# An Unsupervised Masking Objective for
Abstractive Multi-Document News Summarization

Nikolai Vogler Computer Science and Engineering
University of California, San Diego
<EMAIL_ADDRESS>Songlin Li Computer Science and Engineering
University of California, San Diego
<EMAIL_ADDRESS>Yujie Xu Computer Science and Engineering
University of California, San Diego
<EMAIL_ADDRESS>Yujian Mi Computer Science and Engineering
University of California, San Diego
<EMAIL_ADDRESS>Taylor Berg-Kirkpatrick Computer Science and Engineering
University of California, San Diego
<EMAIL_ADDRESS>
###### Abstract

We show that a simple unsupervised masking objective can approach near
supervised performance on abstractive multi-document news summarization. Our
method trains a state-of-the-art neural summarization model to predict the
masked out source document with highest lexical centrality relative to the
multi-document group. In experiments on the Multi-News dataset (Fabbri et al.,
2019), our masked training objective yields a system that outperforms past
unsupervised methods and, in human evaluation, surpasses the best supervised
method without requiring access to any ground-truth summaries. Further, we
evaluate how different measures of lexical centrality, inspired by past work
on extractive summarization, affect final performance.

##  1 Introduction

Multi-document summarization (MDS) aims to condense and combine information
from topically related groups of documents into a single concise yet
comprehensive short text. The news domain, where input documents are article
clusters of varying size covering the same event, is a common challenge
setting for MDS systems (Paul and James, 2004; Owczarzak and Dang, 2011).
While most early approaches to the problem focused on extractive methods that
select and copy relevant sections from the input documents (Radev et al.,
2004; Erkan and Radev, 2004; Mihalcea and Tarau, 2004), advances in neural
text generation have increased both performance of and, subsequently, interest
in abstractive methods, which generate a summary from scratch conditioned on a
combined representation of the encoded source documents.

Figure 1: Proposed unsupervised masking objective for discriminative MDS
models. An input document is selected based on best lexical overlap with the
rest of the input documents (in this example, $C_{i3}$). Then a discrimintive
summarizer is trained to predict the selected (masked) document conditioned on
the rest of the input collection.

While large neural systems have seen success on single-document summarization,
where human summaries are relatively easy to obtain at scale (Napoles et al.,
2012; Hermann et al., 2015), their application to the multi-document task has
been restricted by the difficulty of collecting such data—reading and then
summarizing a _collection_ of documents is a daunting annotation task. We
focus on the task of _unsupervised_ abstractive MDS where the system is only
allowed access to input document collections at training time (Chu and Liu,
2019). Similar to older work on unsupervised extractive summarization (Radev
et al., 2004; Erkan and Radev, 2004; Mihalcea and Tarau, 2004), we propose an
approach that leverages lexical centrality of the document clusters to select
a document to mask out and then predict. This allows for any, black box
discriminative neural abstractive summarization model to be trained without
ground-truth summaries by maximizing the likelihood of the masked out
document, drawing inspiration from masked language modeling (e.g., BERT
(Devlin et al., 2018)). Some recent works have automatically built large-
scale, supervised MDS datasets for news (Fabbri et al., 2019) and Wikipedia
(Liu et al., 2018). We evaluate our unsupervised method on the news domain
using the Multi-News dataset (Fabbri et al., 2019).

##  2 Masked Document Objective

Typically, neural abstractive single-document and multi-document summarization
models consist of an encoder-decoder architecture (Sutskever et al., 2014)
discriminatively trained end-to-end to maximize the likelihood of the
supervised, ground truth summaries given the input document(s). This can be
achieved by minimizing the cross-entropy under the model between the predicted
summary’s word sequence and the true summary’s word sequence. In order to
adapt discriminative training to an unsupervised multi-document setting, we
propose a surrogate objective that maximizes the likelihood of a masked
document from each topically-related cluster of documents as shown in Figure
1. Crucially, this procedure can be applied to any black box neural
summarization model including current state-of-the-art approaches normally
trained on supervised data.

###  2.1 Selecting the Best Masking Candidate

In principle, by training a summarizer to predict randomly chosen masked
documents in each input collection, we implicitly train the model to predict
the centroid of the collection—a feature of successful extractive systems in
past work (Radev et al., 2004; Erkan and Radev, 2004; Mihalcea and Tarau,
2004). However, since document collections may themselves be noisy, we
hypothesize that the summarization model will receive an improved training
signal from documents that are more representative of the cluster topic. By
masking the central-most document instead of a random document, we avoid
learning to predict diverging documents comprised of irrelevant information.
Contrary to auto-encoding based objectives that learn to reconstruct each
input document (Chu and Liu, 2019), our objective encourages overlapping
information to be used to predict a target document it does not see as input.
We discuss overlap metrics in the next subsection and outline the candidate
mask selection procedure in Algorithm 1.

###  2.2 Measuring Document Overlap

Ultimately, since Rouge, a measure based on $n$-gram recall, is
summarization’s quantitative evaluation measure, we choose to use $n$-gram
recall as our overlap metric too. Due to both its simplicity and
effectiveness, $n$-gram overlap measures have historically been a mainstay for
centroid-based extractive approaches (Radev et al., 2004; Erkan and Radev,
2004; Mihalcea and Tarau, 2004), which measure the lexical overlap between
documents of the same topic. We empirically investigate using either the
unigram or bigram recall between each document in a cluster and the rest of
the combined documents in the cluster for the document distance metric,
referred to as $\mathcal{O}$ in Algorithm 1. The document with the highest
score is selected to be masked out and predicted.

##  3 Related Work

Non-neural unsupervised MDS has been approached from both extractive
(Carbonell and Goldstein, 1998; Gillick and Favre, 2009; Haghighi and
Vanderwende, 2009) and abstractive (McKeown and Radev, 1995; Radev and
McKeown, 1998; Barzilay et al., 1999; Ganesan et al., 2010) angles. At a high
level, extractive methods compute a heuristic ranking of source segments and
compose them together. Centroid-based extractive summarization ranks a graph
representation of the input document(s), where edges are drawn between
segments with similar meanings (Radev et al., 2004; Erkan and Radev, 2004;
Mihalcea and Tarau, 2004). Instead of computing centroid-based score functions
to produce an output, in our approach we will use them to select a masked
document candidate as an unsupervised objective for training a neural
abstractive summarization system.

Supervised neural abstractive single-document summarization has received
increased interest lately (Rush et al., 2015; See et al., 2017; Gehrmann et
al., 2018) despite its many shortcomings (Kryściński et al., 2019), such as a
reliance on large human-annotated datasets (Napoles et al., 2012; Hermann et
al., 2015). Additionally, supervised abstractive MDS models have also been
proposed alongside new datasets (Liu and Lapata, 2019; Liu et al., 2018;
Fabbri et al., 2019). Our unsupervised masking objective enables us to train
any of the above neural methods discriminatively without ground truth
summaries.

Research into unsupervised neural abstractive multi-document summarization is
scarcer. Lebanoff et al. (2018) and Zhang et al. (2018) transferred single-
document summarization models to MDS, which required no supervised multi-
document data. Chu and Liu (2019)’s recently proposed MeanSum, which learns to
summarize groups of reviews from a mean-pooled document representation via an
auto-encoder objective, represents the most related approach to our own
method.

##  4 Experiments

###  4.1 Dataset

We use the Multi-News dataset for training and evaluation since it contains
supervised summaries for over 125k total news articles in 50k multi-document
clusters. Following Fabbri et al. (2019), we concatenate each document cluster
into a single 500 token mega-document separated by special document separator
tokens using the same strategy: for each cluster, the first $500/M$ tokens are
added from each of the $M$ documents, with additional tokens being added
iteratively thereafter.111In practice, we also truncate masked documents to
300 tokens for efficiency.

#### Cleaning

Most automatically collected summarization datasets are noisy (Kryściński et
al., 2019) and Multi-News is no exception. We remove articles less than 100
characters, nonsensical articles containing links to tweets, and
duplicated/syndicated articles by thresholding 4-gram overlap. Next, we find
and delete repeated phrases like captions by computing $n$-gram similarity
between sentences within a document. Finally, we use regular expressions to
remove article metadata. This reduces the dataset from 45K / 5.6K / 5.6K train
/ valid / test clusters to 42K / 5.2K / 5.2K.

###  4.2 Unsupervised Baselines

We compare to multiple popular unsupervised extractive approaches222Methods
are configured to output similar length summaries to our system. like Fabbri
et al. (2019) along with a recent neural unsupervised approach.

#### Lede-$k$

We concatenate the first $k$ sentences from each article cluster together to
form the summary, which has proven to be a straightforward, but effective
baseline for summarization systems (See et al., 2017).

#### LexRank

Erkan and Radev (2004)’s LexRank computes sentence importance via eigenvector
centrality in a graph representation of the sentences in each multi-document
cluster.

#### TextRank

TextRank (Mihalcea and Tarau, 2004) is another graph-based method that ranks
sentences for extraction, originally proposed for single-document
summarization.

#### MMR

Maximal Marginal Relevance (Carbonell and Goldstein, 1998) ranks candidate
sentences based on a balance between relevance and redundancy. Top-ranked
sentences are appended to form an extractive summary.

#### MeanSum

MeanSum333https://github.com/sosuperic/MeanSum is a completely unsupervised
method introduced by Chu and Liu (2019) for summarizing Yelp and Amazon
reviews. It is trained to simultaneously auto-encode reviews and produce a
summary that is semantically similar to the input reviews using their mean-
pooled representation. We remove their review rating classifier and adapt it
to the Multi-News dataset.

#### Source Doc with Best Overlap

We simply return the source document with the highest bigram overlap (i.e.,
the document that would be masked out and predicted). This tests how effective
our model is at fusing the information between source documents as opposed to
just generating a new, separate document on the same topic.

Output Length: | 100 Words | 150 Words | 200 Words |
---|---|---|---|---
Method Metric: | R-1 | R-2 | R-1 | R-2 | R-1 | R-2 |  $\mu$ len $\pm$ $\sigma$
Unsupervised Baselines |  |  |
Bigram Overlap Src Doc | 37.35 | 11.39 | 41.02 | 13.08 | 43.29 | 14.25 | 294 $\pm$ 26
Random Src Doc | 36.74 | 11.33 | 39.68 | 12.70 | 40.81 | 13.32 | 203 $\pm$ 71
Lede-3 | 36.84 | 11.15 | 39.73 | 12.14 | 40.83 | 12.54 | 212 $\pm$ 84
Lede-4 | 36.95 | 11.28 | 40.32 | 12.55 | 42.11 | 13.23 | 269 $\pm$ 94
Lede-5 | 36.98 | 11.34 | 40.47 | 12.74 | 42.56 | 13.56 | 318 $\pm$ 98
LexRank | 35.30 | 10.28 | 38.85 | 11.93 | 40.89 | 12.85 | 246 $\pm$ 15
TextRank | 36.29 | 11.10 | 39.49 | 12.44 | 41.33 | 13.23 | 244 $\pm$ 16
MMR | 35.62 | 9.35 | 39.89 | 11.48 | 42.57 | 12.91 | 237 $\pm$ 17
MeanSum | 24.88 | 3.03 | 27.16 | 3.66 | 28.48 | 4.06 | 288 $\pm$ 129
Unsupervised Mask Objective |  |  |
Hi-MAP _Random_ | 36.12 | 10.83 | 40.13 | 12.90 | 42.52 | 14.09 | 233 $\pm$ 53
Hi-MAP _Unigram_ | 37.25 | 11.77 | 41.05 | 13.61 | 43.19 | 14.61 | 243 $\pm$ 56
Hi-MAP _Bigram_ | 37.21 | 11.48 | 41.01 | 13.33 | 43.35 | 14.54 | 247 $\pm$ 53
Hi-MAP _Random_ $\geq 3$ | 36.68 | 11.15 | 40.58 | 13.11 | 42.47 | 13.99 | 213 $\pm$ 46
Hi-MAP _Unigram_ $\geq 3$ | 37.03 | 11.28 | 41.00 | 13.31 | 43.28 | 14.47 | 256 $\pm$ 58
Hi-MAP _Bigram_ $\geq 3$ | 37.58 | 11.60 | 41.45 | 13.63 | 43.75 | 14.82 | 247 $\pm$ 52
Supervised |  |  |
Hi-MAP | 38.47 | 12.33 | 42.15 | 13.98 | 44.47 | 15.09 | 213 $\pm$ 26
Hi-MAP $\geq 3$ | 37.51 | 11.88 | 41.69 | 14.03 | 43.86 | 15.12 | 199 $\pm$ 35
Table 1: Limited-length Rouge scores at different output lengths on the Multi-
News dataset (Fabbri et al., 2019). The outputs from each model were truncated
to meet the length requirement. The distribution of output lengths before
truncation for each model is shown in the right-hand column. Results are
divided into unsupervised baselines, supervised baselines, and our proposed
unsupevised masked objective. The best unsupervised result in each column is
bolded. Models with ‘$\geq 3$’ in their name indicate that they were only
trained on document collections that contained at least three input documents.
‘Bigram Overlap Src Doc’ is a baseline that simply uses the truncated masking
candidate as the output document. ‘Random Src Doc’ selects a random input
document.

###  4.3 Neural Abstractive Summ. Model

While any neural multi-document abstractive summarization model could be used
(e.g., Liu and Lapata (2019); Liu et al. (2018)), we use the Hierarchical MMR-
Attention Pointer-generator network (Hi-MAP) model (Fabbri et al., 2019)
because it was developed on the Multi-News dataset.444https://github.com/Alex-
Fabbri/Multi-News As the name implies, the Hi-MAP model is composed of a
hierarchical pointer-generator network (See et al., 2017) that has an
additional attention mechanism over MMR-ranked (Carbonell and Goldstein, 1998)
input sentences. Thus, sentence-level extractive summarization scores
balancing relevancy and redundancy can be effectively combined with a commonly
used single document neural summarization model capable of learning when to
produce new output or copy from the source vocabulary. We refer the reader to
Fabbri et al. (2019) for hyperparameter settings and implementation details.

##  5 Results

#### Automatic Evaluation

In Table 1, we report Rouge-{1, 2} (Lin, 2004) F-scores for hypotheses555We
use beam search with beam size 4, block repeated trigrams, and apply a length
penalty with $\alpha=0.9$ & $\beta=5$ (Wu et al., 2016). limited to 100, 150,
and 200 words to better compare among uncontrollable neural output lengths
(pointed out in (Sun et al., 2019)), along with their untruncated average
lengths on the test set summaries from the Multi-News dataset. Unsupervised
methods are purely unsupervised since we use only the masked source documents
for the validation set for early stopping. Since clusters range in size from
2–10 documents in Multi-News, we also experiment with filtering our training
set by number of documents per cluster to see if it affects the choice of
masks for each unsupervised method. We report on both the full training set
and the other best training set which excludes all 2 document clusters
(denoted $\geq 3$ in the table). While Rouge-2 performance on $\geq 3$ is
slightly better for the supervised system, filtering out 2-article clusters
gives statistically significant gains between models trained with the
unsupervised objective. Allowing the unsupervised objective to choose the
article to mask and predict among $\geq 3$ articles is clearly important,
which is also suggested by the increase in the difference between scores when
excluding 2-article clusters from training with a randomly selected masked
document and the bigram overlap mask. We perform a paired bootstrap test on
Rouge-{1,2} 200 scores and find that our best method is significantly better
than the baselines ($p\ll 0.05$) and not significantly different from the best
supervised model on Rouge-1 ($p=0.1$). We attribute MeanSum’s poor performance
to it losing supervision from its review rating classifier and the subtle task
differences between its original domain (Yelp reviews) and the news domain.

Evaluation Setting | Info. | Fluency | Non-Red.
---|---|---|---
Ours vs. MeanSum | 99% | 99% | 95%
Ours vs. Sup$\geq 3$ | 59% | 53% | 50%
Table 2: Combined human evaluator preferences for our best system (Bigram$\geq
3$) compared to MeanSum and Supervised.

#### Human Evaluation

We ask two native English speaking human annotators to evaluate 50 randomly
sampled test document clusters and their summaries shuffled between our best
bigram method and MeanSum, and our best method and the best supervised Hi-MAP.
Annotators are shown the input documents and either pair of summaries shown
above and asked to choose which is best and which is worst (Louviere and
Woodworth, 1991) for informativeness, fluency, and non-redundancy, in line
with past work (Fabbri et al., 2019). Annotators almost always selected our
method over MeanSum. Surprisingly, annotators also ranked our method better
than the supervised Hi-MAP system in informativeness (58%) and fluency (53%).

##  6 Conclusion

The unsupervised masking method achieves comparable performance with
supervised abstractive summarizers. Further, the masking candidate itself
proves to be a simple and performant baseline. Similar masking objectives for
long-form text generation might find applications in related domains.

## References

  * Barzilay et al. (1999) Regina Barzilay, Kathleen R McKeown, and Michael Elhadad. 1999.  Information fusion in the context of multi-document summarization.  In _Proceedings of the 37th Annual meeting of the Association for Computational Linguistics_ , pages 550–557.
  * Carbonell and Goldstein (1998) Jaime G Carbonell and Jade Goldstein. 1998.  The use of mmr, diversity-based reranking for reordering documents and producing summaries.  In _SIGIR_ , volume 98, pages 335–336.
  * Chu and Liu (2019) Eric Chu and Peter Liu. 2019.  Meansum: a neural model for unsupervised multi-document abstractive summarization.  In _International Conference on Machine Learning_ , pages 1223–1232.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_.
  * Erkan and Radev (2004) Günes Erkan and Dragomir R Radev. 2004.  Lexrank: Graph-based lexical centrality as salience in text summarization.  _Journal of artificial intelligence research_ , 22:457–479.
  * Fabbri et al. (2019) Alexander Fabbri, Irene Li, Tianwei She, Suyi Li, and Dragomir Radev. 2019.  Multi-news: A large-scale multi-document summarization dataset and abstractive hierarchical model.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 1074–1084, Florence, Italy. Association for Computational Linguistics.
  * Ganesan et al. (2010) Kavita Ganesan, ChengXiang Zhai, and Jiawei Han. 2010.  Opinosis: A graph based approach to abstractive summarization of highly redundant opinions.  In _Proceedings of the 23rd International Conference on Computational Linguistics (Coling 2010)_ , pages 340–348.
  * Gehrmann et al. (2018) Sebastian Gehrmann, Yuntian Deng, and Alexander Rush. 2018.  Bottom-up abstractive summarization.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 4098–4109, Brussels, Belgium. Association for Computational Linguistics.
  * Gillick and Favre (2009) Dan Gillick and Benoit Favre. 2009.  A scalable global model for summarization.  In _Proceedings of the Workshop on Integer Linear Programming for Natural Langauge Processing_ , pages 10–18. Association for Computational Linguistics.
  * Haghighi and Vanderwende (2009) Aria Haghighi and Lucy Vanderwende. 2009.  Exploring content models for multi-document summarization.  In _Proceedings of Human Language Technologies: The 2009 Annual Conference of the North American Chapter of the Association for Computational Linguistics_ , pages 362–370. Association for Computational Linguistics.
  * Hermann et al. (2015) Karl Moritz Hermann, Tomáš Kočiský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. 2015.  Teaching machines to read and comprehend.  In _Advances in Neural Information Processing Systems (NIPS)_.
  * Kryściński et al. (2019) Wojciech Kryściński, Nitish Shirish Keskar, Bryan McCann, Caiming Xiong, and Richard Socher. 2019.  Neural text summarization: A critical evaluation.  _arXiv preprint arXiv:1908.08960_.
  * Lebanoff et al. (2018) Logan Lebanoff, Kaiqiang Song, and Fei Liu. 2018.  Adapting the neural encoder-decoder framework from single to multi-document summarization.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 4131–4141, Brussels, Belgium. Association for Computational Linguistics.
  * Lin (2004) Chin-Yew Lin. 2004.  Rouge: A package for automatic evaluation of summaries.  In _Text summarization branches out_ , pages 74–81.
  * Liu et al. (2018) Peter J Liu, Mohammad Saleh, Etienne Pot, Ben Goodrich, Ryan Sepassi, Lukasz Kaiser, and Noam Shazeer. 2018.  Generating wikipedia by summarizing long sequences.  _arXiv preprint arXiv:1801.10198_.
  * Liu and Lapata (2019) Yang Liu and Mirella Lapata. 2019.  Hierarchical transformers for multi-document summarization.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 5070–5081, Florence, Italy. Association for Computational Linguistics.
  * Louviere and Woodworth (1991) Jordan J Louviere and George G Woodworth. 1991.  Best-worst scaling: A model for the largest difference judgments.  _University of Alberta: Working Paper_.
  * McKeown and Radev (1995) Kathleen McKeown and Dragomir R Radev. 1995.  Generating summaries of multiple news articles.  _Proceedings, ACM Conference on Research and Development in Information Retrieval SIGIR’95_.
  * Mihalcea and Tarau (2004) Rada Mihalcea and Paul Tarau. 2004.  Textrank: Bringing order into text.  In _Proceedings of the 2004 conference on empirical methods in natural language processing_ , pages 404–411.
  * Napoles et al. (2012) Courtney Napoles, Matthew Gormley, and Benjamin Van Durme. 2012.  Annotated Gigaword.  In _Proceedings of the Joint Workshop on Automatic Knowledge Base Construction and Web-scale Knowledge Extraction (AKBC-WEKEX)_ , pages 95–100, Montréal, Canada. Association for Computational Linguistics.
  * Owczarzak and Dang (2011) Karolina Owczarzak and Hoa Trang Dang. 2011.  Overview of the tac 2011 summarization track: Guided task and aesop task.  In _Proceedings of the Text Analysis Conference (TAC 2011), Gaithersburg, Maryland, USA, November_.
  * Paul and James (2004) Over Paul and Yen James. 2004.  An introduction to duc-2004.  In _Proceedings of the 4th Document Understanding Conference (DUC 2004)_.
  * Radev et al. (2004) Dragomir R Radev, Hongyan Jing, Małgorzata Styś, and Daniel Tam. 2004.  Centroid-based summarization of multiple documents.  _Information Processing & Management_, 40(6):919–938.
  * Radev and McKeown (1998) Dragomir R Radev and Kathleen R McKeown. 1998.  Generating natural language summaries from multiple on-line sources.  _Computational Linguistics_ , 24(3):470–500.
  * Rush et al. (2015) Alexander M. Rush, Sumit Chopra, and Jason Weston. 2015.  A neural attention model for abstractive sentence summarization.  In _Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing_ , pages 379–389, Lisbon, Portugal. Association for Computational Linguistics.
  * See et al. (2017) Abigail See, Peter J. Liu, and Christopher D. Manning. 2017.  Get to the point: Summarization with pointer-generator networks.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1073–1083, Vancouver, Canada. Association for Computational Linguistics.
  * Sun et al. (2019) Simeng Sun, Ori Shapira, Ido Dagan, and Ani Nenkova. 2019.  How to compare summarizers without target length? pitfalls, solutions and re-examination of the neural summarization literature.  In _Proceedings of the Workshop on Methods for Optimizing and Evaluating Neural Language Generation_ , pages 21–29.
  * Sutskever et al. (2014) I Sutskever, O Vinyals, and QV Le. 2014.  Sequence to sequence learning with neural networks.  _Advances in NIPS_.
  * Wu et al. (2016) Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. 2016.  Google’s neural machine translation system: Bridging the gap between human and machine translation.  _arXiv preprint arXiv:1609.08144_.
  * Zhang et al. (2018) Jianmin Zhang, Jiwei Tan, and Xiaojun Wan. 2018.  Adapting neural single-document summarization model for abstractive multi-document summarization: A pilot study.  In _Proceedings of the 11th International Conference on Natural Language Generation_ , pages 381–390.


# Resonant subwavelength control of the phase of spin waves reflected from a
ferromagnetic film edge

Krzysztof Sobucki  Faculty of Physics, Adam Mickiewicz University,
Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Paweł Gruszecki  Institute
of Molecular Physics, Polish Academy of Sciences, Mariana Smoluchowskiego 17,
60-179 Poznań, Poland  Faculty of Physics, Adam Mickiewicz University,
Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Justyna Rychły  Institute
of Molecular Physics, Polish Academy of Sciences, Mariana Smoluchowskiego 17,
60-179 Poznań, Poland  Maciej Krawczyk  Faculty of Physics, Adam Mickiewicz
University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Wojciech
Śmigaj  Met Office, FitzRoy Rd, Exeter, EX1 3PB, UK

(August 27, 2024)

###### Abstract

Using frequency-domain finite element calculations cross-checked with
micromagnetic simulations, we demonstrate that the phase of spin waves
reflected from an interface between a permalloy film and a bilayer can be
controlled changing dimensions of the bilayer. Treating the bilayer formed by
the permalloy film and a ferromagnetic stripe as a segment of a multi-mode
waveguide, we show that spin-wave Fabry-Perot resonances of one of its modes
are responsible for the high sensitivity of the phase of reflected waves to
stripe width and the stripe-film separation. The bilayer can be treated as a
magnonic counterpart of a metasurface since it enables manipulation of the
phase of spin waves at subwavelength distances. Knowledge gained from these
calculations might be used to design magnonic devices such as flat lenses for
spin waves or detectors based on surface plasmon resonances phenomenon.

††preprint: APS/123-QED

##  I Introduction

The recent years have been marked by a rapidly growing demand for
interconnected mobile devices. This emerging ecosystem of connected devices,
preferably communicating wirelessly, is referred to as the Internet of Things.
There are estimations that within the next few years, the number of WiFi-
enabled devices will be at least four times larger than the total population
of the world [1]. One of the essential components of the Internet of Things
are small and energetically efficient devices processing signals converted
from and then back to microwaves. In this field, the application of spin waves
(SWs), which are collective disturbances of magnetization propagating in the
same frequence range as microwaves and thus able to couple to them, opens up a
new opportunity to increase the efficiency of these devices. Compared to
existing microwave devices, SW components offer prospects for increased
miniaturization (SWs have wavelengths 3–5 orders of magnitude shorter than
microwaves of the same frequency), easy external control of SW signals,
reprogrammability, and small energy demands due to lack of Joule heating
related with SWs propagation [2, 3, 4].

In order to use any kind of waves as an information carrier, efficient methods
of their excitation and control over their amplitude and phase must be
developed. In modern photonics, arrays of nanostructured antennas absorbing
and reemitting modified electromagnetic waves have recently made a
breakthrough in control the phase (and amplitude) of reflected or transmitted
waves at subwavelength distances [5, 6]. These arrays of nanoantennas, called
metasurfaces, are used to obtain anomalous refraction of incident waves or to
design flat, ultra-narrow lenses able to focus waves. Moreover, such
nanostructured antennas can serve as color filters that can be used to produce
silicon- or metallic-based color pixels for printing purposes as a replacement
of chemical dyes [7, 8].

There are several reports on the coupling of small magnetic elements with an
uniform film. Kruglyak et al. have demonstrated that a narrow ferromagnetic
element placed on top of a magnetic waveguide can be used to emit SWs, to
control the phase of SWs passing below the resonator, and under some
conditions even to absorb the energy of propagating SWs [9, 10, 11]. Yu at al.
have demonstrated the chiral excitation of SWs in the thin film through the
dipolar coupling with a single nanowire and the grating of nanowires excited
by a spatially uniform microwave-frequency magnetic field [12, 13].
Subsequently, the existence of Fano resonances and their influence on the
amplitude and phase of transmitted waves in a single-mode waveguide has been
further studied by Al et al. [14], and Zhang et al. have demonstrated the
application of a single dynamically tunable resonator in zero bias field
placed on top of a waveguide to tune the phase of the transmitted SWs [15]. A
grating coupler made up of an array of resonators has been used to excite
short-wavelength SWs [16, 17], and Graczyk et al. [18] have demonstrated that
dynamical coupling of a homogeneous ferromagnetic film with a periodic array
of ferromagnetic stripes placed underneath can lead to the formation of a
magnonic band structure in the film. However, the effect of the resonator on
the phase of the reflected or transmitted wave has not yet been studied in
magnonics, moreover, the conditions for the existence of Fabry-Perot
resonances and their effect on the SW remain almost unexplored [19], while
both may be key to creating a magnonic metasurface.

In this work, we investigate theoretically the interaction of a narrow,
subwavelength-width stripe placed above the edge of the homogeneously
magnetized film supporting propagating SWs and its influence on the phase of
reflected SWs. Using frequency-domain finite-element calculations and
micromagnetic simulations we find that this shift depends on the width of the
stripe in a non-trivial way: an overall slow and steady increase of the phase
shift with stripe width is repeatedly interrupted by sharp phase jumps by
360$\degree$. Treating the stripe and the underlying film as a non-reciprocal
waveguide supporting two pairs of counter-propagating modes, we formulate a
semi-analytical model that explains this behavior as a consequence of Fabry-
Perot resonances produced by one of these mode pairs. We also show that by
varying the film-stripe separation it is possible to switch between resonances
of different order. Our results point to the importance of Fabry-Perot
resonances appearing in locally bilayered ferromagnetic elements, with
potential applications for the control of SW propagation in magnonic devices.

The paper is organized as follows. In Sec. II, we present the geometry of the
system under consideration and the numerical methods used in our study. In
Secs. III.1–III.3, we analyze the key physical effects occurring in individual
components of the system of interest: SW dynamics in infinitely extended films
and bilayers, SW resonances in finite-width stripes and SW reflection from
edges of truncated films. In Secs. III.4–III.6, which form the core of this
work, we investigate SW reflection at the end of a film coupled to a finite-
width stripe. We show how interactions between parts of the system discussed
in the previous subsections make the phase shift of the reflected wave highly
sensitive to stripe width and the stripe-film separation. Our results are
summarized in Sec. IV.

##  II Structure and methods

###  II.1 Structure under consideration

Figure 1: Geometry of the system used in simulations. A ferromagnetic stripe
of width $w$ and thickness $d_{2}$ is separated from a semi-infinite permalloy
film of thickness $d_{1}$ by a distance $s$. The right edges of both layers
are aligned at $x=w$. The whole system is placed in vacuum in an external
magnetic field $\mu_{0}H_{0}=0.1$ parallel to the $y$ axis. The geometry of
the system is independent from $y$.

We consider a system composed of non-magnetic and ferromagnetic materials. Its
geometry is independent of the $y$ coordinate and piecewise constant along
$x$, as shown schematically in Fig. 1. The system consists of a semi-infinite
permalloy (Py) film of thickness 50 nm and a ferromagnetic stripe of thickness
40 nm and finite width $w$. Both elements are separated by a distance $s$ and
their right edges are aligned to $x=w$. Throughout the paper, we will vary the
width $w$ of the stripe and its separation $s$ from the film. We are
interested in manipulating the phase of the reflected SWs using subwavelength
elements; therefore the width of the stripe will be smaller than or comparable
to the wavelength of SWs in the Py film at the frequency of operation. The
system is magnetized by a uniform in-plane bias magnetic field of magnitude
$\mu_{0}H_{0}=0.1$ T directed along the $y$ axis. In all calculations we have
taken the saturation magnetization of the film to be $M_{\text{S}}=760$ kA/m
and its exchange constant, $A=13$ pJ/m. The stripe is made of a material
(called FM2 from here on) with $M_{\text{S}}=525$ kA/m and $A=30$ pJ/m;
lowering its saturation magnetization and increasing the exchange constant
with respect to Py will allow us to exploit interactions of local resonances
of the stripe with propagating waves in Py at frequencies characteristic for
the dipolar and dipole-exchange SWs. Such a choice of the stripe’s material
shifts the SW spectrum down and flattens the first band for lower wavevectors
with respect to the Py, as will be discussed in the next section. The
gyromagnetic ratio of both ferromagnets is $\gamma=-176$ rad GHz/T.

###  II.2 Governing equations

The magnetization dynamics is described by the Landau-Lifshitz equation:

$\partial_{t}\mathbf{M}=-\frac{|\gamma|\mu_{0}}{1+\alpha^{2}}\left[\mathbf{M}\times\mathbf{H}_{\mathrm{eff}}+\frac{\alpha}{M_{\mathrm{S}}}\mathbf{M}\times(\mathbf{M}\times\mathbf{H}_{\mathrm{eff}})\right],$
(1)

where $\mathbf{M}$ is the magnetization vector, $\mu_{0}$ is the permeability
of vacuum, $\alpha$ is a dimensionless damping parameter, and

$\mathbf{H}_{\mathrm{eff}}=\mathbf{H}_{0}+\mathbf{H}_{\mathrm{m}}+\mathbf{H}_{\mathrm{ex}}$
(2)

is the effective magnetic field. The latter is the sum of the external
magnetic field $\mathbf{H}_{0}$, the magnetostatic field
$\mathbf{H}_{\mathrm{m}}$, and the isotropic Heisenberg exchange field
$\mathbf{H}_{\mathrm{ex}}=\boldsymbol{\nabla}\cdot(l^{2}\boldsymbol{\nabla}\mathbf{M})$,
where $l=\sqrt{2A/(\mu_{0}M_{\mathrm{S}}^{2})}$ is the exchange length. The
magnetostatic field fulfils the magnetostatic Maxwell’s equations

$\displaystyle\boldsymbol{\nabla}\cdot(\mathbf{M}+\mathbf{H}_{\text{m}})$
$\displaystyle=0,$ (3a)
$\displaystyle\boldsymbol{\nabla}\times\mathbf{H}_{\text{m}}$
$\displaystyle=0;$ (3b)

the latter makes it possible to write it as
$\mathbf{H}_{\text{m}}=-\boldsymbol{\nabla}\varphi$, where $\varphi$ is the
magnetic scalar potential.

Assuming a harmonic time dependence [$\exp(-\mathrm{i}\omega t$)], zero
damping ($\alpha=0$) and alignment of the external magnetic field $H_{0}$ with
the $y$ axis, splitting the magnetization $\mathbf{M}$ and magnetostatic field
$\mathbf{H}_{\text{m}}$ into static ($M_{\text{S}}\hat{y},H_{0}\hat{y}$) and
dynamic (radio-frequency) components ($\mathbf{m}=[m_{x},0,m_{z}]$,
$\mathbf{h}_{\mathrm{m}}=[-\partial_{x}\varphi,0,-\partial_{z}\varphi]$),
linearizing the Landau-Lifshitz equation (1) (applicable only in the
ferromagnetic layers) and coupling it with the Gauss law for magnetism, Eq.
(3a) (applicable everywhere), we arrive at the following system of equations
for the magnetic potential $\varphi$ and the dynamic magnetization component
$\mathbf{m}$:

$\displaystyle\partial_{x}(m_{x}-\partial_{x}\varphi)+\partial_{z}(m_{z}-\partial_{z}\varphi)$
$\displaystyle=0,$ (4a)
$\displaystyle\partial_{x}\varphi-\boldsymbol{\nabla}\cdot(l^{2}\boldsymbol{\nabla}m_{x})+\frac{H_{0}}{M_{S}}m_{x}+\frac{\mathrm{i}\omega}{|\gamma|\mu_{0}M_{S}}m_{z}$
$\displaystyle=0,$ (4b)
$\displaystyle\partial_{z}\varphi-\boldsymbol{\nabla}\cdot(l^{2}\boldsymbol{\nabla}m_{z})+\frac{H_{0}}{M_{S}}m_{z}-\frac{\mathrm{i}\omega}{|\gamma|\mu_{0}M_{S}}m_{x}$
$\displaystyle=0.$ (4c)

###  II.3 Numerical methods

We have used three complementary numerical methods to study SW dynamics.
First, we use micromagnetic simulations performed in the open-source
environment mumax3 [20], which solves the full Landau-Lifshitz equation [Eq.
(1)] with the finite-difference time-domain (FDTD) method. We use this method
to calculate the dispersion relations of SWs and steady states obtainable
after long continuous excitation of SWs by a specified source. More details
can be found in Appendix A. The main disadvantage of this method is its high
computational cost. Resonant systems can take a long time to reach steady
state, and the cost of a single time step is pushed up by the need to
discretize the whole system on a uniform grid whose resolution is dictated by
the size of the smallest geometric features.

In order to avoid these limitations, we rely primarily on calculations using
the frequency-domain finite element method (FD-FEM). Its major advantage is
the possibility of refining the mesh locally, e.g. only around small geometric
features, rather than globally. In addition, it allows direct and fast
calculation of the eigenfrequencies and eigenmodes (mode profiles) of the
system, which can be identified with its steady states. To perform the
calculations, we have used the COMSOL Multiphysics software to solve the
linearized Landau-Lifshitz equation coupled with the Gauss law, Eqs. (4), as
described in [21]. At the edges of the computational domain (far from the
ferromagnetic materials) the Dirichlet’s boundary conditions, forcing the
magnetic potential to vanish, are imposed.

In Sec. III.5 we formulate a semi-analytical model dependent on the numerical
values of scattering matrices associated with interfaces separating SW
waveguides with different cross-sections. To calculate these matrices, we used
the finite element modal method. In this method, Eq. (4) is transformed into
an eigenproblem whose solutions are the wavenumbers and profiles of
propagative and evanescent modes of a particular $x$-invariant part of the
system shown in Fig. 1, e.g. the Py film or the Py/FM2 bilayer. This
eigenproblem is then discretized by expanding the magnetic potential and
dynamic magnetization in a finite-element basis and solved numerically. Fields
on each side of the interface are represented as superpositions of the
eigenmodes of the respective parts calculated in the previous step. Finally,
the scattering matrix is obtained by imposing the standard continuity
conditions on these fields and solving the resulting system of equations for
the excitation coefficients of outgoing modes. This method, inspired by
similar techniques used in photonics [22], will be described in detail in a
forthcoming paper [23].

##  III Results

###  III.1 Dispersion relation of bilayers

Figure 2:  (a)–(b) Dispersion relations calculated for two infinite films made
of Py and FM2 separated by non-magnetic gap of width (a) 200 nm and (b) 10 nm.
Colormaps in the background present results obtained by means of micromagnetic
simulations, whereas green points correspond to the results of FD-FEM
calculations. The dashed blue and the dash-dotted red lines are the analytical
dispersion relations of SWs supported by isolated FM2 and Py films,
respectively. The horizontal line marks the frequency $f=11$ GHz used in
further calculations. (c) Dependence of the wavenumber of the slow modes, d
and g, located predominantly in FM2, on the separation between films at
frequency 11 GHz. (d)–(g) Mode profiles of $m_{z}$: (d), (g) slow and (e)–(f)
fast modes, in the Py/FM2 bilayer separated by a 10 nm non-magnetic gap at
frequency 11 GHz [marked in (b)].

Before we study the coupling of a film with a finite-width stripe, let us
first analyze the interaction between modes in infinitely extended films,
i.e., in bilayers composed of an infinite Py film separated by a non-magnetic
gap from another infinite film made of FM2. Two example dispersion relations
for gap widths $s=200$ nm and $s=10$ nm are presented in Fig. 2(a) and (b),
respectively. It is clear that for a 200 nm-wide gap, the SW modes in Py and
stripe almost do not interact with each other: the calculated dispersion curve
coincides with the analytical dispersion curves of isolated Py and FM2 films
calculated using [24, Appendix C.7]:

$\omega^{2}=\omega_{0}\left(\omega_{0}+\omega_{\mathrm{M}}\right)+\frac{\omega_{\mathrm{M}}^{2}}{4}\left[1+\mathrm{e}^{-2kd}\right],$
(5)

where $d$ is the film thickness, $\omega=2\pi f$ is the angular frequency of
SWs ($f$ denotes the frequency), $k$ is the wavenumber,
$\omega_{0}=|\gamma|\mu_{0}(H_{0}+M_{\mathrm{S}}l^{2}k^{2})$111We have
included to the formula from Ref. [24, Appendix C.7] the impact of the
exchange interaction on dispersion by
$\omega_{0}\rightarrow\omega_{0}+M_{\mathrm{S}}l^{2}k^{2}$, and
$\omega_{\mathrm{M}}=|\gamma|\mu_{0}M_{\mathrm{S}}$.

The only visible difference occurs at frequencies above 14 GHz where we can
see a hybridization between the fundamental SW mode and the first mode
quantized across the thickness, a so-called perpendicular standing SW [24,
Appendix C.7]. This hybridization and perpendicular standing SWs, however, are
not considered in the analytical model.

At frequencies below 14 GHz the modes in bilayered structure can be classified
according to their origin and group velocity. The fast modes are related to SW
dynamics in Py [see Fig. 2(e) and (f)] and are characterized by steeper
dispersion (therefore higher group velocity) and longer wavelengths. In
contrast, the slow modes originating in FM2 [see Fig. 2(d) and (g)] have lower
group velocity and shorter wavelengths. It is worth noting that the
wavelengths of SWs in separated layers do not depend on the direction of
propagation, while the dynamic dipolar coupling between SW modes in both
layers combined with the nonreciprocal nature of surface SW modes introduces
asymmetry [26, 27, 28]. For $s=200$ nm the coupling is still very weak and at
the frequency of 11 GHz that is used in further analysis and marked with the
white horizontal line in Fig. 2(a), the fast and slow modes have wavelengths
of 2660 nm and 590 nm, respectively, for both propagation directions.

Reduction of the non-magnetic gap width to $s=10$ nm causes the modes of the
individual films to interact much more strongly. The wavelength of the slow
modes decreases significantly and their dispersion relation becomes strongly
nonreciprocal; at 11 GHz, slow modes propagating leftwards have wavelength 390
nm and those propagating rightwards, 270 nm. Figure 2(c) shows that the
asymmetry of the dispersion relation for slow modes grows as the films are
brought closer together.

###  III.2 Eigenmodes of finite-width stripes

The dependence of the eigenmode spectrum of isolated finite-width FM2 stripe
on their width is displayed in Fig. 3(a). These calculations were made with
FD-FEM for the first thirty modes of stripes with widths up to 2800 nm (note
that SW wavelength in uniform Py films at 11 GHz is 2660 nm). As intuitively
expected, mode frequency decreases with increasing stripe width. The
horizontal dashed line in Fig. 3 marks $f=11$ GHz. It is visible that stripes
of multiple widths support modes at this frequency. Profiles of three
successive modes (for stripes of width 656 nm, 936 nm, and 1200 nm) are shown
in Fig. 3(b)–(d). Successive resonances appear for stripes of widths differing
by approximately half of the wavelength of the eigenmode of a homogeneous FM2
film, i.e., $\Delta w\approx 280\;\mathrm{nm}\approx
0.5\lambda_{\mathrm{FM2}}$.

Lateral mode confinement in a reciprocal medium leads to formation of standing
waves and quantization of the wavenumber. The standing waves have the form
$\exp(\mathrm{i}k_{n}x)+\exp(-\mathrm{i}k_{n}x)$, where $n$ is the mode index,
$k_{n}=r_{n}\pi/w$ and $r_{n}=n+\delta$ ($0\leq\delta\leq 1$). For the
Dirichlet (magnetic wall) boundary conditions, with the dynamic magnetization
vanishing at the edges, we get $r_{n}=n+1$, whereas for “free spins” at the
edges, $r_{n}=n$. However, due to dipolar interactions, in magnetic stripes
neither of these cases is correct and magnetization is partially pinned at the
stripe edges, $0<\delta<1$ [29]. This can also be interpreted as the effective
length of the waveguide being slightly larger than the real one, or in terms
of a non-zero phase shift $\varphi$ being experienced at the stripe edges by
the SWs forming the standing wave. Thus, resonances occur when the following
condition is met:

$kw+\varphi=\pi n,\quad n=1,2,\dots$ (6)

According to this equation, successive resonances at frequency 11 GHz should
appear for stripes of widths differing by ca. $\lambda/2\approx 280$ nm; this
is confirmed by the FD-FEM calculations.

Figure 3: (a) Dependence of the frequency of stripe modes on stripe width. The
dashed line marks the frequency $f=11$ GHz. (b), (c), and (d) Profiles of
modes supported by stripes of width 656 nm, 936 nm, and 1216 nm at frequency
11 GHz.  Figure 4:  (a) Incident and reflected plane waves at an arbitrary
time, and the intensity of the resulting interference pattern, in the zero
phase shift case, i.e., $\varphi=0^{\circ}$. The dotted orange line and the
dashed blue line correspond to the incident and reflected plane waves,
respectively. The solid green line corresponds to the SW intensity (time-
averaged squared dynamic magnetization) of the interference pattern. (b) The
same for a phase shift of $\varphi=230^{\circ}$, i.e., the value obtained for
SW reflection from the edge of a semi-infinite 50-nm-thick Py film.

###  III.3 Phase shift of the reflected SWs

Before studying the influence of the stripe’s presence on the SW reflection,
let us first discuss SW reflection from the edge of an isolated truncated
film. According to Stigloher et al. [30], dynamic dipolar interactions induce
a phase shift between the incident and reflected SWs. This phase shift is a
natural consequence of the previously discussed dipolar pinning occurring at
the boundaries of thin ferromagnetic film [29]. Interestingly, Verba et al.
[31] have recently shown that a phase shift may also be introduced by a
polarization mismatch between incident and reflected SW modes. Regardless of
the physical mechanism responsible for the phase shift, we can extract its
magnitude from steady-state solutions formed far away from the edge.

The phase shift manifests itself in the resulting interference pattern as a
displacement of nodes with respect to the interface from which the waves are
reflected. If the interface is located at $x=x_{0}$ and the reflection
coefficient is $\mathrm{e}^{\mathrm{i}\varphi}$, where $\varphi$ is the phase
shift, the standing wave pattern sufficiently far from the interface (at $x\ll
x_{0}$) will be

$\begin{split}m(t;x)&=\operatorname{Re}\bigl{\\{}A\mathrm{e}^{-\mathrm{i}2\pi
ft}[\mathrm{e}^{\mathrm{i}k(x-x_{0})}+\mathrm{e}^{\mathrm{i}\varphi}\mathrm{e}^{-\mathrm{i}k(x-x_{0})}]\bigr{\\}}\\\
&=a(t)\cos[k(x-x_{0})-\varphi/2],\end{split}$ (7)

where $A$ and $a(t)$ are scaling coefficients independent of $x$. The change
in the standing wave pattern due to varying phase shift is illustrated in
Figs. 4(a) and (b).

In practice, we calculate $\varphi$ by fitting the expression on the right-
hand side of Eq. (7) to a snapshot of $m_{x}$ on the symmetry axis of the Py
film obtained from micromagnetic or FD-FEM simulations. To avoid distortions
caused by evanescent waves excited at the interface, only points lying at
least one stripe width to the left of the interface are taken into account.

The phase shift occurring at the edge of a 50-nm-thick Py film at frequency 11
GHz is found numerically to be $230^{\circ}$. The resulting standing wave
pattern is shown in Fig. 4(b).

###  III.4 Phase shift dependence on stripe width

Figure 5: (a) Phase shift of SWs at frequency 11 GHz as a function of the
stripe width, calculated by the FD-FEM. The stripe-film separation is $s=10$
nm. Resonances (rapid changes of the phase shift, marked with vertical dotted
lines) appear periodically with a period of ca. 160 nm. (b) SW amplification
factor in Py below the stripe, obtained by dividing the maximum of $|m_{x}|$
in the part of the film lying below the stripe by the maximum of $|m_{x}|$ in
the far field, i.e., further than half of width of the Py layer to the left of
the stripe. (c) Snapshot of the dynamic magnetization $m_{x}$ in a plateau
region (stripe width: 1270 nm). (d) Snapshot of the system in resonance at 11
GHz (stripe width: 1350 nm). The red dashed lines in (c) and (d) denote the SW
wavelength in a uniform Py film at the same frequency.

The introduction of a stripe over the Py film edge locally modifies the
environment in which SWs propagate due to dynamic dipolar interactions between
the film and the stripe. In consequence, it influences also the phase shift of
reflected waves. The variation of this phase shift with stripe width at
frequency 11 GHz, calculated using FD-FEM, is plotted in Fig. 5(a). In
general, this phase shift, defined according to Eq. (7) with interface defined
at the left edge of the stripe, at $x_{0}=0$, grows steadily with stripe
width, however with periodic jumps by $360^{\circ}$. These jumps occur
approximately every 160 nm and are accompanied by an increase of the amplitude
of SWs in the Py film underneath the stripe, as shown in Fig. 5(b). In fact,
at these stripe widths SWs are amplified in the whole bilayer, indicating that
a resonant mode of the bilayer is excited. This can be seen by comparing
snapshots of $m_{x}$ for stripes of width 1270 nm (slowly changing phase
shift) and 1350 nm (rapidly changing phase shift) presented in Figs. 5(c) and
(d), respectively.

In addition to the enhancement of SW amplitude in the bilayer, Fig. 5(c) shows
that the SWs in the stripe and in the underlying Py layer have approximately
opposite phases. In view of the profiles of the fast and slow modes shown in
Fig. 2(d)–(g), this indicates that the slow modes dominate. The magnetization
pattern in both layers is more complex than that of a typical standing wave
composed of two counter-propagating waves with the same wavelength; indeed, as
discussed in Sec. III.1, the bilayer modes have an asymmetric dispersion
relation. For such a scenario, we can generalize the resonance condition (6)
to

$(k_{u}+k_{d})w+\varphi_{l}+\varphi_{r}=2\pi n,\quad n=1,2,...,$ (8)

where $k_{u}$ and $k_{d}$ are the wavenumbers of right- and left-propagating
modes, and $\varphi_{l}$ and $\varphi_{r}$ are the phase shifts occurring at
the left and right interfaces of the stripe. For $k_{u}=k_{d}=k$ and
$\varphi_{r}=\varphi_{l}$ this equation reduces to Eq. (6). Substituting here
the wavelengths of the slow modes of a bilayer with $s=10$ nm given in Sec.
III.1, we conclude that successive resonances should occur every 160 nm, which
matches well the results of FD-FEM calculations shown in Fig. 5.

We have cross-checked these results against micromagnetic simulations made
with a finite damping coefficient $\alpha=0.0001$, see Fig. 6. Due to
computational demands, these have been performed for a narrower range of
stripe widths, 0–490 nm, 1000–1650 nm, and 2350–2720 nm, encompassing several
resonances. The obtained results are consistent with those of FD-FEM
calculations: the positions of resonances are the same as obtained by FD-FEM
and the slope of the curve in intermediate regions is virtually identical. The
almost perfect alignment of those results obtained by two different numerical
methods confirms their correctness.

Snapshots of $m_{x}$ in systems with stripes of width 1270 nm and 1350 nm are
displayed in Fig. 6(c) and (d), respectively. The former does not have a
resonance at the chosen frequency, whereas the latter does. The obtained
magnetization patterns are qualitatively similar to those calculated by FD-FEM
[Figs. 5(c) and (d)].

Figure 6: (a) Phase shift of SWs at frequency 11 GHz as a function of stripe
width, obtained with micromagnetic simulations. (b) SW amplification factor in
Py below the stripe, defined as in Fig. 5(b). (c) Snapshot of the dynamic
magnetization $m_{x}$ in a system without a resonance at 11 GHz (stripe width:
1270 nm). (d) The same in a system with a resonance at 11 GHz (stripe width:
1350 nm). The red dashed lines in (c) and (d) denote the SW wavelength in a
uniform Py film at the same frequency.

###  III.5 Two-mode model analysis

As discussed above, the plot of the phase shift vs. stripe width [Fig. 5(a)]
shows areas of slow but steady growth separated by sharp resonances. The
resonances are not identical: some, such as those at $n=7$ and $n=9$, are
broader than others. These and other features can be explained by considering
a semi-analytical model introduced below.

Wave scattering on the interface $x=0$ separating the film and the bilayer
(see Fig. 1) can be described by a scattering matrix $\mathsf{S}$ linking the
complex amplitudes of the incoming and outgoing modes on both sides of the
interface. If both parts are sufficiently long for the amplitudes of all
incoming evanescent modes to be negligible, the amplitudes of the outgoing
propagative modes are given by

$\begin{bmatrix}d_{1}\\\ u_{2}\\\
u_{3}\end{bmatrix}=\mathsf{S}\begin{bmatrix}u_{1}\\\ d_{2}\\\
d_{3}\end{bmatrix}\equiv\begin{bmatrix}S_{11}&S_{12}&S_{13}\\\
S_{21}&S_{22}&S_{23}\\\
S_{31}&S_{32}&S_{33}\end{bmatrix}\begin{bmatrix}u_{1}\\\ d_{2}\\\
d_{3}\end{bmatrix}.$ (9)

Here, $u_{1}$ and $d_{1}$ are the amplitudes of the right- and left-
propagating modes of the Py film, $u_{2}$ and $d_{2}$ are the amplitudes of
the right- and left-propagating slow modes of the bilayer, and $u_{3}$ and
$d_{3}$ are the amplitudes of the right- and left-propagating fast modes of
the bilayer (see dispersion relation shown in Fig. (2). All these amplitudes
are measured at the interface between the film and the bilayer. The elements
of the scattering matrix $\mathsf{S}$ can be calculated using the finite-
element modal method [23]. At 11 GHz, their numerical values are

$\mathsf{S}=\begin{bmatrix}0.100-0.011\mathrm{i}&0.135+0.016\mathrm{i}&0.986+0.008\mathrm{i}\\\
-0.104-0.168\mathrm{i}&-0.290+0.929\mathrm{i}&0.035-0.113\mathrm{i}\\\
0.975-0.023\mathrm{i}&0.119+0.143\mathrm{i}&-0.118-0.0273\mathrm{i}\end{bmatrix}$
(10)

(these values are obtained for modes normalized to carry unit power, with the
phase at the interface chosen so that $m_{x}$ is real and positive on the
symmetry axis of the Py layer). It can be seen that the film mode is coupled
primarily with the fast mode of the bilayer. The slow bilayer mode is strongly
reflected. There is only weak, though non-negligible, coupling between the
fast and slow bilayer modes.

Likewise, the interface $x=w$ between the bilayer and the vacuum can be
described by a scattering matrix $\mathsf{S}^{\prime}$:

$\begin{bmatrix}d_{2}^{\prime}\\\
d_{3}^{\prime}\end{bmatrix}=\mathsf{S}^{\prime}\begin{bmatrix}u_{2}^{\prime}\\\
u_{3}^{\prime}\end{bmatrix}\equiv\begin{bmatrix}S_{22}^{\prime}&S_{23}^{\prime}\\\
S_{32}^{\prime}&S_{33}^{\prime}\end{bmatrix}\begin{bmatrix}u_{2}^{\prime}\\\
u_{3}^{\prime}\end{bmatrix}.$ (11)

Here, $u_{2}^{\prime}$ and $d_{2}^{\prime}$ are the amplitudes of the right-
and left-propagating slow modes of the bilayer, and $u_{3}^{\prime}$ and
$d_{3}^{\prime}$ are the amplitudes of the right- and left-propagating fast
modes of the bilayer, all measured at the bilayer-vacuum interface (hence the
prime, used to distinguish them from the amplitudes measured at the film-
bilayer interface). The numerical values of these scattering coefficients
calculated at 11 GHz are

$\mathsf{S}^{\prime}=\begin{bmatrix}-0.043+0.975\mathrm{i}&-0.103+0.190\mathrm{i}\\\
0.189+0.105\mathrm{i}&-0.561-0.799\mathrm{i}\end{bmatrix}.$ (12)

Both modes are strongly reflected and there is only weak cross-coupling.

Mode amplitudes at the two interfaces are linked by

$\displaystyle u_{i}^{\prime}$
$\displaystyle=\exp(\mathrm{i}k_{iu}w)\,u_{i}\equiv\Phi_{iu}u_{i},$ (13a)
$\displaystyle d_{i}$
$\displaystyle=\exp(-\mathrm{i}k_{id}w)\,d_{i}^{\prime}\equiv\Phi_{id}d_{i}^{\prime},\quad
i=2,3,$ (13b)

where $k_{iu}$ and $k_{id}$ are the wave numbers of the right- and left-
propagating modes, numerically determined to be $k_{2u}=16.2$, $k_{3u}=2.22$,
$k_{2d}=-23.1$ and $k_{3d}=-1.90$ rad/µm.

Together, (9), (11) and (13) form a system of nine equations for as many
unknown mode amplitudes (the amplitude $u_{1}$ of the mode incident from the
input film is treated as known). To obtain an intelligible expression for the
reflection coefficient $r\equiv d_{1}/u_{1}$, it is advantageous to start by
eliminating the amplitudes $u_{3}$, $d_{3}$, $u_{3}^{\prime}$ and
$d_{3}^{\prime}$ of the fast bilayer mode, which is only weakly reflected at
the interface with the Py film and hence will not give rise to strong Fabry-
Perot-like resonances. This mimics the approach taken by Lecamp et al. [32] in
their model of pillar microcavities. This reduces the second row of Eq. (9)
and the first row of Eq. (11) to

$\displaystyle u_{2}$ $\displaystyle=\tilde{S}_{21}u_{1}+\tilde{S}_{22}d_{2},$
(14a) $\displaystyle d_{2}^{\prime}$
$\displaystyle=\tilde{S}_{22}^{\prime}u_{2}^{\prime}+\tilde{S}_{23}^{\prime}\Phi_{3u}S_{31}u_{1},$
(14b)

where

$\displaystyle\begin{split}\begin{bmatrix}\tilde{S}_{21}&\tilde{S}_{22}\end{bmatrix}&\equiv\frac{1}{1-\kappa
S_{23}\Phi_{3d}S_{32}^{\prime}\Phi_{2u}}\\\ &\quad\begin{bmatrix}S_{21}+\kappa
S_{23}\Phi_{3d}S_{33}^{\prime}\Phi_{3u}S_{31}&S_{22}+\kappa
S_{23}\Phi_{3d}S_{33}^{\prime}\Phi_{3u}S_{32}\end{bmatrix},\end{split}$ (15a)
$\displaystyle\begin{bmatrix}\tilde{S}_{22}^{\prime}&\tilde{S}_{23}^{\prime}\end{bmatrix}$
$\displaystyle\equiv\frac{1}{1-\kappa
S_{23}^{\prime}\Phi_{3u}S_{32}\Phi_{2d}}\begin{bmatrix}S_{22}^{\prime}+\kappa
S_{23}^{\prime}\Phi_{3u}S_{33}\Phi_{3d}S_{32}^{\prime}&\alpha
S_{23}^{\prime}\end{bmatrix}$ (15b)

and

$\kappa\equiv(1-S_{33}\Phi_{3d}S_{33}^{\prime}\Phi_{3u})^{-1}.$ (16)

The fast bilayer mode is only weakly reflected at the interface with the film:
$\lvert S_{33}\rvert\approx 0.12\ll 1$. Therefore multiple reflections of the
fast mode at bilayer interfaces do not give rise to strong Fabry-Perot
resonances and the coefficient $\kappa$ remains close to 1 for all bilayer
lengths. Together with the fact that the cross-coupling coefficients $S_{23}$,
$S_{32}$, $S_{23}^{\prime}$ and $S_{32}^{\prime}$ are small, this means we can
expect the scattering coefficients with a tilde defined in Eq. (15) to be well
approximated by

$\begin{bmatrix}\tilde{S}_{21}&\tilde{S}_{22}\\\
\tilde{S}_{22}^{\prime}&\tilde{S}_{23}^{\prime}\end{bmatrix}\approx\begin{bmatrix}S_{21}+S_{23}\Phi_{3d}S_{33}^{\prime}\Phi_{3u}S_{31}&S_{22}\\\
S_{22}^{\prime}&S_{23}^{\prime}\end{bmatrix}.$ (17)

Solving the equations remaining after elimination of the amplitudes of the
fast mode for the amplitudes of the slow mode and substituting the resulting
expressions to the formula for $d_{1}$ in the first row in Eq. (9), we arrive
at the following formula for the reflection coefficient:

$r\equiv d_{1}/u_{1}=(a+\beta b),$ (18)

where

$\displaystyle a$ $\displaystyle\equiv\underline{S_{11}}+S_{13}\Phi_{3d}\kappa
S_{33}^{\prime}\Phi_{3u}S_{31},$ (19a)
$\displaystyle\begin{split}b&\equiv\underline{S_{12}}\Phi_{2d}(\tilde{S}_{22}^{\prime}\Phi_{2u}\underline{\tilde{S}_{21}}+\underline{\tilde{S}_{23}^{\prime}}\Phi_{3u}S_{31})\\\
&\quad+S_{13}\Phi_{3d}\kappa\bigl{[}S_{33}^{\prime}\Phi_{3u}\underline{S_{32}}\Phi_{2d}(\tilde{S}_{22}^{\prime}\Phi_{2u}\underline{\tilde{S}_{21}}+\underline{\tilde{S}_{23}^{\prime}}\Phi_{3u}S_{31})\\\
&\quad+\underline{S_{32}^{\prime}}\Phi_{2u}(\underline{\tilde{S}_{21}}+\tilde{S}_{22}\Phi_{2d}\underline{\tilde{S}_{23}^{\prime}}\Phi_{3u}S_{31})\bigr{]}\end{split}$
(19b)

and $\beta$ represents the effect of multiple reflections of the slow mode:

$\beta\equiv(1-S_{22}\Phi_{2d}S_{22}^{\prime}\Phi_{2u})^{-1}.$ (20)

To facilitate the interpretation of Eqs. (18)–(20), the scattering
coefficients with magnitude much smaller than 1 have been underlined.

It can be seen that the reflection coefficient $r$ is made up of two terms.
The first, $a$, is dominated by the phase shift acquired by the fast mode of
the bilayer during a single round-trip across it. This term produces the slow
but steady increase of the phase shift visible in Fig. 5(a) (also in Fig. 7).
The second term, $\beta b$, is proportional to $b$, which is a superposition
of six small terms, each containing a product of two scattering coefficients
of small magnitude. Therefore $\beta b$ has an appreciable effect on the
reflection coefficient $b$ only when the factor $\beta$, representing the
combined effect of multiple reflections of the slow mode on both ends of the
bilayer, is much greater than 1. This happens at stripe widths $w$
corresponding to Fabry-Perot resonances of the slow mode, where $[\arg
S_{22}+\arg S_{22}^{\prime}+(k_{2u}+k_{2d})w]$ is a multiple of $2\pi$,
justifying the postulated resonance condition Eq. (8). Since $b$ is a
combination of multiple terms of similar magnitude, its dependence on the
stripe width is rather complicated. This explains the variability of the
shapes of individual resonances in Fig. 5(a) [also in Fig. 7(a)].

To confirm this interpretation of the role of the various terms in Eq. (18),
let us visualize and compare the effects of applying successively stronger
approximations to it. In Fig. 7, the black symbols show the variation of the
phase of the reflection coefficient obtained directly from numerical
calculations made with the finite-element modal method [in close agreement
with the FD-FEM results from Fig. 5(a)]. The red solid curve in 7(a) shows the
phase of the reflection coefficient calculated from Eq. (18). The only
approximation made in its derivation was to neglect evanescent coupling
between the left and right end of the bilayer; clearly, this approximation is
very well satisfied everywhere except for stripes narrower than 250 nm. The
blue dashed curve in 7(a) shows the effect of applying the approximation (17)
and setting $\kappa$ to 1 in the formula (19b) for $b$ (but not in the formula
(19a) for $a$). This corresponds to neglecting terms proportional to products
of more than two small scattering coefficients; the resulting curve is almost
indistinguishable from the previous one. Neglecting the second term $\beta b$
in Eq. (18) produces the red solid curve in 7(b). The resonances are gone, but
the long-term increase in phase shift with stripe width is still reproduced
faithfully. Finally, the blue dashed curve in 7(b) shows the result of
approximating $\kappa$ by 1 also in the formula (19a) for $a$. Its small
deviation from the red curve confirms the minor role played by multiple
reflections of the fast mode.

Figure 7: Comparison of the reflection coefficient phase calculated
numerically using the finite-element modal method FD-MM (black points) with
the semi-analytical model from Eq. (18) at varying degrees of approximation
(color lines on both subplots). Details in the plot’s legends and in the text.
Figure 8: (a) Phase shift dependence on the separation between the FM2 stripe
and Py film, calculated by the FD-FEM. (b)–(e) Snapshots of the dynamic
magnetization $m_{x}$ at separations $10$ nm, $28$ nm, $50$ nm and $106$ nm,
corresponding to the resonances marked in plot (a). The dashed red lines mark
the SW wavelength in the Py film.

###  III.6 Phase shift dependence on the layer separation

In Sec. III.1 we observed that the separation influences the strength of the
dynamical dipolar coupling between the infinitely wide stripe and Py film. It
affects the SW dispersion relation, especially the wavelengths of the slow
modes propagating leftwards and rightwards. Therefore, according to Eq. (8),
by varying separation $s$, and hence $k_{u}$ and $k_{d}$, while keeping the
stripe width $w$ constant, it should be possible to sweep over resonances of
different orders $n$. Indeed, we have found multiple resonances in dependence
of the phase shift on separation $s$ for a stripe of width $w=1676$ nm, as
shown in Fig. 8.

Resonances do not appear periodically; the spacing between subsequent Fabry-
Perot resonances increases with the altitude of the stripe, and for the chosen
stripe width the last resonance occurs at the separation $s=106$ nm. This is
because with increasing $s$ the coupling between the Py film and the stripe
weakens and the wavenumbers $k_{u}$ and $k_{d}$ approach their asymptotic
limits. Snapshots of the magnetization at the resonances found at separations
$10$ nm, $28$ nm, $50$ nm and $106$ nm are shown in Figs. 8(c)–(e). These
figures demonstrate a clear enhancement of the SW amplitude below the stripe
and a decrease in the number of nodal point with increasing separation, in
line with the shift of the dispersion relation of the slow mode towards
smaller wavenumbers shown in Fig. 2(c).

Figure 9: Red line: dependence of the resonance index $n$ calculated from Eq.
(8) on the separation $s$, with $\varphi_{\mathrm{l}}+\varphi_{\mathrm{r}}$
set to the fitted value $0.12+2\pi$. Resonances are predicted to occur at
integer values of $n$. Black circles: positions of resonances found in FD-FEM
calculations shown in Fig. 8.

Combining the Fabry-Perot resonance condition, Eq. (8), with the numerically
calculated dispersion relations and setting
$\varphi_{\mathrm{l}}+\varphi_{\mathrm{r}}$ to the fitted value of $0.12+2\pi
n$, we calculate the dependence of $n$ on the separation between the stripe
and the Py film, plotted with the red line in Fig. 9. In the range from 106
down to 5 nm, we find six integer values of $n$, corresponding to successive
resonances. These values agree well with the results of FD-FEM calculations,
where six resonances, marked with black points in Fig. 9, are detected in that
range of separation.

##  IV Conclusions

We have studied theoretically the influence of a narrow ferromagnetic stripe
of subwavelength width placed at the edge of a ferromagnetic film on the phase
shift of reflected SWs. At the considered frequency (11 GHz) the bilayer
formed by the film and the stripe supports two pairs of slow (short
wavelength) and fast (long wavelength) guided SW modes propagating in opposite
directions; these modes couple with the SW mode of the Py film. This allowed
us to interpret the numerical results by modelling the system as a series of
waveguides linked by junctions at which waveguide modes are scattered into
each other.

We have found a strong nonlinear dependence of the phase shift on the stripe
width. We showed that the reflection coefficient, from which the phase shift
can be derived, consists of two terms, each having a different origin. One
produces a slow but steady increase of the phase shift with increasing bilayer
width and is dominated by the phase accumulated by the fast mode during a
single round-trip across the bilayer. The other term has an appreciable effect
on the reflection coefficient only when multiple reflections of the slow mode
on both edges of the bilayer interfere constructively, which corresponds to
Fabry-Perot resonances of this mode. Interestingly, the incoming wave from the
Py film couples strongly only to the fast mode, but at Fabry-Perot resonances,
the phase of the reflected SW is controlled by the weakly coupled slow mode.
These results demonstrate that the bilayered part of the system, whose width
in the examples studied in this paper is much lower than the SW wavelength in
the isolated Py film, can be treated as a subwavelength resonator.

We have also found that the phase shift of the reflected SW passes through a
series of resonances as the separation between the stripe and the Py film is
increased. This unexpected effect originates from the dependence of the
wavelength of the slow SW in the bilayer on the strength of the dipolar
coupling between the two layers. As a result, the bilayer width at which the
resonance Fabry-Perot condition is satisfied changes with the coupling
strength as well, giving rise to the separation-dependent resonances.

Overall, this research shows that SW Fabry-Perot resonators can be used to
modify the phase of reflected SWs in a wide range by tiny changes of the
bilayer part width or stripe-film distance. This is significant for the
further development of magnonic devices where SW phase control is of key
importance, in particular in integrated systems with components smaller than
the SW wavelength. This may include the use of arrays of resonators in tunable
SW optical elements, such as lenses, magnonic metasurfaces and phase shifters,
as well as the sensing applications of magnonics, for example the development
of magnonic counterparts of sensors utilizing surface plasmon resonances.

###  IV.1 Acknowledgments

The research leading to these results has received funding from the Polish
National Science Centre project No. UMO-2015/17/B/ST3/00118. PG acknowledges
support from the National Science Centre of Poland, under OPUS funding
UMO-2019/33/B/ST5/02013. The simulations were partially performed at the
Poznan Supercomputing and Networking Center (Grant No. 398).

##  Appendix A Micromagnetic simulations

Micromagnetic simulations were performed in the mumax3 environment [20] for
the same magnetic parameters and geometry as described in Sec. II and damping
$\alpha=0.0001$. The simulated structure was discretized on a mesh consisting
of regular $5\times 100\times 5$ nm3 unit cells. In order to model a infinite
film along the $y$ axis, we have used the periodic boundary conditions along
the $y$ axis with assumed 1024 repetitions of the system along the $y$ axis.

After stabilizing the system with a magnetic field of value 0.1 T applied
along the $y$ axis, SWs were excited in the film by a local source of
microwave-frequency magnetic field of frequency 11 GHz and amplitude 0.1 mT
placed at 3.6 µm from the right edge. To prevent wave reflections from the
left edge of the film, an absorbing zone with gradually increasing damping was
defined on the left side of the system. A continuous harmonic SW excitation
was maintained for 162.6 ns in order to reach a fully evolved (steady-state)
interference pattern of the incident and reflected waves.

## References

  * Ng and Wakenshaw [2017] I. C. Ng and S. Y. Wakenshaw, The internet-of-things: Review and research directions, Int. J. Res. Mark. 34, 3 (2017).
  * Krawczyk and Grundler [2014] M. Krawczyk and D. Grundler, Review and prospects of magnonic crystals and devices with reprogrammable band structure, J. Phys. Condens. Matter 26, 123202 (2014).
  * Chumak  _et al._ [2015] A. Chumak, V. Vasyuchka, A. Serga, and B. Hillebrands, Magnon spintronics, Nat. Phys. 11, 453 (2015).
  * Chumak [2019] A. V. Chumak, Fundamentals of magnon-based computing (2019), arXiv:1901.08934 [cond-mat.mes-hall] .
  * Yu and Capasso [2014] N. Yu and F. Capasso, Flat optics with designer metasurfaces, Nat. Mater. 13, https://doi.org/10.1038/nmat3839 (2014).
  * Yu  _et al._ [2011] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science 334, 333 (2011).
  * Kumar  _et al._ [2012] K. Kumar, H. Duan, R. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W. Yang, Printing colour at the optical diffraction limit, Nat. Nanotechnol. 7, 557 (2012).
  * Vashistha  _et al._ [2017] V. Vashistha, G. Vaidya, P. Gruszecki, A. E. Serebryannikov, and M. Krawczyk, Polarization tunable all-dielectric color filters based on cross-shaped Si nanoantennas, Sci. Rep. 7, 8092 (2017).
  * Kruglyak  _et al._ [2017] V. Kruglyak, C. Davies, Y. Au, F. Mushenok, G. Hrkac, N. Whitehead, S. Horsley, T. Philbin, V. Poimanov, R. Dost, _et al._ , Graded magnonic index and spin wave Fano resonances in magnetic structures: Excite, direct, capture, in  _Spin Wave Confinement_ (Jenny Stanford Publishing, 2017) pp. 11–46.
  * Au  _et al._ [2012a] Y. Au, M. Dvornik, O. Dmytriiev, and V. Kruglyak, Nanoscale spin wave valve and phase shifter, Appl. Phys. Lett. 100, 172408 (2012a).
  * Au  _et al._ [2012b] Y. Au, E. Ahmad, O. Dmytriiev, M. Dvornik, T. Davison, and V. Kruglyak, Resonant microwave-to-spin-wave transducer, Appl. Phys. Lett. 100, 182404 (2012b).
  * Yu  _et al._ [2019a] T. Yu, Y. M. Blanter, and G. E. Bauer, Chiral pumping of spin waves, Physical Review Letters 123, 247202 (2019a).
  * Yu  _et al._ [2019b] T. Yu, C. Liu, H. Yu, Y. M. Blanter, and G. E. Bauer, Chiral excitation of spin waves in ferromagnetic films by magnetic nanowire gratings, Physical Review B 99, 134424 (2019b).
  * Al-Wahsh  _et al._ [2008] H. Al-Wahsh, E. El Boudouti, B. Djafari-Rouhani, A. Akjouj, T. Mrabti, and L. Dobrzynski, Evidence of Fano-like resonances in mono-mode magnetic circuits, Phys. Rev. B 78, 075401 (2008).
  * Zhang  _et al._ [2020] Z. Zhang, S. Liu, T. Wen, D. Zhang, L. Jin, Y. Liao, X.-L. Tang, and Z. Zhong, Bias-free reconfigurable magnonic phase shifter based on a spin-current controlled ferromagnetic resonator, J. Phys. D 53, 105002 (2020).
  * Yu  _et al._ [2013] H. Yu, G. Duerr, R. Huber, M. Bahr, T. Schwarze, F. Brandl, and D. Grundler, Omnidirectional spin-wave nanograting coupler, Nat. Commun. 4, 2702 (2013).
  * Yu  _et al._ [2016] H. Yu, O. d’ Allivy Kelly, V. Cros, R. Bernard, P. Bortolotti, A. Anane, F. Brandl, F. Heimbach, and D. Grundler, Approaching soft x-ray wavelengths in nanomagnet-based microwave technology, Nat. Commun. 7, 11255 (2016).
  * Graczyk  _et al._ [2018] P. Graczyk, M. Krawczyk, S. Dhuey, W.-G. Yang, H. Schmidt, and G. Gubbiotti, Magnonic band gap and mode hybridization in continuous permalloy films induced by vertical dynamic coupling with an array of permalloy ellipses, Phys. Rev. B 98, 174420 (2018).
  * Mieszczak  _et al._ [2020] S. Mieszczak, O. Busel, P. Gruszecki, A. N. Kuchko, J. W. Kłos, and M. Krawczyk, Anomalous refraction of spin waves as a way to guide signals in curved magnonic multimode waveguides, Phys. Rev. Applied 13, 054038 (2020).
  * Vansteenkiste  _et al._ [2014] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of mumax3, AIP Adv. 4, 107133 (2014).
  * Rychły and Kłos [2017] J. Rychły and J. W. Kłos, Spin wave surface states in 1D planar magnonic crystals, J. Phys. D: Appl. Phys. 50, 164004 (2017).
  * Liu and Chew [1990] Q.-H. Liu and W. C. Chew, Numerical mode-matching method for the multiregion vertically stratified media, IEEE Trans. Antennas Propag. 38, 498 (1990).
  * Śmigaj [2020] W. Śmigaj, Magnetostatic waveguide mode reflection on resonators, unpublished (2020).
  * Stancil and Prabhakar [2009] D. D. Stancil and A. Prabhakar,  _Spin waves theory and application_ (Springer, 2009).
  * Note [1] We have included to the formula from Ref. [24, Appendix C.7] the impact of the exchange interaction on dispersion by $\omega_{0}\rightarrow\omega_{0}+M_{\mathrm{S}}l^{2}k^{2}$.
  * Mruczkiewicz and Krawczyk [2014] M. Mruczkiewicz and M. Krawczyk, Nonreciprocal dispersion of spin waves in ferromagnetic thin films covered with a finite-conductivity metal, J. Appl. Phys. 115, 113909 (2014).
  * Gallardo  _et al._ [2019] R. Gallardo, T. Schneider, A. Chaurasiya, A. Oelschlägel, S. Arekapudi, A. Roldán-Molina, R. Hübner, K. Lenz, A. Barman, J. Fassbender, _et al._ , Reconfigurable spin-wave nonreciprocity induced by dipolar interaction in a coupled ferromagnetic bilayer, Phys. Rev. Appl. 12, 034012 (2019).
  * An  _et al._ [2019] K. An, V. Bhat, M. Mruczkiewicz, C. Dubs, and D. Grundler, Optimization of spin-wave propagation with enhanced group velocities by exchange-coupled ferrimagnet-ferromagnet bilayers, Phys. Rev. Appl. 11, 034065 (2019).
  * Guslienko  _et al._ [2002] K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes, Phys. Rev. B 66, 132402 (2002).
  * Stigloher  _et al._ [2018] J. Stigloher, T. Taniguchi, H. Körner, M. Decker, T. Moriyama, T. Ono, and C. Back, Observation of a Goos-Hänchen-like phase shift for magnetostatic spin waves, Phys. Rev. Lett. 121, 137201 (2018).
  * Verba  _et al._ [2020] R. Verba, V. Tiberkevich, and A. Slavin, Spin-wave transmission through an internal boundary: Beyond the scalar approximation, Phys. Rev. B 101, 144430 (2020).
  * G. Lecamp and Gérard [2005] J. P. H. G. Lecamp, P. Lalanne and J. Gérard, Energy transfer through laterally confined bragg mirrors and its impact on pillar microcavities, IEEE J. Quantum Electron. 41, 1323 (2005).