10pt utf8

# Calliar: An Online Handwritten Dataset for Arabic Calligraphy

Zaid Alyafeai
Department of Computer Science
King Fahd University of Petroleum and Minerals
Dhahran, Saudi Arabia 31261
<EMAIL_ADDRESS>
&Maged S. Al-shaibani
Department of Computer Science
King Fahd University of Petroleum and Minerals
Dhahran, Saudi Arabia 31261
<EMAIL_ADDRESS>
&Mustafa Ghaleb
Department of Computer Science
King Fahd University of Petroleum and Minerals
Dhahran, Saudi Arabia 31261
<EMAIL_ADDRESS>
&Yousif Ahmed Al-Wajih
Department of Systems Engineering
King Fahd University of Petroleum and Minerals
Dhahran, Saudi Arabia 31261
<EMAIL_ADDRESS>

###### Abstract

Calligraphy is an essential part of the Arabic heritage and culture. It has
been used in the past for the decoration of houses and mosques. Usually, such
calligraphy is designed manually by experts with aesthetic insights. In the
past few years, there has been a considerable effort to digitize such type of
art by either taking a photo of decorated buildings or drawing them using
digital devices. The latter is considered an online form where the drawing is
tracked by recording the apparatus movement, an electronic pen for instance,
on a screen. In the literature, there are many offline datasets collected with
a diversity of Arabic styles for calligraphy. However, there is no available
online dataset for Arabic calligraphy. In this paper, we illustrate our
approach for the collection and annotation of an online dataset for Arabic
calligraphy called Calliar that consists of 2,500 sentences. Calliar is
annotated for stroke, character, word and sentence level prediction.

##  1 Introduction

Handwriting is used to convey information by writing on a paper or digital
screen by hand. Every person, while writing the same statement, has a unique
way of drawing letters. Compare that to digital typing which is mostly
invariant and usually conducted using a set of predefined fonts. Processing
handwriting is a difficult task because of its complexity and the different
styles of people’s writings. Add to that, if we consider a complex language
like Arabic which is cursive i.e letters are connected together which causes
different variations in the writing system. For example, the letter س> is
written like that when comes at the end of words, when it comes at the middle
it is written like this ـسـ> . Similarly, the letter ع> is written as ـعـ>
which can be considered as a completely different letter. The complexity is
also even higher when we add diacritics which are used as an alternative for
short vowels in Arabic.

Arabic calligraphy has a long history since it was used for a long time to
decorate houses, mosques and public places. Opposed to regular handwriting,
calligraphy adds some kind of artistic view owing to the freedom of drawing
letters. There are various of calligraphic styles and each one of them has
specific styles and rules. The most popular calligraphy types are Diwani
ديواني>, Thuluth ثلث> , Kufi كوفي> , Farisi فارسي> , Naskh نسخ> and Rekaa
رقعة> . We can consider regular handwriting as a special case of calligraphy
where a specific style is used, more prominently: Naskh نسخ> and Rekaa رقعة>.
Another complexity of writing in calligraphy is the freedom of drawing in a
convoluted and non-linear approach; which might cause people difficulty in
understanding some calligraphic styles. In addition to that, some writers add
some special characters like diacritics and other decoration symbols.

In the literature, there are many studies that collect large datasets for
handwriting in Arabic whether offline or online. Offline writing deals with
handwritten text as an image where we don’t have any idea how each letter is
drawn. On the other hand, online writing records the information of each point
and how to draw each single letter i.e each written character can be redrawn
latter in an animated way. The most popular datasets are KHATT [1] which
consists of offline handwritten text. An example of online dataset is ADAB [2]
which composes of 32,492 online pen traces.

The existing datasets deal only with regular handwriting (as also indicated in
the literature review section). To the best of our knowledge, there is no
online dataset for Arabic calligraphy. Our main contribution is creating a
large online dataset called Calliar for Arabic calligraphy in different
styles. The main outcome is a dataset that allows users to extract stroke,
character, word and sentence information in an easy manner. Moreover, the
dataset consists of different variations of Arabic calligraphy styles like
Diwani, Thuluth, Kufi and Farisi. This paper is organized as follows. Section
2 reviews the current studies and summarizes the existing Arabic datasets.
Section 3 and 4 describe the annotation, visualization and give a summary of
our dataset. Section 5 and 6 emphasize the dataset importance by summarizing
its impact on research, history and society.

##  2 Related Work

In the literature, there are many studies that dealt with collecting a large
dataset for regular handwritten. One of the largest datasets for English that
is used heavily in the literature is the IAM dataset [3] which is an offline
dataset written by 400 different writers. The total number of word instances
in the dataset is 82,227 with a total of 10,841 instances of unique vocabulary
dataset.The Casia [4] (Chinese handwriting databases) dataset consists of 3.9
million samples of 7,356 classes which is a collection of online and offline
datasets. UPTI [5] dataset consists of more than 10,000 synthetic images of
Urdu text written in Nastaleeq font.

In the field of Arabic handwriting recognition systems, there are many
datasets that are offline like KHATT [1], which consists of 1,000 handwritten
forms written by 1,000 different writers from different countries. KHATT was
extended later to online-khatt that consists of 10,040 lines conducted by 623
different writers [6]. ADAB [2] on the other hand is an online dataset that
consists of 32,492 Arabic words handwritten by more than 1000 writers. There
are also multilingual datasets that combine Arabic and English like MAYASTROUN
[7]. It consists of 67,825 samples that were written by 355 writers. The
dataset consists of many variant scripts like words, characters, digits,
mathematical expressions, and signatures.

There are a few studies in the literature that deal with Arabic calligraphy.
Koudja et al. [8] collected 1,685 images and categorized them into 9 classes
of different calligraphic styles like Thuluth, Naskh, Diwani, etc. Allaf and
Al-Hmouz [9] designed a system for recognizing artistic Arabic calligraphy
types. Then, they evaluated the system on three different Arabic handwritten
calligraphy styles which are Thuluth, Reqaa, and Kufi. The total size of the
dataset is 267 images with equal number of images in each class. There are
also challenging datasets that consider historical texts like KERTAS [10]
which composes of around 2,000 images taken from various handwritten Arabic
scripts mainly during the fourteen centuries. AlSalamah et al. [11] collected
1,000 calligraphy images that were scrapped from public websites. The dataset
was manually annotated to extract letter images. Bataineh et al. [12]
collected 700 samples of Diwani, Rekaa, Kufi, Persian, Andalusi, Naskh, and
Thuluth consisting of 100 images, gathered from several sources. In another
research, Bataineh et al. [13] collected a dataset that consists of 14 Arabic
degraded document images. Each image represents one of binarization challenges
such as law contract, seeking ink, multi-color and dirty spots. Allaf et al.
[9] developed an algorithm for Arabic calligraphy types recognition. Then,
they tested their algorithm on a combination of local and public dataset. The
local dataset consists of 18 images of sentences written by a skilled
calligraphist. In their work, words are separated manually from the sentence
images and some of the words are rewritten independently since they could not
split it from the original sentence. The total number of samples in this
dataset is 71 words. There is a also big computer-generated dataset called
APTI (Arabic Printed Text Image) [14]. This dataset was generated utilizing
lexicon of 113,284 words with 20 different fonts and sizes in 4 different
calligraphy styles. Khayyat et al. [15] collected 2,653 images of a historical
Arabic manuscripts. The dataset is categorized into 37 classes with six
handwriting styles. Kaoudja at al. [8] proposed a dataset consisting of 9
calligraphy styles. In total, 1,685 images have been classified under their
inscribed styles. Each calligraphy style consists of around 180 to 195 images.
Adam et al. [16] collected 330 images of isolated Arabic letters that were
extracted from ancient manuscripts. This dataset consists of Rekaa, Diwani,
Kufi, Naskh, and Farsi styles. This dataset has been used to classify Arabic
scripts style based on segmented letters.

As apparent from the aforementioned studies and up to our knowledge, there is
no dataset in the literature that deals with Arabic calligraphy in online
format. In Table 1, we summarize the existing datasets based on four criteria
named number of samples, type, language and either the dataset is public or
private.

Reference | Number of samples | Type | Language | Public
---|---|---|---|---
Allaf et al. [9] | 18 | Sentences | Arabic |
Al-Hmouz [9] | 267 | Sentences | Arabic |
Adam et al. [16] | 330 | Characters | Arabic |
KHATT [1] | 1,000 | Sentences | Arabic | $\checkmark$
online-khatt [6] | 10,040 | Sentences | Arabic | $\checkmark$
AlSalamah et al. [11] | 1,000 | Characters | Arabic | $\checkmark$
Koudja et al. [8] | 1,685 | Sentences | Arabic | $\checkmark$
KERTAS [10] | 2,000 | Historical Documents | Arabic | $\checkmark$
Khayyat et al. [15] | 2,653 | Historical Documents | Arabic |
ADAB [2] | 32,492 | Words | Arabic | $\checkmark$
Slimane et al. [14] | 113,284 | Words | Arabic | $\checkmark$
Bataineh et al. [12] | 700 | Sentences | Arabic/Jawi |
UPTI [5] | 10,000 | Sentences | Arabic/Urdu | $\checkmark$
MAYASTROUN [7] | 67,825 | Sentences | Arabic/English | $\checkmark$
Table 1: Summary of datasets in the literature.

##  3 Annotation, Representation and Exploration

###  3.1 Annotation

The initial phase started by scraping the internet of Arabic calligraphy. We
collected a wide variety of calligraphy like Diwani, Thuluth, Kufi, Farsi and
many others. The annotation was implemented by tracking the calligraphy images
using a digital pen and tablet by four different annotators. In Figure 1, we
present an example of the annotation process. The main canvas in the center
contains the image to be annotated. The different colors used in the
annotation represent different strokes. At the bottom, we have different
buttons to simplify the annotation process like undoing a certain stroke in
addition to being able to clear or skip the current drawing. There are mainly
two input fields at the top of the annotated image. In the second input, we
see the annotation text which is editable so the annotator can write new
annotations or fix any misspelled data instances. In the top input field, we
split up each letter into simple components where each component represents a
stroke. A stroke is defined as a single writing procedure where the writer
doesn’t lift their hand. Hence, a lot of letters will be split into different
components depending on how they are written as shown in Table 2. For example,
letters that contain dots or nekat نقاط> in Arabic are split into the basic
letter without the dot and the dot as a different stroke. We also
differentiate between dots that appear above or below the letter. If the dot
appears above the letter we write it before the letter itself, otherwise it
comes after. This also extends to other special characters like Hamza ء> which
can appear below or above a certain letter. The point behind encoding that in
an explicit format is to guide the annotator how to write each letter and
whether to ignore certain characters like special characters. In our
annotation process diacritics and other characters that are used for
decoration are ignored. We only include Maddah $\sim$ which usually happens
with Alef ا> . In total we have 37 unique letters in our vocabulary.

Figure 1: The annotation process. Letter | Decomposition | Letter | Decomposition
---|---|---|---
أ> |  ا ء> |  آ > |  $\sim$ ا>
إ> |  ء ا> |  ب> |  . ٮ >
ت> |  ٮ . . > |  ث> |  ٮ . . . >
خ> |  ح .> |  ج> |  . ح>
ذ> |  د .> |  ز> |  ر .>
ش> |  س . . . > |  ض> |  ص .>
ط> |  صـ ا> |  ظ> |  صـ ا .>
غ> |  ع .> |  ف> |  ٯ .>
ق> |  ٯ . .> |  ك> |  ل ء >
ن> |  ٮ .> |  ؤ> |  و ء >
ة> |  ه . . > |  ئ> |  ى ء >
Table 2: Decomposition of letters into basic components.

###  3.2 Representation

We tried to come up with a consistent representation such that researchers can
easily reuse the dataset. In this manner, the annotation process is carefully
designed. When the annotator finishes a certain stroke, it is recorded as a
list of $(x,y)$ coordinates representing where the pen traveled while drawing
the stroke. We define a sketch as a list of strokes and each stroke consists
of the annotation i.e what kind of letter was written in addition to the
$(x,y)$ components. Each sketch is saved as json file and can be easily
converted to an image by drawing the coordinates. In order to avoid the
problem of having images with different dimensions, we fix the maximum size of
each dimension i.e width or height to 600 pixels. In order to keep the aspect
ratio of the images and the strokes, the larger dimension was scaled to 600
pixels and the smaller dimension was multiplied by the same factor. As an
example, if an image had a size (400, 200) then we rescale it to the size
(600, 300). In Figure 3, we show 16 images by redrawing the content of the
json files. The dataset is publicly available at GitHub
111https://github.com/ARBML/Calliar. The compressed zip format of the dataset
has size of around 52 MB. We also create a compressed format of size 8.6 MB
which has the same format as QuickDraw [17]. The compressed format doesn’t
have any stroke level annotations and the Ramer-Douglas-Peucker (RDP)
algorithm is applied to reduce the number of points [18]. In Figure 2, we
visualize the raw data of a given word خالد> . Note that for visual purposes
we apply the RDP algorithm. The raw data contains a list of dictionaries where
each dictionary represents the stroke as text and a list of points.

Figure 2: Raw data for the word خالد> . Figure 3: Examples of the visualizing
the json files.

###  3.3 Exploration

Data visualization and exploration is a necessary technique to assess the
quality of the collected dataset. As we saw from Figure 3, there is some kind
of variations in writing the same letter. In Arabic script writing, letters
are written connected together which causes a variation between a letter in
beginning, middle or at the end. Moreover, various calligraphy styles draw
characters differently which might cause a single letter to be confused with
another. This is much more apparent when we try visualizing individual
characters in our datasets. In Figure 4(a), we visualize four different
letters sampled randomly from our dataset. For example, the letter س> , in the
first row is written in different variations. The first image of the character
is interesting because the character is drawn upside down. This type of
drawing mostly happens in Kufic calligraphy. The letter م> is also interesting
because it can be written with or without a loop depending on the style of
calligraphy. The letter ك> is unique because it can appear in either single
stroke or double stroke depending in its position on a given word. Similarly
in Figure 4(b), we see the results of visualizing word level representations.
This is important because this proves that our dataset can be used to extract
word-level features which can be utilized for word level recognition. The most
interesting one is the writing of the word هو> which can be written in many
variations as seen in the third row. One thing that makes this dataset complex
is the freedom of writing in different angles like in the last column of the
third row where the word is written vertically.

In Table 3, we summarize the annotation levels of the dataset. Each level is
annotated and can be extracted easily from the dataset with its corresponding
annotation. The first level represents an annotation for the full drawn
calligraphy, while the second level represents an annotation for the word
level. The third level is an annotation on the character level and finally,
the fourth level is an annotation on the stroke level. Such granularity of
representation allows our dataset to be used in multiple tasks ranging from
stroke and character to word and sentence level prediction. In Figure 5, we
visualize a subset of the characters in the dataset using the t-SNE algorithm
[19] for data projection. We extract 100 images from 18 characters and crop
and resize to around $64\times 64$ then apply the algorithm to reduce the
dimension to 2D space. We see from the projections that the algorithm was able
to cluster some letters like س> and ح>. We also notice that the clusters for
ا> and ل> are close which is expected because they usually have similar
sketches especially in connected letters.

Level 1: full sketch |
---|---
Level 2: word sketches |
Level 3: character sketches |
Level 4: stroke sketches |
Table 3: Levels of drawing a sample image containing the text عيد مبارك>.

(a) Sample character variations.

(b) Sample word variations.

Figure 4: Different variations of drawing characters and words.  Figure 5:
Visualizing top characters using t-SNE algorithm.

In Figure 6 we see the variations of drawing the sentence بسم الله الرحمن
الرحيم>. Which is one of the most frequent sentences that writers like to
write. We see the complexity in style of drawing the same sentence which adds
an extra dimension of artistic view for Arabic calligraphy For a given
calligraphic style we can sketch a sentence in numerous styles. One style that
seems prominent is the alignments of all the Alef characters ا> which adds
some kind of symmetry to the drawing as we see in many examples in the figure.

Figure 6: Visualizing 16x16 grid of the sentence بسم الله الرحمن الرحيم>.

##  4 Dataset Statistics

In Table 4, we summarize the statistics about the collected dataset. We split
the dataset randomly into training, validation and testing and we show the
number of strokes, characters, words and sentences in each split. In Figure 7,
we show the histogram of all characters across all the datasets. Mainly, we
see the characters with high frequency are those which are mostly used in
Arabic writing like ا , ل , م > . Note that most of the characters with low
frequency are composite characters like ؤ ، ئ ، آ > . Along with some
characters that are rarely used in Arabic writing like ظ ، غ ، ڤ > . Note
that, many characters might happen most frequently because of the imbalance in
the representations in some calligraphic images. For example, the sentence بسم
الله الرحمن الرحيم > happen a lot in the writing for calligraphy. The reason
of including the same sentence multiple times is to increase the diversity of
drawing the same sentence as we see in Figure 6.

| Number of Samples | Number of Words | Number of Chars | Number of Strokes
---|---|---|---|---
Train | 2,000 | 6,065 | 24,722 | 36,561
Valid | 250 | 738 | 2,946 | 4,410
Test | 250 | 753 | 3,052 | 4,601
Total | 2,500 | 7,556 | 30,720 | 45,572
Table 4: Data statistics.  Figure 7: Character histograms across all the
dataset.

##  5 Research Impact

The stroke-level annotations of the dataset allow it to be used in a wide
range of research like text classification, optical character recognition
(OCR), and sketch generation. The text classification can range from character
level classification to complete sentence classification and prediction.
Arabic OCR is a difficult task because it contains mostly letters that are
written connected together. Our dataset makes annotations on the stroke and
character level in many variations and styles which allows training efficient
models for such tasks. Such examples of these variations could be found in
Figure 8. The artistic aspect is also important for this dataset. In the
literature, there is no dataset that captures such level of details for Arabic
calligraphy in an online format. Moreover, for OCR for instance, we need to
preprocess the dataset by segmenting individual letters which is not necessary
in our dataset.

In addition to the above-mentioned examples, the dataset could be used for
sketch generation as a creative application. Most of the research in the
literature that deals with sketch drawing and text-to-image, consider English
like GANwriting [20], Scrabble-GANs [21], DF-GANs [22], BézierSketch [23],
sketchRNN [17] and DoodlerGAN [24]. This field of research is very important
because it combines multiple modalities ranging from natural language
processing (NLP) , generative adverserial networks (GANs) and creative
applications. In the literature, there are hundreds of papers published in
each year for each of those fields but due to the lack of proper datasets,
there are no real advancements in making systems that deal with all of them
especially for Arabic. We believe that this dataset can fill this gap.

##  6 Social and Historical Impact

Arabic calligraphy has a long history of being used in different communities
in the Arabic world. Each community embraced a specific style of calligraphy
which resulted in many calligraphic styles across the Arabic world. For
instance, the Moroccan calligraphy style "Al Maghribi" might contain up to
five different types of calligraphy [25], which is considered an important
part of the Moroccan heritage and culture. Without a proper way of making such
calligraphy digital by encoding the specific drawing of each letter as written
by original writers, such calligraphy might disappear in the future. Moreover,
encoding different calligraphic styles can help in deciphering some of the old
calligraphic styles which are used in historical manuscripts.

In the current digital world, a lot of companies are using calligraphy as a
way of delivering information and capturing the attention of possible
customers. Looking at advertisements on social media and on billboards, there
is an increase interest in making use of different calligraphic styles.
Moreover, during holidays and vacations, people use calligraphic written posts
to congratulate each other during different holy events. We believe that our
dataset can be used for automating the process of generating different
calligraphic styles.

(a) Naskh

(b) Rekaa

(c) Wisam

(d) Farisi

(e) Thuluth

(f) Diwani

Figure 8: Different styles of calligraphy.

##  7 Conclusion

In this paper, we introduced Calliar which is an online dataset for Arabic
calligraphy that contains 2,500 sentences and more than 40,000 strokes. The
dataset allows capturing calligraphy in multiple levels ranging from stroke,
character, word to sentence level representation. The granularity level allows
for using the dataset in multiple tasks like classification on the character
or the word level. In addition to that, we can use the dataset for calligraphy
generation and character recognition. The dataset consists of a wide range of
calligraphic styles like Diwani, Thuluth, Farisi, etc. The different styles
make the dataset unique because of the complexity of drawing letters in each
of the styles. As a future work, we are planning to release new versions which
contain multiple variations in addition to increasing the size of the dataset
massively. This is important because training deep networks usually require
thousands of samples, and as the dataset size increases, the quality of the
results increases. Moreover, we would like to balance the number of samples of
each character to increase the probability of some rare characters like ث> and
ض>.

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11institutetext:  Universidad de Concepción, Departamento de Astronomía,
Casilla 160-C, Concepción, Chile 22institutetext: Unidad Mixta Internacional
Franco-Chilena de Astronomía (CNRS UMI 3386), Departamento de Astronomía,
Universidad de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile
33institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
44institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany 55institutetext: Instituto de Astronomía, Universidad
Nacional Autónoma de México, Apdo. Postal 70264, Ciudad de México 04510,
Mexico 66institutetext: School of Physics and Astronomy, University of Exeter,
Exeter, Stocker Road, EX4 4QL, UK 77institutetext: Space Telescope Science
Institute, Baltimore 21218, MD, USA 88institutetext: LESIA, Observatoire de
Paris, PSL Research Univ., CNRS, Univ. Paris Diderot, Sorbonne Paris Cité(c),
UPMC Paris 6, Sorbonne Univ., 5 place Jules Janssen, 92195 Meudon, France
99institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France

# Probing the innermost region of the AU Microscopii debris disc ††thanks:
Based on observations made with ESO telescopes at Paranal and La Silla
observatory under program IDs 0101.C-0218(A), 087.C-0450(A), 087.C-0450(B)
087.C-0750(A), 088.C-0358(A) and 60.A-9165(A).

A. Gallenne 1122 C. Desgrange 3344 J. Milli 33 J. Sanchez-Bermudez 4455 G.
Chauvin 2233 S. Kraus 66 J.H. Girard 77 A. Boccaletti 88 A.M. Lagrange 88 P.
Delorme 99<EMAIL_ADDRESS>
(Received June 09, 2022; accepted June 29, 2022)

###### Abstract

Context. AU Mic is a young and nearby M-dwarf star harbouring a circumstellar
debris disc and one recently discovered planet on an eight-day orbit. Large-
scale structures within the disc were also discovered and are moving outwards
at high velocity.

Aims. We aim to study this system with the highest spatial resolution in order
to probe the innermost regions and to search for additional low-mass
companions or set detection limits.

Methods. The star was observed with two different high-angular resolution
techniques probing complementary spatial scales. We obtained new $K_{s}$-band
sparse aperture masking observations with VLT/SPHERE, which we combined with
data from VLT/NACO, VLTI/PIONIER and VLTI/GRAVITY.

Results. We did not detect additional close companions within the separation
range $0.02-7$ au from the parent star. We determined magnitude upper limits
for companions of $H\sim 9.8$ mag within $0.02-0.5$ au, $K_{s}\sim 11.2$ mag
within $0.4-2.4$ au, and $L\arcmin\sim 10.7$ mag within $0.7-7$ au. Using
theoretical isochrones, we converted these magnitudes into upper limits on the
mass of $\sim 17\,\mathrm{M_{jup}}$, $\sim 12\,\mathrm{M_{jup}}$, and $\sim
9\,\mathrm{M_{jup}}$, respectively. The PIONIER observations also allowed us
to determine the angular diameter of AU Mic, $\theta_{\mathrm{LD}}=0.825\pm
0.033_{\mathrm{stat}}\pm 0.038_{\mathrm{sys}}$ mas, which converts to a linear
radius $R=0.862\pm 0.052\,R_{\odot}$ when combined with the Gaia parallax.

Conclusions. We did not detect the newly discovered planets orbiting AU Mic
($M<0.2\,\mathrm{M_{jup}}$), but we derived upper limit masses for the
innermost region of AU Mic. We do not have any detection with a significance
beyond $3\sigma$, the most significant signal with PIONIER being $2.9\sigma$
and that with SPHERE being $1.6\sigma$. We applied the pyMESS2 code to
estimate the detection probability of companions by combining radial
velocities, multi-band SPHERE imaging, and our interferometric detection maps.
We show that 99 % of the companions down to $\sim 0.5\,\mathrm{M_{jup}}$ can
be detected within 0.02 au or 1 Mjup down to 0.2 au. The low-mass planets
orbiting at $\lesssim 0.11$ au ($\lesssim 11$ mas) from the star will not be
directly detectable with the current adaptive optics (AO) and interferometric
instruments because of its close orbit and very high contrast ($\sim
10^{\mathrm{-10}}$ in $K$). It will also be below the angular resolution and
contrast limit of the next Extremely Large Telescope Infrared (ELT IR) imaging
instruments.

###### Key Words.:

techniques: high angular resolution – stars: individual: AU Mic –planetary
systems, planets and satellites: formation

††offprints: A. Gallenne

##  1 Introduction

AU~Microscopii (AU Mic, HD 197481, $V=8.63$, $K_{s}=4.53$, $L^{\prime}=4.32$)
is a young nearby M-type star, and a member of the 18.5 Myr old $\beta$
Pictoris Moving Group (Miret-Roig et al. 2020) located at $d=9.714\pm 0.002$
pc (Gaia Collaboration et al. 2021; Lindegren et al. 2021; Gaia Collaboration
et al. 2016). It hosts one of the best-studied edge-on debris discs and has
been observed at X-ray to millimetre wavelengths (see e.g. Kalas et al. 2004;
Quillen et al. 2007; Graham et al. 2007; Schneider & Schmitt 2010; Wilner et
al. 2012; MacGregor et al. 2013; Matthews et al. 2015; Boccaletti et al. 2015;
Daley et al. 2019; Esposito et al. 2020) with both space- and ground-based
telescopes. Early modelling suggested the existence of a ring of colliding
parent bodies located at $\sim 40$ au, generating micron-sized dust particles
later put on eccentric orbits through stellar wind and radiation forces, or
migrating inwards through Poynting Robertson drag (Augereau & Beust 2006;
Strubbe & Chiang 2006). Submillimetre ALMA observations were later used to
infer that this birth ring lies between $\sim 8$ and 40 au (MacGregor et al.
2013; Matthews et al. 2015), although the inner radius is poorly constrained.
An additional halo component made of particles down to submicron sizes is seen
both in scattered light and thermal imaging and extends up to $\sim 210$ au
(Kalas et al. 2004; Krist et al. 2005; Liu 2004; Schneider et al. 2014) with a
complex morphology (Fitzgerald et al. 2007). Disc features such as gaps or
asymmetries created by planetesimal or infant planets make debris discs good
laboratories with which to study planet formation and evolution. AU Mic is a
particularly interesting system as several morphological features are present
in the disc. The disc is asymmetrical with the northwest side brighter than
the southwest side, and present some substructures possibly created by larger
unseen orbiting bodies (Liu 2004). Brightness enhancements have been detected
and attributed to dust-density variations in the range $25-40$ au. A prominent
bump at a projected separation of $\sim 13$ au was also identified on the
southeast side above the disc plane. The disc also exhibits time variability
with large-scale arc-like structures located at projected separations of $\sim
5-10$ au on the southeast side. Those arches are moving away from the star
with projected speeds that are inconsistent with a Keplerian velocity, and
some even reach speeds higher than the escape velocity (Boccaletti et al.
2015, 2018). One of the scenarios to explain those features would be an unseen
point-source parent body releasing dust sequentially (Sezestre et al. 2017) or
dust avalanches (Chiang & Fung 2017). Recent 1.3 mm ALMA observations (Daley
et al. 2019) set upper limits on the largest planetesimals stirring the disc,
and ruled out a perturbing body larger than a Neptune-size object within the
outer belt.

Several high-contrast and high-spatial resolution observations have been
reported, either using the Hubble Space Telescope (HST) or extreme adaptive
optics (AO) systems, but none of these have yet been able to directly detect a
planet-size body. Krist et al. (2005) performed the first coronagraphic
observations of AU Mic but only suggested an unseen companion that would
perturb the disc to explain the various detected features. Subsequent
coronagraphic observations revealed more and finer details about the disc
dynamic and dust distribution (see e.g. Wang et al. 2015; Boccaletti et al.
2015; Lomax et al. 2018; Boccaletti et al. 2018), but still no detection of
this hypothetical companion. Gauchet et al. (2016) observed the innermost
region of the AU Mic disc in the $L^{\prime}$ band using the sparse aperture
masking (SAM) technique, which provides a spatial resolution below the
diffraction limit ($\sim\lambda/2D$). These latter authors did not find any
companion brighter than $\Delta L^{\prime}=5.7$ mag ($5\sigma$ limit) within
$30-500$ mas ($0.3-5$ au). This would correspond to a companion mass of larger
than $10\,\mathrm{M_{jup}}$. From angular differential imaging with a vortex
coronagraph, Launhardt et al. (2020) reported a similar $5\sigma$ contrast
limit in the $L^{\prime}$ band between 100 mas and 250 mas (6.2 mag).
Detection limits were also provided by Lannier et al. (2017) who combined
radial velocity (RV) measurements and direct imaging. These authors
constrained the presence of planets with masses $\geqslant
2\,\mathrm{M_{jup}}$ from a fraction of an astronomical unit (au) of to
several tens of au.

Recently, a super-Neptune planet transiting AU Mic was detected by Plavchan et
al. (2020) using NASA’s Transiting Exoplanet Survey Satellite (TESS). This
planet orbits the star in 8.46 days with a semi-major axis of 0.066 au (6.74
mas). Such a small separation partly explains why it was not detected before
with high-contrast imaging, being hidden behind the coronagraph. From RV
analysis, these latter authors inferred an upper limit for the planet of
$0.18\,\mathrm{M_{jup}}$ ($=3.4\,M_{\mathrm{nep}}$). Klein et al. (2021)
confirmed the planet detection at a $3.9\sigma$ level with the SPIROU
instrument, and inferred a mass of $0.054\,\mathrm{M_{jup}}$. Plavchan et al.
(2020) suspected a possible second transiting planet, which was later
confirmed by Martioli et al. (2021) with new TESS observations. This second
planet of $\sim 0.08\,\mathrm{M_{jup}}$ orbits its star with a period of 18.9
d and a semi-major axis of 0.11 au ($\sim 11.3$ mas).

In this paper, we aim at setting detection limits for additional orbiting
companions within the range $0.04-7$ au ($4-730$ mas) using high-angular-
resolution observations, including infrared (IR) long-baseline interferometry
and SAM. Section 2 presents new SAM observations acquired in 2018 with the
SPHERE/IRDIS instrument mounted at the Very Large Telescope (VLT), together
with literature data from the VLTI/GRAVITY and VLT/NACO instruments. We
present the details of our search for a companion and estimate our detection
limits in Sect. 3 using the CANDID algorithm. We then discuss our results and
provide conclusions in Sect. 4.

##  2 Observations and data reduction

###  2.1 SPHERE/SAM observations

Table 1: Log of our SAM observations with SPHERE/IRDIS and long-baseline
interferometry with GRAVITY, together with some information about the
observation of the literature data.

Star | Date | MJD | DIT | NDIT | NEXP | Sp. Chan. | $<\mathrm{Seeing}>$ | $<\tau_{0}>$ | Baselines | #
---|---|---|---|---|---|---|---|---|---|---
|  | (day) | (s) |  |  |  | (″) | (ms) |  |
SPHERE |
AU Mic | 2018-08-12 | 58342.22527 | 1 | 90 | 5 | 1 | 0.51 | 6.1 | – | –
Sky | 2018-08-12 | 58342.23169 | 1 | 10 | 1 | 1 | 0.47 | 6.0 | – | –
AU Mic | 2018-08-12 | 58342.23624 | 1 | 90 | 5 | 1 | 0.53 | 5.5 | – | –
HD 196919 | 2018-08-12 | 58342.24860 | 1 | 40 | 6 | 1 | 0.50 | 6.0 | – | –
Sky | 2018-08-12 | 58342.25213 | 1 | 10 | 1 | 1 | 0.52 | 5.8 | – | –
AU Mic | 2018-08-12 | 58342.26256 | 1 | 30 | 15 | 1 | 0.48 | 5.8 | – | –
Sky | 2018-08-12 | 58342.26847 | 1 | 10 | 1 | 1 | 0.51 | 5.1 | – | –
AU Mic | 2018-08-12 | 58342.27387 | 1 | 30 | 15 | 1 | 0.51 | 5.6 | – | –
HD 196919 | 2018-08-12 | 58342.28662 | 1 | 40 | 6 | 1 | 0.54 | 6.4 | – | –
Sky | 2018-08-12 | 58342.29015 | 1 | 10 | 1 | 1 | 0.51 | 6.4 | – | –
AU Mic | 2018-08-12 | 58342.29770 | 1 | 30 | 12 | 1 | 0.47 | 6.5 | – | –
Sky | 2018-08-12 | 58342.30257 | 1 | 10 | 1 | 1 | 0.41 | 7.2 | – | –
NACO |
AU Mic | 2011-08-31 | 55804.15287 | 0.04 | 600 | 104 | 1 | 0.82 | 3.1 | – | –
PIONIER |
AU Mic | 2011-08-09 | 55782.30522 | 0.21 | 100 | 1 | 3 | 0.64 | 6.9 | A1-G1-I1-K0 | 1
2011-08-09 | 55782.31373 | 0.21 | 200 | 5 | 3 | 0.78 | 5.7 | A1-G1-I1-K0 | 1
2011-08-09 | 55782.32517 | 0.21 | 100 | 1 | 3 | 0.66 | 6.8 | A1-G1-I1-K0 | 1
2011-08-09 | 55782.34457 | 0.21 | 100 | 1 | 3 | 0.63 | 7.1 | A1-G1-I1-K0 | 1
2011-11-03 | 55868.01837 | 0.39 | 100 | 5 | 7 | 0.94 | 2.3 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.03855 | 0.39 | 100 | 5 | 7 | 0.64 | 3.4 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.05373 | 0.39 | 100 | 5 | 7 | 0.97 | 2.3 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.06742 | 0.39 | 100 | 5 | 7 | 1.01 | 2.2 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.07997 | 0.39 | 100 | 5 | 7 | 1.17 | 1.9 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.09390 | 0.39 | 100 | 5 | 7 | 0.81 | 2.8 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.10677 | 0.39 | 100 | 5 | 7 | 0.99 | 2.2 | D0-G1-H0-I1 | 2
2011-11-03 | 55868.12090 | 0.39 | 100 | 5 | 7 | 0.99 | 2.3 | D0-G1-H0-I1 | 2
2013-08-09 | 56513.10329 | 0.39 | 100 | 5 | 3 | 0.92 | 2.5 | A1-B2-C1-D0 | 3
2013-08-09 | 56513.11264 | 0.39 | 100 | 5 | 3 | 0.76 | 3.0 | A1-B2-C1-D0 | 3
2013-08-09 | 56513.12271 | 0.39 | 100 | 5 | 3 | 0.75 | 3.0 | A1-B2-C1-D0 | 3
2013-08-11 | 56515.07933 | 0.39 | 100 | 5 | 3 | 1.24 | 3.5 | A1-B2-C1-D0 | 4
2013-08-11 | 56515.08979 | 0.39 | 100 | 5 | 3 | 1.42 | 4.0 | A1-B2-C1-D0 | 4
2013-08-11 | 56515.09935 | 0.39 | 100 | 5 | 3 | 1.12 | 4.4 | A1-B2-C1-D0 | 4
2014-10-12 | 56942.09731 | 0.38 | 100 | 5 | 3 | 1.42 | 1.7 | A1-B2-C1-D0 | 5
2014-10-12 | 56942.10788 | 0.38 | 100 | 5 | 3 | 1.32 | 1.8 | A1-B2-C1-D0 | 5
2014-10-12 | 56942.11896 | 0.38 | 100 | 5 | 3 | 1.61 | 1.5 | A1-B2-C1-D0 | 5
2014-10-12 | 56942.13064 | 0.38 | 100 | 5 | 3 | 1.48 | 1.6 | A1-B2-C1-D0 | 5
GRAVITY |
AU Mic | 2016-06-16 | 57555.43405 | 30 | 10 | 5 | 1742 | 1.18 | 3.7 | A0-G1-J2-K0 | –
2016-06-16 | 57555.44252 | 30 | 10 | 5 | 1742 | 1.12 | 3.0 | A0-G1-J2-K0 | –
2016-06-16 | 57555.45238 | 30 | 10 | 5 | 1742 | 1.35 | 3.0 | A0-G1-J2-K0 | –

111Sp. Chan. is the number of spectral channel, $<\mathrm{Seeing}>$ is the
average seeing as provided by the Differential Motion Image Monitor (DIMM) at
$0.5\,\mu$m, and $<\tau_{0}>$ the coherence time provided by the MASS-DIMM and
Baselines is the telescope configuration. The calibrators used are HD~197540
(G8III, $V=6.50$ mag, $H=4.32$ mag, $\theta_{\mathrm{LD}}=0.645\pm 0.046$
mas), HD~201865 (K2III, $V=7.14$ mag, $H=4.04$ mag,
$\theta_{\mathrm{LD}}=0.793\pm 0.075$ mas) and HD~202775 (K4/5III, $V=7.49$
mag, $H=4.05$ mag, $\theta_{\mathrm{LD}}=0.844\pm 0.072$ mas) for #1,
HD~202293 (K5III, $V=7.92$ mag, $H=3.74$ mag, $\theta_{\mathrm{LD}}=0.994\pm
0.088$ mas), HD~196387 (K4III, $V=7.28$ mag, $H=3.52$ mag,
$\theta_{\mathrm{LD}}=1.110\pm 0.106$ mas) and HD~202775 for #2, HD~197540,
HD~194432 (K1III, $V=7.16$ mag, $H=4.87$ mag, $\theta_{\mathrm{LD}}=0.555\pm
0.013$ mas) and HD~200750 (K1III, $V=6.81$ mag, $H=4.53$ mag,
$\theta_{\mathrm{LD}}=0.627\pm 0.047$ mas) for #3, HD 197540, HD 194432 and HD
200750 for #4, and HD 196387 for #5.

We observed AU Mic with the extreme AO instrument SPHERE (Spectro-Polarimetric
High-contrast Exoplanet REsearch, Beuzit et al. 2019) and the IRDIS/SAM mode
(Cheetham et al. 2016). This is an interferometric method (Tuthill et al.
2000) which uses a mask with non-redundant holes placed in the pupil plane.
Each pair of subapertures (or baseline) will form an interference fringe
pattern. Observables that are closure phases (sum of the phases between three
baselines forming a triangle) and squared visibilities (fringe contrast) can
then be extracted, allowing us to reach an angular resolution down to
$\lambda/2D$ ($D$ is the diameter of the primary mirror).

The observations were carried out on 2018 August 18 with the IRDIS (Dohlen et
al. 2008) in the $Ks$-band filter ($\lambda=2.10\,\mu$m,
$\Delta\lambda=0.30\,\mu$m) and a seven-hole aperture mask. This provides 21
baselines, that is 21 fringes with a unique spatial frequency and direction.
Data were acquired in a data cube mode with tens of frames of 1 s exposure
time. Similar sequences are executed on an empty sky region for background
subtraction and on a calibrator star. The full log is summarised in Table 1.
AU Mic observations were interleaved with the calibrator star HD~196919
(K4III, $R=6.94$ mag, $K=4.31$ mag, $\theta_{\mathrm{LD}}=0.717\pm 0.023$ mas)
in order to monitor the interferometric transfer function and calibrate the AU
Mic visibilities. HD 196919 was chosen with the SearchCal
software222https://www.jmmc.fr/english/tools/proposal-preparation/search-cal/
to be as close as possible to AU Mic in the sky and with a similar brightness.

Our data reduction steps were as follows. First, each image was corrected for
detector cosmetics using a standard procedure, including sky subtraction,
flat-fielding, and bad-pixel correction. Figure 1 shows an example of the
interferogram and power spectrum of AU Mic. Subwindows of $200\times 200$ px
($2.45\arcsec\times 2.45\arcsec$) of the two images (left and right channels
of the IRDIS detector) in each frame of the cubes were then extracted, stored
in new cubes of 450 interferograms each, and recentred at a subpixel level. To
extract complex visibilities, we used our own Python algorithm based on the
formalism detailed in Tuthill et al. (2000). Briefly, for each cube complex
visibilities were computed frame by frame from the Fourier transform of the
interferograms. Squared visibilities ($V^{2}$) were then calculated from the
squared modulus, while the bispectrum was obtained by multiplying three
complex visibilities corresponding to a triangle of holes in the mask. Closure
phases ($CPs$) were then obtained from the argument of the bispectrum.
Calibrated $V^{2}$ were obtained by dividing the AU Mic visibilities by those
of the calibrator, while calibrated $CPs$ were derived by subtracting those of
the calibrator. The transfer functions are displayed in Fig. 7 and 8. We have
a total of 294 $V^{2}$ and 490 $CPs$ calibrated measurements, which are
displayed in Fig. 9.

Our SAM observations with SPHERE allow us to probe the spatial scale $\sim
0.6-5.3$ au ($60-550$ mas) from the star, with the lower limit set by the
longest baselines and the upper limit set by the bandwidth smearing. To
improve the detection limits on a wider angular scale, we collected additional
high-angular-resolution data from the VLTI/GRAVITY (Eisenhauer et al. 2011),
VLTI/PIONIER (Le Bouquin et al. 2011) and VLT / NACO (Rousset et al. 2003;
Lenzen et al. 2003) instruments.

Figure 1: Interferogram of AU Mic trimmed within $2.45\arcsec\times
2.45\arcsec$ (left image) and its power spectrum (squared visibilities, right
image).

###  2.2 NACO/SAM observations

We retrieved the reduced data published by Gauchet et al. (2016). These are
also SAM mode observations and were made on 2011 August 04 with NACO in the
$L^{\prime}$-band filter ($\lambda=3.80\,\mu$m, $\Delta\lambda=0.62\,\mu$m)
with a seven-hole mask. The observation log from Gauchet et al. (2016) is
reported in Table 1. Data reduction (flat-field, bad-pixels, and sky
subtraction) were performed in the same way as for the SPHERE/SAM observations
(see more details in Gauchet et al. 2016). The direct fringe-fitting method
was applied instead of Fourier transform as we did previously ; however both
methods provide similar results. To calibrate the data, Gauchet et al. (2016)
used HD~197339 (K2III, $R=6.47$ mag, $K=4.47$ mag,
$\theta_{\mathrm{LD}}=0.646\pm 0.021$ mas). These data allow to probe the
spatial scale $\sim 0.7-7$ au ($70-730$ mas). Although these authors already
published detection limits with this data set, we included it so that we can
re-analyse the data with our CANDID tool, and perform a uniform analysis with
the other datasets. The observations contain 546 $V^{2}$ and 910 $CPs$
measurements, and are displayed in Fig. 9.

###  2.3 VLTI/PIONIER observations

PIONIER calibrated data were retrieved from the Optical interferometry
Database (OiDB333http://www.jmmc.fr/english/tools/data-bases/oidb/) developed
by the Jean-Marie Mariotti Center (JMMC444https://www.jmmc.fr). PIONIER (Le
Bouquin et al. 2011) is an interferometric instrument at the VLTI, combining
the light coming from the four auxiliary telescopes (AT) in the $H$ band,
either in a broad band mode or with a low spectral resolution, where the light
is dispersed across a few spectral channels.

The observations were performed on several dates from 2011 to 2014 for
different observing programs. Data before 2014 were collected when PIONIER was
a visitor instrument at the VLTI, and therefore the calibrated data are only
available in the OiDB (not in the ESO archive). We listed these observations
in Table 1. To monitor the instrumental and atmospheric contributions, the
standard observational procedure was used which consists of interleaving the
science target with reference stars. These stars are listed in Table 1 and
taken from SearchCal.

The data were reduced with the pndrs package described in Le Bouquin et al.
(2011, version v2.3 for 2011 data, v2.591 for 2013 and v2.71 for 2014). The
main procedure is to compute squared visibilities and triple products for each
baseline and spectral channel, and to correct for photon and readout noises.
The combination of four telescopes provides six visibility and four closure
phase measurements per spectral channel. Although the PIONIER observations
were performed mostly with small telescope configurations, the baselines range
from 10 m to 128 m, allowing us to probe the spatial scale $0.02-2$ au
($2-200$ mas ; where the interferometric FoV for our data set defined from the
optical path difference separation of the fringe packet of two components).
Data are displayed in Fig. 10.

###  2.4 VLTI/GRAVITY observations

AU Mic was observed with GRAVITY and the Auxiliary Telescopes (ATs) on 2016
June 16 during the ESO science verification run of the instrument. GRAVITY is
a near-IR interferometric beam combiner operating in the $K$-band
($\lambda=1.990-2.450\,\mu$m). The observations were carried out using the
highest spectral resolution ($R\sim 4000$) and the telescope baselines
A0-G1-J2-K0 (projected baselines range from 48 m to 110 m). Three datasets of
300 s each ($10\times 30$ s) interleaved by sky exposures were recorded. To
monitor the interferometric transfer function, the calibrator HD~155276
(K0.5IIICN0.5, $K=3.93$ mag, $\theta_{\mathrm{LD}}=0.799\pm 0.078$ mas) was
observed (taken from SearchCal.). However, HD 155276 was observed about 6 h
before AU Mic, and therefore the calibration of the squared visibilities is
likely unreliable. However, the closure phases can be used as they are
independent of telescope-specific phase shifts induced by the atmosphere or
optics (see e.g. Monnier 2007). Observables (visibilities and closure phases)
were extracted using the provided GRAVITY pipeline (v1.1.2, Lapeyrere et al.
2014). The projected baselines allow us to probe the inner region of
$0.03-2.4$ au ($3.5-250$ mas) from the central star (FoV limited by the fiber
FoV and not the bandwidth smearing thanks to the high spectral resolution).
Data are displayed in Fig. 10.

##  3 Companion search and detection limits

###  3.1 The CANDID algorithm

To search for a possible companion and set detection limits, we used the
interferometric tool CANDID555Available at https://github.com/amerand/CANDID
and https://github.com/agallenne/GUIcandid for a GUI version. (Gallenne et al.
2015). The main function allows a systematic search for companions (point-
sources in this case) performing an $N\times N$ grid of fits, the minimum
required grid resolution of which is estimated a posteriori in order to find
the global minimum in $\chi^{2}$. The tool delivers the binary parameters,
namely the flux ratio $f$, and the relative astrometric separation
$(\Delta\alpha,\Delta\delta)$. The uniform disc (UD) angular diameters of both
components can also be fitted, but in our case we do not expect a spatially
resolved companion. Furthermore, the angular diameter $\theta_{\mathrm{UD}}$
of AU Mic is only resolved by GRAVITY and PIONIER. The significance of the
detection is also given, taking into account the reduced $\chi^{2}$ and the
number of degrees of freedom.

The second main function of CANDID incorporates a robust method to set a
$n\sigma$ detection limit on the flux ratio for undetected components, which
is based on an analytical injection of a fake companion at each point in the
grid. We refer the reader to Gallenne et al. (2015) for more details about
CANDID and its formalism. Two-dimensional detection-limit maps are expressed
in contrast ratio in the corresponding bandpass and spatial range of the
instrument. To have a quantitative estimate of the sensitivity limit with
respect to the separation, we estimated a radial profile using the 90 %
completeness level (i.e. 90 % of all possible positions) from the cumulated
histogram in rings for all azimuths.

###  3.2 Angular diameter of AU Mic

To measure the diameter, we only used the #1 PIONIER observations because this
is the only dataset with long enough baselines to allow a reliable
measurement. In addition, PIONIER operates in the $H$ band and therefore
provides a better spatial resolution than GRAVITY. There is no bright
companion orbiting AU Mic reported, and therefore we do not expect the angular
diameter measurement to be biased by the presence of a low-contrast component.

We first estimated the uniform disc diameter $\theta_{\mathrm{UD}}$ by fitting
the calibrated squared visibilities with the following formula:

$|V(u,v)|^{2}=\left|\dfrac{2\leavevmode\nobreak\ J_{1}(x)}{x}\right|^{2}$ (1)

where $J_{1}(x)$ is the first-order Bessel function,
$x=\pi\theta_{\mathrm{UD}}B/\lambda$, $\lambda$ the wavelength,
$B=\sqrt{u^{2}+v^{2}}$ the baseline length and $(u,v)$ are the spatial
frequencies. We measured $\theta_{\mathrm{UD}}=0.797\pm
0.032_{\mathrm{stat}}\pm 0.038_{\mathrm{sys}}$ mas. The statistical error was
determined using the bootstrapping technique (with replacement) on all
baselines. The given diameter corresponds to the median of the distribution,
while the maximum value between the 16th and 84th percentile was chosen as the
statistical uncertainty. The systematic error is the quadratic sum of two
systematic uncertainties. The first comes from the calibrator uncertainties ;
we took the quadratic average of the calibrator uncertainties used. The second
is a 0.35 % systematic error due to the uncertainty of the wavelength
calibration of PIONIER (Gallenne et al. 2012). We used this value in the
following CANDID analysis. Our measurement is in good agreement with the
$0.78\pm 0.023$ mas reported by White et al. (2015, we assume they report the
uniform disc diameter as there is no reference to any limb-darkening
calculation).

We additionally tested a UD model including an IR emission from the hot dust
in the inner disc. Such emission is usually difficult to detect because the
flux ratio $f_{\mathrm{d}}=f_{\mathrm{disc}}/f_{\star}$ is of the order of 1 %
which needs high-precision visibilities and small baselines. We fitted a
simplified visibility model $(1-2f_{\mathrm{d}})*|V(u,v)|^{2}$ which assumes
the disc is fully resolved resulting in a visibility deficit compared to the
values expected from the star alone. We found $\theta_{\mathrm{UD}}=0.679\pm
0.12$ mas and $f_{\mathrm{d}}=1.09\pm 0.67$ %. The angular diameter is smaller
but within $1\sigma$ with the single star value. In addition, $f_{\mathrm{d}}$
is barely constrained with a $\chi^{2}_{r}$ difference of 0.2 only,
corresponding to a significance level of $1.8\sigma$. We can therefore assume
that the single star model provides an unbiased angular diameter.

We determine the limb-darkened angular diameters, $\theta_{\mathrm{LD}}$.
Assuming a circular symmetry, we fitted the calibrated squared visibilities
following the formalism of Mérand et al. (2015), which consists in extracting
the radial intensity profile $I(r)$ of the spherical SATLAS models (Neilson &
Lester 2013), which was converted to a visibility profile using a Hankel
transform:

$V_{\lambda}(x)=\frac{\int_{0}^{1}I_{\lambda}(r)J_{0}(rx)rdr}{\int_{0}^{1}I_{\lambda}(r)rdr},$
(2)

where $\lambda$ is the wavelength, $x=\pi\theta_{\mathrm{LD}}B/\lambda$, $B$
is the interferometric baseline projected onto the sky, $J_{0}$ the Bessel
function of the first kind, and $r=\sqrt{1-\mu^{2}}$, with $\mu=\cos(\theta)$,
$\theta$ being the angle between the line of sight and a surface element of
the star.

The SATLAS grid models span effective temperatures from 3000 to 8000 K in
steps of 100 K, effective gravities from -1 to 3 in steps of 0.25, and masses
from 0.5 to 20 $M_{\odot}$. We chose the model with the closest parameters,
that is with $T_{\mathrm{eff}}=3700$ K, $\log g=4.5$, and $M=0.5\,M_{\odot}$
(Plavchan et al. 2020, with $\log{g}\sim 4.4$ dex estimated from the mass,
temperature and luminosity). We estimated $\theta_{\mathrm{LD}}=0.825\pm
0.033_{\mathrm{stat}}\pm 0.038_{\mathrm{sys}}$ mas and errors were estimated
in the same way as above. Changing the models with $T_{\mathrm{eff}}\pm 200$
K, $\log g\pm 0.5$, and $M\pm 2.5\,M_{\odot}$ changes the diameters by at most
0.1 %, which we also quadratically added. Combining our estimate with the Gaia
parallax of $102.943\pm 0.023$ mas provides a linear radius of $R=0.862\pm
0.052\,R_{\odot}$. The best-fit solution for the angular diameter is shown in
Fig. 2.

We also checked the angular diameter using the squared visibilities from
GRAVITY. We found $\theta_{\mathrm{LD}}=1.078\pm 0.255_{\mathrm{stat}}\pm
0.078_{\mathrm{sys}}$ mas (systematic error from the uncertainty of the
calibrator). This value is higher than the one determined with PIONIER,
although within $1\sigma$. The squared visibilities are not very accurate or
precise, with a standard deviation of $\sim 10$ % (see Fig. 10). This is
likely linked to a poor calibration, which was somewhat expected as the
calibrator was observed 6 h before. However, as stated previously, the closure
phases are independent of atmosphere seeing, and therefore we use only this
observable for the following analyses.

Figure 2: Best-fit solution of the limb-darkened disc model (red dots) to the
PIONIER squared visibilities of AU Mic (blue dots).

###  3.3 Search for orbiting companion

Figure 3: $\chi^{2}_{r}$ map of the local minima for the SPHERE observations
(left panel), detection level map (middle panel), and the $(u,v)$ plane
coverage of the observations (right panel). The yellow lines represent the
convergence from the starting points to the final fitted position. The maps
were re-interpolated in a regular grid for clarity. The yellow star denotes AU
Mic, while the orange arrow in the middle indicates the disc orientation.

Every instrument has a different minimum and maximum spatial range $r$ within
which we can search for a companion. The loss of coherence caused by spectral
smearing mainly limits the maximum separation for a reliable detection
($\lesssim R\lambda/B$), while the smallest spatial scale is set by the
angular resolution ($\sim\lambda/2B$). The ranges are mentioned in Sect. 2 ;
we determined $r\sim 60-550$ mas for SPHERE, $r\sim 70-730$ mas for NACO,
$r\sim 2-200$ mas for PIONIER and $r\sim 3.5-250$ mas for GRAVITY. In the
following, we searched for companions using $CPs$ only. As explained by
Gallenne et al. (2015), the $CPs$ are more sensitive to faint off-axis
companions (although depending on its location and the $(u,v)$ coverage), and
are also less affected by instrumental and atmospheric perturbations. The
angular diameter is fixed in the fitting process to the uniform disc value
listed above.

In the following, the contrasts have been converted to masses using the AMES-
COND models (Baraffe et al. 2003).

#### SPHERE data:

We did not detect any companion at more than $1.9\sigma$. Figure 3 shows the
2D detection maps given by CANDID. The highest detection is at $\rho\sim 72$
mas and $PA\sim-17^{\circ}$ with a flux ratio $f_{\mathrm{K}}=0.041$ %, which
would correspond to a companion mass of $\sim 7.3\,M_{\mathrm{j}}$. However,
even if the position seems slightly above the disc plane ($PA=128.41^{\circ}$,
MacGregor et al. 2013), this detection is not significant. The $(u,v)$
coverage, that is, the sampling of the Fourier plane, is displayed in Fig. 3.

#### NACO data:

A clear location at $\rho\sim 139$ mas and $PA\sim-156^{\circ}$ is detected in
the $\chi^{2}$ map but the detection level is only $1.6\sigma$ (see maps in
Appendix C). The flux ratio is $f_{\mathrm{L}}=0.07$ % ($\sim
4.2\,M_{\mathrm{j}}$) but its location perpendicular to the disc plane makes
this detection unlikely, although similar in brightness with the previous
highest detection of SPHERE.

#### PIONIER data:

We analysed all dates independently. There is also no detection with a
statistical significance larger than $2.9\sigma$ (see maps in Appendix C). The
highest peak has a flux ratio of $f_{\mathrm{H}}=0.49$ % ($\sim
14.1\,M_{\mathrm{j}}$) and is located at $\rho\sim 44$ mas and
$PA\sim-109^{\circ}$. We can see additional peaks but these are spuriously
produced by an incomplete $(u,v)$ coverage, and their significance is less
than $2.9\sigma$.

#### GRAVITY data:

Only $CPs$ from the science combiner have been used as it offers a much
greater number of spectral channels than the fringe tracker (five channels
only). No additional sources are detected at more than $2.6\sigma$. The
highest peak is at $\rho\sim 14.8$ mas, $PA\sim-128^{\circ}$, and with a flux
ratio of $f_{\mathrm{K}}=0.42$ % ($\sim 13.5\,M_{\mathrm{j}}$). This companion
is unlikely as it is too bright, and the flux ratio is not consistent with the
SPHERE estimate.

We therefore did not significantly detect (i.e. $>5\sigma$) any component
orbiting AU Mic. The highest probable locations given by the PIONIER and
GRAVITY data are unlikely as they would correspond to a brown dwarf. Although
such objects would not have been detected from AO imaging because of the
limited angular resolution and size of the coronagraph, it would have been
detected from the extensive observing RV observations (from Kepler’s law and
assuming our measured projected separation as a lower limit for the angular
semi-major axis, the minimum RV amplitude would be $\sim
0.8\,\mathrm{km\leavevmode\nobreak\ s^{-1}}$).

###  3.4 $5\sigma$ detection limits

CANDID has also implemented a robust method to derive the dynamic range we can
reach with a given set of data. It consists in injecting a fake companion into
the data at each astrometric position with different flux ratios. As we inject
a companion, we therefore know that the binary model should be the true model.
We then compare the $\chi^{2}$ with that of a single star model (uniform disc
model) to obtain the probability of the binary model being the true model. We
set the significance level on the flux ratios at $5\sigma$, meaning that lower
flux ratios are not significantly detected. Doing this for all points in the
grid, we then have a $5\sigma$ detection limit map for the flux ratio. We
refer the reader to Gallenne et al. (2015) for more information about the
method. To obtain a quantitative estimate of the sensitivity limit with
respect to the separation, we estimated a radial profile using the 90 %
completeness level (i.e. 90 % of all possible positions) from the cumulated
histogram in rings for all azimuths. The radial profile for the detection
limit of our SPHERE observations is displayed in Fig. 4.

We list three different values in Table 2: the average limit for the
separation ranges $r<r_{\mathrm{mid}}$, $r_{\mathrm{mid}}<r<r_{\mathrm{max}}$,
and the full range $r<r_{\mathrm{max}}$. All of the final $5\sigma$ contrast
limits, $\Delta m_{5\sigma}$ expressed in magnitude, are conservative as they
correspond to the mean plus the standard deviation for the given radius range.
Upper limit masses, that is, the lowest masses we can safely exclude, were
estimated from the contrast limits calculated for the full separation range.

Figure 4: Contrast limit at $5\sigma$ for our SPHERE observations. Table 2: Average $5\sigma$ contrast limits ($\Delta m_{\lambda}$). | Date | $\Delta m$ | upper limit mass
---|---|---|---
|  | (mag) | $\mathrm{M_{jup}}$
|  |  $0.4<r<1$ au |  $1<r<2.4$ au |  $0.4<r<2.4$ au |
SPHERE ($K_{\mathrm{s}}$) | 2018-08 | 6.67 | 6.72 | 6.68 | 12
|  |  $0.7<r<3.9$ au |  $3.9<r<7$ au |  $0.7<r<7$ au |
NACO ($L^{\prime}$) | 2011-08 | 6.51 | 6.32 | 6.33 | 9
|  |  $0.02<r<0.25$ au |  $0.25<r<0.5$ au |  $0.02<r<0.5$ au |
PIONIER ($H$) | 2011-11 | 4.87 | 5.10 | 5.02 | 17
|  |  $0.03<r<1.4$ au |  $1.2<r<2.4$ au |  $0.03<r<2.4$ au |
GRAVITY ($K$) | 2016-06 | 5.25 | 5.52 | 5.39 | 16

##  4 Discussion and Conclusion

With our new SPHERE observations, we do not detect additional objects brighter
than $K_{s}\sim 11.2$ mag within $0.4-2.4$ au, which would correspond to a
mass limit for a planet of $12.3\pm 0.5\,\mathrm{M_{jup}}$. The mass limit was
estimated using a cubic spline interpolation of the AMES-COND isochrones
encompassing 18.5 Ma (i.e. 9, 10, 20, and 30 Ma). We then interpolated the
mass for the corresponding magnitude limit and age of $18.5\pm 2$ Ma. The best
mass sensitivity is achieved with NACO because the $L^{\prime}$ band is
favourable at young ages, providing an upper limit on the mass of $\sim 8.3\pm
0.8\,\mathrm{M_{jup}}$ within $0.7-7.3$ au. This is similar to the upper mass
limit of $10.1\,\mathrm{M_{jup}}$ determined by Gauchet et al. (2016) with the
same data set, but updated with our more robust CANDID tool. The SAM
observations enable us to reach higher contrast thanks to a better AO
correction and a more stable point spread function. In addition, the light
beam in SPHERE is prone to a higher throughput than the whole VLTI optical
path. Long-baseline interferometric observations enable us to probe smaller
spatial scales but the contrast limit reached is not as good. Within
$0.02-0.5$ au, PIONIER provides a detectable mass limit of $\sim 17.1\pm
1.0\,\mathrm{M_{jup}}$.

We added additional constraints on the mass limits by combining our calculated
$5\sigma$ detection maps with the existing RV data from Lannier et al. (2017)
and SPHERE contrast maps estimated from the data of Boccaletti et al. (2018,
determined using a PCA analysis of the IFS and IRDIS images taken in the
$J,H,H23$, and $K12$ bands). We used the Multi-epoch multi-purpose Exoplanet
Simulation System algorithm (MESS2, Lannier et al. 2017) which generates
populations of synthetic planets through a Monte Carlo simulation for a given
range of mass and separation. MESS2 grids are then compared to our combined
data sets to explore the detection probability. Our contrast limit maps were
converted into masses using an age of 18.5 Ma (Miret-Roig et al. 2020) and the
COND-2003 evolutionary models (Baraffe et al. 2003). We considered two
inclination distributions for the synthetic planets: a uniform distribution
between 0∘ and 90∘, and an inclination of 90$\pm$1∘, that is, assuming the
synthetic planets orbit in the same plane as the resolved disc. We also took
an ad hoc eccentricity distribution with $e<0.6$. The probability maps are
displayed in Fig. 5. As expected, assuming a uniform prior on the inclination
significantly reduces the sensitivity to companions. This is particularly true
within 1 au where the sensitivity is driven by the RV technique, which is more
sensitive to edge-on systems. In the case of co-planar orbits, 99 % of
additional companions with $\gtrsim 4$ Mjup can be excluded within $0.2-1$ au,
and $\gtrsim 2$ Mjup within $1-8$ au. These detection limits are not better
than the ones derived by Lannier et al. (2017) combining RVs and direct
imaging at separation $r>0.2$ au. This is because direct imaging has a better
sensitivity than interferometry at these angular separations. However, for the
innermost region, at $\lesssim 0.2$ au, which is the spatial scale probed by
interferometry, we find that 99 % of the companions down to $\sim 0.5$ Mjup
can be detected. This is actually similar to using RVs alone because of the
lack of high-dynamic range of the current interferometric instruments and our
limited datasets. In this specific case (i.e. for co-planar orbits of the
edge-on disc), interferometry does not bring additional constraints when
precise RVs are available. In a more general case where no precise RVs are
available, interferometry is complementary to direct imaging by providing
better contrast in the innermost region, as we can see in Fig. 6. We also
notice that the coplanarity hypothesis mostly affects the RVs, and has a
limited effect when only interferometry and direct imaging are used.

Figure 5: Detection probability maps of AU Mic assuming coplanarity with the
disc (left panel) or with no hypothesis on the inclination (right panel),
combining RV, interferometric, and direct-imaging observations. Blue colour-
coded areas denote the detectability probability. The red area shows where the
newly detected companions should be located (Martioli et al. 2021).

Figure 6: Same as Fig. 5 but only interferometric and direct-imaging
observations.

The newly detected planets have a mass of $<0.08\,\mathrm{M_{jup}}$ for a
maximum separation of 11.3 mas (0.11 au), well below our contrast limits (red
area in Fig. 5). This would correspond to a lower limit $K$-band magnitude of
$\sim 33$ mag, corresponding to a contrast of $\sim 4\times 10^{-12}$ ($\Delta
K\sim 28.5$ mag, extrapolated from the AMES-COND models). Such dynamic range
is beyond the capabilities of the current imaging and interferometric
instruments, and also beyond future facilities in the near future. Only space-
based coronagraph like Habex or Louvoir (see e.g. Mennesson et al. 2016;
Bolcar et al. 2017) could reach such contrasts but they would not have the
required angular resolution. Direct imaging of these planets with AO is not
possible due to its proximity with the host star. Only long-baseline
interferometry can probe the innermost regions, but the current reachable
contrast is about $\Delta H\sim 6.5$ mag (Gallenne et al. 2015; Roettenbacher
et al. 2015). The dual-field mode of GRAVITY reached a contrast of $\Delta
K=10.7$ mag by directly measuring a spectrum of an exoplanet (Gravity
Collaboration et al. 2019), but this is a special case as the location of the
companion must be known a priory to be able to position the fibre on it. A
’blind search’ was tested on AU Mic with the Unit Telescopes by dithering the
fibre along the southeastern part of the debris disc (GRAVITY Collaboration et
al. 2019). No planet was detected but they achieved a $5\sigma$ dynamic range
of 11 mag at 1.2 au and 13.5 mag at 2.4 au. The astrometric mode of GRAVITY,
which can provide a differential astrometric precision of a few tens of
$\,\mu$as, could also be an alternative used to obtain the astrometric orbit
of the more massive planet ; however the astrometric wobble produced by this
planet on the central star is too small ($\sim 1.7\,\mu$as, taking the upper
limit of $0.08\,\mathrm{M_{jup}}$ at 0.11 au).

The upcoming ELT with a SAM mode will likely also not detect these planets.
Assuming a maximum baseline of 38 m (for the possibly two most distant mask
holes) and observations performed in the $K$ band, planets detectable down to
$\sim 7$ mas would need an optimal AO correction and strongly improved
sensitivity limits (although near the SAM resolution limit). Unfortunately,
even the next generation of extreme-AO instruments for the ELT will not
provide a contrast of better than $10^{-7}$ in such close proximity to the
star.

###### Acknowledgements.

A.G. acknowledges the support of the French Agence Nationale de la Recherche
(ANR), under grant ANR-15-CE31-0012-01 (project UnlockCepheids). A.G.
acknowledges financial support from ”Programme National de Physique Stellaire”
(PNPS) of CNRS/INSU, France. A.G. also acknowledges financial support from the
ANID-ALMA fund No. ASTRO20-0059. The research leading to these results has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 695099 (project CepBin), grant agreement No 951549 (project UniverScale)
and from the National Science Center, Poland grants MAESTRO
UMO-2017/26/A/ST9/00446 and BEETHOVEN UMO-2018/31/G/ST9/03050. A.G.
acknowledges support from the IdP II 2015 0002 64 and DIR/WK/2018/09 grants of
the Polish Ministry of Science and Higher Education, from the BASAL Centro de
Astrofísica y Tecnologías Afines (CATA) AFB-170002, and from the Millennium
Institute of Astrophysics (MAS) of the Iniciativa Científica Milenio del
Ministerio de Economía, Fomento y Turismo de Chile, project IC120009. S.K.
acknowledges support from an ERC Consolidator Grant (Grant Agreement ID
101003096). The GRAVITY data have been recorded as part of the Science
Verification program and we thank the SV team, which is composed of ESO
employees and GRAVITY consortium members, for carrying out the observations
(https://www.eso.org/sci/activities/vltsv/gravitysv.html). This work has made
use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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##  Appendix A Transfer functions of our SPHERE/SAM observations

Figure 7: Transfer function calibration for the squared visibilities.
Calibrator and science target observations are represented with black and blue
dots, respectively. The solid line denotes the interpolation of the transfer
function, while the grey area represents the standard deviation with the
calibrators.

Figure 8: Same as Fig. 7 but for the closure phases.

##  Appendix B Squared visibilities and closure phases for all observations

Figure 9: Squared visibilities and closure phases for the SPHERE (left) and
NACO (right) observations (blue). The red dots correspond to the best model
found with CANDID.

Figure 10: Squared visibilities and closure phases for the PIONIER and GRAVITY
observations (blue). The red dots correspond to the best model found with
CANDID.

##  Appendix C Detection maps for the NACO, PIONIER and GRAVITY observations

Figure 11: $\chi^{2}_{r}$ map of the local minima for the NACO observations
(left panel), detection level map (middle panel), and the $(u,v)$ plane
coverage of the observations (right panel). The yellow lines represent the
convergence from the starting points to the final fitted position. The maps
were re-interpolated in a regular grid for clarity. The yellow star denotes AU
Mic, while the orange arrow in the middle indicates the disc orientation.

Figure 12: Same as Fig. 11 but for the PIONIER observations of 2011 and 2013.

Figure 13: Same as Fig. 11 but for the 2014 PIONIER (top) and GRAVITY (bottom)
observations.