# Deep Video Inpainting Guided by Audio-Visual Self-Supervision

###### Abstract

Humans can easily imagine a scene from auditory information based on their
prior knowledge of audio-visual events. In this paper, we mimic this innate
human ability in deep learning models to improve the quality of video
inpainting. To implement the prior knowledge, we ﬁrst train the audio-visual
network, which learns the correspondence between auditory and visual
information. Then, the audio-visual network is employed as a guider that
conveys the prior knowledge of audio-visual correspondence to the video
inpainting network. This prior knowledge is transferred through our proposed
two novel losses: audio-visual attention loss and audio-visual pseudo-class
consistency loss. These two losses further improve the performance of the
video inpainting by encouraging the inpainting result to have a high
correspondence to its synchronized audio. Experimental results demonstrate
that our proposed method can restore a wider domain of video scenes and is
particularly effective when the sounding object in the scene is partially
blinded.

Index Terms—  audio-visual learning, audio-visual correspondence, audio-visual
network, deep video inpainting

##  1 Introduction

Imagine hearing the sound of a bird singing. You may come up with an image of
a bird flying in the sky or sitting on top of a tree. In this fashion, humans
can easily visualize a scene related to incoming auditory signals [1]. This
natural behavior is empowered by the prior knowledge of semantic mapping
between the visual and auditory modalities learned from ubiquitous audio-
visual events around us. This ability to connect the dots between two
modalities allows humans to restore videos better whose spatial information is
corrupted. In other words, even though the video is partially blinded, humans
can easily imagine what is happening in missing parts by listening to the
corresponding audio. Based on this intuition, our work tries to mimic this
human ability in deep learning models to better solve the following video
inpainting problem: filling in missing visual regions in a video, guided by
the audio signal. Hence, our goal can be articulated into answering the
following question: can machines also learn to restore the visual content of a
video by hearing its corresponding sound?

To achieve such goal, we exploit the audio-visual correspondence learned by
the audio-visual network (AV-Net) [2] to train the video inpainting network
(VI-Net). The AV-Net learns to generate an audio-visual attention map that
highlights visual regions which are corresponding to the synchronized audio
and to capture the pseudo-class of each modality within the audio-visual pair.
In this manner, the AV-Net learns the semantic relationship within the audio-
visual pairs without video labels in a self-supervised manner, without labeled
videos. There have been previous attempts to use this audio-visual
correspondence for several of their unique downstream tasks, such as sounding
object localization [2, 3] and sound source separation [4, 5]. Unlike these
attempts, we aim to leverage the prior knowledge of audio-visual
correspondence for the video inpainting task, which has not been explored yet.

Fig. 1: Overview of our proposed method. We use the audio-visual network (AV-
Net) as a guider of the video inpainting network (VI-Net) that conveys the
prior knowledge of audio-visual correspondence through our proposed audio-
visual losses.

As shown in Fig. 1, the AV-Net guides the VI-Net to use the corresponding
audio signal as an important cue for restoring the corrupted frame. Given the
prior information of audio-visual correlation that AV-Net provides, we propose
two novel audio-visual losses to convey the prior knowledge to the VI-Net:
audio-visual attention loss and audio-visual pseudo-class consistency loss.
Audio-visual attention loss encourages the VI-Net to minimize the disparity of
the audio-visual attention maps between the original and the inpainted frame.
By doing so, the VI-Net solely focuses on restoring areas corresponding to the
sounding object, making the inpainting result semantically more accurate.
Audio-visual pseudo-class consistency loss is designed to indicate that visual
and audio information from the same video should belong to the identical
class. Using auxiliary classifiers, we encourage the VI-Net to learn that the
visual features of inpainted frames and the synchronized audio features should
belong to the same pseudo-classes. This audio-guided class consistency
information can further enhance the video inpainting quality.

Fig. 2: Schematics of (a) Training the audio-visual network (AV-Net) and (b)
Training the video inpainting network (VI-Net) with the audio-visual guidance
from the pretrained AV-Net. Modules with dotted line indicate that their
parameters are frozen in the training time.

In summary, our main contributions are as follows:

  * •

To enhance the video inpainting quality, we propose a novel approach that
utilizes the inherent sound from the video itself.

  * •

Based on the pretrained AV-Net, we propose two novel losses that enable the
VI-Net to utilize the inherent sound of a video for restoring corrupted
frames.

  * •

Experimental results show that our approach is especially effective when
restoring the frame whose sounding object in the scene is partially blinded.

##  2 Related Work

In this section, we briefly discuss two research domains relevant to our work:
deep video inpainting and audio-assisted visual synthesis.

Deep video inpainting. Video inpainting is a challenging problem aiming to
restore missing regions in consecutive frames with spatially and temporally
plausible content [6]. Recent approaches have achieved significant
improvements via deep learning by using encoder-decoder-based architectures
[7]. Along this line, there have been attempts of adopting optical flows [8,
9, 10], novel architectures [11, 12], attention modules [13, 14], and
adversarial mechanisms [15]. Despite these successes, little attention has
been given to employing the audio signal, which is the innate correspondence
prior within a video. Thus, our work takes a pioneering step to demonstrate a
generic method of utilizing audio signals to the video inpainting problem.

Audio-assisted visual synthesis. Audio has been used as an effective prior for
synthesizing images or video frames, but in limited application domains. Such
domains include audio-based adversarial image generation [16], talking face
synthesis [17, 18], and speakers’ face super-resolution [19]. Compared to
these previous approaches, our work has the following distinctions. While [16]
used human-labeled videos to obtain semantic knowledge, our work utilizes an
audio-visual relationship learned from the self-supervised training procedure.
We also consider a broader scope of audio-visual events occurring in the real
world, rather than managing only the video of talking faces as in [17, 18,
19].

##  3 Proposed Method

In this section, we explain our audio-guided video inpainting framework. Fig.
2 shows an overview of this framework consisting of two main parts: the audio-
visual network (AV-Net) and the video inpainting network (VI-Net). We first
review the AV-Net (Sec. 3.1). Then, we provide details on two novel losses
derived from the AV-Net that are used to train the VI-Net: audio-visual
attention loss and audio-visual pseudo-class consistency loss (Sec. 3.2).

###  3.1 The audio-visual network

Let $\mathcal{X}=\left\\{\left(a_{i},v_{j}\right)\mid 1\leq i\leq N,1\leq
j\leq N\right\\}$ denote a set of audio-visual pairs such that a pair
$(a_{i},v_{j})$ is sampled from $N$ number of videos. Here, $a$ and $v$ each
represents the audio signal and the video frame. Given the pair
$(a_{i},v_{j})\in\mathcal{X}$ as input, we aim to obtain the prior information
in two forms: audio-visual attention map and pseudo-class of each input’s
modality. The former considers a pair $(a_{i},v_{j})$ where $a_{i}$ and
$v_{j}$ are each randomly sampled from $N$ videos, while the latter considers
only a pair drawn from the same video (i.e., $i=j$). Our training methodology
of the AV-Net refers to [2, 20].

Audio-visual attention map. As shown in Fig. 2 (a), the AV-Net consists of two
convolutional sub-networks for feature extraction: audio network
$f_{\mathcal{A}}$ and visual network $f_{\mathcal{V}}$. From two sub-networks,
we extract an audio feature $f_{\mathcal{A}}(a_{i})\in\mathbb{R}^{c}$ and a
visual feature map $f_{\mathcal{V}}(v_{j})\in\mathbb{R}^{h\times w\times c}$.
Note that $h\times w$ and $c$ denote spatial and channel dimensions,
respectively. Then, we obtain the similarity map of $\mathbb{R}^{h\times w}$
by computing the scalar product between $L_{2}$-normalized
$f_{\mathcal{A}}(a_{i})$ and $f_{\mathcal{V}}(v_{j})$ along the channel
dimension for each of the spatial units within $f_{\mathcal{V}}(v_{j})$. The
similarity map then describes how strongly each spatial location of
$f_{\mathcal{V}}(v_{j})$ reacts to the audio descriptor
$f_{\mathcal{A}}(a_{i})$. Finally, we apply a sigmoid operation to this
similarity map to obtain the audio-visual attention map
$M(a_{i},v_{j})\in\mathbb{R}^{h\times w}$.

Intuitively, the attention map $M(a_{i},v_{j})$ would show high attention in
the area that semantically corresponds to both the given audio $a_{i}$ and the
video frame $v_{j}$. Based on this intuition, the objective of training the
AV-Net can be formulated into a binary classification problem as follows:

$\mathcal{L}_{att}=\texttt{BCE}\left(y_{corr},\texttt{GMP}(M(a_{i},v_{j}))\right),$
(1)

where $\texttt{BCE}(\cdot,\cdot)$ denotes the binary cross-entropy loss,
$\texttt{GMP}(\cdot)$ denotes the global max-pooling operation, and $y_{corr}$
denotes a binary label that indicates whether the audio-visual pair comes from
the same video. By minimizing the cross-entropy between $y_{corr}$ and the
largest value of the attention map $M(a_{i},v_{j})$, the network is encouraged
to maximize the attention values in regions that correspond to the given audio
$a_{i}$, and to suppress them when audio-visual pairs do not match.

Pseudo-class prediction. Pseudo-labels of audio and visual features are also
used to stabilize the training of the AV-Net. For pseudo-label extraction, we
use matching audio-visual pairs $(a_{i},v_{i})\in\mathcal{X}$. We apply set
threshold to the attention map $M(a_{i},v_{i})$ to obtain a binary mask
$m_{i}\in\left\\{0,1\right\\}^{h\times w}$. Using this attention-based binary
mask, we compute the object representation $o_{i}\in\mathbb{R}^{c}$ from the
visual feature $f_{\mathcal{V}}(v_{i})$ to pick out the area where the audio-
visual event is present. In specific, $o_{i}=\texttt{GAP}\left(m_{i}\odot
f_{\mathcal{V}}(v_{i})\right)$, where $\texttt{GAP}(\cdot)$ and $\odot$ denote
the global average pooling operation and the channel-wise Hadamard product,
respectively. We finally perform a K-means clustering on the set of object
descriptors $\mathcal{O}=\left\\{o_{1},o_{2},\cdots,o_{N}\right\\}$ to assign
each of them a pseudo-label corresponding to the cluster to which it belongs.

With these pseudo-labels set as ground truth, the network and classifiers are
trained by minimizing the following classification objective:

$\mathcal{L}_{cls}=\texttt{CE}\left(y_{p}(o_{i}),\hat{y_{\mathcal{A}}}(a_{i})\right)+\texttt{CE}\left(y_{p}(o_{i}),\hat{y_{\mathcal{V}}}(v_{i})\right),$
(2)

where $\texttt{CE}(\cdot,\cdot)$ denotes the categorical cross-entropy loss
between two logit vectors and $y_{p}(o_{i})$ represents the one-hot pseudo-
label of $o_{i}$. $\hat{y_{\mathcal{A}}}(a_{i})$ and
$\hat{y_{\mathcal{V}}}(v_{i})$ indicate the logit vectors from the linear
classifiers $\hat{y_{\mathcal{A}}}$ and $\hat{y_{\mathcal{V}}}$ given $a_{i}$
and $v_{i}$, respectively.

We train the AV-Net with Eq. 1 and Eq. 2 in an alternate manner, as two
objectives mutually improves the overall performance [21].

###  3.2 Training the video inpainting network

In this sub-section, we describe our two novel losses derived from the AV-Net
to further improve the training of the VI-Net. Suppose a pair of a corrupted
video frame $\bar{v}$ and its ground truth frame $v$. Then, the VI-Net returns
an inpainted frame $\hat{v}$ given the corrupted frame $\bar{v}$.

Audio-visual attention loss. We exploit the ability of the AV-Net to localize
the sounding object in order to design our novel audio-visual attention loss.
The audio-visual network takes video frame $v$ and its paired audio $a$ as
inputs and generates an attention map $M(a,v)$ which highlights the area
matching the given audio $a$. In the same way, the attention map
$M(a,\hat{v})$ can be obtained by replacing $v$ with $\hat{v}$. The key idea
is that if the spatial content of the audio-visual event are successfully
recovered in $\hat{v}$, the attention maps $M(a,\hat{v})$ and $M(a,v)$ should
be identical. Otherwise, $M(a,\hat{v})$ would be vastly different from
$M(a,v)$, especially in the area where the audio-visual event takes place.

From the investigation above, we observe that minimizing the difference
between these two attention maps would reduce the disparity between $v$ and
$\hat{v}$. Hence, we propose the following audio-visual attention loss:

$\mathcal{L}^{AV}_{att}=\frac{1}{hw}\left\lVert
M(a,v)-M(a,\hat{v})\right\rVert^{2}_{2},$ (3)

where $h$ and $w$ are the height and width of $M(\cdot,\cdot)$, respectively.
This objective encourages the VI-Net to reconstruct the corrupted frame in a
way such that the audio-visual attention map of the inpainted frame
$M(a,\hat{v})$ is similar to the attention map of the ground truth frame
$M(a,v)$. As a result, the inpainting network can better restore the missing
part within sound-salient areas by filling it with content or texture that
actively reacts to the given audio feature. This property cannot be found in
common reconstruction losses (e.g., $L_{1}$ loss), which ignore additional
cues from the audio.

Audio-visual pseudo-class consistency loss. To further improve the performance
of the VI-Net, we additionally guide it with the class-consistency information
between the audio and video frame inputs. The audio and visual information
from a synchronized video should semantically belong to the same class. Hence,
by learning that the restored frame $\hat{v}$ should belong to the same class
as the corresponding audio $a$, the VI-Net could better reconstruct $\hat{v}$
such that it is more similar to the ground truth frame $v$.

We inject the audio information to the VI-Net by concatenating the audio
feature $f_{\mathcal{A}}(a)$ to the bottleneck feature from the encoder of the
VI-Net (the upper part of Fig. 2 (b)). Note that we broadcast the audio
feature $f_{\mathcal{A}}(a)$ to the spatial dimension of the bottleneck
feature before the concatenation. As the pretrained AV-Net can already predict
the pseudo-class of the audio $a$, we set this as a guideline to determine
whether the inpainted frame $\hat{v}$ has coherent content. Therefore, we
design the audio-visual pseudo-class consistency loss as follows:

$\mathcal{L}^{AV}_{cls}=\texttt{CE}\left(\hat{y_{\mathcal{A}}}(a),\hat{y_{\mathcal{V}}}(\hat{v})\right),$
(4)

where $\hat{y_{\mathcal{A}}}(a)$ and $\hat{y_{\mathcal{V}}}(\hat{v})$ denote
the logit vectors from the linear classifiers given $a$ and $\hat{v}$,
respectively. Note that the linear classifiers are also pretrained and frozen
as parts of the AV-Net. Audio-visual pseudo-class consistency loss guides the
VI-Net to synthesize a frame $\hat{v}$ that is class-consistent with the
synchronized audio $a$.

Total loss. To train the VI-Net, we use the final loss as follows:

$\mathcal{L}=\lambda_{L_{1}}\mathcal{L}_{L_{1}}+\lambda_{adv}\mathcal{L}_{adv}+\lambda^{AV}_{att}\mathcal{L}^{AV}_{att}+\lambda^{AV}_{cls}\mathcal{L}^{AV}_{cls},$
(5)

where $\mathcal{L}_{L_{1}}$ and $\mathcal{L}_{adv}$ respectively denote the
$L_{1}$ loss and the adversarial loss from T-PatchGAN [15]. Note that these
two losses are borrowed from [13], which is our baseline VI-Net. The VI-Net is
optimized jointly with our proposed losses $\mathcal{L}^{AV}_{att}$ and
$\mathcal{L}^{AV}_{cls}$. Hence, the network learns to consider the audio-
visual consistency while minimizing the visual difference. The weights for
each loss are empirically set as follows: $\lambda_{L_{1}}=1$,
$\lambda_{adv}=0.01$, $\lambda^{AV}_{att}=2$, and $\lambda^{AV}_{cls}=1$.

Fig. 3: Qualitative results of two samples from AVE dataset blinded by I-masks
(left) and S-masks (right). While the baseline STTN without audio signals
shows more artifacts around the object and produces blurry result, our method
produces more realistic and clearer results.

##  4 Experiments

###  4.1 Experimental settings

Datasets. We adopt AVE [22] and MUSIC-Solo [4] dataset to show the
effectiveness of our approach. AVE dataset contains 4113 video clips covering
29 categories of diverse real-life audio-visual events. MUSIC-Solo dataset
contains 493 video clips with 11 categories that exclusively cover solo
performances of diverse musical instruments. We follow the official split of
AVE dataset. On the other hand, we randomly split MUSIC-Solo dataset into
343/50/100 for train/validation/test since there is no designated split.

Moreover, we evaluate our method on two types of maskings: I-mask and S-mask.
I-masks irregularly blind the pixels with random strokes and shapes. We adopt
the subset of NVIDIA Irregular Mask Dataset [23]. For testing, we randomly
pick three masks with a blinding ratio of 20.0%, 27.7%, and 28.4%,
respectively. We also design S-masks to blind the region which corresponds to
the sounding object. We collect S-masks by eroding the object mask $m_{i}$
mentioned in Sec. 3.1 until the spatial area of the masking covers 20% of the
image. This ratio refers to the approximate proportion of the region that the
sounding object occupies in the video scene.

Preprocessing. Given a video clip, we extract video frames at 8 fps and
resample its mono-channel audio at 16 kHz. Then, the video frame is resized to
the spatial size of 256$\,\times\,$256 and then randomly cropped (for
training) or resized (for testing) into 224$\,\times\,$224\. The audio is
sampled by retrieving a 1-second segment, and converted to the log-scale mel
spectrogram with 0.01-second window size, half-window hop length, and 80 mel
bins, finally treated as a single-channel matrix with the spatial dimension of
201$\,\times\,$80.

Audio-visual network We follow [2, 20] to implement the audio-visual network
(AV-Net). For visual and audio sub-networks $f_{\mathcal{V}}$ and
$f_{\mathcal{A}}$, we use ResNet-18-based architectures as in [20].

Video inpainting baseline. We adopt one of the state-of-the-art architectures,
the Spatial-Temporal Transformer Network (STTN) [13] as our baseline model. As
our major interest lies in inpainting videos with audio-visual events, our
choice of video dataset is different from the original work [13]. Therefore,
we train the STTN on the aforementioned datasets from scratch, without audio
signals.

Training details. To train the AV-Net, we adopt Adam optimizer with the
learning rate of 5e-5 for AVE and 1e-4 for MUSIC-Solo dataset. The batch size
is set to 32 for both datasets. Furthermore, we set the threshold value to
0.07 while obtaining binary masks and the number of clusters to 10 while
collecting the pseudo-classes of object representations. While training the
AV-Net for 4 epochs total, the learning rate is decayed by 0.1 after 2 epochs.
Then, the parameters of the pretrained AV-Net are frozen while training the
VI-Net. For AVE dataset, we train the VI-Net using Adam optimizer with the
initial learning rate of 1e-4 decayed by 0.1 for every 100k iterations for a
total of 350k iterations. For MUSIC-Solo dataset, due to the lack of training
data, we fine-tune the VI-Net pretrained on AVE dataset using Adam optimizer
for a total of 100k iterations with the learning rate of 1e-5 for first 50k
iterations, and 1e-6 for the remaining iterations. For both datasets, the
batch size is set to 8.

Evaluation metrics. The quantitative result is reported using three widely-
used metrics: PSNR [9], SSIM [24], and video-based Fréchet Inception Distance
(VFID) [15]. In detail, PSNR and SSIM are standard metrics to assess the
synthesized scenes, whereas VFID quantifies the perceptual difference compared
to the ground truth.

Method | I-mask | S-mask
---|---|---
| $\scriptstyle\mathcal{L}^{AV}_{att}$ | $\scriptstyle\mathcal{L}^{AV}_{cls}$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  VFID$\downarrow$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  VFID$\downarrow$
Baseline | - | - | 30.76 | 93.45 | 3.549 | 26.58 | 91.93 | 5.553
w/ Ours | - | ✓ | 30.81 | 93.55 | 3.356 | 26.83 | 92.21 | 5.305
w/ Ours | ✓ | - | 30.94 | 93.61 | 3.273 | 27.16 | 92.47 | 5.271
w/ Ours | ✓ | ✓ | 31.18 | 93.65 | 3.184 | 27.32 | 92.69 | 4.961

Table 1: Quantative evaluation and ablation study of applying our method on
AVE dataset with two different types of masks. $\uparrow$ indicates that
higher is better and $\downarrow$ means that lower is better.

Method | I-mask | S-mask
---|---|---
| $\scriptstyle\mathcal{L}^{AV}_{att}$ | $\scriptstyle\mathcal{L}^{AV}_{cls}$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  VFID$\downarrow$ |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  VFID$\downarrow$
Baseline | - | - | 29.49 | 93.85 | 4.316 | 26.47 | 91.88 | 5.706
w/ Ours | - | ✓ | 29.60 | 93.84 | 4.191 | 26.95 | 92.38 | 5.205
w/ Ours | ✓ | - | 29.66 | 93.87 | 4.221 | 27.05 | 92.40 | 5.194
w/ Ours | ✓ | ✓ | 29.68 | 93.84 | 4.092 | 27.12 | 92.53 | 4.929

Table 2: Quantative evaluation and ablation study of applying our method on
MUSIC-Solo dataset with two different types of masks. $\uparrow$ indicates
that higher is better and $\downarrow$ means that lower is better.

###  4.2 Result and discussion

We test our method on 4 different experimental setups derived from the
combinations of video and mask datasets mentioned in Sec. 4.1. Table 1 shows
that adopting our proposed audio-visual objectives outperforms the visual-only
baseline on AVE dataset for all suggested metrics. As shown in Table 2, our
method also performs substantially well on MUSIC-Solo dataset with video
scenes strictly related to musical instruments. Performance improvements over
two different video datasets also show that our method is effective not only
in domain-specific videos such as MUSIC-Solo dataset but also in videos with a
broader domain such as AVE dataset. Ablation studies in Table 1 and 2 imply
that our two losses harmoniously give a positive impact on the inpainting
quality, with the audio-visual attention loss showing a bigger influence.

One interesting point is that performance gains on S-masks are larger than
those on I-masks. As shown in Table 1, on AVE dataset masked with I-masks, our
method of applying both audio-visual losses improves the baseline PSNR and
VFID by 0.42 and 0.365, respectively. On the same dataset with S-masks, our
method shows a larger PSNR increase of 0.74 and VFID improvement of 0.592. The
same tendency is shown in Table 2 on the MUSIC-Solo dataset. In the case of
I-masks, PSNR and VFID improvement each shows 0.19 and 0.224 compared to the
baseline. On the other hand, improvements are greater in the case of S-masks,
showing PSNR increase of 0.65 and VFID improvement of 0.777. Recalling that
S-masks are designed to mask audio-visual events, this tendency indicates that
our method indeed effectively restores those regions. This shows that the
audio-visual correspondence given as the prior information allows the video
inpainting model to better restore regions corresponding to audio-visual
events.

Fig. 3 demonstrates that our method produces more pleasing results for both
types of masking. While the baseline model produces blurry artifacts around
the sounding object, our approach can synthesize plausible results.
Particularly, when the audio-visual event is partially deteriorated (by
S-masks), the baseline fails to generate a realistic scene in the blinded
area. In contrast, our method successfully restores the frame with clearer and
comprehensible content while preserving the audio-visual coherency.

##  5 Conclusion

In this paper, we investigate a novel approach to using audio for video
inpainting tasks by employing audio-visual self-supervision. We adopt the
audio-visual network to bridge the gap between visual and audio modality, then
use its functionalities to further guide the video inpainting network through
proposed two novel losses: audio-visual attention loss and audio-visual
pseudo-class consistency loss. Experimental results on two different audio-
visual datasets – AVE and MUSIC-Solo dataset – with two types of masking –
I-mask and S-mask – show that our approach improves the performance of the
video inpainting network compared to the baseline.

## References

  * [1] Relja Arandjelovic and Andrew Zisserman,  “Look, listen and learn,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2017, pp. 609–617.
  * [2] Relja Arandjelovic and Andrew Zisserman,  “Objects that sound,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 435–451.
  * [3] Honglie Chen, Weidi Xie, Triantafyllos Afouras, Arsha Nagrani, Andrea Vedaldi, and Andrew Zisserman,  “Localizing visual sounds the hard way,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021, pp. 16867–16876.
  * [4] Hang Zhao, Chuang Gan, Andrew Rouditchenko, Carl Vondrick, Josh McDermott, and Antonio Torralba,  “The sound of pixels,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 570–586.
  * [5] Ariel Ephrat, Inbar Mosseri, Oran Lang, Tali Dekel, Kevin Wilson, Avinatan Hassidim, William T. Freeman, and Michael Rubinstein,  “Looking to listen at the cocktail party: A speaker-independent audio-visual model for speech separation,”  ACM Transactions on Graphics, vol. 37, no. 4, 2018.
  * [6] Dahun Kim, Sanghyun Woo, Joon-Young Lee, and In So Kweon,  “Deep video inpainting,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 5792–5801.
  * [7] Chuan Wang, Haibin Huang, Xiaoguang Han, and Jue Wang,  “Video inpainting by jointly learning temporal structure and spatial details,”  in Proceedings of the AAAI Conference on Artificial Intelligence (AAAI), 2019, vol. 33, pp. 5232–5239.
  * [8] Jia-Bin Huang, Sing Bing Kang, Narendra Ahuja, and Johannes Kopf,  “Temporally coherent completion of dynamic video,”  ACM Transactions on Graphics, vol. 35, no. 6, pp. 1–11, 2016.
  * [9] Rui Xu, Xiaoxiao Li, Bolei Zhou, and Chen Change Loy,  “Deep flow-guided video inpainting,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 3723–3732.
  * [10] Chen Gao, Ayush Saraf, Jia-Bin Huang, and Johannes Kopf,  “Flow-edge guided video completion,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2020, pp. 713–729.
  * [11] Ya-Liang Chang, Zhe Yu Liu, Kuan-Ying Lee, and Winston Hsu,  “Learnable gated temporal shift module for deep video inpainting,”  in British Machine Vision Conference (BMVC), 2019.
  * [12] Seoung Wug Oh, Sungho Lee, Joon-Young Lee, and Seon Joo Kim,  “Onion-peel networks for deep video completion,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019, pp. 4403–4412.
  * [13] Yanhong Zeng, Jianlong Fu, and Hongyang Chao,  “Learning joint spatial-temporal transformations for video inpainting,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2020, pp. 528–543.
  * [14] Sungho Lee, Seoung Wug Oh, DaeYeun Won, and Seon Joo Kim,  “Copy-and-paste networks for deep video inpainting,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019, pp. 4413–4421.
  * [15] Ya-Liang Chang, Zhe Yu Liu, Kuan-Ying Lee, and Winston Hsu,  “Free-form video inpainting with 3d gated convolution and temporal patchgan,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2019, pp. 9066–9075.
  * [16] Chia-Hung Wan, Shun-Po Chuang, and Hung-Yi Lee,  “Towards audio to scene image synthesis using generative adversarial network,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019, pp. 496–500.
  * [17] Amir Jamaludin, Joon Son Chung, and Andrew Zisserman,  “You said that?: Synthesising talking faces from audio,”  International Journal of Computer Vision (IJCV), vol. 127, no. 11, pp. 1767–1779, 2019.
  * [18] Alexandros Koumparoulis, Gerasimos Potamianos, Samuel Thomas, and Edmilson da Silva Morais,  “Audio-assisted image inpainting for talking faces,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2020, pp. 7664–7668.
  * [19] Givi Meishvili, Simon Jenni, and Paolo Favaro,  “Learning to have an ear for face super-resolution,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2020, pp. 1364–1374.
  * [20] Di Hu, Rui Qian, Minyue Jiang, Xiao Tan, Shilei Wen, Errui Ding, Weiyao Lin, and Dejing Dou,  “Discriminative sounding objects localization via self-supervised audiovisual matching,”  Advances in Neural Information Processing Systems (NeurIPS), vol. 33, 2020.
  * [21] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba,  “Learning deep features for discriminative localization,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2016, pp. 2921–2929.
  * [22] Yapeng Tian, Jing Shi, Bochen Li, Zhiyao Duan, and Chenliang Xu,  “Audio-visual event localization in unconstrained videos,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 247–263.
  * [23] Guilin Liu, Fitsum A. Reda, Kevin J. Shih, Ting-Chun Wang, Andrew Tao, and Bryan Catanzaro,  “Image inpainting for irregular holes using partial convolutions,”  in Proceedings of the European Conference on Computer Vision (ECCV), 2018.
  * [24] Zhou Wang, Alan C. Bovik, Hamid R. Sheikh, and Eero P. Simoncelli,  “Image quality assessment: from error visibility to structural similarity,”  IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600–612, 2004.


11institutetext: Department of Mechanical Engineering, University of
Massachusetts, Lowell, Lowell, Massachusetts 01854, USA

# Exceptional points and scattering of discrete mechanical metamaterials

Weidi Wang  Alireza V. Amirkhizi _email:_<EMAIL_ADDRESS>
(Received: date / Revised version: date)

###### Abstract

Exceptional points (EPs) are complex singularities of parametric linear
operators where two or more eigenvalues and eigenvectors coalesce. EPs are
attracting increasing interest in mechanical metamaterials due to their strong
potentials for wave filtering, cloaking, and sensing applications. This work
studies the band topology and scattering behaviors near EPs, using discrete
models of metamaterial (MM) systems. The questions of existence of EPs and
their physical manifestations will be addressed with particular focus on
symmetry considerations and scattering behavior. Discrete mass-spring models
with adjustable parameters are used here to elucidate the EP-related phenomena
in a fundamental form. The transfer and scattering matrices are analyzed to
provide practical insights on the restrictions associated with reciprocity and
fundamental symmetries. By including complex stiffness in frequency domain as
a representation of non-conservative mechanical loss or gain, the MM arrays
can be tuned to achieve bi-directional transparency or one-way reflection when
operating at the EPs. This analytical study will contribute to the
understandings of EPs in mechanical context and the design of micro-structured
media for novel applications.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

Exceptional points (EPs) were originally introduced Heiss and Sannino (1990)
in quantum mechanics and are defined as the complex branch point singularities
where eigenvectors associated with repeated eigenvalues of a parametric non-
Hermitian operator coalesce. This distinguishes an EP from a degeneracy branch
point where two or more linearly independent eigenvectors exist with the same
eigenvalue. The mathematical aspects of EPs have been discussed in literature
Seyranian, Kirillov, and Mailybaev (2005); Amore and Fernández (2021). EPs can
widely be found in non-Hermitian systems and have been reported in different
physical problems including optics Rüter  _et al._ (2010); Othman, Galdi, and
Capolino (2017) and acoustics Lu and Srivastava (2018); Maznev (2018); Ding
_et al._ (2016). This work aims to use discrete models of mechanical
metamaterials (MMs) to analyze the EPs of two different operators (dynamic
matrix and scattering matrix) and the associated scattering behaviors. The EPs
of the dynamic matrix are shown to lead to bi-directional transparency, which
features zero reflection and unitary transmission with zero phase difference.
On the other hand, the EPs of the scattering matrix are associated with
spontaneously broken parity-time ($\mathcal{PT}$) symmetry and one-way
reflection. These two distinct occurrences of EPs have been reviewed and
discussed in literature for optical and photonic systems Miri and Alù (2019).
However, there has been little discussion on the EPs in mechanical context.
The introduced discrete systems may be considered as reduced order analogs of
continuum micro-structured media and help the conceptual design of these
system by removing all but essential dynamic features.

The EPs of a dynamic matrix can be found in the eigenfrequency study, where
the equations of motion are established for a repeating unit cell (RUC).
Bloch-Floquet condition is embedded in the wavenumber-dependent matrices. Such
a setup enables the computation of the eigenfrequencies and mode shapes of an
infinite periodic array, for any prescribed wavenumber. The eigenfrequency
band structure is of prime importance in the studies of MMs and phononic
crystals (PCs) Liu  _et al._ (2000) as it signifies the overall dispersion of
the micro-structured medium. Due to the coupling effects between degrees of
freedom in a locally resonant structure, the band structure exhibits mode
mixing and frequency band gaps. It has been shown Amirkhizi and Wang (2018)
that, an internal resonator does not necessarily lead to a stop band. The
existence and the width of a stop band are strongly related to the coupling
strength between multiple degrees of freedom. To study this effect with a
quantified coupling strength, a tunable discrete model is developed and
presented here, in which the coupling tunability is achieved using a skewed
resonator. In Section 2, it is shown that a wide band gap is associated with a
large coupling constant, while a decoupled resonator leads to independent
dispersion branches without a band gap. In cases where coupling is weak, the
band gap becomes extremely narrow and the dispersion curves appear to repel
each other to form an avoided crossing. Such a gap could lead to incorrect
sorting of branches, due to its extremely small width and the sharp changes in
mode shapes. This phenomenon is referred to as level repulsion (LR) and has
been studied in literature Lu and Srivastava (2018); Yeh  _et al._ (2016);
Wang, Maznev, and Laude (2016); Amirkhizi and Wang (2018). It is hypothesized
that the sharp mode changes may be utilized for accurate and robust
identification of a perturbative parameter in the operator under study, in
this case wavenumber.

While the frequency dispersion curves (levels) are repelled in the real
wavenumber domain, the two dispersion surfaces intersect each other at an EP
in the complex domain. Lu and Srivastava Lu and Srivastava (2018) introduced a
method based on the mode shape continuity around such points to distinguish
the real vs. avoided crossing points in the band structure. They showed that
the instances of frequency level repulsion in the real wavenumber domain have
their associated Riemann surfaces crossing at an EP in the complex wavenumber
domain. The exotic topology of the eigenvalue surfaces in the vicinity of EPs
has attracted extensive research interest in recent years Ryu  _et al._
(2015); Doppler  _et al._ (2016); Xu  _et al._ (2016); Maznev (2018); Shen,
Zhen, and Fu (2018); Miri and Alù (2019). However, the EPs discussed in
literature usually possess complex parameters (e.g., frequency, wavevector
components) and are studied only in the eigenfrequency analysis. Similarly, in
the first part of the present work, complex wavenumber is used as the
parameter leading to non-Hermiticity of the dynamic matrix and controls the
location of EPs. The question thus rises: how would an EP of the
eigenfrequency band structure affect the scattering of such systems? To answer
this question, one may seek to tune the parameters that can break the
Hermiticity of an elastic system, and then modulate the system so that the
complex singularity point is re-positioned onto the real frequency axis. It is
shown that wavenumber-parameterized systems can be tuned by adding loss and
gain to various spring elements. This could enable moving the location of EP
into the real wavenumber domain as well as making the associated frequency to
be real (while in contrast the EPs in an earlier work Maznev (2018) had
complex frequency). Such a system may be studied in a simple harmonic
scattering (real frequency) numerical experiment. In practice loss and gain
elements may be realized via viscous or other coupled multi-physics (e.g.,
piezoelectric) components. We derive the conditions to make the EP locate on
the real frequency plane and show that stiffness parameters must have certain
compatible loss and gain factors.

With an EP re-positioned to real frequency domain, it is then feasible to
analyze the scattering behavior when operating near such an EP. Using the
transfer matrix method (TMM), which relates the mechanical states on the left
and right boundaries of a finite medium, the scattering coefficients, which
describes the relation of incoming and outgoing waves through a sample, can
then be derived to study the response when operating near EPs. Transfer matrix
method is discussed in depth Nemat-Nasser  _et al._ (2015); Amirkhizi (2017);
Nanda and Karami (2018); Amirkhizi and Alizadeh (2018); Psiachos and Sigalas
(2018, 2019) for wave propagation problems in 1D systems. It is widely used to
determine the band structure and can also be used to compute the reflectance
spectrum Ardakani, Ghasemi, and Golshan (2017). A similar approach to
determine the band gap behavior of permuted PCs is the transfer function
method Al Ba’ba’a, Nouh, and Singh (2017, 2019). In Section 3, the transfer
and scattering matrices are constructed for the presented discrete MM array of
finite length, with adjustable parameters that allow the unit cell to convert
into a monatomic, diatomic, or locally resonant cell. The physical behavior of
a MM crystal near an EP (e.g., scattering of steady state waves off a finite
specimen) can lead to various interesting phenomena. It is illuminating to
summarize the restrictions and simplifications that reciprocity (applicable to
all 1D linear systems) and symmetry considerations (applicable to specific
structures that admit them) provide, particularly when applied to mechanical
systems with loss and gain.

An example of parity symmetric scattering is shown in Section 4. In this
example, the EP of the dynamic matrix is tuned by the loss and gain factors in
the viscous or multi-physical springs to have real frequency and wavenumber,
so that such a singularity point can be accessed in a scattering experiment.
With certain number of unit cells, the MM sample becomes completely invisible
in both directions at the EP frequency of the dynamic matrix, and the energy
is dynamically balanced, i.e., the loss and gain mechanisms perfectly cancel
each other and total mechanical energy is conserved. The discrete modeling
approach helps understand the scattering properties analytically, and can be
easily adapted for various tuning possibilities.

On the other hand, if the sample possesses only the combined parity-time
($\mathcal{PT}$) symmetry, then the _scattering matrix_ can exhibit EPs as
well. To demonstrate this, we show the scattering response of a $\mathcal{PT}$
symmetric system near the EPs of its scattering matrix spectrum in Section
4.2. Non-Hermitian Hamiltonians with $\mathcal{PT}$ symmetry were first
discussed by Bender and Boettcher Bender and Boettcher (1998). A more general
category of pseudo-Hermitian systems in elastodynamics is investigated by
Psiachos and Sigalas Psiachos and Sigalas (2018, 2019). The asymmetric
scattering responses of $\mathcal{PT}$ symmetric media have been investigated
in electronics Sakhdari  _et al._ (2018), photonics Ge, Chong, and Stone
(2012), and acoustics Zhu  _et al._ (2014); Shi  _et al._ (2016); Fleury,
Sounas, and Alu (2016); Achilleos  _et al._ (2017); Fleury and Alu (2014).
These studies have shown that the EPs of scattering matrix correspond to
unidirectional zero reflection and unitary transmission. Moreover, the EPs of
the scattering matrix are associated with spontaneous symmetry breaking and
mark the spectral boundaries between $\mathcal{PT}$ broken and unbroken
phases. The majority of these studies are performed experimentally or
numerically using simulations, which can be time consuming or computationally
expensive. In contrast, using the analytical formulas derived in Section 3 and
appendices, designing these novel artificial media and tuning towards desired
target frequencies can be achieved relatively easily.

The feasibility of implementing gain units (represented by complex-valued
springs in this work), a necessary ingredient of this study, is a major
challenge to experimental realization of such EP-based designs. To this end, a
number of studies have demonstrated implementation of gain units, realized by
electronic devices Popa and Cummer (2014); Fleury, Sounas, and Alu (2016) or
piezoelectric semiconductors Christensen  _et al._ (2016). In a recent study,
Mokhtari et al. Mokhtari  _et al._ (2020) show the possibility of accessing
EPs with fully elastic PCs in real frequency and wave vector domain. With such
a 2D scattering setup, it is then possible to take advantage of the spectral
properties of EPs to design novel sensing devices Wang  _et al._ (2021).

The structure of this paper is as follows. In Section 2, we first study the
dynamic matrix of a discrete resonator system and show the relationship
between coupling strength and level repulsion. This will be followed by an
analytical representation and detailed discussion of the eigenfrequency and
eigenvector behaviors in the vicinity of EPs. Then it will be shown that by
adjusting the stiffness parameters, EPs can be moved to the real frequency and
wavenumber domain. In Section 3, the transfer and scattering matrices for
discrete MM arrays are presented, along with a discussion on the restrictions
reciprocity and fundamental symmetries enforced on these matrices. In Section
4, we examine the scattering behaviors of a parity symmetric system and a
parity-time symmetric system, which feature bi-directional transparency and
one-way reflection near EPs of the dynamic and scattering matrices,
respectively. The paper is concluded with a summary of important results and
potentials for application of EPs resulting from their influence on the
physical response of mechanical metamaterials. The use of the discrete mass-
spring systems provides fundamental insights in MMs and can be easily adapted
for various tuning possibilities.

##  2 Exceptional points in the dynamic matrix eigenspectrum

###  2.1 Level repulsion and coupling of DOFs

The 1D discrete periodic structure studied in this section is represented in
Fig. 1(a). Each cell consists of the main “crystal chain mass” $M^{c}$ and the
“internal resonator mass” $M^{i}$. In each RUC, a linear spring element with
stiffness coefficient $\beta^{i}$ connects the resonator to the crystal. There
is also a spring with stiffness $\beta^{c}$ between every two neighboring
crystal masses. To show the level repulsion and EPs with a simple set up, we
consider a longitudinal wave propagating along the chain. In this analysis,
the crystal masses are constrained to have a single horizontal degree of
freedom (DOF). One can assume that the structure is confined in a tube
parallel to the $x$ axis (with frictionless surfaces). It is also assumed that
the rotational inertia of the masses are high enough to allow one to ignore
the rotational DOFs. The resonator is also constrained to have only one
independent DOF, $u^{i}_{n}$, which makes angle $\theta$ with the horizontal
direction and main chain mass DOF, $u^{c}_{n}$. The coupling constant
$\kappa=\cos\theta$ is defined where the angle $\theta$ is in the range
$[-\pi/2,\pi/2]$. For other values of $\theta$ a simple change of sign in
either of the two DOFs will render the following mathematical description
identically applicable. When $\kappa=1$ this model is identical to the 1D
lattice with resonator model which can be commonly found in literature
Amirkhizi and Wang (2018); Hussein, Leamy, and Ruzzene (2014).

(a)

(b)

Figure 1: (1(a)) Schematic drawing of the studied 1D infinitely periodic
resonator array. (1(b)) Longitudinal wave band structure (real domain) for
different values of coupling constant $\kappa=\cos\theta$. In the example
here, all parameters (mass, stiffness) are normalized to one.

For the $n$-th RUC the DOFs that satisfy Bloch-Floquet periodicity can be
written as:

$\displaystyle u^{c}_{n}$ $\displaystyle=u^{c}\ \exp\left[\mathrm{i}(\omega
t-nQ)\right],$ (1) $\displaystyle u^{i}_{n}$ $\displaystyle=u^{i}\
\exp\left[\mathrm{i}(\omega t-nQ)\right],$ (2)

where $\omega$ is angular frequency, and $n$ is an integer representing cell
location along the chain. The dimensionless wavenumber $Q$ represents the
phase advance between neighbor cells, and it can be calculated as the product
of wavevector component and cell length. In the eigenfrequency study, $Q$ is
usually a prescribed parameter sweeping the Brillouin zone. The complex
amplitudes of displacements in harmonic motion $u^{c}$ and $u^{i}$ are to be
determined. To do this, the equations of motion can be written for the $n$-th
cell and resonator (see Appendix Appendix A):

$\displaystyle M^{c}\frac{\partial^{2}u^{c}_{n}}{\partial t^{2}}$
$\displaystyle=\beta^{c}(u^{c}_{n+1}-2u^{c}_{n}+u^{c}_{n-1})+\kappa\beta^{i}(u^{i}_{n}-\kappa
u^{c}_{n})-(1-\kappa^{2})M^{i}\frac{\partial^{2}u^{c}_{n}}{\partial t^{2}},$
(3) $\displaystyle M^{i}\frac{\partial^{2}u^{i}_{n}}{\partial t^{2}}$
$\displaystyle=\beta^{i}(\kappa u^{c}_{n}-u^{i}_{n}),$ (4)

rendering, for each value of $Q$, an eigenvalue problem:

$\displaystyle[\bm{D}-\lambda\bm{I}]\bm{U}^{R}$ $\displaystyle=\bm{0},$ (5)
$\displaystyle\bm{U}^{L\dagger}[\bm{D}-\lambda\bm{I}]$ $\displaystyle=\bm{0},$
(6)

where $\lambda=\omega^{2}$ is the eigenvalue of the dynamic matrix
$\bm{D}=\bm{M}^{-1}\bm{K}$, $\bm{I}$ is the $2\times 2$ identity matrix,
$\bm{U}^{R}=[u^{c},\ u^{i}]^{\top}$ is the right eigenvector, $\bm{U}^{L}$ is
the left eigenvector, and $\dagger$ denotes complex conjugate transpose.
$\bm{K}$ and $\bm{M}$ are the stiffness and mass matrices of the cell,
respectively:

$\bm{K}=\begin{pmatrix}4\beta^{c}\sin^{2}\dfrac{Q}{2}+\kappa^{2}\beta^{i}&-\kappa\beta^{i}\\\
-\kappa\beta^{i}&\beta^{i}\end{pmatrix},$ (7)
$\bm{M}=\begin{pmatrix}M^{ci}&0\\\ 0&M^{i}\end{pmatrix}.$ (8)

The coupling constant $\kappa=\cos\theta$ quantifies the interaction strength
between the internal resonator and the main crystal chain, and

$M^{ci}=M^{c}+(1-\kappa^{2})M^{i}$ (9)

is defined as the effective mass associated with $u^{c}_{n}$ DOF dynamics.
Solving the characteristic equation $|\bm{D}-\omega^{2}\bm{I}|=0$ yields the
frequency band structure, a representation of which is shown in Fig. 1(b). In
the shown example, all the stiffness and mass values are taken as 1, but the
coordinates of a number of important points and other geometrical features can
be calculated explicitly in terms of the model parameters. As the coupling
constant $\kappa$ approaches 0, level repulsion (LR) becomes more evident and
the dispersion curves appear to approach a crossing point. Only when
$\kappa=0$ the resonator and the cell become fully decoupled leading to an
actual crossing of the branches. Then a topological transition occurs in the
band structure as the frequency gap disappears. In such a case, the right
eigenvectors on the two crossing dispersion curves will stay linearly
independent.

###  2.2 Exceptional points in the complex band structure

For a 2-DOF system like the one shown in Fig. 1(a), normally there are two
eigenfrequencies for each value of wavenumber. There also exist branch points
(BP), potentially in the complex domain, where two frequency solutions match
(degeneracy or frequency coalescence). After solving for the frequency as a
function of $Q$ analytically, the location of branch points can be obtained:

$\displaystyle Q_{\mathrm{BP}}$
$\displaystyle=2\arcsin\left(\dfrac{\omega^{i}}{2\omega^{c}}\left(1+\mathrm{i}\kappa\sqrt{\dfrac{M^{i}}{M^{ci}}}\right)\right),$
(10) $\displaystyle\omega_{\mathrm{BP}}$
$\displaystyle=\omega^{i}\sqrt{1+\mathrm{i}\kappa\sqrt{\dfrac{M^{i}}{M^{ci}}}}.$
(11)

The shown solution is in the region where $\Re Q\geq 0$ and $\Im Q\geq 0$.
Here the symbols $\Re,\Im$ represent the real and imaginary parts of a complex
quantity. A non-zero coupling $\kappa$ will enforce the branch point to be
complex-valued for real parameters in the cell and there are in general eight
possible solutions, namely $(\pm Q_{\mathrm{BP}},\pm\omega_{\mathrm{BP}})$ and
$(\pm Q_{\mathrm{BP}}^{*},\pm\omega_{\mathrm{BP}}^{*})$. At the branch point
Eq. 5 becomes:

$\frac{\kappa\beta^{i}}{M^{ci}M^{i}}\begin{pmatrix}\mathrm{i}\sqrt{M^{ci}M^{i}}&-M^{i}\\\
-M^{ci}&-\mathrm{i}\sqrt{M^{ci}M^{i}}\end{pmatrix}\begin{pmatrix}u^{c}\\\
u^{i}\end{pmatrix}=\bm{0}.$ (12)

When $\kappa=0$ the $2\times 2$ matrix in Eq. 12 becomes a zero matrix (which
means any arbitrary vector in $\mathbb{C}^{2}$ is an eigenvector) and the
branch point is exactly the crossing point shown in Fig. 1(b) residing is in
the real domain. The two frequency solutions are overlapping each other while
two linearly independent eigenvectors exist. Such a case is referred to as a
degeneracy. For non-zero $\kappa$ values, the branch point is referred to as
an exceptional point Heiss and Sannino (1990) (EP) which is usually in the
complex parameter domain. As the $\kappa$ value gets closer to zero, avoided
crossing/level repulsion will be more apparent in the real domain, and the EP
location will have smaller imaginary parts. At the EP there exists only one
non-trivial right eigenvector:

$\bm{U}^{R}_{\mathrm{EP}}=\begin{pmatrix}-\mathrm{i}\\\
\sqrt{\dfrac{M^{ci}}{M^{i}}}\end{pmatrix}.$ (13)

The corresponding left eigenvector of matrix $\bm{D}$ at EP is

$\bm{U}^{L\dagger}_{\mathrm{EP}}=\begin{pmatrix}-\mathrm{i},&\sqrt{\dfrac{M^{i}}{M^{ci}}}\end{pmatrix}.$
(14)

Here we show an example of the complex band structure for $\kappa=0.5$ and
allow the wavenumber $Q$ to be complex. All cell parameters (mass, stiffness)
are set to one for the sake of demonstration. The calculated complex frequency
and right eigenvectors are shown only in region where $\Re Q\geq 0,\Im Q\geq
0$, and $\Re\omega\geq 0$. Figures 2(a) and 2(b) show the real and imaginary
part of the band structure, respectively. The components of the right
eigenvector are shown in Figs. 2(c) and 2(e). For this configuration the EP is
located at $Q_{\mathrm{EP}}=1.36218+\mathrm{i}0.63297$ and
$\omega_{\mathrm{EP}}=1.01711+\mathrm{i}0.18580$, as represented by the solid
black dot. The two modes are separated based on the continuity of branches in
any complex $Q$ disk around the origin that does not include $Q_{\mathrm{EP}}$
and are shown in different colors. The corresponding right eigenvector
components are shown in Figs. 2(c) and 2(e). The complex eigenvectors
associated with each mode are normalized by a complex factor in such a way
that
$\mathinner{\\!\left\lVert\bm{U}^{R}\right\rVert}=\sqrt{u^{c*}u^{c}+u^{i*}u^{i}}=1$,
and $u^{i}\in\mathbb{R}$. It is important to normalize both the amplitude and
complex phase of the eigenvector in such a way to ensure consistency
throughout the analysis. To keep $u^{i}$ on the real axis, both components of
eigenvector are rotated together in the complex plane keeping their ratio
unchanged. The major benefit of phase normalization is that the eigenvector
components can be shown in a continuous manner (see Figs. 2(c) to 2(e)) even
in the vicinity of the EPs, thus making it easier to understand the mode shape
behavior. It is clear that both the frequencies and eigenvectors form Riemann
sheet structures in the vicinity of the EP, and all the complex quantities can
be made to behave continuously (when properly normalized) with respect to $\Re
Q$ and $\Im Q$, at any simply connected neighborhood that does not include
$Q_{\mathrm{EP}}$. For the sake of presentation quality of 3D figures, we show
only four representative cuts of these Riemann sheets at $\Im Q=0,\ 0.3,\ \Im
Q_{\mathrm{EP}},$ and 1. When $\Im Q=0$ the frequencies are real. The two
branches are clearly distinguished by considering $\Re\omega$ and $\angle
u^{c}$. As $\Re Q$ increases, the amplitudes of eigenvector components
increase or decrease monotonically due to local resonance. There is an inverse
correlation between the coupling strength (which is quantified as $\kappa$ in
this case) and the abruptness of such change in amplitude. Weaker coupling
(smaller but non-zero $\kappa$) results in more evident frequency level
repulsion and sharper changes in displacement amplitudes. The exact $\pi$
difference between the two lines in Fig. 2(d) indicates a sign difference
between the $u^{c}$ of acoustic and optical branches, with $u^{i}$ normalized
to be real and positive. A detailed discussion on eigenfrequency and mode
shape behaviors in the LR region in real $Q$ domain can be found in our
previous work Amirkhizi and Wang (2018).

As $\Im Q$ increases from 0, all the presented quantities show similar trends
in terms of continuity. The two branches remain continuous, as long as $\Im
Q<\Im Q_{\mathrm{EP}}$. When $\Im Q=\Im Q_{\mathrm{EP}}$, all these quantities
coalesce at the EP. Continuity with respect to $\Re Q$ can not be used as the
basis of branch selection at this point due to the coalescence. In other
words, branch sorting becomes ambiguous at $\Im Q=\Im Q_{\mathrm{EP}}$. If one
seeks to extend the continuous branches (for $\Im Q<\Im Q_{\mathrm{EP}}$)
beyond $\Im Q_{\mathrm{EP}}$, by maintaining continuity along $\Im Q$, the
resulting choices will be discontinuous along $\Re Q$ when $\Im Q>\Im
Q_{\mathrm{EP}}$. If one wishes to maintain the continuity along $\Re Q$, the
branches will be discontinuous along $\Im Q$, when $\Re Q>\Re
Q_{\mathrm{EP}}$, see for example the slice at $\Im Q=1$ in Figs. 2(a) to
2(e). In general, it can be seen that both the eigenvalues and the
eigenvectors maintain analyticity, except at the EP where the Taylor series
expansion fails.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Four representative cross sectional cuts (at four different $\Im Q$
values) of the Riemann sheets showing (2(a)) real and (2(b)) imaginary parts
of the frequency, (2(c)) amplitude and (2(d)) complex argument of main crystal
chain DOF $u^{c}$, and (2(e)) resonator DOF $u^{i}$ (when eigenvectors are
normalized for it to be real). The two modes are sorted based on branch
continuity along $\Re Q$. (2(f)) Inner product of left and right eigenvectors
which belong to same ($\alpha=\beta$) and different ($\alpha\neq\beta$) modes.

The calculated inner products of the normalized left and right eigenvectors
are shown in Fig. 2(f), where it can be seen that the left and right
eigenvectors corresponding to the different eigenfrequencies are orthogonal to
each other, except at the EP. The bi-orthogonality relation of the non-
Hermitian system reads:

$\langle\bm{U}^{L}_{\alpha},\bm{U}^{R}_{\beta}\rangle=\begin{cases}0&\text{at
EP;}\\\ \delta_{\alpha\beta}&\text{elsewhere;}\end{cases}$ (15)

where $\delta_{\alpha\beta}$ is the Kronecker delta, and subscripts
$\alpha,\beta=1,2$ denote the first or second mode. The self-orthogonality
Moiseyev (2011) at the EP implies a defect of the Hilbert space Kato (1995);
Rotter (2003).

###  2.3 Tuning an EP into real frequency and wavenumber domain

All previous results are based on an EP with complex $Q$ and $\omega$ values.
It is possible to look for EPs with real $Q$ using complex stiffness $\beta$.
For physical realization of such systems, see Section 1. Associated Riemann
sheets for the band structure and eigenvectors may be found following the
procedure discussed earlier. A similar analysis on bifurcation in the vicinity
of EPs in such systems with damping (complex stiffness) can be found in
Ref.Maznev (2018), where the EP has complex frequency and real wavenumber. If
one wishes to study the scattering properties around an EP, one may try
instead to design a system including an EP that has both real frequency and
$Q$. Such a system may be interrogated experimentally through scattering of
harmonic waves off a finite specimen. Given the EP location in Eq. 10 and Eq.
11, it is possible to make both $Q_{\mathrm{EP}}$ and $\omega_{\mathrm{EP}}$
real if the complex arguments of stiffness parameters are such that:

$\angle\beta^{c}=-\angle\beta^{i}=\angle\left[1+\mathrm{i}\kappa\sqrt{\frac{M^{i}}{M^{ci}}}\right].$
(16)

It is inevitable that such process will make one stiffness parameter lossy
while the other requires gain (i.e., has negative imaginary part), which is
feasible as discussed in literaturePopa and Cummer (2014); Fleury, Sounas, and
Alu (2016); Christensen  _et al._ (2016). With these adjustments in stiffness
values, Eqs. 10 to 14 are still valid, and the eigenfrequency and eigenvector
behaviors are still qualitatively the same as those shown in Fig. 2, except
the location of EP is now adjustable. The behavior of such structures will be
further studied in Section 4.

The measurable response of the micro-structured media is, in fact, not just
based on their dispersion surfaces, but rather more thoroughly understood from
the scattering off finite specimens. In the following sections, the influence
of EPs of the band structure, and independently, those of the scattering
matrix, on the overall response of finite specimens are studied.

##  3 Transfer and scattering matrices

The analysis of steady state waves (real frequency) traveling in an infinite
homogeneous domain and interacting with a finite sized specimen of a 1D MM
array (with finite number of unit cells) can be solved easily using the
transfer matrix (TM) of such structures. This is different from eigenfrequency
analysis which analyzes an infinite periodic array of unit cells, though the
eigenfrequency band structure are also essentially associated with the
eigenvalues of the transfer matrix. In either case, the TM is the matrix form
of the linear relationship between the physical states on the left and right
boundaries of a control volume or the unit cell. To utilize the transfer
matrix method, the unit cell is selected as the part in the dashed rectangle
shown in Fig. 3. It is selected in such a way that the springs connecting
crystals are cut in the middle. To be more general, the springs at the left
and the right sides of a crystal chain atoms are allowed to be different, as
denoted by stiffnesses $\beta^{p}$ and $\beta^{q}$ in Fig. 3.

Figure 3:  Control volume selection for transfer matrix analysis. Note that
the cell is no longer inherently symmetric.

Therefore, the left and right halves of the main crystal chain springs are
$2\beta_{p}$ and $2\beta_{q}$. This setup allows the analysis of non-
periodic/permuted MM samples. The derivation of the TM can be found in
Appendix Appendix B. Applying Bloch-Floquet periodicity in an infinitely
periodic array, the governing equation reads:

$[\mathsf{T}^{cell}-\mathrm{e}^{-\mathrm{i}Q}\bm{I}]\begin{pmatrix}v^{l}\\\
N^{l}\end{pmatrix}=\bm{0},$ (17)

where $\mathsf{T}^{cell}$ is the transfer matrix of a unit cell, $Q$ is the
phase advance, and $v^{l},N^{l}$ are the velocity and internal axial force at
the left boundary. For any desired frequency, solving the characteristic
equation of Eq. 17 for the normalized wavenumber $Q$ yields the band
structure. This is complementary to eigenfrequency calculation, where one
would solve for frequencies given a prescribed normalized wavenumber (phase
advance), $Q$.

Figure 4:  Scattering set up of $J$ cells between two semi-infinite domains.

For finite structures, once the TM of one arbitrary cell or a number of them
is obtained, one can retrieve the scattering coefficients of the model as
shown in Fig. 4. In the scattering experiment, the sample is set between two
homogeneous semi-infinite domains and contains $J$ cells. The cells of the
sample are in general allowed to be different. Here the semi-infinite domains,
without loss of generality, are modeled as circular bars. The Young’s modulus,
mass density, and cross sectional radius of the two identical cylindrical bars
are denoted by $E_{0}$, $\rho_{0}$, and $r_{0}$, respectively. We generally
equate the measurement/de-embedding locations with the boundaries of the
sample, i.e., $x^{a}=x^{l}$ and $x^{b}=x^{r}$. Note that the superscripts $l$
and $r$ here represent locations with respect to the entire sample and their
distinction from the cell faces earlier should be clear from the context. The
displacements at these locations are assumed to have the form
$A^{(a,b)(+,-)}e^{\mathrm{i}\omega t}$ (harmonic waves), where the
superscripts $+$ and $-$ represent the waves propagating in positive and
negative $x$ directions in the bars (in the sense of phase advance or flux
direction). The scattering matrix

$\tilde{\mathsf{S}}=\begin{pmatrix}\mathsf{S}_{ba}&\mathsf{S}_{bb}\\\
\mathsf{S}_{aa}&\mathsf{S}_{ab}\end{pmatrix}$ (18)

relates the complex displacement or velocity amplitudes of outgoing ($A^{b+}$
and $A^{a-}$) and incoming ($A^{a+}$ and $A^{b-}$) waves at measurement
locations $x^{l}$ and $x^{r}$:

$\bm{A}_{out}=\begin{pmatrix}A^{b+}\\\
A^{a-}\end{pmatrix}=\tilde{\mathsf{S}}\begin{pmatrix}A^{a+}\\\
A^{b-}\end{pmatrix}=\tilde{\mathsf{S}}\bm{A}_{in}.$ (19)

The derivation of the scattering matrix can be found in Appendix Appendix C.

If one considers $\mathinner{\\!\left\lvert\mathsf{T}\right\rvert}=1$ and
$Z^{a}=Z^{b}$, then Eqs. 47 to 51 will be simplified and identical to the
scattering coefficients derived in literature Amirkhizi (2017), and the
scattering matrix $\tilde{\mathsf{S}}$ given by Eq. 18 has the eigenvalues:

$\sigma_{1,2}=\mathsf{S}_{ab}\pm\sqrt{\mathsf{S}_{aa}\mathsf{S}_{bb}},$ (20)

since $\mathsf{S}_{ab}=\mathsf{S}_{ba}$. The eigenvectors are

$\bm{\mu}_{1,2}=\begin{pmatrix}\pm\sqrt{\mathsf{S}_{bb}}\\\
\sqrt{\mathsf{S}_{aa}}\end{pmatrix}.$ (21)

Coalescence of the $\tilde{\mathsf{S}}$ eigenspectrum occurs if and only if
one of the reflection coefficients becomes zero. An EP of the scattering
matrix is thus related to one-way reflection phenomenon.

###  3.1 Effect of reciprocity

Reciprocity is commonly considered Herrero-Durá  _et al._ (2019); Horsley  _et
al._ (2014); Fleury  _et al._ (2015); Deák and Fülöp (2012) as a property
observed in scattering measurements, i.e., the equivalence of transmission
coefficients when a source and a detector exchange their positions. It is
shown in Ref Alizadeh and Amirkhizi (2021) that elastodynamic reciprocity
imposes certain restrictions on the constitutive tensors of layered media.
Here, we consider reciprocity as a fundamental property of linear elastic
structures (and a broad class of linear viscoelastic systems, e.g., those that
can be represented by simple networks of 2-force spring or dashpot, or
continuum elements with isotropic viscoelasticity), described by Betti’s
reciprocity theorem. It can be seen from Eq. 44 that this transfer matrix has
a determinant of 1, i.e.,
$\mathinner{\\!\left\lvert\mathsf{T}^{cell}\right\rvert}=1$. The unimodularity
of the TM is equivalent to the reciprocity of the 1D medium. It is worth to
study the proof and understand the assumptions that are sometimes implicitly
included. Consider an arbitrary control volume in a 1D linear time-invariant
system. Assuming absence of body forces, Betti’s reciprocity theorem in
frequency domain states that:

$F^{l}_{\alpha}u^{l}_{\beta}+F^{r}_{\alpha}u^{r}_{\beta}=F^{l}_{\beta}u^{l}_{\alpha}+F^{r}_{\beta}u^{r}_{\alpha},$
(22)

where superscript $l$ or $r$ represent the left or right boundary of the
control volume and subscripts $\alpha$ and $\beta$ represent two different
states, and $u$ and $F$ represent displacement and applied force on the
boundaries. The axial traction force (positive in the positive $x$ direction)
are $F^{l}=-N^{l}$ and $F^{r}=N^{r}$, and $N^{l,r}$ are assumed to be tensile
positive. The theorem is generally proved in elastic systems based on the
existence of the strain energy density function and the subsequent major
symmetry of the elasticity tensor Achenbach (2006). In linear viscoelastic
systems, isotropic material response will also allow for a proof in frequency
domain. Analytical considerations have been used to address this in
viscoelastic materials Rogers and Pipkin (1963); Day (1971); Matarazzo (2001).
For the systems studied here (among a large class of discrete structures), the
reciprocity can be proven explicitly. The general proof is omitted, but we
show that the reciprocity of the systems considered here is equivalent to the
$\mathinner{\\!\left\lvert\mathsf{T}\right\rvert}=1$. Assuming harmonic
velocities $v^{l,r}=\mathrm{i}\omega u^{l,r}$, Eq. 22 becomes:

$v^{r}_{\alpha}N^{r}_{\beta}-v^{r}_{\beta}N^{r}_{\alpha}=v^{l}_{\alpha}N^{l}_{\beta}-v^{l}_{\beta}N^{l}_{\alpha}\leavevmode\nobreak\
.$ (23)

Define matrix $\bm{J}$ as

$\bm{J}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\leavevmode\nobreak\ ,$ (24)

and

$\bm{\psi}=\begin{pmatrix}v\\\ N\end{pmatrix}\leavevmode\nobreak\ .$ (25)

Then the reciprocity condition Eq. 23 can be written as

$(\mathsf{T}\bm{\psi}^{l}_{\alpha})^{\top}\bm{J}(\mathsf{T}\bm{\psi}^{l}_{\beta})=(\bm{\psi}^{l}_{\alpha})^{\top}\bm{J}\bm{\psi}^{l}_{\beta}\leavevmode\nobreak\
.$ (26)

Since the L.H.S. of Eq. 26 is identical to
$(\bm{\psi}^{l}_{\alpha})^{\top}(\mathsf{T}^{\top}\bm{J}\mathsf{T})\bm{\psi}^{l}_{\beta}$,
the structure is reciprocal if and only if TM is symplectic, i.e.,

$\mathsf{T}^{\top}\bm{J}\mathsf{T}=\bm{J},$ (27)

which is equivalent to $\mathinner{\\!\left\lvert\mathsf{T}\right\rvert}=1$
for $2\times 2$ matrices.

The transfer matrix of any array of cells may be constructed by multiplying
their individual transfer matrix in reverse order since, $\bm{\psi}^{r}$ of
one cell is the same as $\bm{\psi}^{l}$ of the cell to its right. Therefore,
the transfer matrix of any array of such cells, will also be reciprocal and
have a determinant of 1. This would be true regardless of whether either the
parity symmetry (i.e., $\beta^{p}=\beta^{q}$) or time-reversal symmetry (i.e.,
$\\{\beta^{p},\ \beta^{q}\\}\subset\mathbb{R}$) of any of the cells is broken
or not. Since $\mathinner{\\!\left\lvert\mathsf{T}\right\rvert}=1$, the two
eigenvalues of $\mathsf{T}$ are inverse of each other, which yields
$Q^{-}(\omega)=-Q^{+}(\omega)$, where the superscripts $+$ and $-$ represent
up to two possible solutions, presumably forward and backward traveling waves
in the sense of phase advance or power flux. Note that at least for 1D media,
either $\mathcal{P}$ or $\mathcal{T}$ symmetry will inherently lead to
reciprocity, but a system that only admits combined $\mathcal{PT}$ symmetry
can potentially be non-reciprocal.

###  3.2 Symmetry considerations

The effect of symmetries of the sample are best represented in the transfer
matrix formulation and, similar to the reciprocity consideration, they are
independent of the environment. However, the effect on scattering response
will include the environment properties. The summary of symmetry restrictions
is listed in Table 1. Here,

$\bm{F}=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix},\leavevmode\nobreak\
\bm{P}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

The detailed proof is omitted, but can be derived following similar processes
presented in literature Jin  _et al._ (2016); Ge, Chong, and Stone (2012);
Mostafazadeh (2014). Coalescence of the $\tilde{\mathsf{S}}$ eigenvectors and
its inherent one-way reflection phenomenon can not be achieved by a parity
symmetric system (including the identical bars on either side) because
$\mathcal{P}$ symmetry simply imposes strong condition that enforces equal
reflections, i.e., $\mathsf{S}_{aa}=\mathsf{S}_{bb}$. A system that has time
reversal $\mathcal{T}$ symmetry does not support the coalescence of
$\tilde{\mathsf{S}}$ eigenvectors either because it leads to
$|\mathsf{S}_{aa}|=|\mathsf{S}_{bb}|$, which again precludes one-way
reflection. However, a $\mathcal{PT}$ symmetric system that lacks individual
$\mathcal{T}$ and $\mathcal{P}$ symmetries may demonstrate non-trivial
coalescence of $\tilde{\mathsf{S}}$ eigenvectors and the one-way reflection
phenomenon. While $\mathcal{PT}$ symmetry is usually studied with
electromagnetic/optical setups Hu, Chen, and Hu (2017); Suneera and Subha
(2018), the conclusions here apply as well.

Table 1: Effects of symmetries on the transfer and scattering matrices. Notice that the $\mathsf{T}$ restrictions of the $\mathcal{P}$ symmetry implicitly leads to reciprocity through $|\mathsf{T}|=1$ as well, which is why it is not shown in the table, but in case of $\mathcal{T}$ it is an additional conclusion. Symmetry |  $\mathsf{T}$ restrictions |  $\tilde{\mathsf{S}}$ restrictions
---|---|---
$\mathcal{P}$ | $\mathsf{T}=\bm{F}\mathsf{T}^{-1}\bm{F}$ | $\tilde{\mathsf{S}}=\bm{P}\tilde{\mathsf{S}}\bm{P}$
$\mathcal{T}$ |  $\mathsf{T}=\bm{F}\mathsf{T}^{*}\bm{F}$, $|\mathsf{T}|=1$ | $\tilde{\mathsf{S}}=\bm{P}(\tilde{\mathsf{S}}^{*})^{-1}\bm{P}$
$\mathcal{PT}$ | $\mathsf{T}=(\mathsf{T}^{*})^{-1}$ | $\tilde{\mathsf{S}}=(\tilde{\mathsf{S}}^{*})^{-1}$

The summary here is applicable for 1D linear media. The excitation frequencies
are prescribed to be real. Although this paper mainly uses discrete systems as
examples, the conclusions here are independent from the discrete setup and are
applicable for 1D continuum systems as well. One can use these relations to
simplify the process and reduce the number of needed experiments in real
scattering experiments Aghighi, Morris, and Amirkhizi (2019); Abedi and
Amirkhizi (2020). In the following, we choose $Z^{a}=Z^{b}=Z_{0}=\pi
r_{0}^{2}\sqrt{E_{0}\rho_{0}}$ so that we can focus on the symmetry properties
of the MM samples only.

##  4 Examples of scattering responses

In the following illuminating examples the same setup in Section 3 is used,
where the bars have Young’s modulus
$E_{0}=$69\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$, density
$\rho_{0}=$2710\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$, and
radius $r_{0}=$1\text{\,}\mathrm{m}\mathrm{m}$$. The unit cell length is
$d=$0.1\text{\,}\mathrm{m}$$. Different cases are demonstrated by varying the
number and properties of the unit cells.

###  4.1 Scattering at the EP of dynamic matrix spectrum ($\mathcal{P}$
symmetric system)

(a)

(b)

(c)

(d)

(e)

Figure 5: Scattering response of the sample with 4 cells. The exceptional
point belongs to the eigenfrequency band structure and is moved to real
frequency domain
($\omega_{\mathrm{EP}}\approx$338.1\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$)
by using suitable loss and gain springs. The band structure shown in (5(e)) is
calculated using TMM. The scattering amplitudes, phases, and power fluxes (as
well as net power loss) are shown in (5(a)), (5(b)) and (5(c)), respectively.
The eigenvalues of $\tilde{\mathsf{S}}$, denoted by $\sigma$, are shown in
(5(d)).

In this section, we examine the scattering properties at the EP of the dynamic
matrix eigenspectrum derived in Section 2. To study the scattering of steady
state harmonic waves, the EP is located in the real domain. In this section
the unit cell is set to be symmetric (i.e., $\beta^{p}=\beta^{q}=\beta^{c}$,
see Fig. 3 and Eq. 44). The main chain crystal and resonator masses are chosen
to be equal $M^{c}=M^{i}=$0.1\text{\,}\mathrm{k}\mathrm{g}$$ and the coupling
constant is selected as $\kappa=0.5$. The stiffness values are
$\beta^{p}=\beta^{q}=\beta^{i*}\approx$10+3.78\text{$\mathrm{i}$}\text{\,}\mathrm{k}\mathrm{N}\mathrm{/}\mathrm{m}$$,
based on Eq. 16. The band structure of such a unit cell can be obtained using
Eq. 17, and is shown in Fig. 5(e). The two solutions are denoted by $Q^{\pm}$.
The EP is calculated based on Eqs. 10 to 11 and is located at
$\omega_{\mathrm{EP}}\approx$338.1\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$,
and $Q_{\mathrm{EP}}^{\pm}=\pm\pi/2$, which are both real due to the choice of
$\beta$ values. As pointed out by Maznev Maznev (2018), no branch bifurcation
can be observed due to the fact that the wavenumber $Q$ is solved as a complex
function of real frequency in the scattering analysis. This band structure is
associated with an infinitely periodic array and therefore agnostic to the
number of cells in the scattering analysis. However, the actual scattering
responses of such structures are affected by the number of unit cells.

To demonstrate this a sample consisting of $J=4$ unit cells is analyzed. The
amplitude, phase, and associated power flux scattering coefficients are shown
in Figs. 5(a) to 5(c). The eigenvalues $\sigma$ of the scattering matrix are
shown in Fig. 5(d). At the frequency of the EP, unitary transmission and zero
reflection can be observed. The energy is dissipated at most frequencies due
to the lossy nature of cell springs $\beta^{p}$ and $\beta^{q}$. The net
absorption (loss) of energy per unit time is shown by in Fig. 5(c). At the EP
frequency, the power generated by the gain in resonator springs compensates
exactly for the loss from main chain springs. This fact can also be seen in
the eigenvalue plot Fig. 5(d) as only at the EP frequency, the two eigenvalues
of $\tilde{\mathsf{S}}$ reach the unit circle (shown as the dashed circle),
indicating the amplitude of incoming and outgoing waves are equal.

In the cases studied here, the phases of the stiffness parameters follow the
relation in Eq. 16, while $\Re\beta^{i}=\Re\beta^{c}$ and $M^{i}=M^{c}$. Then
based on Eq. 10, Eq. 11 and Eq. 44 it can be found that at the EP the sample
transfer matrix becomes

$\mathsf{T}(\omega_{\mathrm{EP}})=(-\bm{I})^{N/2},$ (28)

when $N$ is even. Under such circumstances, the transmission coefficients
$\mathsf{S}_{ab}$ and $\mathsf{S}_{ba}$ will have unitary amplitudes, and the
reflection coefficients $\mathsf{S}_{aa}$ and $\mathsf{S}_{bb}$ will vanish.
This bi-directional reflectionlessness at $\omega_{\mathrm{EP}}$ is not
affected by the outside material (assuming two bars are identical), as all the
impedance terms in the S-parameters will be canceled out. The frequency-
dependent reflectionless scattering property can lead to design of wave
filtering devices that only allow waves with certain frequency to transmit.
According to Eq. 28, the sample becomes fully invisible at
$\omega_{\mathrm{EP}}$ if cell number is $J=4,8,12,\cdots$, as if the two
boundaries $x^{l}$ and $x^{r}$ are directly connected, irrespective of the
outside bar material. However, for $J=2,6,10\cdots$ cells, an extra phase of
$\pi$ will be added to each scattering coefficient, with the amplitude being
the same as observed also in $J=4,8,12,\cdots$ cases.

At the frequency of the dynamic matrix EP, the sample with even $N$ happens to
satisfy an apparent overall $\mathcal{T}$ symmetry condition for
$\tilde{\mathsf{S}}$, with balanced energy gain and loss in the system.
Multiple application scenarios could arise with such exotic properties of EPs.
For example, the bi-directional reflectionless features of EPs can be used for
acoustic camouflage. Since the reflected waves will be suppressed at the EPs’
frequencies, a target object covered by a properly designed micro-structured
medium will be undetectable by a sonar-based sensor. The scattering matrix has
repeating eigenvalues at this frequency, but the two eigenvectors remain
linearly independent and, not surprisingly, there is no reason for the
scattering matrix to have an EP related to that of the dynamic matrix. The EP
and coalescence of the scattering matrix eigenspectrum shall be analyzed next.

###  4.2 Scattering at the EP of scattering matrix spectrum ($\mathcal{PT}$
symmetric system)

The scattering matrix may have an EP only when the system exhibits
$\mathcal{PT}$ symmetry but not individual $\mathcal{P}$ or $\mathcal{T}$
symmetries. Since the EP of the scattering matrix is generally unrelated to
the eigenfrequency band structure of locally resonant systems and to simplify
further exposition, we remove the internal resonator so that $M^{i}=0$.
Consider two spring constants $\beta^{q}=\beta^{p*}$ with $\beta^{p}$ in the
first quadrant (lossy). A cell with $\beta^{q}$ on right and $\beta^{p}$ on
the left and mass $M^{g}$ is referred to as $g$. The transfer matrix of the
$g$ cell is:

$\mathsf{T}^{(g)}=\begin{pmatrix}1-\dfrac{M^{g}\omega^{2}}{2\beta^{q}}&\mathrm{i}\omega\dfrac{2\beta^{p}+2\beta^{q}-M^{g}\omega^{2}}{4\beta^{p}\beta^{q}}\\\
\mathrm{i}M^{g}\omega&1-\dfrac{M^{g}\omega^{2}}{2\beta^{p}}\end{pmatrix}\leavevmode\nobreak\
.$ (29)

A cell with $\beta^{p}$ on the right and $\beta^{q}$ on the left and mass
$M^{l}$ is referred to as $l$. The transfer matrix of the $l$ cell is denoted
as $\mathsf{T}^{(l)}$, which can be derived by changing the superscript $p$
into $q$ and vice versa in Eq. 29 as well as changing $M^{g}$ to $M^{l}$. We
construct a sample consisting of five cells, for which total transfer matrix
is
$\mathsf{T}=\mathsf{T}^{(g)}\mathsf{T}^{(l)}\mathsf{T}^{(g)}\mathsf{T}^{(l)}\mathsf{T}^{(g)}$.
A numerical example based on $\beta^{p}=(10+1\mathrm{i})\ \mathrm{kN/m}$ and
$\beta^{q}=(10-1\mathrm{i})\ \mathrm{kN/m}$ with masses
$M^{g}=$0.12\text{\,}\mathrm{k}\mathrm{g}$$ and
$M^{l}=$0.10\text{\,}\mathrm{k}\mathrm{g}$$ is studied here.

(a)

(b)

(c)

(d)

(e)

Figure 6: Scattering response near two EPs of the scattering matrix for a
$\mathcal{PT}$ symmetric system. (6(a)) Amplitudes of the scattering
coefficients in logarithmic scale. (6(b)) Amplitudes of the transmission
coefficients in linear scale for clarification. (6(c)) the scattering
coefficient phases, (6(d)) the eigenvalues of $\tilde{\mathsf{S}}$ matrix, and
(6(e)) the second components of the eigenvectors Eq. 30 of
$\tilde{\mathsf{S}}$ matrix.

The scattering responses are calculated and shown in Fig. 6, where the
frequency range is $[320,\ 420]\ \mathrm{rad/s}$. Figure 6(a) shows the
amplitudes of scattering coefficients in the logarithmic scale, where two
poles of $\mathsf{S}_{aa}$ can be found at
$\omega_{\mathrm{EP1}}\approx$348.7\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$
and
$\omega_{\mathrm{EP2}}\approx$405.4\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$$.
At these two frequencies, the left reflection coefficient $\mathsf{S}_{aa}$
becomes essentially zero, indicating one-way reflection. Based on Eqs. 20 and
21, the $\tilde{\mathsf{S}}$ matrix exhibits coalescing eigenvalues and
eigenvectors at $\omega_{\mathrm{EP(1,2)}}$, as shown in Figs. 6(d) and 6(e).
The eigenvectors are re-normalized as

$\bm{\mu}=\begin{pmatrix}1\\\ \mu\end{pmatrix},$ (30)

and only the second component is shown in Fig. 6(e). The two EPs are labelled
as EP1 and EP2, and they correspond to the phase transition thresholds where
the $\mathcal{PT}$ symmetry of the eigenvectors is spontaneously broken. At
these EPs, the transmission amplitude becomes one, as shown in Fig. 6(b). Both
$\sigma$ and $\mu$ bifurcate at the EPs. For the frequency smaller than
$\omega_{\mathrm{EP1}}$ or larger than $\omega_{\mathrm{EP2}}$, the system is
in the $\mathcal{PT}-$unbroken phase, where the two reflection coefficients
have the same phase (see Fig. 6(c)) and the transmission coefficient has
amplitude smaller than one. The non-degenerate eigenvalues are both unimodular
but different in phase, and the eigenvectors are real. The frequency range
$(\omega_{\mathrm{EP1}},\omega_{\mathrm{EP2}})$ represents the
$\mathcal{PT}-$broken phase. In the $\mathcal{PT}-$broken phase, the
eigenvalues have same phases but inverse amplitudes, i.e.,
$|\sigma_{1}\sigma_{2}|=1$. The second component of the eigenvector becomes
purely imaginary. The reflection coefficients have exactly $\pi$ difference in
their phases. The transmission amplitude exceeds one, and all single-sided
incident waves will be amplified. This single-sided reflection is most easily
observed in the the transmission and reflection amplitudes shown in Fig. 6(a).

Due to the defectiveness of the scattering matrix, prescribed or measured
states at $\omega_{\mathrm{EP(1,2)}}$ can not be decomposed into the
eigenvectors of $\tilde{\mathsf{S}}$. On the other hand, a scattering state
can always be decomposed into the two eigen-modes when operating in the
$\mathcal{PT}-$broken or unbroken phases. In the symmetry-unbroken phase, the
two basis modes have purely real components and are invariant under
$\mathcal{PT}$ reversal. In the symmetry-broken phase, the two basis vectors
no longer maintain the symmetry due to the imaginary components. Nevertheless,
the $\mathcal{PT}$ symmetry conditions in Table 1 are always satisfied.

##  5 Conclusions

Using a simple tunable discrete model setup, the exceptional points (EP) of
the dynamic and scattering matrices of monatomic, diatomic, or locally
resonant mechanical systems are analyzed. It is shown that the eigenfrequency
band structure of a micro-structured medium can possess EPs as complex
singularities or defects of the associated linear operators. Various phenomena
associated with wave propagation in mechanical materials can be categorized
and analyzed with this tool set. To summarize, the highlights of this work
are:

  * •

Elucidation of EP-related phenomena such as level repulsion, mode coalescence,
mode switching and self-orthogonality in a simple yet physical setup,

  * •

Summary of the transfer and scattering matrix properties for general 1D
(discrete and continuous) systems,

  * •

Demonstration of unique scattering behavior at the EPs of the dynamic and
scattering matrices (bi-directional reflectionless and single-sided
reflection, respectively).

This study of discrete metamaterial systems contributes to fundamental
understanding of the EPs in mechanical micro-structured media, and will be of
interest to novel applications such as robust sensing and filtering. The
complex valued springs (especially the ones with gains) used in this paper
represent some practical challenges. However, they are helpful in the
theoretical investigation of the topological and spectral properties of EPs,
and ideas for their realization are already presented in literature.
Furthermore, the discrete modeling approach can be utilized for conceptual
design of novel devices as well as transferring the knowledge from EM and
photonics domain into mechanical counterpart devices. See for example, the
potential lasing mechanism Zhang, Wang, and Song (2019) and prototype EP-
enabled lasing devices studied in optics Peng  _et al._ (2016).

ACKNOWLEDGEMENTS
The authors acknowledge NSF grant #1825969 to the University of Massachusetts,
Lowell.

DATA AVAILABILITY
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

ACKNOWLEDGEMENTS
The Version of Record of this article is published in The European Physical
Journal Plus, and is available online at
https://doi.org/10.1140/epjp/s13360-022-02626-6

##  Appendix A Equations of motion derivation

To derive the equations of motion (EOMs) of a unit cell in Section 2, an
illustration is shown in Fig. 7. The cell springs $\beta^{c}$ and neighbor
cells are not shown here.

Figure 7:  Illustration of the forces and displacement of the local resonator.
The subscript $n$ is omitted here.

In one unit cell, the forces acting on the internal resonator include two
components. The spring force $F^{is}$ is parallel to the $u^{i}$ direction,
and is defined positive if spring is in tension. The rigid crystal mass wall
applies a force $F^{in}$ normal to $u^{i}$ (parallel to component
$u^{c}\sin\theta$) since all contacting surfaces are frictionless. The crystal
mass is kept from vertical motion by frictionless walls above and below it and
therefore that DOFs does not enter the kinematics or dynamics equations. The
net length increased in the resonator spring is $\Delta
l^{i}=u^{i}-u^{c}\cos\theta=u^{i}-\kappa u^{c}$. The tensile spring force is

$F^{is}=\beta^{i}\Delta l^{i}=\beta^{i}(u^{i}_{n}-\kappa u^{c}_{n}).$ (31)

The EOM of the $u^{i}$ DOF is

$M^{i}\frac{\partial^{2}u^{i}_{n}}{\partial t^{2}}=-F^{is}=\beta^{i}(\kappa
u^{c}_{n}-u^{i}_{n}),$ (32)

The resonator also has a dependent DOF normal to $u^{i}$, which is simply
$u^{c}\sin\theta$. The acceleration in this direction is caused by the wall
force $F^{in}$. Therefore, we have

$M^{i}\sin\theta\frac{\partial^{2}u^{c}_{n}}{\partial t^{2}}=F^{in}.$ (33)

For the cell mass $M^{c}$, its motion is allowed only in the horizontal
direction. Therefore, its EOM is

$M^{c}\frac{\partial^{2}u^{c}_{n}}{\partial
t^{2}}=\beta^{c}(u^{c}_{n+1}-2u^{c}_{n}+u^{c}_{n-1})+F^{is}\cos\theta-F^{in}\sin\theta.$
(34)

The first term on the R.H.S. of Eq. 34 is the force applied by neighbor cells
(not shown in Fig. 7). The second and third terms are the forces supplied by
the resonator, projected onto the horizontal direction. Substituting Eq. 31
and Eq. 33 into Eq. 34 yields

$M^{c}\frac{\partial^{2}u^{c}_{n}}{\partial
t^{2}}=\beta^{c}(u^{c}_{n+1}-2u^{c}_{n}+u^{c}_{n-1})+\kappa\beta^{i}(u^{i}_{n}-\kappa
u^{c}_{n})-(1-\kappa^{2})M^{i}\frac{\partial^{2}u^{c}_{n}}{\partial t^{2}},$
(35)

where $\kappa=\cos\theta$ and $1-\kappa^{2}=\sin^{2}\theta$.

##  Appendix B Unit cell transfer matrix

At the boundaries of the cell, the state vectors are:

$\bm{\psi}^{l,r}=\begin{pmatrix}v^{l,r}\\\ N^{l,r}\end{pmatrix},$ (36)

where $l$ or $r$ denotes left or right, $v^{l,r}=\mathrm{i}\omega u^{l,r}$ is
the particle velocity in $x$ direction, and $N^{l,r}$ is the internal normal
traction force in the springs applied at the boundary (tensile positive,
relating to normal stress component in a continuum system). The spring
constitutive equations are:

$\displaystyle N^{l}$ $\displaystyle=2\beta^{p}(u^{c}-u^{l}),$ (37)
$\displaystyle N^{r}$ $\displaystyle=2\beta^{q}(u^{r}-u^{c}).$ (38)

The equation of motion for the main crystal chain mass DOF is:

$N^{r}-N^{l}+\kappa\beta^{i}(u^{i}-\kappa u^{c})+\omega^{2}M^{ci}u^{c}=0,$
(39)

where $\kappa$ and $M^{ci}$ are quantities defined in the main text. The
equation of motion for the internal resonator is:

$\beta^{i}(u^{i}-\kappa u^{c})=\omega^{2}M^{i}u^{i}.$ (40)

Combining Eqs. 39 to 40, the crystal displacement can be written as:

$u^{c}=\frac{N^{l}-N^{r}}{K_{T}},$ (41)

where

$K_{T}=\dfrac{\kappa^{2}M^{i}\omega^{2}}{1-(\omega/\omega^{i})^{2}}+M^{ci}\omega^{2},$
(42)

which would also simplify to $K_{T}=M^{c}\omega^{2}$ in the limit when
$M^{i}=0$. Substituting Eq. 41 into Eqs. 37 to 38, the state vectors at the
boundaries of a unit cell can be written as:

$\begin{pmatrix}v^{r}\\\
N^{r}\end{pmatrix}=\mathsf{T}^{cell}\begin{pmatrix}v^{l}\\\
N^{l}\end{pmatrix},$ (43)

with

$\mathsf{T}^{cell}=\begin{pmatrix}1-\dfrac{K_{T}}{2\beta^{q}}&\mathrm{i}\omega\dfrac{2\beta^{p}+2\beta^{q}-K_{T}}{4\beta^{p}\beta^{q}}\\\
\dfrac{\mathrm{i}K_{T}}{\omega}&1-\dfrac{K_{T}}{2\beta^{p}}\end{pmatrix}\leavevmode\nobreak\
.$ (44)

The eigenvectors of a non-defective transfer matrix span $\mathbb{C}^{2}$ and
therefore form a basis for any possible state. Thus any state vector
$\bm{\psi}$ observed can be decomposed into a superposition of two eigenmodes,
i.e., one can identify and separate (any wave or any linear combination of)
the forward and backward wave components in this 1D case into eigenmodes of
the transfer matrix of the finite specimen.

##  Appendix C Scattering matrix

The state vectors at locations $x^{a}$ and $x^{b}$ are simply derived using
superposition:

$\bm{\psi}^{a,b}=\mathrm{i}\omega e^{\mathrm{i}\omega t}\begin{pmatrix}1&1\\\
-Z^{a,b}&Z^{a,b}\end{pmatrix}\begin{pmatrix}A^{(a,b)+}\\\
A^{(a,b)-}\end{pmatrix},$ (45)

where the impedance for bar $a$ or $b$ is $Z^{a,b}=-N^{(a,b)+}/v^{(a,b)+}=\pi
r_{0}^{2}\sqrt{E_{0}\rho_{0}}$. For a continuum Amirkhizi (2017), the
impedance is simply $Z=-\sigma/v=\sqrt{E_{0}\rho_{0}}$. For the continuum-
discrete interfaces here, one needs to include the bar cross-sectional area in
calculation. Since it was chosen that $x^{a,b}=x^{l,r}$ (associated with the
left and right boundaries of the full system) then

$\bm{\psi}^{b}=\mathsf{T}\bm{\psi}^{a}=\begin{pmatrix}\mathsf{T}_{11}&\mathsf{T}_{12}\\\
\mathsf{T}_{21}&\mathsf{T}_{22}\end{pmatrix}\bm{\psi}^{a},$ (46)

due to the construction of
$\mathsf{T}=\mathsf{T}^{(j)}\cdots\mathsf{T}^{(2)}\mathsf{T}^{(1)}$ as the
transfer matrix of entire sample (in total $J$ cells) from $x^{l}$ to $x^{r}$,
where and $\mathsf{T}^{(j)}$ is the TM of $j$-th cell counting from the left
interface $x^{l}$, and $\bm{\psi}^{a,b}=\bm{\psi}^{l,r}$. Substituting Eq. 45
into Eq. 46, the scattering coefficients in Eq. 19 can be obtained
analytically:

$\displaystyle\mathsf{S}_{aa}$
$\displaystyle=\frac{-\mathsf{T}_{21}+\mathsf{T}_{22}Z^{b}-\mathsf{T}_{11}Z^{a}+\mathsf{T}_{12}Z^{a}Z^{b}}{\Delta},$
(47) $\displaystyle\mathsf{S}_{bb}$
$\displaystyle=\frac{-\mathsf{T}_{21}+\mathsf{T}_{11}Z^{a}-\mathsf{T}_{22}Z^{b}+\mathsf{T}_{12}Z^{a}Z^{b}}{\Delta},$
(48) $\displaystyle\mathsf{S}_{ba}$
$\displaystyle=\frac{2Z^{b}}{\Delta}\mathinner{\\!\left\lvert\mathsf{T}\right\rvert},$
(49) $\displaystyle\mathsf{S}_{ab}$ $\displaystyle=\frac{2Z^{a}}{\Delta},$
(50) $\displaystyle\Delta$
$\displaystyle=\mathsf{T}_{21}+\mathsf{T}_{11}Z^{a}+\mathsf{T}_{22}Z^{b}+\mathsf{T}_{12}Z^{a}Z^{b},$
(51)

## References

  * Heiss and Sannino (1990) W. D. Heiss and A. L. Sannino, J. Phys. A. Math. Gen. 23 (1990).
  * Seyranian, Kirillov, and Mailybaev (2005) A. P. Seyranian, O. N. Kirillov, and A. A. Mailybaev, J. Phys. A. Math. Gen. 38 (2005).
  * Amore and Fernández (2021) P. Amore and F. M. Fernández, The European Physical Journal Plus 136, 133 (2021), arXiv:1911.00452 .
  * Rüter  _et al._ (2010) C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6 (2010).
  * Othman, Galdi, and Capolino (2017) M. A. Othman, V. Galdi, and F. Capolino, Phys. Rev. B 95 (2017).
  * Lu and Srivastava (2018) Y. Lu and A. Srivastava, J. Mech. Phys. Solids 111 (2018).
  * Maznev (2018) A. A. Maznev, J. Appl. Phys. 123 (2018).
  * Ding  _et al._ (2016) K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Phys. Rev. X 6 (2016).
  * Miri and Alù (2019) M.-A. Miri and A. Alù, Science 363 (2019).
  * Liu  _et al._ (2000) Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, Science 289 (2000).
  * Amirkhizi and Wang (2018) A. V. Amirkhizi and W. Wang, J. Appl. Phys. 124 (2018).
  * Yeh  _et al._ (2016) S. L. Yeh, Y. C. Lin, Y. C. Tsai, T. Ono, and T. T. Wu, Ultrasonics 71 (2016).
  * Wang, Maznev, and Laude (2016) Y.-F. Wang, A. Maznev, and V. Laude, Crystals 6 (2016).
  * Ryu  _et al._ (2015) J. W. Ryu, W. S. Son, D. U. Hwang, S. Y. Lee, and S. W. Kim, Phys. Rev. E 91 (2015).
  * Doppler  _et al._ (2016) J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537 (2016).
  * Xu  _et al._ (2016) H. Xu, D. Mason, L. Jiang, and J. G. Harris, Nature 537 (2016).
  * Shen, Zhen, and Fu (2018) H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. 120 (2018).
  * Nemat-Nasser  _et al._ (2015) S. Nemat-Nasser, H. Sadeghi, A. V. Amirkhizi, and A. Srivastava, Mech. Res. Commun. 68 (2015).
  * Amirkhizi (2017) A. V. Amirkhizi, Mech. Mater. 114 (2017).
  * Nanda and Karami (2018) A. Nanda and M. Karami, J. Sound Vib. 424 (2018).
  * Amirkhizi and Alizadeh (2018) A. V. Amirkhizi and V. Alizadeh, Wave Motion 83 (2018).
  * Psiachos and Sigalas (2018) D. Psiachos and M. M. Sigalas, J. Appl. Phys. 123 (2018).
  * Psiachos and Sigalas (2019) D. Psiachos and M. M. Sigalas, Phys. Rev. B 99 (2019).
  * Ardakani, Ghasemi, and Golshan (2017) A. G. Ardakani, Z. Ghasemi, and M. M. Golshan, Eur. Phys. J. Plus 132, 206 (2017).
  * Al Ba’ba’a, Nouh, and Singh (2017) H. Al Ba’ba’a, M. Nouh, and T. Singh, J. Sound Vib. 410 (2017).
  * Al Ba’ba’a, Nouh, and Singh (2019) H. Al Ba’ba’a, M. Nouh, and T. Singh, Proc. R. Soc. A Math. Phys. Eng. Sci. 475 (2019).
  * Bender and Boettcher (1998) C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80 (1998).
  * Sakhdari  _et al._ (2018) M. Sakhdari, M. Hajizadegan, Y. Li, M. M. C. Cheng, J. C. Hung, and P. Y. Chen, IEEE Sens. J. 18 (2018).
  * Ge, Chong, and Stone (2012) L. Ge, Y. D. Chong, and A. D. Stone, Phys. Rev. A 85 (2012).
  * Zhu  _et al._ (2014) X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, Phys. Rev. X 4 (2014).
  * Shi  _et al._ (2016) C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, Nat. Commun. 7 (2016).
  * Fleury, Sounas, and Alu (2016) R. Fleury, D. L. Sounas, and A. Alu, IEEE J. Sel. Top. Quantum Electron. 22 (2016).
  * Achilleos  _et al._ (2017) V. Achilleos, G. Theocharis, O. Richoux, and V. Pagneux, Phys. Rev. B 95 (2017).
  * Fleury and Alu (2014) R. Fleury and A. Alu, in  _2014 8th Int. Congr. Adv. Electromagn. Mater. Microwaves Opt._, August (IEEE, 2014).
  * Popa and Cummer (2014) B.-I. Popa and S. A. Cummer, Nat. Commun. 5 (2014).
  * Christensen  _et al._ (2016) J. Christensen, M. Willatzen, V. R. Velasco, and M. H. Lu, Phys. Rev. Lett. 116 (2016).
  * Mokhtari  _et al._ (2020) A. A. Mokhtari, Y. Lu, Q. Zhou, A. V. Amirkhizi, and A. Srivastava, Int. J. Eng. Sci. 150 (2020).
  * Wang  _et al._ (2021) W. Wang, A. A. Mokhtari, A. Srivastava, and A. V. Amirkhizi,  (2021), arXiv:2108.12080 .
  * Hussein, Leamy, and Ruzzene (2014) M. I. Hussein, M. J. Leamy, and M. Ruzzene, Appl. Mech. Rev. 66 (2014).
  * Moiseyev (2011) N. Moiseyev,  _Non-Hermitian Quantum Mechanics_ , 1st ed. (Cambridge Univeristy Press, New York, 2011).
  * Kato (1995) T. Kato,  _Perturbation Theory for Linear Operators_, Classics in Mathematics, Vol. 132 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995).
  * Rotter (2003) I. Rotter, Phys. Rev. E 67 (2003).
  * Herrero-Durá  _et al._ (2019) I. Herrero-Durá, A. Cebrecos Ruíz, L. M. García-Raffi, and V. Romero-García, Model. Sci. Educ. Learn. 12 (2019).
  * Horsley  _et al._ (2014) S. A. R. Horsley, J.-H. Wu, M. Artoni, and G. C. L. Rocca, Am. J. Phys. 82 (2014).
  * Fleury  _et al._ (2015) R. Fleury, D. Sounas, M. R. Haberman, and A. Alu, Acoust. Today 2 (2015).
  * Deák and Fülöp (2012) L. Deák and T. Fülöp, Ann. Phys. 327 (2012).
  * Alizadeh and Amirkhizi (2021) V. Alizadeh and A. V. Amirkhizi, Mech. Mater. 160 (2021).
  * Achenbach (2006) J. D. Achenbach, Appl. Mech. Rev. 59 (2006).
  * Rogers and Pipkin (1963) T. G. Rogers and A. C. Pipkin, Zeitschrift Für Angew. Math. Und Phys. ZAMP 14 (1963).
  * Day (1971) W. A. Day, Q. J. Mech. Appl. Math. 24 (1971).
  * Matarazzo (2001) G. Matarazzo, Mech. Res. Commun. 28 (2001).
  * Jin  _et al._ (2016) L. Jin, X. Z. Zhang, G. Zhang, and Z. Song, Sci. Rep. 6 (2016).
  * Mostafazadeh (2014) A. Mostafazadeh, J. Phys. A Math. Theor. 47 (2014).
  * Hu, Chen, and Hu (2017) S. Hu, H. Chen, and W. Hu, Eur. Phys. J. Plus 132, 374 (2017).
  * Suneera and Subha (2018) T. P. Suneera and P. A. Subha, Eur. Phys. J. Plus 133 (2018).
  * Aghighi, Morris, and Amirkhizi (2019) F. Aghighi, J. Morris, and A. V. Amirkhizi, Mech. Mater. 130 (2019).
  * Abedi and Amirkhizi (2020) R. Abedi and A. V. Amirkhizi, Int. J. Solids Struct. 200-201 (2020).
  * Zhang, Wang, and Song (2019) K. L. Zhang, P. Wang, and Z. Song, Phys. Rev. A 99 (2019).
  * Peng  _et al._ (2016) B. Peng, . K. Ozdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, Proc. Natl. Acad. Sci. 113 (2016).