# Origin of bound states in the continuum in a grating waveguide:
Parity symmetry breaking due to mode crossing

C.B. Reynolds, Vl.V. Kocharovsky, V.V. Kocharovsky
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843, USA

###### Abstract

A mechanism leading to the formation of bound states in the continuum (BICs)
in a planar grating waveguide is revealed. BICs appear when dispersion curves
of the leaking even-parity and non-leaking odd-parity infinite-grating
eigenmodes intersect. These intersections engender a parity symmetry break of
the leaking eigenmode, causing it to decouple from a spatial Fourier harmonic
that is emitted out of the waveguide. As a result, the otherwise low-Q
waveguide eigenmode, comprised of those two aforementioned and some other
infinite-grating eigenmodes, acquires a very high quality factor in a narrow
vicinity of the mode-crossing frequency. Implementation of such a mechanism
can be instructive for designing BICs in other photonic crystals and
structures.

##  I Introduction: Hunting a bound state in the continuum as a resource for a
solitary high-Q resonance

By means of the analytical theory of the infinite-grating eigenmodes [1], we
disclose a distinct, explicit mechanism leading to the formation of the bound
states in the continuum (BICs) and an entire hierarchy of BICs in a planar
grating waveguide, Fig. 1. Namely, a grating-waveguide eigenmode becomes a BIC
due to a parity symmetry breaking of a leaking even-parity infinite-grating
eigenmode at a crossing frequency (a degeneracy point) where its dispersion
curve intersects the dispersion curve of a non-leaking odd-parity infinite-
grating eigenmode. The aforementioned two and some other infinite-grating
eigenmodes are coupled into the BIC grating-waveguide eigenmode due to
reflection at the borders of the waveguide grating layer. However, only the
aforementioned even-parity one is leaking (i.e., providing radiation losses)
since, due to an in-plane Bragg coupling, the losses are limited to just one
radiation channel of emission of the zeroth spatial Fourier harmonic outside
the grating waveguide into the cover and substrate. All other higher-order-
Bragg Fourier harmonics cannot be emitted into the cover and substrate because
they are evanescent there. This phenomenon results in a complete decoupling of
the waveguide eigenmode from the radiation-loss channel. It is of general
nature and could exist in a number of other photonic structures and crystals.

Figure 1: The planar grating waveguide. (a) Geometry: the grating of thickness
$L$ and period $\lambda_{g}$ sandwiched between the substrate and cover, with
permittivities $\varepsilon^{-}$ and $\varepsilon^{+}$, respectively. (b) The
wave vectors associated with the propagating or evanescent eigenmodes inside
the grating and the plane waves (Fourier harmonics) in the substrate and
cover.

The BIC waveguide eigenmode demonstrates an extremely narrow, solitary high-Q
resonance despite the fact that a continuum of plane waves in a wide range of
frequencies and wave vectors around the resonant ones can freely escape from
the waveguide into the cover and substrate. The latter distinguishes the
phenomenon of BICs from a conventional trapping of fields inside waveguides or
cavities which is based on a complete prevention (say, by means of reflection)
of the entire continuum of waves from propagating outside the waveguide or
cavity. The search, analysis, design, and applications of the BICs constitute
a very active field of modern research in a variety of wave systems,
especially in photonics, despite the BIC having first been suggested for
matter waves in quantum mechanics [2, 3, 4]. The mechanisms of formation and
properties of various BICs in photonic structures had been widely discussed in
the literature (see, for example, reviews [5, 6, 7, 8, 9] and papers [10, 11,
12, 13, 14, 15, 16, 17, 18, 19]). In particular, the BIC in the grating layer
sandwiched by the cover and substrate had been demonstrated via numerical
simulations for a particular set of parameters in [10]. Yet, despite the fact
that the planar lamellar (that is, uniform along $y$ axis) grating waveguide
is one of the most basic, simple examples of photonic structures supporting
BICs, a full analytical picture of the specific mechanism leading to BIC in
the grating waveguide has been missing until now.

Apparently, the latter stems from the fact that most works on BICs are based
on ad hoc numerical simulations dominating an analysis of photonic-crystal
structures in the modern literature. They employ various software based on
finite-element modeling like COMSOL [20], finite-difference time-domain (FDTD)
method [21], codes for the layered periodic structures [22], finite element
method (FEM) [15], etc.

Another reason is that the BIC-formation mechanism in such a simple, basic
photonic structure as the planar lamellar-grating waveguide turns out to be so
special that it simultaneously merges almost all other known mechanisms
leading to the BIC formation (see reviews [5, 6, 7, 8, 9]), including
mechanisms of the symmetry-protected BIC, accidental BIC, single-resonance
parametric BIC, coupled multiple resonances (Friedrich–Wintgen, but not
Fabry–Pérot) BIC, interference-based BIC through parameter tuning, and
topologically-protected BIC. A more detailed discussion of this fact requires
results presented in sections II–V and is postponed to sect. VI. It is
remarkable how merely splitting the central dielectric layer of the planar
waveguide into two alternating sections of lengths $d_{1},d_{2}$ (with fill
factor $\rho=d_{1}/(d_{1}+d_{2})$) and different permittivities
$\varepsilon_{1},\varepsilon_{2}$ converts an elementary, trivially soluble
problem of the planar dielectric waveguide [23, 24, 25] into a rich, complex
problem demonstrating many generic features of optical crystals.

Optical gratings and related waveguides are also interesting by themselves,
irrespective to BICs, since they have found numerous applications and been
studied for decades (see, for instance, [23, 24, 26, 27, 28, 29, 30, 31, 32,
33, 34] and references therein). One of the most elegant methods of their
studies is based on the analysis of eigenmodes that is very fruitful both for
the analytical theory and numerical simulations (see [1, 25, 26, 27, 28] and
references therein). Another, more straightforward method is the Fourier modal
method which prevails in the literature and is known as the rigorous coupled
wave theory in the diffractive optics community [26, 27, 11] or the
scattering-matrix approach in the photonic-crystal community [29, 30, 31, 32,
33, 34].

We employ the eigenmode approach which, in the present case of BICs in the
grating waveguide shown in Fig. 1, is based on the hierarchy of the following
three field constituents: (i) the plane waves, (ii) the infinite-grating
eigenmodes, (iii) the waveguide eigenmodes. The plane waves are the simplest
field configurations possessing a quasi-harmonic temporal-spatial profile
$\propto e^{ik_{zp}z+i(k_{x}+pk_{g})x-i\omega t}$ satisfying a dispersion law
of a uniform medium
$k_{zp}^{2}+(k_{x}+pk_{g})^{2}=\varepsilon\omega^{2}/c^{2}$. The latter means
that their phase speed depends on a permittivity $\varepsilon$ of the
dielectric medium in which they propagate. The prefix ’quasi’ indicates that
the frequency $\omega$ and the $z$-wavenumber $k_{zp}$, in general, are
complex-valued. For a given $x$-wavenumber within the first Brillouin zone
$k_{x}\in[-k_{g}/2,k_{g}/2]$ of the lamellar grating of the period
$\lambda_{g}\equiv 2\pi/k_{g}=d_{1}+d_{2}$, they are present in the waveguide
always as an entire series of spatial Fourier harmonics coupled to each other
via Bragg scattering on the grating and enumerated by an integer $p=0,\pm
1,\pm 2,\ldots$ .

Figure 2: Hierarchy of dispersion curves $k_{zn}^{2}(\omega)$, Eq. (22), and
their intersections (degenerate points) for the infinite-grating eigenmodes in
the titanium-oxide grating;
$\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307;k_{x}=0$. A symbol
$D_{n,n+1}^{l}$ labels the $l$-th intersection of the $n$-th and $(n+1)$-th
eigenmodes and signifies the associated BIC.

The infinite-grating eigenmodes are the field configurations possessing a
quasi-harmonic temporal and spatial $z$-profile $\propto e^{ik_{zn}z-i\omega
t}$, but a nontrivial spatial $x$-profile $f_{n}(x)$ such that it remains
invariant in the course of propagation inside the infinite grating despite a
persistent in-plane Bragg scattering. Such an invariance occurs only for a
discrete set of eigen $z$-wavenumbers $k_{zn}$ enumerated by an integer
$n=1,2,\ldots$ (see Fig. 2). Each $n$-th infinite-grating eigenmode lives
inside the infinite grating independently of others, is a superposition of
many spatial Fourier harmonics (partial plane waves), and represents a photon
dressed via Bragg scattering.

At last, the waveguide eigenmodes are the field configurations which follow a
quasi-harmonic evolution $\propto e^{-i\omega_{m}t}$ with a complex-valued
eigenfrequency $\omega_{m}=\omega^{\prime}_{m}-i\omega^{\prime\prime}_{m}$,
enumerated by a composite integer $m$, while possessing a stationary
(invariant in time) spatial structure. Each, $m$-th waveguide eigenmode is a
superposition of many infinite-grating eigenmodes, coupled to each other via
mutual reflections at the borders between the grating layer and the cover and
substrate, and represents a real, physical, fully dressed photon living in the
waveguide comprising the grating layer of a finite thickness $L$.

In the above hierarchy of waves and eigenmodes, the plane waves (spatial
Fourier harmonics), as well as the infinite-grating eigenmodes, are partial,
non-physical waves. The problem of hunting BICs consists of finding the
waveguide eigenmodes of vanishing decay, leakage rate
$\omega^{\prime\prime}_{m}$. The eigenmode approach has a great advantage over
other approaches since it operates with the most natural, living constituents
in the photonic structure and, hence, minimizes the number of couplings to
account for. Another great advantage of the eigenmode approach is that it
elucidates the physical mechanism and interpretation of the BIC phenomenon.
The difference between the eigenmode approach and other approaches (see, for
instance, [2, 3, 4, 5, 6, 7, 8, 9, 10]) is similar to the difference between
the eigenmode method and the rigorous coupled-wave (Fourier-modal) method in
the theory of optical gratings mentioned above.

In the present paper, we consider a particular example of a waveguide
eigenmode that turns out to be a BIC due to the phenomenon outlined in the
first paragraph. The in-plane $x$-wavenumber of that BIC waveguide eigenmode
is chosen to be zero, that is, at the center of the first Brillouin zone of
the grating, $k_{x}=0$. (We assume that $k_{y}=0$ since we consider only the
standard, invariant in the $y$-direction, grating problem when fields are
uniform along the $y$ axis and propagate in the $xz$-plane. So, we leave the
conical case of diffraction [35] aside.) This choice is predetermined by
strongly enhanced backscattering coupling in this exceptional high-symmetry
point in the momentum space so that the corresponding eigenmode looks more
like a cavity eigenmode rather than a waveguide eigenmode. At wavenumbers away
from $k_{x}=0$, the waveguide eigenmode radiates and forms a leaky resonance.
The latter had been illustrated in [10] by a numerical example and won’t be
tackled here.

Also, we assume that just the central, $p=0$, spatial Fourier harmonic is
emitted out of the planar grating layer into the cover and substrate, while
all higher-order, $p\neq 0$, Fourier harmonics are evanescent and, hence, do
not contribute to the radiation losses. According to the plane-wave dispersion
relation, the $p$-th Fourier harmonic becomes evanescent in the cover
(superscript “+”) or substrate (superscript “-”) when its $z$-wavenumber
squared turns negative:

$(k_{zp}^{\pm})^{2}=\frac{\varepsilon^{\pm}\omega^{2}}{c^{2}}-(k_{x}+pk_{g})^{2}<0.$
(1)

Here $\varepsilon^{+}$ or $\varepsilon^{-}$ is the permittivity of the cover
or substrate, respectively. For modes at the center of the first Brillouin
zone, $k_{x}=0$, the $p$-th Fourier harmonic ceases to provide a radiation-
loss channel when

$\varepsilon^{\pm}<\Big{(}\frac{2\pi cp}{\omega\lambda_{g}}\Big{)}^{2}.$ (2)

Below we consider the case in which the BIC originates from the degenerate
point $D_{2,3}^{0}$ (see Fig. 2) when the condition (2) is satisfied for all
nonzero orders $p=\pm 1,\pm 2,\ldots$. In such case, the infinite-grating
eigenmode whose $x$-profile is odd with respect to the center of each uniform
segment has vanishing contribution from the zeroth Fourier harmonic and thus
minimal radiation losses. At the same time, the infinite-grating eigenmode
whose $x$-profile is even with respect to the center of each uniform segment
has significant contribution from the zeroth Fourier harmonic and thus leaks
out of the grating waveguide.

Note that in the same region of parameters, there exists a waveguide eigenmode
comprising the same two, but anti-phased infinite-grating eigenmodes. In
general, it has a very different spatial structure and a different
eigenfrequency. So, it does not interfere with the considered BIC and will be
discussed elsewhere.

The content of the paper is as follows. In sect. II we present the necessary
analytical formulae describing the infinite-grating eigenmodes as well as the
waveguide eigenmodes. In sect. III we disclose the universality of the
behavior of the infinite-grating eigenmodes within the vicinity of a mode
crossing point and a remarkable connection between the dispersion degeneracy
and geometry of the eigenmode spatial profiles (their odd or even spatial
parity). In sect. IV we explain how to design a waveguide manifesting a BIC at
a given frequency. It is done on the basis of a numerical example related to a
waveguide based on titanium oxide (TiO2) – an optical material known for many
applications in photonics. The mechanism behind the emergence of a BIC is
explained and detailed in sect. V. It involves a parity symmetry breaking,
that occurs in accord with the above universality, and a decoupling from a
radiation-loss channel due to mode crossing. Conclusions, discussion of the
simultaneous manifestation of other known BIC mechanisms in the formation of
the above BIC, and other comments make up sect. VI.

##  II Analytical theory of grating-waveguide eigenmodes

In the present paper we study the high-Q transverse-electric (TE) eigenmodes
of a planar grating waveguide. They can be represented by two column
$N$-vectors ${\bf F^{\pm}}=(\\{F_{n}^{\pm}|n=1,...,N\\})^{T}$, each containing
some $N$ complex amplitudes of co- or counter-propagating infinite-grating
eigenmodes, respectively. (A superscript $T$ denotes the transpose operation.)
The electric field of the TE waveguide eigenmode is linearly polarized along
the $y$ axis and is a superposition of the infinite-grating eigenmodes:

$E_{y}(x,z)e^{-i\omega
t}=\sum_{n=1}^{N}\big{[}F_{n}^{+}e^{ik_{zn}z}+F_{n}^{-}e^{ik_{zn}(L-z)}\big{]}f_{n}(x)e^{-i\omega
t},$ (3)

where the non-negative factor, $\omega^{\prime\prime}\geq 0$, in the imaginary
part of the frequency $\omega=\omega^{\prime}-i\omega^{\prime\prime}$
describes a mode decay and $k_{zn}$ is the $z$-wavenumber of the $n$-th
infinite-grating eigenmode. The $x$-profile of the $n$-th infinite-grating
eigenmode is a superposition of two counter-propagating waves,

$f_{n}(x)=\frac{B_{q}(x)}{\sqrt{\langle B_{q}^{\dagger},B_{q}\rangle}},\
B_{q}(x)=b_{q}^{+}e^{ik_{q}(x-d_{1})}+b_{q}^{-}e^{ik_{q}(d_{1}-x)},$ (4)

where $k_{q}=\big{[}\varepsilon_{q}\omega^{2}/c^{2}-k_{zn}^{2}\big{]}^{1/2}$.
The norm $\langle B_{q}^{\dagger},B_{q}\rangle^{1/2}$ is defined in Eq. (17)
via adjoint eigenfunctions, Eq. (14). The complex amplitudes of the waves in
Eq. (4),

$\begin{split}&b_{1}^{\pm}=\pm 2e^{-ik_{x}\lambda_{g}/2}\big{[}\pm
ik_{1}\sin(k_{2}d_{2})-k_{2}\cos(k_{2}d_{2})\\\ &\qquad+k_{2}e^{\pm
ik_{1}d_{1}+ik_{x}\lambda_{g}}\big{]},\\\ &b_{2}^{\pm}=\pm
2e^{ik_{x}\lambda_{g}/2}\big{[}\pm
ik_{2}\sin(k_{1}d_{1})+k_{1}\cos(k_{1}d_{1})\\\ &\qquad-k_{1}e^{\mp
ik_{2}d_{2}-ik_{x}\lambda_{g}}\big{]},\end{split}$ (5)

are specified by the index $q$ equalling 1 or 2 depending on whether $x$ is
inside the grating section of length $d_{1}$ or $d_{2}$ and permittivity
$\varepsilon_{1}$ or $\varepsilon_{2}$, respectively (see Fig. 1).

Below, we choose the number $N$ of the contributing infinite-grating
eigenmodes based on a required accuracy of the waveguide-eigenmode
representation. Usually it is enough to include just all propagating and a few
lower-order evanescent eigenmodes. The number of propagating infinite-grating
eigenmodes is finite. All higher-order evanescent eigenmodes exponentially
decay on a path from one border to the other border of the planar waveguide.
Hence, they couple very weakly to the Fourier harmonics escaping from the
waveguide, while having minimal contribution to the electric field of the
waveguide eigenmode inside the grating as well. At the same time, the number
of the involved spatial Fourier harmonics, coupled to the essential infinite-
grating eigenmodes due to Bragg scattering on the grating, also turns out to
be limited to a few central harmonics. Such a low, finite dimensionality of
the eigenmode approach is the main reason for its tremendous efficiency and
advantage over other approaches for analysis of relatively thick gratings or
other photonic structures.

The waveguide eigenmodes can be found as the eigenvectors of an equation
expressing a self-reproduction of the eigenmode amplitudes ${\bf F^{+}}$ after
a round trip over the grating layer along the $z$ axis: (i) propagation from
the grating-substrate border at $z=0$ to the grating-cover border at $z=L$,
(ii) reflection into the counter-propagating eigenmodes ${\bf F^{-}}$ from the
border at $z=L$, (iii) propagation of the eigenmode ${\bf F^{-}}$ back to the
border at $z=0$, and (iv) reflection back into the co-propagating eigenmodes
${\bf F^{+}}$ from the border at $z=0$. Propagation through the grating layer
along the $z$ axis is described by the diagonal transfer matrix $\Lambda={\rm
diag}\\{e^{ik_{zn}L}|n=1,...,N\\}$ since each eigenmode propagates
independently on the other eigenmodes in accord with its own eigen wavenumber
$k_{zn}$. Reflections at the grating-layer borders are described by the matrix
of reflectance coefficients $R^{\pm}=\\{(R^{\pm})_{n}^{n^{\prime}}\\}$ which
depends on the permittivity $\varepsilon^{\pm}$ of the adjacent dielectric
medium - cover or substrate. Thus, the $m$-th TE eigenmode of the planar
grating waveguide,

$\begin{split}&E_{y}^{(m)}(x,z)e^{-i\omega_{m}t}=\sum_{n=1}^{N}\Big{[}F_{n}^{(m)}e^{ik_{zn}z}\\\
&+\Big{(}\sum_{n^{\prime}=1}^{N}(R^{+})_{n}^{n^{\prime}}F^{(m)}_{n^{\prime}}\Big{)}e^{ik_{zn}(L-z)}\Big{]}f_{n}(x)e^{-i(\omega_{m}^{\prime}-i\omega_{m}^{\prime\prime})t},\end{split}$
(6)

is determined by the $m$-th eigenvector ${\bf F^{(m)}}$ associated with the
unity eigenvalue of a matrix $M=(M_{n}^{n^{\prime}})$ equal to the product of
the aforementioned four matrices:

$\begin{split}&M{\bf F^{(m)}}={\bf F^{(m)}};\quad M=R^{-}\Lambda
R^{+}\Lambda,\\\
&M_{n}^{n^{\prime}}=\sum_{n^{\prime\prime}=1}^{N}(R^{-})_{n}^{n^{\prime\prime}}(R^{+})_{n^{\prime\prime}}^{n^{\prime}}e^{i(k_{zn^{\prime}}+k_{zn^{\prime\prime}})L}.\end{split}$
(7)

This generating matrix $M(\omega)$ fully described the properties of the
planar grating waveguide and nontrivially depends on the complex frequency
$\omega$. Hence, the complex eigenfrequencies $\omega_{m}$ of the waveguide
eigenmodes are determined by the roots of the characteristic equation

$D(\omega_{m})=0;\qquad
D(\omega)\equiv\det\Big{[}M(\omega)-\mathbbm{1}\Big{]}.$ (8)

Hereinafter, the symbol $\mathbbm{1}$ stands for the identity matrix.

The reflectance matrix can be found from the boundary conditions of continuity
of the tangential component of the electric and magnetic fields at the borders
$z=0,L$ of the grating layer. These two conditions can be written in terms of
the reflectance, $R^{\pm}$, and transmittance, $T^{\pm}$, matrices as well as
the basis-transformation matrix $C=(c_{n}^{p})$ (defined below in Eq. (10)):

$R^{\pm}=CT^{\pm}-\mathbbm{1},\qquad
C\hat{k}_{z}^{\pm}T^{\pm}=\hat{k}_{z}(\mathbbm{1}-R^{\pm}).$ (9)

Here $\hat{k}_{z}={\rm diag}\\{k_{zn}|n=1,2,...\\}$ and
$\hat{k}_{z}^{\pm}={\rm diag}\\{k_{zp}^{\pm}|p=0,\pm 1,\pm 2,...\\}$ are the
diagonal matrices of the $z$-wavenumbers $k_{zn}$ of the infinite-grating
eigenmodes and $z$-wavenumbers
$k_{zp}^{\pm}=[\varepsilon^{\pm}\omega^{2}/c^{2}-(k_{x}+pk_{g})^{2}]^{1/2}$ of
the plane waves which propagate in the cover (superscript ”+”) or substrate
(superscript ”-”); $c$ is the speed of light in vacuum. The longitudinal
wavenumber of the latter $p$-th plane wave, $k_{x}^{(p)}=k_{x}+pk_{g}$, is the
sum of the wavenumber $k_{x}$ within the first Brillouin zone and the $p$-th
Bragg harmonic of the grating wavenumber $k_{g}$. The transmittance matrix
$T^{\pm}=\\{(T^{\pm})_{p}^{n}\\}$ is required for the evaluation of the
radiation losses. Its entry $(T^{\pm})_{p}^{n}$ gives the amplitude,
$G_{p}^{\pm}=(T^{\pm})_{p}^{n}F_{n}^{\pm}$, of the $p$-th spatial Fourier
harmonic $G_{p}^{\pm}e^{i(k_{x}+pk_{g})x+k_{zp}^{\pm}[\pm i(z-L/2)-iL/2]}$
emitted from the grating layer into the cover or substrate by the infinite-
grating eigenmode $F_{n}^{\pm}e^{k_{zn}[\pm i(z-L/2)-iL/2]}$ incident onto the
cover ($z=L$) or substrate ($z=0$) border from inside the grating,
respectively.

In the general case of the complex permittivities $\varepsilon_{q}$ and
complex frequncy $\omega=\omega^{\prime}-i\omega^{\prime\prime}$, the
coefficients in the Fourier expansions of (i) a spatial Fourier harmonic via
the infinite-grating eigenfunctions,

$e^{i(k_{x}+p^{\prime}k_{g})x}=\sum_{n=1}^{\infty}c^{p^{\prime}}_{n}f_{n}(x),\
c^{p^{\prime}}_{n}=\langle
f_{n}^{\dagger},e^{i(k_{x}+p^{\prime}k_{g})x}\rangle,$ (10)

and (ii) an infinite-grating eigenfunction via the spatial Fourier harmonics,

$f_{n}(x)=\sum_{p=-\infty}^{\infty}{\tilde{c}}^{n}_{p}e^{i(k_{x}+pk_{g})x},\
{\tilde{c}}^{n}_{p}=\langle e^{i(k_{x}+pk_{g})x},f_{n}\rangle,$ (11)

are related to each other via the straightforward substitution $\\{k_{q}\to
k_{q}^{\dagger}|\,q=1,2\\}$:

$(c_{n}^{p})^{*}=\tilde{c}_{p}^{n}(\\{k_{q}\to k_{q}^{\dagger}\\}),\quad
k_{q}^{\dagger}=\Big{[}\Big{(}\frac{\varepsilon_{q}\omega^{2}}{c^{2}}\Big{)}^{*}-(k_{zn}^{2})^{*}\Big{]}^{1/2}.$
(12)

It includes the following substitution of the amplitudes of the eigenmode’s
exponential components

$b_{q}^{\pm}\to b_{q}^{\pm\dagger}\equiv b_{q}^{\pm}(\\{k_{q}\to
k_{q}^{\dagger}\ |q=1,2\\}).$ (13)

The set of the infinite-grating eigenfunctions $\\{f_{n}(x)|n=1,2,...\\}$ and
their adjoint eigenfunctions

$f_{n}^{\dagger}(x)=\frac{B_{q}^{\dagger}(x)}{\sqrt{\langle
B_{q},B_{q}^{\dagger}\rangle}},\
B_{q}^{\dagger}=b_{q}^{+\dagger}e^{ik_{q}^{\dagger}(x-d_{1})}+b_{q}^{-\dagger}e^{ik_{q}^{\dagger}(d_{1}-x)},$
(14)

constitutes a biorthogonal basis, so that the product of the matrices
${\tilde{C}},C$ in any order equals the identity matrix:

${\tilde{C}}C=C{\tilde{C}}=\mathbbm{1}\leftrightarrow\sum_{n=-\infty}^{\infty}{\tilde{c}}^{n}_{p}c_{n}^{p^{\prime}}=\delta_{p,p^{\prime}},\sum_{p=-\infty}^{\infty}c_{n}^{p}{\tilde{c}}^{n^{\prime}}_{p}=\delta_{n,n^{\prime}},$
(15)

where $\delta_{n,n^{\prime}}$ is the Kronecker delta. (Note that here we
define normalization of the entries $c_{n}^{p}$ and $\tilde{c}_{p}^{n}$
slightly different from that in [1]. Namely, we move one of the norms $\langle
B_{q}^{\dagger},B_{q}\rangle^{1/2}$ from the $c_{n}^{p}$ to the
$\tilde{c}_{p}^{n}$.) The norm squared is given by the standard inner product,

$\langle
B_{q}^{\dagger},B_{q}\rangle=\frac{1}{\lambda_{g}}\int_{0}^{\lambda_{g}}B_{q}^{\dagger*}(x)B_{q}(x)dx.$
(16)

The latter can be calculated explicitly by employing the biorthonormal basis
of the eigenfunctions $f_{n}(x)$ and the adjoint eigenfunctions
$f_{n}^{\dagger}(x)$ as follows

$\begin{split}&\langle B_{q}^{\dagger},B_{q}\rangle\\\
&=\sum_{q=1,2}\frac{d_{q}}{\lambda_{g}}\\{b_{q}^{+}[b_{q}^{+}(k_{q}\to
k_{q}^{\dagger})]^{*}\nu_{1}[i(-1)^{q}d_{q}(k_{q}-k_{q}^{\dagger*})]\\\
&\qquad\quad+b_{q}^{-}[b_{q}^{-}(k_{q}\to
k_{q}^{\dagger})]^{*}\nu_{1}[-i(-1)^{q}d_{q}(k_{q}-k_{q}^{\dagger*})]\\\
&\qquad\quad+b_{q}^{+}[b_{q}^{-}(k_{q}\to
k_{q}^{\dagger})]^{*}\nu_{1}[i(-1)^{q}d_{q}(k_{q}+k_{q}^{\dagger*})]\\\
&\qquad\quad+b_{q}^{-}[b_{q}^{+}(k_{q}\to
k_{q}^{\dagger})]^{*}\nu_{1}[-i(-1)^{q}d_{q}(k_{q}+k_{q}^{\dagger*})]\\}.\end{split}$
(17)

Here the elementary function $\nu_{1}(u)=(e^{u}-1)/u$ behaves analytically and
tends to unity, $\nu_{1}(0)=1$, when $u\to 0$. The matrix
$\tilde{C}=(\tilde{c}_{p}^{n})$ can be written explicitly as follows

$\begin{split}&{\tilde{c}}^{n}_{p}=\frac{ie^{-\frac{i}{2}k_{x}\lambda_{g}}}{\lambda_{g}}\\\
&\times\sum_{q=1}^{2}\frac{[(b_{q}^{-}-b_{q}^{+})k_{q}-(b_{q}^{-}+b_{q}^{+})k_{x}^{(p)}](e^{-ik_{q}d_{q}}-e^{-ik_{x}^{(p)}d_{q}})}{\langle
B_{q}^{\dagger},B_{q}\rangle^{1/2}[(k_{x}^{(p)})^{2}-k_{q}^{2}]e^{i(1-q)(k_{q}d_{q}-k_{x}^{(p)}d_{1})}}.\end{split}$
(18)

We find the following explicit solutions to Eq. (9) for the transmittance and
reflectance matrices,

$T^{\pm}=2(\mathbbm{1}+\tilde{C}\hat{k}_{z}^{-1}C\hat{k}_{z}^{j})^{-1}\tilde{C},\
R^{\pm}=2(\mathbbm{1}+\hat{k}_{z}^{-1}C\hat{k}_{z}^{j}\tilde{C})^{-1}-\mathbbm{1},$
(19)

suitable for their computing in the case of arbitrary finite numbers $N$ and
$S=2p^{*}+1$ of the infinite-grating eigenmodes and spatial Fourier harmonics
$p=0,\pm 1,...,\pm p^{*}$, respectively, chosen to approximate the field of a
given waveguide eigenmode. In such an approximation the matrices
$C=(c_{n}^{p})$ and $\tilde{C}=(\tilde{c}_{p}^{n})$ are the rectangular
$N\times S$ and $S\times N$ matrices, respectively, and their inverse matrices
do not exist, except the square-matrix case $N=S$. The square $N\times N$ and
$S\times S$ diagonal matrices

$\begin{split}&\hat{k}_{z}={\rm diag}\\{k_{zn}|n=1,...,N\\},\\\
&\hat{k}_{z}^{j}={\rm
diag}\\{k_{zp}^{j}|p=-p^{*},...,-1,0,1,...,p^{*}\\}\end{split}$ (20)

of the $z$-wavenumbers $k_{zn}$ of the infinite-grating eigenmodes inside the
grating layer and $z$-wavenumbers
$k_{zp}^{j}=[\varepsilon^{j}\omega^{2}/c^{2}-(k_{x}+pk_{g})^{2}]^{1/2}$ of the
spatial Fourier harmonics emitted into the cover ($j=+$) or substrate ($j=-$),
respectively, can be easily inverted:

$\hat{k}_{z}^{-1}=\rm{diag}\\{1/k_{zn}\\},\quad(\hat{k}_{z}^{j})^{-1}=\rm{diag}\\{1/k_{zp}^{j}\\}.$
(21)

The analytical formulae for the eigenmodes of an infinite lamellar grating had
been found in [1] for a real-valued frequency $\omega=\omega^{\prime}$.
However, radiative losses of the high-Q waveguide eigenmode due to emission
into the cover and/or substrate of the planar grating waveguide lead to an
appearance of a decay rate, $\omega^{\prime\prime}$, contributing with a small
negative imaginary part to the waveguide-eigenmode frequency
$\omega=\omega^{\prime}-i\omega^{\prime\prime},\
\omega^{\prime\prime}/\omega^{\prime}\ll 1$. The corresponding first-order
correction $\Delta k_{zn}$ to the infinite-grating eigenmode $z$-wavenumber
$k_{zn}(\omega^{\prime\prime})=k_{zn}(\omega^{\prime\prime}=0)+\Delta k_{zn}$
is easy to find from the characteristic equation for the infinite-grating
eigenmodes,

$\begin{split}&\cos(k_{1}d_{1})\cos(k_{2}d_{2})-\sin(k_{1}d_{1})\sin(k_{2}d_{2})\frac{k_{1}^{2}+k_{2}^{2}}{2k_{1}k_{2}}\\\
&=\cos(k_{x}\lambda_{g});\qquad
k_{q}=\Big{(}\frac{\varepsilon_{q}\omega^{2}}{c^{2}}-k_{zn}^{2}\Big{)}^{1/2}.\end{split}$
(22)

Suppose the zeroth-order values of the $x$-wavenumber of a monochromatic
eigenmode plane wave in a grating medium of the permittivity
$\varepsilon_{q},\ q=1,2$, are $k_{q}=k_{q}(\omega^{\prime\prime}=0)$. Keeping
only the linear corrections to them in the characteristic equation (22), we
find

$k_{zn}(\omega^{\prime\prime})=k_{zn}(\omega^{\prime\prime}=0)+\Delta k_{zn},\
\Delta
k_{zn}\approx\frac{i\omega^{\prime}\omega^{\prime\prime}(A_{1}/A_{0})}{c^{2}k_{zn}(\omega^{\prime\prime}=0)};$
(23)
$\begin{split}&A_{s}=\frac{\varepsilon_{2}^{s}k_{1}^{2}+\varepsilon_{1}^{s}k_{2}^{2}}{k_{1}k_{2}}\big{[}\cos(k_{x}\lambda_{g})-\cos(k_{1}d_{1})\cos(k_{2}d_{2})\big{]}\\\
&\qquad+(\varepsilon_{1}^{s}+\varepsilon_{2}^{s})\sin(k_{1}d_{1})\sin(k_{2}d_{2})\\\
&\qquad+\Big{(}k_{2}\varepsilon_{1}^{s}d_{1}+\varepsilon_{2}^{s}d_{2}\frac{k_{1}^{2}+k_{2}^{2}}{2k_{2}}\Big{)}\sin(k_{1}d_{1})\cos(k_{2}d_{2})\\\
&\qquad+\Big{(}k_{1}\varepsilon_{2}^{s}d_{2}+\varepsilon_{1}^{s}d_{1}\frac{k_{1}^{2}+k_{2}^{2}}{2k_{1}}\Big{)}\cos(k_{1}d_{1})\sin(k_{2}d_{2}).\end{split}$
(24)

This result is valid for all infinite-grating eigenmodes everywhere except
within a vicinity of points in the parameter space where $A_{0}=0,A_{1}=0$. At
those points, the overall contribution of linear terms in Eq. (22) is zero and
$\Delta k_{zn}$ should be found from the characteristic equation with the
quadratic corrections. Such a case is considered below and is described by a
nontrivial result in Eq. (29).

##  III Spatial parity at mode crossing and universality of its establishment

The vicinity of each degenerate point $D_{n,n+1}^{l}$, where the dispersion
curves of the $n$-th and $(n+1)$-th infinite-grating eigenmodes intersect,
requires an exceptionally detailed analysis. The reason for the latter is not
just the dispersion degeneracy, but the fact that those two intersecting
eigenmodes acquire, at this point, a certain odd or even spatial parity of the
$(m_{x1},m_{x2})$-th order (see Eq. (4)). Namely, in the case of the
propagating eigenmodes and $k_{x}=0$, the phase accumulations over each of the
grating sections along the $x$ axis become multiples of $\pi$,

$k_{q}d_{q}=m_{xq}\pi;\ q=1,2;\ m_{x1}-m_{x2}=2l,\ l=0,\pm 1,\ldots,$ (25)

where $k_{q}=[\omega^{2}\varepsilon_{q}/c^{2}-k_{zn}^{2}]^{1/2}$;
$m_{x1},\,m_{x2}=1,2,3,\dots$ . According to Eq. (25), the $z$-wavenumber and
the critical frequency at the crossing point of the spatial-parity order
$(m_{x1},m_{x2})$ are given by the following explicit formulae

$\begin{split}&\Big{(}\frac{k_{zn}c}{\omega_{c}}\Big{)}^{2}=\varepsilon_{2}+\frac{\varepsilon_{1}-\varepsilon_{2}}{1-\big{[}\frac{(1-\rho)m_{x1}}{\rho
m_{x2}}\big{]}^{2}}\quad,\\\
&\Big{(}\frac{\omega_{c}\lambda_{g}}{c}\Big{)}^{2}=\frac{\pi^{2}}{\varepsilon_{1}-\varepsilon_{2}}\Big{[}\Big{(}\frac{m_{x1}}{\rho}\Big{)}^{2}-\Big{(}\frac{m_{x2}}{1-\rho}\Big{)}^{2}\
\Big{]}.\end{split}$ (26)

The remarkable connection (25) between the dispersion degeneracy and geometry
(spatial parity) of the $x$-distributions of the intersecting infinite-grating
eigenmodes can be proven as follows. The intersection of the two dispersion
curves, $k_{zn}=k_{z(n+1)}$, at some critical frequency $\omega_{c}$ means
that the characteristic equation (22) admits an existence of two different
derivatives $dk_{zn}/d\omega$ at this point. The latter is possible only if
the Taylor expansion of the left hand side of the characteristic equation (22)
over the variables $\Delta k_{zn}=k_{zn}-k_{zn}(\omega_{c})$ and
$\Delta\omega=\omega-\omega_{c}$ does not contain linear terms in the vicinity
of this point. Hence, the corresponding partial derivatives should be zero,
that is, $A_{0}=0$ and $A_{1}=0$ (see Eq. (24)). It is easy to see that, in
the general case, the multiple of $\pi$ phase accumulations in Eq. (25)
provide the required solution to the equations $A_{0}=0$ $\&$ $A_{1}=0$.

Therefore, the leading asymptotic corrections are quadratic and constitute a
quadratic equation for $\Delta k_{zn}$. As a result, it yields exactly two
solutions which describe two intersecting dispersion curves with different
slopes, as it should be at the point of mode crossing.

Figure 3: Dispersion, $k_{zn}^{2}(\omega)$, of the infinite-grating eigenmodes
in the titanium-oxide grating
($\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307$), $k_{x}=0$. The
first three eigenmodes are propagating ($k_{zn}^{2}>0$) while the higher order
ones are evanescent ($k_{zn}^{2}<0$) within the selected range of frequencies.

In the vicinity of a degenerate point, the two intersecting infinite-grating
eigenmodes show the following remarkable universality in their behavior when
approaching such a point. Namely, if one approaches this point along a path
parameterized by a function $\nu$ of the complex frequency
$\omega=\omega^{\prime}-i\omega^{\prime\prime}$ and grating parameters
$\\{\varepsilon_{q},\ d_{q}|\,q=1,2\\}$ reaching a value $\nu=\nu_{0}$ at this
point, then the ratio of the related derivatives of the above phase
accumulations acquires only two possible values,

$\frac{d(k_{1}d_{1})/d\nu}{d(k_{2}d_{2})/d\nu}\Big{|}_{\nu=\nu_{0}}=-\Big{(}\frac{d_{1}}{d_{2}}\Big{)}^{\eta_{n}},\qquad\eta_{n}=\pm
1,$ (27)

determined by a purely geometrical parameter – the ratio between the lengths
of the grating sections, $d_{1}/d_{2}$. The proof of the fact stated in Eq.
(27) follows from the second derivative of the characteristic equation (22)
with respect to the parameter $\nu$ at $\nu=\nu_{0}$.

It is then easy to see that the leading contribution to the amplitudes
$b^{\pm}_{q}$ of each of the two infinite-grating eigenmodes in Eqs. (4), (5)
tends to zero in terms of the small deviation $\nu-\nu_{0}\to 0$ either
linearly or quadratically:

$\begin{split}&b^{\pm}_{q}\sim\frac{db^{\pm}_{q}}{d\nu}\Big{|}_{\nu=\nu_{0}}(\nu-\nu_{0})\qquad\textrm{if}\qquad\eta_{n}=+1,\\\
&b^{\pm}_{q}\sim\frac{1}{2}\frac{d^{2}b^{\pm}_{q}}{d\nu^{2}}\Big{|}_{\nu=\nu_{0}}(\nu-\nu_{0})^{2}\qquad\textrm{if}\qquad\eta_{n}=-1.\end{split}$
(28)

The two intersecting infinite-grating eigenmodes have different critical
indices $\eta_{n+1}\neq\eta_{n}$, either +1 or -1.

Figure 4: Phase accumulations, $k_{q}d_{q},\,q=1,2$, of the $n=2$ (solid
curves) and $n=3$ (dashed curved) infinite-grating eigenmodes over the grating
section $d_{1}$ or $d_{2}$ along the $x$ axis as the functions of the
dimensionless frequency $\omega\lambda_{g}/c$. All of them become equal to
$\pi=3.14\ldots$ at the mode-crossing point,
$\omega=\omega_{c}=4c/\lambda_{g}$;
$\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,k_{x}=0$.

The result (27) leads to the linear correction $\Delta
k_{zn}=k_{zn}(\omega+\Delta\omega)-k_{zn}(\omega)$ to both infinite-grating
eigenmode $z$-wavenumbers in the vicinity of the degenerate point:

$\Delta
k_{zn}=\frac{\varepsilon_{1}+\varepsilon_{2}(d_{2}/d_{1})^{(2-\eta_{n})}}{1+(d_{2}/d_{1})^{(2-\eta_{n})}}\frac{\omega\
\Delta\omega}{c^{2}k_{zn}(\omega)},\ \eta_{n}=\pm 1.$ (29)

Note that the correction (29), albeit being linear, is different from Eq. (23)
and therefore cannot be given by Eq. (23) since the latter does not apply due
to the degeneracy $A_{0}=A_{1}=0$.

In the present paper we consider the intersection $D_{2,3}^{0}$,

$k_{z2}(\omega)=k_{z3}(\omega)\qquad\textrm{at}\quad\omega=\omega_{c},$ (30)

between the dispersion curves of the $n=2$ and $n=3$ infinite-grating
eigenmodes (see Fig. 3) which have the critical indices $\eta_{2}=-1$ and
$\eta_{3}=+1$ and possess the even and odd, respectively, $x$-spatial parity
of the main order $(m_{x1}=1,\,m_{x2}=1)$ as per Eq. (25) (see Fig. 4).

In fact, as is exemplified in Fig. 4 and also below in sect. V, Fig. 16, the
leading asymptotics (28) provides not just the scaling, but a full analytical
description of the parity symmetry breaking, i.e., the restructuring of the
spatial $x$-profile of the infinite-grating eigenmodes, in the vicinity of the
mode-crossing point $D_{2,3}^{0}$ (Figs. 2, 3).

For the evaluation of the radiation losses, we need to know the amplitude of
the contribution from the zeroth spatial Fourier harmonic to the infinite-
grating eigenmode $f_{n}$ at the center of the Brillouin zone, $k_{x}=0$. It
can be found from Eq. (18) in the following, very simple form

$\begin{split}&\tilde{c}_{0}^{n}=\frac{8\big{[}\cos(k_{1}d_{1})-\cos(k_{2}d_{2})\big{]}}{\lambda_{g}k_{1}k_{2}\langle
B_{q}^{\dagger},B_{q}\rangle^{1/2}}\\\
&\times\Big{[}k_{2}^{2}\sin\Big{(}\frac{k_{1}d_{1}}{2}\Big{)}e^{-ik_{1}d_{1}/2}+k_{1}^{2}\sin\Big{(}\frac{k_{2}d_{2}}{2}\Big{)}e^{ik_{2}d_{2}/2}\Big{]},\end{split}$
(31)

where the relations
$b_{1}^{+}-b_{1}^{-}=4k_{2}[\cos(k_{1}d_{1})-\cos(k_{2}d_{2})]$ and
$b_{2}^{+}-b_{2}^{-}=4k_{1}[\cos(k_{1}d_{1})-\cos(k_{2}d_{2})]$ were employed.
Since
$\cos\alpha_{1}-\cos\alpha_{2}=2\sin\frac{\alpha_{1}-\alpha_{2}}{2}\sin\frac{\alpha_{1}+\alpha_{2}}{2}$,
we can approximate the contribution $\tilde{c}_{0}^{n}$ of the zeroth spatial
Fourier harmonic to the $n=2$ and $n=3$ eigenmodes in the vicinity of the
degenerate point $D_{2,3}^{0}$, where $|k_{q}d_{q}-\pi|\ll 1,q=1,2,$ as
follows

$\begin{split}&\tilde{c}_{0}^{n}\approx\frac{16i(\varepsilon_{2}-\varepsilon_{1})\omega^{2}}{\lambda_{g}k_{1}k_{2}c^{2}\langle
B_{q}^{\dagger},B_{q}\rangle^{1/2}}\\\
&\times\sin\Big{(}\frac{k_{2}d_{2}-k_{1}d_{1}}{2}\Big{)}\sin\Big{(}\frac{k_{2}d_{2}+k_{1}d_{1}}{2}\Big{)}.\end{split}$
(32)

By the way, in the vicinity of the higher-order $(m_{x1}=2,\,m_{x2}=2)$
degenerate point $D_{4,5}^{0}$ (see Fig. 2 and Eq. (25)), where
$|k_{q}d_{q}-2\pi|\ll 1,q=1,2,$ the contribution $\tilde{c}_{0}^{n}$ of the
zeroth spatial Fourier harmonic to the $n=4$ and $n=5$ eigenmodes tends to
zero even faster,

$\begin{split}&\tilde{c}_{0}^{n}\approx-\frac{8\big{[}k_{2}^{2}\sin(k_{1}d_{1}-2\pi)+k_{1}^{2}\sin(k_{2}d_{2}-2\pi)\big{]}}{\lambda_{g}k_{1}k_{2}\langle
B_{q}^{\dagger},B_{q}\rangle^{1/2}}\\\
&\qquad\times\sin\Big{(}\frac{k_{2}d_{2}-k_{1}d_{1}}{2}\Big{)}\sin\Big{(}\frac{k_{2}d_{2}+k_{1}d_{1}}{2}\Big{)},\end{split}$
(33)

that suggests an existence of even narrower BIC in the corresponding region of
waveguide parameters.

##  IV Designing a waveguide that supports a bound state in the continuum at a
given frequency

Below we employ the analytical formulae of sections II and III to exemplify
the design, properties and mechanism of formation of the BIC originating from
the degenerate point $D_{2,3}^{0}$ which is the first in the hierarchy of
degenerate points shown in Fig. 2. As is stated in sect. I, we consider a
particular case of the waveguide eigenmode with the zero in-plane wavenumbers,
$k_{x}=k_{y}=0$, in the grating waveguide with just one central, zeroth
spatial Fourier harmonic leaking into the cover and substrate.

Remarkably, such a BIC is supported by the waveguide with any set of
parameters so long as the intersection $D_{2,3}^{0}$ occurs at the critical
frequency $\omega_{c}$ satisfying the condition in Eq. (2),

$\omega_{c}\lambda_{g}/c<2\pi/\sqrt{\varepsilon^{\pm}}.$ (34)

The latter leaves open just one radiation-loss channel occurring through
emission of the $p=0$ spatial Fourier harmonic out of the grating layer into
the cover and substrate, but prohibits radiation of all higher-order ($p\neq
0$) spatial Fourier harmonics, i.e., diffraction orders.

Recall that there are six dimensionless parameters of the waveguide (see Fig.
1). First, there are three dimensionless parameters of the lamellar grating:
The high, $\varepsilon_{1}$, and low, $\varepsilon_{2}$, dielectric constants
of the two grating sections and the fill factor of the high dielectric
constant section, $\rho=d_{1}/\lambda_{g}$. All spatial dimensions are scaled
by the period of the grating, $\lambda_{g}=d_{1}+d_{2}$. Second, there are
three other dimensionless parameters which set a global design of the planar
grating waveguide: The dielectric constants of the cover, $\varepsilon^{+}$,
and substrate, $\varepsilon^{-}$, materials enclosing the grating layer and
the ratio of the thickness of the grating layer to the grating period,
$L/\lambda_{g}$.

To get the BIC at a given resonant frequency, it suffices to adjust any one or
combination of the grating parameters (permittivities
$\varepsilon_{1},\varepsilon_{2}$, fill factor $\rho$, and period
$\lambda_{g}$) so that the critical frequency of the mode-crossing point
$D_{2,3}^{0}$ would coincide with that given frequency. This is because the
BIC eigenfrequency $\omega_{m}$ is predetermined to be very close to the
critical frequency $\omega_{c}$. (Its dimensionless counterpart,
$\omega_{m}\lambda_{g}/c$, is also scaled by the $\lambda_{g}$.) It can be
achieved by tuning just one of the phase accumulations over the grating
sections in Eq. (25), say, $k_{1}d_{1}$, to $\pi$ since the second one,
$k_{2}d_{2}$, will also become equal to $\pi$ automatically:

$k_{1}d_{1}=\pi,\quad k_{2}d_{2}=\pi\qquad\textrm{at}\quad\omega=\omega_{c}.$
(35)

For instance, let us take titanium oxide (TiO2) as the material for the first
section of the grating, that is, take the high dielectric constant to be
$\varepsilon_{1}=6.25$, and choose the low dielectric constant of the second
grating section to be, say, $\varepsilon_{2}=3.9$. Suppose we want to get the
BIC with the dimensionless resonant frequency equalling exactly four, that is,
$\omega_{c}\lambda_{g}/c=4$, like in Fig. 3. We immediately achieve this goal
by adjusting the fill factor to the value $\rho=0.3918307$ that tunes the
phase accumulation $k_{1}d_{1}$ at this frequency to $\pi$ as per Eqs. (26),
(35).

Next, we need to select the other three waveguide parameters – the thickness
$L$ of the grating layer and the permittivities $\varepsilon^{\pm}$ of the
cover and substrate. There is just one global constraint (34) on the
permittivities: They should be less than a certain threshold value above which
the $p=\pm 1$ spatial Fourier harmonic starts to radiate out of the grating
layer and into the cover or substrate. Such radiation would mean opening a
radiation-loss channel, additional to the one provided by the zeroth Fourier
harmonic, that would degrade the BIC to just a low-Q leaky resonance. For the
chosen resonant frequency $\omega_{c}\lambda_{g}/c=4$, the constraint (34)
requires $\varepsilon^{\pm}<(\pi/2)^{2}$.

Figure 5: The absolute value of the characteristic function
$\det(M-\mathbbm{1})$ of waveguide eigenmodes, Eq. (8), as a function of the
dimensionless grating layer thickness $L/\lambda_{g}$ and the permittivity of
the substrate $\varepsilon^{-}$;
$\varepsilon^{+}=1,\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,\omega^{\prime}=\omega_{c}=4c/\lambda_{g},\omega^{\prime\prime}=1.25\times
10^{-11}\omega_{c},k_{x}=0$.

The only nontrivial constraint on the waveguide parameters is a fulfillment of
the characteristic equation (8) for the considered BIC, that is, the waveguide
eigenmode comprising the $n=2$ and $n=3$ infinite-grating eigenmodes and
demonstrating high Q factor near the selected frequency $\omega_{c}$. Suppose
one sets the dielectric constant of the cover to be some reasonable number,
say, $\varepsilon^{+}=1$. Then the choice of the other two parameters, $L$ and
$\varepsilon^{-}$, can be guided by the corresponding 3D plot of the absolute
value of the waveguide characteristic function
$D(\omega=\omega_{c}-i\omega^{\prime\prime})=\det[M(\omega=\omega_{c}-i\omega^{\prime\prime})-\mathbbm{1}]$
in Eq. (8) considered as a function of the variables $L$ and $\varepsilon^{-}$
at the critical frequency and a relatively small decay rate, say,
$\omega^{\prime\prime}=1.25\times 10^{-11}\omega_{c}$ (see Fig. 5).

The center line of the black branches on this plot corresponds to the curve on
the $(L,\varepsilon^{-})$-plane of parameters where the determinant in Eq. (8)
approaches zero. In other words, the black branches indicate the narrow ranges
of the waveguide parameters for which the waveguide supports the high-Q
waveguide eigenmodes with the eigenfrequency close to the chosen one
$\omega_{c}-i\omega^{\prime\prime}$. Note that the entries of the generating
matrix $M$ in Eq. (7) are calculated by means of the aforementioned explicit,
analytical formulae via the $z$-wavenumbers $k_{zn}(\omega^{\prime\prime})$ of
the infinite-grating eigenmodes which now include the first-order correction
$\Delta k_{zn}\propto\omega^{\prime\prime}$, Eq. (29), along with the zeroth-
order solution $k_{zn}(\omega^{\prime\prime}=0)$ to the transcendental
characteristic equation (22) for the real-valued critical frequency
$\omega_{c}$.

Figure 6: Distribution of the absolute value of the electric field
$|E_{y}^{(m_{z})}(x,z)|$ as per Eq. (6) in the $xz$-plane of the cover (top),
grating (middle) and substrate (bottom) for the $m_{z}=1$ waveguide eigenmode.
The grating layer thickness is $L=1.82551\lambda_{g}$. The permittivity of the
cover and substrate is $\varepsilon^{+}=1$ and $\varepsilon^{-}=2$,
respectively;
$\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,k_{x}=0$. Figure 7:
The same as in Fig. 6, but for the $m_{z}=2$ waveguide eigenmode. The grating
layer thickness is $L=3.92824\lambda_{g}$.

In this paper we consider the BIC waveguide eigenmode associated with the
arch-like branches shown in the lower half of Fig. 5. We enumerate them by the
index $m_{z}=1,2,...$ starting from the most left branch. In fact, the integer
$m_{z}$ indicates approximately the number of half-wavelengths, stacked along
the grating layer of thickness $L$ in the $z$ direction, in the spatial
eigenmode’s electric field distribution inside the grating layer. The
corresponding distribution of the absolute value of the electric field
$|E_{y}^{(m_{z})}(x,z)|$ in the $xz$-plane of the planar grating waveguide,
computed by means of Eq. (6), is shown in Fig. 6 and Fig. 7 for the $m_{z}=1$
and $m_{z}=2$ waveguide eigenmodes with the grating layer thickness equal
$L=1.82551\lambda_{g}$ and $L=3.92824\lambda_{g}$, respectively. A white
vertical line across the middle part of Figs. 6, 7 separates the two sections
of the lamellar grating: the left section of length $d_{1}=\rho\lambda_{g}$
and permittivity $\varepsilon_{1}$ and the right section of length
$d_{2}=\lambda_{g}-d_{1}$ and permittivity $\varepsilon_{2}$. In both cases
the dielectric constants of the cover and substrate are chosen to be
$\varepsilon^{+}=1$ and $\varepsilon^{-}=2$, respectively. The latter choice
ensures, according to Eq. (34), that all Fourier harmonics in the cover and
substrate are evanescent except for the $p=0$ Fourier harmonic. Thus, only the
zeroth Fourier harmonic freely propagates in the cover and substrate and,
hence, provides the only channel of radiation losses for the BIC waveguide
eigenmode. Figs. 6, 7 show that the BIC slightly stands out from the grating
layer due to the coupling of its infinite-grating eigenmodes with the
evanescent spatial Fourier harmonics via the boundary conditions. Obviously,
the fields penetrate inside the substrate deeper than inside the cover since
the permittivity of the substrate is larger,
$\varepsilon^{-}=2>\varepsilon^{+}=1$.

Fig. 5 clearly shows that the BIC waveguide eigenmodes associated with the
arch-like branches cease to exist as high-Q modes due to the opening of the
second radiation-loss channel via the $p=\pm 1$ Fourier harmonic above the
aforementioned cutoff value of the cover/substrate permittivity
$\varepsilon^{\pm}=(\pi/2)^{2}\approx 2.467$. (Note that the low-Q waveguide
eigenmodes are not described within the linear approximation in Eq. (29) which
is valid only if $\omega^{\prime\prime}\ll\omega^{\prime}$.)

Hereinafter, all numerical calculations are easy and fast to perform since
they are based on the explicit analytical formulae and a sufficient accuracy
is achieved already in the approximation involving just a very few, namely,
first five infinite-grating eigenmodes ($n=1,2,3,4,5;\,N=5$) and Fourier
harmonics ($p\,=\,0,\,\pm 1,\,\pm 2;\ S=5$). In other words, only very small
sized $5\times 5$ matrices are involved.

Figure 8: Absolute value of the characteristic function $\det(M-\mathbbm{1})$
of the waveguide eigenmodes, Eq. (8), as a function of the dimensionless real
frequency detuning $(\omega^{\prime}-\omega_{c})\lambda_{g}/c$,
$\omega_{c}=4c/\lambda_{g}$, in the vicinity of the BIC $m_{z}=1$ resonance at
the fixed decay rate and grating layer thickness: (a)
$\omega^{\prime\prime}=\omega^{\prime\prime}_{m}=5\times
10^{-7}c/\lambda_{g},L\approx 1.82645\lambda_{g}$, (b)
$\omega^{\prime\prime}=\omega^{\prime\prime}_{m}=8\times
10^{-6}c/\lambda_{g},L\approx 1.82289\lambda_{g}$. The other waveguide
parameters are the same as in other figures:
$\varepsilon^{+}=1,\varepsilon^{-}=2,\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,k_{x}=0$.
The zero values at the detunings $-0.65\times 10^{-6}$ and $10^{-5}$ point to
the eigenfrequencies of the BIC waveguide eigenmode $\omega_{m}=(4-6.5\times
10^{-7}-i5\times 10^{-7})c/\lambda_{g}$ and $\omega_{m}=(4+10^{-5}-i5\times
10^{-6})c/\lambda_{g}$ corresponding to the respective points on the left and
right wings of the Q-factor resonance at $Q\approx 4\times 10^{6}$ and
$Q\approx 2.5\times 10^{5}$ in Fig. 10.

By knowing the exact location of a given waveguide eigenmode in the parameter
space via the guidance provided by a 3D plot, such as in Fig. 5, of the
determinant of the waveguide eigenmode problem, it is straightforward to
calculate the real and imaginary parts of the waveguide eigenmode frequency
$\omega_{m}=\omega^{\prime}_{m}-i\omega^{\prime\prime}_{m}$ as functions of
the grating layer thickness $L$ or other waveguide parameters. For instance,
for any given set of waveguide parameters, one can use a 3D plot of the
absolute value of the waveguide characteristic function
$|\det[M(\omega^{\prime}-i\omega^{\prime\prime})-\mathbbm{1}]|$ over the
complex $(\omega^{\prime},\omega^{\prime\prime})$-plane, magnify a small
region around a predetermined position of the eigenfrequency, and then
determine its value
$\omega_{m}=\omega^{\prime}_{m}-i\omega^{\prime\prime}_{m}$ corresponding to
the zero of the characteristic function, $\det(M-\mathbbm{1})=0$. In this way,
the complex eigenfrequency can be calculated quite accurately.

Fig. 8 represents a cross section $\omega^{\prime\prime}=const$ of the above
3D plot, that is, the $|\det(M-\mathbbm{1})|$ as a function of the real
frequency detuning $\Delta\omega=\omega^{\prime}-\omega_{c}$, for two sets of
the values of the decay rate $\omega^{\prime\prime}$ and thickness $L$. The
detuning $\Delta\omega_{m}$ at which $|\det(M-\mathbbm{1})|=0$ corresponds to
the eigenfrequency of the BIC waveguide eigenmode
$\omega_{m}=\omega_{c}+\Delta\omega_{m}-i\omega^{\prime\prime}_{m}$. For
example, the detunings $-0.65\times 10^{-6}$ and $10^{-5}$ in Figs. 8a and 8b
point, respectively, to the eigenfrequencies of the BIC waveguide eigenmode
$\omega_{m}=(4-6.5\times 10^{-7}-i5\times 10^{-7})c/\lambda_{g}$ and
$\omega_{m}=(4+10^{-5}-i8\times 10^{-6})c/\lambda_{g}$ corresponding to the
respective points on the left and right wings of the Q-factor resonance at
$Q\approx 4\times 10^{6}$ and $Q\approx 2.5\times 10^{5}$ in Fig. 10 below.

Alternatively, one can plot a set of curves showing the absolute value of the
waveguide characteristic function $|\det(M-\mathbbm{1})|$ as a function of one
of the waveguide parameters, say, the grating layer thickness $L$, for a set
of the real and imaginary parts of the frequency
$\omega^{\prime}-i\omega^{\prime\prime}$ in the vicinity of the BIC resonance.
A curve that approaches zero at some value of the waveguide parameter (say,
the thickness $L$) signifies the presence of the BIC of the corresponding
eigenfrequency $\omega^{\prime}-i\omega^{\prime\prime}$ in the vicinity of
this value of the waveguide parameter.

Figure 9: Absolute value of the characteristic function $\det(M-\mathbbm{1})$
of the waveguide eigenmodes, Eq. (8), as a function of the dimensionless
grating layer thickness $L/\lambda_{g}$ at a particular value of the complex
frequency $\omega^{\prime}-i\omega^{\prime\prime}=(4-i5\times
10^{-10})c/\lambda_{g}$ close to the critical frequency
$\omega_{c}=4c/\lambda_{g}$. Touching down the zero level at a discrete set of
thicknesses definitively points to the series of waveguide eigenmodes of
different spatial patterns corresponding the black branches along the cross
section $\varepsilon^{-}=2$ of the 3D plot in Fig. 5.

Such a function plotted for the critical frequency $\omega_{c}$ and a given
small decay rate $\omega^{\prime\prime}$ within a wider range of the waveguide
parameter, like the one in Fig. 9, reveals an entire set of high-Q BICs and
leaky waveguide eigenmodes. Each sharp tip that points to zero indicates an
approximate value of the waveguide parameter for which the waveguide supports
a high-Q eigenmode. In fact, Fig. 9 is a cross section $\varepsilon^{-}=2$ of
the 3D contour plot in Fig. 5 and reveals the set of waveguide eigenmodes that
is represented by black branches in Fig. 5. The Figs. 8 and 9 clearly
demonstrate how robust, definitive and accurate the determination of the
waveguide eigenmode frequency by means of the characteristic equation (8) is.

Figure 10: The Q factor, $Q=\omega^{\prime}_{m}/(2\omega^{\prime\prime}_{m})$,
of the $m_{z}=1$ waveguide eigenmode shown in Fig. 6 as a function of
frequency detuning from the critical frequency,
$(\omega^{\prime}_{m}-\omega_{c})\lambda_{g}/c$, within the range where the
grating layer thickness $L$ varies from $1.822\lambda_{g}$ to
$1.827\lambda_{g}$.

In this way, by varying any particular parameter of the waveguide, say, the
grating layer thickness $L$, one can easily calculate the Q factor of the BIC
waveguide eigenmode, $Q=\omega^{\prime}_{m}/(2\omega^{\prime\prime}_{m})$, as
a function of that parameter and find the profile of the corresponding high-Q
resonance. Its particular example is shown in Fig. 10 as a function of
frequency detuning from the critical value, $\omega^{\prime}_{m}-\omega_{c}$,
within the range where the grating layer thickness $L$ varies from
$1.822\lambda_{g}$ to $1.827\lambda_{g}$. The dots stand for the numerically
computed points, while the continuous curve is a fit $Q\approx\alpha
c/(|\omega^{\prime}_{m}-\omega_{c}|\lambda_{g})$ approximating the Q factor at
the wings of the resonance. Within the depicted range of detuning
$(\omega^{\prime}_{m}-\omega_{c})\lambda_{g}/c\sim 10^{-5}-10^{-6}$, the
fitting parameter $\alpha$ is about 2.5. The latter dependence describes a
reciprocal, very steep increase in the Q factor when approaching the BIC.
Ideally the BIC Q factor tends to infinity at the center of the resonance.

In reality the Q factor achieves a finite maximum value determined by the
finite size of the grating in the $x$ and $y$ directions, the impurities,
defects and inhomogeneity of the waveguide material, roughness of the
waveguide, cover and substrate surfaces, as well as other similar effects.
Such a behavior is described by saying that in reality the BIC appears as a
quasi-BIC [5, 14].

##  V Bound-state-in-the-continuum origin: Parity symmetry breaking and
decoupling from the radiation-loss channel due to mode crossing

The origin of the BIC described above can be explained as follows. The
eigenvector in Eq. (7) for the BIC waveguide eigenmode $m_{z}=1$ at the
resonance, say, at the frequency $\omega_{m}$ $=(4-i1.25\times
10^{-11})c/\lambda_{g}$ and waveguide parameters $L=1.82551\lambda_{g},\
\varepsilon^{-}=2,\
\varepsilon^{+}=1,\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307$
(see Figs. 6–10), is equal to

${\bf F^{(m)}}=(1.06\times 10^{-9},0.703,0.703,0.113,1.36\times 10^{-9})^{T}.$
(36)

It reveals an equal presence of the 2-nd and 3-rd infinite-grating eigenmodes
and a minor admixture of the 4-th one, while the rest of the infinite-grating
eigenmodes, the first, fourth, and higher, don’t contribute significantly to
this waveguide eigenmode. The amplitudes of the Fourier harmonics constituting
the $n$-th infinite-grading eigenmode for the chosen complex frequency and
waveguide parameters are given by the $n$-th row of the $C$ matrix

$\\{c_{n}^{p}\\}=\left[{\begin{array}[]{ccccc}0.0295&0.239&0.940&0.239&0.0295\\\
0.0785&0.703&2.16\times 10^{-8}&0.703&0.0785\\\ 0.0785&0.703&2.16\times
10^{-8}&0.703&0.0785\\\ 0.702&0.0791&1.72\times 10^{-16}&0.0791&0.702\\\
0.704&0.0450&0.0210&0.0450&0.704\\\ \end{array}}\right],$ (37)

calculated by means of the explicit analytical formulae in Eqs. (10)–(12),
(17), (18). The p-th Fourier harmonic is represented by the p-th column. The
columns are enumerated from left to right by the integer $p=-2,-1,0,1,2$. The
degeneracy between the infinite-grating eigenmodes $f_{2}$ and $f_{3}$ at the
critical frequency leads to degenerate values for the 2-nd and 3-rd rows, that
is, $c^{p}_{2}=c^{p}_{3}$ for all $p$. Also, $k_{x}=0$ implies that the
harmonic’s $x$-wavenumbers $k_{x}^{(p)}=k_{x}+pk_{g}$ are symmetric about
$p=0$, and thus the $C$ is likewise symmetric about $p=0$, that is,
$c_{n}^{p}=c_{n}^{-p}$ for all $n$ and all $p$.

Figure 11: A contour plot of the amplitude $c_{2}^{0}$ of the $p=0$ spatial
Fourier harmonic within the $n=2$ infinite-grating eigenmode over the complex
$(\omega^{\prime},\omega^{\prime\prime})$-plane in the vicinity of the
critical frequency $\omega_{c}=4c/\lambda_{g}$. White dots represent the track
of the $m_{z}=1$ BIC eigenfrequency $\omega_{m}(L)$ on the complex plane when
the grating layer thickness $L$ varies from $1.824\lambda_{g}$ to
$1.827\lambda_{g}$; $\varepsilon^{-}=2,\
\varepsilon^{+}=1,\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,k_{x}=0$.

Remarkably, the central column shows that the $p=0$ Fourier harmonic has a
vanishing presence in all three ($n=2,3,4$) infinite-grating eigenmodes
constituting the $m_{z}=1$ BIC waveguide eigenmode (36). This means that close
to the center of the resonance, the BIC decouples from the only available
radiation channel, provided by the $p=0$ Fourier harmonic, and acquires a very
high Q factor shown in Fig. 10. The couplings $c_{n}^{0}$ with the radiation
channel are given by explicit formulae (10), (31), (32). Radiation losses are
proportional to the amplitude of the zeroth Fourier harmonics, emitted out of
the grating layer into the cover, $g_{0}^{(m)+}$, and the substrate,
$g_{0}^{(m)-}$. They demonstrate the same narrow dip near the BIC resonance as
the one in the coupling $c_{2}^{0}$ shown in Figs. 11, 12 because the latter
is responsible for the coupling of the $n=2$ infinite-grating eigenmode with
the only available radiation-loss channel. Figs. 11 and 12 prove that detuning
the waveguide parameters, say, the grating layer thickness, away from the BIC
resonance greatly, by many orders of magnitude, increases this coupling
$c_{2}^{0}$. The same detuning changes the coupling $c_{3}^{0}$ of the $n=3$
infinite-grating eigenmode with the $p=0$ Fourier harmonic only by about 2
times and keeps the other couplings $c_{n}^{p}$ and the BIC composition ${\bf
F^{(m)}}$ almost intact. Even at the far wings of the resonance, say, at the
detuning $\omega_{m}^{\prime}-\omega_{c}=10^{-4}\omega_{c}$, the composition
is almost the same:

${\bf F^{(m)}}=(0.015,0.705,0.706,0.055,0.002)^{T}.$ (38)

Figure 12: The amplitude $c_{2}^{0}$ of the $p=0$ spatial Fourier harmonic
within the $n=2$ infinite-grating eigenmode as the function of (a) the
frequency detuning $(\omega^{\prime}-\omega_{c})\lambda_{g}/c$ at the fixed
decay rate $\omega^{\prime\prime}\lambda_{g}/c=3\times 10^{-7}$ and (b) the
decay rate $\omega^{\prime\prime}\lambda_{g}/c$ at the fixed real-valued
frequency $\omega^{\prime}=\omega_{c}$; $L=1.82551\lambda_{g},\
\varepsilon^{-}=2,\
\varepsilon^{+}=1,\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,k_{x}=0$.

This striking resonant interplay between infinite-grating eigenmodes and
spatial Fourier harmonics inside the BIC waveguide eigenmode is readily
explained by the restructuring of the $x$-profiles (shown in Figs. 13, 14, 15)
for the $n=2$ and $n=3$ infinite-grating eigenmodes caused by the intersection
of their dispersion curves (see Fig. 3). Away from the mode crossing point,
the $n=2$ and $n=3$ infinite-grating eigenmodes have $x$-profiles of almost
pure opposite parities by construction, in virtue of the $\pi$-phase
accumulations over both grating sections at the mode-crossing point as per
Eqs. (25), (35). Namely, at large detunings, the $x$-profiles of the $n=2$ and
$n=3$ infinite-grating eigenmodes are even and odd, respectively, on each
uniform segment of the grating relative to the center of this segment.
However, due to the degeneracy of those two eigenmodes at the mode-crossing
point, this parity symmetry gets inevitably broken. It happens only very close
to the degenerate point, namely, when the coupling $c_{2}^{0}$ of the $n=2$
infinite-grating eigenmode with the zeroth Fourier harmonic drops below the
couplings $c_{2}^{p},p\neq 0$, with the other Fourier harmonics (see Fig. 12).

Figure 13: Parity symmetry breaking of the $n=2$ infinite-grating eigenmode
via nontrivial restructuring of the (a) imaginary $\rm{Im}[f_{2}(x)]$ and (b)
real $\rm{Re}[f_{2}(x)]$ $x$-profiles in the complex plane from the even
(relative to the center of each grating segment $d_{1}/\lambda_{g}=\rho\approx
0.4$, $d_{2}/\lambda_{g}=1-d_{1}/\lambda_{g}\approx 0.6$) pure imaginary
$x$-profile at large detunings to the complex-valued $x$-profile without
constant spatial component at small detunings as the frequency
$\omega^{\prime}$ crosses the critical frequency $\omega_{c}=4c/\lambda_{g}$;
$\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307$,
$\omega^{\prime\prime}=10^{-4}c/\lambda_{g}$. Figure 14: Parity symmetry
breaking of the $n=2$ infinite-grating eigenmode shown via nontrivial
restructuring of the absolute value of its complex $x$-profile, $|f_{2}(x)|$,
as the frequency $\omega^{\prime}$ crosses the critical frequency
$\omega_{c}=4c/\lambda_{g}$;
$\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307$,
$d_{1}/\lambda_{g}=\rho\approx 0.4$,
$d_{2}/\lambda_{g}=1-d_{1}/\lambda_{g}\approx 0.6$,
$\omega^{\prime\prime}=10^{-4}c/\lambda_{g}$.

The latter means that the zeroth Fourier harmonic (i.e., the constant spatial
component) of the $n=2$ infinite-grating eigenmode, scaled by the norm in Eq.
(16), vanishes and this eigenmode ceases to be even on each uniform segment of
the grating relative to the center of this segment. It is worth noting that
the parity symmetry breaking occurs gradually by expanding into the 2D,
complex-valued $x$-profile functions, that is, via a gradually increasing
admixture of the real-valued odd function $\rm{Re}[f_{2}(x)]$ to the
imaginary-valued even function $\rm{Im}[f_{2}(x)]$, as is clearly illustrated
in Fig. 13. The parity symmetry breaking can also be described by a smooth
nontrivial reshaping of the absolute value of the $x$-profile complex
function, $|f_{2}(x)|$, shown in Fig. 14.

Figure 15: The odd (relative to the center of each grating segment
$d_{1}/\lambda_{g}=\rho\approx 0.4$ and
$d_{2}/\lambda_{g}=1-d_{1}/\lambda_{g}\approx 0.6$) $x$-profile of the $n=3$
infinite-grating eigenmode, represented by (a) its imaginary part
$\rm{Im}[f_{3}(x)]$ and (b) a small admixture of its real part
$\rm{Re}[f_{3}(x)]$, stays almost intact (apart from a trivial change in sign)
as the frequency $\omega^{\prime}$ crosses the critical frequency
$\omega_{c}=4c/\lambda_{g}$;
$\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307$,
$\omega^{\prime\prime}=10^{-4}c/\lambda_{g}$.

At the same time, the $n=3$ infinite-grating eigenmode also moves a little bit
towards its crossing counterpart (i.e., the $n=2$ eigenmode) in the parameter
space and increases the presence of the zeroth Fourier harmonic in a tendency
of losing its pure odd parity. Yet, it turns out that the oscillating, $p\neq
0$ Fourier harmonics always prevail in the composition of the $n=3$ infinite-
grating eigenmode (see Fig. 15), so that it stays almost entirely non-leaking.

The reason for such dramatically different behavior of the $n=2$ and $n=3$
infinite-grating eigenmodes is that they belong to two different universality
classes $\eta_{2}=-1$ and $\eta_{3}=+1$ of the $x$-profile scaling when
approaching the degenerate point $D_{2,3}^{0}$ as per Eq. (28). As a result,
the $n=2$ infinite-grating eigenmode is restructured more radically and
substantially than the $n=3$ eigenmode which remains essentially intact.

The explicit power-law asymptotics (28) of the exact solution (5) for the
amplitudes $b_{q}^{\pm},\,q=1,2$, of the waves in the $x$-profile of the
infinite-grating eigenmodes, Eq. (4), gives a full detailed description of the
universal scaling of the parity symmetry breaking in the vicinity of the
degenerate point $D_{2,3}^{0}$, that is, near the BIC high-Q resonance. The
nontrivial universal behavior of the amplitudes $b_{q}^{\pm}$ for both
sections of the grating ($q=1,2$) possesses a scaling determined by the decay
rate $\omega^{\prime\prime}$ and is illustrated in Fig. 16 for the $n=2$
infinite-grating eigenmode, whose critical index is $\eta_{2}=-1$. In Figs.
13, 14, 15, and 16, we intentionally choose a large enough decay rate,
$\omega^{\prime\prime}\lambda_{g}/c=10^{-4}$, to clearly demonstrate the
details of the $x$-profile restructuring in the course of the parity symmetry
breaking. Note a perfect matching of the exact result in Eq. (5) (shown by the
solid curves) by the leading asymptotic approximation in Eq. (28),
$b_{1}^{\pm}=(\pm
a_{1}+ic_{1})(\omega^{\prime}-\omega_{c}-i\omega^{\prime\prime})^{2},b_{2}^{\pm}=(\pm
a_{2}+ic_{2})(\omega^{\prime}-\omega_{c}-i\omega^{\prime\prime})^{2}$ (shown
by the dashed curves). The fitting parameters in Fig. 16 are $a_{1}\approx
0.4,c_{1}\approx-1.06,a_{2}\approx 0.61,c_{2}\approx-1.06$.

Figure 16: Universal behavior (28) of the amplitudes $b_{q}^{\pm},q=1,2,$ of
the waves in the $x$-profile, Eq. (4), of the $n=2$ infinite-grating
eigenmode, $\eta_{2}=-1$, is illustrated by their real part
$\rm{Re}(b_{q}^{\pm})$ as a function of the dimensionless frequency
$\omega^{\prime}\lambda_{g}/c$ in the vicinity of the critical frequency
$\omega_{c}\lambda_{g}/c=4$. The scaling and, in particular, intersection
points are determined by the decay rate which is chosen large enough,
$\omega^{\prime\prime}\lambda_{g}/c=10^{-4}$, to clearly illustrate the
details; $\varepsilon_{1}=6.25,\varepsilon_{2}=3.9,\rho=0.3918307,k_{x}=0$.
The exact numerical result in Eq. (5) (solid curves) is perfectly fitted (for
details, see sect. V) by the leading asymptotics in Eq. (28) (dashed curves).

The result of this restructuring of the infinite-grating eigenmodes near the
degenerate point on the radiative losses and Q factor of the $m$-th BIC is
determined by the amplitudes of the zeroth Fourier harmonic, $g_{0}^{(m)\pm}$,
emitted out of the grating layer into the cover and substrate. These
amplitudes are given by the $p=0$ component of the Fourier-harmonic column
$S$-vectors, ${\bf g^{(m)\pm}}=(\\{g_{p}^{(m)\pm}|p=0,\pm 1,...,\pm
p^{*}\\})^{T}$, each containing $S=2p^{*}+1$ amplitudes of the co- or counter-
propagating Fourier harmonics emitted from the cover ($z=L$) or substrate
($z=0$) borders, respectively. The Fourier harmonic vectors are given by the
transmittance matrix in Eq. (19) applied to the waveguide eigenmode vector
${\bf F^{(m)}}$, Eq. (7), of the infinite-grating eigenmodes:

${\bf g^{(m)+}}=T^{+}\Lambda{\bf F^{(m)}},\qquad{\bf g^{(m)-}}=T^{-}\Lambda
R^{+}\Lambda{\bf F^{(m)}}.$ (39)

The radiation losses of the $m_{z}=1$ BIC, as measured by the amplitudes
$g_{0}^{(m)\pm}$ of the outgoing into the cover and substrate zeroth Fourier
harmonic, closely follow the resonance in the contribution $c_{2}^{0}$ of the
zeroth Fourier harmonic to the $n=2$ infinite-grating eigenmode (compare Figs.
11, 12 against Fig. 10).

There are no other high-Q BICs in the range of parameters associated with the
aforementioned narrow resonance. In particular, this is a consequence of the
fact that the $n=2$ infinite-grating eigenmode is generally present in any
waveguide eigenmode due to reflection at the boundaries with the cover and
substrate, but any waveguide eigenmode associated with the second infinite-
grating eigenmode has a low Q factor everywhere outside this resonance due to
the strong presence of the freely leaking (i.e., propagating from the grating
layer into the cover and substrate) zeroth Fourier harmonic.

##  VI Conclusions

We unfold a general mechanism for the formation of a BIC in a planar waveguide
with a lamellar grating. The BIC emerges near the degenerate frequency
$\omega_{c}$ at which (i) the dispersion curve of a leaking infinite-grating
eigenmode meets/crosses the dispersion curve of a non-leaking infinite-grating
eigenmode and (ii) changes the spatial symmetry in such a way that it becomes
orthogonal to the spatial Fourier components/harmonics leaking out of the
grating waveguide layer and into the cover and/or substrate. As a result, in
the narrow vicinity of such a mode-crossing point, the waveguide eigenmode
dominated by those two coupled infinite-grating eigenmodes disconnects from
the radiation-loss channel and becomes a BIC with a very high Q factor.

In the ideal case when the waveguide parameters are tuned precisely to the
values making the eigenfrequency of the BIC waveguide eigenmode equal to the
critical, crossing frequency, $\omega_{m}=\omega_{c}$, the BIC’s leakage rate
becomes zero, $\omega_{m}^{\prime\prime}=0$. Hence, the ideal BIC becomes
completely decoupled from radiating waves, i.e., invisible from outside the
waveguide, but at the same time it cannot be excited by continuum waves
incident from outside. (Of course, a laser or other source of continuum waves
located inside the grating layer will always excite the field in the BIC.)
However, if the thickness of the grating layer or/and other waveguide
parameters are detuned from those ideal values, then the BIC acquires nonzero
coupling with the radiating waves and a leakage rate
$\omega_{m}^{\prime\prime}\neq 0$. Such a perturbed, leaky BIC becomes visible
and excitable by continuum waves incident from outside the grating waveguide.
This leaky BIC, as a usual damped harmonic oscillator, has a standard response
to the excitation by an incident monochromatic wave in the form of the
Lorentzian resonance curve. It should not be mixed with the BIC’s hallmark -
the sharp resonant profile of the BIC Q factor shown in Fig. 10. The latter
describes a certain internal property of the leaky BIC expressed by the
function
$Q(\omega_{m}^{\prime})=\omega_{m}^{\prime}/(2\omega_{m}^{\prime\prime})$,
namely, the relation between corresponding perturbations in the real and
imaginary parts of the BIC eigenfrequency
$\omega_{m}=\omega_{m}^{\prime}-i\omega_{m}^{\prime\prime}$, and has nothing
to do with the BIC’s excitation by the external source.

The existence of such a resonance phenomenon is explicitly verified by
numerical plots of the analytical formulae for the high-Q $m_{z}=1$ BIC (see
Fig. 5) originated from the degenerate point $D_{2,3}^{0}$ (see Fig. 2) in the
planar grating waveguide based on a well-known optical material, titanium
oxide (TiO2). The sharp resonance is illustrated in Fig. 10 by the Q-factor
profile calculated within the $5\times 5$ approximation of the analytical
waveguide-eigenmode theory. Remarkably, such a simple low-dimensional
approximation (explained in the beginning of sect. II and after Eq. (19)),
which includes just five ($N=5$) first infinite-grating eigenmodes and five
($S=5$) spatial Fourier harmonics, provides an accurate description of the BIC
and all details of the mechanism of its formation. The point is that all of
the higher-order infinite-grating eigenmodes and Fourier harmonics are
evanescent and contribute very little to the selected BIC. In particular, we
verified that numerical results within the $7\times 7$ approximation, which
includes two more infinite-grating eigenmodes and two more Fourier harmonics,
do not lead to any appreciable corrections.

Various BICs and mechanisms of their formation in different photonic
structures had been widely discussed in the literature (see, for example, [11,
12, 13, 10, 5, 9, 4, 14, 15, 16, 17, 18, 19] and references therein). The
formation mechanism revealed in the present paper for the BIC in the grating
waveguide shares features of most those mechanisms. The reasons for that are,
first, an insight provided by the eigenmode approach (which addresses the
ultimate physical entities living in the photonic system and, hence, is the
most logical physically) and, second, a basic nature of the planar lamellar-
grating waveguide. This waveguide is the simplest photonic structure (among
nontrivial ones) which shares many common principal features, including the
existence of BICs, with general photonic crystals, but still admits an
analytical solution via eigenmodes and is just one further step of
complication from the trivial case of the planar dielectric waveguide which
has an elementary solution in terms of plane waves [23, 24, 25] and does not
support BICs.

Below we briefly comment on these standard (marked below in italic) BIC
formation mechanisms and related physical interpretations. Of course, the BIC
is connected with an interference between partial modes of different structure
or nature [3, 4, 5, 10, 14, 15, 16, 11]. In the present case, those partial
modes are the two infinite-grating eigenmodes of the opposite, even and odd,
spatial parity which are both responsible for the phenomenon of parity
symmetry breaking due to an intersection of their dispersion curves and, for
the majority of the BIC’s content, formed due to their mutual reflection and
interference. Besides, the interference due to Bragg diffraction on the
grating, which is infinite in the $xy$-plane, is responsible for
”discretization” of the radiation continuum into a discrete set of radiation-
loss channels (each of them being the leaking Fourier harmonic of the
diffraction order $p=0,\pm 1,\ldots$) as well as the very formation of the
infinite-grating eigenmodes. In the end, the destructive interference of
radiation for different partial waves constitutes a physical explanation for
the BIC’s existence and, in particular, is the ultimate reason for emission
cancellation of the zeroth Fourier harmonic at the resonant frequency
$\omega_{c}$.

Similar to the BIC in a photonic crystal slab at the center of the first
Brillouin zone, the BIC in the grating waveguide at the origin of the momentum
space, $k_{x}=0$, is symmetry-protected since it exists due to a certain
spatial symmetry (parity) of the infinite-grating eigenmodes at the mode-
crossing point. Yet, this BIC can also be called accidental since it occurs
due to the vanishing of the eigenmode coupling with the radiation continuum as
per Figs. 11, 12.

At the same time, as is discussed in sect. IV and illustrated in Fig. 5, the
BIC formation is achieved through parameter tuning (with coupled partial
infinite-grating eigenmodes, or with a single resonance if one employs a
waveguide-eigenmode point of view) and such a BIC is known as an accidental
BIC. From the waveguide-eigenmode point of view, the grating-waveguide BIC
appears as a so-called single-resonance parametric BIC.

Moreover, since the BIC is formed as the waveguide eigenmode out of more than
one partial infinite-grating eigenmodes via their coupling, its mechanism is
reminiscent of the Friedrich–Wintgen mechanism of the BIC formation [4] that
also acts near frequency crossings of the uncoupled resonances. A Fabry–Pérot
BIC, originated from spatially separate cavity modes coupled through a
semitransparent mirror, is so special that it is not directly relevant to the
grating-waveguide BIC.

Note also that some analogy can be traced with the BICs in the conventional
open cavities and waveguides [7], outside the realm of photonic crystals. For
instance, the accidental or single-resonance parametric BIC can exist in the
open Sinai billiard in which a cavity eigenmode gets decoupled from the
continuum of the waveguide by an appropriate smooth adjusting of the position
of the intracavity disc. In the open cavities, the BIC frequency is also
located in a vicinity to the degenerate-point frequency of eigenmodes of the
closed cavity [7, 36].

There is a principle feature of the BIC clearly seen in Figs. 6, 7. The BIC
slightly extends out from the grating layer into the cover and substrate
because of the coupling with the evanescent Fourier harmonics due to boundary
conditions. Such coupling with the evanescent waves is crucially important for
the very existence of the BIC.

The presence of the degenerate, mode-crossing points $D_{n,n+1}^{l}$ (see Fig.
2) and the corresponding spatial parity, Eqs. (25), (30), is a topological,
robust property associated with the map of the solutions to the characteristic
equation (22) over the space of the waveguide parameters and frequency. The
existence and qualitative properties of the infinite-grating eigenmodes as
well as the topology of the intersection of their dispersion curves
predetermine how perturbations in the grating, cover, substrate and waveguide
parameters affect the BIC. Such perturbations continuously, smoothly modify,
but do not destroy neither the eigenmodes and mode crossings nor the BIC. In
that sense, the BIC is topologically protected (cf. [5]).

The BIC shows a very sharp (compared to that in a conventional optical cavity)
increase in the Q factor that ideally tends to infinity in a vicinity of the
mode-crossing, i.e. critical, frequency, Fig. 10. However, such an ideal
behavior implies a 1D or 2D infinite size of the photonic structure that
should be reminiscent of an infinite diffraction lattice discretizing the
radiation continuum $\omega=ck/\sqrt{\varepsilon^{\pm}}$ into the diffraction
orders (spatial Fourier harmonics). The point is a rigorous theorem [37] that
forbids appearance of a BIC in a finite-size dielectric structure. In reality,
the Q factor is saturated at some maximum value that is determined by finite-
size effects, structural disorder, fabrication imperfections, defects,
impurities and losses in the waveguide materials, roughness of the layers’
surfaces, and similar scattering and dissipation effects. They say that the
ideal BIC actually manifests itself as a quasi-BIC due to the above factors
[5, 9, 14, 15, 11].

A general analysis of the emergence of BICs via the mechanism of parity
symmetry breaking due to mode crossing as well as an analysis of the entire
spectrum of all BICs and leaky resonances in the grating waveguide and similar
photonic structures will be given elsewhere. It is based on the hierarchy of
the eigenmode-crossing points $D_{n,n+1}^{l}$ in the infinite grating
illustrated in Fig. 2. In particular, the row of crossing points
$D_{2,3}^{0},D_{4,5}^{0},D_{6,7}^{0}$, $D_{8,9}^{0},\ldots$ corresponds to the
intersections $D_{2m_{x},2m_{x}+1}^{0}$ between the dispersion curves of the
$n=2m_{x}$ and $n+1=2m_{x}+1$ infinite-grating eigenmodes of the spatial
parity of the order $(m_{x},m_{x}),\,m_{x}=1,2,\ldots$. All such mode-crossing
points have the same $z$-wavenumber (see Eqs. (25),(26)) which is exactly the
one emerged in Figs. 2 and 3, $(k_{zn}c/\omega_{c})^{2}\approx 2.232$. A
sequence of the crossing points $D_{n,n+1}^{l}$ for a given pair of the
intersecting infinite-grating eigenmodes $n=2m_{x}$ and $n+1=2m_{x}+1$ is
finite and runs over the spatial parities of order
$(m_{x}+l,m_{x}-l),\,l=0,1,\ldots,m_{x}-1$. For example, the series
$D_{6,7}^{0},D_{6,7}^{1},D_{6,7}^{2}$ in Fig. 2 contains three crossing points
of spatial parities $(3,3),(4,2),(5,1)$ corresponding to phase accumulations
(25) over the grating sections $d_{1},d_{2}$ equal to
$(3\pi,3\pi),(4\pi,2\pi),(5\pi,\pi)$, respectively.

To demonstrate the BIC within the model of the planar grating waveguide
studied in the present paper, one has to specify the values of eight physical
quantities. They include four parameters of the lamellar grating (the high,
$\varepsilon_{1}$, and low, $\varepsilon_{2}$, dielectric constants of the two
grating sections of a length $d_{1}$ and $d_{2}$, the grating period
$\lambda_{g}=d_{1}+d_{2}$, and the fill factor of the high dielectric constant
section, $\rho=d_{1}/\lambda_{g}=1-d_{2}/\lambda_{g}$), three parameters of
the waveguide (the thickness of the grating layer $L$ and the dielectric
constants of the cover, $\varepsilon^{+}$, and substrate, $\varepsilon^{-}$),
and the resonant frequency $\omega_{m}$. The corresponding design procedure is
explained in sect. IV.

The general conclusion is that the planar grating waveguide supports BICs for
a very wide range of parameters if three simple, physically logical conditions
are met. They are related to (i) the emergence of the parity symmetry breaking
due to mode crossing at the corresponding critical, crossing frequency
$\omega_{c}$, (ii) turning off the radiation-loss channels via undue spatial
Fourier harmonics by making them evanescent, and (iii) choosing the waveguide
parameters ensuring an existence of the solution to the characteristic
equation (8) for the high-Q waveguide-eigenmode frequency close to the
critical frequency $\omega_{c}$ of a mode-crossing point $D_{n,n+1}^{l}$.

In the case of the BIC at the mode-crossing point $D_{2,3}^{0}$, those
conditions are explained in sect. IV. The first of them requires the existence
of crossing of the dispersion curves of two neighboring, $n=2$ and $n=3$,
infinite-grating eigenmodes with the opposite, even and odd respectively,
spatial parities. In fact, the existence of crossings is a generic property of
the solutions to the characteristic equation (22) for the infinite-grating
eigenmodes as is illustrated in Fig. 2. The second condition, stated in Eq.
(34), limits the number of radiation-loss channels to just one provided by the
zeroth spatial Fourier harmonic propagating into the cover and substrate. The
only nontrivial, third condition is the existence of the solution to the
characteristic equation (8),
$\det\,[M(\omega^{\prime}_{m}-i\omega^{\prime\prime}_{m})-1]=0$, for the
waveguide-eigenmode frequency
$\omega_{m}=\omega_{m}^{\prime}-i\omega_{m}^{\prime\prime}$ with a very small
decay rate $\omega_{m}^{\prime\prime}\ll\omega^{\prime}_{m}\approx\omega_{c}$.
The solution can be conveniently traced by plotting the contour plots such as
in Fig. 5. Fortunately, the position of the BIC eigenfrequency is known in
advance since it coincides with the critical frequency $\omega_{c}$ of the
mode-crossing point $D_{2,3}^{0}$ given by Eq. (26).

Thus, most of the eight grating and waveguide parameters remain free.
Obviously, for implementing BICs in more complex photonic waveguides, crystals
or structures the number of free parameters could be even larger.

Anyway, understanding of the BIC formation mechanism described above and the
presence of such free parameters could help targeting various specific
applications of the BIC phenomenon in laser cavities [14, 38], sensors [39,
40, 41, 42, 43], filters [44, 45, 46, 47, 48, 49], polarizers [50], reflectors
[51, 52, 53, 54, 55], photodetectors [56], spectroscopy, precision measurement
and control in photonics and metamaterial fabrication technologies, optical
communications, e.g., low-loss fibres, etc. For a discussion of such
applications, we refer to a recent review [9] and references therein.

## Acknowledgements

We acknowledge the support by the Texas A$\&$M Triads for Transformation
program, Round Four T3 project 1750, for C. B. Reynolds.

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