Crowdsourcing Airway Annotations in Chest Computed Tomography Images

Veronika Cheplygina1*, Adria Perez-Rovira2,3, Wieying Kuo3,4, Harm A. W. M.
Tiddens3,4, Marleen de Bruijne2,5

1 IMAG/e, Department of Biomedical Engineering, Eindhoven University of
Technology, Eindhoven, The Netherlands

2 Biomedical Imaging Group Rotterdam, Department of Radiology and Nuclear
Medicine, Erasmus MC - University Medical Center Rotterdam, Rotterdam, The
Netherlands

3 Department of Pediatric Pulmonology and Allergology, Erasmus MC - Sophia
Children’s Hospital, Rotterdam, The Netherlands

4 Department of Radiology and Nuclear Medicine, Erasmus MC - University
Medical Center Rotterdam, Rotterdam, The Netherlands

5 Machine Learning Section, Department of Computer Science, University of
Copenhagen, Denmark

*<EMAIL_ADDRESS>
## Abstract

Measuring airways in chest computed tomography (CT) scans is important for
characterizing diseases such as cystic fibrosis, yet very time-consuming to
perform manually. Machine learning algorithms offer an alternative, but need
large sets of annotated scans for good performance. We investigate whether
crowdsourcing can be used to gather airway annotations. We generate image
slices at known locations of airways in 24 subjects and request the crowd
workers to outline the airway lumen and airway wall. After combining multiple
crowd workers, we compare the measurements to those made by the experts in the
original scans. Similar to our preliminary study, a large portion of the
annotations were excluded, possibly due to workers misunderstanding the
instructions. After excluding such annotations, moderate to strong
correlations with the expert can be observed, although these correlations are
slightly lower than inter-expert correlations. Furthermore, the results across
subjects in this study are quite variable. Although the crowd has potential in
annotating airways, further development is needed for it to be robust enough
for gathering annotations in practice. For reproducibility, data and code are
available online: http://github.com/adriapr/crowdairway.git.

## Introduction

Chest computed tomography (CT) can be used to quantify structural
abnormalities in the lungs, such as bronchiectasis, air trapping and
emphysema, which in turn can be used for diagnostic or prognostic purposes.
For example, the airway-to-artery ratio (AAR) is an objective measurement of
bronchiectasis which is sensitive to detect early lung disease [13, 7]. Other
promising measurements are the wall-area percentage (WAP) and the wall
thickness ratio (WTR) which characterize the ratio of the airway wall to the
airway lumen [9]. Unfortunately, manual measurements of the airways and
vessels suffer from intra- and inter-observer variation and are time-consuming
(8-16 hours per chest CT) [3]. Machine learning techniques such as [6, 4, 11]
can be an alternative, but may require a large amount of annotated data to be
able to generalize to all situations.

In various applications, crowdsourcing has been proposed as an alternative for
tasks where annotated data is scarce. Crowdsourcing refers to outsourcing
tasks (often referred to as human intelligence tasks or HITs) to a group of
online users (often referred to as knowledge workers or KWs). This strategy
has also been quite effective in medical image analysis - [8] surveys over 50
papers where results have been mostly positive. One of these papers is our
earlier study [2] where we described our experiences with crowdsourcing airway
measurements. We found that 67.8% of the collected results were not valid,
i.e. the airway measurements could not be extracted. However, after filtering
out such results, strong correlations between the crowd and expert were
observed. Although these experiences were encouraging, they only concerned a
single chest CT image, and it was unclear whether they could be generalized to
other scans.

In this paper we describe crowdsourced airway measurements collected shortly
thereafter for a larger set of 24 chest CT images, and with a slightly updated
crowdsourcing procedure. With this follow-up study we aim to answer the
following questions:

  * •

Does the crowd create valid results?

  * •

What is the quality of the crowd compared to a trained expert, after combining
different results per task?

  * •

Can we predict the quality of the crowd results, given a particular scan?

## Materials and Methods

### Chest CT scans

We used inspiratory pediatric CT scans from a cohort of 24 subjects [10, 5],
collected at the Erasmus MC - Sophia Children’s Hospital. In each scan, a
number of airways were annotated by an expert. The expert localized an airway,
outlined the airway lumen (inner airway boundary) and airway wall (outer
airway boundary) in a plane approximately perpendicular to the airway center
line, and recorded the measurements of the areas.

### Generating Airway Images

Fig. 1 shows a global overview of our method. The first step is to create a
crowdsourcing task for each airway, which requires extracting 2D image slices
from a 3D volume. This requires having a 3D location and orientation of the
airway. Normally this localization would be done by the expert, however in
this study we assume that localization was already done, and focus only on
outlining the airway in the image.

Figure 1: Overview of the method. A 3D image is annotated by experts. The
locations and orientations of the airways are then used to generate 2D slices
of the airways, which are then annotated by the workers.

More specifically, we used 3D voxel coordinates, at which experts have
previously outlined airways using the MyrianTM software. We generated 2D
slices of $50\times 50$ voxels which we reviewed by one observer (APR) to
retain only the images with a visible airway that was cut approximately
perpendicularly. There were 1026 such images, which are further analysed here.
We used cubic interpolation and an intensity range between -950 and 550
Hounsfield units for better contrast, as recommended by the experts. Each
image slice was rescaled to $500\times 500$ pixels for annotation purposes.

### Annotating Airway Images

Each of the generated airway images is a crowdsourcing task. A worker assigned
to a task creates a result, consisting of one or more annotations (outlines)
placed in the image.

To gather these results, we used Amazon Mechanical Turk [1]. All decisions
regarding Amazon Mechanical Turk (MTurk) were based on consultation with
colleagues who had used MTurk in the past. Apart from updating the
instructions to workers, we used the same settings as in our preliminary
study, which we repeat here for completeness. All results were collected in
2016.

The annotation interface was integrated into the platform by supplying a
dynamic webpage, built with HTML5 and Javascript. This custom-made interface
had an ellipse tool, which resembled the tool used by the experts more closely
than the default annotation tools available on MTurk. The details of our HIT,
which the workers could see when searching for HITs, are shown in Table 1. The
workers were instructed to draw two ellipses outlining the airway lumen and
the airway wall, or to place a small circle in the top right corner of the
image, if no airway is visible. Following our experiences in the preliminary
experiments, we revised our instructions, placing more emphasis on the need to
draw two ellipses. A screenshot is shown in Fig. 2.

Title |  Save lives by annotating airways!
---|---
Description |  Draw two contours to annotate an airway (dark circle or ellipse) in image from a lung scan
Keywords |  image, annotation, contour, draw, drawing, segmentation, medical
Table 1: Details of HIT on Amazon Mechanical Turk Figure 2: First set of
instructions provided to the workers. One more set of instructions was
available, but this was the set that all workers would see.

We randomly created HITs with 10 images per HIT. A worker could request a HIT,
annotate 10 images, and then submit the HIT. The workers were paid $0.10 per
completed HIT. Only workers who had previously done at least 100 HITs with an
acceptance rate of 90% could request the HITs.

We collected 20 results per image, because with 10 results per image as in the
preliminary experiment, some images did not have valid annotations. For each
result, we recorded an anonymized ID of the worker and the coordinates of the
annotations. The data collection was done in 2016, shortly after our
preliminary study [2].

### Measuring Crowd Annotations

We applied a simple filtering step to filter out invalid results. The
following results were excluded:

  * •

number of ellipses not equal to 2

  * •

not resized ellipses (default size of the tool is a circle)

  * •

not overlapping ellipses

After filtering, we measured the areas of the inner ($a_{i}$) and outer
($a_{o}$) ellipse, and calculated the wall thickness ratio (WTR) and wall area
percentage (WAP). The WTR is the wall thickness divided by the outer diameter:

$WTR=\frac{WT}{d_{o}}=\frac{(d_{o}-d_{l})/2}{d_{o}}$ (1)

where $d_{o}$ is the diameter of the outer ellipse and $d_{i}$ is the diameter
of the inner ellipse, and the wall thickness $WT$ is defined as:

$WT=\frac{d_{o}-d_{i}}{2}.$ (2)

The WAP is the percentage of the total airway area that is airway wall:

$WAP=\frac{a_{o}-a_{i}}{a_{o}}\times 100.$ (3)

### Quality of Crowd Measurements

Before measuring how good the crowd is on each task, we need to combine the
results per task. We used three different strategies for this:

Median

Taking the inner/outer areas of all valid results, and combining them with the
median function. WAP and WTR are then calculated based on these median values.
This is the strategy used in our preliminary study.

Random

Selecting a random valid result per task. This gives an indication of how good
the crowd could be, if each task was assigned to only one worker, and gives a
pessimistically biased indication of how good the crowd could be.

Best

Taking the valid result that is closest to the expert measurement, based on
the inner and outer measurements. This is an optimistically biased indication
of how good the crowd could be, if we only selected the best workers.

Additionally, we can choose to exclude tasks that have less than $v$ valid
results. This will reduce the number of tasks for which a combined result is
available, but will presumably increase the quality of the result.

After combining the results per task, we use the Pearson’s correlation
coefficient, $\rho$, between the crowd measurement and the expert measurement.
Correlation coefficients are interpreted as follows: weak correlation for
$0\leq\rho<0.3$, moderate correlation for $0.3\leq\rho<0.5$, strong
correlation for $0.5\leq\rho<1$. Note that, if a task has had no valid
results, it will be excluded from the analysis.

### Predicting Crowd Quality

Lastly, we investigate whether any factors contribute to the crowd’s
performance across different scans in our data. We use the inner airway after
median combining as a proxy for the quality.

We then look at the relationship between the quality and the following
characteristics:

  * •

Whether or not the subject has cystic fibrosis (CF)

  * •

Forced expiration volume in 1 second (FEV1), which measures how much air a
participant can exhale in 1 second.

  * •

Forced vital capacity (FVC), which measures the total volume of air a
participant can exhale.

  * •

Number of airways as indicated by the expert.

  * •

Average airway generation, which indicates the number of bifurcations between
the current branch and the trachea. Higher generations correspond to smaller
airways and vice versa.

We use the Spearman correlation to investigate the relationship of these
characteristics, because we cannot assume a linear relationship between them
(in particular, the CF status variable is binary). We report the correlation
coefficient and the p-value from a two-sided hypothesis test, where the null
hypothesis is that the characteristics are not correlated. We use a
significance threshold of 0.05. Since we perform five comparisons in total,
after adjusting for multiple comparisons the threshold becomes 0.01.

## Results

### Validity of Crowdsourcing Annotations

In total we collected 20520 results for 1026 tasks. A few typical examples are
shown in Fig. 3. Of these 11742 results (57.2%) were classified as invalid,
and 624 (3.0%) contained multiple pairs of ellipses per image, which we
excluded to simplify the analysis.

Of the 11742 invalid results, 8809 tasks only had one annotation. This could
indicate not seeing an airway, which was the case for 2641 of the results. A
further 2933 results had signs of the worker trying to annotate the image
(placing ellipses on top of airways), but not following the instructions of
outlining two ellipses.

Figure 3: Example results acquired for the same task: valid result with two
annotations, and two invalid results: a worker who indicates not seeing an
airway, and a worker who detects the airway but does not outline it.

The results were created by 577 workers in total, who made as little as 1 or
as many as 2313 results. Similar to the observations in [12], most workers
only created a few results, and a few workers were responsible for a lot of
the results, as shown in Fig. 4 (left). Fig. 4 (right) shows the number of
valid and invalid results made by each worker. Overall there is a tendency for
workers to create more valid than invalid results. However, there are a few
workers who have created a lot of results overall, and who tend to create more
invalid results. They contribute to 57.2% of the invalid results. Finally,
there are no workers that created only invalid results.

Figure 4: (Left) Cumulative results made by the workers. Many workers
contribute a few results, and a few workers contribute the most results.
(Right) Valid results vs invalid results by each worker. The blue line shows
the fit to the data, and that workers create tend to create more valid results
in general.

### Quality of Airway Measurements

When considering each result independently (without combining the results per
task), there is a correlation of 0.803 for the inner airway and 0.697 for the
outer airway.

Additionally, we found moderate correlations for the ratio based measures,
0.426 for the WAP and 0.424 for the WTR.

| Subjects 1-24
---|---
Combining | Inner | Outer | WAP | WTR
None | 0.803 | 0.697 | 0.426 | 0.424
Random | 0.803 | 0.701 | 0.421 | 0.418
Median | 0.844 | 0.769 | 0.572 | 0.565
Best | 0.858 | 0.896 | 0.585 | 0.590
Expert-Expert | 0.964 | 0.925 | 0.701 | 0.687
Table 2: Pearson correlations between the expert and the crowd with different
combining methods, and between two experts.

The airway measurements and correlations after combining the results across
workers are shown in Table 2, as well as Fig.5, 6, 7 and 8. Combining improves
all correlations, and for the ratios the correlations can be categorized as
strong for median and “best” combining. “Best” combining gives the highest
correlations, although the difference with median combining is rather small
for the inner airway, WAP and WTR. For the outer airway, the difference is
more pronounced (0.769 vs 0.896), suggesting that the task is more difficult,
leading to more variation in the crowd.

Overall, since the “best” combining method is optimistically biased due to
access to ground truth, our results suggest median combining is a good choice
for this data.

Figure 5: Measurements of the inner airway, comparing expert 1 (x-axis) and to
three combining methods and expert 2 (y-axis). Figure 6: Measurements of the
outer airway, comparing expert 1 (x-axis) and to three combining methods and
expert 2 (y-axis). Figure 7: Measurements of the WTR, comparing expert 1
(x-axis) and to three combining methods and expert 2 (y-axis). Figure 8:
Measurements of the WAP, comparing expert 1 (x-axis) and to three combining
methods and expert 2 (y-axis).

Median combining simply combines all (between 1 and 20) the valid results
available for a particular task. To understand how the number of valid results
affects the correlations, we investigated combining only for tasks where at
least a certain number of valid results must be available.

The correlations are shown in Fig. 9. There is almost no effect on the
correlations for the inner and outer airway, and the correlations for WAP and
WTR steadily improve as more valid results are combined. This could also
indicate that the tasks with more valid results, are in general easier images
to annotate.

Figure 9: Correlation of all four measures between the expert and crowd
(median combining), for tasks with at least a specific number of valid
results.

To summarize, the crowd can create good annotations, and combining annotations
using the median helps to improve the quality, although not to the quality of
the expert. For median combining strong correlations for the inner and outer
airways (0.844, 0.769), but moderate to strong correlations for the ratios are
observed (0.572, 0.565). It is important to note that a similar trend is
noticeable in the expert-to-expert correlations: the correlations for the
airway dimensions are much higher (0.964, 0.925) than correlations of WAP and
WTR (0.701, 0.687).

### Predicting Crowdsourcing Quality

Next, we look at the correlations per subject, and whether this correlation
can be predicted based on subject characteristics. The individual subject
characteristics and correlations (between expert and median combining) are
shown in Table 3. Overall we can see high variability across subjects.
Correlations range between 0.64 and 1.00 for inner airway, 0.60 and 0.98 for
outer airway, 0.07 and 0.75 for WAP, and 0.14 and 0.68 for WTR. Note that
these correlations are based on smaller (and different) numbers of tasks
(column “n”).

Table 3: Characteristics of the subject: ID (not used in modeling), whether a subject has CF (1 = yes), FVC1, FEV (as percentage of predicted value), number of airways (n), and correlations between the crowd (median combining) and the expert. Horizontal lines inserted for legibility. | Subject characteristics | Correlation crowd-expert
---|---|---
Index | CF | FVC1 | FEV | n | inner | outer | WAP | WTR
0 | 1 | 109.20 | 119.20 | 77 | 0.93 | 0.97 | 0.52 | 0.42
1 | 0 | 111.50 | 95.40 | 89 | 0.93 | 0.98 | 0.33 | 0.36
2 | 1 | 112.20 | 99.00 | 60 | 0.64 | 0.92 | 0.07 | 0.14
3 | 0 | 82.70 | 78.60 | 168 | 0.78 | 0.94 | 0.35 | 0.36
4 | 0 | 110.00 | 105.10 | 73 | 0.90 | 0.90 | 0.61 | 0.59
5 | 1 | 118.60 | 94.50 | 68 | 0.76 | 0.71 | 0.75 | 0.75
6 | 0 | 95.60 | 85.40 | 173 | 0.90 | 0.87 | 0.45 | 0.43
7 | 1 | 94.20 | 73.40 | 82 | 0.91 | 0.95 | 0.45 | 0.43
8 | 0 | 99.40 | 73.20 | 143 | 0.75 | 0.97 | 0.39 | 0.37
9 | 0 | 120.30 | 123.80 | 122 | 0.93 | 0.97 | 0.60 | 0.61
10 | 1 | 70.60 | 75.30 | 75 | 0.82 | 0.86 | 0.22 | 0.21
11 | 0 | 104.20 | 100.10 | 134 | 0.77 | 0.90 | 0.50 | 0.51
12 | 1 | 73.40 | 56.10 | 47 | 0.91 | 0.91 | 0.51 | 0.44
13 | 1 | 70.60 | 78.10 | 18 | 0.65 | 0.60 | 0.58 | 0.59
14 | 0 | 98.90 | 95.90 | 145 | 0.93 | 0.92 | 0.56 | 0.56
15 | 0 | 73.40 | 66.70 | 278 | 0.87 | 0.93 | 0.43 | 0.39
16 | 1 | 128.60 | 80.60 | 66 | 0.81 | 0.98 | 0.42 | 0.45
17 | 1 | 91.90 | 79.10 | 40 | 0.86 | 0.98 | 0.37 | 0.35
18 | 1 | 96.90 | 91.10 | 27 | 1.00 | 0.89 | 0.48 | 0.53
19 | 1 | 109.00 | 105.80 | 104 | 0.94 | 0.98 | 0.67 | 0.68
20 | 0 | 110.10 | 104.10 | 32 | 0.88 | 0.98 | 0.54 | 0.55
21 | 1 | 64.30 | 69.60 | 64 | 0.87 | 0.94 | 0.24 | 0.26
22 | 0 | 109.60 | 82.00 | 151 | 0.86 | 0.93 | 0.42 | 0.44
23 | 0 | 77.00 | 71.20 | 144 | 0.91 | 0.94 | 0.63 | 0.61

Lastly we looked at the relationship between the quality of the crowd (here
represented by the inner airway correlation) and five subject characteristics.
The Spearman correlations and corresponding p-values are shown in Table 4.
There is a weak negative correlation between the subject having CF and the
crowd quality, however this correlation is not significant for the adjusted
alpha level of 0.01. The other characteristics show almost no correlation with
the crowd quality.

Characteristic | Correlation | p-value
---|---|---
Has CF | -0.265 | 0.211
Generation | 0.003 | 0.987
FVC | -0.038 | 0.859
FEV1 | 0.090 | 0.677
Number of airways: | -0.038 | 0.861
Table 4: Spearman correlation between the crowd quality (measured by the
crowd-expert correlation of the inner airway) and five subject
characteristics.

These results suggest that other factors, not investigated here, are more
important. We suspect that these factors are related to the difficulty of
individual tasks (for example depending on size, shape, and contrast of the
airway and its proximity to vessels or other structures), and/or assignment of
workers to different tasks.

## Discussion

This paper describes a follow-up study of [2]. In that study we concluded that
workers try to annotate airways in the images but often do not create valid
results. After filtering out the invalid results, the correlations between the
crowd and the expert were 0.69 for the inner and 0.75 for the outer airway,
and could be further improved by combining the results. As follow-up steps, we
revised our instructions to the crowd, increased the number of workers from 10
to 20 per slice, and collected annotations for all 24 subjects in the cohort.

The current study (increased from 1 to 24 subjects) shows that despite
revising the instructions, the number of invalid annotations is still high
(57.2). This can happen when a worker does not see an airway, sees an airway
but annotates it incorrectly, or due to spam (workers who submit random
results just to get the reward). Our analysis shows that most workers create
both valid and invalid results. An improvement to increase the number of valid
results would be to perform checks (such as requiring one ellipse to be inside
the other) inside the annotation interface.

After removing the invalid results, we examined the correlations between the
crowd and the expert. Without combining, the correlations were strong for the
inner and outer airway, and moderate for WAP and WTR. Combining results across
tasks improved correlations, so that all four measures had strong
correlations. However, all correlations were still lower than correlations
between two experts (see Table 3).

We used simple combining methods to combine the results of the crowd. Here
many alternatives are possible, such as weighting the workers by their
estimated quality. Instead we tried to estimate an “upper limit” for the crowd
with a method which selected the best available result for each task. This
method did indeed lead to the highest correlations, but simple median
combining was a close second. We conclude that median combining is suitable
for this data.

Overall we conclude that the crowd is capable of producing good-quality, but
not expert-quality, results. As such, in its current form the proposed method
is not robust enough for gathering measurements “in the wild”. In our
experience this is primarily due to the difficulty of converting a clinical
problem into a crowdsourcing problem, such as figuring out how to display
parts of a 3D image in a 2D interface, explaining the task to the workers, and
dealing with the constraints of the crowdsourcing platform.

There are a number of important lessons from this study, which could be
valuable for other researchers doing similar studies. Firstly, our interface
was custom built by a crowdsourcing start-up, in the context of a pilot for
academic groups. This allowed us to use an ellipse tool that is similar to the
tool used by the experts. A disadvantage of this approach is that we could not
easily access the interface after the pilot ended, and thus would not be able
to collect additional data.

Secondly, although we did a test run of the task (collecting results described
in [2]), we did not gather feedback from the workers about their experiences.
This is possible through various online groups such as
https://www.turkernation.com, and could have reduced possible
misunderstandings of the instructions. Furthermore, we used crowd
qualifications (such as acceptance rate) and rewards that are outdated by
today’s standards, so we would recommend other researchers to consult the
latest crowdsourcing literature before setting up such a study.

For reproducibility of the results and any follow-up analyses, we made the
airway images, crowd results and our code available via
http://github.com/adriapr/crowdairway.git.

## Conclusions

We conducted a follow-up study of crowdsourcing airway measurements in slices
of chest CT scans. In a previous, preliminary study we examined airways of one
subject and found moderate to strong correlations with the expert, motivating
this follow-up study. Overall we observed similar or better results, with
strong correlations for the inner and outer airway dimension measurements, and
moderate to strong correlations for ratios of these structures. Combining
results across different workers improved the correlations, but correlations
were still lower than between two experts. We conclude that with appropriate
processing, the crowd has potential in measuring airways, but that our method
is not yet robust enough for use in practice.

## References

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# Design of free-space couplers for suspended triangular nano-beam waveguides

J.P. Hadden1,2, Cobi Maynard1, Daryl M. Beggs1, Robert A. Taylor3, Anthony J.
Bennett1,2 1 School of Physics and Astronomy, Cardiff University, Queen’s
Buildings, Cardiff CF24 3AA, United Kingdom  2 School of Engineering, Cardiff
University, Queen’s Buildings, Cardiff CF24 3AA, United Kingdom  3 Clarendon
Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford
OX1 3PU, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Photonic waveguides with triangular cross section are being investigated for
material systems such as diamond, glasses and gallium nitride, which lack easy
options to create conventional rectangular nanophotonic waveguides. The design
rules for optical elements in these triangular waveguides, such as couplers
and gratings, are not well established. Here we present simulations of
elements designed to couple light into, and out of, triangular waveguides from
the vertical direction, which can be implemented with current angled-etch
fabrication technology. The devices demonstrate coupling efficiencies
approaching $50\text{\,}\%$ for light focused from a high numerical aperture
objective. The implementation of such couplers will enable fast and efficient
testing of closely spaced integrated circuit components.

##  1 Introduction

Triangular waveguides have been used to study integrated photonics with colour
centres hosted in diamond [1, 2], silicon carbide[3] and also with rare earth
ions in glass [4]. Angled etching of these materials creates high-index-
contrast waveguides, and thus tight confinement of the optical field, albeit
with geometry that is less well studied than conventional rectangular or ridge
waveguides. Two methods have been used to create triangular waveguides: (1)
Faraday cage-assisted dry-etching, which deflects the ions in a inductively
coupled plasma etcher away from the surface normal, creating under-cut
structures in a single step by etching from both sides simultaneously as shown
by Burek $et$ $al$. [2], and (2) angled focused-ion-beam etching (FIB), where
sequential etching from one side, then the other, fully releases the waveguide
from the surrounding material as shown by Bayn $et$ $al$. [1]. Both methods
are useful for materials systems without a selective anisotropic etch that
that may otherwise be used to undercut. The latter can offer in-situ control
of the waveguide position and shape, but the former is best suited to mass-
manufacture of many devices in one step.

Figure 1: a) Illustration of waveguide device with input and output couplers
of Type A - Grating coupler with mode converter. b) Type B - Wedge mirror with
Bragg reflector, and c) Type C - Wedge mirror with mode converter.

Recently, the fabrication of suspended triangular nano-beams in gallium
nitride (GaN) has been demonstrated using angled Faraday cage-assisted etching
[5]. GaN is a wide band-gap compound semiconductor underpinning high-power
electronics and UV-visible light sources. It has also been shown to host
single photon emitters with wavelengths spanning the visible to near infra red
spectrum - which is a key requirement for optical quantum technologies [6, 7,
8]. New fabrication technology can open the way towards integrated optical
devices based on this versatile electro-optic material, such as low-footprint
nano-lasers [9], non-linear optical light sources [10] and high speed optical
switches [11, 12] with potential applications as on-chip optical
interconnects.

Figure 2: a) Refractive index of triangular waveguides as a function of top
waveguide width. b) Electric field intensity of TE and c) TM -like modes for a
waveguide with top width of $765\text{\,}\mathrm{nm}$

Light may be coupled into and out of waveguides using grating couplers based
on periodic modulation of the effective refractive index. Such vertical
couplers have important applications in the testing and development of
photonic integrated circuit components, allowing for fast and efficient
testing of closely spaced components. In ridge waveguides, such as silicon on
insulator, this is commonly achieved by etching periodic trenches into the
waveguide. The effective refractive index is dependent on the etch depth and
length of the etched sections, reducing the effective refractive index
$n_{\mathrm{eff}}$ of the waveguide mode in these regions [13]. For triangular
waveguides, one may modify the refractive index by modulating the top
waveguide width. Figure 2 a) shows the effective refractive indices of the
first four modes in an equilateral triangular waveguide as a function of the
waveguide top width width $w$, at $1550\text{\,}\mathrm{nm}$ wavelength,
calculated using Ansys Lumerical Mode [14]. The first two modes may be
categorised as TE-like and TM-like, while the subsequent modes appear to show
rotation symmetry of order three. The nominal target waveguide size is chosen
to be $765\text{\,}\mathrm{nm}$, the largest size that prevents modes 3 and 4
being guided [15]. Field intensity plots of the TE and TM modes for a
waveguide with top width of $765\text{\,}\mathrm{nm}$ are shown in Figure 2 b)
and c).

Although there has already been considerable effort developing optimised fibre
to chip couplers in conventional rectangular waveguide geometries [16], there
has been much less on the on the problem of coupling light from microscope
objectives into nanophotonic waveguides in or in triangular waveguide
geometries. In this work we present designs for free-space couplers based on
the triangular nano-beam geometry which couple light focused from above to a
Gaussian spot by a high NA microscope objective. We present three classes of
designs as illustrated in figure 1: Type A - grating coupler with mode
converter, Type B - wedge mirror with Bragg reflector, and Type C - wedge
mirror with mode converter, with performance of the designs is evaluated and
optimised using Ansys Lumerical FDTD [14]. We focus on gallium nitride as the
host material. Although the refractive indices may be different, the design
concepts and fabrication processes will be transferable to other material
systems such as glass, diamond, silicon, silica and other compound
semiconductors.

##  2 Type A coupler design - Grating coupler with mode converter

Figure 3: Type A coupler design a) Top and b) side view with main design
parameters marked: grating coupler widths ($w_{1}$ and $w_{2}$) and lengths
($l_{1}$ and $l_{2}$), mode converter length ($l_{\mathrm{MC}}$) and waveguide
width ($w_{\mathrm{wg}}$). c) Calculated coupling efficiency for a Gaussian
source focused from above into the TE-like waveguide mode as function of duty
cycle ($l_{1}/(l_{1}+l_{2})$) and grating pitch ($l_{1}+l_{2}$) for a grating
coupler with $w_{1}=$2000\text{\,}\mathrm{nm}$$ and
$w_{2}=$400\text{\,}\mathrm{nm}$$ The pitch and duty cycle ($\mathrm{DC}$)
which satisfy the 1st and 2nd order vertical grating (VG) and Bragg reflector
(BR) phase matching conditions are marked. d) Side view cross section field
intensity distribution for coupler with light input from the top and e)
through the waveguide with TE-like polarisation. f) Far field intensity
distribution with TE waveguide input source. g) Simulated coupling efficiency
as a function of wavelength for light input from the top (labelled ”Top”) and
through the waveguide (labelled ”WG”) in TE and TM-like polarisations.

The top and side profiles of a Type A vertical grating coupler designed to
couple light focused from above using a 0.9 NA objective lens are shown in
Figure 3 a) and b). The grating region consists of repeating sections of
thicker (width $w_{1}=$2000\text{\,}\mathrm{nm}$$, length $l_{1}$) and thinner
(width $w_{2}=$400\text{\,}\mathrm{nm}$$, length $l_{2}$) sections, with a
mode converter (length
$l_{\mathrm{MC}}=$7.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) allowing
adiabatic transition between the thicker grating region, and the
$w_{\mathrm{wg}}=$765\text{\,}\mathrm{nm}$$ waveguide output. The phase
matching condition for a grating coupler is
$n_{\mathrm{eff}}-n_{\mathrm{air}}\sin(\theta)=m~{}\lambda/a$, where
$n_{\mathrm{eff}}$ is the effective index of the optical mode in the grating
region, $n_{\mathrm{air}}$ is the refractive index of air,
$\theta\approx$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$ is the target grating
diffraction angle, $m$ is the grating order number, $\lambda$ is the
wavelength of the light, and $a=l_{1}+l_{2}$ is the grating pitch. The
effective refractive index of the optical mode in the grating region may be
estimated using
$n_{\mathrm{eff}}=\sqrt{\mathrm{DC}~{}{n_{1}}^{2}+((1-\mathrm{DC})~{}{n_{2}}^{2}}$
for TE Modes [17], where $n_{1}$ and $n_{2}$ are the effective refractive
indices for waveguide modes in the thicker ($w_{1}$) and thinner ($w_{2}$)
sections respectively, and $\mathrm{DC}=l_{1}/(l_{1}+l_{2})$ is the duty cycle
of the thicker ($w_{1}$) section. The coupling efficiency for a Gaussian
source focused from above into the TE-like waveguide mode is calculated as a
function of the grating duty cycle and grating pitch, as shown in Figure 3 c).
The efficiency is defined as the overlap of the normalised field intensity
with the nano-beam’s TE waveguide mode. The pitch and duty cycle which satisfy
the 1st and 2nd order vertical grating and Bragg reflector phase matching
conditions are marked as a dotted line in c).

The most efficient design is achieved with duty cycle $\mathrm{DC}=0.35$, and
grating pitch $a=$ $1.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$, leading to a
coupling efficiency of $24\text{\,}\%$. This is modest compared to grating
couplers optimised for single mode optical fibre to chip coupling in silicon
on silica platforms, where theoretical coupling efficiencies can exceed
$90\text{\,}\%$ [18]. However, such optical fibre grating couplers typically
have dimensions comparable to the single mode fibres - orders of magnitude
larger than the $\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$
focussed spot considered here, and require long tapering regions to convert
from the grating coupler to the waveguides. Compact grating couplers designed
for microscope collection from silicon nitride waveguides show more comparable
theoretical coupling efficiencies of up to $40\text{\,}\%$ (or $60\text{\,}\%$
when including a reflective mirror underneath) [19].

Interestingly, this design does does not appear to satisfy the grating coupler
phase matching condition. Designs which do are limited to around
$15\text{\,}\%$. We believe the reason that this design outperforms grating
designs that satisfy the phase matching condition is that the size of focused
the Gaussian spot is only large enough to cover one period of the grating, and
thus, the device is exhibiting a Mie resonance which is enhancing the coupling
- as recently shown in silicon nano-wire super-lattices [20]. Grating couplers
which demonstrate out-coupling from a single period have also been
demonstrated in gallium arsenide for out coupling quantum dot emission [21].

The grating coupler design was further optimised using a particle swarm
algorithm, varying the grating pitch, duty cycle, and the offset of the
focused source TE polarised source. The resulting solution was then evaluated
using four different input scenarios: Top (TE) focused Gaussian input with TE-
like polarisation - where the electric field is perpendicular to the
waveguide, Top (TM) focused Gaussian input with TM-like polarisation - where
the electric field parallel is to the waveguide, WG (TE) input through the
waveguide TE-like mode, WG (TM) input through the waveguide TM-like mode. This
allows us to test how the optimised coupler performs to couple light from
free-space into the waveguide, and conversely from the waveguide into free-
space in both TE and TE-like polarisations. Figure 3 d) shows the field
intensity of a cross section of the optimised grating coupler when illuminated
from above with a TE source, demonstrating how the field is coupled from a
focused Gaussian source above the grating region, to the waveguide output on
the right hand side, while Figure 3 d) shows the shows the field intensity
when the waveguide is excited through the waveguide mode, while Figure 3 f)
shows a far field projection of the output of the coupler in the vertical
direction. We can estimate the collection efficiency of this output into a
microscope objective by integrating the field intensity within the angular
cone defined by a 0.9 NA collection objective
($\theta=$64\text{\,}\mathrm{\SIUnitSymbolDegree}$$). The performance of the
coupler for both TE and TM-like modes from the four different input scenarios
over a wavelength range of $200\text{\,}\mathrm{nm}$ around the
$1550\text{\,}\mathrm{nm}$ design wavelength is shown in Figure 3 g). It can
be seen that the device performs well for TE-like modes with peak efficiencies
of $23\text{\,}\%$ for top illumination, $28\text{\,}\%$ for waveguide
illumination with a bandwidth of around $200\text{\,}\mathrm{nm}$, however for
TM polarised light efficiencies peak at less than $2\text{\,}\%$.

##  3 Type B coupler design: Wedge mirror with Bragg reflector

Figure 4: Type B coupler design a) Top and b) side view with main design
parameters marked: grating coupler widths ($w_{1}$ and $w_{2}$) and lengths
($l_{1}$ and $l_{2}$), mode converter length ($l_{\mathrm{MC}}$) and waveguide
width ($w_{\mathrm{wg}}$). c) Calculated efficiency of coupling a Gaussian
source focused from above into the TE-like waveguide mode as function of duty
cycle ($l_{1}/(l_{1}+l_{2})$) and grating pitch ($l_{1}+l_{2}$) for a Bragg
reflector with $w_{1}=$3000\text{\,}\mathrm{nm}$$ and
$w_{2}=$400\text{\,}\mathrm{nm}$$ The pitch and duty cycle which satisfy the
vertical grating and Bragg reflector phase matching conditions are marked. d)
Side view cross section field intensity distribution for coupler with focused
Gaussian TE input and e) with TE waveguide input sources. f) Far field
intensity distribution with TE waveguide input. g) Simulated coupling
efficiency as a function of wavelength for light input from the top (Top) and
through the waveguide (WG) in TE and TM-like polarisations.

During optimisation of the type A design it was discovered that for certain
lengths of $l_{\mathrm{MC}}$ the mode converter section of the waveguide
actually performs as an angled mirror, allowing light to be reflected through
total internal reflection from the vertical direction to the horizontal.
Rectangular total internal reflection mirror couplers have previously been
demonstrated in diamond through focused ion beam milling [22]. However, the
triangular geometry investigated here presents a tapered ’wedge mirror’ shape.
When combined with a Bragg reflector, it is possible to design a device which
reflects and directs light focused from above into the the waveguide mode.

Figure 4 a) and b) show the top and side profiles of a Type B device combining
a wedge mirror and a Bragg reflector. Bragg reflectors based on modulating the
waveguide width can be implemented in a similar way to the grating couplers,
however in this case the phase matching condition becomes
$n_{\mathrm{eff}}=2(m+1)~{}\lambda/2a$. The grating region consists of
repeating sections of thicker (width $w_{1}=$3000\text{\,}\mathrm{nm}$$,
length $l_{1}$) and thinner (width $w_{2}=$400\text{\,}\mathrm{nm}$$, length
$l_{2}$) sections, with a wedge mirror (length
$l_{\mathrm{MC}}=$2.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) allowing total
internal reflection of the focused beam from above, and directing the
reflected light into the $w_{\mathrm{wg}}=$765\text{\,}\mathrm{nm}$$ waveguide
output.

Figure 4 c) shows the coupling efficiency of a Gaussian source focused from
above into the TE-like waveguide mode as a function of the grating duty cycle,
and grating pitch. The pitch and duty cycle which satisfy the vertical grating
and Bragg reflector phase matching conditions are marked as a dotted line in
c). It can be seen that the most efficient designs run along the Bragg
reflector first order line as expected, with the best design parameters
$DC=0.5$ and grating pitch $a=$0.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
showing efficiency of $30\text{\,}\%$. It is notable that this is higher than
the best Type A grating coupler design even though it combines two reflective
elements.

Following a similar procedure as for the previous design, we optimise the
coupling efficiency by varying the grating pitch, duty cycle, wedge mirror
length and the offset of the device with respect to the focused source TE
polarised source. We then evaluate the performance of the optimised coupler
device as before for TE and TM polarised light for in-coupling a focused
Gaussian beam from above into the waveguide mode (Top (TE) and Top (TM)), and
for out-coupling the waveguide modes into a 0.9 NA collection cone
representing a microscope objective (WG (TE) and WG (TM)). Figure 4 d), and 4
e), show the field intensity cross sections for the Top (TE) and WG (TE)
scenarios respectively, while 4 f) shows the far field projection of the field
intensity for the out coupled mode. Figure 4 g) shows the performance of the
device over a $200\text{\,}\mathrm{nm}$ wavelength range around the design
wavelength.

The peak efficiency of this Type B design is $35\text{\,}\%$ for top
illumination, and $29\text{\,}\%$ for waveguide illumination for TE polarised
light, which is better than the Type A design. It also performs better than
the Type A design for TM polarised light, with peak efficiencies of
$17\text{\,}\%$ and $14\text{\,}\%$ respectively for Top (TM) and WG (TM)
input scenarios, and spatially it is much more compact - with a total length
of $4.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ as compared to
$13.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for design A, however the
efficiency profile over the wavelength range, and the output mode are both
more complex, indicating several output modes, which is less desirable for
some applications - for example where the output needs to be coupled into a
single mode fibre.

##  4 Type C coupler design: Wedge mirror with mode converter

Figure 5: Type C coupler design a) Top and b) side view with main design
parameters marked: wedge mirror and mode converter widths ($w_{1}$, $w_{2}$,
$w_{3}$) and lengths ($l_{1}$, $l_{2}$) and waveguide width
($w_{\mathrm{wg}}$). c) Calculated efficiency of coupling a Gaussian source
focused from above into the TE-like waveguide mode as function of mode
converter length and wedge mirror width $w_{1}=$2000\text{\,}\mathrm{nm}$$ and
$w_{2}=$2000\text{\,}\mathrm{nm}$$ d) Side view cross section field intensity
distribution for coupler with focused Gaussian TE input and e) with TE
waveguide input sources. f) Far field intensity distribution with TE waveguide
input source. g) Coupling efficiency of coupler as a function of wavelength
with focused Gaussian, and waveguide TE and TM like input sources.

The third design we consider, Type C, consists of a wedge mirror combined with
a mode converter. This design uses only wedged mirrors and mode converters,
with geometry controlled by their relative widths. Figure 5 a) and b) show the
top and side profiles of the Type C device, combining a wedge mirror with
widths ($w_{1}=$400\text{\,}\mathrm{nm}$$ and
$w_{2}=$3000\text{\,}\mathrm{nm}$$) and length $l_{1}$ and a mode converter
with widths ( $w_{2}=$3000\text{\,}\mathrm{nm}$$ and
$w_{3}=$765\text{\,}\mathrm{nm}$$) and length $l_{2}$. Figure 5 c) shows the
coupling efficiency of a Gaussian source focused from above as the mode
converter length ($l_{2}$) and wedge mirror/mode converter thicker width
($w_{2}$) length are varied. The highest efficiency design is obtained with
$l_{2}=$3.75\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$w_{2}=$2.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

Subsequently, we optimise the design, allowing the wedge mirror output width
($w_{2}$), and length ($l_{1}$), and mode converter output width ($w_{2}$) and
length ($l_{2}$), along with the offset of the device with respect to the
focused Gaussian TE polarised source to be varied. This yields a device with a
peak coupling efficiency of $49\text{\,}\%$ for a focused Gaussian TE beam.
Interestingly, the highest efficiency design presents a step discontinuity
between the mode converter and the output waveguide. This device shows good
performance both for in-coupling (Figure 5 d) and out coupling (Figure 5 e),
with a concentrated far field projection field intensity distribution (Figure
5 f). It can also be seen in Figure 5 g) that although there are some spectral
resonances visible, the device works relatively well for both TE and TM
polarised light, in both in-coupling and out-coupling configurations, with
coupling efficiencies greater than $26\text{\,}\%$ for all configurations
across the range.

##  5 Conclusion

In this work, we present the design of free space couplers for suspended
triangular nano-beam waveguides. We investigate three different designs which
demonstrate grating coupler, Bragg reflector, and total internal reflection
elements, using the suspended triangular waveguide geometry. The optimised
device designs demonstrate simulated coupling efficiencies approaching
$50\text{\,}\%$ for light focussed from a high numerical aperture objective.
The development of such couplers will enable fast and efficient testing of
closely spaced integrated circuit components.

## Acknowledgements

This work was performed using the computational facilities of the Advanced
Research Computing @ Cardiff (ARCCA) Division, Cardiff University. We
acknowledge financial support provided by EPSRC via Grant No. EP/T017813/1 and
EP/P006973/1.

## Data Availability

The data that support the findings of this study are openly available in the
Cardiff University Research Portal at
http://doi.org/10.17035/d.2022.0214949311.

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