prefer larger energy positions. Thus, the gap between the two energies is

reduced. Although the field separates electrons and holes, their separation is in the order of

the well width, which typically is smaller than the size of the excitonic wave function.

Hence, the electron-hole separation has no significant impact on the electron-hole

interaction, e.g. electro-optic transitions. For large fields, however, this does not hold

anymore and the absorption quenches.

Figure 3-13. Visualization of the band structure giving rise to the quantum-confined Stark effect (QCSE). With

no potential applied to the double heterostructure (left), the wavefunctions of electron and hole overlap

strongly. Their energetic gap is homogeneous across the well. However, the electronic states get tilted along

the y-axis when a potential drops across the structure (right). The respective wavefunctions of electron and

hole are displaced such that the energetic difference between the states becomes smaller. This gives rise to a

shift of the optical absorption spectrum.

Due to the strong polarization sensitivity of the momentum matrix elements associated with

the heavy hole- and light hole transitions, the QCSE is generally also strongly polarization

dependent. We saw that for compressively strained wells, the heavy hole transition has the

lowest energy. But since the associated momentum matrix element vanishes for TM

polarized light, it only contributes to absorption of TE-polarized light. A quantitative analysis

of the shift of the bands Δ𝐸 due to an electric field 𝐹 was given by Bastard in 1983 [68]. In

literature, one mostly only finds the small-field approximation for thin wells, infinitely high

barriers and small fields. [69]–[71]. Thin wells means around 3 nm [68], while the field is

considered small if:

𝑒𝐹𝑡𝑤≪𝐸1=ℏ2𝜋2

2𝑚∗𝑡𝑤

2 (3.34)

I.e. the potential dropping across a well is small compared to the ground state 𝐸1 in the

well. In this regime, the shift is simply a quadratic function in the electric field:

𝛥𝐸=−𝑚∗𝑒2𝐹2𝑡𝑤

24ℏ2𝜋4(𝜋2−15) (3.35)

The shift of the band edge then simply becomes Δ𝐸𝑔=Δ𝐸𝑐+Δ𝐸𝑣, calculated with the

respective masses in the sub bands. Because the shift is proportional to the effective mass,

the heavy hole band gap shifts more than the light hole gap. So if one were to make a

QCSE-based modulator operating around the light hole gap (e.g. to make a TM modulator),

it would intrinsically suffer from a lower modulation depth.

The quadratic field dependence of (3.35) breaks down in the case of strong fields. A closed

form description for ΔE versus any arbitrary field is not possible, but Bastard et al. gave the

eigenenergy equation that is to be minimized in [68]:

𝐸(𝛽)=𝐸1

(

1+𝛽2

4𝜋2+𝜙(1𝛽+2𝛽

4𝜋2+𝛽2−1

2𝑡𝑎𝑛ℎ(𝛽2))

)

(3.36)

with the minimization parameter 𝛽 and the dimensionless electrostatic energy

𝜙=𝑒𝐹𝑡𝑤

𝐸1 (3.37)

The equation is implemented in python where it can be solved numerically for arbitrary

fields. Simulated Stark shifts obtained by this model are shown in figure 3-14.

Figure 3-14. Simulated Stark shifts for the respective carriers versus electric field. The MQW structure has a

net photoluminescence of 1.53 µm. The plot is generated by numerically solving equation (3.36). Therfore it

holds only for infinitely high barriers, but for arbitrarily large fields. Because tunneling does not occur in this

case, the strength of the shifts shows a clear dependence on the effective carrier mass.

For finite barrier heights, however, the situation is further complicated by tunneling effects

and the increased spatial separation of electrons and holes induced by the electric field.

Bastard et al. pointed out that the effect of finite barrier heights is rather drastic, reporting

a 28x to 38x increase of the Stark shift [68]. For weak fields, the Stark shift enhancement

can be expressed by Ω2/Ω∞

2 with

𝛺=𝐴(13+𝑠𝑖𝑛𝑘0

𝑘0+2𝑐𝑜𝑠𝑘0

𝑘02−2𝑠𝑖𝑛𝑘0

𝑘03+2

𝑞0(1+2

𝑞0+2

𝑞02)𝑐𝑜𝑠2(𝑘0

2)) (3.38)

where

𝐴= 1

1+𝑠𝑖𝑛𝑘0

𝑘0+2

𝑞0𝑐𝑜𝑠(𝑘0

2) ;𝑞02=2𝑚∗𝑡𝑤

ℏ2(𝑉0−𝐸1) ; 𝑘02=2𝑚∗𝑡𝑤

2𝐸1

ℏ2 ;𝛺∞=13−2

𝜋2 (3.39)

𝐴 is a normalization constant for the continuity of the wave function, 𝑞0 is the wave

function’s decay constant into the barrier of finite height 𝑉0 and 𝑘0 is the dimensionless

wave vector with no field applied. Naturally, Ω→𝛺∞ for 𝑉0→∞. Note that to calculate 𝑘0,

the ground state energy for a finite barrier height is needed rather than the infinite case of

(3.34). It is calculated with section 3.2.2.1.

Figure 3-15. Simulated Stark shifts for the respective carriers versus electric field. The MQW structure has a

net photoluminescence of 1.53 µm. The plot is generated by numerically solving equation (3.36) and takes

into account the enhancement factor for finite barrier heights. Therfore it holds only for small fields. The

critical fields 𝐹𝑐𝑟𝑖𝑡=𝐸1

𝑒𝐿𝑤 at which the approximation brakes down are depicted as dashed lines.

Given that the enhancement factor is invalid for practical fields (50 kV/cm correspond to a

bias of -1 V), we will assume the infinite barrier case in the following. Therefore, the

calculated Stark shifts will underestimate the real shifts. The path towards a complete

treatment of finite barriers and large fields is pointed towards in [68] and [72].

The results of the QCSE theory will be used in the following to simulate the polarization

dependent behavior of the EAM. This will ultimately be used to enable performance

estimations of the full DP EAM structure.

3.2.3 EAM Design

3.2.3.1 Modes, birefringence

The EAM is implemented in the MQW-loaded waveguide of HHI’s generic integration

technology. It uses a ridge waveguide design with an n-doped guiding layer underneath the

MQW stack. A cross sectional view of the refractive index distribution is given in figure

3-16. The waveguide supports TE and TM modes.

Figure 3-16. Cross-sectional view of the refractive index profile (left) and mode distribution (right) for TE

polarized light (top) and TM polarized light (bottom).

To verify that an EAM based on this waveguide really only modulates TE-polarized light, the

complex effective index is simulated as a function of MQW PL. This effectively emulates

modulation in a real device, as modulation is achieved by means of the QCSE-induced band

gap shift. The result is shown in figure 3-17. Indeed, the TM mode modulation is negligible

compared to the TE modulation.

Figure 3-17. MODE simulations of the EAM waveguide operated at a wavelength 𝜆𝑝ℎ of 1565 nm for TE (left)

and TM (right). Plotted are the waveguide absorption and effective index versus the MQW photoluminescence

(PL). The as-grown PL is 1530 nm, corresponding to 𝛥𝐸𝑔=0 in the upper axis.

Because the integration platform relies on iron-doped (i.e. semi-insulating) substrates, the

n-region has to be contacted from the top. To bring the n-contact as close to the active

layer as possible, a new form of n-contact is used in the generic technology. Normally, the

n-region is contacted 15 µm away from the ridge waveguide and only on one side of the

waveguide. A closer and double-sided contact would be desirable, but is prohibited by the

electro-plating process. To solve this, the platinum heater, that is also part of the prcess

design kit (PDK) is used. The configuration is shown in figure 3-18.

Figure 3-18. EAM cross-section (left) and the layout top view (right). It uses the platinum process of HHI’s

generic technology, which is normally used for heaters. For the EAM, it is used to bring the n-side contact as

close to the active region as possible, while also using both sides of the waveguide for contacting. Most of the

layers in the layout view are left out for clarity. The dark grey layer corresponds to the electro plating. The

plated gold forms bridges over the waveguide and the platinum stripes.

The lower boundary for the separation of the platinum to the waveguide is given by

plasmonic losses. If the platinum is too close, the optical mode will experience excess loss.

The propagation loss increases significantly for an edge-to-edge separation between

waveguide and platinum of less than 1 µm, as shown in figure 3-19. To allow some error

margin for the lithography, the separation in the EAM is chosen as 2.5 µm.

Figure 3-19. Propagation loss in the EAM versus edge-to-edge distance from waveguide to platinum stripe.

Both TE (left) and TM (right) experience significant excess loss for separations below 1 µm.

3.2.3.2 Electrical properties

The EAM can be thought as an equivalent circuit as in figure 3-20. In reverse bias, the pin-

junction acts as a capacitor. To simulate the structure, Lumerical’s DEVICE solver is used

[41]. It solves the Poisson equation to calculate the electric field and charge distribution,

taking into account the device band structure and doping profile. The purpose of this

simulation is to ultimately extract the junction capacitance, which is critical for the RF

performance of the EAM.

Cjunction

Rn/2 Rn/2

Figure 3-20. Equivalent circuit drawn on top of the cross section as used in DEVICE simulations. The roughness

on some interfaces is an artifact caused by the export and not present in the actual simulations.

Figure 3-21. Electric fields along x (right) and y (left) for no bias (top) and -2 V bias (bottom). The built-in field

across the MQW region due to carrier depletion is around 50 kV/cm along the y-axis.

Neglecting carrier life- and transit-times in the active region, the EAM frequency response is

RC limited:

𝑓3𝑑𝐵=1

2𝜋𝑅𝐶=1

2𝜋(𝑅𝑛+𝑅𝑝)𝐶𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (3.40)

The resistances of the n- and p-regions are simply given by:

𝑅=𝐿

𝜎𝐴=𝐿

(𝑛𝜇𝑒+𝑝𝜇𝑝)𝑒𝐴 (3.41)

The capacitance can be approximated by a parallel plate capacitor or more accurately via

𝐶=𝑑𝑄

𝑑𝑉. Since DEVICE can simulate the charge distribution as a function of bias voltage, this

enables the simulation of the capacitance as a function of voltage. The simulated excess

charge density distribution Δ𝑄=Δ𝑛+Δ𝑝 at -2 V is shown in figure 3-22. The voltage

dependent capacity is plotted in figure 3-23.

Figure 3-22. Charge density distribution accumulated by applying a bias of -2 V. The distribution at zero bias is

subtracted from the total charge density, so only the excess charge density is shown. Charge accumulated at

the interfaces to the intrinsic MQW region.

Figure 3-23 also shows the average field inside the MQW region, which is the field

responsible for modulation. It is generally the voltage across the MQW stack divided by the

thickness of the space charge region (SCR). Since the SCR widens with bias voltage, the

field can be written as:

𝐸𝑦,𝑀𝑄𝑊=𝑉𝑀𝑄𝑊

𝑡𝑆𝐶𝑅(𝑉𝑏𝑖𝑎𝑠)=𝑉𝑏𝑖𝑎𝑠−𝑉𝑑

𝑡𝑀𝑄𝑊−𝑎𝑉𝑏𝑖𝑎𝑠 (3.42)

where 𝑎 is the linear dependence of the SCR on the bias voltage and 𝑉𝑑 is the built-in

diffusion potential. As can be seen in figure 3-23, the fit gives a good approximation. 𝑉𝑑 is

0.7 V, while the SCR widening is found to be 𝑎≈20 nm/V.

Figure 3-23. Simulated capacitance per unit length as a function of bias voltage (left). Increasing the bias

widens the depletion zone in the junction, leading to a decrease of the capacitance. The widening of the

depletion zone can also be seen in the sub-linear dependence of the field across the MQWs versus voltage

(right). It can be approximated by 𝐸𝑦=𝑉−𝑉𝑑

𝑡𝑀𝑄𝑊−𝑎𝑉.

3.2.3.3 Electro-optic properties

With the fields calculated in the previous section and the index and QCSE models from

sections 3.2.2.3 and 3.2.2.4, the actual EAM modulation can be studied. As discussed

before, the QCSE model in the simulations can underestimate the real shifts by a factor of

up to 28 because it assumes infinite barriers. Effectively, this lead to simulations that

suggest too large modulation voltages for a given extinction. Nevertheless, figure 3-24

demonstrates that the EAM as designed in the generic technology only modulates TE

polarized light.

Figure 3-24. Modulation of the TE mode (left) and TM mode (right) in the EAM model. The TM modulation is

negligible compared to the TE modulation.

The curves from figure 3-24 encapsulate the entire static optoelectronic behavior of the

EAM. They are used in the following section to perform system level analysis.

3.2.4 DP EAM Simulations

So far, polarization-resolved simulations of the individual EAMs are obtained. Together with

a low-pass filter derived from the equivalent circuit, we use these results to perform large

signal simulations of the DP EAM. To do so, Lumerical’s INTERCONNECT tool is used [41].

The EAM is modelled electrically as a low-pass filter. Optically, it is modelled as two

independent optical paths for TE and TM polarized light, respectively (see figure 3-25). At

the EAM input (output), the two paths are split (combined) by ideal polarization splitters

(combiners). In the two parallel branches the modulation of the complex indices for TE and

TM from chapter 3.2.3 are implemented. Effectively, each branch represents the

modulation of one of the polarizations. This model is encapsulated into a composite

building block that is then used for the full system simulation as it is shown in figure 3-26.

Figure 3-25. Equivalent circuit to model the EAM in INTERCONNECT. The voltage dependent complex indices

𝑛𝑇𝐸(𝑉) and 𝑛𝑇𝑀(𝑉) are taken from the QCSE calculations in chapter 3.2.3. Because the software does not

include a modulator model with anisotropy, the EAM is modelled as two modulators for the respective

polarizations which are split and combined accordingly.

Figure 3-26. The simulated dual-polarization modulator circuit as used in INTERCONNECT. A TE-polarized

continuous wave (CW) optical source undergoes polarization rotation by 45°. Two EAMs follow which are

interconnected by a polarization rotator that rotates by another 90° in the ideal case. The EAMs themselves

are circuits as in Figure 3-25. A noise loading stage consisting of an attenuator, an amplifier and additive

white Gaussian noise (AWGN) is used to set arbitrary OSNR values. A polarization beam splitter is used with

two receivers to analyze the modulated signals in the two polarization states.

Example eye diagrams of the two received polarizations are shown in figure 3-27. The PR in

between the two EAMs is intentionally set to not rotate by exactly 90° but rather 72.5°, 81°

and 84.5°. This corresponds to a PER of 10 dB, 15 dB and 20 dB, respectively. The OSNR is

set to 20 dB in all cases. In the example of the 15 dB case, the eyes are still open but the

signal transmitted by the first EAM is impaired. This can be understood with the fact that

15 dB conversion efficiency means that 0.13 dB are left in the original TE polarization state.

This part of the modulated signal then gets modulated again by the second EAM, leading

to cross modulation. The 10 dB PER case gives a closed eye transmitted by the first EAM

while for 20 dB PER, the first eye has seemingly the same quality as the second EAM’s eye.

PER=20 dB

PER=15 dB

PER=10 dB

Figure 3-27. Received eye diagrams in the two polarizations for an OSNR of 20 dB. With decreasing PER, the

eye corresponding to the first EAM deteriorates.

By calculating the optimum decision thresholds and the 𝜇 and 𝜎 values for the one and

zero levels, INTERCONNECT can calculate the bit error ratios (BER) of the two received

signals. BER versus OSNR curves are shown in figure 3-28 for different PERs of the critical

PR. At a PER of around 20 dB, the BER converges to a constant value. At a BER of 10−9, the

OSNR penalty introduced by the non-ideal PER is plotted in figure 3-29. The penalty is

below 1 dB for a PER above 16 dB. In terms of rotational angle, this leaves a corridor of ±9°

around the ideal 90°.

Figure 3-28. Bit error ratio (BER) of the dual-polarization transmitter simulated for different PERs of the rotator

in between the two EAMs. The total BER is calculated as the sum of the two individual BERs.

Figure 3-29. Penalty in terms of receiver OSNR for different PER values. The penalty converges to 0 dB for

large PERs. It stays below 1 dB (yellow line) for a PER greater than 15 dB.

4 Experiments

4.1 Experimental Methodology – Fiber Based Stokes Measurements

With the mathematical convenience of the Jones formalism comes the price of

experimental impracticalities. Jones vectors are representations of perfectly polarized

coherent light. Most laboratory equipment cannot measure the absolute phase of optical

signals though, so coherent measurements become an issue. Further, a Jones matrix cannot

describe the process of depolarization and a Jones vector cannot describe unpolarized light.

As was seen in chapter 2.3, both issues are tackled by moving to the Stokes/Müller

Formalism. The goal of this chapter is the extraction and analysis of the Müller matrix MPIC

of any given integrated device. Except the absolute phase acquired during propagation, the

Müller matrix contains a complete picture of the polarization properties of the element it

represents. To leverage mature equipment and enable automated measurements, a

completely fiber-based setup is pursued.

It should be noted that measurements of integrated optical devices are typically not done in

Stokes space in literature. Rather, extinction ratios between TE and TM polarization are

recorded at the in- and outputs of the device under test [16], [51], [73]–[75]. As a result,

measurements cannot be done without going through free space optics such as lenses and

polarizers. When measuring pure TE/TM extinction ratios, the relative phase information

gets lost, so the birefringence or helicity of devices remains inaccessible. Some authors used

distinct optical polarizations at the input, while only measuring the total power at the

output [76], [77]. Effectively, this yields only the first row of the Müller matrix, but it is

sufficient to determine the dependence of transmittance on polarization, i.e. PDL. There are

reports on interferometric schemes that effectively measure the Jones matrix of a given

device, sometimes referred to as optical vector network analyzer [78]–[80]. Apart from

depolarization, these methods yield the full Müller matrix, but they require at least one

interferometer in the fiber setup. This method is not pursued here to avoid the long-term

stability issues that come with fiber based interferometers. To the author’s best knowledge,

no non-coherent method for full Müller matrix measurements of PICs has been shown yet.

Dong et al. did show a method to measure the Müller matrix of fibers, however [81].

4.1.1 Experimental Setup

4.1.1.1 Setup Overview

The schematic of the complete setup is shown in figure 4-1. Its key components are a

polarization controller and a polarimeter, both computer controlled. On the control

computer, python is used for the communication with the measurement equipment and all

necessary calculations.

ECL PC

𝜃𝐿𝑃, 𝜃 / , 𝜃 / PM

𝑠 𝑒𝑐𝑙 𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛 𝑀𝑓𝑖𝑏𝑒𝑟,𝑜𝑢𝑡

𝑠 𝑖𝑛 𝑠 𝑜𝑢𝑡

𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2)

𝑠 𝑚𝑒𝑎𝑠

Figure 4-1. Schematic overview of the fiber-based setup to measure the polarization effects in integrated

devices. Light of an external cavity lasers (ECL) of Stokes vector 𝑠𝑒𝑐𝑙 goes through a commercial computer-

controlled polarization controller (PC). The PC consists of a cascade of a linear polarizer, 𝜆/4- and 𝜆/2-

waveplate. All elements can be rotated by arbitrary angles. The PC is followed by a lensed SSMF for efficient

chip coupling. The same kind of fiber is used to couple light out of the chip and to route it to a polarimeter

(PM) that measures 𝑠𝑚𝑒𝑎𝑠. Both input and output fibers are described by their respective Müller matrices

𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛 and 𝑀𝑓𝑖𝑏𝑒𝑟,𝑜𝑢𝑡. 𝑠𝑖𝑛 and 𝑠𝑜𝑢𝑡 are the Stokes vectors of the light coupling into and out of the integrated

device, which is characterized by its Müller matrix 𝑀𝑃𝐼𝐶.

As will be seen in chapter 4.1.2, control over sin and knowledge of sout enables the

measurement of MPIC. The polarization controller can be used to change sin. The

polarimeter measures the Stokes vector smeas, which is determined by sout. However, in

both cases an optical fiber complicates the relationships between the respective vectors.

The complete setup uses standard single-mode fiber (SSMF). A reliable polarization model

for SSMF is therefore needed. The model is then used to undo the effects of the fibers to

extract both sin and sout.

It might seem a natural choice to use polarization maintaining fibers (PMF) instead of SSMF.

But PMF only work properly if the polarization injected into the fiber is exactly aligned with

either the slow or the fast axis of the PMF. If not, the PMF does in fact rotate the

polarization much more rapidly than SSMF. This is because PMF has strong built-in

birefringence due to its non rotationally symmetric cross section. But since we want to set

arbitrary 𝑠𝑖𝑛 and measure the corresponding 𝑠𝑜𝑢𝑡, PMF are actually ill suited for this task.

Another practical issue would be to accurately align the fast or slow axis of the PMF with

the substrate of the chip.

4.1.1.2 Polarization in Optical Fibers

Any piece of SSMF induces an arbitrary rotation of the incident SOP. A practical model for

polarization rotation in SSMF is therefore needed.

As has been shown by Walker and Walker [82], [83], mapping between arbitrary SOPs can

be achieved by three linear retardation elements at angles 𝜃𝑖 with retardation 𝛿𝑖. Without

changing the retardation, any given SOP can be mapped to any desired SOP by adjusting

the 𝜃𝑖. This only holds, however, for the condition

𝜋2−(|𝛿1′|+|𝛿2′|+|𝛿3′|)≥0 (4.1)

with

𝛿𝑖′={𝜋2−𝛿𝑖, 0≤𝛿𝑖≤𝜋

−𝜋2−𝛿𝑖, −𝜋≤𝛿𝑖≤0 (4.2)

One solution to this is 𝛿1=𝛿3=𝜋2 and 𝛿2=𝜋, i.e. quarter- half- and quarter-wave plate.

This is the traditional choice for fiber-based polarization controllers where fiber loops of N,

2N and N windings are used (“Mickey Mouse Ears”). As has been commented in [82], this

is not the only solution and also not the ideal solution in some cases. Three quarter-wave

plates are the obvious solution to (4.1) and (4.2) and offer the highest tolerance with

respect to the elements retardation, namely ±𝜋6. Further, the order of the waveplates is not

significant. Since we are only interested in a good mathematical model for arbitrary SOP

mapping, we stick to the well known model (𝜋2,𝜋,𝜋2). The resulting Müller matrix Mfiber

describes how an input SOP gets mapped to an output SOP:

𝑀𝑓𝑖𝑏𝑒𝑟=𝑀𝜆/4(𝜃3)𝑀𝜆/2(𝜃2)𝑀𝜆/4(𝜃1)(4.3)

Since SSMF is dispersive, the angles 𝜃𝑖 will generally also be dispersive. An experimental

example is shown in figure 4-2. Because the 𝜃𝑖 are continuous versus wavelength, they can

be smoothly interpolated between sampling points. In the following chapters, we will use

the (𝜋2,𝜋,𝜋2) model for all fibers involved.

Figure 4-2. Experimental values of the equivalent retarder angles of some meters of SSMF. The retardations

are 𝜋2,𝜋 and 𝜋2, i.e. the classical quarter-, half- and quarter-wave plate arrangement as used in loop-based

polarization controllers.

4.1.1.3 Setup Modelling

With the setup as shown in figure 4-1, one can write the important relationships for sin and

sout: 𝑠𝑖𝑛=𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆4,𝜃𝜆2)𝑠𝑒𝑐𝑙 (4.4)

𝑠𝑚𝑒𝑎𝑠=𝛼𝐹2𝐹𝑀𝑓𝑖𝑏𝑒𝑟,𝑜𝑢𝑡𝑠𝑜𝑢𝑡 (4.5)

𝛼𝐹2𝐹 is the fiber-to-fiber loss. The polarization controller is comprised of three elements: a

linear polarizer, a quarter waveplate and a half waveplate. For higher accuracy of the

model, they are modelled as an elliptical diattenuator and two elliptical retarders (see

appendix A)), each rotated by the angle set in the controller. Further, the PDL of the

controller is modelled by another elliptical diattenuator. The PDL is specified by the manual

with ±0.03 dB. So, we can write the Müller matrix of the PC as:

𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆4,𝜃𝜆2)=𝑀𝜆2(𝜃𝜆2 )𝑀𝜆4(𝜃𝜆4 )𝑀𝐿𝑃(𝜃𝐿𝑃 )(4.6)

Here, 𝑀 denotes non-ideal elements with some residual ellipticity and retardation offset. In

the example of a non-ideal half waveplate, it is defined as follows: the eigenvector 𝑢

󰇍



of the

non-rotated waveplate should be (1 1 0 0)𝑇, or 𝜓𝑢

󰇍



=0 and 𝜒𝑢

󰇍



=0. The actual

eigenvector will be slightly rotated, so that 𝜓𝑢

󰇍



=Δ𝜓𝜆2 and 𝜒𝑢

󰇍



=Δ𝜒𝜆2. Also, the retardation

should ideally be 𝛿=𝜋, but it is actually slightly off so that 𝛿=𝜋+Δ𝛿𝜆2. The quarter

waveplate is completely analogous. Also, the polarizer diattenuation vector 𝐷

󰇍



is slightly

rotated by a Δ𝜓𝐿𝑃 and Δ𝜒𝐿𝑃.

Equation (4.5) can simply be inverted to yield sout for a given 𝑠𝑚𝑒𝑎𝑠. Setting a desired sin is

a matter of finding the correct parameters (𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2). Both tasks, however, require the

knowledge of Mfiber,in, Mfiber,out, secl as well as all the residual errors in the PC. What

follows is a discussion of how we extract all those three objects to move on with actual

device measurements.

4.1.1.4 Setup Calibration

In the optical back-to-back case (no device under test present), sout=sin and therefore

(4.4) and (4.5) give: 𝑠𝑚𝑒𝑎𝑠=𝛼𝐹2𝐹𝑀𝑓𝑖𝑏𝑒𝑟,𝑜𝑢𝑡𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆4,𝜃𝜆2)𝑠𝑒𝑐𝑙 (4.7)

This poses the issue of differentiating Mfiber,out from Mfiber,in, since only their product

appears in the back-to-back case. This makes an additional measurement necessary that

introduces a modification to the setup as in figure 4-1. The modified setup is shown in

figure 4-3. We can write the power measured at the photodetector as:

𝑃𝐹𝑆=(1 0 0 0)𝑠𝐹𝑆 (4.8)

With some loss 𝛼𝐹𝑆 associated to the free space path, the Stokes vector in free space is

given by: 𝑠𝐹𝑆=𝛼𝐹𝑆𝑀𝐿𝑃𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆4,𝜃𝜆2)𝑠𝑒𝑐𝑙 (4.9)

With (4.7) and (4.9), we have the two important equations to calibrate the setup, i.e. to

determine Mfiber,in, Mfiber,out, secl as well as all the residual errors in the PC. The flow of the

calibration procedure is depicted in figure 4-4. First, the setup is brought to free-space

configuration (as in figure 4-3). A set of measurements of 𝑃𝐹𝑆 is recorded, with 5∙5∙3=

75 different settings for (𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2) at each wavelength to be calibrated. The setup is

then brought to its fiber-to-fiber configuration (figure 4-1 without a device). The same 75

settings for (𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2) are set, but now s𝑚𝑒𝑎𝑠 is recorded. Again, this is done for each

wavelength that is to be calibrated. As a result, 150 equations per wavelength are obtained

(75 per (4.7) and 75 per (4.9)), with only Mfiber,in, Mfiber,out, secl and the PC’s residuals

unknown. Each fiber is determined by only three parameters (see 4.1.1.2) and secl has two

degrees of freedom (two spherical coordinates, its power is known). Hence, the 150

equations can be used to extract the unknown parameters with enough degrees of

freedom left to implement a least-square fit. It uses the same python-based methodology

of a target function and its minimization that has already been used for the device

optimization in chapter 2.5.4. The result of the minimization is numerical values for the

matrices of the two fibers as well as secl. Additionally, the residuals of the PC are also

computed.

ECL PC

𝜃𝐿𝑃, 𝜃 / ,𝜃 / Lens Linpol Det

𝑠 𝑒𝑐𝑙

𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2)

𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛 𝑀𝑙𝑖𝑛𝑝𝑜𝑙

polarizer / /

𝑠 𝑖𝑛

Figure 4-3. Schematic of the setup in its free space configuration. A lens is used to collimate the light exiting

the input fiber. The beam is analyzed by a linear polarizer that transmits only TE light followed by a

photodetector. The detector only measures 𝑃𝐹𝑆.

Figure 4-4. Flow of the setup calibration process. 70 triples (𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2) are used at the polarization

controller (PC), for each triple the free space power 𝑃𝐹𝑆 (1) and the in-fiber stokes vector 𝑠𝑚𝑒𝑎𝑠 (2) at the

polarimeter (PM) are recorded. This data is used in 3) to fit all unknowns in the model: 𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛, 𝑀𝑓𝑖𝑏𝑒𝑟,𝑜𝑢𝑡,

𝑠𝑒𝑐𝑙 and the imperfections at the PC. The entire mechanism is carried out on a per-wavelength basis, with

wavelengths from 1465 to 1575 nm in 5 nm steps. Spline fits are used to fill in the calibration data for a

continuous wavelength range.

4.1.1.5 Calibration Validation

The validation of the calibrated setup is done two-fold. First, the triple (𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2) is

scanned and the measured values for smeas and 𝑃𝐹𝑆 are compared with what can be

calculated using (4.7) and (4.9). The comparison at different wavelengths is shown in

figure 4-5 and figure 4-6.

As a second step, the actual task of the setup is emulated. Rather than blindly scanning

(𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2), 𝑠𝑖𝑛 is set and 𝑠𝑜𝑢𝑡 is deduced. To do so, (4.4) and (4.5) are used. (4.5) can

easily be solved for 𝑠𝑜𝑢𝑡 after measuring 𝑠𝑚𝑒𝑎𝑠. The appropriate (𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2) for a

desired 𝑠𝑖𝑛 can be calculated numerically. For the validation, the setup is operated in the

back-to-back regime, so 𝑠𝑖𝑛=𝑠𝑜𝑢𝑡. This way, the deviation between the desired

polarization ellipse and what is measured can be specified. The deviation is expressed in

terms of the central angle between the corresponding points on the Poincaré sphere. The

experimental results are summarized in figure 4-7.

ECL PC

𝜃𝐿𝑃, 𝜃 / , 𝜃 / Lens Linpol Det

𝑠 𝑒𝑐𝑙 𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛

𝑠 𝑖𝑛 𝑃𝐹𝑆

Figure 4-5. Validation of the setup calibration for the input path. The angles of the three elements of the

polarization controller are swept: the linear polarizer (top right), the quarter waveplate (bottom left) and the

half waveplate (bottom right). The solid lines show the predicted values after (4.9), the markers indicate

actually measured values. The errors between measurement and model stay below 5% across all wavelengths

and angles.

𝑠 𝑖𝑛=𝑠 𝑜𝑢𝑡 𝜓,𝜒

𝑀𝑓𝑖𝑏𝑒𝑟,𝑜𝑢𝑡

ECL PC

𝜃𝐿𝑃, 𝜃 / , 𝜃 /

𝑠 𝑒𝑐𝑙 𝑀𝑓𝑖𝑏𝑒𝑟,𝑖𝑛

𝑀𝑃𝐶(𝜃𝐿𝑃,𝜃𝜆/4,𝜃𝜆/2)

polarizer / /

Figure 4-6. Validation of the setup calibration for the output path. The setup is operated in the back-to-back

configuration (top left). The three elements in the PC are rotated, the angles of the polarization ellipse at the

polarimeter 𝜓,𝜒 are recorded. The solid lines show the values predicted by the calibrated model, the markers

represent the actual measurements. The plots show the sweeps of the linear polarizer (top right), quarter

waveplate (bottom left) and half waveplate (bottom right). In each plot, the other two angles are set to 0°.

Directly after calibration

Two Weeks after calibration

Figure 4-7. Validation of the setup core functionality: setting 𝑠𝑖𝑛 (characterized by 𝜓=𝜓𝑖𝑛 and 𝜒=𝜒𝑖𝑛) and

measuring 𝑠𝑜𝑢𝑡. The setup is operated back-to-back, so ideally 𝑠𝑖𝑛=𝑠𝑜𝑢𝑡. The measure for the deviation

between 𝑠𝑖𝑛 and 𝑠𝑜𝑢𝑡 is their central angle 𝛥𝜉 on the Poincaré sphere. The heatmap on the left shows the

deviation across all angles at a wavelength of 1.55 µm. Slices along 𝜒=0 and 𝜓=0 are shown in the middle

and right plots, respectively (including wavelength dependency). The position on the Poincaré sphere is

accurate within 2°, or 30 dB in terms of PER across the whole C-band. Two weeks after calibration the

performance is slightly degraded, but the errors are still below 5° or better than 21 dB PER.

As can be seen in figure 4-7, the overall angular error is below 2° across the entire C-band

and for all target angles. By carefully fixating all fibers to the optical bench and avoiding

long paths, the calibration accuracy can be maintained over more than a week. The setup is

located in an air-conditioned grey room area. The deviations between model and

measurement (see figure 4-7) stays well below 5° over the course of two weeks in the

entire C-band and beyond.

4.1.2 Solving the Polarimetric Measurement Equation

With 4.1.1, it is possible to set any arbitrary polarization ellipse at the input of a PIC and to

measure the polarization ellipse at its output. I.e., it is possible to control sin and to

measure sout. In general, the PIC is characterized by its Müller matrix as follows:

𝑠𝑜𝑢𝑡=𝑀𝑃𝐼𝐶𝑠𝑖𝑛 (4.10)

To determine 𝑀PIC, at least four different sin have to be set and sout has to be recorded for

each of those. Since each measurement yields four linear equations from (4.10), the four

measurements yield 16 equations which allow solving for all elements of the 4x4 matrix

𝑀PIC. Suppose we carry out 𝑁 measurements, each with an input polarization sin,q, with

𝑞=0,1,…,15. We record each output polarization sout,q. By defining 𝑁 analyzer vectors 𝑎𝑞,

𝑁 measured powers are defined as:

𝑃𝑞= 𝑎𝑞𝑇∙𝑠𝑜𝑢𝑡,𝑞=𝑎𝑞𝑇𝑀𝑃𝐼𝐶𝑠𝑖𝑛,𝑞 (4.11)

Note that because we always measure all four elements of sout,q, we really only do 𝑁/4

measurements but still get 𝑁 different resulting powers. When re-writing MPIC as a vector

𝑚

󰇍



, we can write the powers as

𝑃𝑞=𝑤

󰇍



𝑞∙𝑚

󰇍



(

𝑎𝑞,0𝑠𝑖𝑛,𝑞,0

𝑎𝑞,0𝑠𝑖𝑛,𝑞,1

𝑎𝑞,0𝑠𝑖𝑛,𝑞,2

𝑎𝑞,0𝑠𝑖𝑛,𝑞,3

𝑎𝑞,1𝑠𝑖𝑛,𝑞,0

⋮

𝑎𝑞,3𝑠𝑖𝑛,𝑞,3

)

∙

(

𝑚00

𝑚01

𝑚02

𝑚03

𝑚10

⋮

𝑚33

)

(4.12)

Finally, all powers can be absorbed in one vector 𝑃

󰇍



and the 𝑤

󰇍



𝑞 are interpreted as rows of a

matrix 𝑊: 𝑃

󰇍



=𝑊𝑚

󰇍



(4.13)

This result is in literature commonly referred to as the polarimetric measurement equation,

[26], [84]. 𝑊 is the polarimetric measurement matrix. A necessary condition for this method

to correctly yield MPIC is that 𝑊 has to be of rank 16. To achieve this, we use the following

analyzer vectors 𝑎𝑖:

𝑎0,…,𝑎3=(1000); 𝑎4,…,𝑎7=(0100); 𝑎8,…,𝑎11=(0010); 𝑎12,…,𝑎15=(0001) (4.14)

Four measurements are performed with the input polarizations:

𝑠𝑖𝑛,0,𝑠𝑖𝑛,4,𝑠𝑖𝑛,8,𝑠𝑖𝑛,12=(1100); 𝑠𝑖𝑛,1,…,𝑠𝑖𝑛,13=(1010); 𝑠𝑖𝑛,2,…,𝑠𝑖𝑛,14=(1001); 𝑠𝑖𝑛,3,…,𝑠𝑖𝑛,15=(1

−1

00) (4.15)

Again, even though this gives 16 independent “measured” powers, only four physical

measurements are carried out. Each one gives four virtual measurements by virtue of four

different analyzer vectors. With this, it can be shown that the polarimetric measurement

equation simplifies and can be solved for 𝑚

󰇍



(

𝑚0,0

𝑚1,0

𝑚2,0

𝑚3,0

𝑚0,1

𝑚1,1

𝑚2,1

𝑚3,1

𝑚0,2

𝑚1,2

𝑚2,2

𝑚3,2

𝑚0,3

𝑚1,3

𝑚2,3

𝑚3,3

)

=12

(

𝑠𝑜𝑢𝑡,0,0+𝑠𝑜𝑢𝑡,3,0

𝑠𝑜𝑢𝑡,0,0−𝑠𝑜𝑢𝑡,3,0

−𝑠𝑜𝑢𝑡,0,0+2𝑠𝑜𝑢𝑡,1,0−𝑠𝑜𝑢𝑡,3,0

−𝑠𝑜𝑢𝑡,0,0+2𝑠𝑜𝑢𝑡,2,0−𝑠𝑜𝑢𝑡,3,0

𝑠𝑜𝑢𝑡,0,1+𝑠𝑜𝑢𝑡,3,1

𝑠𝑜𝑢𝑡,0,1−𝑠𝑜𝑢𝑡,3,1

−𝑠𝑜𝑢𝑡,0,1+2𝑠𝑜𝑢𝑡,1,1−𝑠𝑜𝑢𝑡,3,1

−𝑠𝑜𝑢𝑡,0,1+2𝑠𝑜𝑢𝑡,2,1−𝑠𝑜𝑢𝑡,3,1

𝑠𝑜𝑢𝑡,0,2+𝑠𝑜𝑢𝑡,3,2

𝑠𝑜𝑢𝑡,0,2−𝑠𝑜𝑢𝑡,3,2

−𝑠𝑜𝑢𝑡,0,2+2𝑠𝑜𝑢𝑡,1,2−𝑠𝑜𝑢𝑡,3,2

−𝑠𝑜𝑢𝑡,0,2+2𝑠𝑜𝑢𝑡,2,2−𝑠𝑜𝑢𝑡,3,2

𝑠𝑜𝑢𝑡,0,3+𝑠𝑜𝑢𝑡,3,3

𝑠𝑜𝑢𝑡,0,3−𝑠𝑜𝑢𝑡,3,3

−𝑠𝑜𝑢𝑡,0,3+2𝑠𝑜𝑢𝑡,1,3−𝑠𝑜𝑢𝑡,3,3

−𝑠𝑜𝑢𝑡,0,3+2𝑠𝑜𝑢𝑡,2,3−𝑠𝑜𝑢𝑡,3,3

)

(4.16)

In this form, all 16 elements 𝑚𝑖,𝑗 of 𝑀PIC can be obtained directly from the four

measurements. We use (4.16) for all measurements in this chapter.

To summarize, four the different states from (4.15) are launched into the PIC, namely TE-,

+45° linearly-, right-hand circularly and TM-polarized light. From the measured output

states, 𝑀PIC is obtained via (4.16). Then, the complete polarization behavior of the PIC is

known.

4.1.3 Müller Matrix Decomposition

Even though with the previous chapter we can in principle fully describe the polarization

properties of a PIC, it remains unclear how to extract tangible properties like birefringence,

polarization rotation or PDL. The reason is that the physical interpretation of an

experimental Müller matrix is generally not obvious. One way of interpreting a given matrix

is to decompose it into simple factors which can then be understood individually. Several

authors investigated the possibilities of decomposing experimental Müller matrices [85]–

[89]. We will follow Lu’s and Chipman’s method of polar decomposition [85]. They have

shown that any experimental 𝑀 can be written as:

𝑀=𝑀𝐷𝑀𝛥𝑀𝑅 (4.17)

Where MΔ represents depolarization and 𝑀𝐷 and 𝑀𝑅 are elliptical diattenuators and

retarders, respectively. They can be constructed from the diattenuation vector 𝐷

󰇍



, average

transmission 𝑇 and the retardation vector 𝑅

󰇍



=𝛿𝑢

󰇍



as given in appendix A). From the

experimental 𝑀, 𝑇 and 𝐷

󰇍



can be directly obtained [85]:

𝑇=𝑚00 𝐷

󰇍



𝑚00(𝑚01

𝑚02

𝑚03) (4.18)

From this, we can calculate 𝑀𝐷. If we assume non-depolarizing elements, MΔ becomes the

identity. Then, by calculating 𝑀𝐷−1, equation (4.17) yields 𝑀𝑅, which in turn yields 𝑅

󰇍



(see

appendix A)). Note that 𝑅

󰇍



is along fast axis of the retarder. The third component of 𝑅

󰇍



vanishes for a linear retarder and we get the case from chapter 2.3.3. Any polarization

conversion in a given DUT will express itself as an 𝑅

󰇍



that is neither parallel nor anti-parallel

to the 𝑠1-axis. The former corresponds to a fast axis parallel to the substrate (along TE),

while the latter corresponds to a fast axis perpendicular to the substrate (along TM, the

usual case for ridge waveguides). A DUT whose 𝑅

󰇍



and 𝐷

󰇍



are parallel is said to be

homogeneous [85].

In the general case, MΔ might not be the identity, i.e. the PIC might be depolarizing. An

SOA would be an example, as ASE has a stochastic polarization and its phase noise will be

imprinted on the relative phase between the two principal polarizations. In this case, the

calculation is slightly more tedious, effectively getting extended by the polarizance vector 𝑃

󰇍



The decomposition in the depolarizing case is given in [85]. The depolarization power Δ

gets defined by:

𝛥=1−|𝑡𝑟𝑎𝑐𝑒(𝑀𝛥)−1|

3 , 0≤𝛥≤1 (4.19)

4.1.4 Software Implementation

As mentioned before, the numerics of the Müller/Stokes measurements are implemented in

python. Using the bokeh library [90], a web-based interface is written that visualizes in- and

output polarization and displays the Müller matrix of a given sample. A screenshot is shown

in figure 4-8. It also allows the decomposition of the matrix into its retarding, diattenuating

and depolarizing parts, as described in the previous section.

Figure 4-8. Screenshot of the graphical interface for the Stokes measurements. The input polarization can be

controlled on the left using two sliders for 𝜓 and 𝜒. The Müller matrix can be measured and is displayed with

normalized values in the middle. The right panel shows the current output polarization.

With this, the measurement setup is ready to perform wavelength dependent Müller matrix

measurements of any integrated device with just fiber coupling.

4.2 Polarization Rotator Results

With the experimental methodology in place, measurements of actual devices can be

carried out. Optical photographs of fabricated PRs are shown in figure 4-9. A scanning

electrode microscope picture of the asymmetric waveguide used in the PRs is shown in

figure 4-10.

Figure 4-9. Photograph showing an overview of a fabricated 90° PR (top). The two zooms at the bottom show

the area around the 100 µm long in- and output taper (bottom left) and the 25 µm long taper in the mid

section (bottom right). The dummy etch window at the in- and outputs leads to a more homogeneous

etching along the PR. The waveguides are fabricated with a SiN mask.

Figure 4-10. Scanning electron microscope (SEM) picture of the asymmetric waveguide cross section. The top

width of the waveguide shown is 1 µm. The green inset shows that the sidewall is circular (with a diameter of

2.3 µm).

Results for 90° PRs and 45° PRs are presented in the following two sections.

4.2.1 90° Polarization Rotators

The fabricated polarization rotators are measured on the Stokes space setup as described in

chapter 4.1. Therefore, the full Müller matrix is retrieved as a function of wavelength.

Because the rotators are fabricated on bars of fixed length, the actual device under test

consists of two waveguides with the PR in between. Therefore we can write:

𝑀𝑃𝐼𝐶=𝑀𝑊𝐺,𝑜𝑢𝑡∙𝑀𝑃𝑅∙𝑀𝑊𝐺,𝑖𝑛 (4.20)

We assume that the in- and output waveguides are linear retarders with no rotation.

Further, we write the PR itself as a rotated linear retarder with some polarization dependent

loss (i.e. diattenuation): 𝑀𝑃𝑅=𝑀𝛿𝑃𝑅(𝜃𝑃𝑅)∙𝑀𝑃𝐷𝐿 (4.21)

A first simple measure for the performance is the PER of the exiting light for incident TM-

polarized light. It can easily be shown that

𝑃𝐸𝑅=𝑃𝑇𝐸

𝑃𝑇𝑀|𝑠𝑖𝑛=𝑇𝑀=𝑠𝑜𝑢𝑡,0+𝑠𝑜𝑢𝑡,1

𝑠𝑜𝑢𝑡,0−𝑠𝑜𝑢𝑡,1|𝑠𝑖𝑛=𝑇𝑀 (4.22)

So, by exciting the measured Müller matrix with a TM-polarized input vector, the PER of the

device can be calculated. The results in this framework for several devices are shown in

figure 4-12.

Figure 4-11. PER and loss for designs with different widths at a wavelength of 1.53 µm (left) and 1.55 µm

(right). The device with w=1.55 µm has an epitaxial defect in the input waveguide, giving around 8 dB of

excess loss. The design width refers to the width of the PR’s first section, all other widths (second section,

tapers) scale linearly with this width.

Figure 4-12. Measured PER and insertion loss of the same device from two different wafers. The PER is above

10 dB on both wafers. Both devices have a PDL below 1 dB, with an average loss around 1 dB.

To analyze the PRs further, we want to understand the origins of the PER. We decompose

MPIC into its elliptical retardation and diattenuation components as outlined in section

4.1.3. Thus, we retrieve the retardation vector 𝑅

󰇍



that emerges from the combination of in-

and output waveguides and the actual PR. Since the in- and output waveguides are

symmetric, their fast axis will be at 90° with the substrate. So, in the absence of a PR, 𝑅

󰇍



will

be along −𝑠1.

It can be shown geometrically that in the case of an ideal PR (i.e. full polarization

conversion), the fast axis 𝑅

󰇍



of MPIC will lie in the 𝑠2𝑠3-plane. In other words, the fast axis

will be at 45° with the substrate in real space or 90° with the 𝑠1-axis in Stokes space. The

total rotation around that axis, 𝛿, should be 180°. Therefore, we take the following

quantities to specify the PR performance: the angle enclosed by 𝑅

󰇍



and the 𝑠1-axis, and 𝛿.

The former is a measure for the rotation of the principal axes, the latter gives the phase

retardation between the principal axes. The relationships are illustrated in figure 4-13.

Figure 4-13. Principle of operation of an ideal integrated polarization rotator on incident TM light (𝑠). The PR

is preceded and followed by non-rotating waveguides of different lengths, i.e. different phase retardations.

Left: zero retardation before and behind the PR. The overall PIC then simply rotates around the 𝑠2 axis, i.e. its

fast axis is with 45° to the PIC substrate. Middle: 20° phase retardation before the PR, zero behind. The PIC

now rotates around an axis that is not along 𝑠2 anymore, but still at 90° to the 𝑠1 axis. Even when an

additional retardation behind the PR is present (50°, right graph), the rotational axis is still at 90° to the 𝑠1

axis, i.e. in the 𝑠2𝑠3 plane.

The fast axis and retardation of the same devices as in figure 4-12 is shown in figure 4-14.

The retardance 𝛿 is around 25° too small. This can be explained by the fact that for the

design of the devices, the birefringence of the asymmetric waveguide was overestimated.

The rational function Δ𝑛(𝑤) describing the width dependence did not sample accurately

enough around the minimum, which is where the PR operates. Thus, the design can be

improved by increasing its length accordingly.

Figure 4-14. Measured retardance 𝛿 and fast axis angle ∠(𝑅

󰇍



,𝑠𝑇𝐸) of the same device from two different

wafers. The ideal angles for retardance and fast axis are depicted as dashed lines.

Figure 4-15. Measured dependence of PER and loss (left) and fast axis angle and retardance (right) on

temperature for 90° PRs.

Figure 4-16. Measured dependence of PER and loss (left) and fast axis angle and retardance (right) on input

power for 90° PRs.

4.2.2 45° Polarization Rotators

Using the same approach as in the previous section, devices designed for 45° rotation are

characterized. Note that the PER should ideally be 0 dB for these devices, as both TE and

TM incident light ought to be mapped to a 50/50 mix of TE and TM. ±1 dB PER

corresponds to ±3.3° error around the ideal 45°.

Figure 4-17. Experimental results for 45° polarization rotators. The mask widths are different by 150 nm for

the two devices shown. The left device has narrower waveguides. Apart from the widths, the design for both

devices is identical, and they are processed on the same wafer. The losses are normalized to straight

symmetric waveguides of the same length. For some polarizations and wavelengths, the rotators have less loss

than the symmetric waveguides, which is expressed in positive loss values in the plots.

An analysis of the mask width dependence is shown in figure 4-18. The PER stays within

±1 dB for a mask width deviation of at least 150 nm.

Figure 4-18. Dependence of PER and loss on mask width for 45° PRs. The left shows the dependence for

1.53 µm wavelength, the right for 1.55 µm.

Following the same logic as for the 90° rotators, the Müller matrices of the devices are

decomposed to calculate the retardances and fast axes. The result for the same devices as

in figure 4-17 are shown in figure 4-19.

Figure 4-19. Measured retardance 𝛿 and fast axis angle ∠(𝑅

󰇍



,𝑠𝑇𝐸) of 45° rotator devices. The mask widths are

different by 150 nm in for two devices shown. The ideal angles for retardance and fast axis are depicted as

dashed lines (both 90°).

4.3 EAM Results

4.3.1 DC Measurements

The fabricated EAMs are measured the same way as the PRs, with voltage as an additional

parameter. The decomposition of the measured Müller matrices again allows the

quantification of retardance and diattenuation. Because the slow axis of EAM as well as in-

and output waveguides should be along the TM-polarization, this should also be the axis of

the retardance and diattenuation vectors. An optical photograph of an exemplary EAM

structure is shown in figure 4-20.

Figure 4-20. Photograph of an entire bar of fabricated EAMs (left) and zoom on a single EAM (right). The bar

is anti-reflection coated on both sides. EAMs of different active lengths are fabricated. The right shows a

picture of an EAM with an active length of 100 µm.

The absolute birefringence of an EAM is hard to measure because the devices always

comprise butt-jointed passive waveguide sections on both sides of the EAM. The measured

birefringence cannot be readily divided into the EAM and passive waveguide contributions,

respectively. However, applying a voltage to the EAM leaves the passive waveguide

birefringence unchanged, so that the birefringent change in the EAM versus voltage can be

obtained directly. So, we define a relative birefringence such that it vanishes for zero bias.

The corresponding result is shown for two different wavelengths in figure 4-21.

Figure 4-21. Experimental birefringence and PDL for an EAM with 200 µm active length. Voltage dependences

are shown for two different wavelengths: 1.565 µm (left) and 1.57 µm (right). The chip substrate is at 20°C,

3 dBm are launched into the device.

A summary of the EAM measured wavelength dependence is shown in figure 4-22. A slight

dependence on input power is also found, as shown in figure 4-23. Higher input powers

tend to increase the achievable extinction. This can be attributed to the larger

photocurrents that are flowing, corresponding to a higher concentration of heavy holes in

the active layer. Heavy holes can participate in intraband absorption. The photocurrent can

also heat up the active region to such an extent that the bandgap shrinks, leading to higher

absorption too.

Figure 4-22. Wavelength dependence of birefringence (top) and PDL (bottom). The input power is 3 dBm, the

substrate temperature is 20°C.

Figure 4-23. Input power dependence of birefringence (top) and PDL (bottom). The wavelength is 1.57 µm,

the substrate temperature is 20°C.

4.3.1.1 Extraction of Chirp

As suggested by theory in chapter 3.2, the previous section proves that the effective index

of the EAM TM mode depends much less on reverse bias than the TE index. The same is

true for the absorption of the two modes. Mathematically:

𝜕𝑛𝑇𝐸

𝜕𝑉 −𝜕𝑛𝑇𝑀

𝜕𝑉 ≈𝜕𝑛𝑇𝐸

𝜕𝑉 ; 𝜕𝛼𝑇𝐸

𝜕𝑉 −𝜕𝛼𝑇𝑀

𝜕𝑉 ≈𝜕𝛼𝑇𝐸

𝜕𝑉 (4.23)

So, it follows that: 𝜕𝑛𝑇𝐸

𝜕𝛼𝑇𝐸≈𝜕𝛥𝑛

𝜕𝑃𝐷𝐿 (4.24)

In other words, the measured birefringence and PDL offer a good approximation for the

chirp parameter. We use the definition that is sometimes referred to as Henry factor or

simply chirp, which we then may rewrite as follows:

𝜈=2𝜕𝜑

𝜕𝛼≈4𝜋

𝜆𝜕𝛥𝑛

𝜕𝑃𝐷𝐿 (4.25)

With the PDL in natural logarithmic scale. So, in the approximation of (4.23), the chirp can

be simply deduced from Müller measurements. This may be useful in many circumstances

since (4.23) is a good approximation for most typical MQW-based devices. The extracted

chirp parameter of a 200 µm long EAM is given in figure 4-24.

Figure 4-24. EAM chirp extracted via (4.25). The chirp parameter crosses zero at around 2.8 V for most

wavelengths. At bias-wavelength combinations that yield too low output power, the Müller matrix and

therefore the chirp could not be determined due to the polarimeter sensitivity.

The measured chirp shows typical behavior, starting with positive values for small voltages

and crossing zero at higher voltages. The crossing moves to lower voltages with lower

wavelengths, as the photon energy moves closer to the band gap.

4.3.2 Small Signal Measurements

To determine the electro-optic bandwidth of the EAMs, a lightwave component analyzer

(LCA) is used. It is essentially a vector network analyzer (VNA) with a calibrated electro-optic

frontend to characterize the frequency-dependent scattering matrix of electro-optic

components. A schematic of the setup that was used for the measurements is shown in

figure 4-25.

DUT

LCA

TLS PC

Figure 4-25. Setup used for the electro-optic small signal measurements. The LCA consists of a VNA and a

calibrated photodetector. A reverse bias voltage is applied via an external bias-T.

Measured frequency responses for a 200 µm long EAM are shown in figure 4-26. A

modulated electrical power of -8 dBm (100 mV peak-to-peak) is applied to the EAM. The

figure includes an overview of the dependence of bandwidth on bias, wavelength,

temperature and optical power. To allow a reliable determination of the 3 dB bandwidth

𝑓3𝑑𝐵 even when the power received by the LCA is low (i.e. the signal is noisy), a moving

average filter is used. It averages across 10 points, 1600 points are recorded by the LCA.

The electro-optic bandwidth is found to be highest at a wavelength 1.569 nm, a

temperature of 20°C and around 6 dBm input power. The optimum bias is around -3.5 V.

Figure 4-26. Top left: measured electro-optic frequency responses of a 200 µm long EAM at different bias

voltages. The EAM is operated at a wavelength of 1.569 µm, 20°C and 6 dBm input power. The color maps

show the 3 dB bandwidth dependence on wavelength (top right), temperature (bottom left) and input power

(bottom right).

The highest achievable bandwidth is around 17 GHz. To compare this to the simulations of

the previous chapter, we recall that figure 3-23 suggest 1.5 nF/m at biases above -3 V. So a

200 µm long device should give 0.3 pF. Together with the measured series resistance of

25 Ω, the theoretical RC limited bandwidth is 21 GHz.

4.3.3 Large Signal Measurements

The setup used for the small signal measurements is extended to enable NRZ OOK. An

electrical PRBS source is used to generate a 39 Gbit/s PRBS of length 231−1 bits. The

electrical signal from the PRBS is slightly impaired, as shown in figure 4-27. The EAM is

operated at 1565 nm, 20°C and a bias voltage of -2.5 V. A clean eye opening is obtained

as seen in figure 4-28, but the double-rails from the electrical signal get enhanced. The

extinction ratio is 4.3 dB. The eye is recorded with a high speed photodetector and a digital

communications analyzer (DCA).

Figure 4-27. Electrical eye diagram at 39 Gbit/s generated by the PRBS. The eye shows double-rail behavior.

Figure 4-28. Measured eye diagram generated by an EAM at 39 Gbit/s. The double-rail features stem from the

slightly impaired electrical signal that is generated by the PRBS source.

4.4 DP EAM Results

A SEM picture of a fabricated DP EAM is shown in figure 4-29. It is colorized for clarity.

2 mm

0.25 mm

Figure 4-29. Colorized SEM photograph of a manufactured DP EAM. The blue boxes are the windows for the

PR wet etch, the red line indicates the waveguide. The active areas are covered in BCB (green), the Au

contacts (yellow) reside on SiNx (purple) for electrical isolation.

4.4.1 DC Measurements

For the DC characterization, the fabricated device is again characterized via the

Stokes/Müller formalism. The Müller matrix of the integrated DP EAM is recorded as a

function of the two bias voltages of the respective EAMs.

As a first analysis, the DP EAM Müller matrix is decomposed according to 4.1.3 into its

retarding and diattenuating constituents. A detailed visualization of 𝐷

󰇍



is shown in figure

4-30.

Figure 4-30. Dependence of the normalized diattenuation vector 𝐷

󰇍



of the DP EAM on the bias voltages. The

color map on the left shows the dependence of the vector longitude 𝜓𝐷

󰇍



. The left plot shows the position

of 𝐷

󰇍



on the Poincaré sphere. Green points indicate that the second bias is zero, purple points indicate that

the first bias is zero. The smaller blue points correspond to both EAMs being biased. Each pixel on the left as a

corresponding dot on the right Poincaré sphere. The wavelength is 1.575 µm, the input power 8 dBm and the

substrate temperature is 20°C.

For a certain incident Stokes vector, the resulting output polarization as a function of the

two bias voltages is calculated via:

𝑠𝑜𝑢𝑡=𝑀𝑃𝐼𝐶(𝑉1,𝑉2)∙𝑠𝑖𝑛 (4.26)

Ideally, 𝑠𝑜𝑢𝑡 should be along [−1 0 0]𝑇 and [1 0 0]𝑇 in the two respective bias

conditions (𝑉1=0,𝑉2=−3 𝑉) and (𝑉1=−3 𝑉,𝑉2=0). Indeed, when 𝑠𝑖𝑛≈

[1 0.9 0.05 −0.42]𝑇, the DP EAM operates this way, as shown in figure 4-31.

Figure 4-31. Operation of the DP EAM when 𝑠𝑖𝑛≈[1 0.9 0.05 −0.42]𝑇. Left: relative power of the exiting

signal as a function of the two bias voltages. Without bias, the TE fraction is around 50%, so its relative

power is around -3 dB. Increasing one bias suppresses TE, increasing the other increases the relative TE power

by virtue of suppressing the TM fraction. Right: output polarization on the Poincaré sphere for the voltages on

the left. Green dots show the SOPs when only sweeping the first EAM, purple dots when only sweeping the

second EAM. Blue dots correspond to both EAMs being biased.

With the results in figure 4-31, it is shown experimentally that the device can indeed

modulate two orthogonal polarization states independently. The next section shows how

this can be used for PDM communication channels. The DP EAM has an on-chip insertion

loss of around 14 dB. Most of this loss originates from the butt-joint transitions between

active and passive waveguides, which are measured to be around 3 dB per interface, so

12 dB in total.

4.4.2 Transmission Experiments

4.4.2.1 On-Off Keying

A PDM transmission setup as shown in figure 4-32 is used to demonstrate non return-to-

zero (NRZ) OOK transmission in a PDM scheme. A “Mickey mouse”-type polarization

controller is used to set the SOP at the DP EAM input. A second one is used to align the

modulated PDM signal with the coordinate system of a polarization beam splitter (PBS). A

1 dB coupler is used to tap 20% of the modulated signal an monitor it on an optical

spectrum analyzer (OSA). Two high speed photodiodes at the two outputs of the PBS are

used do analyze the eye diagrams on a digital communication analyzer (DCA). An erbium-

doped fiber amplifier (EDFA) is used in conjunction with an etalon wavelength filter on the

receiver side of the setup.

DUT 20/80

OSA

EDFA PBS

DCA

TLS

PRBS

D1D2

Figure 4-32. Transmission setup used for PDM-OOK. A PRBS source generates two data signals for the two

respective EAMs of the DP EAM. Polarization controllers (PC) are used to set the input polarization and to

align the output polarization with the PBS coordinate system on the receiver side.

The received optical power is set to 3 dBm. Wavelength is 1570 nm, the TEC is at 20°C.

Figure 4-33. Recorded eye diagrams after 32 Gbit/s modulation in the DP EAM, so a total data rate of

64 Gbit/s. The upper eye diagram corresponds to the first EAM, the lower to the second. The extinction ratio

is above 6 dB in both polarizations.

4.4.2.2 Quaternary Pulse Amplitude Modulation

To demonstrate 100 Gbit/s transmission, the setup is modified to allow PAM-4

transmission. The new setup is shown in figure 4-34.

To allow offline digital signal processing (DSP), a coherent receiver is used. The receiver is

located in another laboratory at HHI of the photonic networks (PN) department, so an in-

house fiber link is used to interconnect the two laboratories. The link transmission loss is

measured to be 0.9 dB. PN also provided an arbitrary waveform generator (AWG) that is

used to generate two PAM-4 signals. A variable optical attenuator (VOA) or an 80 km fiber

coil is used to emulate a real transmission link. The receiver itself consists of an integrated

coherent receiver (ICR) module together with a 4-channel digital sampling oscilloscope

(DSO) that is used to record the received waveforms.

VOA

Off li ne

DSP

DUT

OSA

EDFA

PBS

AWG

D1D2

I/V

1/99 PM

PN Lab

TLS

PN Lab

ICR DSO

PC Lab

Back-to-back

80 km

DCA

Figure 4-34. Setup used for PDM-PAM4 transmission. Two laboratories are involved in the experiment, one of

HHI’s photonic component department (PC, top) and one of the photonic networks department (PN, bottom).

The two labs are 6 floors apart and are interconnected by a SSMF link with 0.9 dB insertion loss.

The DSP is used for chromatic dispersion compensation, phase estimation and data-aided

channel estimation [91]. The signal is resampled to two samples per symbol. The symbol

rate per EAM is 28 GBaud, giving a total 2x2x28 =112 Gbit/s gross data rate.

Constellation diagrams for back-to-back and 80 km transmission are shown in figure 4-35.

In both cases, bit error ratios of 2.2×10−3 are measured, giving error-free transmission of

100 Gbit/s with hard-decision forward-error correction, assuming 7% overhead [92, p. 1]

and allowing up to 5% protocol overhead. The EAMs are driven with 2 V peak-to-peak

modulation around a bias of 2.5 V.

B2B

80 km

Figure 4-35. Received constellation diagrams of 28 Gbaud PDM-PAM4 in the back-to-back case (top) and

after transmission over 80 km of SSMF (bottom).

4.5 DP EML Results

The most complex PIC that is fabricated as part of this work is a complete dual-polarization

externally modulated laser (DP EML). Besides the DP EAM, it also monolithically integrates a

45° PR and a distributed feedback (DFB) laser source. For the DFB laser, the standard

building block in the integration platform is used. It has an active length of 200 µm. The

45° PR is the device that is characterized in section 4.2.2. An optical photograph of the

fabricated DP EML is shown in figure 4-36. It has a footprint of around 1.5 mm² of InP.

3.2 mm

EAM EAM

DFB

MD 45° PR 90° PR

200 µm

out

Figure 4-36. Photograph of the Dual Polarization Externally Modulated Laser (DP EML). From left to right: a

monitor diode (MD), followed by a DFB. The DFB emits TE polarization. The first polarization rotator (PR)

rotates polarization by 45°. The following DP EAM is the same as in the previous section. The output facet is

anti-reflection coated. The PIC width of 200 µm is dominated by the RF pads, the length is 3.2 mm.

4.5.1 DC Measurements

Just as the previous devices, the DP EML is characterized using the setup from section 4.1.

In the case of the DP EML, the laser source is integrated on chip. So, rather than measuring

the Müller matrix of the device, only the Stokes vector exiting the device can be measured.

For practical purposes, the DP EML EAMs are contacted with RF probes also for the DC

measurements. Only the DFB laser is probed with DC needles. A picture of a contacted DP

EML is shown in figure 4-37. An emission spectrum of the unmodulated DP EML is shown

in figure 4-38. The DFB has a side mode suppression ratio (SMSR) of over 45 dB, its series

resistance is about 25 Ω and it has a threshold current of 20 mA.

Figure 4-37. Optical photograph of a contacted bar containing DP EMLs. The DFB laser is on the left and is

contacted by two probe needles. The two EAMs are contacted by GSG high-frequency probes that contain an

integrated AC-coupled 50 𝛺 resistor. The fiber on the right for optical coupling has a tapered tip and a mode

field diameter (1/e²) of 3.5 µm.

Figure 4-38. Output spectrum of the DP EML. No bias voltages are applied to the EAMs and the TEC is set to

20°C. The DFB is biased with a CW current of 95 mA. The side mode suppression ratio exceeds 45 dB.

The output SOP of the DP EML as a function of the two EAM biases is measured. Due to

the 3 mm long output waveguide with non-zero birefringence, the output SOP for zero

bias is arbitrary. For understanding the SOP behavior, however, the ability to measure the

complete output Stokes vector becomes very powerful: we can rotate the output SOP in

our model using any arbitrary retarder. For clarity, we rotate the output SOP to 45° linearly

polarized light, i.e. 𝑠45°=[1 0 1 0]𝑇. The result is illustrated in figure 4-39. The

strategy to find the (generally elliptical) retarder that achieves this rotation is as follows: we

define a retardation vector 𝑅

󰇍



 using the unbiased (and normalized) output SOP 𝑠0 as:

𝑅

󰇍



=𝑠0×𝑠45° (4.27)

This way, 𝑅

󰇍



is orthogonal to both SOPs, so rotating around it can map one onto the other.

The rotation angle 𝑅 then simply is the spherical distance between 𝑠0 and 𝑠45°:

𝑅=∠(𝑠0,𝑠45° )(4.28)

For the spherical distance see appendix D1). For the details of general elliptical retarders see

A). The output SOPs at the DP EML facet as well as the rotated SOPs are shown in figure

4-39. The DFB laser is biased with 130 mA and the PIC substrate temperature is set to

20°C. Ideally, the trajectories corresponding to the two respective EAMs should go along

the equator of the Poincaré sphere, but in opposite directions, i.e. towards pure TE and TM

respectively. After rotation, the measurement clearly show that the two EAMs do indeed

cause trajectories in opposite directions, but they have some residual movement along the

𝑠3-axis. This is the effect of the EAM chirp. The retarder necessary for the virtual rotation

has a retardation of 75° around and axis with 𝜓=0° and 𝜒=−20°, so it is purely circular.

The dependence of TE- and TM-polarized output power after the virtual SOP rotation is

shown in figure 4-40. The colormaps also include contour lines, showing that the

100

modulation of the two polarizations is indeed orthogonal with respect to the two EAMs.

The powers in the two polarizations are obtained from the measured Stokes vectors via:

𝑃𝑇𝐸=𝑠0+𝑠1

2; 𝑃𝑇𝑀=𝑠0−𝑠1

2 (4.29)

ex-facet rotated

Figure 4-39. Output state of polarization (SOP) of the DP EML for different bias voltages. The laser current is

130 mA and the TEC is set to 20°C. Left: measured SOPs of the light exiting the facet. Right: SOPs rotated

such that the unbiased DP EML emits 45° linearly polarized light. The rotation is done after the measurement

in python, emulating a circular retarder with retardance 75°, 𝜓=0° and 𝜒=−20°. The retarder axis is

indicated as 𝑅

󰇍



. Green dots show the SOPs when only sweeping the first EAM, purple dots when only

sweeping the second EAM. Blue dots correspond to both EAMs being biased. The EAMs bias voltages are

between 0 and -5 V.

Figure 4-40. Colormaps and contour plots of the output power carried by the TE polarization (left) and TM

polarization (right) versus EAM bias voltages. Ideally, the contours should be perpendicular in the two plots,

corresponding to an ideal 90° PR and consequently no crosstalk between the two polarizations.

The total fiber-coupled output power of the DP EML is -25 dBm with zero bias. This can be

broken down as follows: 0 dBm is emitted from one side of the DFB, experiencing 1.5 dB

loss per PR on this particular wafer, 9 dB per EAM (including the two epitaxial butt-joint

101

interfaces) and 4 dB coupling loss to a tapered single mode fiber. The output power of

future devices could be improved by 10 dB by lowering the butt-joint interface coupling

loss between PR and EAM sections by 2.5 dB. Another ~5 dB could be gained by reducing

the DFB series resistance to below 10 Ω and mounting the chip on a proper heat sink.

4.5.2 Large Signal Measurements

For a large signal modulation experiment, a setup as shown in figure 4-41 is used. Just like

in the case of the DP EAM in the previous section, a polarization controller together with a

beam splitter and two high speed photodetectors and an oscilloscope form the receiver.

Two 20 Gbit/s PRBS signals of length 231−1 are generated and launched via the RF probes

onto the two EAMs.

DPEML EDFA PBS

PRBS

D1D2

DCA

Figure 4-41. Transmission setup used for PDM-OOK. A PRBS source generates two data signals for the two

respective EAMs of the DP EML. A polarization controllers (PC) is used to align the output polarization with

the PBS coordinate system on the receiver side. A digital communication analyzer record the eye diagrams and

dynamic extinction ratios.

For the generation of the 40 Gbit/s PDM-OOK signal, the bar substrate is set to 10°C, and

the two EAMs are biased via bias-Ts at -4.6 and -4.7 V respectively. The 20 Gbit/s signals

are amplified to a peak-to-peak voltage of 2 Vpp. The DFB current is 130 mA. The resulting

eye diagrams are shown in figure 4-42. The eye corresponding to the first EAM has a

strong DC fraction, limiting its dynamic ER to 2.5 dB. The ER corresponding to the second

EAM is 7 dB. The DC fraction can be attributed to a sub-optimal PR in this particular device,

resulting in an incomplete conversion of the light that is unmodulated in the first EAM. Due

to this incomplete conversion, some part of this light does not get modulated by the

second EAM either, giving rise to a DC fraction. The eye diagrams are noise limited due to

the comparably low output power of the DP EML. To ensure that the two optical eyes

indeed correspond to the two EAMs, respectively, we turned off one of the electrical

amplifiers feeding the EAMs. As expected, one of the eye diagrams vanished with the other

one remaining.

102

Figure 4-42. Recorded eye diagrams after 2x 20 Gbit/s modulation in the DP EML, corresponding to a total

data rate of 40 Gbit/s. The upper eye diagram corresponds to the first EAM, the lower to the second. The

extinction ratios (ER) are 2.5 and 7 dB, respectively. The ER of the upper eye is DC-limited. The modulated

power in the fiber is -26.5 dBm.

103

5 Conclusion & Outlook

5.1 Conclusion

The focal point of this thesis is polarization multiplexing in photonic integrated circuits.

Integrated polarization rotators are identified as the key enabler to achieve PDM

functionality on both the transmit- and receive side. To this end, two major milestones are

achieved in this thesis: first, bringing polarization rotators into the HHI InP photonic

integration technology. Second, exploiting the PRs to implement a dual-polarization

transmitter for 100 Gbit/s. With the iSTOMP, the theory for a completely new PDM/Stokes

receiver is also given.

A new design methodology for PRs is developed [93] and verified. It is based on the Jones

formalism and combines this fomalism with modern numerical optimization techniques to

achieve maximum fabrication tolerance. PRs with a conversion efficiency up to 24 dB are

demonstrated experimentally. No active tuning is necessary. The conversion stays above

10 dB across the entire C-band even among different wafers. The PRs are implemented as

part of HHI’s generic InP photonic integration technology.

Using the PRs, a new DP EA modulator configuration is proposed and implemented,

capable of error-free 100 Gbit/s PDM-PAM4 transmission [94], [95]. The DP EA modulator

concept is also shown in a fully integrated DP EML PIC, integrating a DFB laser source, two

PRs and two EAMs to give monolithic PDM transmitter [96]. To realize those PICs, EAMs are

introduced into HHI’s generic InP photonic integration technology as part of this work.

Extensive calculations of the MQW-based active layer are done to feed a circuit simulator.

Circuit simulations show that the PR PER should be above 15 dB for an OSNR penalty of the

transmitter below 1 dB.

All newly developed components (PR, EAM, DP EAM) are now part of HHI’s process design

kit (PDK). They can readily be used in multi-project wafers.

With the iSTOMP, an integrated version of a classical polarimeter is presented. It provides

PDM- or Stokes receiver functionality. Various ways of implementing such a device are

pointed out, which can be pursued with relative ease now since the only critical part are the

PRs.

Based on the Müller/Stokes formalism, a new measurement methodology is demonstrated.

It enables an analysis of the newly developed devices with unprecedented detail because it

goes beyond classical extinction measurements. By measuring the Müller matrix of any

given device, it allows insights into a device birefringence, mode hybridization and

depolarization. To the author’s best knowledge, no Müller matrix measurements of PICs

have been done before. The implemented measurement setup is accurate within 2° on the

entire Poincaré sphere and across the whole C-band. It is also stable for more than a week.

5.2 Outlook

The design methodology used to derive the PR design optimizes the tolerance versus one

parameter only, namely width deviations. As the methodology is proven, multidimensional

104

optimization might be done, to optimize the performance versus several parameters. These

might include wavelength to make the PR more broadband. Also epitaxial deviations

(thickness and material composition) can be taken into account.

Even though the presented DP EML is the first integrated PDM transmitter of its kind,

several further developments can be perceived. The demonstrated design was made with

the assumption that a 45° PR after the laser is the ideal component. Given the PDL along

the transmitter, this might not be true, however. In this case, it might be desirable to have

less ore more conversion. The lasing wavelength is also not optimized yet. The results on

single EAM devices indicate that a wavelength of around 1.565 µm should give better

performance over the now demonstrated 1.575 µm. Regarding output power, improving

the butt-joint interface coupling loss from between PR and EAM sections by 2.5 dB per

interface should be achievable, giving 10 dB more output power. Also improving the

parasitic resistance of the active sections would improve the DFB output power, possible

giving another 5 dB when the device is properly packaged. This would also benefit the EAM

frequency response. The output power could be improved even further by adding SOA

sections to both EAMs. Finally, integrating an phase shifter with strong polarization

dependence in the same serial fashion would allow movement along on the entire Poincaré

sphere, making the device a true Stokes space transmitter and/or polarization tunable laser

source. For the latter, a slow phase modulator based on carrier injection would be sufficient

(see Kazi et al. in [97])

The extensive experimental analysis of the EAMs will be made available as part of HHI’s PDK

for external users of HHI’s generic photonic technology. The frequency response of the first

devices is RC-limited, mostly due to excessive p-side series resistances. Test runs showed

that an optimized processing can give more than two times lower resistances. Using these

optimized processes, the EAM bandwidth in future runs should reach values above 30 GHz,

at least if the carrier transit times prove to be fast enough. Then, 56 GBaud signaling

becomes feasible.

With the iSTOMP, a new receiver scheme is proposed that can be used together with the

DP EAM to avoid coherent reception. MMIs and photodetectors are already part of HHI’s

PDK and the necessary PRs can be implemented following the presented methodology.

Some custom MMI design might be required, however. To make use of the devices in real

transmission systems, the required DSP complexity to retrieve the Stokes vector from the

measured signals remains to be analyzed.

The measurement of the full Müller matrix of an integrated device can enable new insights

into device characteristics even beyond what is demonstrated so far. The depolarization of

devices is encapsulated in its Müller matrix, potentially giving insight into noise mechanisms

such as spontaneous emission in SOAs, non-coherent scattering or nonlinear processes [98].

5.3 Acknowledgements

I would like to thank Prof. Dr. Martin Schell for allowing me to pursue the topics his young

PhD student was so keen on investigating. In modern science, however, a single person

105

cannot achieve much on his or her own. I am especially indebted to my supervisor Dr.

Francisco M. Soares. His tireless advice and technological expertise was invaluable. Special

thanks go to Dr. Ute Troppenz for many fruitful discussions during late hours and thorough

proof reading. I would also like to thank my group leader, Dr. Martin Moehrle, for his

patience and trust, as well as for sharing the knowledge of the highly skilled and

experienced expert that he is. Without Dr. Norbert Grote much of this work would have

been impossible as it is based on the large number of projects he acquired for HHI. I owe

thanks to the entire clean room team at HHI, whose excellence shines through in the

resulting devices.

106

6 Appendices

A) Müller Matrix of Elliptical Diattenuators and Retarders

The most general diattenuators and retarders are elliptical. They are used in this work to

accurately model the elements in the polarization controller of the experimental setup (see

4.1.1.3) and for the decomposition of measured Müller matrices (see 4.1.3). We use the

same definitions as Lu and Chipman in [85].

For a given diattenuation vector of length 𝐷

𝐷

󰇍



=(𝐷𝐻

𝐷45

𝐷𝐶)=𝐷(𝑑1

𝑑2

𝑑3) (6.1)

and the unpolarized transmission 𝑇, we can write the diattenuator Matrix as

𝑀𝐷=𝑇(1 𝐷

󰇍



𝑇

𝐷

󰇍



𝑚𝐷) (6.2)

With the 3x3 kernel:

𝑚𝐷=√1−𝐷2 +(1−√1−𝐷2)

𝐷2𝐷

󰇍



𝐷

󰇍



𝑇 (6.3)

For a given retardation vector with total retardation 𝑅

𝑅

󰇍



=(𝑅𝐻

𝑅45

𝑅𝐶)=𝑅(𝑎1

𝑎2

𝑎3)=𝑅𝑅

󰇍



(6.4)

We can write the retarder matrix as

𝑀𝑅=(1 0

󰇍



𝑇

󰇍



𝑚𝑅) (6.5)

And, with the Kronecker symbol 𝛿 and the Levi-Civita symbol 𝜖:

[𝑚𝑅]𝑖𝑗=𝛿𝑖𝑗𝑐𝑜𝑠𝑅+𝑎𝑖𝑎𝑗(1−𝑐𝑜𝑠𝑅)+∑𝜖𝑖𝑗𝑘𝑎𝑘𝑠𝑖𝑛𝑅

𝑘=1 (6.6)

The retardation vector 𝑅

󰇍



can also be obtained from a given 𝑀𝑅 as follows:

𝑅=𝑐𝑜𝑠−1(𝑡𝑟𝑎𝑐𝑒(𝑀𝑅)

2−1)

𝑎𝑖=1

2𝑠𝑖𝑛𝑅∑𝜖𝑖𝑗𝑘

𝑗,𝑘=1 [𝑚𝑅]𝑗𝑘 (6.7)

107

B) Material Models

With the values from Table 2, any parameter 𝑝 of the quarternary material In1-xGaxAsyP1-y

can be interpolated in between the values of the four binary materials using the

stoichiometric fractions (Vegard’s law, [99]):

𝑝(𝑥,𝑦)=(1−𝑥)(1−𝑦)𝑝𝐼𝑛𝑃+(1−𝑥)𝑦𝑝𝐼𝑛𝐴𝑠+𝑥𝑦𝑝𝐺𝑎𝐴𝑠+𝑥(1−𝑦)𝑝𝐺𝑎𝑃 (6.8)

The stoichiometric fractions for a desired band gap can be deduced from the condition of

lattice matching to InP.

Parameter

InP

InAs

GaAs

GaP

Hydrostatic

deformation

potential, a

[eV]

-6.16

-5.79

-8.68

-9.76

Shear

deformation

potential, b

[eV]

-1.6

-1.8

-1.7

-1.5

Elastic

stiffness

coefficient,

[GPa]

1022

833

1188

1412

Pressue

coefficient

of Eg,

[GPa]

576

453

538

625

Lattice

constant [

Å]

5.8687

6.0583

5.6533

5.4505

Band gap Eg

[eV]

1.35

1.42

0.36

2.74

𝒎𝒆∗

0.077𝑚𝑒

0.021𝑚𝑒

0.063𝑚𝑒

0.33𝑚𝑒

𝒎𝑯𝑯

∗

0.56𝑚𝑒

0.517𝑚𝑒

0.5𝑚𝑒

0.54𝑚𝑒

𝒎𝑳𝑯

∗

0.12𝑚𝑒

0.024𝑚𝑒

0.088𝑚𝑒

0.16𝑚𝑒

Table 2. Material parameters of binary alloys after Li [100]. 𝑚𝑒 is the electron rest mass, 𝑚∗ denotes the

effective masses of electrons and holes.

For passive waveguides (i.e. with a core band gap much larger than the photon energy), we

use the refractive index model after [101]:

𝑛2=1+𝐸𝑑

𝐸0+𝐸𝑑𝐸

𝐸03+𝜂𝐸4

𝜋𝑙𝑛(2𝐸02−2𝐸𝑔2−𝐸2

𝐸𝑔2−𝐸2) (6.9)

Where 𝐸𝑔 is the band gap and 𝐸 is the photon energy. The oscillator parameters are (in eV):

𝐸0=0.595𝑥2(1−𝑦)+1.626𝑥𝑦−1.891𝑦+0.524𝑥+3.391 (6.10)

108

𝐸𝑑=(12.36𝑥−12.71)𝑦+7.54𝑥+28.91 (6.11)

C) Simulation Parameters

Meshsize

25 nm

Wavelength

1.55 µm

Threads

Table 3: MODE FDE Parameters

Meshsize

25 nm

Wavelength

1.55 µm

CVCS

Yes

Number of modes

Threads

Table 4 MODE EME Parameters

Meshsize

55 nm

Mesh Refinement

Threads

Table 5 FDTD Parameters

Figure 6-1. Convergence of EME simulations for tapered PR designs. The number of cells per taper should be

at least 10 if one wishes to calculate proper transmission losses. The number of modes should be larger than

8 to model the loss accurately. The PER is much less sensitive.

109

D) Formulas

D1) Orthodromic distance

To calculate the central angle Δ𝜉 between two points (2𝜓1,2𝜒1) and (2𝜓2,2𝜒2) on the

Poincaré sphere, we use the definition of the inner product and compute it in spherical

coordinates:

𝑐𝑜𝑠𝛥𝜉=( s2𝜓1 s2𝜒1

𝑠𝑖𝑛2𝜓1 s2𝜒1

𝑠𝑖𝑛2𝜒1)∙( s2𝜓2 s2𝜒2

𝑠𝑖𝑛2𝜓2 s2𝜒2

𝑠𝑖𝑛2𝜒2)

= s2𝜓1 s2𝜒1 s2𝜓2 s2𝜒2+𝑠𝑖𝑛2𝜓1 s2𝜒1𝑠𝑖𝑛2𝜓2 s2𝜒2+𝑠𝑖𝑛2𝜒1𝑠𝑖𝑛2𝜒2

= s2𝜒1 s2𝜒2( s2𝜓1 s2𝜓2+𝑠𝑖𝑛2𝜓1𝑠𝑖𝑛2𝜓2)+𝑠𝑖𝑛2𝜒1𝑠𝑖𝑛2𝜒2

= s2𝜒1 s2𝜒2𝑐𝑜𝑠(2𝜓1−2𝜓2)+𝑠𝑖𝑛2𝜒1𝑠𝑖𝑛2𝜒2(6.12)

Consequently: 𝛥𝜉=𝑎𝑟𝑐𝑐𝑜𝑠(𝑠𝑖𝑛2𝜒1𝑠𝑖𝑛2𝜒2+𝑐𝑜𝑠2𝜒1𝑐𝑜𝑠2𝜒2𝑐𝑜𝑠(2𝜓1−2𝜓2)) (6.13)

The distance 𝑑 between the points along the sphere of radius 𝑟 then writes as:

𝑑=𝑟𝛥𝜉 (6.14)

E) Abbreviations

Alternating current

Direct current

DFB

Distributed feedback

DOP

Degree of polarization

Dual polarization

DP EML

Dual-polarization externally modulated

laser

DUT

Device under test

EAM

Electro-absorption modulator

iSTOMP

Integrates Stokes mapper

LCA

Lightwave component analyzer

MQW

Multi quantum well

NRZ

Non return to zero

OOK

On-off keying

PAM-4

Quaternary pulse amplitude modulation

PBS

Polarization beam splitter

Polarization controller

PDM

Polarization division multiplexing

Polarization rotator

QAM

Quadrature-amplitude modulation

QCSE

Quantum-confined Stark effect

SCR

Space charge region

SMSR

Side mode suppression ratio

SOP

State of polarization

TEC

Temperature controller

110

7 Published Work

Parts of this work have already been published:

M. Baier et al., “Highly fabrication tolerant polarization converter for generic photonic

integration technology,” in 2016 Compound Semiconductor Week (CSW) [Includes 28th

International Conference on Indium Phosphide Related Materials (IPRM) 43rd International

Symposium on Compound Semiconductors (ISCS), 2016, pp. 1–2.

Moritz Baier, Francisco M. Soares, Martin Moehrle, Norbert Grote, and Martin Schell, “A

New Approach to Designing Polarization Rotating Waveguides,” presented at the European

Conference on Integrated Optics 2016, Warsaw, 2016.

M. Baier et al., “112-Gb/s PDM-PAM4 Generation and 80-km Transmission Using a Novel

Monolithically Integrated Dual-Polarization Electro-Absorption Modulator InP PIC,” in Proc.

43rd European Conference on Optical Communication (ECOC), 2017, p. Th.1.C.3.

M. Baier et al., „Fully Integrated Serial Dual-Polarization Electro-absorption Modulator PIC

in InP“, in Proceedings of the 19th European Conference on Integrated Optics, 2017.

M. Baier et al., “64 Gbit/s Generation from a Fully Integrated Serial 0.25 x 2.0 mm2 Dual-

Polarization Electroabsorption Modulator PIC in InP,” presented at the Compund

Semiconductor Week 2017, Berlin, 2017, vol. C1.2.

M. Baier und F. Soares, „Verfahren zum Herstellen eines Polarisationskonverters,

Polarisationskonverter und Polarisationskonverterelement“, Patent DE102016202634A1,

24-Aug-2017.

M. Baier, F. M. Soares, T. Gaertner, A. Schoenau, M. Moehrle, and M. Schell, “New

Polarization Multiplexed Externally Modulated Laser PIC,” in 2018 European Conference on

Optical Communication (ECOC), 2018, pp. 1–3.

M. Baier, F. M. Soares, T. Gaertner, M. Moehrle, and M. Schell, “Fabrication Tolerant

Integrated Polarization Rotator Design Using the Jones Calculus,” J. Light. Technol., May

2018.

M. Baier, F. Soares, und M. Schell, „Modulatoranordnung und Verfahren zum Modulieren

von Licht“, Patent DE102016224615, 14-Juni-2018.

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