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# Subgraph2vec: A random walk-based algorithm for embedding knowledge graphs

Elika Bozorgi1,∗ 1School of Computing, The University of Georgia, Athens, GA,
USA
{elika.bozorgi, saber.s, sakher.a, hra<EMAIL_ADDRESS>Saber Soleimani1
1School of Computing, The University of Georgia, Athens, GA, USA
{elika.bozorgi, saber.s, sakher.a, hra<EMAIL_ADDRESS>Sakher Khalil
Alqaiidi1 1School of Computing, The University of Georgia, Athens, GA, USA
{elika.bozorgi, saber.s, sakher.a, hra<EMAIL_ADDRESS>Hamid Reza Arabnia1
1School of Computing, The University of Georgia, Athens, GA, USA
{elika.bozorgi, saber.s, sakher.a, hra<EMAIL_ADDRESS>Krzysztof Kochut1
1School of Computing, The University of Georgia, Athens, GA, USA
{elika.bozorgi, saber.s, sakher.a, hra<EMAIL_ADDRESS>
###### Abstract

Graph is an important data representation which occurs naturally in the real
world applications [9]. Therefore, analyzing graphs provides users with better
insights in different areas such as anomaly detection [18], decision making
[8], clustering [25], classification [27] and etc. However, most of these
methods require high levels of computational time and space. We can use other
ways like embedding to reduce these costs. Knowledge graph (KG) embedding is a
technique that aims to achieve the vector representation of a KG. It
represents entities and relations of a KG in a low-dimensional space while
maintaining the semantic meanings of them. There are different methods for
embedding graphs including random walk-based methods such as node2vec,
metapath2vec and regpattern2vec. However, most of these methods bias the walks
based on a rigid pattern usually hard-coded in the algorithm. In this work, we
introduce subgraph2vec for embedding KGs where walks are run inside a user-
defined subgraph. We use this embedding for link prediction and prove our
method has better performance in most cases in comparison with the previous
ones.

Keywords: Representation Learning, Information Engineering, Link Prediction,
Deep Learning, Graph Embedding

##  I Introduction

Knowledge graphs play a crucial role in organizing, understanding, and
leveraging information in various domains. Hence, they become increasingly
popular in different areas due to their valuable features. For example, they
are widely used in recommendation systems in E-commerce to model the
relationships between users and items. Next, they enable personalized
recommendations by leveraging knowledge about user interactions and item
characteristics [13]. As another example, they are widely used in healthcare
to support practitioners for disease diagnosis by knowledge discovery from
patients’ personal health repositories [24]. In addition, they provide a
structured way of connecting entities with relationships, which creates a
network to organize and represent information. Thus, it is easier to discover
relevant information either manually, by navigating through nodes to discover
visions, or by using machine learning and AI to make predictions and generate
insights into data. In addition, Knowledge graphs facilitate querying and
discovery of complex patterns within data [26] and also capture semantic
connections which leads to more accurate queries.

Although knowledge graphs offer many benefits, they have several issues. For
instance, Knowledge graphs are often incomplete [3], as they are frequently
populated using various external resources which are often incomplete as well.
Also, these outside resources have different structure and format which makes
integrating data more difficult. In addition, knowledge graphs usually grow in
size and complexity which makes them inefficient and therefore special
infrastructure and algorithms are needed to make them more scalable [20]. This
complexity also makes interpreting patterns and drawing insights more
challenging and time-consuming. Moreover, building and maintaining knowledge
graphs often need domain expertise to ensure the accuracy of the represented
data.

A reasonable number of such issues can be addressed using Artificial
Intelligence (AI). In fact, different methods of AI can be used for resolving
the previous mentioned problems with knowledge graphs as well as: knowledge
graph completion [4], node representation learning [31], semantic search [21],
question answering [22], anomaly detection [17], quality assessment [12], etc.
However, applying AI algorithms on different types of Knowledge graphs
requires a significant number of adjustments due to the high number of the
dimensions of the input data. In fact, in many cases, these methods cannot be
applied on Knowledge graphs directly. Therefore, it is desired to reduce the
number of the dimensions of the input data or knowledge graph by embedding it
with different AI methods. Knowledge graph embedding is the act of translating
the high-dimensional data to a low-dimensional space, while trying to maintain
the semantic meanings of the KG elements. The embedding of each element in the
dataset is a unique vector-representation of that element. The resulted
embedding of the Knowledge graph can be used for different purposes, such as
link prediction, entity classification, semantic search, and others. There
exist different types of embedding methods for Knowledge graphs based on
supervision, which include supervised methods such as Graph Convolutional
Networks (GCNs) [30], Unsupervised methods like TransE [2] and DeepWalk and
Hybrid ones such as GraphSAGE [11]. In this paper, we introduce an
unsupervised algorithm based on random walks for embedding Knowledge graphs.
There are previous random walk-based methods for embedding knowledge graphs,
such as node2vec, metapath2vec and regpattern2vec. However, these methods come
with challenges. For example, in node2vec the random walks are biased to
highly visible types of nodes, the ones with a dominant number of paths. On
the other hand, in metapath2vec and regpattern2vec the walks are biased by a
series of relationships (or node types) or a fixed regular pattern of
relationships (or node types), respectively. In our method, the user enters an
arbitrary pattern which defines a schema subgraph in the actual knowledge
graph and the walk is done within this subgraph. In the next section, we will
compare our method with the previous related ones.

##  II Preliminaries

In this section, we will explain some primitive concepts which are fundamental
to the understanding of our method and the previous ones, starting with the
explanation of the Knowledge graphs (or heterogeneous networks).

Knowledge graph: A knowledge graph is a data set that represents real-world
facts and semantic relationships in the form of triplets, where the triplets
are represented as a graph with edges as relations and nodes as entities [1].
Mathematically, consider $G=(V,E)$ where $G$ represents the knowledge graph
and $V$ are the nodes or entities and $E$ represent the relations. Walk: A
walk is a finite sequence of edges which join a sequence of vertices. In
$G=(V,E)$ where $G$ is a knowledge graph and $V$ represents the nodes and $E$
represents the edges, a finite walk is a sequence of edges
$(e_{1},e_{2},...,e_{n-1})$ for which there is a sequence of vertices
$(v_{1},v_{2},...,v_{n})$ such that $\phi(e_{i})=(v_{i},v_{i+1})$ for
$i=1,2,...,n-1$. $(v_{1},v_{2},...,v_{n})$ is the vertex sequence of the walk.
The walk is closed if $v_{1}=v_{n}$, and it is open otherwise [28]. Path: A
path is a walk on a graph where the vertices are repeated. Subgraph: Graph
${S=(V_{S},E_{S})}$ is considered as a subgraph of ${G=(V_{G},E_{G})}$ if and
only if its vertex set $(V_{S})$ and edge set $(E_{S})$ are subsets of those
of G. In other words: $V_{S}\in V_{G}$ and $E_{S}\in E_{G}$. Schema: Given a
knowledge graph $G$, an edge with a relation type $R$ connects source nodes of
type $S$ and target nodes of type $T$ defines a meta edge
$S\xrightarrow{R}\hbox{$T$}$ . A schema graph (aka known as meta-template) for
$G$ is a set of all such meta edges. In fact, a schema graph is a directed
graph defined over node types $T$, with edges from $R$, denoted as
$G_{S}=(T,R)$ [23]. Heterogeneous Network [29]: or HIN, is a graph denoted as
$G=(V,E,T)$, where each $v\in V$ and $e\in E$ has a mapping function
$\Phi(V)=V\rightarrow T_{v}$ and $\phi(E)=E\rightarrow T_{e}$ and $T_{v}$ and
$T_{e}$ denote sets of node and relation types where $|T_{v}|+|T_{e}|>2$. In
simple words, in these networks, nodes and edges can belong to different
types, and the connections between nodes can have various semantic meanings.

##  III Related works

DeepWalk [19], is a graph embedding method which aims to learn continuous
representations of nodes in a graph. For each walk, it begins by generating
the walks from a random starting node and moves to the next random node on the
graph. Each random walk sequence is treated as a sentence, in which the nodes
are considered as the words of the sentence. Next, the word2vec [6] model from
NLP is applied to learn embeddings for the nodes. The resulting sentences are
embedded by giving them as an input to a Skip-gram model, a variant of the one
used in the word2Vec model. After the skip-gram is trained, it predicts the
context nodes based on its input and the output is the embedding of the nodes.
Node2vec [10], is a technique for learning node embeddings in a graph. In
fact, it is an extension of the word2vec model which is a method for word
embedding in textual data. It learns embeddings of the nodes in a graph by
using a neighborhood sampling strategy which captures both local and global
structures. Node2vec generates random walks which are balanced between
breadth-First Search (BFS) and depth-First search (DFS) and applies it on the
input data. After generating random walks, node2vec learns embeddings using
the Skip-gram model. The Skip-gram model predicts the context (neighboring
nodes) of a target node based on its embedding. Node2vec optimizes the
obtained embeddings by maximizing the likelihood of observing neighboring
nodes within the context window of each target node. Metapath2vec [7], is a
representation learning method designed specifically for HINs to learn
embeddings of the nodes and captures both semantic and structure of the
network. In this method, meta-path guided random walks are used to make a
sentence of the nodes. The meta-paths are created by domain experts based on
nodes according to the dataset. For example, on a DBLP computer science
bibliographic dataset [15], the created meta-paths are : APA, APVPA, OAPVPAO,
where A represents the author, P the paper, O the organization and V the
venue. Consider APA as an example. The first node to choose must be of type A,
the second node must be of type P and the third node must be of type A. This
means that at each step the next node to visit is chosen according to a pre-
defined meta-path (APA in this example), ensuring that the walk follows a
meaningful path in the network. After generating a large number of meta-path
based random walks, these sequences of nodes are used as training data for the
Skip-gram model. Finally, these embeddings are aggregated to obtain
comprehensive representations for each node in the network beside capture the
diverse relationships and semantic meanings associated with each node.
Regpattern2vec [14], is a method for embedding KGs which samples a large
knowledge graph to learn node embedding while capturing the semantic
relationships among the nodes of the graph. In this method, the walk is biased
by a fixed pattern H[^T] + HT which is based on edges. The walk starts at a
random node from a set of given nodes known as source nodes $(S)$ on the
knowledge graph. After choosing the source node, the walk chooses an edge of
type $H$ and moves to a randomly chosen neighbor creating a path at each time.
Next, the walk chooses a random edge of type ^T and moves to the next random
node. The walk follows the pattern H[^T] + HT and continues the walk according
to the a parameter called walk length. We can control the number of the walks
in each path by walk length and when the number of the walks reaches to walk
length, the walk is complete. Once the first walk is complete, we can start
another walk by choosing a random node from $S$. The number of the times we
start a new walk is based on a parameter called number of walk. Choosing a
node from the set of source nodes each time as the starting node and creating
random walks will result in several paths of random walks. To obtain the
embedding of these nodes, the resulted paths are fed into a modified version
of a skip-gram model. This method of embedding is an unsupervised method and
it can be applied on any given raw text corpus or document. In regpattern2vec,
the algorithm runs the walks on a rigid regular expression H[^T] + HT which is
defined by domain experts and is hard-coded in the algorithm and cannot be
changed.

Contribution: In the previously mentioned methods, the algorithms are either
based on a pre-defined sequence of node/edge types or a pattern designed by
domain experts or is biased toward specific nodes by experts. It means that
they are not generic and the user does not have any role in guiding the walks.
In our algorithm, however, we define a method in which the algorithm runs on
any arbitrary random walk path inside a user-defined schema subgraph based on
edges. The user enters a schema subgraph by entering integers where each
integer represents an edge in the knowledge graph. This schema subgraph
defines the subgraph. After the subgraph is defined, we choose the first
random node as the source node inside this subgraph and continue the walk
based on the walk length parameter and move to any random nodes via any random
edges. The walks are valid only if they are within the subgraph and invalid
otherwise. The advantage of using a subgraph is that it is more permissive;
since we can run the walks totally randomly inside the user-defined subgraph
rather than having biased walks based on a rigid pattern like the previous
mentioned methods. The walks are valid as long as they are within the subgraph
and invalid otherwise.

##  IV Methodology

Our algorithm runs with any user-given schema subgraph ($s$) based on the
edges. The schema subgraph is entered in the form of integers where each
integer denotes an edge of the KG. This schema subgraph which defines the
subgraph ($S$) is actually a part of the original graph ($G$). Let’s assume
the user enters a schema subgraph such as $s$ = ’$x_{1},x_{2},x_{3}$’ where
$x_{i=1,2,3}$ is an integer representing an edge based on the dataset.

This schema subgraph defines subgraph $S=(V^{\prime},E^{\prime})\in G=(V,E)$
where $G$ is the primary knowledge graph. The algorithm chooses the first node
inside $S$ randomly and from there, uses the below equation to calculate the
probability of each neighbor edge based on its type ($t$). To calculate the
probability of moving to the next edge based on subgraph $S$, we use this
equation:

$\sum^{l}_{i=1+1}P(r^{i+1}|r^{i},S)=$

$\displaystyle\begin{cases}\frac{1}{r_{t_{i}}}\times\frac{1}{\sum^{n}_{i=1}t_{i}}\;\;\;\;\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(r\in
S\right)\\\ \\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \left(r^{i},\ r^{i},\ r^{i+1}\right)\notin\
G^{\prime}\end{cases}$

where $t_{i}$ denotes each type of edges connected to the current node and
$r_{t_{i}}$ denotes the number of the edges of each type. We choose the type
of the next edge based on its probability. An important thing to consider is
that we have a hierarchy of edges in our knowledge graph; which means that a
type of edge might have different sub-types. With that being said, all the
sub-types of a specific type should be considered as the same type.

In addition, we have a parameter called number of walks in our code, which
defines the number of the walks that should be walked in each path. The
default number for it is set as 40, which the user can adjust on their own
interest in the code.

(a) ROC on YAGO Dataset, predicted link:is citizen of

(b) ROC on YAGO Dataset, predicted link:Located In

(c) ROC on YAGO Dataset, predicted link:Leader of

(d) ROC on NELL Dataset, predicted link:competes with

(e) ROC on NELL Dataset, predicted link:plays against

Figure 1: Comparing ROC of different links of Subgraph2vec, Regpattern2vec and
Metapath2vec on NELL and YAGO.

###  IV-A Walk-based Embedding

After the user has entered the schema subgraph, the subgraph ($S$) is defined
and we are able to conduct the walk. We start the walk by choosing the first
node randomly within the sub-graph. Then we will choose an edge connected to
this node randomly and if it is inside the schema-graph, the chosen edge is
valid which means we choose the next random edge. Otherwise, we will delete
that edge type from the neighbor of the node and choose another random edge
and check its validation. We have decided to conduct the walk based on 40
steps (walk_length $=$ 40). We should repeat this scenario for 40 walks
(number_of_walks $=$ 40) and these walks are written to a file. Both walk-
length and number_of_walks can be modified by the user. After traversing all
the walks and writing them on the file, we obtain the walk file which contains
all the walks and their steps and then we should embed these walks. Embedding,
in simple words is translating high-dimensional vectors to a low-dimensional
space. To embed the walks of the walk file, we can consider each walk as a
sentence of the words in which each node is considered as one word. Therefore,
we can get help from the word2vec model in which a neural network learns node
embedding from a corpus.

In this paper, we are using a modified version of the skip gram model which
captures the similarity of the walks based on their types. Skip gram is from
the word2vec family and all of the word2vec models are consisting of two-layer
neural networks used for word embedding. In a general sense, in the skip gram
architecture, the model uses the current word (input) to predict the
surrounding window -usually of size 5 to 10- of the context words (output). In
fact, a skip gram is trying to find a semantic similarity between the words in
a context by learning a meaningful representation of each word (embedding) in
the document. After feeding the walk file as an input to the skip gram model
and getting the embedding file, we are able to use it. The embedding file can
be used for various tasks such as link prediction, node classification,
community detection and etc. In this work, we use the embedding file for link
prediction.

###  IV-B Application

In this paper, we decided to move forward with the link prediction task. For
conducting link prediction, we need to train our model first. We do it by
using the vector representation of the current edges in the graph which is
considered as the positive example. Therefore, for negative examples, we can
consider combining pairs of edges in the graph that are not connected. Both of
the positive and negative examples are used to train the classification model.
We use Logistic Regression as the classifier, which can be used for link
prediction as well.

##  V Experiments

In this section, we will evaluate the conductance of the subgraph2vec method
by running it on different datasets.

###  V-A Dataset

We use two different datasets to evaluate our model.

(i) The first dataset is YAGO39K [16], which includes data from Wikipedia,
WordNet and GeoNames and is a subset of the YAGO knowledge base [5]. It
contains 123,182 unique entities and 1,084,040 unique edges with 37 different
relation types. (ii) The second dataset used was NELL, which is built from the
Web via an intelligent agent and contains 49,869 unique nodes 296,013 edges
and 827 relation types.

TABLE I: Statistics of split data (MST method) based on the relation to be predicted [14] Dataset | Relations | Train est | Test set
---|---|---|---
NELL | CompetesWith | $9,154$ | $1,070$
PlaysAgainst | $2,945$ | $2,225$
YAGO | isLocatedIn | $44,542$ | $44,541$
isCitizenOf | $3,128$ | $342$
| isLeaderOf | $855$ | $106$

###  V-B Experimental Setup

To apply random walks in Subgraph2vec, we set the number_of_walks = 40 and
maximum walk_length = 40. Also, the logistic regression parameters are
constant for all the datasets.

For both of our datasets, we split the dataset in order to train and test for
any relation we want to predict in either of our datasets. Note that in our
model, it is necessary to have the pair of the nodes we want to do prediction
on in both the training and the test dataset. However, the relation between
that pair is different in each of the train and test dataset. To achieve this
goal, we apply minimum spanning tree method to take the minimum possible nodes
from the graph to prepare the test dataset.

We have split the dataset for each of the relations we want to predict
individually. Table 1 illustrates the number of the rows for each dataset
(split with MST) based on the relation to be predicted.

###  V-C Link Prediction

To explain the link prediction, we will give a brief explanation for each
dataset. For the YAGO dataset, we decided to predict these relations:
’isLocatedIn’, ’isCitizenOf’, ’isLeaderOf’. Here is $S$ defined for these
relations:

For ’isLeaderOf’, $S$ consists of these edges: ’PlayisIn’, ’isLeaderOf’ and
’isLocatedIn’. For ’isCitizenOf’, $S$ consists of these edges: ’isCitizenOf’,
’isLocatedIn’ and ’isLocatedIn’. For ’isLeaderOf’, $S$ consists of these
edges: ’isLeaderOf’, ’isLocatedIn’ and ’wasBornIn’.

For the NELL dataset, we chose two relations of interest: ’competesWith’ and
’playsAgainst’. For ’competeswith’, $S$ consists of :’Competeswith’,
’hasofficeincity’ and ’cityhascompanyoffice’. For ’playsAgainst’, $S$ consists
of :’teamplaysagainstteam’ or in short ’playsagainst’ which is the link to be
predicted, ’teamplaysinleague’ and ’sportsgameteam’.

Figure 1 shows the prediction results on the test datasets. Here, we are
comparing the results of our method to the results we obtained from running
regpattern2vec and metapath2vec algorithms using Nell and YAGO data sets. Our
results imply that our method outperforms the regpattern2vec and metapath2vec
methods in most cases. That is due to being capable of choosing the
nodes/edges randomly within the subgraph rather than choosing them based on a
regular expression. Figure 1 illustrates the ROC curve from each of the
algorithms.

##  VI Conclusion and future work

In this paper, we present Subgraph2vec, a random walk-based method in which a
subgraph is used for limiting random walks on a knowledge graph in a generic
fashion. There are different random walk-based methods for embedding knowledge
graphs such as node2vec, metapath2vec and regpattern2vec which were discussed
earlier. In the previous methods, random walks are biased based on different
algorithms such as BFS and DFS in node2vec or fixed patterns in metapath2vec
and regpattern2vec. However, our goal is to implement an algorithm which runs
the random walks inside on a user-given subgraph; which makes the embedding
algorithm more broad. The generated walk file is embedded using skipgram and
the resulted embedding files can be used for various purposes such as node
classification, link prediction, community detection and etc. In this work, we
decided to use it for link prediction. Our results on NELL and YAGO datasets
prove our method outperforms other methods such as regpattern2vec and
metapath2vec in most cases. In the future work, we will use the obtained
embedding file for the mentioned tasks and also in another work we will add
weights to the edges to make our model more customizable.

## References

  * [1] Antoine Bordes, Jason Weston, Ronan Collobert and Yoshua Bengio  “Learning structured embeddings of knowledge bases”  In _Proceedings of the AAAI conference on artificial intelligence_ 25.1, 2011, pp. 301–306
  * [2] Antoine Bordes et al.  “Translating embeddings for modeling multi-relational data”  In _Advances in neural information processing systems_ 26, 2013
  * [3] Xuelu Chen, Ziniu Hu and Yizhou Sun  “Fuzzy logic based logical query answering on knowledge graphs”  In _Proceedings of the AAAI Conference on Artificial Intelligence_ 36.4, 2022, pp. 3939–3948
  * [4] Zhe Chen et al.  “Knowledge graph completion: A review”  In _Ieee Access_ 8 IEEE, 2020, pp. 192435–192456
  * [5] PA Chirita, S Costache, W Nejdl and S Handschuh  “WWW’07: Proceedings of the 16th International Conference on World Wide Web”  ACM New York, NY, USA:, 2007
  * [6] Kenneth Ward Church  “Word2Vec”  In _Natural Language Engineering_ 23.1 Cambridge University Press, 2017, pp. 155–162
  * [7] Yuxiao Dong, Nitesh V Chawla and Ananthram Swami  “metapath2vec: Scalable representation learning for heterogeneous networks”  In _Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining_ , 2017, pp. 135–144
  * [8] Fan Fan et al.  “A Graph Neural Network Model with a Transparent Decision-Making Process Defines the Applicability Domain for Environmental Estrogen Screening”  In _Environmental Science & Technology_ 57.46 ACS Publications, 2023, pp. 18236–18245
  * [9] Palash Goyal and Emilio Ferrara  “Graph embedding techniques, applications, and performance: A survey”  In _Knowledge-Based Systems_ 151 Elsevier, 2018, pp. 78–94
  * [10] Aditya Grover and Jure Leskovec  “node2vec: Scalable feature learning for networks”  In _Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining_ , 2016, pp. 855–864
  * [11] Will Hamilton, Zhitao Ying and Jure Leskovec  “Inductive representation learning on large graphs”  In _Advances in neural information processing systems_ 30, 2017
  * [12] Elwin Huaman  “Steps to Knowledge Graphs Quality Assessment”  In _arXiv preprint arXiv:2208.07779_ , 2022
  * [13] Zan Huang, Wingyan Chung and Hsinchun Chen  “A graph model for E-commerce recommender systems”  In _Journal of the American Society for information science and technology_ 55.3 Wiley Online Library, 2004, pp. 259–274
  * [14] Abbas Keshavarzi, Natarajan Kannan and Krys Kochut  “RegPattern2Vec: link prediction in knowledge graphs”  In _2021 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS)_ , 2021, pp. 1–7  IEEE
  * [15] Michael Ley and Patrick Reuther  “Maintaining an Online Bibliographical Database: The Problem of Data Quality.”  In _EGC_ , 2006, pp. 5–10  Citeseer
  * [16] Xin Lv, Lei Hou, Juanzi Li and Zhiyuan Liu  “Differentiating concepts and instances for knowledge graph embedding”  In _arXiv preprint arXiv:1811.04588_ , 2018
  * [17] Rongrong Ma, Guansong Pang, Ling Chen and Anton Hengel  “Deep graph-level anomaly detection by glocal knowledge distillation”  In _Proceedings of the fifteenth ACM international conference on web search and data mining_ , 2022, pp. 704–714
  * [18] Xiaoxiao Ma et al.  “A comprehensive survey on graph anomaly detection with deep learning”  In _IEEE Transactions on Knowledge and Data Engineering_ 35.12 IEEE, 2021, pp. 12012–12038
  * [19] Bryan Perozzi, Rami Al-Rfou and Steven Skiena  “Deepwalk: Online learning of social representations”  In _Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining_ , 2014, pp. 701–710
  * [20] Axel Polleres et al.  “How does knowledge evolve in open knowledge graphs?”  In _Transactions on Graph Data and Knowledge_ 1.1, 2023, pp. 11–1
  * [21] Ridho Reinanda, Edgar Meij and Maarten Rijke  “Knowledge graphs: An information retrieval perspective”  In _Foundations and Trends® in Information Retrieval_ 14.4 Now Publishers, Inc., 2020, pp. 289–444
  * [22] Apoorv Saxena, Aditay Tripathi and Partha Talukdar  “Improving multi-hop question answering over knowledge graphs using knowledge base embeddings”  In _Proceedings of the 58th annual meeting of the association for computational linguistics_ , 2020, pp. 4498–4507
  * [23] Michael Sipser  “Introduction to the Theory of Computation. Cengage Learning”  In _International edition_ , 2012
  * [24] Xiaohui Tao et al.  “Mining health knowledge graph for health risk prediction”  In _World Wide Web_ 23 Springer, 2020, pp. 2341–2362
  * [25] Anton Tsitsulin, John Palowitch, Bryan Perozzi and Emmanuel Müller  “Graph clustering with graph neural networks”  In _Journal of Machine Learning Research_ 24.127, 2023, pp. 1–21
  * [26] Ruben Verborgh et al.  “Triple pattern fragments: a low-cost knowledge graph interface for the web”  In _Journal of Web Semantics_ 37 Elsevier, 2016, pp. 184–206
  * [27] Yiwei Wang et al.  “Mixup for node and graph classification”  In _Proceedings of the Web Conference 2021_ , 2021, pp. 3663–3674
  * [28] Wikipedia contributors  “Path (graph theory) — Wikipedia, The Free Encyclopedia” [Online; accessed 16-February-2024], https://en.wikipedia.org/w/index.php?title=Path_(graph_theory)&oldid=1191420024, 2023
  * [29] Carl Yang et al.  “Meta-graph based hin spectral embedding: Methods, analyses, and insights”  In _2018 IEEE International Conference on Data Mining (ICDM)_ , 2018, pp. 657–666  IEEE
  * [30] Liang Yao, Chengsheng Mao and Yuan Luo  “Graph convolutional networks for text classification”  In _Proceedings of the AAAI conference on artificial intelligence_ 33.01, 2019, pp. 7370–7377
  * [31] Muhan Zhang et al.  “Labeling trick: A theory of using graph neural networks for multi-node representation learning”  In _Advances in Neural Information Processing Systems_ 34, 2021, pp. 9061–9073


figure table

# Metasurface-based Toroidal Lenslet Array Design for Addressing Laser Guide
Star Elongation

Josephine Munro  ARC Centre of Excellence for Transformative Meta-Optical
Systems (TMOS), Department of Electronic Materials Engineering, Research
School of Physics, Australian National University, Canberra, ACT 2600,
Australia  Sarah E. Dean  ARC Centre of Excellence for Transformative Meta-
Optical Systems (TMOS), Department of Electronic Materials Engineering,
Research School of Physics, Australian National University, Canberra, ACT
2600, Australia  Neuton Li  ARC Centre of Excellence for Transformative Meta-
Optical Systems (TMOS), Department of Electronic Materials Engineering,
Research School of Physics, Australian National University, Canberra, ACT
2600, Australia  Israel J. Vaughn  Research School of Astronomy and
Astrophysics, Australian National University, Weston Creek, ACT 2611,
Australia  Andrew W. Kruse  Research School of Astronomy and Astrophysics,
Australian National University, Weston Creek, ACT 2611, Australia  Tony
Travouillon  Research School of Astronomy and Astrophysics, Australian
National University, Weston Creek, ACT 2611, Australia  Dragomir N. Neshev
ARC Centre of Excellence for Transformative Meta-Optical Systems (TMOS),
Department of Electronic Materials Engineering, Research School of Physics,
Australian National University, Canberra, ACT 2600, Australia  Robert Sharp
Research School of Astronomy and Astrophysics, Australian National University,
Weston Creek, ACT 2611, Australia  Andrey A. Sukhorukov  ARC Centre of
Excellence for Transformative Meta-Optical Systems (TMOS), Department of
Electronic Materials Engineering, Research School of Physics, Australian
National University, Canberra, ACT 2600, Australia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Giant Magellan Telescope will use laser tomography adaptive optics to
correct for atmospheric turbulence using artificial guide stars created in the
sodium layer of the atmosphere (altitude $\approx$ 95km). The sodium layer has
appreciable thickness ($\approx$ 11km) and this results in the laser guide
star being an elongated cylinder shape. Wavefront sensing with a Shack-
Hartmann is challenging, as subapertures located further away from the laser
launch position image an increasingly elongated perspective of the laser guide
star. Large detectors can be used to adequately pack and sample the images on
the detector, however, this increases readout noise and limits the design
space available for the wavefront sensor. To tackle this challenge, we propose
an original solution based on nano-engineered meta-optics tailored to produce
a spatially varying anamorphic image scale compression. We present meta-
lenslet array designs that can deliver $\approx$100% of the full anamorphic
image size reduction required for focal lengths down to 8 mm, and greater than
50% image size reduction for focal lengths down to 2 mm. This will allow
greatly improved sampling of the available information across the whole
wavefront sensor, while still being a viable design within the limits of
current-generation fabrication facilities.

###### keywords:

meta-lenslet array, Shack-Hartmann wavefront sensor, laser guide star,
anamorphic image scale compression

*Equal contribution from lead authors Josephine Munro,
and Sarah Dean,

##  1 Introduction

With a primary mirror diameter of 24.5 m, The Giant Magellan Telescope (GMT
[1]) is one of three next-generation ground-based Extremely Large Telescopes
(ELTs) scheduled to begin operations in the next decade. As with all large
ground-based optical telescopes, Adaptive Optics (AO) will be used to correct
atmosphere-induced aberrations to ensure the ELTs deliver images at or near
the diffraction limit. AO requires a sufficiently bright star in close angular
proximity to the science target of interest to provide a reference for
accurate wavefront correction [2]. Although natural guide stars offer a high
quality of correction, there is not always a bright star sufficiently close to
the target of interest, which restricts sky coverage for what can and cannot
be observed with AO systems [3]. This can be overcome with the creation of an
artificial laser guide star (LGS) close to the science target. A laser with a
wavelength resonant to the spectral doublet, known as the Sodium D-lines, is
launched from near the telescope to excite sodium atoms at the altitude of the
mesosphere sodium layer ($\approx$ 90km) [4]. The excited sodium atoms
resonance fluoresce; releasing a 589nm photon as they relax from the excited
$3p$ into the lower $3s$ energy state [5]. This luminescent emission is the
artificial LGS. Owing to their ability to drastically increase sky coverage,
LGS facilities will be available at all three extremely large telescopes.

The GMT will have three AO modes [6]; Natural Guide Star Adaptive Optics,
Laser Tomography Adaptive Optics (LTAO), and Ground Layer Adaptive Optics. Of
particular interest to this study is the LTAO mode which uses an asterism of
six LGS and one off-axis natural guide star to achieve $<$ 290nm root-mean-
square (RMS) wavefront error with $>$ 50% sky coverage at the galactic pole
for near-infrared diffraction-limited imaging [6]. The six laser launch
locations are peripherally positioned around the primary mirror. The wavefront
sensor for the LTAO system will be a Shack-Hartmann design [7], which is the
standard choice of wavefront sensors for ELTs due to their simplicity and
well-characterized operation [2]. Despite this, Shack-Hartmann wavefront
sensors (SHWFS) face substantial challenges when trying to sample the
wavefront optimally due to laser guide star elongation; a phenomenon that will
especially affect ELTs.

###  1.1 Laser Guide Star Elongation

The atmospheric sodium layer is not infinitely thin; the vertical extension
(thickness) is approximately 1020 [4, 8]. The volumetric shape of the LGS is a
column of illumination along the laser beam axis, as opposed to an ideal point
source. This has the effect of the artificial star appearing elongated when
observed from a laterally offset position with respect to the laser launcher
[2], as geometrically depicted in Figure 1. The angular image elongation,
$\varepsilon$, can be approximated for each subaperture position as

$\varepsilon(r)\approx r\frac{\Delta H}{H^{2}}\cos(\theta),$ (1)

where $r$ is the distance of the subaperture (imposed on the primary mirror)
from the laser launch position, $\Delta H$ is the thickness of the sodium
layer, $H$ is the height of the sodium layer, and $\theta$ is the on-sky angle
subtended by the LGS in the subaperture [2].

Figure 1: Schematic of laser guide star elongation due to the finite thickness
of the sodium layer in the atmosphere. The perspective elongation in each
subaperture increases as its distance from the laser launch position
increases. Similarly, the angle of the elongated axis is azimuthally dependent
on each subaperture’s position with respect to the laser launch position.

Subapertures with larger lateral displacement to the laser launcher have a
larger image elongation. On the GMT, for example, the maximum elongation will
be $\varepsilon\approx$ 9.7arcsec for a subaperture on the opposite edge of
the pupil to the laser launch position. The image size in the directional
orthogonal to the elongation is considered to be 1arcsec [7]. The maximum
elongation ratio of the LGS image for ELTs is between 1:9 to 1:13 depending on
the telescope’s diameter and the atmospheric statistics of each telescope
site.

Perspective elongation is challenging because it makes packing the images onto
the detector difficult. Along with the normal design-space constraints of
sensitivity and dynamic range, further consideration needs to be taken to
avoid the overlap of neighbouring images and avoid excessive truncation of the
most elongated images. The standard solution to address image elongation has
been to compromise the detector plate scale. Normally, the plate scale can be
chosen such that the trade-off between the sampling and the field of view
(FOV) still satisfies system requirements. But to combat elongation, the FOV
should be maximized to avoid excessive truncation, simultaneously the sampling
of non-elongated images must be adequate for the centroiding algorithm used,
given the noise characteristics of the system. This is achieved by using a
large number of pixels to sample a large FOV in each subaperture, resulting in
large-area detectors [9]. However, this approach generally increases the
readout noise, the readout time, the computational time and the power
required. Furthermore, a large amount of the detector is unused; sampling
blank sky around the less elongated spots and in the transverse direction to
elongated ones. Fixing a minimum subaperture FOV to limit the bias that is
introduced by truncation of the elongated images reduces the available design
space of the wavefront sensor. For example, the compromise of the plate scale
for GMT’s LTAO leads to 0.71 arcsec per pixel in a system expecting 1 arcsec
non-elongated LGS image size [7].

This problem has attracted multiple proposals that aim to minimize the effect
of LGS elongation, with solutions that are computational, detector, and
optically based. Computational-based solutions include discarding or weighing
the elongation-axis wavefront information from each LGS in an asterism and
calculating the missing information from a combination of the other signals
[10]. Detector-based approaches include variable gains for different regions
of interest [11], and CCDs with custom pixel morphology (where pixels in each
subaperture are radially aligned to the elongation axis) [12]. However, this
approach still requires a large number of pixels to limit truncation to an
acceptable level.

Optical domain solutions have been previously suggested, including
combinations of multiple arrays of custom optics, free-form solutions [13],
and tilting prism arrays [14], but have not been investigated beyond
theoretical and modelling stages of development. A viable optical solution has
to be able to address the elongation in each subaperture while meeting
requirements for fabrication, integration and alignment. There has been recent
renewed interest in finding an optical domain solution to the elongation
problem with aspirations of increasing the sensitivity, accuracy and
efficiency of AO systems for ELTs. In the context of the Thirty Meter
Telescope, Lombini et al., [2] detailed the design and simulation results of a
bi-prism lenslet array, originally described by Schreiber et al. [15]. Each
subaperture bi-prism is aligned along the axis of elongation, effectively
acting as a pyramid prism with only two faces. The elongated signal is divided
into two images of the pupil, where the difference in intensity between the
two images is proportional to the local tip/tilt in the orthogonal axis to the
elongation. This method is insensitive to the aberrations occurring in the
elongation axis. A recombination algorithm using other LGS signals could be
used to reconstruct the elongation axis aberrations. The recombination process
is not trivial and is another source of wavefront error. Notwithstanding, the
bi-prism array is a promising optical solution as simulations have shown it
allows for reducing the size of the detector; from 14-20 pixels2 per
subaperture to 2-6 pixels2 per subaperture.

Combating the negative effects of elongation is an important, though
challenging, problem. It would be ideal to set a variable plate scale across
the detector to best sample the available information in each subaperture.
Anamorphic compression is widely done in the optics community with a pair of
cross cylindrical lenses. However, parfocal operation across an array
requiring variable anamorphic compression with only a pair of crossed
cylindrical lenslets is not possible since either the back focal distance or
the air gap between the cylindrical lenslets would need to vary across the
array. Taking the principle of a toroidal lens, which has different focal
lengths in orthogonal axes, two separate layers of lenslets arrays have enough
degrees of freedom to achieve parfocal operation and satisfy a large variation
in anamorphic compression. To the authors’ knowledge, there has been no
significant progress in the fabrication of a variably anamorphic lenslet array
comprised of crossed cylinders, prism pairs, or toroidal lenses using
conventional optics. There have been preliminary fabrication efforts towards
3D diamond machining of free-form micro-optics designs which includes working
towards spatially varying lenslet sizes and dithered lenslets. However, it is
noted that there are significant limitations to the capability of the
fabrication technique due to the finite tool size setting a limit for the size
of discontinuities in the workpiece [16].

Conversely, a variable plate scale is readily achievable with the meta-lenslet
approach presented here. We propose a meta-lenslet array that is designed to
individually address the elongation ratio and azimuthal angle in each
subaperture, as depicted in Figure 2(b). The field of metaoptics has grown
rapidly, with instrumentation applications evolving alongside further
fundamental nanophotonics research [17]. Nanostructured flat metaoptics have a
wide range of applications that are particularly suited to optical systems
with constraints on size, weight, and power requirements. In the literature,
there are examples of toroidal meta-lenses in a zoom telescope system that
have been shown to enable variable magnification when switching between
orthogonal axes [18, 19]. However, metaoptics have never been explored for LGS
elongation correction before. We show how this meta-lenslet array solution is
viable, with the parfocal operation across a broad range of anamorphic
compression ratios achieved with two layers (a bilayer) of metasurfaces
separated by an air gap.

Figure 2: (a) Laser guide star image elongation on a WFS detector with a
conventional spherical lenslet array. The images in subapertures that are
laterally displaced further away from the laser launch position have a larger
elongation ratio. (b) Anamorphically compressed images on a WFS detector with
our proposed meta-lenslet array. Two layers of metasurfaces (bilayer) are
fabricated on a fused silica substrate and separated by an air gap. With the
shrinking of the image size across the whole aperture, the number of pixels
required in each subaperture can be reduced, reducing the size of the
detector.

###  1.2 Meta-lenslet Approach

Conventional optics rely on propagation through bulk media to manipulate the
optical properties of the incoming light, whereas metasurfaces induce an
abrupt change in optical properties at an ultra-thin surface [20].
Metasurfaces are artificially designed and fabricated arrays of sub-wavelength
sized structures that are purpose-designed to manipulate the incoming
wavefront to achieve a desired output wavefront, for example with a particular
amplitude, phase, or polarisation [21]. In infrared and visible spectral range
of operation, the size of the sub-wavelength-sized structures is on the order
of tens to hundreds of nanometres, leading to the term ‘nanostructures’.
Geometric parameters such as the size, shape and orientation of each
nanostructure determine the local effect on the wavefront. By varying a
geometric parameter across the metasurface, the output wavefront can be
controlled. The manipulation of the incoming light can replicate the functions
of conventional optics, but can also go beyond what conventional bulk optics
can achieve, especially for multi-mode and multi-functional capabilities. The
use of dielectric instead of metallic materials was an important advancement
in the field; as materials such as silicon (Si) and numerous silicon
compounds, gallium nitride ($\mathrm{GaN}$), and titanium dioxide
($\mathrm{TiO_{2}}$) have significantly less absorption loss than metallic
materials, making them suitable for use with metasurfaces in the infrared and
visible regions. Additionally, the high refractive index of dielectric
materials allows for better light confinement and reduced near-field coupling.
The behaviour of the dielectric structures can be described by the
interference of supported Mie-like resonances within the structure [22].
Metasurfaces can be fabricated using existing micro and nanofabrication
technologies commonly used in the semiconductor manufacturing industry.

Both cylindrical and spherical focusing meta-lenses are commonly designed by
determining the phase profile required to manipulate an incident wavefront,
and then arranging nanoresonator elements from a library with predetermined
phase transmissions to match the target phase profile. To focus a planar
wavefront at normal incidence, a metasurface should impose a hyperbolic phase
delay on the incoming wavefront to achieve aberration-free focusing [23].
Similarly, a cylindrical metalens also imposes a hyperbolic phase delay but is
only spatial dependent on one axis. This concept is easily extended to
anamorphic behaviour by introducing coefficients ${a_{1}}$ and ${a_{2}}$ for
$x$ and $y$, respectively, as follows:

$\psi(x,y)=\frac{2\pi}{\lambda}\left(f-\sqrt{a_{1}x^{2}+a_{2}y^{2}+f^{2}}\right),$
(2)

where $f$ is the focal length of the metalens, and ${a_{1}}={a_{2}}$ is the
special case of spherical focusing and ${a_{1}}=0$ or ${a_{2}}=0$ is the
special case of cylindrical focusing. Applications for spherical and
cylindrical meta-lenses include imaging and microscopy [24, 25, 26, 27].
Additional numerical optimization is often used to correct aberrations or
improve performance. The metasurface materials, thickness, and resonator
shapes are all tailored to the operating wavelengths and behaviours desired
from the metasurface [23]. A more complex design was demonstrated by Li et al.
[28], where a metasurface for polarization-dependent beam steering utilized
anamorphic metalenses. The anamorphic lens components were also aspheric to
properly function as part of the beam-steering doublet, and were implemented
by extracting the phase profiles from a designed geometric doublet and
discretizing it appropriately for individual nanoresonators. The maximum ratio
of focal lengths between the $x$ and $y$ directions for the designed
anamorphic lenses demonstrated by Li et al. [28] was approximately $1:3.5$.

Metasurface-based replication of lenslet arrays has focused on instrument
advances towards miniaturization [29], or additional functionality such as
polarisation sensitivity for beam profiling [30]. For the application to LGS
elongation, metasurfaces with polarisation independence will be desirable.
Polarisation independence is typically achieved by using nanoresonators with
three or higher degrees of symmetry in the $x,y$ plane, such as cylindrical
nanopillars, which avoids the shape birefringence. The lattice arrangement
should also have a high degree of symmetry to avoid lattice-induced
birefringence. Polarisation-insensitive metasurfaces are well-represented in
previous literature, with several examples existing specifically for
metalenses [26, 31, 32]. A polarisation independent 60x60 meta-lenslet array
was fabricated for long-wave infrared (10.6), with a pitch size of 100 and
focal length of 100, with good agreement between simulations of focal spot
size and laboratory measurements [29]. A significant drop from the simulated
efficiency of 43% to the measured efficiency of 34% was noted. The authors
postulate that reflection from the substrate backside and the absorbance of
the silicon substrate at 10.6 are major contributing factors to the efficiency
loss, and so the efficiency could be improved by employing an anti-reflective
coating, and choosing a substrate material with a higher long-wave infrared
transparency.

The above examples of metasurface-based lenslet arrays have been designed for
infrared wavelengths, however, our LGS application will be specifically for a
wavelength of 589nm in the visible region. Metasurfaces operating at visible
wavelengths use materials such as $\mathrm{TiO_{2}}$ [25] and GaN [33] for
increased efficiency compared to silicon. However, a material analysis study
showed crystalline silicon (c-Si), for a 500nm thick metasurface structure,
performs similarly to $\mathrm{TiO_{2}}$ for a $None$ deflection meta-grating
at 550nm [34].

An analytical study quantified the cross-talk prevalence for small pitch-sized
$\mathrm{TiO_{2}}$ on fused silica substrate meta-lenslet arrays at 532nm.
Their findings showed pitch sizes larger than 8 and dead-space between meta-
lenslets larger than 0.5 resulted in negligible cross-talk between meta-
lenslets [35]. This robust numerical study provides a sound technical
reference to designing a meta-lenslet array for visible wavelengths, with the
caveat that experimental results are required to confirm the characteristic
optical behaviour and to characterise the effect of noise and other system
errors that are otherwise ignored for simplicity in simulations. To the
authors’ knowledge, this study presents the first design of a metasurface-
based array to directly and uniquely minimize the effect of LGS elongation for
SHWFS.

##  2 Meta-lenslet Optical System Modelling

The lenslet parameter values of the LTAO WFS for the GMT [7] were used as the
design reference for the meta-lenslet optical design, and are given in Table
1.

Table 1: Meta-lenslet design values Parameter | Value
---|---
Effective focal length |  2mm
Pitch size (meta-lenslet diameter) |  400
Wavelength |  589nm
Range of elongation ratios | 1:1 - 1:10

Maintaining the effective focal length for the non-elongated axis preserves
the WFS characteristics in the GMT’s LTAO design. The effective focal length
in the non-elongated axis (hereafter denoted $f_{\alpha}$) is held constant
across all meta-lenslets in the array. To achieve anamorphic compression in
paraxial conditions, the effective focal length in the elongated axis
(hereafter denoted as $f_{\beta}$) is smaller than $f_{\alpha}$ by a factor
equal to the anamorphic ratio. $f_{\beta}$ therefore varies across the meta-
lenslet array. If we consider paraxial conditions for the case of 1:10
compression, and $f_{\alpha}$ = 2mm, then $f_{\beta}$ = 0.2mm, which gives a
numerical aperture of 1 in the elongated axis. Large numerical apertures are
difficult to achieve as they require sophisticated full-scale optimization
designs [36]. We limited our $\mathrm{2\pi}$ phase shift gradient to 3 linear
distance on the meta-lenslet to not exceed the typical fabrication
capabilities. Consequently, the shortest focal length achievable in our
simulations will be $\approx$ 0.8mm (numerical aperture $\approx$ 0.25). This
is not to imply the problem is unsolvable by metasurfaces; much higher
numerical apertures have been successfully demonstrated but are more complex
to fabricate, and have trade-offs with increasing aberrations [37].

The paraxial numerical aperture in the elongated axis is tabulated in Table 2
for our design target of $f_{\alpha}$ = 2mm, as well as longer focal lengths
up to $f_{\alpha}$ = 10mm. It is clear from Table 2 that most of the
anamorphic ratios for $f_{\alpha}$ = 2mm will require a numerical aperture in
the elongation axis above our achievable limit. We, therefore, expect to have
less than 100 reduction in the image size in subapertures with numerical
apertures in the elongation axis larger than 0.25. It should be noted that
even a reduction of 50 in image size would still significantly improve the LGS
WFS. GMT’s LTAO design value of $f_{\alpha}$ = 2mm is unusually low to
maximize FOV to avoid excessive truncation. This is why we also modelled
effective focal lengths up to 10mm, investigating a representative range of
common lenslet array focal lengths. With ten individual meta-lenslets designed
with anamorphic ratios from 1:1–1:10 for each focal length between 2-10mm; a
total of 90 meta-lenslets were designed.

Table 2: Paraxial solution numerical aperture in the elongated axis for each meta-lenslet bilayer. $f_{\alpha}$ (mm) | Numerical Aperture in Elongated Axis
---|---
| Input Anamorphic Ratio
| 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | 1:8 | 1:9 | 1:10
2 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.0
3 | 0.07 | 0.13 | 0.20 | 0.27 | 0.33 | 0.40 | 0.47 | 0.53 | 0.60 | 0.67
4 | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50
5 | 0.04 | 0.08 | 0.12 | 0.16 | 0.20 | 0.24 | 0.28 | 0.32 | 0.36 | 0.40
6 | 0.03 | 0.07 | 0.1 | 0.13 | 0.17 | 0.20 | 0.23 | 0.27 | 0.30 | 0.33
7 | 0.03 | 0.06 | 0.09 | 0.11 | 0.14 | 0.17 | 0.20 | 0.23 | 0.26 | 0.29
8 | 0.03 | 0.05 | 0.08 | 0.10 | 0.13 | 0.15 | 0.18 | 0.20 | 0.23 | 0.25
9 | 0.02 | 0.04 | 0.07 | 0.09 | 0.11 | 0.13 | 0.16 | 0.18 | 0.20 | 0.22
10 | 0.02 | 0.04 | 0.06 | 0.08 | 0.10 | 0.12 | 0.14 | 0.16 | 0.18 | 0.20

The meta-lenslets were modelled in OpticsStudio (Zemax). The layout consisted
of two metasurfaces directly in contact with fused silica substrates, facing
away from each other, as depicted in Figure 2(b). There are four variable
parameters: the phase profile of metasurface 1, the bilayer separation
distance (air gap), the phase profile of metasurface 2, and the distance to
the back focal plane. Since the 1:10 case is the most challenging to optimize
for, the air gap and the back focal length were optimized for the 1:10 case.
These distances were set as non-variable for all other anamorphic ratios
between 1:1 - 1:9, so that each meta-lenslet in an array would satisfy
parfocal operation.

The $x$ and $y$ axes were chosen to be the elongated and the non-elongated
axes, respectively. The anamorphic ratio was represented by the ratio of field
angles in $x$ and $y$. For example, the 1:10 anamorphic ratio was represented
by field angles of (0.3,0) and (0,0.03). To optimize for RMS spot size and
anamorphic compression, the optimization routine included setting the centroid
positions in the focal plane of the elongated and non-elongated fields
(operands CENX and CENY) to be equidistant, but a higher weighting was given
to requiring the RMS radius to be smaller than the Airy radius. The image size
is quoted as the centroid position of the elongated field angle plus the
radius of the RMS spot size.

The metasurfaces were modelled using ‘Binary 1’ surfaces, which impart a phase
delay according to ${x^{2}}$ and ${y^{2}}$ polynomial coefficients, as set out
in Equation 2. The normalized limits on the ${x^{2}}$ and ${y^{2}}$
coefficients are set to -250 to 250, reflecting the 3 per $\mathrm{2\pi}$
phase shift limitation. Note that the polynomial is normalized by the radius
of the meta-lenslet (200) and is not expressed in terms of $\mathrm{2\pi}$.

To make a comparison with conventional optics, a spherical design was also
modelled in OpticsStudio. The design parameters are held constant for both the
spherical and meta-lenslet designs between each test to allow for a fair
comparison. The spherical lenslet was optimized for the radius of the plano-
convex lens and the back focal distance for the same fields as mentioned
above, and the same 589 nm wavelength. The thickness of the silica substrate
was set to 1.2 mm, a common thickness found in commercially available lenslet
arrays. The diameter of the spherical lenslet was set to 400, the same as for
the meta-lenslet.

###  2.1 Optical System Design Results

For our design target case of $f_{\alpha}$ = 2mm, the image size is reduced by
the meta-lenslet compared to the image size produced by the spherical lens
across all input anamorphic ratios. Figure 3(a) shows a graph of the image
size for each input anamorphic ratio for the meta-lenslet and spherical
lenslet. Figures 3(b) and (c) show the image sizes for the spherical lenslet
and the meta-lenslet for the 1:10 input anamorphic ratio, respectively.

Figure 3: (a,d) Graph of image size versus input anamorphic ratio for a meta-
lenslet and a spherical lenslet. The focal length is (a,b,c)
$\mathrm{f_{\alpha}}$ = 2mm and (d,e,f) $\mathrm{f_{\alpha}}$ = 10mm.
(b,c,e,f) Focal plane field angle positions and image footprint size at 1:10
anamorphism for (b,e) spherical lenslet and (c,f) meta-lenslet. The image size
is reduced by (c) 58% and (f) 98% for a meta-lenslet relative to a spherical
lenslet, as defined in Eq. (3).

The spherical 1:1 anamorphic case defines the minimum image size, thereby the
reduction in image size for the 1:10 anamorphic case is calculated as:

$\text{Reduction (\%)}=\frac{\text{meta-lenslet image size}-\text{minimum
image size}}{\text{spherical image size}-\text{minimum image size}}*100.$ (3)

This scales the result so that a 100% image size reduction corresponds to the
elongated image being compressed down to the minimum image size, instead of
zero. The image size is reduced by 94% for the smallest anamorphic ratio of
1:2, and is between 58% and 64% for anamorphic ratios from 1:3 to 1:10. As
predicted by the paraxial solutions to each case in Table 2, 1:3 to 1:10 were
not able to be fully compressed due to the higher numerical apertures needed.
There is inherently greater RMS spot spread in the elongation axis as the
anamorphic ratio increases due to the increasing required numerical aperture.

Since full image size reduction cannot be achieved for $f_{\alpha}$ = 2mm,
counter-intuitively, it may be beneficial to enlarge the smallest images to be
similar in size to most of the array to attain a more favourable plate scale;
to test this assertion, the effect on the SNR variation and wavefront error
would need to be quantified.

Comparatively, for the longer focal length of $\mathrm{f_{\alpha}}$ = 10mm,
full image size reduction is achieved across all input anamorphic ratios, as
shown in Figure 3(d). Figures 3(e) and (f) show the image sizes for a
spherical lenslet and the meta-lenslet for the 1:10 input anamorphic ratio,
respectively. The greater RMS spot spread in the elongation axis as the
anamorphic ratio increases, resulting in the meta-lenslet image size
increasing from 10.4 to 11.2 for input anamorphic ratios 1:1 and 1:10,
respectively.

In every case, the meta-lenslet was able to reduce the image size by
compressing the image scale in the elongation axis. As predicted by the
paraxial approximation, the effective focal length and anamorphic ratio
combinations that required numerical apertures above 0.25 in the elongation
axis were unable to be fully compressed. Table 3 gives the percentage image
size reduction for all cases.

Table 3: Percentage image size reduction for each meta-lenslet bilayer, with varying input anamorphic ratio and effective focal length. $f_{\alpha}$ (mm) | Image Size Reduction (%)
---|---
| Input Anamorphic Ratio
| 1:1 | 1:2 | 1:3 | 1:4 | 1:5 | 1:6 | 1:7 | 1:8 | 1:9 | 1:10
2 | - | 98 | 64 | 59 | 60 | 58 | 60 | 59 | 59 | 58
3 | - | 103 | 91 | 78 | 76 | 77 | 76 | 77 | 76 | 76
4 | - | 102 | 97 | 94 | 91 | 81 | 84 | 86 | 85 | 84
5 | - | 101 | 99 | 97 | 95 | 95 | 92 | 86 | 90 | 88
6 | - | 101 | 99 | 98 | 97 | 96 | 96 | 94 | 92 | 91
7 | - | 100 | 100 | 99 | 98 | 98 | 97 | 95 | 93 | 94
8 | - | 100 | 100 | 99 | 99 | 99 | 98 | 97 | 97 | 97
9 | - | 100 | 100 | 100 | 99 | 99 | 99 | 98 | 98 | 98
10 | - | 100 | 100 | 100 | 100 | 99 | 99 | 99 | 98 | 98

##  3 Meta-lenslet Nanostructure Modelling

Our nanostructure design consists of $\mathrm{TiO_{2}}$ cylindrical pillars on
a 460 thick fused silica substrate. The unit cell is one cylindrical pillar on
a defined area of substrate. To meet Nyquist sampling requirements, the unit
cell size should be less than $\mathrm{{\lambda}/{(2\,NA)}}$ [21]. Given our
design wavelength of 589nm, and taking a numerical aperture of 1, the unit
cell should be less than 295nm. However, as shown in Section 2, we are
expecting to only utilize maximum numerical apertures of $\mathrm{NA}\sim$
0.3. Therefore, we obtain a corresponding unit cell size of less than 982nm.
In our design, we chose a unit cell size of 350nm, corresponding to a
numerical aperture of 0.8, thereby giving adequate sampling well above our
expected maximum numerical aperture while also having acceptable transmission
[35].

To simulate the electric and magnetic field components inside the
nanostructures, we use a rigorous coupled wave analysis implemented in the
RETICOLO software package [38]. We calculate the phase shift and transmission
response for a periodic arrangement of cylinders of heights varying in the
range of 300650nm, with 50nm increments. The diameter of the cylinders was
also varied between 0350nm with approximately 4nm increments.

As shown in Figure 4(a), the unit cell was modelled as a $\mathrm{TiO_{2}}$
cylindrical pillar with a 350nm $\times$ 350nm $\times$ 460 fused silica
substrate. The effect of the cylindrical pillar height on the transmission
(normalised intensity) and phase delay results are shown in Figures 4(b) and
(c), respectively. Results of cylinder diameters below 50nm are shown for
completeness, but they are not included in the full $\mathrm{2\pi}$ phase
coverage considerations, as pillars of these sizes cannot be adequately
fabricated. Although a full $\mathrm{2\pi}$ phase shift is accessible for all
of the cylinder heights tested, there is significant transmission loss for the
short cylinders with heights $<400$ nm.

Figure 4:  (a) Diagram of the metasurface unit cell, with a 350nm square fused
silica substrate, and a $\mathrm{TiO_{2}}$ cylindrical pillar. (b)
Transmission (normalised intensity) and (c) phase shift calculated for varying
diameters and cylinder heights. (d) Phase shift and transmission (normalised
intensity) versus diameter of the 600nm high pillars. The grey shaded areas
indicate regions of pillar diameter that have acceptable transmission and are
above the minimum diameter required for fabrication. There is a full
$\mathrm{2\pi}$ shift accessible in these regions.

As a result of these parameter maps, we select a design based on a cylinder
with a height of 600nm. This choice of height gives a full $\mathrm{2\pi}$
phase shift in regions of high transmission for cylinders with diameters
between $\sim$100260nm, as shown by the grey regions in Figure 4(d). A
metasurface within these design constraints is expected to be comfortably
within acceptable levels of fabrication accuracy and repeatability using
current nanofabrication technology.

We then construct the meta-lenslets by searching for the meta-atom in our
library that produces the smallest phase offset from the target phase at each
spatial location. Since each of our meta-lenslets have mirror symmetry in both
the $x$ and $y$ axes, only one quadrant of the device is constructed, which is
then translated and mirrored to build the full device. This approach saves
computation time by a factor of four.

The meta-lenslet device design for the $\mathrm{f_{\alpha}}$ = 2mm, 1:10
anamorphic ratio case is shown in Figure 5. The first metasurface imparts a
mostly cylindrical phase profile [Figs. 5(b,c)], while the second metasurface
imparts a slightly toroidal profile with stronger focusing in the $x$ axis
than the $y$ axis [Figs. 5(d,e)]. The edges of the second metasurface in this
design are the steepest changes in phases of all the devices simulated. It can
be seen that even in this most extreme example, there are approximately eight
pillars covering a full $\mathrm{2\pi}$ change, supporting the feasibility of
the design.

Figure 5:  (a) Optical layout of a meta-lenslet bilayer with
$\mathrm{f_{\alpha}}$ = 2mm, and a 1:10 anamorphic ratio, as modelled in
OpticsStudio. (b,d) Phase profile generated from the OpticsStudio model for
(b) metasurface 1, and (d) metasurface 2. (c,e) $\mathrm{TiO_{2}}$ cylinders
of varying diameters impart the required phase profile for (c) metasurface 1
and (e) metasurface 2. The zoomed-in areas show the varying diameters of the
pillars in an area of large phase shift.

##  4 Conclusion and Future Work

Prior studies have noted the importance of decreasing the detector size for
LGS wavefront sensing for ELTs. This study explored the feasibility of using a
meta-lenslet array to anamorphically compress the variable LGS image size on a
wavefront sensor to allow a decrease in detector size.

We have shown a conceptual design for a meta-lenslet array with an effective
focal length of $\mathrm{f_{\alpha}}$ = 2mm to match the current design
parameters for GMT’s LTAO. The meta-lenslets do not compress the higher
anamorphic ratios fully at this focal length, and consequently have some
residual laser guide star image elongation, but still offer significant
improvement over the static plate-scale problem as a reduction of 50 in image
size for the largest anamorphic compressions was achieved. However, we have
also shown that effective focal lengths $\mathrm{f_{\alpha}\gtrsim}$ 8mm would
be capable of complete anamorphic compression across a range of 1:1-1:10
anamorphic ratios. Noting that the current design for GMT’s LTAO is to
compromise the pixel scale to adequately sample smaller images and avoid
excessive truncation of larger LGS images, relaxing the focal length
requirements to achieve full anamorphic compression across the whole SHWFS
could be considered.

Future work will focus on fabricating a prototype to verify the physical
parameters and performance of meta-lenslets with a range of anamorphic
compression ratios. A design validation prototype can be made with silicon, as
the fabrication process is mature and reliable at the expense of throughput
efficiency at the target visible wavelength. Further designs can employ
$\mathrm{TiO}_{2}$ to gain a higher throughput efficiency.

### Disclosures

The authors have no conflicts of interest to declare.

### Code, Data, and Materials Availability

Code and data can be made available upon reasonable request to the
corresponding author.

### Acknowledgments

We acknowledge the funding support of the Australian Research Council
(NI210100072, CE200100010). This research was initially presented as a poster,
”Meta-lenslet array for laser guide star anamorphic compression”, by co-lead
author S.E. Dean, Paper 12990-108, at SPIE Photonics Europe, Strasberg,
France, 7-11 April 2024.

## References

  * [1] J. Fanson, R. Bernstein, G. Angeli, D. Ashby, B. Bigelow, G. Brossus, A. Bouchez, W. Burgett, A. Contos, R. Demers, F. Figueroa, B. Fischer, F. Groark, R. Laskin, R. Millan-Gabet, M. Pi, and N. Wheeler, “Overview and status of the Giant Magellan Telescope project,” in Ground-based and Airborne Telescopes VIII, H. K. Marshall, J. Spyromilio, and T. Usuda, Eds., 11445, 114451F, International Society for Optics and Photonics, SPIE (2020).  doi:10.1117/12.2561852.
  * [2] M. Lombini, L. Schreiber, E. Diolaiti, and F. Cortecchia, “Optical solution to spot elongation in laser guide stars wavefront sensors for Extremely Large Telescopes,” Monthly Notices of the Royal Astronomical Society 510, 3876–3884 (2021).  doi:10.1093/mnras/stab3762.
  * [3] B. Ellerbroek and D. Andersen, “Sky coverage estimates for the natural guide star mode of the TMT facility AO system NFIRAOS,” in Adaptive Optics Systems, N. Hubin, C. E. Max, and P. L. Wizinowich, Eds., 7015, 70155W, International Society for Optics and Photonics, SPIE (2008).  doi:10.1117/12.788053.
  * [4] Pfrommer, T. and Hickson, P., “High resolution mesospheric sodium properties for adaptive optics applications,” A&A 565, A102 (2014).  doi:10.1051/0004-6361/201423460.
  * [5] L. Lihang, H. Wang, W. Hua, Y. Ning, and X. Xu, “Fluorescence enhancing mechanism of optical repumping in sodium atoms for brighter laser guide star,” Opt. Express 24, 6976–6984 (2016).  doi:10.1364/OE.24.006976.
  * [6] A. H. Bouchez, D. S. Acton, R. Biasi, R. Conan, B. Espeland, S. Esposito, J. Filgueira, D. Gallieni, B. A. McLeod, E. Pinna, F. Santoro, G. Trancho, and M. A. van Dam, “The Giant Magellan Telescope adaptive optics program,” in Adaptive Optics Systems IV, E. Marchetti, L. M. Close, and J.-P. Véran, Eds., 9148, 91480W, International Society for Optics and Photonics, SPIE (2014).  doi:10.1117/12.2057613.
  * [7] R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” in Adaptive Optics Systems III, B. L. Ellerbroek, E. Marchetti, and J.-P. Véran, Eds., 8447, 84473P, International Society for Optics and Photonics, SPIE (2012).  doi:10.1117/12.924865.
  * [8] B. Neichel, C. D’Orgeville, J. Callingham, F. Rigaut, C. Winge, and G. Trancho, “Characterization of the sodium layer at Cerro Pachón, and impact on laser guide star performance,” Monthly Notices of the Royal Astronomical Society 429, 3522–3532 (2013).  doi:10.1093/mnras/sts631.
  * [9] T. Fusco, G. Agapito, B. Neichel, S. Oberti, C. M. Correia, P. Haguenauer, C. Plantet, F. P. Bustos, Z. Ke, A. Costille, P. Jouve, L. Busoni, and S. Esposito, “Key wavefront sensors features for laser-assisted tomographic adaptive optics systems on the Extremely Large Telescope,” Journal of Astronomical Telescopes, Instruments, and Systems 8(2), 021514 (2022).  doi:10.1117/1.JATIS.8.2.021514.
  * [10] T. Fusco, B. Neichel, C. Correia, L. Blanco, A. Costille, K. Dohlen, F. Rigaut, E. Renaud, A. Bonnefoi, Z. Ke, K. El-Hadi, J. Paufique, S. Oberti, F. Clarke, I. Bryson, and N. Thatte, “A story of errors and bias: The optimization of the LGS WFS for HARMONI,” in AO4ELT6, (Québec, Canada) (2019).  URL: https://hal.science/hal-02614095.
  * [11] M. Downing, J. Kolb, P. Balard, B. Dierickx, A. Defernez, B. Dupont, P. Feautrier, M. Fryer, J.-L. Gach, C. Guillaume, N. Hubin, O. Iwert, P. Jerram, P. Jorden, A. Pike, J. Pratlong, J. Reyes, and A. Walker, “Backside-illuminated, high qe, 3e-ron, fast 700fps, 1760x1680 pixels cmos imager for ao with highly parallel readout,” Proceedings of SPIE - The International Society for Optical Engineering 8453 (2012).  doi:10.1117/12.925616.
  * [12] J. W. Beletic, S. Adkins, B. Burke, R. Reich, B. Kosicki, V. Suntharalingham, C. Bleau, R. DuVarney, R. Stover, J. Nelson, and F. Rigaut, “The ultimate ccd for laser guide star wavefront sensing on extremely large telescopes,” Experimental Astronomy 19(1), 103–109 (2005).  doi:10.1007/s10686-005-9012-3.
  * [13] W. Jahn, E. Hugot, T. Fusco, B. Neichel, M. Ferrari, C. Correia, L. Pueyo, K. Dohlen, S. Pascal, P. Vola, J.-F. Sauvage, K. E. Hadi, and J. L. Gach, “Laser guide star spot shrinkage for affordable wavefront sensors,” in Adaptive Optics Systems V, E. Marchetti, L. M. Close, and J.-P. Véran, Eds., 9909, 99096L, International Society for Optics and Photonics, SPIE (2016).  doi:10.1117/12.2232367.
  * [14] E. Gendron, “Optical solutions for accommodating ELT LGS wave-front sensing to small format detectors,” in Adaptive Optics Systems V, E. Marchetti, L. M. Close, and J.-P. Véran, Eds., 9909, 99095Z, International Society for Optics and Photonics, SPIE (2016).  doi:10.1117/12.2234288.
  * [15] L. Schreiber, M. Lombini, I. Foppiani, E. Diolaiti, J.-M. Conan, and E. Marchetti, “An optical solution to the LGS spot elongation problem,” in Adaptive Optics Systems, N. Hubin, C. E. Max, and P. L. Wizinowich, Eds., 7015, 70151O, International Society for Optics and Photonics, SPIE (2008).  doi:10.1117/12.789390.
  * [16] J. Schmoll, D. J. Robertson, and D. A. Ryder, “Design and manufacture of micro-optical arrays using 3D diamond machining techniques,” in Optomechanical Technologies for Astronomy, E. Atad-Ettedgui, J. Antebi, and D. Lemke, Eds., 6273, 62731T, International Society for Optics and Photonics, SPIE (2006).  doi:10.1117/12.671228.
  * [17] D. N. Neshev and A. E. Miroshnichenko, “Enabling smart vision with metasurfaces,” Nat. Photon. 17, 26–35 (2023).  doi:10.1038/s41566-022-01126-4.
  * [18] N. Bregenzer, M. Bawart, and S. Bernet, “Zoom system by rotation of toroidal lenses,” Opt. Express 28, 3258–3269 (2020).  doi:10.1364/OE.383090.
  * [19] F. Yang, H.-I. Lin, M. Y. Shalaginov, K. Stoll, S. An, C. Rivero-Baleine, M. Kang, A. Agarwal, K. Richardson, H. Zhang, J. Hu, and T. Gu, “Reconfigurable parfocal zoom metalens,” Advanced Optical Materials 10(17), 2200721 (2022).  doi:10.1002/adom.202200721.
  * [20] H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Reports on Progress in Physics 79, 076401 (2016).  doi:10.1088/0034-4885/79/7/076401.
  * [21] M. Khorasaninejad and F. Capasso, “Metalenses: Versatile multifunctional photonic components,” Science 358(6367), eaam8100 (2017).  doi:10.1126/science.aam8100.
  * [22] K. Koshelev and Y. Kivshar, “Dielectric resonant metaphotonics,” ACS Photonics 8(1), 102–112 (2021).  doi:10.1021/acsphotonics.0c01315.
  * [23] M. Pan, Y. Fu, M. Zheng, H. Chen, Y. Zang, H. Duan, Q. Li, M. Qiu, and Y. Hu, “Dielectric metalens for miniaturized imaging systems: progress and challenges,” Light: Science & Applications 11(1), 195 (2022).  doi:10.1038/s41377-022-00885-7.
  * [24] E. Bayati, R. Pestourie, S. Colburn, Z. Lin, S. G. Johnson, and A. Majumdar, “Inverse designed metalenses with extended depth of focus,” ACS Photonics 7(4), 873–878 (2020).  doi:10.1021/acsphotonics.9b01703.
  * [25] M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352(6290), 1190–1194 (2016).  doi:10.1126/science.aaf6644.
  * [26] E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with meta-molecules,” Optica 3, 628–633 (2016).  doi:10.1364/OPTICA.3.000628.
  * [27] D. Wen, F. Yue, M. Ardron, and X. Chen, “Multifunctional metasurface lens for imaging and fourier transform,” Scientific Reports 6(1), 27628 (2016).  doi:10.1038/srep27628.
  * [28] H. Li, C. Zhou, W.-B. Lee, D.-Y. Choi, and S.-S. Lee, “Flat telescope based on an all-dielectric metasurface doublet enabling polarization-controllable enhanced beam steering,” Nanophotonics 11(2), 405–413 (2022).  doi:10.1515/nanoph-2021-0609.
  * [29] M. Liu, Q. Fan, L. Yu, and T. Xu, “Polarization-independent infrared micro-lens array based on all-silicon metasurfaces,” Opt. Express 27, 10738–10744 (2019).  doi:10.1364/OE.27.010738.
  * [30] Z. Yang, Z. Wang, Y. Wang, X. Feng, M. Zhao, Z. Wan, L. Zhu, J. Liu, Y. Huang, J. Xia, and M. Wegener, “Generalized hartmann-shack array of dielectric metalens sub-arrays for polarimetric beam profiling,” Nature Communications 9(1), 4607 (2018).  doi:10.1038/s41467-018-07056-6.
  * [31] S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technology Letters 26(13), 1375–1378 (2014).  doi:10.1109/LPT.2014.2325947.
  * [32] M. Khorasaninejad, A. Y. Zhu, C. Roques-Carmes, W. T. Chen, J. Oh, I. Mishra, R. C. Devlin, and F. Capasso, “Polarization-insensitive metalenses at visible wavelengths,” Nano Letters 16(11), 7229–7234 (2016).  doi:10.1021/acs.nanolett.6b03626.
  * [33] S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nature Nanotechnology 13(3), 227–232 (2018).  doi:10.1038/s41565-017-0052-4.
  * [34] J. Yang and J. A. Fan, “Analysis of material selection on dielectric metasurface performance,” Opt. Express 25, 23899–23909 (2017).  doi:10.1364/OE.25.023899.
  * [35] A. Özdemir, N. Yilmaz, S. A. Alboon, Y. Takashima, and H. Kurt, “Analysis of the focusing crosstalk effects of broadband all-dielectric planar metasurface microlens arrays for ultra-compact optical device applications,” OSA Continuum 1, 506–520 (2018).  doi:10.1364/OSAC.1.000506.
  * [36] T. Phan, D. Sell, E. W. Wang, S. Doshay, K. Edee, J. Yang, and J. A. Fan, “High-efficiency, large-area, topology-optimized metasurfaces,” Light: Science & Applications 8(1), 48 (2019).  doi:10.1038/s41377-019-0159-5.
  * [37] H. Liang, A. Martins, B.-H. V. Borges, J. Zhou, E. R. Martins, J. Li, and T. F. Krauss, “High performance metalenses: numerical aperture, aberrations, chromaticity, and trade-offs,” Optica 6, 1461–1470 (2019).  doi:10.1364/OPTICA.6.001461.
  * [38] J. P. Hugonin and P. Lalanne, “RETICOLO software for grating analysis,” arXiv 2101.00901 (2021).  doi:10.48550/arXiv.2101.00901.


# M31N 1926-07c: A Recurrent Nova in M31 with a 2.8 Year Recurrence Time

Allen W. Shafter Department of Astronomy, San Diego State University, San
Diego, CA 92182, USA  Kamil Hornoch Astronomical Institute of the Czech
Academy of Sciences, Fričova 298, CZ-251 65 Ondřejov, Czech Republic  Hana
Kučáková Astronomical Institute of the Czech Academy of Sciences, Fričova 298,
CZ-251 65 Ondřejov, Czech Republic  Research Centre for Theoretical Physics
and Astrophysics, Institute of Physics, Silesian University in Opava,
Bezručovo nám. 13, CZ-74601 Opava, Czech Republic  Jingyuan Zhao (赵经远)
Xingming Observatory, Mount Nanshan, Xinjiang, China  Mi Zhang (张宓) Xingming
Observatory, Mount Nanshan, Xinjiang, China  Xing Gao (高兴) Xingming
Observatory, Mount Nanshan, Xinjiang, China  John Della Costa Department of
Astronomy, San Diego State University, San Diego, CA 92182, USA  William A.
Burris Department of Astronomy, San Diego State University, San Diego, CA
92182, USA  J. Grace Clark Department of Astronomy, San Diego State
University, San Diego, CA 92182, USA  Marek Wolf Astronomical Institute,
Charles University, Prague, V Holešovičkách 2, CZ-180 00 Praha 8, Czech
Republic  Petr Zasche Astronomical Institute, Charles University, Prague, V
Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic

###### Abstract

The M31 recurrent nova M31N 1926-07c has had five recorded eruptions. Well-
sampled light curves of the two most recent outbursts, in January of 2020
(M31N 2020-01b) and September 2022 (M31N 2022-09a), are presented showing that
the photometric evolution of the two events were quite similar, with peak
magnitudes of $R=17.2\pm 0.1$ and $R=17.1\pm 0.1$, and $t_{2}$ times of
$9.7\pm 0.9$ and $8.1\pm 0.5$ days for the 2020 and 2022 eruptions,
respectively. After considering the dates of the four most recent eruptions
(where the cycle count is believed to be known), a mean recurrence interval of
$\langle P_{\mathrm{rec}}\rangle=2.78\pm 0.03$ years is found, establishing
that M31N 1926-07c has one of the shortest recurrence times known.

Cataclysmic variable stars (203) – Novae (1127) – Recurrent Novae (1366) –
Andromeda Galaxy (39) – Time Domain Astronomy (2109)

††facilities: Ondřejov Observatory (OND) 0.65-m, Ningbo Bureau of Education
and Xinjiang Observatory Telescope (NEXT) 0.6-m, Burke-Gaffney Observatory
(BGO) 0.6-m, Mount Laguna Observatory (MLO) 1.0-m.

##  1 Introduction

The nova M31N 1926-07c was first recognized to be recurrent by Shafter et al.
(2015) who discovered that two subsequent eruptions of the nova had been
observed: one in October 1997 (M31N 1997-10f) and the other in August 2008
(M31N 2008-08b). Since the publication of that study just seven years ago, two
additional eruptions have now been reported, the first in 2020 January (M31N
2020-01b), and the second in 2022 September (M31N 2022-09a). As pointed out by
Hornoch et al. (2022a, b), the latest two eruptions are significant in that
they have revealed the recurrence time of this nova to be much shorter than
previously realized.

##  2 The Mean Recurrence Time

It is difficult to unambiguously establish the mean recurrence time for M31N
1926-07c due to the large gap between the first recorded eruption in July of
1926 and October of 1997. However, if we restrict our analysis to the times of
peak brightness for the four most recent observed eruptions, which have taken
place over just the past $\sim$25 years, a conservative estimate for the mean
recurrence time is found to be $\langle P_{\mathrm{rec}}\rangle=2.78\pm 0.03$
years. If we include the initial eruption from July of 1926 in the analysis,
we find that either $\langle P_{\mathrm{rec}}\rangle=2.75\pm 0.01$ or $\langle
P_{\mathrm{rec}}\rangle=2.83\pm 0.01$ years, depending on whether 25 or 24
eruptions were missed between July 1926 and October 1997. The shorter mean
recurrence time, $\langle P_{\mathrm{rec}}\rangle=2.75\pm 0.01$ years, gives a
marginally better fit to the data, and is slightly favored.

Of all known recurrent novae, only M31N 2008-12a, with its $0.99\pm 0.02$ year
mean inter-outburst interval (Darnley & Henze, 2020), and possibly M31N
2017-01e (= 2019-09d, 2022-03d) – a recently-identified M31 recurrent nova
with a likely mean eruption interval of $2.55\pm 0.13$ years (Shafter et al.
2022, in preparation) – have been observed to erupt more frequently.

Figure 1: The light curves for the 2020 January (M31N 2020-01b, top panel) and
2022 September (M31N 2022-09a, middle panel) eruptions of the recurrent nova
M31N 1926-07c. The bottom panel shows both eruptions superimposed after
adopting times of maximum of JD 2,458,877.048 and 2,459,830.725 for the 2020
and 2022 eruptions, respectively. Different symbols represent data taken with
different telescopes as shown in the key (the 2020 data are shown as open
symbols in the bottom panel). The downward-facing triangles represent lower
limits on the magnitudes: NEXT (red) and OND (black). The BGO (NEXT) data for
the 2022 eruption were taken through a luminance filter (no filter), and were
calibrated using $R$-band secondary standard stars. The resulting data are
given as $CR$ magnitudes. Only measurements taken within 14 days of maximum
light were used to determine the best-fitting peak magnitudes and fade rates
(shown as dotted lines) for each eruption. The photometric data are available
in the appendix, Table A1.

##  3 The Light Curves

Figure 1 shows the light curves for 2020-01b and 2022-09a, the last two
observed eruptions of M31N 1926-07c. It is clear that the light curves of the
two eruptions are quite similar, with peak observed magnitudes of $R=17.2\pm
0.1$ and $CR=16.9\pm 0.1$111 Since the BGO $CR$ photometry is $\sim$0.2
magnitudes brighter on average compared with our $R$ band data, we have
estimated that the peak $R$-band magnitude of the 2022 eruption is given by
$R=17.1\pm 0.1$. being reached during the 2020 and 2022 eruptions,
respectively. The fade rates also appear to be reasonably consistent, however
the 2022 eruption seems to have faded somewhat more quickly at late times
($\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}8$ days post peak).

If we restrict our analysis to the first two weeks after peak brightness when
the photometry was most reliable, linear least-squares fits to the declining
branches of the 2020 and 2022 light curves give fade rates of $0.1909\pm
0.0145$ and $0.1959\pm 0.0094$ mag per day, which correspond to
$t_{2}(R)=10.5\pm 0.8$ and $t_{2}(R)=10.2\pm 0.5$ days, respectively, in
excellent agreement. The predicted peak magnitudes are also very similar,
where fits to the 2020 and 2022 light curves give, respectively,
$R_{\mathrm{peak}}=17.4\pm 0.06$ and $R_{\mathrm{peak}}=17.5\pm 0.04$.

Given that the best-fitting peak magnitudes slightly underestimate the
observed peak brightnesses, it may be that our measured fade rates slightly
overestimate the $t_{2}$ times. In keeping with the precise definition of
$t_{2}$ – the time for a nova to fade 2 magnitudes below peak brightness – we
have also determined $t_{2}$ following this prescription, finding slightly
shorter times of $t_{2}(R)=9.7\pm 0.9$ and $t_{2}(R)=8.1\pm 0.5$ days for the
2020 and 2022 eruptions, respectively.

##  4 Discussion

Recurrent novae with very short intervals between successive eruptions
($t_{\mathrm{recur}}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}3$ yr) are thought to harbor massive white
dwarfs ($M_{\mathrm{WD}}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}1.2~{}M_{\odot}$) that accrete at a high
rate ($\dot{M}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}3\times
10^{-7}~{}M_{\odot}~{}\mathrm{yr}^{-1}$) before achieving the accreted mass
($\sim 10^{-6}~{}M_{\odot}$) necessary to trigger a thermonuclear runaway
(e.g., see Kato et al., 2014). The low ejected mass is expected to result in a
rapid photometric evolution characterized by a short $t_{2}$ time. For
example, M31N 2008-12a with its one year recurrence time has a $t_{2}$ time of
only $\sim$2 days (Darnley et al., 2015).

Although M31N 1926-07c is considered a “fast” nova, its $t_{2}$ time of
$\sim$9 days is perhaps longer than one would predict for a recurrent nova
with such a short inter-outburst interval. Spectroscopic observations of the
2008-08b eruption (Di Mille et al., 2008, 2010) show a He/N spectrum that is
typical of recurrent novae, but the lines were reported to be unusually narrow
(FWHM $\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower
4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}850$ km s-1), perhaps consistent with the
longer than expected $t_{2}$ time.

M31N 1926-07c with its five observed outbursts is one of the handful of
recurrent novae in M31 with more than three detected eruptions. Only M31N
1963-09c and 2008-12a with their 6 and 14 outbursts, respectively, have been
seen to erupt more frequently. Based on the times of previous outbursts, the
next eruption of M31N 1926-07c is predicted for June 2025. It will be
important to secure both photometric and spectroscopic observations at that
time in order to further characterize the properties of this intriguing
recurrent nova.

## References

  * Darnley & Henze (2020) Darnley, M. J., & Henze, M. 2020, Advances in Space Research, 66, 1147, doi: 10.1016/j.asr.2019.09.044
  * Darnley et al. (2015) Darnley, M. J., Henze, M., Steele, I. A., et al. 2015, A&A, 580, A45, doi: 10.1051/0004-6361/201526027
  * Di Mille et al. (2008) Di Mille, F., Ciroi, S., Orio, M., et al. 2008, The Astronomer’s Telegram, 1703, 1
  * Di Mille et al. (2010) Di Mille, F., Orio, M., Ciroi, S., et al. 2010, Astronomische Nachrichten, 331, 197, doi: 10.1002/asna.200911326
  * Hornoch et al. (2022a) Hornoch, K., Shafter, A. W., Zasche, P., & Kucakova, H. 2022a, The Astronomer’s Telegram, 15609, 1
  * Hornoch et al. (2022b) —. 2022b, The Astronomer’s Telegram, 15608, 1
  * Kato et al. (2014) Kato, M., Saio, H., Hachisu, I., & Nomoto, K. 2014, ApJ, 793, 136, doi: 10.1088/0004-637X/793/2/136
  * Shafter et al. (2015) Shafter, A. W., Henze, M., Rector, T. A., et al. 2015, ApJS, 216, 34, doi: 10.1088/0067-0049/216/2/34

Table A1: Nova Photometry (JD - 2,450,000) | FilteraaR – Johnson $R$; L – Luminance; C – Clear (no filter) | Mag | ObservatorybbOND: Ondr̂ejov – 0.65-m; NEXT: Ningbo Bureau of Education and Xinjiang Observatory Telescope – 0.6-m; BGO: Burke Gaffney Observatory – 0.6-m; MLO: Mount Laguna Observatory – 1.0-m
---|---|---|---
M31N 2020-01b
8876.088 | R | $>20.2$ | NEXT
8876.225 | R | $>19.2$ | OND
8877.048 | R | $17.2\pm 0.1$ | NEXT
8877.206 | R | $17.5\pm 0.2$ | OND
8878.086 | R | $17.6\pm 0.1$ | NEXT
8878.234 | R | $17.4\pm 0.2$ | OND
8878.250 | R | $17.5\pm 0.2$ | OND
8879.066 | R | $17.8\pm 0.2$ | NEXT
8879.460 | R | $18.0\pm 0.3$ | OND
8880.108 | R | $18.1\pm 0.2$ | NEXT
8880.334 | R | $18.1\pm 0.2$ | OND
8881.074 | R | $18.2\pm 0.2$ | NEXT
8882.045 | R | $18.1\pm 0.2$ | NEXT
8883.068 | R | $18.6\pm 0.2$ | NEXT
8885.067 | R | $19.4\pm 0.3$ | NEXT
8885.247 | R | $18.8\pm 0.3$ | OND
8886.067 | R | $19.0\pm 0.4$ | NEXT
8887.061 | R | $19.0\pm 0.5$ | NEXT
8888.220 | R | $19.2\pm 0.3$ | OND
8888.244 | R | $19.1\pm 0.3$ | OND
8892.274 | R | $19.7\pm 0.4$ | OND
8895.299 | R | $>19.6$ | OND
M31N 2022-09a
9830.376 | R | $>20.0$ | OND
9830.426 | R | $19.4\pm 0.3$ | OND
9830.450 | R | $18.2\pm 0.2$ | OND
9830.472 | R | $18.0\pm 0.2$ | OND
9830.493 | R | $17.5\pm 0.1$ | OND
9830.725 | L | $16.9\pm 0.1$ | BGO
9831.277 | C | $17.6\pm 0.1$ | NEXT
9831.308 | R | $17.8\pm 0.2$ | OND
9831.593 | R | $17.7\pm 0.1$ | OND
9832.334 | C | $17.8\pm 0.1$ | NEXT
9832.749 | L | $17.7\pm 0.2$ | BGO
9833.273 | C | $18.1\pm 0.2$ | NEXT
9833.654 | L | $18.0\pm 0.2$ | BGO
9834.292 | R | $18.4\pm 0.2$ | OND
9834.308 | R | $18.3\pm 0.1$ | OND
9834.343 | R | $18.1\pm 0.2$ | OND
9834.571 | L | $18.0\pm 0.2$ | BGO
9835.301 | C | $18.1\pm 0.2$ | NEXT
9835.437 | R | $18.4\pm 0.2$ | OND
9835.462 | R | $18.4\pm 0.2$ | OND
9835.678 | L | $18.2\pm 0.3$ | BGO
9836.314 | R | $18.5\pm 0.1$ | OND
9836.335 | R | $18.5\pm 0.1$ | OND
9837.579 | L | $18.6\pm 0.3$ | BGO
9837.754 | R | $18.8\pm 0.2$ | MLO
9838.258 | C | $19.3\pm 0.4$ | NEXT
9839.758 | R | $19.6\pm 0.2$ | MLO
9840.337 | R | $19.4\pm 0.3$ | OND
9840.774 | R | $19.7\pm 0.3$ | MLO
9841.759 | R | $19.9\pm 0.3$ | MLO
9843.746 | R | $19.9\pm 0.2$ | MLO
9844.422 | R | $19.9\pm 0.5$ | OND
9845.267 | R | $20.1\pm 0.4$ | OND
9845.761 | R | $21.4\pm 0.5$ | MLO
9845.313 | R | $20.5\pm 0.4$ | OND
9846.741 | R | $21.9\pm 0.5$ | MLO
9847.739 | R | $21.9\pm 0.5$ | MLO
9848.734 | R | $22.2\pm 0.7$ | MLO