IN DEGREE PROJECT TECHNOLOGY,FIRST CYCLE, 15 CREDITS

, STOCKHOLM SWEDEN 2019

Coupling of Light Into a Silicon-on-Silica Strip WaveguideQuantum Photonic Integrated Circuit Simulations

LUDVIG DOESER

ERIK RYDVING

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

INOM EXAMENSARBETE TEKNIK,GRUNDNIVÅ, 15 HP

, STOCKHOLM SVERIGE 2019

Koppling av ljus in i en bandvågledare av kiselnitratSimuleringar av integrerade kvantfotoniska kretsar

LUDVIG DOESER

ERIK RYDVING

KTHSKOLAN FÖR TEKNIKVETENSKAP

Coupling of Light Into a Silicon-on-Silica Strip Waveguide

Quantum Photonic Integrated Circuit Simulations

Ludvig Doeser – doeser@kth.seErik Rydving – erydving@kth.se

SA114X Degree Project in Engineering Physics, First LevelDepartment of Applied Physics

KTH Royal Institute of TechnologySupervisor: Klaus Jöns

Co-supervisor: Samuel Gyger

June 2, 2019

Acknowledgement

We would like to thank our supervisor Klaus Jöns and our co-supervisor Samuel Gyger for

their guidance and input during the course of the project. Their support has truly been help-

ful and appreciated. We would also like to thank Arthur Branny for participating in discus-

sions and forwarding relevant images and references, and Altay Dikme for reviewing the first

draft of the report and coming with instructive feedback. Finally, we would like to direct our

gratitude to our fellow students Martin Brunzell, Theodor Staffas and Sebastian Johnsen for

participating in meetings and discussion, making the project a highly enjoyable experience.

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Abstract

Recent technological advances have made it possible to miniaturize and integrate optical com-ponents in quantum circuits. The connection between different components is enabled bywaveguides, which support the propagation of the information carrier, a single-photon. Aprerequisite for functioning quantum photonic chips is the efficient coupling of non-classicallight into the circuit. In this work, this coupling efficiency from an on-chip single-photonsource, approximated by a dipole, into a waveguide has been simulated. The high refractiveindex material silicon nitride Si3N4 has been used as a strip waveguide, placed on top of asilicon oxide SiO2 wafer with surrounding air. To solve Maxwell’s equations in the structures,the finite difference time-domain (FDTD) method has been used through software by Lumer-ical. It is shown that for the light spectrum with wavelengths 750 to 800 nm a waveguidewith cross section dimensions 600x250 nm supports the fundamental transversal electric (TE)and transversal magnetic (TM) modes. The coupling efficiency is shown to reach 7 % in eachdirection when the dipole is placed on top of the waveguide. Having the dipole on in frontof the waveguide, however, results in over 50 % coupling in the forward direction. Addition-ally, it is shown that in-plane 2D-material single-photon emitters, approximated by in-planedipoles, give better results than out-of-plane dipoles for most of the tested configurations. Inconclusion, these results present evidence for a substantially higher coupling efficiency from2D-material quantum dots than have been achieved in experiments.

ii

Sammanfattning

Den senaste teknologiska utvecklingen har gjort det möjligt att miniatyrisera optiska kom-ponenter ner på ett fotoniskt integrerat kretskort. Sammankopplingen mellan olika kompo-nenter möjliggörs av vågledare, i vilka informationsbäraren, en enskild foton, propagerar. Enförutsättning för fungerande fotoniska chip är effektiv koppling av icke-klassiskt ljus in i kret-sen. I detta arbete har denna kopplingseffektivitet från en fotonkälla, approximerad som endipol, in i en vågledare simuleringats. Ett vanligt förekommande material med högt bryt-ningsindex, kiselnitrat Si3N4, har använts som bandvågledare, placerad på ett underlag avkiseloxid SiO2 och med luft runtomkring. För att lösa Maxwells ekvationer i vågledarstruk-turen har finita differensmetoden i tidsdomän (FDTD) använts i mjukvara från Lumerical.Det visas att för ljusspektrat med våglängder från 750 till 800 nm och en vågledare medtvärsnitt 600x250 nm, propagerar de fundamentala transversella elektriska (TE) och trans-versella magnetiska (TM) moderna. Kopplingseffektiviteten visas nå värden på 7 % i varjerikting när dipolen placeras ovanpå vågledaren. Om dipolen däremot placeras framför vå-gledaren nås en kopplingseffektivitet på över 50 %. Vidare visas det att 2D-fotonkällor somligger i vågledarens plan, approximerade av dipoler i vågledarens plan, ger bättre resutlat änmotsvarande ur dipolens plan. Dessa resultat presenterar sålunda bevis för att det är möjligtatt nå avsevärt större koppling av ljus från kvantprickar av 2D-material än vad som åstad-kommits experimentellt.

iii

Contents

1 Introduction 11.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theoretical Background 32.1 Quantum Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Single-photon Emitters and Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Manufacturing and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Lumerical Finite Difference Eigensolver (FDE) Simulations . . . . . . . . . . . . 82.7 Lumerical Finite Difference Time Domain (FDTD) Simulations . . . . . . . . . . 92.8 Coupling efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Convergence Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Method 123.1 Lumerical – MODE Solutions (2D) . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Convergence test for mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Sweep of waveguide width and height . . . . . . . . . . . . . . . . . . . 14

3.2 Lumerical – FDTD Solutions (3D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Convergence tests for meshes and monitors . . . . . . . . . . . . . . . . 153.2.2 Sweeps of dipole position and waveguide dimensions . . . . . . . . . . 163.2.3 Coupling schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Result & Analysis 184.1 2D Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Convergence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.2 2D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 3D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2.1 Convergence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.2 Dipole direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.3 Transmission analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2.4 Optimal dipole position and waveguide dimensions . . . . . . . . . . . 26

4.3 Coupling Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Pillar on top of waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.2 Dipole in front of waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 324.3.3 Tapered waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.4 Slot waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Discussion and Conclusion 35

Appendix A – MODE Solutions Code 39

Appendix B – FDTD Solutions Code 41

Appendix C – Example code for sweep 45

1 Introduction

In the course of the last century, the field of information technology has exploded and the use

of computers and the internet has become incorporated into everyday life for the vast ma-

jority of people. As modern information systems and devices were developed the speed and

size were of great importance. The revolution came with the invention and development of

the integrated circuit [1], commonly known as a chip, which is a thin piece of semiconducting

material, normally silicon, with electric components such as transistors on top. Since the birth

of integrated circuits in the 1950s, the number of transistors on a square centimeter of silicon

has doubled every two years as Moore anticipated [2], increasing by a factor of one million

over the last 50 years [3]. Electronic integrated circuits are used everywhere today. However,

in some cases, one could benefit from starting to use photonic integrated circuits instead.

Both the energy efficiency and the information speed could be improved by using photonic

integrated circuits [4]. These circuits differ from the microelectronic ones in that optical de-

vices such as optical amplifiers and phase shifters are used instead of transistors, resistors,

and capacitors. Also, the information carriers are no longer streams of electrons but particles

of light – photons – enabling quantum properties to be utilized. Therefore, apart from the

improvements that could be made in efficiency and speed, the photonic circuit is of interest in

the field of quantum technology where it could be a vital part for making the use of quantum

mechanics for practical applications possible [5].

The field of quantum photonic integrated circuits, which lie at the interface between quantum

optics and photonics, deals with miniaturization and integration of quantum optical elements

such as beam splitters, phase shifters, and single-photon emitters on chips. Because of its

stability and scalability, quantum photonics is destined to have a central role in future tech-

nologies. There are already existing photonic quantum circuits on silicon chips, and quantum

logic gates have been achieved with high fidelity by fabricating silica-on-silicon waveguide

quantum circuits [5]. The fact that quantum mechanical behavior is built into the system also

means that photonic circuits can be used as a non-general quantum computer to simulate

larger quantum systems which can be extremely memory consuming on conventional com-

puters. By hard-coding a quantum problem onto a photonic chip, complex simulations such

as boson sampling can be achieved by simply observing how the system evolves [6].

1.1 Purpose

Similar to how the number of transistors fitting onto a chip used in computers has increased,

the number of optical devices on a chip will hopefully follow a similar path. However, there is

1

Fig. 1. Miniaturization of the optical set-up [left] requires nanometer sized waveguides [right].These waveguides would be used to guide light from one optical component to another on a chip.

a fundamental limit due to the wavelength of the photon; this sets a lower bound on the size

and bending radii of a waveguide for supporting light propagation. As of now, quantum pho-

tonics experiments are done on large optical tables, taking up several square meters of floor

space. If these experiments instead could be manufactured onto a chip, it would enable a ma-

jor improvement in both size efficiency and reproducibility. For this to happen more research

has to be done. The work in this project is done in collaboration with the Quantum Nano

Photonics group of the Quantum and Biophotonics division at Alba Nova, KTH Royal Insti-

tute of Technology, with the goal to find optimal waveguide structures to achieve maximum

coupling efficiency of novel 2D material single-photon emitters [7, 8, 9] into the fundamental

waveguide mode .

Silicon nitride (Si3N4) waveguides on top of a silicon oxide (SiO2) platform would be a vi-

tal part of the photonic integrated circuit, as the waveguide would be used to transport light

from one part of the chip to another. The use of silicon nitride is mainly motivated by its

high refractive index and low absorption at the target wavelength (700 to 800 nm), properties

that enhance light propagation. There is, therefore, a need to study how waveguide geome-

tries, as well as different positions of a light source relative the waveguide, affect the coupling

efficiency. Due to the enormous parameter space and numerous approaches to integrate 2D

materials on chips, it is beneficial to simulate different structures prior to nanofabrication. Our

simulations will help in finding better waveguide structures for coupling 2D emitters.

1.2 Problem Formulation

• Investigate how the geometry of a strip waveguide affects the number of propagating

modes and find waveguide dimensions for supporting only the fundamental modes.

• Optimize the coupling of light from a single-photon emitter, approximated as a dipole,

into the strip waveguide.

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2 Theoretical Background

2.1 Quantum Simulations

Quantum mechanics is inherently hard; not only because of its unintuitive and often surpris-

ing results, but also simply because the math is complicated. For conventional computers,

those working with ones and zeros, another challenge in simulating quantum systems is the

fact that the properties of quantum object in many cases are in superpositions of several dif-

ferent states. This means that the required memory scales exponentially with the number of

particles in a simulated system, making large scale simulations impractical, or even impos-

sible, for anything less than a supercomputer [10]. So if a problem arises where the math is

too hard and the computer simulations are too demanding, the simplest solution might be

to build a model of the problem and observe how it evolves. One way to do hard-coded

simulations like this is to use photonic circuits, with photons as the quantum particle [11, 12].

2.2 Single-photon Emitters and Dipoles

To allow functionalities in photonic applications mentioned above one needs to go beyond

the use of classical light. Instead, light sources taking advantage of the quantum proper-

ties of light are required. In recent years single-photon sources, which uses the quantum light

states of individual photons, in the form of emissions from semiconductor quantum dots have

made remarkable progress [13]. For this project, however, we want to work with a novel type

of quantum dots originating from 2D materials, such as 2D transition metal dichalcogenides

[14]. These can be used for generating single photon sources by strain [15, 16] and are close

to becoming successfully integrated on waveguides [17], for instance on top of the waveguide.

A 2D material quantum emitter can be thought of as a two-level system. Energy can be sent

into the system to make it go from the ground state to the excited state. This state can then

relax back to the ground state and emit the same amount of energy in the form of a photon.

The decay rate, i.e. the time between excitation and spontaneous relaxation, in the quantum

description of this two-level system can be associated with a dipole moment which can be

described classically [18]. Consequently, one can approximate the single-photon emitter with

a classical dipole (see Fig. 2) instead of having to simulate one photon at a time. Moreover,

most 2D materials working as emitters are best modeled by dipoles in the same plane as the

top of the waveguide, so-called in-plane excitons, even though out-of-plane orientations have

also been shown to exist [19, 20].

To simulate a single frequency dipole is impossible in the time domain. This is because a

single frequency plane wave has no start or end in time domain. If one then wants to simulate

3

Fig. 2. Quantum dot approximations. A 2D-material working as a single-photon source can be ap-proximated as a two-level system, which in turn can be represented by a dipole in simulations.

a dipole emitting light for an amount of time one would need to have the light as a wave

packet. To get a single frequency simulation one needs the wave packet to have its peak at

the wanted frequency, and then only look at the results in a small frequency span around

the peak. This is a compromise between having a well-defined frequency and a well-defined

time span for the emitted light. A broadband simulation, i.e. a simulation with light of several

frequencies, can be achieved by having a wave packet with relatively flat frequency peak such

that the wanted frequencies have similar intensities. Although real emitters have spectra only

a few nanometers wide [9] the use of a broader spectrum is beneficial since several frequencies

can be simulated at once. In Fig. 3 one can see an example of the emitted wave packet when

the emitted light has wavelengths in the interval [750, 800] nm.

Fig. 3. A dipole and it’s symbol in the simulation environment Lumerical [top right] emitting lightin the frequency spectrum [374, 400] GHz [bottom left], corresponding to the wavelength spectrum[750, 800] nm [top left]. Note that the spectra peak at these intervals. In the time domain one can seehow the light intensity oscillates in order to achieve this frequency spectrum [bottom right].

2.3 Waveguides

To guide photons on a circuit board one needs a way to trap them in all but one direction

so they can propagate in the desired path. This can be done using waveguides with dimen-

sions in the order of a few hundred nanometers, of which there are several kinds. The most

4

common structures are shown in Fig. 4, and one of the simplest types is the rectangular strip

waveguides, shown in b).

Fig. 4. Different common structures of waveguides. a) Channel waveguide, b) Strip waveguide, c)Slot waveguide. Here n1, n2 and n3 are the refractive indices of the waveguide (red), the substrate(grey) and the cladding (white) respectively

A strip waveguide consists of a strip of material with a relatively high refractive index com-

pared to its surrounding, trapping the light inside it. The strip lays on top of another material

called the substrate, and can also have another material on top called cladding, usually with

lower refractive index than the substrate. The data for how the refractive indices depend on

frequency can be taken from an online refractive index database [21] for the cases silicon ni-

tride and air, while Lumerical’s1 built-in data can be used for silica. As for this work, the light

frequencies of interest lay in the interval [374, 400] GHz, corresponding to light wavelengths

of [750, 800] nm, since the 2D emitters work in this range [16, 22]. The associated refractive

indices are shown in Table 1.

Wavelength [nm] \ Refractive indices n1 (Si3N4) n2 (SiO2) n3 (Air)

750� 800 2.02597 1.45378 1.00028

Table 1: The refractive indices for the waveguide, substrate and cladding for the wavelength interval750� 800 nm. The highest refractive index is the waveguide’s, trapping the light within it.

2.4 Manufacturing and Limitations

By using nano-fabrication techniques waveguides such as the ones shown in Fig. 4 can be

manufactured onto thin slices of silicon wafers, which are used as a platform for integrated

circuits. One common method for waveguide fabrication is based on the main processes spin

coating, electron beam lithography, and dry etching [23, 24]. In short, the wafer used in the

process consists of three layers: silicon (bottom), silicon oxide SiO2 (middle) and silicon ni-

tride Si3N4 (top). The aim of the methods above is to partly remove the Si3N4 until only the

1see Sections 2.6 and 2.7 for more information about Lumerical

5

a) b) c) d)

Fig. 5. The top-down nanofabrication process in four simplified steps. The white layer represents thesilicon oxide (SiO2), while the bottom silicon layer cannot be seen. The red layer is the silicon nitride(Si3N4), i.e. the waveguide material, and the blue background represents the cladding of air. a) A layerof resist (green) has been applied by spin coating on top of the Si3N4. b) Then e-beam lithography isused for exposing the resist, as can be seen by the black region. c) The regions of the resist that havebeen marked, here in dark green, will be the only ones not removed during the developing processes.The Si3N4 is then etched away, leaving only the waveguide structure with marked resist on top. d)Finally, the remaining resist is chemically removed and the waveguide is done.

waveguide structure is left. One important aspect to have in mind is that these processes use

a top-down approach (as shown in Fig. 5). Thus, strip waveguides (as in Fig. 4b) are easily

fabricated, whilst a waveguide with non-uniform height, e.g. a ramp, would be much harder.

During the development of realistic waveguide structures, this fabrication limitation has to be

taken into consideration. In addition, for fully integrated quantum photonic circuits a non-

classical light source has to be integrated on the chip. This can be achieved by depositing a

semiconductor 2D material flake on top of the waveguide, forming a strain-induced single-

photon source. In simulations, this can be modeled as a dipole on top of the waveguide, as

described in Section 2.2.

2.5 Modes

Light traveling through a normal optical fiber is constrained by a phenomenon called total

internal reflection. This means that light encountering an intersection between two materials

with different refractive indices at a sufficiently low angle of incidence can be reflected with

no losses as seen in Fig. 6. When the diameter of the fiber becomes sub-wavelength in size, as

in the waveguides described above, this description breaks down, even though it is a helpful

way of thinking about it. For a more correct description, the concept of modes needs to be

introduced.

In a rectangular waveguide, the electromagnetic waves will only be able to travel in certain

types of propagation, called modes. How these behave depend on the dimensions of the

waveguide, the frequency of the light and the refractive indices of the different materials of the

waveguide structure. Mathematically, one can derive the propagating modes from Maxwell’s

6

Fig. 6. Total internal reflection in a waveguide. Even though light in a sub-wavelength strip waveg-uide due to its dimensions cannot be modeled with classical ray optics, it conceptually and intuitivelydisplays a way of thinking about the propagation. For a correct description an electromagnetic analysisfrom Maxwell’s equation has to be done, resulting in a discrete number of possible intensity profiles ofthe propagating light – the modes.

source-free equations

r · E = 0 r · B = 0

r⇥ E = �@B@t

r⇥ B =1

c2@E@t

,(1)

where E and B are the electric- and magnetic fields respectively. The common theoretical

derivation assumes a hollow waveguide, i.e. a frame of perfect electric conductor (PEC) be-

ing filled with air. By using PEC boundary conditions Maxwell’s equations can be turned

into two wave equations, which are eigenvalue problems with the magnetic and electric z-

components as eigenvectors. The discrete number of solutions will be standing waves con-

fined to the waveguide, each representing a mode. In fact, since the square of the electric field

is proportional to the light intensity, the modes essentially display how much of the light that

propagates in different regions of the waveguide.

Furthermore, the modes can be categorized as TE, transversal electric, or TM, transversal

magnetic. TE modes only have electric field components in the transverse direction, i.e. per-

pendicular to the directions of propagation, which means Ez = 0. Similarly, TM modes have

Bz = 0, meaning that the magnetic field components are in the transverse direction [25].

The size of the waveguide will affect the number of supported modes. In terms of light inten-

sity, the fundamental modes in TE and TM have one maximum, the first order modes have

two maxima and so on, as shown in Fig. 7. However, in this project only waveguides support-

ing the fundamental modes are desired. The reason is that having higher order modes also

means having all the lower modes. This could lead to the light being in either of the modes

or even in a mixture of them, making it hard to know which state the light is in. Addition-

ally, certain components on a chip such as filters or directional couplers only work with single

modes and, thus, the fundamental modes are required.

7

Fig. 7. The light intensity profile for different modes propagating in a waveguide. The top rowshows TE modes while the bottom row shows TM modes. For the fundamental modes [left] the lightintensity maximum is centered in the waveguide. This means the photons have the highest probabilityof being found there, which is a wanted feature in applications. For higher order modes [middle,right] there are several light intensity maxima. The images have been made by use of finite differenceeigensolver in MODE Solutions.

In the majority of cases, when the hollow waveguide approximation isn’t applicable, it’s not

possible to analytically calculate the mode fields and the propagation constants [26]. As in

the case of light propagating in a solid silicon nitride waveguide, the propagating modes will

be quasi-TE and quasi-TM modes, with their electric field having a main component in the

x-direction and y-direction respectively. For brevity, however, we will leave out "quasi" and

denote these as TE and TM modes. To calculate these, one has to rely on semi-analytical or

numerical approaches [25]. One of these numerical methods is based on the finite difference

time domain, FDTD.

Each mode also has an associated effective refractive index neff which is a measure of what the

phase delay of light propagating in this mode throughout the waveguide would be compared

to what it would be for light in a vacuum. A mode will propagate if its effective refractive

index is higher than the refractive index of the substrate, otherwise it will escape down into

the substrate.

2.6 Lumerical Finite Difference Eigensolver (FDE) Simulations

To run 2D simulations on a cross section of the waveguide we have used the MODE Solu-

tions software from Lumerical, and its FDE – finite difference eigensolver – method. It solves

8

Maxwell’s equations by formulating them into a matrix eigenvalue problem and uses sparse

matrix techniques to calculate the spatial profiles of the possible propagating modes [27]. This

algorithm is implemented in the software MODE Solutions by Lumerical Inc.

A simulation cannot be run without defining a finite simulation region. Therefore, the choice

of boundary conditions on the edges become important. One needs to choose conditions such

that they display realistic behavior. A common type of boundary is the PML boundary con-

dition, which stands for perfectly matched layer and is used to simulate an open boundary.

This is done by absorbing all light hitting the boundary so that no light is reflected. The PML

boundary makes it possible to truncate the simulation region such that only the area of im-

portance needs to be simulated. Caution, however, needs to be taken so the region doesn’t

become too small, since the PML boundaries thus will absorb light even though this isn’t

intended.

2.7 Lumerical Finite Difference Time Domain (FDTD) Simulations

To run the 3D simulations, we have used the software FDTD Solutions from Lumerical. FDTD

stands for finite difference time domain and is a method of simulation where both space and

time are discretized. The software divides the simulation region into a mesh and solves the

time-dependent Maxwell’s equations

r · D = ⇢ r · B = 0

r⇥ E = �@B@t

r⇥ H = J +@D@t

.(2)

Compared to the ones in Eq. (1), these equations include sources and have incorporated the

constitutional relations D = ✏E and B = µH, assuming no polarization or magnetization,

where ✏ and µ are the permitivity and permeability respectively. The software discretizes the

simulation region into mesh cells and solves Eq. (2) for each mesh cell before taking a step in

time. The mesh cells are rectangular blocks and can be of different sizes in different parts of

the simulation region to account for local finer details in the simulated structure. Equation (2)

can be solved by discretizing the vector fields as

E(x, t) = E((i+ 1/2)�x, (n+ 1/2)�t),

H(x, t) = H((i+ 1/2)�x, n�t),(3)

where i and n are integers. This is done in the same way for D and B respectively. This means

the magnetic and electric fields are offset by half a time step in relation to each other in a leap-

frog manner, such that only one of them is calculated at each half time step, and this is used to

9

calculate the other one at the next half time step. The derivatives in Eq. (2) are approximated

by the central difference method. Because of the way Maxwell’s equations look – with the

time derivative of one field depending on the spatial derivatives of the other – FDTD presents

an intuitive and straight-forward way of solving them.

2.8 Coupling efficiency

The built-in monitors for 3D simulations in Lumerical Frequency-domain field and

power monitors and Mode expansion monitors are useful for identifying how much

of the original emitted light that is propagating in a waveguide. Since the first one collects data

(the electromagnetic field) in the frequency domain this monitor can calculate the transmis-

sion from several light frequencies in one simulation. For coupling efficiency the vital result is

the transmission T. The use of the Mode expansion monitors requires it to be connected

with a Frequency-domain field and power monitor since it essentially works in the

same way, except that it calculates how much light that has been transmitted in a particular

mode. This mode transmission is given the name T forward in Lumerical.

When calculating the transmission we need to consider the Purcell factor, which accounts for

the enhancement in emission rate of a spontaneous light emitter. The values given by T and

T forward are, in fact, the power measured at the monitors divided by the power radiated

from a dipole in a homogeneous environment, which is called the sourcepower. In reality,

however, the power emitted by the dipole – when taking the effect from other structures in

the simulation environment into account – is given by the dipolepower. The Purcell factor

p is simply given by

p =dipolepowersourcepower

, (4)

and has to be used to correctly calculate the transmission values. Denoting the real transmis-

sion by Treal and the measured power by M one has

Treal =M

dipolepower(4)=

M

sourcepower · p =Tforward

p. (5)

Since Treal 2 [0, 1] the coupling efficiency in percent, denoted by �, will be given by

� = Treal · 100 [%]. (6)

2.9 Convergence Testing

When a numerical technique such as FDTD is used one ought to check if the solution, e.g. the

number of modes propagating in the waveguide or the transmission factor, has converged.

10

This is to make sure that the values obtained are accurate and trustworthy. To do this one

can make a convergence test, which in this case means that the mesh cell size is reduced

until the solution does not change value even after additional refinement. This change can be

quantified by

��(i) =

s(�i � �i�1)2

�2i

, i = 1, . . . , N, (7)

where � represents a measurable quantity, called the figure of merit, for the ith refinement of

the mesh cell size and the relative error ��(i) is the change in percent. The figure of merit can

be for instance the effective refractive index or the coupling efficiency. It’s desirable to have a

relative error in the simulations that’s lower than the relative error during the manufacturing

process [28], which can be expected to be approximately 5 %. Thus, this determines an ac-

ceptance level of the error, i.e. an upper bound on ��(i). However, another important aspect

of simulations to take into account is the simulation time. Desirably, it should be as short as

possible, all the while receiving results which are as accurate as possible. As stated in [29] the

relationship

ttot ⇠✓

�

�x

◆4

(8)

holds for FDTD in 3D, where ttot is the total simulation time, � is the wavelength of the present

light in the simulation and �x is the spatial size of a single mesh cell, which here is assumed

to be uniform, i.e. same in all directions (�x = �y = �z). To minimize ttot one can use con-

vergence testing of the mesh cell size to determine the largest �x that satisfies the demanded

accuracy. In a similar way one can try to minimize the simulation region without affecting the

results and in this case, when the mesh is uniform, the simulation time scales as

ttot ⇠ V, (9)

where V is the volume of the simulation region, which in 2D becomes the area. To minimize

the simulation time, however, one should apply a non-uniform mesh. This is done so that

areas free from complicated structures can be given a coarse grid, while a fine mesh will be

applied only where it is needed, i.e. in the waveguide structure where light propagates. This

will result in a mesh gradient, meaning that the mesh will continuously become more coarse

as the distance from the waveguide increases, which will decrease the simulation time.

11

3 Method

To investigate the optimal properties for a strip waveguide and, hence, the highest efficiency

in light coupling, two simulation softwares from Lumerical Inc have been used. Firstly, the

2D-simulator MODE Solutions was used to characterize the number of propagating modes.

The main simulator, however, was the 3D-simulator FDTD Solutions which was used to sim-

ulate light propagation in different structures.

In all simulations, the coordinate system has been defined as shown in Fig. 8, i.e. with x

and y defining the cross-section of the waveguide and z the direction of the propagating light.

The origin of the coordinate system is situated centered at the bottom of the waveguide, di-

rectly below the dipole in the Figure. In 2D the mesh used in simulation will be on a cross

section as shown in Fig. 8, while in 3D the mesh will be extended into the third dimension to

cover the whole waveguide. During 3D simulations the light source – the dipole – will be put

on one edge of the waveguide and the monitor on the other as illustrated in Fig. 8.

Fig. 8. The basic layout used in all 2D and 3D simulations. The coordinate system used in all simula-tions is here shown together with the waveguide in red, dipole as a blue dot, mesh (in 2D) and monitor.The blue background represents air and the white box under the waveguide is the SiO2 substrate, i.e.part of the silicon chip.

12

3.1 Lumerical – MODE Solutions (2D)

A model was designed and built in MODE Solutions – a software with layout as shown in

Fig. 9 – by using a script and not the GUI that comes with the program. This meant the model

could be quickly rebuilt by simply running the script, and variables such as waveguide di-

mensions could easily be altered in the Script Editor (bullet 9 in Fig. 9). The script used for

2D-simulations can be found in Appendix A.

Fig. 9. The Lumerical software, which looks the same in MODE Solutions (2D-simulations) as in FDTDSolutions (3D-simulations). Picture owned by Lumerical Inc [29].

The model built in the script consisted of a substrate of silicon dioxide SiO2, also called silica,

a rectangular waveguide with initial cross section dimensions of 600x250 nm of silicon nitride

Si3N4 (since waveguides with these dimensions were available at KTH), and cladding of air,

as can be seen in Fig. 10a. The refractive indices used are described in Section 2.3. As the FDE

solver used in MODE Solutions is 2-dimensional and perpendicular to the direction of light

propagation, here the z-direction, the z-span of the simulated objects were not of importance

and were arbitrarily set to 600 nm by the software. The largest dimension of the waveguide

in the solver plane would be 2000x300 nm, so the solver region – the inside of the orange

rectangle in Fig. 10b – was set to 4500x3000 nm to be sufficiently large, i.e. such that the light

intensity had gone to zero at the boundaries. The boundaries of the FDE region shown by the

orange frame in Fig. 10b was set to PML to absorb the scattered light and to only support the

physical real modes.

The goal of the simulations in MODE Solutions was to find appropriate dimensions for a

rectangular waveguide such that only the desired modes would propagate. In this case, the

sought after modes were the fundamental TE mode and the fundamental TM mode (see Sec-

tion 2.5). Before this could be done, however, some steps were needed to ensure the results

13

Fig. 10. The layout in MODE Solutions for 2D-simulations. a) The waveguide and its dimensionstogether with the different materials. b) The layout as seen in MODE Solutions. The inside of the largeorange rectangle is the simulation region and its thick frame represent the PML boundaries. The meshoverride region is the inner orange area, while the innermost red rectangle is the waveguide.

from the simulations could be trusted. The FDE solver automatically applies a mesh to dis-

cretize the simulation region, but to make sure that the results would be reliable a mesh over-

ride region was added. This is the inner orange region in Fig. 10b around the waveguide.

The size of the mesh override region was set to 2500x700 nm and centered on the waveguide

so that the waveguide would be fully enclosed in the mesh. Noteworthy is that the fineness

of the mesh override region in MODE Solutions2 extends throughout the region despite the

defined size, making the mesh dense in almost all of the simulation region, as seen in Fig. 10b.

3.1.1 Convergence test for mesh

The decision on the mesh cell size, that is the fineness of the mesh, for the override region was

based on a convergence test. The effective indices neff for the fundamental TE and TM modes

were used as the figures of merit, that is, the parameters that ought to converge. The mesh cell

size for which the effective index would not change more than 1 % upon further refinement

would be chosen as the cell size to be used for the simulations.

3.1.2 Sweep of waveguide width and height

To find the appropriate waveguide dimensions for light propagation only in the fundamental

TE and TM modes, the waveguide’s width and height were swept, i.e. changed incrementally.

The width was swept from 100 nm to 2000 nm with 50 nm steps, while the height was swept

from 50 nm to 300 nm with 10 nm steps. For each increment, the number of supported modes

and their classification (TE or TM) was found. This was done for light of the wavelengths 750

nm and 775 nm.

2This is also the case in FDTD Solutions.

14

3.2 Lumerical – FDTD Solutions (3D)

The simulations in FDTD Solutions were run with a similar model3 as the one in MODE So-

lutions, but now in 3D. The silicon nitride waveguide was situated on top of a silica substrate

with air above. In the direction of propagation, here the z-direction, the materials spanned

a few micrometers, but would later be shortened to 1500 nm as a larger simulation vol-

ume was unnecessary (see Section 3.2.1). The FDTD solver region then had the dimensions

4500x3000x1500 nm in x, y and z respectively.

A dipole source was now introduced to approximate a single-photon source (see Section 2.2).

This was initially placed 10 nm above the waveguide, 500 nm from the edge of the simulation

region so that edge effects of the PML boundary would not be present. Using a broadband

simulation, that is with wavelengths spanning from 750 nm to 800 nm for the emitted light,

one simulation could be run to get results for several wavelengths. Also, monitors were intro-

duced to enable the measurement of coupling efficiency, with the layout as in Fig. 8.

The goal of the simulations in FDTD Solutions was to optimize the coupling of light from

a dipole source into a strip waveguide. Before this could be done, however, some steps were

needed to ensure the results from the simulations could be trusted. Several convergence tests

were run to ensure the accuracy of the results.

3.2.1 Convergence tests for meshes and monitors

As in the 2D model, a mesh override region was added to the 3D model to ensure accurate re-

sults. Firstly, a mesh of size 1000x1000 nm in the xy-plane around the waveguide and a z span

to cover the whole waveguide was introduced, called mesh1. The aim of this finer mesh was

to support the propagation of the correct number of modes. Because the dipole was situated

only 10 nm above the waveguide in the basic layout, a smaller but finer mesh was introduced,

called mesh2. This override region would ensure that the region between the dipole and the

waveguide would be fine enough for trustworthy simulations. This mesh override region was

given the shape of a cube with dimensions 20x20x20 nm. Several convergence test were run

with the mesh1 cell size going from 100 nm to 5 nm and different cell sizes for mesh2. This

made it possible to check both the convergence of the larger mesh, mesh1, as well as of the

smaller mesh, mesh2.

To decide how large the simulation region needed to be, the structure was extended in the

z-direction to a length of 6000 nm. Mode expansion monitors were placed at a distance of 1,

3The script used in 3D can be found in Appendix B.

15

4, 7, 10 and 13 wavelengths away from the dipole source to quantify the losses, i.e. to see how

much of the coupled light would disperse as it traveled down the waveguide.

3.2.2 Sweeps of dipole position and waveguide dimensions

To find the coupling efficiency’s dependence on the position and direction of the dipole as well

as on the wavelength of its emitted light, several sweeps were made. The aim was to find the

optimal position of the dipole, but also to see how much deviation from the optimal position

would affect the coupling. The sweeps were executed in the same order as they appear in the

list below and can graphically be seen in Fig. 11. First of all, in order to see the qualitative

behavior of the positioning, 1D-sweeps was done in x and y before a 2D sweep was run. The

dipole was swept ...

a) in x in the interval [�400, 400] nm, i.e. across the width of the waveguide, with 5 nm

steps, while being at a fixed distance 10 nm above the waveguide. This was done for all

different dipole directions.

b) in y in the interval [252, 264] nm ([2, 14] nm above the waveguide) with 2 nm steps as

to change the distance to the waveguide while being positioned in the middle of the

waveguide, at x = 0. Again, this was done for all different dipole directions.

c) in y in the interval [25, 275] nm with 25 nm steps when positioned outside the waveg-

uide, at x = 310 nm. Again, this was done for all different dipole directions.

d) in 2D in the interval [0, 320] in x (instead of [�320, 320] thanks to symmetry of the struc-

ture and layout) with 5 nm steps and for the heights [0.1, 1, 2, 3, .., 10] nm for y (above

waveguide). This was done only for the x� and z�directions of the dipole due to the

large number of simulations and lower probability of a quantum emitter being approx-

imated by a y-dipole (see Section 2.2).

Thereafter, sweeps were done on the dimension of the waveguide. The results from MODE

Solutions were used for determining the possible dimension of a waveguide supporting only

the fundamental modes. The sweeps done were

e) the width of the waveguide in the interval [270, 620] nm (for a dipole in the x-direction

the interval [290, 620] was used) with the height fixed to 250 nm.

f) the height of the waveguide in the interval [180, 270] nm (for a dipole in the x-direction

the interval [120, 270] was used) with the width fixed to 600 nm.

16

Fig. 11. The different sweeps done in FDTD Solutions for finding the optimal dipole position. Thedifferent sweeps a)-f) are explained in section 3.2.2. In all images the direction of light propagation isout of the paper plane.

3.2.3 Coupling schemes

Even though the strip waveguide is the simplest one from a fabrication point of view, other

kinds of structures can also be made. These could be of interest as they have the potential to

improve the coupling efficiency. Four other structures apart from the strip waveguide have

therefore been considered and built in Lumerical’s FDTD Solutions. The ones built are shown

in Fig. 12. In fact, there are three different variants of case a), which are presented alongside

the result, with different ways of having a dipole in front of the strip waveguide.

Fig. 12. Different structures that could improve coupling efficiency have also been analyzed. a)Dipole in front of a waveguide (also see section 4.3.2). b) A waveguide with a tapered end. c) Awaveguide with a 100x100x100 nm pillar on top with one side at x = 0 having the dipole still centeredand 10 nm above the pillar. d) A slot waveguide.

17

4 Result & Analysis

The presentation of the main results has been divided in accordance with the methods into

two parts: MODE and FDTD Solutions. Each part is structured to first present convergence

tests for defending the accuracy of the main results before presenting them.

4.1 2D Simulations

4.1.1 Convergence test

With the layout explained in Section 3.1 a convergence test of the cell size of the mesh was

executed. The results are shown in Fig. 13, where the relative difference in the effective index

found using a certain cell size and the effective index using the previous cell size is plotted, in

accordance with Eq. 7 with � = neff. This is plotted as a function of the number of mesh cells

per micrometer, that is 1/(mesh cell size) [µm]. This means the mesh gets finer as the x value

gets larger. With regards to limitations in fabrication a relative error of at most 5 % is wanted.

However, thanks to short simulation times in 2D there was, in this case, no disadvantage

of going even further to a perhaps unnecessary low relative error. An upper bound of 1 %

was chosen and the exponential decay of the relative error was taken into account, meaning

that it preferably ought to have a small slope. This made 40 cells per micrometer, i.e. mesh

0 10 20 30 40 50 60 70 80 90

0

1

2

3

4

5

0 25 50 75 100

1.6

1.7

1.8

Fig. 13. Convergence test for the cell size of the mesh override region for simulations in MODESolutions in 2D. a) The relative error �neff between the effective refractive indices of two adjacentmesh cell sizes for the fundamental TE in blue and TM mode in orange. Here plotted against thenumber of mesh cells per micrometer. b) The effective refractive indices for the fundamental TE andTM modes. As the mesh becomes finer, the values settle and the relative error goes to zero.

18

cells with �x = �y = �z = 25 nm, a suitable choice, which thenceforth was used for simula-

tions in MODE Solutions in 2D.

Also, one can note in Fig. 13b that the effective refractive indices are approximately 1.72 and

1.62 for the fundamental TE and TM modes respectively. Both these are higher than the re-

fractive index n2 = 1.45 of the SiO2. Indeed, the modes will thus propagate instead of leaking

down to the substrate.

4.1.2 2D Simulation

The results from the sweeps in height and width of the waveguide (see Fig. 14) in terms of

number of supported modes are shown in Fig. 15 and Fig. 16 for light with a wavelength of

750 nm and 775 nm respectively. The color indicates the number of supported modes for each

waveguide dimension, as shown by the color bar to the right of the heat map. For further 3D

simulations, the desired waveguide should have the fundamental TE and TM modes propa-

gating, so its dimensions should be in the blue area with two modes. As the quantum optics

lab at Alba Nova already has wafers for producing waveguides of height 250 nm, doing the

3D simulations with this height would make it possible to experimentally verify the simula-

tion results in the future. The dimensions that were settled on were a width of 600 nm and a

height of 250 nm.

In Fig. 17 and Fig. 18 the percentage of TE modes, and indirectly the percentage of TM modes

(note that e.g. 0 % TE-modes means 100 % TM-modes), for the mentioned wavelengths are

shown. What can be seen is that if the ratio between the width and height is large the TE-

modes dominate (the yellow regions). On the other hand, if the height is larger than the

width, TM-modes dominate, as can be seen by the dark green area. Since we only want the

fundamental TE and TM modes we can have neither a too wide nor too high waveguide.

Fig. 14. Sweep of waveguide dimensions in 2D. The image, taken in the xy-plane, shows the mini-mum and maximum cross section dimensions of the waveguide during the width and height sweep.

19

Fig. 15. Number of supported modes depending on waveguide dimensions, with � = 750 nm. It’sdesirable to choose a waveguide supporting only the two fundamental modes (TE and TM, shown inthe red circle). One waveguide fulfilling this has width = 600 nm and height = 250 nm (red dot).

Fig. 16. Number of supported modes depending on waveguide dimensions, with � = 775 nm. Thereis only a small difference to the � = 750 nm case. The transition lines between different number ofsupported modes become slightly shifted.

20

Fig. 17. The percentage of TE-modes (and indirectly TM-modes) for light of wavelength 750 nm.The classification of TE/TM has been made from all propagating modes in Fig. 15. The wider thewaveguide the more TE-modes will propagate (yellow region). The greener the area the more TM-modes propagate. The red circle shows the desired area with only the fundamental TE and TM modes.

Fig. 18. The percentage of TE-modes (and indirectly TM-modes) for light of wavelength 770 nm.The classification of TE/TM has been made from all propagating modes as shown in Fig. 16. Thesame color rules as in the figure above apply here; the wider the waveguide the more TE-modes willpropagate, shown by the yellow region and the greener an area is the more TM-modes will propagate.

21

4.2 3D Simulation

With the layout described in Section 3.2 and, in addition, using a movie monitor as shown in

Fig. 19, execution of the simulation in Lumerical FDTD Solutions yields light propagation as

shown in Fig. 20. It is evident that the waveguide supports propagation of the light emitted

by the dipole, but the value of the coupling efficiency still needs to be investigated. Before

this question can be answered by sweeping different configurations of waveguide and dipole,

convergence tests are required.

Fig. 19. A movie monitor captures how the light intensity evolves in time as it propagates down thewaveguide. The montior is placed at a cross section in the xz-plane at y equal to half of the waveguide’sheight. This movie monitor produces the images in Fig. 20.

|E|2

(a.

u.)z Dipole

Fig. 20. Light propagation as seen from a cross section at y = wg_height/2 with direction of propa-gation in z (downwards). Redder means higher light intensity I (since I is proportional to the electricfield squared |E|2). In this top-down view, the vertical lines are the walls of the waveguide. a) The reddot represents the recently activated dipole and its position in the simulation region. b) Even thoughsome light emitted by the dipole scatter out and away from the waveguide most of it seems to becomeguided. c) The light propagation in the strip waveguide. Note that guided modes arise in both thewanted direction (z) and in the opposite (�z, upwards).

22

4.2.1 Convergence test

To find the most appropriate cell sizes for these meshes, convergence tests as described in Sec-

tion 3.2.1 were run. The results can be seen in Fig. 21 and shows that a mesh1 cell size of 10

nm gives a relative error in coupling efficiency below the desired 1 %, regardless of whether

mesh2 was applied. This might seem odd since the insertion of mesh2 would ensure that a

sufficient number of mesh cells were placed between the waveguide and dipole, but the result

tells us that 1 cell is enough. The reason behind this is that the FDTD algorithm distributes

different percentages of light into the 4 adjacent mesh cells to the dipole in such a way that

it becomes equivalent to having smaller mesh cells. As a consequence, the simulations were

thenceforth run with mesh1’s cell size set to 10 nm, but without mesh2, since this advanta-

geously meant shorter simulation times.

0102030405060708090100

0

10

20

30

40

50

60

70

51015202530

0

1

2

3

4

5

Fig. 21. Convergence tests for cell sizes of mesh1 and mesh2. a) The relative error in coupling effi-ciency �� for different cell sizes of the two meshes. The x-axis shows variations in mesh1’s cell sizewhile the different curves show variations in mesh2’s cell size. b) A closer look at the region below30 nm in cell size shows that for a cell size of 10 nm the relative errors are below 1 % for all the testedmesh2 cell sizes.

Deciding on the number of monitors to use and at what distances also required convergence

testing. The result for the layout with 5 monitors as shown in Fig. 22a is shown in Table 2

and they all display the same coupling within a 1 % margin. This result was expected since

23

the material data in terms of refractive indices found in [21] did not include any data about

propagation loss in silicon nitride. The use of several monitors was concluded to be unnec-

essary, so from that point on only one monitor at a distance of one wavelength away from

the dipole, as in Fig. 22b was needed. This made the simulation region smaller and, thereby,

saved simulation time.

Fig. 22. View of the simulation region, seen from above, with a different number of monitors. Thesimulation region could be reduced from 4.5x3x13 µm with 5 monitors as in a) to 4.5x3x3.5 µm with 1monitor as in b), hence making simulation times considerably shorter.

Number of �s away from dipole z 1 4 7 10 13

Difference in [%] for y dipole = 5 nm 0 0.37 0.29 0.30 0.29

Difference in [%] for y dipole = 50 nm 0 0.44 0.33 0.32 0.34

Table 2: Difference in % in coupling efficiency for 5 monitors, compared to the first monitor. Thedata is taken from a simulation with mesh1’s cell size being 20 nm and the dipole positioned 5 and 50nm above the waveguide.

4.2.2 Dipole direction

As mentioned, the fundamental modes in TE or TM were the ones of interest. To understand

which modes were propagating depending on dipole direction, simulations were run with

the basic layout (see Fig. 8) for the three different dipole orientations at different positions

relative the waveguide. It could be concluded that the x-oriented dipole couples virtually

all light into TE, while TM is predominant for the y- and z-oriented dipoles as shown in Fig.

23. All the following results will therefore be showing the coupling into the respective modes.

Another important aspect to consider is which dipole direction that the real quantum emitters

have. When a 2D-material single-photon source sends out a photon, its corresponding dipole

approximation in most likely to be in the plane of the waveguide, that is in the x or z direction.

24

Fig. 23. Percentage of light propagating in TE and TM normalized to the light coupled to the waveg-uide from that position. Depending on the direction of the dipole it emits light into either the funda-mental TE or TM mode. At all locations, the x-oriented dipole emits practically everything into TE,while TM is the predominant mode for the y- and z-oriented dipoles. The total amount of coupledlight at the edges are close to zero and should therefore not be given too much attention.

4.2.3 Transmission analysis

Which directions the light from the dipole scatters is an important question to consider. The

original layout with a 600x250 nm waveguide and a centered dipole placed 10 nm above it was

used. Several transmission monitors were placed to record the scattering light. The results

for a dipole in x-direction and z-direction respectively are shown in Fig. 24. It’s interesting

to notice that in both cases approximately 90 % of the light is emitted down towards the

waveguide. This can be compared to the 50/50-ratio of light going up/down being the case

if the dipole would be placed in vacuum. The reason for this has to do with the evanescent

near-field of the emitted light near a high refractive index material [25]. Unfortunately, even

though 90 % of the emitted light goes down towards the waveguide, around 70 % of the

original light goes right through the Si3N4 to the SiO2. Only about 7 % of the light couples

into the waveguide in the wanted direction. In total, however, 14 % is coupled to the guided

modes of the waveguide if considering both directions, although this is usually not wanted.

Fig. 24. Transmission analysis for a dipole above the waveguide. a) With the dipole in z-directionas much as 90 % of the emitted light is sent downwards, but only 14 % (7 % in each direction) iscoupled into the modes, while 75 % (of the original light) is scattered below the waveguide. b) Havingthe dipole in x-direction gives similar results. This means most of the emitted light goes into thewaveguide, but only a fraction is coupled into the modes and starts propagating.

25

4.2.4 Optimal dipole position and waveguide dimensions

To find the optimal position of the dipole and get as much coupling as possible, its position

was swept in two directions, one at the time, as described in Section 3.2.2. Each sweep was

performed three times, with the dipole oriented in the x-, y-, and z-directions respectively.

Fig. 25 shows the coupling efficiency for different x positions of the dipole and for several

wavelengths, with a fixed y-position of 10 nm above the waveguide. The curves are com-

pletely symmetric around the plane x = 0, as can be expected from the symmetric layout.

It seems that a higher wavelength, i.e. lower frequency, represented by the red line, gives a

higher coupling of light into the desired modes.

-400 -200 0 200 400

0

2

4

6

8

750 755 760 765 770 775 780 785 790 795 800

-400 -200 0 200 400

0

2

4

6

8

-400 -200 0 200 400

Fig. 25. Sweep of dipole’s x-position a) The x-sweep was done in the interval [�400, 400] nm with5 nm steps, while being at a fixed distance 10 nm above the waveguide. The three different dipoledirections x, y, z in b), c) and d) respectively. For all directions, the emitted light spectrum from thedipole contains wavelengths in [750, 800] nm.

26

In Fig. 26 the dipole’s height has been swept for all three dipole directions and the wave-

lengths 750 nm, 775 nm, and 800 nm. The curves show an almost linear dependence of the

coupling efficiency on the distance between the waveguide and the dipole. From these two re-

sults, it seems that the best dipole position would be centered in the middle above the waveg-

uide, as close to it as possible.

2 4 6 8 10 12 14

5

6

7

8

2 4 6 8 10 12 14

5

6

7

8

2 4 6 8 10 12 14

Fig. 26. Sweep of dipole’s y-position. a) The y-sweep was done in the interval [2, 14] nm above thesubstrate with 2 nm steps and the dipole being at x = 0. Coupling efficiency � for the three differentdipole directions x, y, z in b), c) and d) respectively. For all dipole directions, the wavelengths 750, 775and 800 nm have been used.

The height of the dipole was also swept with the dipole being positioned on the side of the

waveguide instead of above it. The coupling efficiencies are, as already found in Fig. 25 for

|x| > 300 nm, quite low. It can be seen in Fig. 27 that the best position seems to be about 100

nm above the substrate for the x-direction of the dipole, while the z-dipole has no coupling

at this height. Nonetheless, these coupling efficiencies � are so small that it would not be of

much interest to put the dipole on the side.

27

0 50 100 150 200 250 300

0

0.5

1

1.5

2

2.5

Fig. 27. Sweep of dipole’s y-position outside waveguide. a) The y-sweep was done in the interval[25, 275] nm with 25 nm steps being at x = 310, that is outside the waveguide. b) Coupling efficiency� for the three different dipole directions x, y, z. The low values of � < 2.5% tells us that placing asingle-photon source on the side of the waveguide isn’t so beneficial.

A sweep with the dipole position changing in both x and y was then run to produce a heat

map of the region above the waveguide. The layout is shown in Fig. 28. The further away

from the center, and the further away from the waveguide, the worse the coupling gets. The

smaller zoomed in plots, Fig. 29b, d, and f, show that an elliptical pattern emerges around

the point x = 0, y = 250. with maximum coupling around 8% (or 16% if taking both direc-

tions into account). One can conclude that the absolute maximum coupling efficiency seems to

be 8 %, in one direction, for a dipole that’s in-plane (in the xz-plane) on top of the waveguide.

Fig. 28. The layout for the sweep in both x and y resulting in the heat maps in Fig. 29. The greenarea shows to scale the different positions the dipole was given. The actual simulation was only runfor the right half, but since the layout is symmetric around the plane x = 0, the results were extendedto the area shown here.

28

-300 -200 -100 0 100 200 300

0

2

4

6

8

10

-50 0 50

0

5

10

-300 -200 -100 0 100 200 300

0

2

4

6

8

10

-50 0 50

0

5

10

-300 -200 -100 0 100 200 300

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8

-50 0 50

0

5

10

6 7 8

Fig. 29. Heat maps showing the coupling efficiency � for different positions of the dipole above thewaveguide. All plots show the average of the coupling of the x-oriented and z-oriented dipoles. a) Thetransmission for a wavelength of 750 nm, with b) showing a zoomed in and slightly rescaled version. c)and d) are equivalent but with a wavelength of 775 nm. e) and f) are equivalent but with a wavelengthof 800 nm. With b), d), and f) side by side, one can see how a longer wavelength seems to give bettercoupling. The optimal position is apparently centered in the middle, as close to the waveguide aspossible. The color bar shows the coupling efficiency � in percent.

29

Consequently, one could ask if the width and height of the waveguide are at optimum in

the acceptable region of the two fundamental modes shown by the red circles in Fig. 15 and

16. Firstly, a width-sweep was done for the waveguide, keeping the height fixed at 250 nm.

The results in Fig. 30 indicate that a waveguide with a width of 350 nm would be better; this

optimum is evident for the in-plane dipole directions x and z by averaging over the coupling

efficiencies in these directions. However, the improvement would only be of a few percents.

300 400 500 600

4

6

8

10

300 400 500 600

4

6

8

10

300 400 500 600

Fig. 30. Sweep of waveguide width for constant height 250 nm. a) The layout and sweep interval, aswell as the position of the dipole, is shown when the dipole is directed in x. The coupling efficiency �for the x�,y� and z�direction of the dipole in b), c) and d) respectively. The dipole was always placedcentered 10 nm above the waveguide.

On the other hand, if one would change the height at a constant width of 600 nm, as shown in

Fig. 31, the coupling could increase, although only for the x-direction. Indeed, it seems that

no modes will be able to propagate if the width is less than 180 nm for dipoles orientated in y

and z. As the real-world direction of an in-plane dipole is random, the 15 % for the x-directed

dipole at 120 nm does not present a major improvement; averaging with the z-dipole’s 0 %

gives only 7.5 % coupling. In conclusion, to support only the fundamental TE and TM modes

(remember Fig. 15 and 16) a 600x120 nm waveguide would be optimal if a quantum emit-

ter’s dipole could be intentionally oriented in the x-direction, while a 400x200 nm waveguide

would be better if taking all dipole directions into account.

30

150 200 250

5

10

15

160 180 200 220 240 260 280

4

5

6

7

8

160 180 200 220 240 260 280

Fig. 31. Sweep of waveguide height for constant width 600 nm. a) The layout and sweep interval, aswell as the position of the dipole, is shown when the dipole is directed in x. The coupling efficiency �for the x�, y� and z�direction of the dipole in b), c) and d) respectively. Note that the scales for c) andd) differs from the one in b). The dipole was always placed centered 10 nm above the waveguide.

4.3 Coupling Schemes

4.3.1 Pillar on top of waveguide

To get as much coupling as possible, the geometry of the waveguide was changed in several

ways as described in Section 3.2.3. The addition of a non-centered pillar with dimensions

100x100x100 nm on top of the waveguide resulted in a decrease in the coupling efficiency. The

initial simulation done for a dipole placed in x = 0 and positioned 10 nm above one corner of

the pillar gave under 1 % coupling. It was, therefore, decided not to do any parameter sweeps

for this configuration. The result, however, is in accordance with what one could expect, as

the waveguide-dipole distance has increased, meaning that light first would have to enter and

be transmitted through the pillar before entering the waveguide. In conclusion, this coupling

scheme is not to prefer.

31

4.3.2 Dipole in front of waveguide

One could imagine putting the dipole in front of the strip waveguide (without the pillar) in-

stead of on top. In reality, this would represent letting a 2D-material be hung over the edge

of the waveguide. The light wavelength used was 750 nm, emitted from a dipole 10 nm away

from the structure since this is the place it probably will be formed.

Three different schemes were considered and their results are presented in Fig. 32 for waveg-

uides with cross sections 600x250 and 400x200 respectively in a) and b). At optimum for

this coupling scheme, over 50 % of the light will go into the propagating modes. Complex

structures such as the one in c) require additional fabrication steps after the top-down manu-

facturing approach in Fig. 5. Using a method called wet chemical etching [30] it is possible to

selectively etch away the SiO2 without etching the Si3N4. If the process is stopped in time the

etch won’t remove all of the SiO2 and the rest of the waveguide can still lay on top of it.

0 50 100 150 200 250 3000

20

40

60

0 50 100 150 200 250 3000

20

40

60

Fig. 32. Coupling efficiency � for a dipole placed in front of the waveguide. Two different crosssection dimensions for the waveguide has been used, shown in a) and b) respectively. Three differentschemes were simulated for light with a wavelength of 750 nm. The dipole y-position was swept infront of the waveguide with c) substrate underneath and d) air underneath (for the 70 nm outermostpart). e) Dipole in front of the waveguide with waveguide material Si3N4 under the dipole. The sweepwas here done for the height of this additional layer, always keeping the dipole 10 nm above it (thefirst point when ydipole = 10 nm was, thus, done without the extra layer).

32

4.3.3 Tapered waveguide

In the section above the end of the waveguide was a flat rectangle, but one could also have a

tapered end to the waveguide. This means the width of the waveguide is reduced gradually

such that the end gets a triangular shape, here with an outer angle of 104 �. As one can see

in Fig. 33 the results for the tapered end are similar as to the rectangular end, but has slightly

lower coupling for a dipole in the x- and z-directions. In the y-direction, i.e. in-line with

the edge of the taper, however, the coupling is a few percentage points higher than the flat

counterpart.

0 50 100 150 200 250 300

0

10

20

30

40

50

Fig. 33. Coupling efficiency � for a dipole in front of a tapered (104� at outer edge) 600x250 [nm]waveguide. a) The height of the dipole was swept from 25 to 300 nm above the substrate, positionedat the edge of the taper, with a waveguide height of 250 nm. b) The coupling efficiency is highest whenthe dipole sits at half the height of the waveguide. The result is similar to that in Fig. 32, but slightlyhigher or lower depending on the dipole orientation.

4.3.4 Slot waveguide

In a slot waveguide light propagation will be supported in between two stripes of waveguide

material. The slot width, i.e. the air gap, was swept from 10 to 70 nm with 5 nm steps with

the dipole positioned centered at half the height of the waveguide. The total width and height

were kept at 600x250 nm, so a change in slot width would not change the outer perimeters

of the waveguide. Light with a wavelength of 750 nm was used and the result is presented

in Fig. 34. It seems the smaller the slot width, the better the coupling. Also, a dipole be-

ing x-orientated gives the most coupling, which is reasonable since this means that light will

emerge from the dipole into mainly the y- and z-directions, the latter of which is the direction

of propagation. In the real world, one could imagine laying down a 2D-material on top of

the slot waveguide like a carpet, hanging down in the air gap, forming a quantum emitter

in the simulated dipole position. The coupling efficiency is better than having the dipole on

top of the strip waveguide, although it is worse than having it in front of the strip waveguide.

33

10 20 30 40 50 60 70

0

5

10

15

20

25

30

Fig. 34. Coupling efficiency � for a dipole centered in a slot waveguide. a) In a slot waveguide,the mode is propagating in the space between two strips of waveguide material. Here the slot widthwas changed from 10 nm to 70 nm and the dipole was positioned centered at half the height of thewaveguide. b) The smaller the slot width, the better the coupling of light, and the x-directed dipolegives the highest results.

34

5 Discussion and Conclusion

In this work simulations of light coupling into a waveguide have been performed. The first

aim of the project was to investigate how the geometry of a strip waveguide affects the num-

ber of propagating modes and to find for which waveguide geometries only the fundamental

TE and TM modes would propagate. It was shown that the wider and taller the waveguide is,

the more modes propagate. Waveguides of dimensions 400x200 as well as 600x250 nm were

shown to support the desired modes. The results did not differ much between light of wave-

length 750 and 775 nm.

The second aim was to optimize the coupling efficiency by varying the light source’s position

and by using different coupling schemes. For a dipole on top of the waveguide, the opti-

mal position was found to be in the center and as close as possible, with a maximal coupling

efficiency of about 8 % in each direction for a 600x250 nm waveguide and slightly higher for

smaller waveguides. The possible directions of the dipole, x, y and z, were also of importance.

A dipole in the x- or z-direction, being in-plane with the waveguide and therefore the most

likely to occur from a 2D-material single-photon source, gave the best coupling as compared

to the out-of-plane y-directed dipole. As for wavelength, the simulated span of 750 nm to 800

nm varied in giving the highest coupling efficiency, but for the 600x250 nm waveguide, the

longer wavelengths were somewhat more efficient.

Several coupling schemes were tested, the best being the one having the dipole in front of

one end of the waveguide. This gave a maximal coupling of over 50 % when the dipole was

centered in both height and width of the waveguide, and directed in the x- or y-directions,

i.e. in-plane with the waveguide’s cross section. Having a tapered end to the waveguide with

an outer angle of 104 �gave slightly worse results for the x- and z-directed dipole, but signif-

icantly better for a dipole in the y-direction. Furthermore, a slot waveguide presented results

better than a strip waveguide with a dipole on top; this is especially the case for a x-directed

dipole, but also marginally for a y-dipole.

The results obtained in this work are comparable to the ones in [22] where a dipole on top

would send around 7 % in either direction. In addition, [31] shows coupling efficiencies of

over 50 % by having single-photons from dibenzoterrylene (DBT) molecules embedded into

thin matrices of crystalline anthracene on top of the waveguide. This result originates from a

slightly different design with the dipole being placed within a structure, but it, nevertheless,

indicates in accordance with our results that a coupling efficiency of up to 50 % is feasible. We

will intently wait for experimental verification.

35

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38

Appendix A – MODE Solutions Code

Build Model⌥ ⌅1 # #############################################################2 # Waveguide MODE S o l u t i o n s3 #4 # wg width : 600 nm5 # wg height : 250 nm6 # wg m a t e r i a l : Si3N4 � Luke7 # s u b s t r a t e m a t e r i a l : SiO2 � P a l i k8 # clad m a t e r i a l : Air � Peck9 #

10 # Tags : i n t e g r a t e d o p t i c s s t r i p waveguide11 #12 # #############################################################13 swi tchto layout ;14 d e l e t e a l l ;15 c l e a r ;1617 # Set v a r i a b l e s18 wg_width = 0 . 6 e�6;19 wg_height = 0 . 2 5 e�6;20 wg_material = " Si3N4 � Luke " ;21 s u b s t r a t e _ m a t e r i a l = " SiO2 ( Glass ) � P a l i k " ;22 c l a d _ m a t e r i a l = " Air � Peck " ;2324 # Set s u b s t r a t e25 addrect ;26 s e t ( "name" , " s u b s t r a t e " ) ;27 setnamed ( " s u b s t r a t e " , " x " , 0 ) ;28 setnamed ( " s u b s t r a t e " , " x span " ,7 e�6) ;29 setnamed ( " s u b s t r a t e " , " y max" , 0 ) ;30 setnamed ( " s u b s t r a t e " , " y min " , �2.5e�6) ;31 setnamed ( " s u b s t r a t e " , " m a t e r i a l " , s u b s t r a t e _ m a t e r i a l ) ;3233 # Set c lad34 addrect ;35 s e t ( "name" , " c lad " ) ;36 setnamed ( " c lad " , " x " , 0 ) ;37 setnamed ( " c lad " , " x span " ,7 e�6) ;38 setnamed ( " c lad " , " y max" , 2 . 5 e�6) ;39 setnamed ( " c lad " , " y min " , 0 ) ;40 setnamed ( " c lad " , " m a t e r i a l " , c l a d _ m a t e r i a l ) ;4142 # Set waveguide

39

43 addrect ;44 s e t ( "name" , " waveguide " ) ;45 setnamed ( " waveguide " , " x " , 0 ) ;46 setnamed ( " waveguide " , " x span " , wg_width ) ;47 setnamed ( " waveguide " , " y max" , wg_height ) ;48 setnamed ( " waveguide " , " y min " , 0 ) ;49 setnamed ( " waveguide " , " m a t e r i a l " , wg_material ) ;5051 # Add FDE52 addfde ;53 s e t ( " x " , 0 ) ;54 s e t ( " x span " , 4 . 5 e�6) ;55 s e t ( " y " , 0 ) ;56 s e t ( " y span " ,3 e�6) ;57 s e t ( " x min bc " , "PML" ) ;58 s e t ( " x max bc " , "PML" ) ;59 s e t ( " y min bc " , "PML" ) ;60 s e t ( " y max bc " , "PML" ) ;61 s e t a n a l y s i s ( " wavelength " , 0 . 7 5 e�6) ;6263 # Add mesh64 addmesh ;65 s e t ( "name" , "mesh" ) ;66 s e l e c t ( "mesh" ) ;67 s e t ( " x " , 0 ) ;68 s e t ( " x span " , 2 . 5 e�6) ;69 s e t ( " y " , 0 . 2 e�6) ;70 s e t ( " y span " , 0 . 7 e�6) ;71 s e t ( " z span " , 0 ) ;7273 s e t ( " overr ide x mesh" , 1 ) ;74 s e t ( " s e t maximum mesh step " , 1 ) ;75 s e t ( " dx " , 0 . 0 2 5 e�6) ;76 s e t ( " overr ide y mesh" , 1 ) ;77 s e t ( " s e t maximum mesh step " , 1 ) ;78 s e t ( "dy " , 0 . 0 2 5 e�6) ;79 s e t ( " overr ide z mesh" , 0 ) ;⌃ ⇧

40

Appendix B – FDTD Solutions Code⌥ ⌅

1 # #############################################################2 # Waveguide FDTD S o l u t i o n s3 #4 # wg width : 600 nm5 # wg height : 250 nm6 # wg m a t e r i a l : Si3N4 � Luke7 # s u b s t r a t e m a t e r i a l : SiO2 � P a l i k8 # clad m a t e r i a l : Air � Peck9 #

10 # Tags : i n t e g r a t e d o p t i c s s t r i p waveguide11 #12 # #############################################################1314 swi tchto layout ;15 d e l e t e a l l ;16 c l e a r ;1718 # Set v a r i a b l e s19 wg_width = 0 . 6 e�6;20 wg_height = 0 . 2 5 e�6;21 wg_material = " Si3N4 � Luke " ;22 s u b s t r a t e _ m a t e r i a l = " SiO2 ( Glass ) � P a l i k " ;23 c l a d _ m a t e r i a l = " Air � Peck " ;24 wavelength = 750e�9;25 z_length = 3 . 5 e�6;2627 # Set s u b s t r a t e28 addrect ;29 s e t ( "name" , " s u b s t r a t e " ) ;30 s e t ( " x " , 0 ) ;31 s e t ( " x span " ,7 e�6) ;32 s e t ( " y max" , 0 ) ;33 s e t ( " y min " , �2.5e�6) ;34 s e t ( " z min " , 0 ) ;35 s e t ( " z max" , z_length ) ;36 s e t ( " m a t e r i a l " , s u b s t r a t e _ m a t e r i a l ) ;3738 # Set c lad39 addrect ;40 s e t ( "name" , " c lad " ) ;41 s e t ( " x " , 0 ) ;42 s e t ( " x span " ,7 e�6) ;43 s e t ( " y max" , 2 . 5 e�6) ;44 s e t ( " y min " , 0 ) ;

41

45 s e t ( " z min " , 0 ) ;46 s e t ( " z max" , z_length ) ;47 s e t ( " m a t e r i a l " , c l a d _ m a t e r i a l ) ;4849 # Set waveguide50 addrect ;51 s e t ( "name" , " waveguide " ) ;52 s e t ( " x " , 0 ) ;53 s e t ( " x span " , wg_width ) ;54 s e t ( " y max" , wg_height ) ;55 s e t ( " y min " , 0 ) ;56 s e t ( " z min " , 0 ) ;57 s e t ( " z max" , z_length ) ;58 s e t ( " m a t e r i a l " , wg_material ) ;5960 # Add FDTD61 addfdtd ;62 s e t ( " x " , 0 ) ;63 s e t ( " x span " , 4 . 5 e�6) ;64 s e t ( " y " , 0 ) ;65 s e t ( " y span " ,3 e�6) ;66 s e t ( " z min " , 1e�6) ;67 s e t ( " z max" , z_length�1e�6) ;68 s e t ( " x min bc " , "PML" ) ;69 s e t ( " x max bc " , "PML" ) ;70 s e t ( " y min bc " , "PML" ) ;71 s e t ( " y max bc " , "PML" ) ;72 s e t ( " z min bc " , "PML" ) ;73 s e t ( " z max bc " , "PML" ) ;74 s e t ( "mesh type " , " custom non�uniform " ) ;7576 # Add source77 adddipole ;78 source_height = wg_height+10e�9;79 s e t ( "name" , " dipole " ) ;80 s e t ( " x " , 0 ) ;81 s e t ( " y " , source_height ) ;82 z_dipole = 1 . 5 e�6;83 s e t ( " z " , z_dipole ) ;84 s e t ( " t h e t a " , 9 0 ) ;85 s e t ( " phi " , 0 ) ;86 s e t ( " wavelength s t a r t " , 750e�9) ;87 s e t ( " wavelength stop " , 750e�9) ;8889 ## MONITORS: From now on only one monitor but works with s e v e r a l90 number_of_monitors = 1 ;

42

91 dist_between_monitors = 3⇤wavelength ;9293 name_output = c e l l ( number_of_monitors ) ;94 name_expansion = c e l l ( number_of_monitors ) ;9596 f o r ( i =1 : number_of_monitors ) {97 name_output { i } = " output_ "+num2str ( i ) ;98 name_expansion { i } = " mode_expansion_ "+num2str ( i ) ;99 }

100101 f o r ( i =1 : number_of_monitors ) {102 # Add monitor a t X lambdas away103 addpower ;104 s e t ( "name" , name_output { i } ) ;105 s e t ( " monitor type " , " 2D Z�normal " ) ;106 s e t ( " x " , 0 ) ;107 s e t ( " x span " ,4⇤wg_width ) ;108 s e t ( " y " , wg_height /2) ;109 s e t ( " y span " ,6⇤wg_height ) ;110 s e t ( " z " , z_dipole + wavelength + ( i �1)⇤dist_between_monitors ) ;111112 # Add mode expancion a t X lambdas away113 addmodeexpansion ;114 s e t ( "name" , name_expansion { i } ) ;115 s e t ( " monitor type " , " 2D Z�normal " ) ;116 s e t ( " x " , 0 ) ;117 s e t ( " x span " ,4⇤wg_width ) ;118 s e t ( " y " , wg_height /2) ;119 s e t ( " y span " ,6⇤wg_height ) ;120 s e t ( " z " , z_dipole + wavelength + ( i �1)⇤dist_between_monitors ) ;121 # s e t ( " mode s e l e c t i o n " , " fundamental TE mode " ) ;122 updatemodes ( 1 : 2 ) ;123 setexpansion ( " input " , name_output { i } ) ;124 }125126 #Add mesh to s a t i s f y convergence t e s t127 addmesh ;128 o v e r r i d e _ t o t a l _ l e n g t h = 1e�6;129 mesh_cel l_ length = 0 .010 e�6;130131 s e t ( "name" , " mesh1 " ) ;132 s e l e c t ( " mesh1 " ) ;133 s e t ( " x " , 0 ) ;134 s e t ( " x span " , o v e r r i d e _ t o t a l _ l e n g t h ) ;135 s e t ( " y " , wg_height /2) ;136 s e t ( " y span " , o v e r r i d e _ t o t a l _ l e n g t h ) ;

43

137 s e t ( " z min " , 0 ) ;138 s e t ( " z max" , z_length ) ;139140 s e t ( " overr ide x mesh" , 1 ) ;141 s e t ( " s e t maximum mesh step " , 1 ) ;142 s e t ( " dx " , mesh_cel l_ length ) ;143 s e t ( " overr ide y mesh" , 1 ) ;144 s e t ( " s e t maximum mesh step " , 1 ) ;145 s e t ( " dy " , mesh_cel l_ length ) ;146 s e t ( " overr ide z mesh" , 1 ) ;147 s e t ( " s e t maximum mesh step " , 1 ) ;148 s e t ( " dz " , mesh_cel l_ length ) ;⌃ ⇧

44

Appendix C – Example code for sweep⌥ ⌅

1 ### EXAMPLE SWEEP2 ## By changing ( 1 ) and ( 2 ) the other sweeps can be obtained .3 ## Here : Sweeping x�p o s i t i o n of dipole4 swi tchto layout ;56 # ################################7 ### ( 1 ) Define sweeping parameter :89 # Here : Dipole x p o s i t i o n s

10 dipole_x = [ �400 :5 :400]⇤1 e�9;11 x_points = length ( dipole_x ) ;1213 # ################################1415 ## Pre�sweep ( same f o r a l l sweeps )1617 # Change dipole d i r e c t i o n :1819 d i p o l e _ d i r e c t i o n _ v = c e l l ( 3 ) ;20 d i p o l e _ d i r e c t i o n _ v { 1 } = ’ x ’ ;21 d i p o l e _ d i r e c t i o n _ v { 2 } = ’y ’ ;22 d i p o l e _ d i r e c t i o n _ v { 3 } = ’ z ’ ;2324 # Set dipole source to 750�800 nm25 s e l e c t ( " dipole " ) ;26 s e t ( " wavelength s t a r t " ,750 e�9) ;27 s e t ( " wavelength stop " ,800 e�9) ;2829 # Set monitors to c o r r e c t frequency poni ts30 f r e q _ p o i n t s = 1 1 ;31 se tg loba lmoni tor ( " frequency points " , f r e q _ p o i n t s ) ;32 se tg loba lmoni tor ( " use l i n e a r wavelength spacing " , 1 ) ;3334 s e l e c t ( " mode_expansion_1 " ) ;35 s e t ( " overr ide g loba l monitor s e t t i n g s " , 0 ) ;3637 # Progress38 progress = 0 ;39 progress_s tep = 100/( x_points ⇤3) ;4041 # ################################4243 # Changing dipole d i r e c t i o n44

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45 f o r ( d = 1 : 3 ) {4647 d i p o l e _ d i r e c t i o n = d i p o l e _ d i r e c t i o n _ v { d } ;4849 # A l l o c a t i o n50 T_forward_v = matrix ( f req_points , x_points ) ;51 p u r c e l l _ f a c t o r = matrix ( f req_points , x_points ) ;5253 # D i r e c t i o n of dipole54 s e l e c t ( ’ dipole ’ ) ;55 i f ( d i p o l e _ d i r e c t i o n == ’ x ’ ) {56 s e t ( " t h e t a " , 9 0 ) ;57 s e t ( " phi " , 0 ) ;58 }59 e l s e i f ( d i p o l e _ d i r e c t i o n == ’y ’ ) {60 s e t ( " t h e t a " , 9 0 ) ;61 s e t ( " phi " , 9 0 ) ;62 }63 e l s e i f ( d i p o l e _ d i r e c t i o n == ’ z ’ ) {64 s e t ( " t h e t a " , 0 ) ;65 s e t ( " phi " , 0 ) ;66 }6768 ## Sweeping f o r each dipole d i r e c t i o n6970 f o r ( i =1 : x_points ) {7172 # ################################73 ### ( 2 ) Update sweeping parameter7475 # Here : Fix new x�p o s i t i o n f o r dipole76 new_x = dipole_x ( i ) ;77 s e l e c t ( " dipole " ) ;78 s e t ( " x " , new_x ) ;79 # ################################8081 run ;8283 r e s u l t = g e t r e s u l t ( " mode_expansion_1 " , " expansion f o r input .

T_forward " ) ;84 f i r s t _ e x p a n s i o n = r e s u l t . T_forward ;8586 # Get appropriate TE or TM�r e s u l t87 # (1 = TE f o r x )88 # (2 = TM f o r y & z )89 i f ( d i p o l e _ d i r e c t i o n == ’ x ’ ) {

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90 T_forward_v ( : , i ) = f i r s t _ e x p a n s i o n ( : , 1 ) ;91 } e l s e {92 T_forward_v ( : , i ) = f i r s t _ e x p a n s i o n ( : , 2 ) ;93 }9495 # Get p u r c e l l f a c t o r96 p = g e t r e s u l t ( ’ dipole ’ , ’ p u r c e l l . p u r c e l l vs lambda/f ’ ) ;97 purce l l_v = p . p u r c e l l ;98 p u r c e l l _ f a c t o r ( : , i ) = purce l l_v ( [ [ 1 0 0 0 : � 1 0 0 : 1 0 0 ] ; 1 ] ) ;99

100 # P r i n t progress101 swi tchto layout ;102 progress = progress + progress_s tep ;103 p r i n t_ p ro g re s s = num2str ( progress ) +"%" ;104 ? p r i n t_ p ro g re s s ;105106 # Export data to MATLAB107108 i f ( d i p o l e _ d i r e c t i o n == ’ x ’ ) {109 matlabsave ( " 0503 _sweep_example_xdir " , p u r c e l l _ f a c t o r , dipole_x ,

T_forward_v ) ;110 }111 e l s e i f ( d i p o l e _ d i r e c t i o n == ’y ’ ) {112 matlabsave ( " 0503 _sweep_example_ydir " , p u r c e l l _ f a c t o r , dipole_x ,

T_forward_v ) ;113 }114 e l s e i f ( d i p o l e _ d i r e c t i o n == ’ z ’ ) {115 matlabsave ( " 0503 _sweep_example_zdir " , p u r c e l l _ f a c t o r , dipole_x ,

T_forward_v ) ;116 }117118 }119120 # P l o t work f o r v e r i f i c a t i o n d i r e c t l y in lumer ica l121 f o r ( k =1: f r e q _ p o i n t s ) {122 p l o t ( dipole_x , T_forward_v ( k , : ) , ’ x pos i t ion ’ , ’ Transmission ’ , ’

Transmission f o r d i f f e r e n t dipole x pos i t ions ’ ) ;123 holdon ;124 }125 holdof f ;126127 }⌃ ⇧

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