11institutetext: Institute of Science and Technology, Taikang Insurance Group
11email<EMAIL_ADDRESS>

# MeDiaQA: A Question Answering Dataset on Medical Dialogues

Huqun Suri  Qi Zhang  Wenhua Huo  Yan Liu  Chunsheng Guan

###### Abstract

In this paper, we introduce MeDiaQA, a novel question answering(QA) dataset,
which constructed on real online Medical Dialogues. It contains 22k multiple-
choice questions annotated by human for over 11k dialogues with 120k
utterances between patients and doctors, covering 150 specialties of diseases,
which are collected from haodf.com and dxy.com. MeDiaQA is the first QA
dataset where reasoning over medical dialogues, especially their quantitative
contents. The dataset has the potential to test the computing, reasoning and
understanding ability of models across multi-turn dialogues, which is
challenging compared with the existing datasets. To address the challenges, we
design MeDia-BERT, and it achieves 64.3$\%$ accuracy, while human performance
of 93$\%$ accuracy, which indicates that there still remains a large room for
improvement.

###### Keywords:

Machine reading comprehension Deep learning Question answering dataset.

##  1 Introduction

Recently, telemedicine is becoming an important complement to traditional
face-to-face medicine practiced in hospitals, since it can delivering patient
care remotely, there are many advantages including increasing access to
medical care especially for people living in communities that are in shortage
of hospitals, and reducing healthcare costs. The studies in (Pande and Morris,
2015) and (Wootton et al., 2017) also show telemedicine can improve the
satisfaction level of patients. And the fact that most of the telemedicine
services are based on conversations between doctors and patients in text
suggests that understanding user-generated medical dialogues is a significant
task.

The research of Question answering (QA), to build a system to understand
provided reading materials and answer corresponding questions, has attracted
many researchers’ attention in past years. And many various QA datasets have
been built to study how to make a QA system to understand a specific passage,
the common sense knowledge and etc. However, although these datasets are based
on different domains, they share virtually the same design pattern, most
questions in these datasets could be answered by extracting a few relevant
sentences so that the model only needs to search and determine a small set of
supporting evidences, and the order of these supporting materials does not
affect the final answer, which means these questions can be solved purely by
considering a fixed document. But when it comes to the real-world questions, a
good model needs more to give answers, including external professional
knowledge, the ability of doing mathematics calculations and recognizing
different characters in text, and so on. For instance, the question “does the
patient have to receive operation?” could not be answered before analyzing the
whole conversation since the answer may continuously change during the
discussion between doctor and patient on the condition, and sometime there
even have no conclusion appeared in the dialogue, which means the answer will
be given both on the condition and external medical knowledge. Another
question “how many tablets the patient need to take in a week?” need the model
to do mathematics calculation since the amounts provided in the dialogue maybe
are for a day or a month.Therefore, with the analysis above, we think there is
little work on question answering over medical dialogues, and the lack of
available datasets is one of the main factor.

Whatsmore, the questions in our real world often come with mathematics
calculation, including multiplication, addition and conversion. To give
correct answers, the model need to find and extract corresponding text and do
calculation on the evidence.

For all these purposes, we built a QA dataset MeDiaQA, a set of Chinese
telemedicine dialogues from haodf.com and dxy.com, which are the main-stream
websites for online medical consultation in China. On these websites, the
patients first show their electric health records, then the doctors will
discuss with the patients about their conditions and give advice. Figure 1
shows several examples in the MeDiaQA dataset. To solve the first two
questions, the ability of mathematical calculation is required, while the last
two needs reasoning ability.

To conclude, comparing with the existing datasets, our MeDiaQA has three
distinct features. First, to answer the questions, a model need to understand
medical knowledge. Second, since the dialogues are based on two characters,
the model needs to track and judge the information given by different person.
Third, in this dataset, to get the correct answer, the model not only need to
extract corresponding evidence from the dialogue, but also do mathematics
calculation.The analysis and experimental results show how this dataset can
effectively examine how a QA system can handle in medical dialogue
understanding and mathematical calculation which is not covered by previous
datasets.

Figure 1: Question Examples from the MeDiaQA dataset.

The rest parts of this paper are organized as follows: In section 2, we give a
brief introduction of current research on medical QA and multiple-choice QA.
Section 3 describes how we constructed the dataset. And Section 4 shows
statistics of the dataset and analyzes the medical term understanding and
computational inference in the data. Section 5 introduces the MeDia-BERT and
evaluates the results of experiments and in Section 6, we concludes the paper.

##  2 Related Works

Machine reading comprehension (MRC) aims to answer questions by comprehending
evidence from passages. This direction has recently drawn much attention due
to the fast development of deep learning techniques and large-scale datasets.
And researchers propose even more challenging datasets that require QA within
dialogue or conversational context (Reddy et al., 2018; Choi et al., 2018). It
is worth mentioning that in almost all previously developed datasets, the
passages are from Wikipedia, news articles or fiction stories, which are
considered as the formal language. Yet, there is little effort on QA over
informal one like medical dialogues. And according to the features of our
dataset, we introduce related works in two categories.

Medical QA: The medical domain poses a challenge to existing approaches since
the questions may be more challenging to answer. BioASQ (Tsatsaronis et al.,
2012, 2015) is one of the most significant community efforts made for
advancing biomedical question answering (QA) systems. SeaReader (Zhang et al.,
2018) is proposed to answer questions in clinical medicine using documents
extracted from publications in the medical domain. Yue et al. (2020) conduct a
thorough analysis of the emrQA dataset (Pampari et al., 2018) and explore the
ability of QA systems to utilize clinical domain knowledge and to generalize
to unseen questions. Jin et al. (2019) introduce PubMedQA where questions are
derived based on article titles and can be answered with its respective
abstracts. Recently, pre-trained models have been introduced to medical domain
(Lee et al., 2020; Beltagy et al., 2019; Huang et al., 2019a). They are
trained on unannotated biomedical texts such as PubMed abstracts and have been
proven useful in biomedical question answering.

Multiple-choice QA:Datasets such as HotpotQA (Yang et al., 2018), Natural
Questions (Kwiatkowski et al., 2019), ShARC (Saeidi et al., 2018) and BioASQ
(Tsatsaronis et al., 2015) contain multiple-choice questions as well as other
types of questions. BoolQ (Clark et al., 2019) specifically focuses on
naturally occurring yes/no questions, and those questions are shown to be
surprisingly difficult to answer. Recent breakthroughs of pre-trained language
models like ELMo (Peters et al., 2018) and BERT (Devlin et al., 2018) show
significant performance improvements on NLI tasks. In this work, we use a
modified versions of them to set baseline performance on MeDiaQA. The Figure 2
shows some examples in MeDiaQA dataset, which will be discussed in the
following sections.

##  3 Dataset

In this section, we mainly introduce the three-step data collection process of
MeDiaQA: dialogues crawling, data processing and question-answer writing.

###  3.1 Data Crawling

In haodf.com and dxy.com, all the dialogues between patients and doctors has a
unique ID connected with their URLs. We first collect the IDs from
telemedicine pages of the websites. And then we use tool to make access to all
the dialogues by changing IDs in the URLs.

###  3.2 Data Processing

The raw data seems like a complete record of the process of telemedicine,
including the name of doctor and hospital, the date and other information to
introduce the background of patient. Since we only need to focus on the
conversation itself, we remove all the irrelevant text and keep the
description of condition from the patient and dialogue. And due to the data is
collected randomly from the websites, the quality of the dialogues are
various. To guarantee there is enough quantity of information for question
writing, we only keep the dialogues which have more than 6 turns in the
dataset.

Figure 2: The different types of proposed questions.

###  3.3 Question-Answer Writing

We then co-work with the data labeling team to write question-answer pairs for
the processed data, and for each dialogue, we write two question-answer pairs
and each question has two candidate answers. To guarantee the quality and
avoid trivial questions that can be simply answered by superficial text
matching methods, the question writing is instructed by following items:

  * *

The question should be answered by reasoning.

  * *

Medical background knowledge should be considered in the question.

  * *

No other domain questions should be asked.

  * *

The data used for computational questions should be located directly in the
dialogues.

##  4 Dataset Statistics

Table 1: The details of statistics of the dataset. Element | Count | Split | Dialogues(Num) | Questions(Num)
---|---|---|---|---
Dialogues | 11524 | Train | 8524 | 17048
Utterances in total | 123632 | Val | 1340 | 2680
Questions in total | 23048 | Test | 1660 | 3320

The Table 1 shows the statistics of our dataset. The MeDiaQA dataset contains
11,524 dialogues, each of which has 11 turns and 2 question-answer pairs on
average. And then we analyze the questions from the dataset, according to
different methods for giving answers, the questions could be roughly
classified in two types. The first type can be solved by extracting
information from dialogues. Such type occupies 71.2$\%$ of all the questions.
The second type can be replied not only by summarizing all the information
from the given dialogues, but also need external medical knowledge and it
occupies the rest 28.8$\%$.

Since the questions are associated with medical knowledge and numerical data,
we classify these questions based on how the data is used to give the answers.
Three types of operations are generally used including: Judgement, Calculation
and Inference. The following subsections introduce the details of each
classified questions.

###  4.1 Judgement

To answer the judgement questions, we usually need to find the corresponding
evidences and statements. For instance, the question may ask does the patient
need to take operation immediately. The second row of Figure 3 shows the
Judgement questions.

Generally, the type of questions can be solved by finding the statements from
doctors in the dialogues. It is likely to obtain such statements at the end of
the dialogues while the doctors give suggestions for the patients. However,
most of the statements don’t directly answer the questions, for example, the
doctor may say “You should stay at home and take some medicine.” instead of
“You don’t need take operation.”, which means the reasoning ability is
necessary to the final answer.

Figure 3: The examples of different types of questions

###  4.2 Calculation

To solve the Calculation questions, the respondent should find and extract the
corresponding figures from the dialogues and calculate the sum or difference.
There are many different kinds of figures in the dialogues, including drug
dose, the duration of disease, the date of diagnosis or operation, etc.

Since all these figures are scattered, the respondent should look for the
information correctly and efficiently, or the type and direction of the
direction of calculation will be misjudged which leads to a wrong answer.
Whatsmore, to give a correct answer, the respondent is required to have the
ability of unit conversion, for example, a question may be “Can the patient
recover in one week?”, while the doctor may say “The treatment will last at
least half a month.”. A respondent should convert half a month into two weeks
and compare it with “one week” in the question then get the answer. Similar to
the Judgement questions, the Calculation questions still mainly depend on the
corresponding text where the figures(evidence) are found. The third row in
Table 3 shows some examples.

###  4.3 Reasoning

This type of questions requires the ability of combining information and
reasoning. Generally, it needs a respondent to infer based on the collected
information through the dialogue.

For example, a question may be “Can the patient play basketball after taking
the operation?”. Apparently, the doctor will not say “You can play basketball
when recovered” or “You can’t play basketball anymore.”. The respondent may
get the answer from “Your bones and joints will be completely recovered after
the operation.”, which indicates that after the operation, the patient can do
any kind of exercises including basketball.

Besides, some of the questions not only require the ability to utilize the
information obtained from the dialogues, but also the ability to use external
medical ability. For instance, “Are the symptoms of the patient relevant to
diabetes?”. To get the answer, a respondent should have the medical knowledge
of diabetes which is not mentioned in the given dialogues. Therefore, this
kind of questions becomes the most challenging part of the dataset.

##  5 Experiments

Based on the constructed dataset, we could define the QA task: the dataset is
comprised of triple instances _(D, Q, A)_ consisting of a dialogue _D_ , a
natural language question _Q_ and a corresponding answer _A_. _D_ is a set of
several natural language sentences, the question _Q_ proposes a fact to be
answered against the content in the dialogue _D_ , and the answer _A_ gives a
response which could be a word, a number, a phrase, or even a natural language
sentence. During training, the model and the learning algorithm are presented
with _K_ instances like $(D,Q,A)_{k=1}^{K}$ from the training set. In the
testing stage, the model is presented with $(D,Q)_{k=1}^{K^{\prime}}$ and
supposed to predict the answer as Â. The performance is measured by the
prediction accuracy _Acc_ =
$\frac{1}{K^{\prime}}\sum_{1}^{K^{\prime}}\left(\hat{A}_{k}=A_{k}\right)$ on
the test set. We describe the proposed model and compared models as follows.

###  5.1 MeDia-BERT

In this approach, we view the dialogue QA task as a two-sequence binary
classification problem since there are two candidate answers for each
question. We put the dialogue _D_ into a sequence and treating the QuestionQ
as another sequence. Due to the characteristics of dialogue compared with
common text, to encode it as a sequence, we propose a hierarchical method.

Let $D=(u_{1},u_{2},...,u_{T})$ denote a dialogue, where
$C=(u_{1},u_{2},...,u_{T-1})$ is the dialogue context and $u_{T}$ is the
response. Each $u_{i}=(w_{1}^{i},w_{2}^{i},...,w_{u_{i}}^{i})$ in _C_ is an
utterance and $w_{k}^{i}$ is the $k$-th word in $u_{i}$. In order to make the
representation of _C_ better, we utilize a transformer (Vaswani et al. 2017)
encoder with architecture. As shown in Figure 1, two transformer encoders are
hierarchically nested: an utterance encoder to transform each utterance in _C_
to a vector and a context encoder to learn the representations of utterances
and their neighbors in the context. We feed MeDiaQA questions and contexts,
separated by the special [SEP] token, to pre-trained BERT (Devlin et al.,
2019). The answers are predicted using the [CLS] token embedding using a
softmax function. Cross-entropy loss of predicted and true answer distribution
is denoted as _L_. We finetune the model $\theta$ to minimize _L_ on the
training set.

###  5.2 Compared Methods

Dominant: Since more than 70$\%$ of questions are the kind of ‘yes or no’,
where about 60$\%$ has the answer ‘no’, we choose ‘no’ as the default answer.
And for the rest part, we choose the first option.

BiLSTM: We simply concatenate the dialogues and questions with learnable
segment embeddings appended to the biomedical word2vec embeddings (Pyysalo et
al., 2013) of each token. The concatenated sentence is then fed to a BiLSTM,
and the final hidden states of the forward and backward network are used for
giving the predicted answer.

Human Performance: To make a fairly evaluation of the difficulty of our
dataset, we also hold out a small test set with 1K samples for human
evaluation, where we distribute each (dialogue, question) pair to 3 different
annotators to approximate human performance based on their majority voting.

###  5.3 Experiments Results

Table 2:  The results of different baseline models on the test set. Model | Accuracy (val) | Accuracy (test)
---|---|---
Dominant | 56.6$\%$ | 57.1$\%$
BiLSTM | 54.2$\%$ | 55.1$\%$
MeDiaBERT | 64.3$\%$ | 63.8$\%$
Human Performance | 93.2$\%$ | 93.6$\%$

The experiment results of four baseline models are shown with the accuracy on
the test set in Table 4. The model which achieves the best performance is the
proposed model MeDia-BERT, and the accuracy is 63.8$\%$ while the Dominant
method achieves 57.1$\%$. However, the neural-network models, which used to
perform well on other QA dataset, don’t perform better than the default answer
method, and the BiLSTM achieves 55.1$\%$. And compared with human performance
which achieves 93.6$\%$ on the test set, the performance of QA model seems
relatively poor. In sum, the experiment results show that the MeDiaQA dataset
is really challenging for the existing QA models.

##  6 Conclusion

In this paper, we construct MeDiaQA, a novel dataset for QA on medical
dialogues and propose a method MeDia-BERT based on the state-of-the-art pre-
trained language model. The proposed dataset shows the distinctiveness of
medical dialogues compared with other normal domains in the context of QA. Due
to the unique characteristics, it is challenging for existing QA models
compared with human performance. We hope our dataset will lead a deeper
research on machine reading comprehension in medical domain.

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# Real transmission and reflection zeros of periodic structures
with a bound state in the continuum

Lijun Yuan<EMAIL_ADDRESS>Mingyang Zhang  College of Mathematics and
Statistics, Chongqing Technology and Business University, Chongqing, China
Chongqing Key Laboratory of Social Economic and Applied Statistics, Chongqing
Technology and Business University, Chongqing, China  Ya Yan Lu  Department of
Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China

###### Abstract

For lossless periodic structures with a proper symmetry, the transmission and
reflection spectra often have peaks and dips that are truly $100\%$ and $0\%$,
respectively. The full peaks and zero dips typically appear near resonant
frequencies, and they are robust with respect to structural perturbations that
preserve the required symmetry. However, current theories on the existence of
full peaks and zero dips are incomplete and difficult to use. For periodic
structures with a bound state in the continuum (BIC), we present a new theory
on the existence of real transmission and reflection zeros that correspond to
the zero dips in the transmission and reflection spectra. Our theory is
relatively simple, complete, and easy to use. Numerical examples are presented
to validate the new theory.

##  I Introduction

For periodic structures sandwiched between two homogeneous media, the
transmission and reflection spectra often have interesting and useful
features. The peaks and dips can be very sharp. A peak and a dip may appear
close to each other and form an asymmetric line shape. The study on
“anomalous” transmission and reflection has a long history [1, 2]. It is
widely accepted that the rapid change from a peak to a dip, the “anomaly”
first observed by Wood [1], is in fact a particular case of Fano resonance [2,
3, 4, 5]. The asymmetric line shape is formed from the interference between
the resonant and non-resonant wave field components [5]. The resonant wave
field component is caused by the excitation of an eigenmode of the periodic
structure satisfying an outgoing radiation condition. The eigenmode is either
a resonant mode with a complex frequency or a leaky mode with a complex
propagation constant. The non-resonant field component exists in a direct
passway. In case there is no direct passway, the spectra exhibit a Lorentzian
line shape with only a single peak or dip [5].

For lossless periodic structures, the peaks and dips in the transmission and
reflection spectra can actually be $100\%$ and $0\%$, respectively. Popov et
al. [4] first realized that structural symmetry is important to the appearance
of full peaks and zero dips. As functions of frequency and wavenumber, the
reflection and transmission coefficients vanish at their corresponding zeros
(which are complex in general). For lossless periodic structures with a proper
symmetry, the transmission/reflection zeros are either real or form complex
conjugate pairs [4, 6, 7, 8, 9, 10]. A real transmission zero corresponds to a
zero dip in the transmission spectrum and a full peak in the reflection
spectrum. Since a simple zero cannot be turned to a complex conjugate pair by
a small perturbation, the zero dips and full peaks in the
transmission/reflection spectra are robust with respect to small structural
perturbations that preserve the required symmetry [10]. However, even for
structures with the required symmetry, the appearance of a real
transmission/reflection zero (near a resonance) is not guaranteed. To the best
of our knowledge, there is no general theory on the existence of full peaks or
zero dips.

Shipman and Tu [11] developed a theory on the existence of full peaks and zero
dips for symmetric lossless periodic structures with a bound state in the
continuum (BIC). A BIC is a guided mode that decays exponentially in the
homogeneous media surrounding the periodic structure, and it exists in the
radiation continuum, namely, there are propagating plane waves in the
homogeneous media having the same frequency and wavenumber as the BIC [12, 13,
14, 15, 16]. Importantly, a BIC is a special point in a band of resonant
modes. The theory of Shipman and Tu is applicable to resonant modes near a
BIC. It is mathematically rigorous, but the technical conditions are specified
on quantities that are difficult to calculate. They identified a generic
condition under which a real transmission zero and a real reflection zero
likely exist near the frequency of the BIC, and they also studied a non-
generic case.

It should be pointed that the temporal coupled-mode theory (TCMT) [5, 17, 18]
can approximate full peaks and zero dips in transmission and reflection
spectra. In a recent work [19], we showed that a direct approximation to the
exact scattering matrix gives the same approximate transmission/reflection
spectra as the TCMT. However, the approximate formulas have limitations in
accuracy, are valid only under proper conditions, and do not provide a
rigorous justification for the existence of real transmission and reflection
zeros.

In this paper, we present a new theory on the existence of real transmission
and reflection zeros. Similar to the work of Shipman and Tu [11], our theory
is applicable to lossless periodic structures with a BIC, but the symmetry
requirement in our study is less restrictive. Moreover, our results are
relatively simple and more complete, and the technical conditions are only
specified on directly computable quantities. The rest of this paper is
organized as follows. In Sec. II, we recall the definitions and basic
properties for scattering matrices, resonant modes, and BICs. In Sec. III, we
give a brief summary for the theories of Popov et al [4] and Shipman and Tu
[11]. In Sec. IV, we present our new theory with details on the assumptions,
derivations, and results, and also discuss the difference with existing
theories. To valid our new theory, we present numerical examples in Sec. V.
The paper is concluded with a brief discussion in Sec. VI.

##  II Background

We consider a lossless two-dimensional (2D) periodic structure that is
invariant in $z$, periodic in $y$ with period $L$, and sandwiched between two
homogeneous media given for $x>D$ and $x<-D$, respectively. The dielectric
function $\varepsilon(x,y)$ of the structure is real and periodic in $y$, and
$\varepsilon(x,y)=\varepsilon_{0}\geq 1$ for $|x|>D$. In the homogeneous media
for $|x|>D$, we specify two time-harmonic $E$-polarized plane incident waves
with a positive angular frequency $\omega$ and real wave vectors
$(\pm\alpha,\beta)$ satisfying

$\displaystyle-\frac{\pi}{L}<\beta\leq\frac{\pi}{L},$ (1)
$\displaystyle|\beta|<\frac{\omega}{c}\sqrt{\varepsilon_{0}}<\frac{2\pi}{L}-|\beta|,$
(2) $\displaystyle\alpha=\sqrt{(\omega/c)^{2}\varepsilon_{0}-\beta^{2}},$ (3)

where $c$ is the speed of light in vacuum and $\alpha>0$. The $z$ component of
the total electric field, denoted as $u$, can be expanded as

$\displaystyle u(x,y)=b_{1}^{+}e^{i[\beta y+\alpha(x+D)]}+b_{1}^{-}e^{i[\beta
y-\alpha(x+D)]}$ $\displaystyle\qquad+\sum_{j\neq
0}b_{1j}e^{i\beta_{j}y+\tau_{j}(x+D)},\quad x<-D,$ (4) $\displaystyle
u(x,y)=b_{2}^{+}e^{i[\beta y-\alpha(x-D)]}+b_{2}^{-}e^{i[\beta
y+\alpha(x-D)]}$ $\displaystyle\qquad+\sum_{j\neq
0}b_{2j}e^{i\beta_{j}y-\tau_{j}(x-D)},\quad x>D,$ (5)

where $b_{1}^{+}$ and $b_{2}^{+}$ are the amplitudes of the given incident
waves in the left and right homogeneous media, respectively, $b_{1}^{-}$ and
$b_{2}^{-}$ are the amplitudes of the outgoing plane waves,

$\beta_{j}=\beta+2\pi
j/L,\quad\tau_{j}=\sqrt{\beta_{j}^{2}-(\omega/c)^{2}\varepsilon_{0}}$ (6)

for $j\neq 0$, $\tau_{j}$ is positive, $b_{1j}$ and $b_{2j}$ are the
amplitudes of the evanescent waves. The scattering matrix $S$ satisfies

$S\begin{bmatrix}b_{1}^{+}\cr
b_{2}^{+}\end{bmatrix}=\begin{bmatrix}b_{1}^{-}\cr b_{2}^{-}\end{bmatrix}$ (7)

for any $b_{1}^{+}$ and $b_{2}^{+}$. We write down the entries of $S$ as

$S=\begin{bmatrix}r&\tilde{t}\ \cr t&\tilde{r}\end{bmatrix},$ (8)

where $r$ and $t$ are the reflection and transmission coefficients for the
left incident wave, $\tilde{r}$ and $\tilde{t}$ are those of the right
incident wave.

Clearly, the scattering matrix $S$ depends on both $\omega$ and $\beta$. The
definition of $S$ can be extended to complex $\omega$ by analytic
continuation. For a real $\beta$, $S$ satisfies

$\displaystyle S^{-1}(\omega,\beta)$ $\displaystyle=$ $\displaystyle
S^{*}(\overline{\omega},\beta),$ (9) $\displaystyle S^{\sf T}(\omega,\beta)$
$\displaystyle=$ $\displaystyle S(\omega,-\beta),$ (10)

where $\overline{\omega}$ is the complex conjugate of $\omega$, $S^{\sf T}$ is
the transpose of $S$, and $S^{*}(\overline{\omega},\beta)$ is the conjugate
transpose of $S(\overline{\omega},\beta)$ [4, 10]. Equations (9) and (10) are
related to energy conservation and reciprocity, respectively. Notice that if
$\omega$ is real, $S$ is unitary; and if $\beta\neq 0$, $S$ is typically non-
symmetric. If the periodic structure has a proper symmetry, the scattering
matrix can be further simplified [4]. Specifically, we have three cases

  1. (a).

If $\epsilon(x,y)=\epsilon(-x,-y)$, then $r=\tilde{r}$;

  2. (b).

If $\epsilon(x,y)=\epsilon(x,-y)$, then $t=\tilde{t}$;

  3. (c).

If $\epsilon(x,y)=\epsilon(-x,y)$, then $t=\tilde{t}$ and $r=\tilde{r}$.

Without incident waves, the periodic structure can support Bloch eigenmodes
given by

$u(x,y)=e^{i\beta y}\phi(x,y)$ (11)

where $\beta$ is the Bloch wavenumber and $\phi$ is periodic in $y$ with
period $L$. For $x\to\pm\infty$, an eigenmode should either decay
exponentially or radiate out power to infinity (i.e. satisfy the outgoing
radiation condition). For a real $\beta$ satisfying (1) and the real part of
$\omega$ satisfying (2), expansions (II) and (II) are applicable to an
eigenmode, provided that we set $b_{1}^{+}=b_{2}^{+}=0$. Notice that
$b_{1}^{-}$ and $b_{2}^{-}$ are the coefficients of outgoing waves. Since the
structure is passive and non-absorbing (i.e. $\varepsilon$ is real and
positive), the existence of outgoing waves is only possible when $\omega$ has
a nonzero imaginary part, so that the eigenmode decays with time. Such an
eigenmode with a real $\beta$ and a complex $\omega$, and satisfying the
outgoing radiation condition, is a resonant mode (also called a resonant state
or quasi-normal mode) [20, 21]. The resonant modes form bands. Each band is
given by a dispersion relation $\omega=\omega_{\star}(\beta)$, where
$\omega_{\star}(\beta)$ is a complex-valued continuous function of a real
variable $\beta$. Notice that $\alpha$ and $\tau_{j}$ given in (3) and (6) are
complex for the resonant modes. Under special circumstances, both $b_{1}^{-}$
and $b_{2}^{-}$ are zero, then $\omega$ is real, the eigenmode decays
exponentially as $x\to\pm\infty$, and it is a guided mode above the light
line, also called a bound state in the continuum (BIC). Note that a BIC is a
special point in a band of resonant modes. If we denote the wavenumber and
frequency of a BIC by $\beta_{\dagger}$ and $\omega_{\dagger}$, respectively,
then $\omega_{\dagger}=\omega_{\star}(\beta_{\dagger})$.

Since the inverse of the scattering matrix $S$ maps
$[b_{1}^{-},b_{2}^{-}]^{\sf T}$ to $[b_{1}^{+},b_{2}^{+}]^{\sf T}$, and
$b_{1}^{+}=b_{2}^{+}=0$ for a resonant mode, we have

$S^{-1}(\omega_{\star},\beta)\begin{bmatrix}b_{1}^{-}\cr
b_{2}^{-}\end{bmatrix}=S^{*}(\overline{\omega}_{\star},\beta)\begin{bmatrix}b_{1}^{-}\cr
b_{2}^{-}\end{bmatrix}=\begin{bmatrix}0\cr 0\end{bmatrix}.$ (12)

This above implies that $\omega_{\star}$ is a pole of $S$, and
$\overline{\omega}_{\star}$ is a zero of $S$. The entries of $S$ can also
vanish at special values of $\omega$ and $\beta$. We assume the left
transmission coefficient $t=t(\omega,\beta)$ has a zero set given by a
function $\omega=\omega_{t}^{\circ}(\beta)$. Namely,
$t(\omega_{t}^{\circ}(\beta),\beta)=0$ for any $\beta$. Similarly, the left
reflection coefficient $r$ has a zero set given by a function
$\omega=\omega_{r}^{\circ}(\beta)$. For a fixed $\beta$, we call
$\omega_{t}^{\circ}(\beta)$ a transmission zero and
$\omega_{r}^{\circ}(\beta)$ a reflection zero. In general, for a real $\beta$,
the transmission and reflection zeros are complex. We are concerned with the
existence of real transmission and reflection zeros.

##  III Existing theories

The possible existence of real transmission and reflection zeros in symmetric
periodic structures was first investigated by Popov et al. [4]. These authors
identified the three different symmetries (listed in Sec. II) for which
$t=\tilde{t}$, $r=\tilde{r}$, or both $t=\tilde{t}$ and $r=\tilde{r}$,
respectively; and showed that if $t=\tilde{t}$, then either a transmission
zero is real or there is a pair of complex conjugate transmission zeros. The
case for $r=\tilde{r}$ is similar. This implies that a real transmission or
reflection zero for a symmetric periodic structure is robust with respect to
structural perturbations that preserve the symmetry. However, the possibility
of complex conjugate pairs cannot be ruled out, and it remains unclear why
real transmission and reflection zeros are widely observed in practice.

If for a real $\beta$, the periodic structure has a high-$Q$ nondegenerate
resonant mode with a complex frequency $\omega_{\star}=\omega_{0}-i\gamma$
where $\gamma\ll\omega_{0}$, and there is no other resonant modes near
$\omega_{\star}$, then the reflection and transmission coefficients can be
written as

$r(\omega)=\frac{a(\omega)}{\omega-\omega_{\star}},\quad
t(\omega)=\frac{b(\omega)}{\omega-\omega_{\star}},$ (13)

where $a$ and $b$ are analytic functions of $\omega$ near $\omega_{\star}$.
Since $t$ and $r$ are bounded by $1$ in magnitude, if $\omega$ is close to
$\omega_{0}$, $|a|/\omega_{0}$ and $|b|/\omega_{0}$ must also be small, but
this does not imply that $a$ or $b$ (thus $r$ or $t$) must have a single zero
in a domain that is symmetric about the real axis and contains
$\omega_{\star}$. Consequently, it is not possible to predict the existence of
real transmission and reflection zeros by considering only the symmetry of the
structure and the resonant modes.

Shipman and Tu [11] considered periodic structures with a reflection symmetry
in $x$ [case (c) of Sec. II], and showed that the existence of real
transmission and reflection zeros is a generic phenomenon. Their theory is
restricted to structures with a BIC and involves technical conditions are
rather difficult to verify. More specifically, Shipman and Tu defined an
operator $A$ so that the diffraction problem of Sec. II becomes

$Au=p,$ (14)

where both $A$ and $p$ depend on $\omega$ and $\beta$, and $p$ is related to
the incident wave. They further considered the linear eigenvalue problem

$Av=\lambda v,$ (15)

where $\lambda$ is an eigenvalue depending on both $\omega$ and $\beta$, and
$\lambda$ vanishes at the BIC point $(\omega_{\dagger},\beta_{\dagger})$. The
technical conditions are specified using $\lambda$, $\lambda r$ and $\lambda
t$ and their partial derivatives at $(\omega_{\dagger},\beta_{\dagger})$.

The theory of Shipman and Tu is difficult to use in practice, because $A$ is
abstractly defined without an explicit expression, and the eigenvalue problem
Eq. (15) does not have a clear physical interpretation. Moreover, since
Shipman and Tu only considered case (c), it is not clear whether their method
can be extended to cases (a) and (b). In our view, a simpler and more
intuitive theory on the existence of real transmission and reflection zeros is
highly desirable.

##  IV New theory

In this section, we present a new theory to clarify the conditions for the
existence of real transmission and reflection zeros. We consider a lossless
periodic structure as described in Sec. II, and assume there is a BIC with
frequency $\omega_{\dagger}$ and Bloch wavenumber $\beta_{\dagger}$ satisfying
conditions (1) and (2). The BIC is supposed to be a special point in a band of
nondegenerate resonant modes with dispersion relation
$\omega=\omega_{\star}(\beta)=\omega_{0}(\beta)-i\gamma(\beta)$, where
$\omega_{0}$ and $-\gamma$ are the real and imaginary parts of
$\omega_{\star}$, and they satisfy $\gamma(\beta_{\dagger})=0$,
$\omega_{0}(\beta_{\dagger})=\omega_{\star}(\beta_{\dagger})=\omega_{\dagger}$,
and $\gamma(\beta)>0$ for $\beta$ near but not equal to $\beta_{\dagger}$.

Our theory depends on the analyticity of a few functions. First of all,
$\omega_{\star}$ is an analytic function of real variable $\beta$ [11].
Therefore, $\omega-\omega_{\star}(\beta)$, as a function of complex variable
$\omega$ and real variable $\beta$, is analytic in $\omega$ and $\beta$. It is
known that the reflection and transmission coefficients, $r$ and $t$, as
functions of $\omega$ and $\beta$, are not continuous at
$(\omega_{\dagger},\beta_{\dagger})$ [11]. However, we claim that the two
functions $a$ and $b$ given by

$\displaystyle a(\omega,\beta)$ $\displaystyle=$
$\displaystyle[\omega-\omega_{\star}(\beta)]r(\omega,\beta),$ $\displaystyle
b(\omega,\beta)$ $\displaystyle=$
$\displaystyle[\omega-\omega_{\star}(\beta)]t(\omega,\beta),$

are analytic in $\omega$ and $\beta$ in a neighborhood of the BIC point
$(\omega_{\dagger},\beta_{\dagger})$. This implies that the singularity of $r$
and $t$ at $(\omega_{\dagger},\beta_{\dagger})$ can be removed by multiplying
$\omega-\omega_{\star}(\beta)$. Notice that
$a(\omega_{\dagger},\beta_{\dagger})=b(\omega_{\dagger},\beta_{\dagger})=0$.
The above proposition is different from our statement about Eq. (13), where
$\beta$ is fixed, the dependence on $\beta$ is suppressed,
$\mbox{Im}(\omega_{\star})\neq 0$, and $a$ and $b$ are analytic in a single
complex variable $\omega$. The analyticity of $a$ and $b$ can be established
following the proof for the analyticity of $\lambda t$ and $\lambda r$ by
Shipman and Tu [11], where $\lambda$ is the eigenvalue defined in Eq. (15).

Similarly, for the right reflection and transmission coefficients, the two
functions $\tilde{a}=[\omega-\omega_{\star}(\beta)]\tilde{r}$ and
$\tilde{b}=[\omega-\omega_{\star}(\beta)]\tilde{t}$ are analytic in $\omega$
and $\beta$ near $(\omega_{\dagger},\beta_{\dagger})$. The scattering matrix
can be written as

$S(\omega,\beta)=\frac{1}{\omega-\omega_{\star}(\beta)}\begin{bmatrix}a(\omega,\beta)&\tilde{b}(\omega,\beta)\cr
b(\omega,\beta)&\tilde{a}(\omega,\beta)\end{bmatrix}$ (16)

for $\omega\neq\omega_{\star}(\beta)$, where $a$, $b$, $\tilde{a}$ and
$\tilde{b}$ are analytic in $\omega$ and $\beta$, and vanish at
$(\omega_{\dagger},\beta_{\dagger})$.

Next, we consider the function

$f(\omega,\beta)=\frac{\omega-\omega_{\star}(\beta)}{\omega-\overline{\omega}_{\star}(\beta)}\,\det
S(\omega,\beta).$ (17)

For a fixed $\beta\neq\beta_{\dagger}$, $\omega_{\star}=\omega_{\star}(\beta)$
is complex with a nonzero imaginary part. Since the resonant mode is
nondegenerate, $\omega_{\star}$ is a simple pole of $\det S$ and
$\overline{\omega}_{\star}$ is a simple zero of $\det S$. Therefore, for the
fixed $\beta$, $f$ given in Eq. (17) is an analytic function of $\omega$. For
$\beta=\beta_{\dagger}$ and $\omega\neq\omega_{\dagger}$, we have
$f(\omega,\beta)=\det S(\omega,\beta_{\dagger})$. It is known that for a fixed
$\beta$, the reflection and transmission coefficients are continuous in
$\omega$. Therefore, we define $f(\omega_{\dagger},\beta_{\dagger})$ by

$f(\omega_{\dagger},\beta_{\dagger})=\lim_{\omega\to\omega_{\dagger}}\det
S(\omega,\beta_{\dagger})=\det S(\omega_{\dagger},\beta_{\dagger}).$ (18)

Numerical results suggest that $f(\omega,\beta)$ is an analytic function of
$\omega$ and $\beta$ for $\omega$ near $\omega_{\dagger}$ and $\beta$ near
$\beta_{\dagger}$. Unfortunately, a formal mathematical proof is currently not
available.

Using Eq. (9), we can easily show that

$\overline{f}(\overline{\omega},\beta)f(\omega,\beta)=1,$ (19)

where $\overline{f}(\overline{\omega},\beta)$ is the complex conjugate of
$f(\overline{\omega},\beta)$. In particular, if $\omega$ is real, then
$|f(\omega,\beta)|=1$. Since $|f(\omega_{\dagger},\beta_{\dagger})|=1$, we can
choose a small neighborhood (denoted as $\Omega$) of
$(\omega_{\dagger},\beta_{\dagger})$, and choose a proper branch cut to define
a complex square root function, so that
$g(\omega,\beta)=\sqrt{f(\omega,\beta)}$ is nonzero and analytic (in $\omega$
and $\beta$) on $\Omega$. With this function $g$, we can rewrite the
scattering matrix as

$S(\omega,\beta)=\frac{g(\omega,\beta)}{\omega-\omega_{\star}(\beta)}\begin{bmatrix}R(\omega,\beta)&\tilde{T}(\omega,\beta)\cr
T(\omega,\beta)&\tilde{R}(\omega,\beta)\end{bmatrix},$ (20)

where $(R,T,\tilde{R},\tilde{T})=(a,b,\tilde{a},\tilde{b})/g$. Clearly, $R$,
$T$, $\tilde{R}$ and $\tilde{T}$ are analytic functions of $\omega$ and
$\beta$ on $\Omega$, and they vanish at $(\omega_{\dagger},\beta_{\dagger})$.
These functions are introduced, because they have useful properties. Using Eq.
(9), we can show that

$\displaystyle\tilde{T}(\omega,\beta)$ $\displaystyle=$
$\displaystyle-\overline{T}(\overline{\omega},\beta),$ (21)
$\displaystyle\tilde{R}(\omega,\beta)$ $\displaystyle=$
$\displaystyle\overline{R}(\overline{\omega},\beta).$ (22)

In addition, it is also easy to verify that

$\frac{\partial
R}{\partial\omega}(\omega_{\dagger},\beta_{\dagger})=\frac{r_{\dagger}}{g_{\dagger}},\quad\frac{\partial
T}{\partial\omega}(\omega_{\dagger},\beta_{\dagger})=\frac{t_{\dagger}}{g_{\dagger}},$
(23)

where $r_{\dagger}=r(\omega_{\dagger},\beta_{\dagger})$,
$t_{\dagger}=t(\omega_{\dagger},\beta_{\dagger})$ and
$g_{\dagger}=g(\omega_{\dagger},\beta_{\dagger})$.

For case (a) of Sec. II, we have $r=\tilde{r}$, and thus

$R(\omega,\beta)=\overline{R}(\overline{\omega},\beta).$ (24)

This implies that $R(\omega,\beta)$ is a real analytic function of two real
variables $\omega$ and $\beta$ near $(\omega_{\dagger},\beta_{\dagger})$.
Since $R(\omega_{\dagger},\beta_{\dagger})=0$, we can analyze the zero set of
$R$ using the partial derivatives of $R$ at
$(\omega_{\dagger},\beta_{\dagger})$.

  1. 1.

$\partial_{\omega}R\neq 0$ at $(\omega_{\dagger},\beta_{\dagger})$. According
to Eq. (23), this condition is equivalent to $r_{\dagger}\neq 0$. In the
$\omega$-$R$ plane, $R=R(\omega,\beta_{\dagger})$ is a curve passing through
zero at $\omega_{\dagger}$ with a nonzero slope. When $\beta$ is slightly
changed, the curve is slightly shifted. The new curve, given by
$R=R(\omega,\beta)$, still passes through zero near $\omega_{\dagger}$. More
precisely, according to the implicit function theorem, when
$\partial_{\omega}R$ is nonzero at $(\omega_{\dagger},\beta_{\dagger})$,
$R(\omega,\beta)=0$ can be uniquely solved near $\beta_{\dagger}$, and the
solution is a function $\omega=\omega_{r}^{\circ}(\beta)$, such that
$\omega_{r}^{\circ}(\beta_{\dagger})=\omega_{\dagger}$ and
$R(\omega_{r}^{\circ}(\beta),\beta)=0$ for $\beta$ near $\beta_{\dagger}$.
Therefore, if $r_{\dagger}\neq 0$ and $\beta$ is close to $\beta_{\dagger}$,
there is a real reflection zero $\omega_{r}^{\circ}(\beta)$ near
$\omega_{\dagger}$. This is the generic case and $r_{\dagger}\neq 0$ is the
generic condition.

  2. 2.

At $(\omega_{\dagger},\beta_{\dagger})$, $\partial_{\omega}R=0$,
$\partial^{2}_{\omega}R\neq 0$, $\partial^{2}_{\beta}R\neq 0$,
$\partial^{2}_{\omega}R$ and $\partial^{2}_{\beta}R$ have the same sign. This
is a non-generic case. The condition $\partial_{\omega}R=0$ implies
$r_{\dagger}=0$, and thus $\partial_{\beta}R=0$ at
$(\omega_{\dagger},\beta_{\dagger})$. This is the case for which
$(\omega_{\dagger},\beta_{\dagger})$ is a local extremum of $R$. The function
$R(\omega,\beta)$ is nonzero for real $(\omega,\beta)$ near but not equal to
$(\omega_{\dagger},\beta_{\dagger})$. There is no real reflection zero for
$\beta$ near but not equal to $\beta_{\dagger}$.

  3. 3.

At $(\omega_{\dagger},\beta_{\dagger})$, $\partial_{\omega}R=0$,
$\partial^{2}_{\omega}R\neq 0$, $\partial^{2}_{\beta}R\neq 0$,
$\partial^{2}_{\omega}R$ and $\partial^{2}_{\beta}R$ have the opposite sign.
This is another non-generic case. In the $\omega$-$R$ plane,
$R=R(\omega,\beta_{\dagger})$ obtains a local minimum (or maximum) at
$\omega_{\dagger}$ and is exactly zero at $\omega_{\dagger}$. For $\beta$ near
$\beta_{\dagger}$, the minimum (or maximum) of the curve $R=R(\omega,\beta)$
is negative (or positive), and the curve passes zero at two values of $\omega$
near $\omega_{\dagger}$. On the surface given by $R=R(\omega,\beta)$,
$(\omega_{\dagger},\beta_{\dagger})$ is a saddle point. For $\beta$ near but
not equal to $\beta_{\dagger}$, there are real two reflection zeros near
$\omega_{\dagger}$.

There are other non-generic cases. For example, when $\partial_{\omega}R=0$,
$\partial_{\omega}^{2}R\neq 0$, we can consider the case
$\partial_{\beta}^{2}R=0$ and $\partial_{\beta}^{3}R\neq 0$. This leads to one
real reflection zero for $\beta<\beta_{\dagger}$ (or $\beta>\beta_{\dagger}$)
and no real reflection zero for $\beta>\beta_{\dagger}$ (or
$\beta<\beta_{\dagger}$). We can also consider new cases assuming
$\partial_{\omega}R=\partial^{2}_{\omega}R=0$ and $\partial^{3}_{\omega}R\neq
0$. However, these additional non-generic cases are too special and difficult
to find in practice.

For case (b), we have $t=\tilde{t}$, thus

$T(\omega,\beta)=-\overline{T}(\overline{\omega},\beta).$ (25)

Clearly, if $\omega$ is real, $T$ is pure imaginary. Following the approach
for analyzing $R$, we also identify the generic case and two main non-generic
cases as follows.

  1. 1.

$\partial_{\omega}T\neq 0$ at $(\omega_{\dagger},\beta_{\dagger})$, equivalent
to $t_{\dagger}\neq 0$. This is the generic case and the generic condition is
$t_{\dagger}\neq 0$. For $\beta$ near $\beta_{\dagger}$, there is one
transmission zero $\omega_{t}^{\circ}(\beta)$ near $\omega_{\dagger}$.

  2. 2.

At $(\omega_{\dagger},\beta_{\dagger})$, $\partial_{\omega}T=0$,
$\partial_{\omega}^{2}T\neq 0$, $\partial_{\beta}^{2}T\neq 0$, and
$(\partial_{\omega}^{2}T)(\partial_{\beta}^{2}T)<0$. This is a non-generic
case. $\mbox{Im}(T)$ has a local extremum at
$(\omega_{\dagger},\beta_{\dagger})$. There is no real transmission zero for
$\beta$ near $\beta_{\dagger}$ and $\beta\neq\beta_{\dagger}$.

  3. 3.

At $(\omega_{\dagger},\beta_{\dagger})$, $\partial_{\omega}T=0$,
$\partial_{\omega}^{2}T\neq 0$, $\partial_{\beta}^{2}T\neq 0$, and
$(\partial_{\omega}^{2}T)(\partial_{\beta}^{2}T)>0$. This is another non-
generic case. On the surface given by $\mbox{Im}(T)$ as a function of real
variables $\omega$ and $\beta$, $(\omega_{\dagger},\beta_{\dagger})$ is a
saddle point. For any $\beta$ near $\beta_{\dagger}$ and
$\beta\neq\beta_{\dagger}$, there are two real transmission zeros.

For case (c), we have both $t=\tilde{t}$ and $r=\tilde{r}$. Therefore, the
above results on transmission and reflection zeros for cases (a) and (b),
respectively, are valid for case (c).

##  V Numerical examples

To validate our theory, we present several numerical examples involving
periodic arrays of dielectric cylinders. In Fig. 1,

Figure 1: Two periodic arrays of cylinders with period $L$ in the $y$
direction: (a) triangular cylinders with side length $L_{t}$; (b) circular
cylinders with radius $a$.

we depict two periodic arrays with triangular and circular cylinders,
respectively. The arrays are periodic in $y$ with period $L$. The triangular
cylinders have one surface parallel to the $y$ axis, and their cross sections
are equilateral triangles with side length $L_{t}$. The radius of the circular
cylinders is $a$. All cylinders have the same dielectric constant
$\varepsilon_{1}=10$ and are surrounded by air (with dielectric constant
$\varepsilon_{0}=1$).

Our theory is for periodic structures with a BIC. It is well known that many
different BICs may exist in a periodic array of dielectric cylinders [22, 23,
24, 25, 26, 27, 28, 29, 30]. Our first example is for the array of triangular
cylinders shown in Fig. 1(a). For $L_{t}=0.45L$, we found a BIC with frequency
$\omega_{\dagger}=0.6875(2\pi c/L)$ and wavenumber $\beta_{\dagger}=0$. Since
$\beta_{\dagger}$ is zero and the electric field is an odd function of $y$,
this BIC is an anti-symmetric standing wave. The transmission coefficient at
$(\omega_{\dagger},\beta_{\dagger})$ satisfies $|t_{\dagger}|=0.7530$. Since
the periodic array has a reflection symmetry in $y$ and $t_{\dagger}\neq 0$,
there should be a real transmission zero $\omega_{t}^{\circ}$ for $\beta$ near
$\beta_{\dagger}=0$.

Figure 2: Transmission and reflection near a BIC in a periodic array of
triangular cylinders with $L_{t}=0.45L$: (a) the phase of $f$ in unit of
$\pi$, (b) $\mbox{Im}(T)$, (c) $\mbox{Re}(R)$ and $\mbox{Im}(R)$, (d)
$|t|^{2}$, (e) $|t|^{2}$ for $\beta=0.01(2\pi/L)$ (with a logarithmic plot in
the inset). (f): $|r|^{2}$ for $\beta=0.01(2\pi/L)$ (in a logarithmic scale).
The solid red curves in (b) and (c) correspond to $T=0$, $\mbox{Re}{(R)}=0$
and $\mbox{Im}{(R)}=0$, respectively. The circle in (d) corresponds the BIC.

In Fig. 2(a)-(d), we show the phase $f$, $T$, $R$ and $|t|^{2}$ for
$(\omega,\beta)$ near $(\omega_{\dagger},\beta_{\dagger})$. Our theory relies
on the analyticity of function $f$ in $\omega$ and $\beta$. Since
$|f(\omega,\beta)|=1$ for real $(\omega,\beta)$, we show the phase of $f$ (in
unit of $\pi$) in Fig. 2(a). The periodic array has a symmetry corresponding
to case (b) of Sec. II, thus $T$ is pure imaginary and $R$ is complex for real
$(\omega,\beta)$. In Fig. 2(b) and (c), we show $\mbox{Im}(T)$, $\mbox{Re}(R)$
and $\mbox{Im}(R)$, and highlight their zero sets by the solid red curves.
Notice that the two curves for $R$ touch tangentially at
$(\omega_{\dagger},\beta_{\dagger})$. In Fig. 2(d), we show transmittance
$|t|^{2}$ as a function of $\omega$ and $\beta$, where the BIC point
$(\omega_{\dagger},\beta_{\dagger})$ is marked by a small circle. To show
peaks and dips more clearly, we plot transmission and reflection spectra
($|t|^{2}$ and $|r|^{2}$ as functions of $\omega$) for $\beta=0.01(2\pi/L)$ in
Fig. 2(e) and (f), respectively. There is indeed a zero dip (corresponding to
the real transmission zero) in the transmission spectrum, but the dip in the
reflection spectrum is not zero, and the peak in the transmission spectrum is
not $100\%$.

The second example is for a periodic array of circular cylinders with
$a=0.3L$.

Figure 3: Transmission and reflection near a BIC in a periodic array of
circular cylinders with radius $a=0.3L$: (a) the phase of $f$ in unit of
$\pi$, (b) $\mbox{Im}(T)$, (c) $R$, (d) $|t|^{2}$ with the BIC marked by
“$\circ$”. (e) $|t|^{2}$ for $\beta=0.21(2\pi/L)$, (f) $|r|^{2}$ for
$\beta=0.21(2\pi/L)$. The solid red curves in (b) and (c) correspond to $T=0$
and $R=0$, respectively. Insets in (e) and (f) are logarithmic plots.

In this array, there is a propagating BIC with
$\beta_{\dagger}=0.2206(2\pi/L)$ and $\omega_{\dagger}=0.6173(2\pi c/L)$. The
reflection coefficient at the BIC point satisfies $|t_{\dagger}|=0.5568$. The
symmetry of the structure corresponds to case (c) of Sec. II. Since
$t_{\dagger}\neq 0$ and $r_{\dagger}\neq 0$, both transmission and reflection
coefficients have one real zero for $\beta$ close to $\beta_{*}$. In Fig.
3(a)-(d), we show the phase of $f$, $\mbox{Im}(T)$, $R$ and $|t|^{2}$ as
functions of real $\omega$ and $\beta$. The real transmission and reflection
zeros form curves in the $\omega$-$\beta$ plane and shown as the solid red
curves in Fig. 3(b) and (c), respectively. The transmittance $|t|^{2}$ is
shown in Fig. 3(d) for $(\omega,\beta)$ near
$(\omega_{\dagger},\beta_{\dagger})$. To show the peaks and dips more clearly,
we plot transmission and reflection spectra for $\beta=0.21(2\pi/L)$ in Fig.
3(e) and (f), respectively. The zero dips in the spectra correspond to the
real transmission and reflection zeros.

The first two examples cover only generic cases with $t_{\dagger}\neq 0$ and
$r_{\dagger}\neq 0$. The third example is designed to illustrate a non-generic
case. The structure is still a periodic array of circular cylinders. Many BICs
in the array exist continuously with respect to radius $a$. For $a=0.2074L$,
there is a BIC (an anti-symmetric standing wave) with $\beta_{\dagger}=0$,
$\omega_{\dagger}=0.5835(2\pi c/L)$, and $r_{\dagger}=0$. In Fig. 4,

Figure 4: Transmission and reflection near a BIC in a periodic array of
circular cylinders with radius $a=0.2074L$: (a) the phase of $f$ in unit of
$\pi$, (b) $\mbox{Im}(T)$, (c) $R$, (d) $|t|^{2}$ with the BIC marked by
“$\circ$”, (e) transmission spectrum and (f) reflection spectrum for
$\beta=0.01(2\pi/L)$. The solid red curve in (b) corresponds to $T=0$. The
sets in (e) and (f) are logarithmic plots.

we show the phase of $f$, imaginary part of $T$, $R$ and $|t|^{2}$ as
functions of $\omega$ and $\beta$, and transmission and reflection spectra for
$\beta=0.01(2\pi/L)$. Since $|t_{\dagger}|=1$, there is a real transmission
zero for each $\beta$ near $\beta_{\dagger}=0$, and it corresponds to the red
curve in Fig. 4(b). On the other hand, $(\omega_{\dagger},\beta_{\dagger})$ is
a local maximum point of $R$ as shown in Fig. 4(c), and there are no real
reflection zeros for $\beta$ near $\beta_{\dagger}$. For $\beta=0.01(2\pi/L)$,
the transmission spectrum, shown in Fig. 4(e), has an inverted Lorentzian line
shape and a zero dip. The reflection spectrum, shown in Fig. 4(f), has a full
peak and no dip in the frequency range.

The fourth example is also designed to exhibit a non-generic case. It is well
known that a periodic array of circular cylinders can support propagating BICs
(with a nonzero Bloch wavenumber) that depend on the radius $a$ continuously.
For radius $a=0.3087L$, we found a propagating BIC with
$\beta_{\dagger}=0.2018(2\pi/L)$ and $\omega_{\dagger}=0.5994(2\pi c/L)$, and
$t_{\dagger}=0$. In Fig. 5(a)-(d),

Figure 5: Transmission and reflection near a BIC in a periodic array of
circular cylinders with radius $a=0.3087L$: (a) the phase of $f$ in unit of
$\pi$, (b) $\mbox{Im}(T)$, (c) $R$, (d) $|t|^{2}$, (e) $|t|^{2}$ for
$\beta=0.19(2\pi/L)$, (f) $|r|^{2}$ for $\beta=0.19(2\pi/L)$. The solid red
curves in (b) and (c), and the dashed red curves in (d), are the real zero
sets of $T$ (or $t$) and $R$. The insets in (e) and (f) are logarithmic plots.

we show the phase of $f$, $\mbox{Im}(T)$, $R$, and $|t|^{2}$, respectively, as
functions of real $(\omega,\beta)$. It is clear that
$(\omega_{\dagger},\beta_{\dagger})$ is saddle point of $\mbox{Im}(T)$. For
each $\beta$ near $\beta_{\dagger}$, the transmission coefficient has two real
zeros. As shown in Fig. 5(b), the zero set of $T$ form two intersecting curves
in the $\omega$-$\beta$ plane. The real transmission zeros are also shown in
Fig. 5(d) as the red dashed curves. In Fig. 5(e) and (f), we show the
transmission and reflection spectra, respectively, for $\beta=0.19(2\pi/L)$.
As shown clearly in the logarithmic plot (in the inset), there are two zero
dips in the transmission spectrum. Since $|r_{\dagger}|=1$, there is one real
reflection zero for each $\beta$ near $\beta_{\dagger}$. The zero set of $R$
is shown as the red curve in Fig. 5(c). The reflection spectrum of Fig. 5(f)
shows clearly a single zero dip.

The two examples above illustrate the non-generic cases where
$(\omega_{\dagger},\beta_{\dagger})$ is an extremum point of $R$ and a saddle
point of $\mbox{Im}(T)$, respectively. We have also found numerical examples
(still for a periodic array of circular cylinders) where
$(\omega_{\dagger},\beta_{\dagger})$ is an extremum point of $\mbox{Im}(T)$ or
a saddle point of $R$. Since the results are quite similar, we skip these
examples.

##  VI Conclusion

For lossless periodic structures with a proper symmetry, the transmission and
reflection spectra often have peaks and dips that are truly $100\%$ and zero,
respectively. When the transmission and reflection coefficients are considered
as functions of the frequency $\omega$, they vanish at their corresponding
zeros, but the zeros are complex in general. A zero dip in the
transmission/reflection spectrum corresponds to a real zero of the
transmission/reflection coefficient. Existing theories on real
transmission/reflection zeros have limitations and may be difficult to use [4,
11]. In this paper, a relatively simple theory on the existence (and
nonexistence) of real transmission/reflection zeros is developed. The key step
is to scale the transmission and reflection coefficients, $t$ and $r$, to $T$
and $R$, such that for structures with a proper symmetry, $T$ is a pure
imaginary analytic function and $R$ is a real analytic function of real
$\omega$ and $\beta$. We identified the generic case and two non-generic
cases, for which the number of real transmission/reflection zeros is 1, 0, or
2, respectively. The non-generic cases appear when the reflection or
transmission coefficient at $(\omega_{\dagger},\beta_{\dagger})$ (the BIC
frequency and wavenumber) vanishes, and when
$(\omega_{\dagger},\beta_{\dagger})$ is either an extremum point or a saddle
point of $R$ or $\mbox{Im}(T)$. Our theory is validated by numerical examples
involving periodic arrays of dielectric cylinders.

It should be pointed out that our theory relies on the analyticity of function
$f$ given in Eq. (17), but a rigorous mathematical proof is not available. In
addition, the theory is only applicable to periodic structures with a BIC. The
transmission and reflection spectra are those for incident waves with a
wavenumber $\beta$ near $\beta_{\dagger}$. For resonant scattering problems
without wavenumber $\beta$, for example, the scattering problem for a local
defect in a closed waveguide, the transmission and reflection spectra can be
approximated using the transmission and reflection coefficients at the real
resonant frequency [19]. However, it is not clear whether this approximate
theory can be used to establish the existence of real transmission/reflection
zeros rigorously.

## Acknowledgments

The authors acknowledge support from the Natural Science Foundation of
Chongqing, China (Grant No. cstc2019jcyj-msxmX0717), the Graduate Student
Research Innovation Project of Chongqing, and the Research Grants Council of
Hong Kong Special Administrative Region, China (Grant No. CityU 11304619).

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