# Parametric dependence of bound states in the continuum: a general theory

Amgad Abdrabou<EMAIL_ADDRESS>School of Mathematical Sciences, Zhejiang
University, Hangzhou 310027, China  Lijun Yuan  School of Mathematics and
Statistics, Chongqing Technology and Business University, Chongqing 400067,
China  Wangtao Lu  School of Mathematical Sciences, Zhejiang University,
Hangzhou 310027, China  Ya Yan Lu  Corresponding author<EMAIL_ADDRESS>Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong,
China

###### Abstract

Photonic structures with high-$Q$ resonances are essential for many practical
applications, and they can be relatively easily realized by modifying ideal
structures with bound states in the continuum (BICs). When an ideal photonic
structure with a BIC is perturbed, the BIC may be destroyed (becomes a
resonant state) or may continue to exist with a slightly different frequency
and a slightly different wavevector (if appropriate). Some BICs are robust
against certain structural perturbations, but most BICs are nonrobust. Recent
studies suggest that a nonnegative integer $n$ can be defined for any generic
nondegenerate BIC with respect to a properly defined set of structural
perturbations. The integer $n$ is the minimum number of tunable parameters
needed to preserve the BIC for perturbations arbitrarily chosen from the set.
Robust and nonrobust BICs have $n=0$ and $n\geq 1$, respectively. A larger $n$
implies that the BIC is more difficult to find. If a structure is given by $m$
real parameters, the integer $n$ is the codimension of a geometric object
formed by the parameter values at which the BIC exists in the $m$-dimensional
parameter space. In this paper, we suggest a formula for $n$, give some
justification for the general case, calculate $n$ for different types of BICs
in two-dimensional structures with a single periodic direction, and illustrate
the results by numerical examples. Our study improves the theoretical
understanding on BICs and provides useful guidance to their practical
applications.

##  I Introduction

In an open wave system, a bound state in the continuum (BIC) is an eigenmode
with a localized wave field (localized in the open spatial directions) and a
frequency inside the radiation continuum [1, 2]. An ideal BIC can be regarded
as a resonant state with an infinite quality factor ($Q$-factor) [3], and it
gives rise to high-$Q$ resonances when the structure and/or solution
parameters (such as the wavevector) are perturbed [4, 5, 6, 7, 8]. In recent
years, many applications of BICs have been realized in photonics [9, 10]. Most
of these applications rely on field enhancement [11, 12, 13, 14] or sharp
features in scattering spectra [15, 16, 17, 18] caused by the BIC-induced
high-$Q$ resonances. From a theoretical point of view, it is important to
understand how BICs are formed and what properties they possess. Existing
studies have identified a number of physical mechanisms under which BICs can
be found [1, 2]. Rigorous proofs for the existence of BICs are available for
symmetry-protected BICs [19, 20, 21]. In structures that are extended to
infinity in one or two spatial directions, a BIC is characterized by its
frequency and wavevector. It is known that some special BICs are surrounded by
resonances whose $Q$-factors diverge with unusually high rates [6, 8, 22].
Moreover, BICs in a periodic structure are polarization singularities in
momentum space and they can be characterized by a topological charge defined
using the polarization vector of the surrounding resonant states [23, 24, 25,
26, 27].

Another important theoretical question is concerned with the effect of
structural perturbations on BICs. For a symmetry-protected BIC, it is clear
that the BIC should still exist if the perturbation preserves the symmetry.
This means that the BIC is robust with respect to symmetry-preserving
perturbations. On the other hand, for a typical symmetry-breaking perturbation
of magnitude $\delta$, the BIC is destroyed and becomes a resonant state with
an $Q$-factor proportional to $1/\delta^{2}$ [7]. However, for special
symmetry-breaking perturbations, the BIC may become a resonant state with a
larger $Q$-factor (proportional to $1/\delta^{4}$, or $1/\delta^{6}$, …) [8],
and it can even remain as a BIC. Some BICs unprotected by symmetry are also
robust with respect to certain structural perturbations. As examples, we
mention propagating BICs in periodic structures with both the up-down mirror
symmetry and the in-plane inversion symmetry [28, 29], and BICs in optical
waveguides with lateral leakage channels [30]. Even though these BICs are
unprotected by symmetry, their robust existence still depends crucially on
symmetry. Similar to the case of symmetry-protected BICs, if the structural
perturbation breaks the required symmetry, the BIC is usually but not always
destroyed.

In recent works [31, 32], a new approach was developed to characterize
nonrobust BICs. The idea is to determine the minimum number of tuning
parameters needed to continuously follow the BIC. More precisely, assuming a
BIC exists in a structure with dielectric function $\varepsilon_{*}({\bm{r}})$
where ${\bm{r}}=(x,y,z)$ is the position vector, we try to find the smallest
integer $n$, so that the BIC persists in the perturbed structure with
dielectric function

$\varepsilon({\bm{r}})=\varepsilon_{*}({\bm{r}})+\delta
F({\bm{r}})+\gamma_{1}G_{1}({\bm{r}})+...+\gamma_{n}G_{n}({\bm{r}}),$ (1)

where $F$, $G_{1}$, …, $G_{n}$ are arbitrary perturbation profiles, $\delta$
is an arbitrary small real number, $\gamma_{1}$, …, $\gamma_{n}$ are tuning
parameters determined together with the BIC. If $n=0$, i.e., no tuning
parameters are necessary, then the BIC is robust. If $n\geq 1$, the BIC is
nonrobust. A larger $n$ implies that the BIC is more difficult to find. The
number $n$ also describes how the BIC depends on generic structural parameters
[32]. If the structure depends on $m$ parameters, we can consider the
geometric object $P_{\rm BIC}$ formed by the parameter values at which the BIC
exists in the $m$-dimensional parameter space, then $n=m-\dim(P_{\rm BIC})$ is
the codimension of $P_{\rm BIC}$. For example, if $n=1$, the BIC forms a curve
in the plane of two parameters and a surface in the 3D space of three
parameters.

However, the existing studies have covered only the cases $n=1$ and $n=2$ for
some BICs in 2D structures with a single periodic direction [31, 32]. In this
paper, we present a general theory for generic nondegenerate BICs. The rest of
this paper is organized as follows. In Sec. II, we present a general theory
including a formula for $n$ and a brief justification. In Sec. III, we apply
the general theory to BICs in 2D structures that are translationally invariant
in one spatial direction and periodic in another direction. Some details are
given for cases that have not been analyzed in previous works [31, 32].
Numerical examples for the new cases of Sec. III are presented in Sec. IV. The
paper is concluded with a brief discussion in Sec. V.

##  II General theory

To present the theory in a more concrete setting, we assume an electromagnetic
BIC of frequency $\omega_{*}$ exists in a lossless non-magnetic structure
given by scalar dielectric function $\varepsilon_{*}({\bm{r}})$. The structure
may be translationally invariant and/or periodic in one or two directions,
thus the BIC may possess a wavevector ${\bm{\alpha}}_{*}$ which contains at
most two nonzero components. The electric field of the BIC can be written as

${\bm{E}}_{*}({\bm{r}})={\bm{\Phi}}_{*}({\bm{r}})e^{i{\bm{\alpha}}_{*}\cdot{\bm{r}}}$
(2)

where ${\bm{\Phi}}_{*}$ has the same invariant and/or periodic directions as
$\varepsilon_{*}$. For example, if the structure is a photonic crystal (PhC)
slab with 2D periodicity in the $xy$-plane, then
${\bm{\alpha}}_{*}=(\alpha_{*},\beta_{*},0)$, where $\alpha_{*}$ and
$\beta_{*}$ are the Bloch wavenumbers.

The definition of BIC requires the existence of at least one radiation
channel. This implies that the structure must have at least one open direction
where outgoing and incoming waves with the same frequency $\omega_{*}$ and
wavevectors compatible with ${\bm{\alpha}}_{*}$ can propagate to and from
infinity, respectively. For a PhC slab surrounded by air, the $z$ variable
(perpendicular to the slab) provides the open directions (as $z\to\pm\infty$),
and the propagating diffraction orders (compatible with ${\bm{\alpha}}_{*}$)
serve as the radiation channels. For each radiation channel, we have at least
one and at most two linearly independent scattering solutions. The case of two
solutions in one radiation channel arises when the incident waves can have
different polarizations. Let us denote the electric fields of the independent
scattering solutions by
${\bm{E}}_{k}^{(s)}({\bm{r}})={\bm{\Psi}}_{k}({\bm{r}})e^{i{\bm{\alpha}}_{*}\cdot{\bm{r}}}$
for $k=1$, 2, …, $N_{\rm ss}$, where $N_{\rm ss}$ is the total number of such
solutions.

In order to discuss the robustness of a BIC and determine the codimension $n$
of nonrobust BICs, we must specify the conditions (most importantly, the
symmetries) satisfied by the perturbation profiles $F({\bm{r}})$ and
$G_{j}({\bm{r}})$, $1\leq j\leq n$. The original structure, given by the
dielectric function $\varepsilon_{*}({\bm{r}})$, may have more symmetries than
the perturbation profiles. For example, a PhC slab may have $C_{4}$ or $C_{6}$
symmetry, but robustness can be discussed for perturbations with $C_{2}$
symmetry only, where $C_{p}$ is the rotation by $2\pi/p$ about the $z$-axis.
The symmetries in the original structure and the perturbations give rise to
related symmetries in the BIC and the scattering solutions. Typically, this
requires a proper scaling. For example, in a PhC slab with $C_{2}$ symmetry
[same as the in-plane inversion symmetry, namely
$\varepsilon({\bm{r}})=\varepsilon(-x,-y,z)$], the BIC and the scattering
solutions can be scaled such that

$\begin{bmatrix}\overline{E}_{x}({\bm{r}})\cr\overline{E}_{y}({\bm{r}})\cr\overline{E}_{z}({\bm{r}})\end{bmatrix}=\begin{bmatrix}E_{x}(-x,-y,z)\cr
E_{y}(-x,-y,z)\cr-E_{z}(-x,-y,z)\end{bmatrix},$ (3)

where $E_{x}$ is the $x$-component of ${\bm{E}}_{*}$ or ${\bm{E}}^{(s)}_{k}$,
$\overline{E}_{x}$ is its complex conjugate, etc [29]. Notice that
${\bm{\Phi}}_{*}$ and ${\bm{\Psi}}_{k}$ also satisfy Eq. (3).

For each scattering solution, we define an integer index. First, we consider
the linear operator ${\cal L}$, such that the BIC and the scattering solutions
(with $e^{i{\bm{\alpha}}_{*}\cdot{\bm{r}}}$ removed) satisfy

${\cal L}{\bm{\Phi}}_{*}={\bm{0}},\quad{\cal L}{\bm{\Psi}}_{k}={\bm{0}},$ (4)

for $k=1$, …, $N_{\rm ss}$. The inhomogeneous equation

${\cal L}{\bm{u}}={\bm{f}}$ (5)

usually has outgoing solutions, since ${\bm{f}}$ in the right hand side serves
as a source. If we insist that Eq. (5) has a solution that decays to zero
rapidly in the open direction(s), then

$\int_{\Omega}\overline{\bm{\Psi}}_{k}\cdot{\bm{f}}\,d{\bm{r}}=\int_{\Omega}\overline{\bm{\Psi}}_{k}\cdot{\cal
L}{\bm{u}}\,d{\bm{r}}=\int_{\Omega}\overline{{\cal
L}\bm{\Psi}_{k}}\cdot{\bm{u}}\,d{\bm{r}}=0,$ (6)

where $\Omega$ covers one period in the periodic direction and a unit length
in the invariant direction. The index $I_{k}$ is defined as the number of real
constraints on ${\bm{f}}$ for the left hand side above being zero. Since
${\bm{\Psi}}_{k}$ is complex, we usually have $I_{k}=2$. However, if
${\bm{f}}$ has some useful symmetry related to the symmetry of the BIC and the
perturbation profiles, $I_{k}$ may be reduced to $1$ or $0$. Moreover, if
there are two scattering solutions in the same radiation channel, they should
be chosen so that the constraints are independent.

If a BIC exists in the perturbed structure with the dielectric function given
in Eq. (1), it should have a frequency near $\omega_{*}$ and a wavevector near
${\bm{\alpha}}_{*}$. The degrees of freedom $N_{\rm wv}$ of the BIC
wavevector, is the number of components in ${\bm{\alpha}}_{*}$ that will vary
independently. Usually $N_{\rm wv}=2$, but if the BIC is generic and
${\bm{\alpha}}_{*}={\bm{0}}$, then, as we will show in Sec. III, the perturbed
BIC also has a zero wavevector, thus $N_{\rm wv}=0$. If ${\bm{\alpha}}_{*}$
has just one nonzero component, the wavevector of the perturbed BIC may have
one or two nonzero components depending on the symmetry, thus $N_{\rm wv}$ can
be 1 or 2, respectively.

Based on the above definitions, we believe that the integer $n$ of a generic
nondegenerate BIC is

$n=\sum_{k=1}^{N_{\rm ss}}I_{k}-N_{\rm wv}.$ (7)

To find $n$, we actually need to show that a BIC exists in the perturbed
structure with the minimum of $n$ tunable parameters. To do so, we expand the
desired BIC [with frequency $\omega$, wavevector ${\bm{\alpha}}$, electric
field
${\bm{E}}({\bm{r}})={\bm{\Phi}}({\bm{r}})e^{i{\bm{\alpha}}\cdot{\bm{r}}}$] and
the tunable parameters in power series of $\delta$:

$\displaystyle{\bm{\Phi}}={\bm{\Phi}}_{*}+\delta{\bm{\Phi}}_{1}+\delta^{2}{\bm{\Phi}}_{2}+\cdots,$
(8)
$\displaystyle\omega=\omega_{*}+\delta\omega_{1}+\delta^{2}\omega_{2}+\cdots,$
(9)
$\displaystyle{\bm{\alpha}}={\bm{\alpha}}_{*}+\delta{\bm{\alpha}}_{1}+\delta^{2}{\bm{\alpha}}_{2}+\cdots,$
(10)
$\displaystyle\gamma_{j}=\delta\gamma_{j,1}+\delta^{2}\gamma_{j,2}+\cdots,\quad
j=1,...,n.$ (11)

The leading order gives ${\cal L}{\bm{\Phi}}_{*}={\bm{0}}$, i.e., the original
equation satisfied by the BIC. For $l\geq 1$, the ${\cal O}(\delta^{l})$
equation can be written as

${\cal L}{\bm{\Phi}}_{l}={\bm{f}}_{l}$ (12)

where ${\bm{f}}_{l}$ is related to $\omega_{l}$, ${\bm{\alpha}}_{l}$,
$\gamma_{j,l}$, ${\bm{\Phi}}_{*}$, …, ${\bm{\Phi}}_{l-1}$, $\varepsilon_{*}$
and the perturbation profiles.

If Eq. (12) has a solution that decays rapidly in the open direction(s) and
has the same symmetry as the original BIC, we obtain $\sum_{k=1}^{N_{\rm
ss}}I_{k}$ constraints on ${\bm{f}}_{l}$ from the $N_{\rm ss}$ scattering
solutions. In addition, similar to Eq. (6), we have

$\int_{\Omega}\overline{\bm{\Phi}}_{*}\cdot{\bm{f}}_{l}\,d{\bm{r}}=0.$ (13)

The above always gives one real constraint. Therefore, we have the total of
$1+\sum_{j=1}^{N_{\rm ss}}I_{j}$ real equations for solving $\omega_{l}$,
${\bm{\alpha}}_{l}$ and $\gamma_{j,l}$ for $1\leq j\leq n$. The total number
of real unknowns is $1+N_{\rm wv}+n$. We choose $n$ to satisfy Eq. (7), so
that the total number of equations is exactly the total number of unknowns.
Since the unknowns appear in ${\bm{f}}_{l}$ linearly with coefficients only
related to $\varepsilon_{*}$, ${\bm{\Phi}}_{*}$, $F$ and $G_{j}$, we have a
real square matrix $A$ and a real vector ${\bm{b}}_{l}$, such that

$A\begin{bmatrix}\omega_{l}\cr{\bm{\alpha}}_{l}\cr\gamma_{1,l}\cr\vdots\cr\gamma_{n,l}\end{bmatrix}={\bm{b}}_{l},$
(14)

where ${\bm{\alpha}}_{l}$ includes only $N_{\rm wv}$ components. The matrix
$A$ depends on the BIC, the scattering solutions and the perturbation
profiles, but it does not depend on the previous iterations ${\bm{\Phi}}_{s}$
for $1\leq s<l$. Importantly, Eq. (6), with ${\bm{f}}$ replaced by
${\bm{f}}_{l}$, and Eq. (13) are sufficient conditions for Eq. (8) to have a
solution that decays in the open direction(s) and preserves the symmetry of
the original BIC [32, 29]. Therefore, if $A$ is invertible, we can solve
$\omega_{l}$, ${\bm{\alpha}}_{l}$ and $\gamma_{j,l}$ from Eq. (14), then solve
${\bm{\Phi}}_{l}$ from Eq. (12). This implies that we can iteratively find the
BIC and the tuning parameters through the power series.

The theory is applicable to generic BICs defined as those for which the matrix
$A$ is invertible. The perturbation profiles should be arbitrary except for
some specified conditions (such as the symmetry). Even for a generic BIC, if
the perturbations are improperly chosen, the matrix $A$ can be non-invertible,
then a BIC may not exist in the perturbed structure. On the other hand, there
are also non-generic BICs for which the matrix $A$ is always non-invertible
for any choice of the perturbation profiles.

##  III BICs in 2D periodic structures

In this section, we consider BICs in 2D lossless dielectric structures that
are invariant in $x$, periodic in $y$ with period $L$, bounded in $z$ by
$|z|<d$, and surrounded by air. The dielectric function $\varepsilon(y,z)$ of
such a structure is real and satisfies

$\displaystyle\varepsilon(y+L,z)=\varepsilon(y,z),\quad\forall\
(y,z)\in\mathbb{R}^{2},$ (15) $\displaystyle\varepsilon(y,z)=1,\quad|z|>d.$
(16)

BICs in 2D structure with 1D periodicity have been investigated by many
authors [33, 34, 35, 36, 37, 38, 39]. Very often, one assumes that the
structure has the following additional symmetry:

$\displaystyle\varepsilon(y,z)=\varepsilon(-y,z),\quad\forall(y,z)\in\mathbb{R}^{2},$
(17)
$\displaystyle\varepsilon(y,z)=\varepsilon(y,-z),\quad\forall(y,z)\in\mathbb{R}^{2}.$
(18)

In recent works [31, 32], the codimension $n$ for some BICs in 2D periodic
structures with the up-down mirror symmetry, i.e., Eq. (18), has been
determined. In the following, we apply the general theory of Sec. II to all
cases with or without the symmetry conditions (17) and (18).

Let $\varepsilon_{*}(y,z)$ be the dielectric function of a specific periodic
structure [satisfying Eqs. (15) and (16)], in which there is a nondegenerate
BIC with frequency $\omega_{*}$ and wavevector
${\bm{\alpha}}_{*}=(\alpha_{*},\beta_{*},0)$, where $\alpha_{*}$ and
$\beta_{*}$ are real wavenumbers in the $x$ and $y$ directions, respectively.
Due to the periodicity in $y$, we can assume $\beta_{*}\in(-\pi/L,\pi/L]$. For
simplicity, we consider BICs with $\omega_{*}$ satisfying

$\sqrt{\alpha_{*}^{2}+\beta_{*}^{2}}<\frac{\omega_{*}}{c}<\sqrt{\alpha_{*}^{2}+\left(\frac{2\pi}{L}-|\beta_{*}|\right)^{2}},$
(19)

where $c$ is the speed of light in vacuum. The above condition implies that
only the zeroth diffraction order is propagating and all other diffraction
orders (corresponding to the $y$-wavenumber $\beta_{*}+2m\pi/L$ for $m\neq 0$)
are evanescent.

To apply the theory of Sec. II, we first consider the degrees of freedom
$N_{\rm wv}$ for the BIC wavevector. If both $\alpha_{*}$ and $\beta_{*}$ are
nonzero, a BIC in the perturbed structure should have wavenumbers $\alpha$
near $\alpha_{*}$ and $\beta$ near $\beta_{*}$, and thus $N_{\rm wv}=2$. If
$\alpha_{*}\neq 0$ and $\beta_{*}=0$, we claim that the perturbed BIC must
have $\beta=0$ and thus $N_{\rm wv}=1$. This is true only when the BIC in the
unperturbed structure is generic, so that there is only one BIC (near the
original one) can be found in the perturbed structure. Recall that a BIC is a
special point in a band of resonant states with a dispersion relation
$\omega=\omega_{\dagger}(\alpha,\beta)$, where $\omega_{\dagger}$ is a
complex-valued function of $\alpha$ and $\beta$. Due to reciprocity and the
reflection symmetry in $x$, $\omega_{\dagger}$ satisfies

$\displaystyle\omega_{\dagger}(\alpha,\beta)=\omega_{\dagger}(-\alpha,-\beta),$
(20)
$\displaystyle\omega_{\dagger}(\alpha,\beta)=\omega_{\dagger}(-\alpha,\beta).$
(21)

The above two equations lead to
$\omega_{\dagger}(\alpha,\beta)=\omega_{\dagger}(\alpha,-\beta)$. Therefore,
if there is a BIC in the perturbed structure with a small nonzero $\beta$,
there must be another one with wavenumber $-\beta$. Since the original BIC is
generic, this is not possible. Similarly, if the original BIC has
$\alpha_{*}=0$ and $\beta_{*}\neq 0$, then the perturbed BIC must also have
$\alpha=0$, and thus $N_{\rm wv}=1$. Finally, if the unperturbed structure has
a generic BIC with $\alpha_{*}=\beta_{*}=0$, then the perturbed BIC also have
$\alpha=\beta=0$, and $N_{\rm wv}=0$.

In Table 1,

Table 1: Codimension of BICs in 2D periodic structures with different symmetry. $(\alpha_{*},\beta_{*})$ | $(\square,\square)$ | $(\square,0)$ | $(0,\square)$ | $(0,0)$
---|---|---|---|---
No symmetry | 6 | 3 | 3 | 2
$y\leftrightarrow-y$ | 2 | 1 | 1 | 0
$z\leftrightarrow-z$ | 2 | 1 | 1 | 1
$y\leftrightarrow-y$, $z\leftrightarrow-z$ | 0 | 0 | 0 | 0

we list the codimension $n$ for different kinds of BICs according to the
symmetry of the structure and the zero pattern of the wavevector. In the first
column of Table 1, the reflection symmetries in $y$ and $z$ are denoted as
$y\leftrightarrow-y$ and $z\leftrightarrow-z$, respectively. The first row
shows the zero pattern of the wavevector $(\alpha_{*},\beta_{*})$, where
$\square$ denotes a nonzero entry. The last two rows summarize the results
already obtained in previous works [31, 32]. In the following, we give some
justification for the new results listed in the table.

First, we consider the case of no symmetry. The perturbation profiles $F$ and
$G_{j}$ for $1\leq j\leq n$, do not need to satisfy Eq. (17) or (18), but they
must be real functions of $y$ and $z$, periodic in $y$ with period $L$, and
vanish for $|z|>d$. Since the BIC in the unperturbed structure satisfies
condition (19), there are two radiation channels (below and above the periodic
layer, respectively) corresponding to the zeroth diffraction order.

For a generic BIC with $\alpha_{*}\neq 0$ and $\beta_{*}\neq 0$, we need to
consider both polarizations in each radiation channel. Thus, the total number
of independent scattering solutions is $N_{\rm ss}=4$. The constraint for each
scattering solution is a complex condition, thus the index for each scattering
solution is $I_{k}=2$. Therefore, the codimension of the BIC is $n=4\times
2-2=6$.

If $\alpha_{*}\neq 0$ and $\beta_{*}=0$, the BIC propagates along the $x$
axis, has a vectorial field ${\bm{E}}_{*}={\bm{\Phi}}_{*}e^{i\alpha_{*}x}$
with ${\bm{\Phi}}_{*}$ depending on $y$ and $z$ only. Moreover, the BIC can be
scaled such that the $x$ component of ${\bm{\Phi}}_{*}$ is pure imaginary and
the $y$ and $z$ components of ${\bm{\Phi}}_{*}$ are real [32]. Similarly, we
can scale the scattering solutions
${\bm{E}}_{k}^{(s)}={\bm{\Psi}}_{k}e^{i\alpha_{*}x}$, so that the $y$ and $z$
components of ${\bm{\Psi}}_{k}$ are real and the $x$ component of
${\bm{\Psi}}_{k}$ is pure imaginary. As a result, the integral condition
$\int_{\Omega}{\bm{\Psi}}_{k}\cdot{\bm{f}}_{l}\,d{\bm{r}}=0$ gives one real
equation, and $I_{k}=1$. Since $\Omega$ is a 3D domain with a unit length in
$x$, the above integral on $\Omega$ is identical to the integral on the 2D
cross section of $\Omega$ given by $0<y<L$ and $-\infty<z<\infty$. Since
$N_{\rm ss}=4$ and $N_{\rm wv}=1$, we have $n=4\times 1-1=3$.

The BIC with $\alpha_{*}=0$ and $\beta_{*}\neq 0$ is a scalar mode in the $E$
or $H$ polarization. The electric or magnetic field has only one nonzero
component (the $x$ component). Since the fields for opposite polarizations are
orthogonal, we can consider only the scattering solutions in the same
polarization as the BIC. Therefore, $N_{\rm ss}=2$. Since $I_{k}=2$ and
$N_{\rm wv}=1$, we have $n=2\times 2-1=3$.

If the BIC is a standing wave with $\alpha_{*}=\beta_{*}=0$, we have $I_{k}=1$
and $N_{\rm ss}=2$ for the same reasons given above. Therefore, $n=2\times
1-0=2$.

Next, we consider the case with reflection symmetry in $y$. Notice that we
assume $\varepsilon_{*}$, $F$ and $G_{j}$ all satisfy symmetry condition (17).
In addition, $F$ and $G_{j}$ must be real and periodic as before. Since the
structure is invariant in $x$, Eq. (17) is identical to the in-plane inversion
symmetry $\varepsilon(x,y,z)=\varepsilon(-x,-y,z)$ or $C_{2}$ symmetry
(rotation by $180^{\circ}$ about the $z$ axis). As we have mentioned in Sec.
II, this symmetry allows us to scale the BIC and the scattering solutions, so
that their $x$ and $y$ components are ${\cal PT}$-symmetric and their $z$
components are anti-${\cal PT}$-symmetric, as in Eq. (3) [32]. For
$x$-invariant structures, ${\bm{\Phi}}_{*}$ and ${\bm{\Psi}}_{k}$ are
functions of $y$ and $z$ only, then the ${\cal PT}$ and anti-${\cal PT}$
symmetry conditions are

$\begin{bmatrix}\overline{P}_{x}(-y,z)\cr\overline{P}_{y}(-y,z)\cr-\overline{P}_{z}(-y,z)\end{bmatrix}=\begin{bmatrix}P_{x}(y,z)\cr
P_{y}(y,z)\cr P_{z}(y,z)\end{bmatrix}$ (22)

where $P_{x}$, $P_{y}$ and $P_{z}$ are the components of ${\bm{\Phi}}_{*}$ or
${\bm{\Psi}}_{k}$ and $\overline{P}_{x}$ is the complex conjugate of $P_{x}$.

For a generic BIC with $\alpha_{*}\neq 0$ and $\beta_{*}\neq 0$, we have
$N_{\rm ss}=4$ and $N_{\rm wv}=2$ as before. The ${\cal PT}$ and anti-${\cal
PT}$ symmetry reduces the index $I_{k}$ from 2 to $1$. Therefore, the
codimension of the BIC is $n=4\times 1-2=2$.

For a BIC with $\alpha_{*}\neq 0$ and $\beta_{*}=0$, we have $N_{\rm ss}=4$
and $N_{\rm wv}=1$. Since the BIC and the scattering solutions can be scaled
such that their $x$ components are pure imaginary and their $y$ and $z$
components are real, we have

$\begin{bmatrix}P_{x}(y,z)\cr P_{y}(y,z)\cr
P_{z}(y,z)\end{bmatrix}=\begin{bmatrix}iQ_{x}(y,z)\cr P_{y}(y,z)\cr
P_{z}(y,z)\end{bmatrix},$ (23)

where $Q_{x}$, $P_{y}$ and $P_{z}$ are real. The ${\cal PT}$ and anti-${\cal
PT}$ symmetry, i.e., Eq. (22), is obtained under a different scaling which can
be compensated by multiplying a constant $C$ of unit magnitude to the left
hand side of Eq. (22). Therefore,

$\begin{bmatrix}iQ_{x}(y,z)\cr P_{y}(y,z)\cr
P_{z}(y,z)\end{bmatrix}=C\begin{bmatrix}-iQ_{x}(-y,z)\cr P_{y}(-y,z)\cr-
P_{z}(-y,z)\end{bmatrix}.$ (24)

The above implies that $C$ is real and can only be $1$ or $-1$. Among the four
scattering solutions, two have $C=1$ and the other two have $C=-1$. If the
scattering solution has a different value of $C$ with the BIC, then the index
is $I_{k}=0$. Therefore, $n=1+1+0+0-1=1$.

The BIC with $\alpha_{*}=0$ and $\beta_{*}\neq 0$ is a scalar mode, thus
$N_{\rm ss}=2$. Using the scaling for ${\cal PT}$ and anti-${\cal PT}$
symmetry, we have $I_{k}=1$. Therefore $n=2\times 1-1=1$.

The BIC with $\alpha_{*}=\beta_{*}=0$ is a scalar standing wave. If it is
generic, it has the opposite even/odd parity in $y$ with the scattering
solutions of the same polarization. Therefore, the BIC is symmetry-protected
and $n=0$.

In Refs. [31, 32], detailed derivations are given for the results listed in
the last two rows of Table 1. For the two cases considered in this section (no
symmetry and reflection symmetry in $y$), similar derivations can be worked
out following the brief discussions above. In Table 1, the results are shown
only for BICs satisfying condition (19). Using the general formula (7), it is
not difficult to calculate the codimension $n$ for BICs that fail to satisfy
(19).

##  IV Numerical results

To validate the theory developed in the previous sections, we use a highly
accurate boundary integral equation method [40] to analyze two periodic
structures with broken up-down mirror symmetry. The first example is a slab
with a periodic array of air holes shown in Fig. 1(a).

(a)  (b)

(c)  (d)

(e)  (f)

Figure 1: (a) One period of the slab with an array of distorted air holes (for
$\delta=0.1$ and $\gamma=-0.1$) breaking the reflection symmetry in $y$. (b)
Electric field pattern of a propagating BIC in a slab with circular air holes
of radius $a=0.3L$. (c) Parameter curve of the BIC. (d) and (e) Wavenumber
$\beta$ and frequency $\omega$ of the BIC depending on parameters $\delta$ and
$\gamma$. (f) Electric field pattern of the BIC at $\delta=0.1$ and
$\gamma\approx-0.07$.

The thickness and dielectric constant of the slab are $2d=L$ and
$\varepsilon_{1}=11.56$, respectively, where $L$ is period in the $y$
direction. The medium above and below the slab (i.e. for $|z|>d$) is air. The
boundary of the air hole is perturbed from a circle of radius $a$, such that
the reflection symmetry in $y$ is preserved and the reflection symmetry in $z$
is broken. The perturbed boundary consists of two independent real parameters
$\delta$ and $\gamma$ which correspond to the height (relative to $a$) of the
bump and the dents shown in Fig. 1(a). The precise formula of the boundary is

$y=a\rho(s)\cos(s),\ z=a\rho(s)\sin(s),\quad 0\leq s<2\pi,$ (25)

where $\rho(s)=1+\delta g(s)+\gamma[g(s-0.2)+g(s+0.2)]$, and
$g(s)=\exp(-100|s-\pi/2|^{2})$. For $a=0.3L$ and $\delta=\gamma=0$, the air
holes are circular and the structure has an $E$-polarized propagating BIC with
$\alpha_{*}=0$, $\beta_{*}=0.21728(2\pi/L)$ and $\omega_{*}=0.44746(2\pi
c/L)$. The electric field distribution (real part of the $x$ component) of the
BIC is shown in Fig. 1(b).

According to the theory of Sec. III, for perturbations that preserve the
reflection symmetry in $y$, the scalar propagating BIC should have codimension
$n=1$. Therefore, for the fixed $a=0.3L$, the BIC should form a curve in the
$\delta$-$\gamma$ plane. This is confirmed by the numerical result shown in
Fig. 1(c). The Bloch wavenumber $\beta$ and frequency $\omega$ of this BIC are
shown as functions of $\delta$ and $\gamma$ in Figs. 1(d) and 1(e),
respectively. For $\delta=0.1$, the BIC is obtained with $\gamma\approx-0.07$
for $\beta=0.21789(2\pi/L)$ and $\omega=0.44754(2\pi c/L)$. Its electric field
distribution (the real part of $E_{x}$) is shown in Fig. 1(f). Moreover, since
its codimension is 1, this BIC should form a surface in the space of three
parameters. To illustrate this, we simply allow $a$ to be the third parameter.
In Fig. 2,

Figure 2: A surface in the 3D space of $\delta$, $\gamma$ and $a$, for a BIC
in a slab with an array of distorted air holes breaking the reflection
symmetry in $y$.

we show a surface of this BIC in the $\delta$-$\gamma$-$a$ space for $0.28\leq
a/L\leq 0.36$. Notice that this surface includes the vertical axis at
$\delta=\gamma=0$. It simply means that if the air holes are circular, the BIC
is robust with respect to changes in the radius.

The second example is also a slab with a periodic array of air holes, but the
boundaries of the air holes are perturbed to break both reflection symmetries
in $y$ and in $z$. The slab has the same thickness $2d=L$ and the same
dielectric constant $\varepsilon_{1}=11.56$. The boundary of the air hole is
also given by Eq. (25), but $\rho(s)$ is now given by

$\displaystyle\rho(s)=1+\delta
g(s)+\sum_{j=1}^{3}\gamma_{j}\left[g\left(s-\frac{j\pi}{10}\right)+g\left(s+\frac{j\pi}{10}\right)\right],$

and $g(s)$ is given by $g(s)=\exp(-100|s-2\pi/3|^{2})$. Since the codimension
of a scalar propagating BIC in a periodic structure without symmetry is $n=3$,
we have introduced three tunable parameters $\gamma_{j}$ for $1\leq j\leq 3$.
However, if the BIC is a standing wave (in such a structure without symmetry),
the codimension is $n=2$. In Fig. 3(a),

(a)  (b)

(c)  (d)

Figure 3: (a) One period of a slab with an array of distorted air holes
breaking both reflection symmetries in $y$ and $z$. (b) Electric field pattern
of a standing wave in a slab with circular air holes of radius $a=0.3L$. (c)
Tuned parameters $\gamma_{1}$ and $\gamma_{2}$ as functions of $\delta$ for a
standing wave in the slab with distorted air holes. (d) Electric field pattern
of the standing wave for $\delta=0.1$ and corresponding values of $\gamma_{1}$
and $\gamma_{2}$.

we show a unit cell of the periodic structure for $a=0.3L$, $\delta=0.1$,
$\gamma_{1}=0.06456$, $\gamma_{2}=-0.00524$, and $\gamma_{3}=-0.006$.

In the structure with circular air holes of radius $a=0.3L$, there is a BIC (a
symmetry-protected standing wave) with frequency $\omega_{*}=0.52044(2\pi
c/L)$. Its electric field is shown in Fig. 3(b). For $\delta>0$, we keep
$\gamma_{3}=0$ and tune $\gamma_{1}$ and $\gamma_{2}$ to preserve this
standing wave. The resulting $\gamma_{1}$ and $\gamma_{2}$ are shown in Fig.
3(c) as functions of $\delta$ for $0\leq\delta\leq 0.1$. For $\delta=0.1$, the
frequency of the BIC is $\omega=0.52053(2\pi c/L)$ and its electric field is
shown in Fig. 3(d).

To preserve a propagating BIC in such a periodic structure without symmetry,
three tuning parameters are needed. Starting from the same propagating BIC
studied in the first example (for circular air hole with radius $a=0.3L$), we
calculate $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ to follow the BIC as
$\delta$ is increased from $0$, and show the numerical results in Fig. 4.

Figure 4:  (a), (b) and (c) Tuned parameters $\gamma_{1}$, $\gamma_{2}$ and
$\gamma_{3}$ as functions of $\delta$ for a propagating BIC in a slab with
distorted air holes breaking both reflection symmetries in $y$ and $z$. (d)
Wavenumber $\beta$ and frequency $\omega$ of the BIC as functions of $\delta$.
(e) $Q$ factor of the resonant modes near the BIC for $\delta=0.1$. (f)
Electric field pattern of the BIC for $\delta=0.1$.

The three parameters as shown as functions of $\delta$ in Figs. 4(a), 4(b) and
4(c), respectively. The frequency and Bloch wavenumber of the BIC are shown in
Fig. 4(d). For $\delta=0.1$, the BIC has $\beta=0.21673(2\pi/L)$ and
$\omega=0.44735(2\pi c/L)$, and is obtained at $\gamma_{1}=-0.04357$,
$\gamma_{2}=-0.00952$, and $\gamma_{3}=0.01385$. In Fig. 4(e), we show the
quality factor, $Q=-0.5\mathrm{Re}(\omega)/\mathrm{Im}(\omega)$, of nearby
resonant modes. The electric field of the BIC (the real part of $E_{x}$) is
shown in Fig. 4(f).

##  V Conclusion

For theoretical interest and practical applications, it is important to find
out what will happen to a BIC when the structure is perturbed. Although it
typically becomes a resonant state of finite $Q$-factor, the BIC may persist
either because it is robust against a class of perturbations, or the
perturbation contains a sufficient number of tunable parameters. We have
proposed a general formula for the minimum number $n$ of tunable parameters
needed to preserve a generic nondegenerate BIC, and calculated $n$ for BICs in
2D structures that are invariant in one spatial direction and periodic in
another. The integer $n$ is only defined when conditions (typically some
symmetry) on structural perturbations are properly specified. For $n=0$ and
$n\geq 1$, the BIC is robust and nonrobust, respectively. A larger $n$ means
that the BIC is difficult to find. A different point of view is to consider
structures depending on a number of parameters. The set of parameter values at
which a BIC exists form a geometric object in the parameter space, and the
codimension of that geometric object is exactly $n$.

Although we have only verified our results for BICs in 2D structures with a
single periodic direction, the general formula (7) is proposed for all generic
nondegenerate electromagnetic BICs. In fact, we believe the general formula
for $n$ is valid for any classical or quantum wave system. However, the
current theory is only applicable to generic BICs that guarantee $n$ can be
defined. Under a structural perturbation, a generic BIC either becomes a
resonant state or continues its existence with slightly different frequency
and wavevector. In contrast, a non-generic BIC has the additional possibility
of splitting into two or more generic BICs. It is worthwhile to extend our
theory to non-generic BICs, since they have additional interesting properties
and valuable applications.

## Acknowledgments

L.Y. acknowledges support from Natural Science Foundation of Chongqing, China
(Grant No. cstc2019jcyj-msxmX0717). W.L. is partially supported by NSFC Grant
12174310 and by the NSF of Zhejiang Province for Distinguished young scholars
LR21A010001. Y.Y.L. acknowledges support from the Research Grants Council of
Hong Kong Special Administrative Region, China (Grant No. CityU 11307720).

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