# Analytical non-Hermitian description of Photonic Crystals
with arbitrary Lateral and Transverse symmetry

X. Letartre1<EMAIL_ADDRESS>S. Mazauric1 S. Cueff1 T. Benyattou1 H
S. Nguyen1,2 P. Viktorovitch1 1Univ Lyon, Ecole Centrale de Lyon, INSA Lyon,
Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, Ecully 69130,
France  2Institut Universitaire de France (IUF)

(August 30, 2024)

###### Abstract

We propose a general theoretical approach to the modelling of complex
dispersion characteristics of leaky optical modes operating in photonic
crystal slab composed of two coupled high-index contrast gratings. Our
analytical model, based on a non-Hermitian Hamiltonian, allows for a unified
description of the wide family of optical modes which may be generated within
uni-dimensional photonic crystals. Our theory stands for a variety of
illustrative examples relating to the manipulation of bound states in the
continuum and exceptional points, and can be used as a powerful enabler for
the discovery of novel photonic species. Finally, as proof-of-concept, we
demonstrate experimentally the formation of a Dirac point at the merging of
three bound states in the continuum that is the most achieved photonic specie
discussed in this work.

##  I  Introduction

Taming losses has always been the main challenge for the development of
photonic devices in modern history: nurturing lasing emission from leaky
channel of an optical cavity [1], minimizing attenuation of guided light in
integrated circuit [2], confining photons to interact strongly with quantum
emitters [3], sharpening photonic resonances for high-sensitivity optical
sensing [4], engineering emission pattern of light-emitting diodes [5], to
cite a few examples. As a matter of fact, most photonic phenomena are dictated
by the complex energy-momentum dispersion characteristic of which the
imaginary part corresponds to system losses and the real part corresponds to
light frequency. The maturity of nano-fabrication technologies nowadays offers
unprecedented degree-of-freedom for dispersion engineering of light via
molding periodic arrangement of materials with different permittivity and
geometry. More importantly, since the past few years, modern photonics has
entered a new paradigm for the research of non-conservative optical systems
exploiting non-Hermiticity notions [6, 7, 8]. In this approach, the complex
energy-momentum dispersions, theoretically described by non-Hermitian
Hamiltonians, reveal unique features with no Hermitian counterparts. The most
famous example is the non-Hermitian extension of topological band theory which
is originally built for lossless band structure from condensed matter physics
[9, 10, 11]. Two distinctive objects of non-Hermitian topology are Bound
states In the Continuum (BICs) and Exceptional Points (EPs). The first one,
BICs, are lossless states resulting from destructive interference of coupled
lossy photonic resonances [12, 13]. They are topological charges pinned at
polarization vortices in momentum space [14, 15]. Topological manipulations
(merging, splitting..) of BIC charges [16] propose unique way to modify
farfield radiation[17], robustness [18] and threshold [19] of light emitting
devices. The second one, EPs, where photonic complex resonances coalesce, are
degeneracy points of non-Hermitian physics [20]. Fundamentally, they take the
role of Dirac points as singularities in non-Hermitian topological band theory
[9]. For devices applications, they offer novel concepts for making optical
sensors with high sensitivities [21, 22] and lasers with intriguing properties
[23, 24, 25]. Interestingly, in light of non-Hermiticity, “text-book”
dispersion characteristics may gain or evolve with new behaviors: half of
photonic band-edges at the centre of the Brillouin zones are BICs in one
dimensional lattice [26], Dirac points transform into EP rings in two
dimensional lattice [27] … Thus strategies used for Hermitian photonics may
suggest fruitful scenarii to investigate and study complex dispersion
engineering in non-Hermitian context.

We have recently proposed an analytical approach of One Dimensional (1D)
Photonic Crystals (PCs) with broken transverse symmetry, which were shown to
provide a new degree of freedom for the design of optical dispersion, and
hence for the control of spatial and spectral characteristics of light [28].
The 1D PCs were composed of two high-index-contrast subwavelength dielectric
gratings of same period and in close near field proximity. We demonstrated
that breaking transverse symmetry opens the way to the generation of any local
density of photonic states from zero (Dirac cone) to infinity (flatband of
zero curvature), as well as any constant density over an adjustable spectral
range for the same photonic band. At this stage, the considered photonic modes
were assumed to operate inside the light cone and were fully preserved from
radiating into the continuum, thus the system were perfectly described by a
Hermitian Hamiltonian.

In the present work, we generalize our theoretical approach to the modelling
of complex dispersion characteristics of leaky modes operating above the light
cone in 1D PCs. Opening access of wave-guided resonances to free space
continuum provides large amount of extra degrees of freedom for mode coupling
engineering. Not only can the two gratings communicate via near field
coupling, but they are also allowed to couple via the propagating radiated
field. A general non-Hermitian Hamiltonian is proposed to capture both
coupling schemes. In particular, we show that the lateral and transverse
symmetry, both described by phase parameters in the non-Hermitian Hamiltonian,
play crucial roles in the radiative coupling processes. Our approach allows
for a unified description of the wide family of optical modes which may be
generated within an arbitrary 1D PC. Remarkably, through a variety of
illustrative examples, we show that our theoretical approach provides a
simplified categorization of these modes, and it is also a powerful enabler
for the discovery of novel photonic species. Finally, as a proof-of-concept,
we demonstrate experimentally the formation of a Dirac point at the merging of
three BICs that is the most achieved photonic specie discussed in this work.

This paper is organized as follows. The next section II is devoted to the
presentation of the non-Hermitian Hamiltonian $H$. This general Hamiltonian is
the back-bone of the present work and makes it possible the derivation of a
physically insightful analytical model of the dispersion characteristics.
Figure 1 provides a schematic view of the variety of coupled grating
structures, which we propose to handle, using our theoretical approach. Since
the number of parameters involved for full generality is rather large, it is
appropriate to implement simplified versions of the general Hamiltonian,
encompassing a wide variety of specific practical cases.

In sections III to VI, we select a few exemplifying typical grating
configurations to assess the validity and illustrate the effectiveness of our
analytical model, whose results are confronted to RCWA (Rigorous Coupled-Wave
Analysis) and FEM (Finite Element Method) numerical simulations 111for all
simulations the gratings are made in silicon (n=3.5) embedded in silica
(n=1.5). This selection is directly related to specific symmetry
characteristics of coupled grating structures : in sections III to V, we
consider aligned grating structures successively with full lateral and
transverse symmetry III , with broken lateral symmetry IV, and finally with
broken transverse symmetry V. The section VI focuses on a specific case of
misaligned grating structures, so called “fish-bone” structure formed by two
misaligned, identical and fully symmetrical gratings. The “fish-bone” is a
unique specific case of 1D periodic structure, where both the lateral and
transverse symmetries are broken.

Figure 1: Schematic view of the different types of coupled grating structures
investigated in this paper: with full lateral and transverse symmetry (a),
with broken lateral symmetry (b), with broken transverse symmetry (c), “fish-
bone” structure, a specific case where both the lateral and transverse
symmetry are broken (d).

We will show that our analytical model is particularly suited as a predictive
qualitative design tool for the production of a great variety of photonic
states. This includes a simplified and accurate classification of BICs, which
have been the matter of intense investigation during recent years [30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 13, 14, 15, 18, 16, 40, 41, 19]. Our model
provides the clues for the generation of BICs, - with adjustable (from very
flat to Dirac like) complex dispersion characteristics - as well as the
generation of such fascinating photonic species as EPs [20] in the complex
dispersion curves; particular attention is paid on the physical impact of
transverse as well as lateral symmetry of the photonic structures. A specific
emphasis will be placed on the demonstration of two unique photonic species,
called thereafter (i) “triple BIC”, exhibiting unprecedented low optical
radiation losses over a very large k-vector range of the Brillouin zone and
(ii) “Dirac point at triple BIC”, originating from the degeneracy of one
triple and one double BICs at the $\Gamma$ point. Since the latter may be
considered as the most achieved photonic specie based on the combination and
interaction of BICs, we present our experimental demonstration of “Dirac point
at triple BIC” in section VII as a proof-of-concept. In the last section VIII,
we point out the great potential of photonic crystals with controlled lateral
and transverse symmetry, as generic building blocks for a range of new
practical applications as well as for original physical studies.

Additional specific developments, complementary to the main text, are provided
in the Appendix.

##  II  Theoretical approach

###  II.1  Non-Hermitian Hamiltonian

Properties of the eigenmodes in PC slabs are dictated by two symmetry
categories [42]: the “transverse symmetry” (or vertical symmetry), defined by
the z-reflection operation $\sigma_{z}$, and the “lateral symmetry” (or in-
plan symmetry), defined by the x-reflection operation $\sigma_{x}$ in the case
of 1D PC (see Appendix A for details of these operators). Leaky optical
resonances in PC slabs are generally described as complex eigenvalues of
2$\times$2 non-Hermitian Hamiltonian; this approach has been successfully
applied to leaky resonances exhibiting peculiar states such as EP and BIC [27,
13]. However this approach, although attractive in terms of simplified
mathematical resolution, provides a rather partial description of the
structures. It ignores explicitly the transverse dimension of the structures.
The 2$\times$2 Hamiltonian cannot therefore accounts for the impact of a
transverse symmetry breaking nor can it describe properly the transverse
radiative properties, for example the occurrence of resonance trapped BICs
[31, 43], named transverse BICs thereafter. To fully consider this symmetry, a
more complete description with non-Hermitian Hamiltonian of higher dimension
is required.

In the case of 1D PC slabs, the transverse dimension and symmetry can be taken
into account by using a description based on two coupled grating structures,
with no defined transverse dimensions (ie bilayer 1D PC [28]). This approach
might also be considered as a simplified description of a real grating
structure, naturally provided with a transverse dimension; however, switching
from one single grating, with no transverse dimension, to a pair of coupled
gratings allows for taking into account number of factors related to the
transverse dimension of a real grating, despite the transverse dimensionless
nature of each of the two individual coupled gratings. The reader may find
complementary explanation of the rationale and phenomenology of our model in
the appendix section B.1.

Figure 2: Schematic presentation of the coupled grating structure (a) and
detailed cross-sectionnal view of the unit cell (b); $\delta$ is the lateral
off-set between the two grating of period $\Lambda$. $\psi$ is the transverse
phase shift. (c) Different coupling mechanisms between the four guided modes
$a_{1\pm}$,$a_{2\pm}$.

A sketch of the two coupled gratings (with common period $\Lambda$) is
presented in Fig.2(a). The gratings are depicted with finite thickness,
although this parameter is not explicitly accounted for in the analytical
model, nor is explicit the geometrical distance between the two structures.
Indeed, these two parameters are merged into a single one, the phase parameter
$\psi$, shown in Fig.2(b). This phase is the key parameter describing the
transverse dimension of the structure, and is further discussed later in the
main text (see II.4) and in the appendix section B.1. The transverse symmetry
of the system of the bilayer PC is partly defined by the characteristics of
each individual grating.

The lateral symmetry of the system is partly set by the lateral symmetries of
the grating unit cells. In addition, both symmetries are also controlled by
the lateral offset $\delta\times\Lambda$ between the two gratings [see
Fig.2(b)]. For the sake of simplifying the writing of the equations of the
analytical model, the origin of the lateral x-coordinate is taken at mid
lateral offset between the two gratings. Referring the gratings to this origin
is straightforward, if both have laterally symmetric unit cells. If this is
not the case, it is appropriate to define the anchorage position of each
grating at the x-centroid of the lateral distribution of the effective
dielectric constant.

In our model, the $\omega(k_{x})$ dispersion characteristics are derived from
different coupling processes undergone by the forward
$\left(a_{1+},a_{2+}\right)$ and backward $\left(a_{1-},a_{2-}\right)$
fundamental zero-order waves of the two planar waveguides of effective
refractive index which are assumed to be single-mode in the spectral range of
interest. The dispersion engineering is focused in the vicinity of the second
Brillouin zone boundary, where second-order diffractive coupling processes
between backward and forward wave components as well as first order coupling
processes with the radiation continuum (in the vicinity of the $\Gamma$ point)
take place. A phenomenological description of coupling processes within the
two gratings as well as of cross coupling processes occurring between them can
be obtained using the coupled mode theory formalism. In the basis formed by
$\left(a_{1+},a_{1-},a_{2+},a_{2-}\right)$, the equations of coupled mode
theory end up in an eigenequation of a $4\times 4$ non-Hermitian Hamiltonian,
given by:

$H=\left(\begin{matrix}H_{11}&H_{12}\\\ H_{21}&H_{22}\\\ \end{matrix}\right),$
(1)

in which $H_{ij}$ are 2$\times$2 matrices, which describe the optical
interactions intra each gratings ($H_{ij=i}$) and inter the two gratings
($H_{ij\neq i}$). The general expressions of $H_{ij}$ are given in the
Appendix section B.2. As pointed out in the introductory section, the
Hamiltonian is not meant to be exploited in its full generality, but instead
to be broken down into a variety of simplified versions suited to specific
practical cases. The parameters chosen to build up the various matrix element
expressions (given in the Appendix section B.2) allow for comprehensive
analytical description of leaky 1D PC slabs. They are categorized and
scrutinized into different coupling strengths and phases, and will be
discussed in the following.

###  II.2  Uncoupled modes

The dispersion characteristics of the uncoupled modes (i.e. $a_{1\pm}$ for the
upper grating and $a_{2\pm}$ for the lower grating) are given by the
pulsations $\omega_{1,2}$ of the two non-corrugated waveguides at
$2\pi/\Lambda$ wave vector, and their corresponding group velocities
$v_{1,2}$.

In the rest of the manuscript, we denote $k$ the variation of the wave vector
in the vicinity of the second Brillouin zone boundary (i.e. $2\pi/\Lambda$).
The dispersion relationship of uncoupled modes can be written as
$\omega_{i\pm}=\omega_{i}\pm v_{i}k$ with $i=1,2$.

###  II.3  Coupling rates

The coupling mechanisms between modes in the basis
$\left(a_{1+},a_{1-},a_{2+},a_{2-}\right)$, as depicted in Fig.2(c), include
the coupling between guided modes from the same membrane (intra-layer
coupling) or from separated ones (inter-layer coupling), and the coupling
between these guided modes with the radiative continuum. The corresponding
coupling rates are:

  * •

For intra-layer diffractive coupling rates (expressed in intra-layer $H_{ii}$
sub-matrices) :

    * –

$\kappa_{1,2}$:diffractive coupling rates between counter-propagating guided
waves in the same grating (1 or 2), due to their own corrugation

    * –

$\gamma_{1,2}$:diffractive coupling rates between guided waves in grating 1
and 2 and the radiation continuum. The two gratings are supposed to be zero
order in the spectral range of interest ($\frac{\Lambda}{\lambda}<1$) around
the $\Gamma$ point.

    * –

$\beta_{1,2}$:evanescent relative weight of grating 2(1) to the total
diffractive coupling rate of grating 1(2). These coefficients quantify the
amount of diffractive coupling rates within one membrane layer, induced
evanescently by the grating of the other membrane layer. It is assumed that
the relative weight is the same, whether it is applied to $\kappa_{1,2}$ or
$\gamma_{1,2}$. When the two gratings are far apart, $\beta_{1,2}$ tend to
zero, and $H_{ii}$ sub-matrices describe intra-layer interactions within each
individual grating, taken alone

  * •

For inter-layer diffractive coupling rates (expressed in inter-layer $H_{ij}$
sub-matrices) :

    * –

$\kappa_{d}$: evanescent coupling rate between co-directional guided waves
from different gratings, with light transfer from grating 1 to grating 2 and
reciprocally.

    * –

$\kappa_{c}$: diffracto-evanescent coupling rate between counter-directional
guided waves from different gratings, with light transfer from one grating to
the other, then switching the propagation direction.

When the two gratings are far apart, $\kappa_{d}$ and $\kappa_{c}$ tend to
zero; however, the two gratings keep on interacting via radiative coupling,
which does not vanish, as expressed by inter-layer $H_{ij}$ sub-matrices. This
interaction is delayed by the light time of flight between the two gratings,
which is quantified by the transverse phase shift $\psi$ introduced in the
upcoming section II.4 and in Fig. 2. This key phase parameter is further
commented extensively in this paper (see, for example, section III.1 and
appendix sections B.1 and B.2).

The basis is chosen so that $\kappa_{1,2}$ and $\gamma_{1,2}$ are real
positive numbers. However, $\kappa_{d}$ and $\kappa_{c}$ are complex in
general, and they are real numbers if the two gratings are identical.

###  II.4  Coupling phases

While most of the coupling rates presented in the previous subsection (II.3)
are real numbers, they are associated with different phases when implemented
in the Hamiltonian construction:

  * •

$\phi=2\pi\delta$: phase parameter corresponding to different diffraction
order phase-shifts occurring in each gratings. It results from their
respective off-set x-coordinate $\pm\frac{\delta}{2}\times\Lambda$ [see Fig.
2(b)]. The phase shifts are, respectively, $\pm\phi$ for the second order
diffractive coupling between forward and backward guided waves (in the
vicinity of the second Brillouin zone boundary), and $\pm\phi/2$ for the first
order diffractive coupling between guided waves and waves of radiation
continuum (in the vicinity of the $\Gamma$ point). $\phi$ is therefore central
in the diffractive coupling processes mentioned above.

  * •

$\psi$: transverse phase shift built up in free continuum by radiated photons,
along a one way trip between the two gratings. $\psi$ includes the phase
shifts accumulated within the gratings and in between; it is related to the
effective transverse optical distance $L_{opt}$ between the wave-guided
resonances. Therefore, $\psi$ is not solely related to the geometrical
distance between the two gratings, but is also impacted by the transverse
field distribution. $\psi$ is a central phenomenological parameter of our
model, which conveys in itself significant amount of physical significance. An
analytical expression of this parameter will be derived later in this paper
(Eq. (7) in section III.1).

  * •

$\left(\varphi_{1+},\varphi_{2+},\varphi_{1-},\varphi_{2-}\right)$: phases
related to the coupling step of forward (+) and backward (-) guided waves to
the radiation continuum in gratings 1, 2. If the grating unit cell is
laterally symmetric $\varphi_{i+}=\varphi_{i-}$. If the two gratings are
identical $\varphi_{1\pm}=\varphi_{2\pm}$.

Phases $\psi$ and $\varphi_{i\pm}$ play a major role in the radiative coupling
processes, which are accounted for by a phenomenological description of the
radiative channel connecting the two gratings. The radiative channel includes
successively 3 steps, (i) the extraction of the wave-guided field from one
grating to free space continuum, (ii) the propagation of the radiated field in
free space continuum and finally (iii) the resonant insertion of the latter
into the other wave-guide grating. Phases $\varphi_{i\pm}$ are involved in the
first and third steps, while the second step is controlled by the phase
$\psi$. Implementation of the different phases in the Hamiltonian construction
is given in the appendix section B.2.

The reader is invited to refer to this section of the appendix for additional
explanation about the derivation of the Hamiltonian matrix elements.

###  II.5  Approximation of the model

In order to establish a useful and tractable Hamiltonian, two main
approximations are made. First of all, as we will see in the upcoming
sections, the diagonalization of $H$ gives rise to an implicit equation for
the complex eigenvalues $\omega=\omega_{R}-i\omega_{I}$ (see e.g. Eqs. (4) to
(6)). Indeed, some parameters in $H$ are functions of the phase $\psi$ (see
e.g. Eqs. (7)). The latter is related to the propagation of photons in the
transverse direction and obviously depends on $\omega$. In the following we
will assume that $\psi$ depends on the sole real part $\omega_{R}$. This
remains valid when the time spent by photons for propagating in the transverse
direction (called $\tau_{rad}$ in the appendix B.1; see also Eq (12b) in
section III.3.2) is much smaller than $1/\omega_{I}$ . As we will focus in
this article on modes with high Q-factors, the condition $\tau_{rad}\ll
1/\omega_{I}$ is always met.

We also assume that the propagation of light in the vertical direction can be
taken into account by the sole dephasing $\psi$ and is not affected by the
interfaces between the dielectric layers. It can be shown that the qualitative
behaviours described in this article remain valid, even though some
corrections must be made on the eigenvalues, mainly on their imaginary parts.
These approximations are more extensively discussed in the appendix (section
B.1).

##  III  Symmetrical and aligned gratings

Figure 3: Schematic presentation of the fully symmetrical coupled grating
structure, along both the transverse and lateral directions.

In this section, we will discuss and analyze in detail the configuration of
coupled gratings structure which is fully symmetrical along both the
transverse and lateral directions (see Fig. 3). This is the simplest
configuration, yet revealing to be very generic in terms of complex dispersion
characteristics, as demonstrated in the following.

###  III.1  Analytical expression of Hamiltonian

The general Hamiltonian is considerably simplified; first, the phase
parameters, except for the phase $\psi$, are no longer relevant: $\phi=0$,
since the two gratings are aligned, and $\varphi_{1\pm}=\varphi_{2\pm}$,
together with $\varphi_{1,2+}=\varphi_{1,2-}$, since they are identical and
formed with laterally symmetric unit cells. Second, the coupling coefficients
are identical, $\kappa_{1,2}=\kappa$, $\gamma_{1,2}=\gamma$,
$\beta_{1,2}=\beta$, together with the kinetic parameters,
$\omega_{1,2}=\omega_{0}$ and $v_{1,2}=v_{0}$. Finally, the coupling
parameters $\kappa_{c,d}$ are real. As a result the Hamiltonian can be written
along the following simplified version:

$H(k)=\left(\omega_{0}-iG\right)\mathbb{1}_{4}+\left(\begin{matrix}v_{0}k&K-iG&\kappa_{d}-iGe^{i\psi}&\kappa_{c}-iGe^{i\psi}\\\
K-iG&-v_{0}k&\kappa_{c}-iGe^{i\psi}&\kappa_{d}-iGe^{i\psi}\\\
\kappa_{d}-iGe^{i\psi}&\kappa_{c}-iGe^{i\psi}&v_{0}k&K-iG\\\
\kappa_{c}-iGe^{i\psi}&\kappa_{d}-iGe^{i\psi}&K-iG&-v_{0}k\\\
\end{matrix}\right),$ (2)

where $K=\kappa\left(1+\beta\right)$ and
$G=\gamma\left(1+\sqrt{\beta}\right)^{2}$. The complex eigenvalues of Eq. 2
can be expressed in a general form:

$\omega(k)=\omega_{R}(k)-i\omega_{I}(k).$ (3)

Here $\omega_{R}(k)$ and $\omega_{I}(k)$ are given by the real and imaginary
part of $\omega(k)$ and represent the optical frequency and the radiative loss
respectively. Explicitly, the diagonalization of $H$ provides four complex
eigenvalues which can be regrouped into two couples corresponding to two
opposite parities $\eta=\pm 1$ of the transverse symmetry:

$\omega_{\eta}^{\pm}=\omega_{0}+\eta\kappa_{d}-iG_{\psi}\pm\frac{S(k)}{2}.$
(4)

Here the “complex radiation rate” $G_{\psi}$ is given by:

$G_{\psi}=G\left(1+\eta e^{i\psi}\right),$ (5)

and the “complex splitting” of each couple is determined by:

$S(k)=2\sqrt{v_{0}^{2}k^{2}+\left(K+\eta\kappa_{c}-iG_{\psi}\right)^{2}}.$ (6)

The first couple of Eq. (4), given by $\eta=1$, corresponds to two
fundamental/even (along the transverse direction) modes. The second one, given
by $\eta=-1$, corresponds to two excited/odd (along the transverse direction)
modes. We note that the parameter $\kappa_{d}$ is negative, since the
fundamental guided mode of the coupled membranes is obviously even. The even
and odd branches of the complex eigenvalues ignore each other, since the
corresponding modes have opposite parity. Therefore, possible crossing of
these branches is not avoided.

As pointed out in the previous section, the phase $\psi$ is central in our
model; this parameter expresses the transverse phase shift built up in free
continuum by radiated photons within the gratings and in between, along a one
way trip between the two gratings. It is related to the effective transverse
optical distance $L_{opt}$ between the wave-guided resonances. The analytical
expression of the phase $\psi$ is thus simply given by
$\psi=\frac{k_{\perp}}{\overline{n}}L_{opt}$ where $k_{\perp}$ is the
transverse component of the wave-vector and $\overline{n}$ is the average
refractive index perceived by radiated photons along their trip in the
continuum. Neglecting the reflection at the membrane interfaces (see sections
II.5 and B.1 ), it can be written:

$\psi(k)=\sqrt{\left[\frac{\omega_{R}(k)}{c}\right]^{2}-\left(\frac{k}{\overline{n}}\right)^{2}}L_{opt},$
(7)

where $c$ is the light celerity in vacuum.

The phase $\psi$ can be controlled by adjusting the transverse optical
distance $L_{opt}$ between the wave-guided resonances in each membrane, which
is set mainly by the thickness $h$ of the gratings, and by the distance $D$
between them:

$L_{opt}\approx n.h+n_{0}.D.$ (8)

Here $n$ n is the effective transverse refractive index of the grating and
$n_{0}$ is the refractive index of the spacer.

We point out that the real part of the eigenvalue $\omega$ depends on $\psi$,
through Eq. (4), as well as $\psi$ depends on $\omega$ through Eq. (7).
Therefore different specific values of $\psi$ apply to the Eq. (4), except
when degeneracy of $\omega$ occurs between two eigenvalues. As it is
illustrated in the following discussions of this section III, Eq. (7) is
particularly relevant when it comes to provide a quantitative description of
the real eigenvalues for symmetrical structures, featuring such distinctive
characteristics as, for example, double transverse BICs (section III.3.2) or
Dirac point at triple BIC (section III.3.4). It provides also a faith-full
qualitative prediction of the behaviour of imaginary eigenvalues, especially
regarding the variations of the imaginary eigenvalues versus momentum $k$
around the $\Gamma$ point.

Fully analytical resolution of this system of equations is accessible at
(close to) the $\Gamma$ point, while numerical assistance is required further
away, as this will be illustrated in the following. Numerical resolution
procedure will be particularly used, whenever it comes to validate our model
against results of RCWA/FEM numerical simulations.

###  III.2  General features of the complex eigenvalues

####  III.2.1 At the $\Gamma$ point: Lateral BIC

At the $\Gamma$ point (i.e. $k=0$), the expressions given by Eqs. (4),(5) and
(6) are greatly simplified. One can easily show that $\omega_{\eta}^{-}(k=0)$
for both parities have no imaginary part. They are thus two “dark modes” which
are free of losses and are systematically observed at the $\Gamma$ point in
structures being symmetric along the lateral direction. Both of them belong to
the category of “symmetry protected BICs” [44, 13]. This protection arises
from the symmetry of the mode which turns to be odd with respect to the unit
cell of the grating which is symmetrical along the lateral direction. As a
result it cannot radiate at the $\Gamma$ point since coupling to plane waves
is prevented, unlike the case of the bright mode which symmetry is even. This
type of BIC has been widely documented in the literature [44, 13]. In this
work, we will refer to it as “lateral BICs”, since it is controlled by the
lateral symmetry of the grating structure. This point will be addressed in
more details in the section IV, where it is shown that breaking the lateral
symmetry prevents the formation of lateral BICs at the $\Gamma$ point.

On the other hand, Eq. (4) at $k=0$ indicates that $\omega_{\eta}^{+}(k=0)$
for both parities are generally “bright modes” with non-zero radiative losses
(i.e. imaginary part of the eigenvalues). From now on we denote « dark
branches » the two branches (one per parity) of dispersion characteristics
showing a lateral BIC at the $\Gamma$ point and “bright branches”, the other
branches. Note that, for the latter, one may observe more than one branch per
parity, since they are solutions of the system of two equations (4) and (7),
which are not unique in general. This is particularly the case when $L_{opt}$
is significantly larger than the wavelength, as further discussed in section
III.3.2 and Fig. 7.

####  III.2.2 Out of the $\Gamma$ point: Tranverse BIC

Equations (4) to (6) indicate that the four eigenmodes are bright out of the
$\Gamma$ point. Lateral symmetry is broken for $k\neq 0$, and no more room is
left to lateral BICs. It can be shown, from expansion of equations around the
$\Gamma$ point, that the imaginary part of the dispersion characteristics of
the dark branches varies like $k^{2}$ (see appendix section C.2) .

However, for any $k$ value and for each of the four eigenmodes (but not for
all of them at a time, in a given structure), optical losses are inhibited
when $\psi=\pi(0)$ (mod $2\pi$) for the two even (odd) branches: the bright
mode turns into a BIC, occurring accidentally, as a result of destructive
interferences in the transverse direction. Occurrence of a BIC may therefore
take place on any of the two couples of above denoted dark/ bright branches of
even/odd modes. It can be shown from Eqs. (4) to (6) that the imaginary part
of the dispersion characteristics varies like $(k-k_{BIC})^{2}$ around the
wave-vector $k_{BIC}$ corresponding to the occurrence of the transverse BIC
(see appendix section C.3 ). This type of BIC may occur for specific
geometrical / optical characteristics of the photonic structure; it is also
called resonance trapped BIC [31, 43]. We classify this BIC within a common
family which we name transverse BICs, since it is controlled by the transverse
symmetry of the grating structure. This point will be addressed in more
details in section V, where it is shown that breaking the transverse symmetry
prevents the formation of transverse BICs at and apart from the $\Gamma$
point.

We may point out at this stage that the simplified classification of BICs into
the two categories, transverse and lateral BICs, constitutes an attractive
outcome of our model; we will show indeed that this classification naturally
applies to a variety of BICs occurring in the generic structure formed by two
coupled gratings and revealed by our model, in agreement with results of
numerical simulation. Even more, we will show that these two basic BIC
building blocks can be combined and result in structures featuring remarkable
characteristics (in the subsection III.3 and appendix C).

All the theoretical predictions of the model described above are fully
confirmed by RCWA simulations of a fully symmetrical coupled grating structure
as reported in Fig. 4(a) ($\Lambda=1\ \mu m,\ h=0.3\ \mu m,\ L=0.8\ \mu m$ and
$D=0.29\ \mu m$). Indeed, from the angle-resolved reflectivity spectra, we
distinguish four resonances corresponding to the four branches previously
discussed. These are Fano resonances resulting from the coupling between
incident plane-wave and the Bloch resonances. Since BICs correspond to a
perfect uncoupling of Bloch resonances from the radiative continuum, one may
identify easily BICs at local vanishings of these Fano resonances. We observe
two lateral BICs at the $\Gamma$ point of A and B branches that are the two
dark branches; and a transverse BIC in oblique direction on the bright branch
C. More details are written in the caption of Fig. 4. Finally, Fig. 4(b)
depicts the quality factor of the bright branch C in the vicinity of the
transverse BIC, extracted from FEM simulation. We observe indeed a quadratic
law as predicted by the analytical theory.

Figure 4: (a) Angle-resolved reflectivity spectra obtained by RCWA
simulations, showing the two couples of dark/bright branches of even/odd
modes. The two systematic lateral BICs, even (A) and odd (B), occurring at the
$\Gamma$ point can be observed. A transverse BIC (black arrow) is also
observed in oblique direction on the bright branch of the even mode (C) ; (b)
Log-log plot of the quality factor of the branch C in the vicinity of the
wave-vector $k_{BIC}$ corresponding to the occurrence of the transverse BIC in
oblique direction; the imaginary part of the eigenfrequency varies like
$(k-k_{BIC})^{2}$. The black squares are numerical results obtained by FEM
simulations. The red line is a polynomial fit. The parameters are: $\Lambda=1\
\mu m,\ h=0.3\ \mu m,\ L=0.78\ \mu m$ and $D=0.29\ \mu m$.

###  III.3  Some selective cases with distinctive characteristics

####  III.3.1  Band Inversion and Double Exceptional Point

Further analytical treatment of the complex eigenvalues of even modes in the
vicinity of the $\Gamma$ point or for small values of $k$, where they can be
expanded, is given in section C.2 of the appendix. Results can be summed up as
follows. In general, the complex eigenfrequencies are a quadratic function of
$k$ in the vicinity of the $\Gamma$ point. The real curvatures change in sign
when the quantity $K+\kappa_{c}+G\sin(\psi_{0})$, where $\psi_{0}=\psi(k=0)$,
changes in sign as well, strictly speaking for the dark branch, and in first
approximation for the bright branch. For $K+\kappa_{c}+G\sin(\psi_{0})>0$, it
is observed that the upper (lower) branch is bright (dark) with upward
(downward) curvature and the other way around for
$K+\kappa_{c}+G\sin(\psi_{0})<0$. This band inversion behaviour, referred to
as band flip in [26], is shown in RCWA simulations of Fig. 5. For
$K+\kappa_{c}+G\sin(\psi_{0})=0$, degeneracy of the real eigenvalue of the
dark and bright branches occurs, which means that the overall diffractive
coupling processes between wave-guided resonances cancel, and the dark branch
is flat at the $\Gamma$ point (see Eq. (82) of the appendix (C.2)). RCWA
simulations show that the principal controlling parameter of the band
inversion phenomena is the filling factor $FF=\frac{L}{\Lambda}$ of the
grating structure. Our model does not include explicitly an ab initio
parameter to account for FF; the latter is implicitly included through the
factor $K+\kappa_{c}+G\sin(\psi_{0})$, which expresses the overall diffractive
processes encountered by wave-guided resonances. The reader may find
complementary physical insights in the appendix C.6, which details the
practical design rules of a Dirac point at triple BIC.

Conditions for full degeneracy of the complex eigenvalues of the dark and
bright branches can be met out of the $\Gamma$ point at double exceptional
point for $k=k_{EP}$. Section C.4 of the appendix provides a detailed account
of this occurrence, based on our analytical model.

Figure 5: Angle-resolved reflectivity spectra obtained by RCWA simulations,
showing band inversion phenomenon resulting from a sign inversion of the
factor $K+\kappa_{c}+G\sin(\psi_{0})$. At the onset of the band inversion, the
dark branch is flat and degeneracy occurs between the real eigenvalues of the
dark and the bright branches. Other than the lateral BIC at $k=0$ of the dark
branch, we distinguish also two transverse BIC pinned at $\sim\pm 2$ degrees
in the upper branch of both configurations. The band inversion is obtained by
varying the filling factor: (a) FF=0.35, (b) FF=0.36 and (c) FF=0.37. The
other parameters are $\Lambda=1\mu m$, $h=0.25\mu m$ and $D=0.3\mu m$.

####  III.3.2  Double transverse BIC

Let us concentrate on even modes, given that conclusions are qualitatively
similar for odd modes. At the $\Gamma$ point of the bright branch, the mode is
generally bright except when conditions are met for the formation of a
transverse BIC, that is for $\psi=\pi$ (mod $2\pi)$, as explained in the
previous section. This particular circumstance corresponds to the merging of
two ordinary transverse BICs belonging to the bright branch and occurring at
$\pm k_{BIC}$ vectors, when $k_{BIC}\rightarrow 0$. We call this particular
transverse BIC, double transverse BIC (see appendix sections C.2 and C.3 for
complementary detailed analysis). The generation of a double transverse BIC in
fully symmetrical coupled gratings is illustrated in RCWA simulation of Fig. 6
.

Figure 6: (a) Angle-resolved reflectivity spectra obtained by RCWA
simulations, showing the occurrence of a double transverse BIC at the $\Gamma$
point (black arrow) of the bright branch (here dielectric branch). The
ubiquitous lateral BIC shows up at the $\Gamma$ point of the dark branch. (b)
Log-log plot of the quality factor of the bright branch versus $k$ vector in
the vicinity of the $\Gamma$ point. Black squares are numerical results
obtained by FEM simulations. Red line is a polynomial fit. The slope $-3.95$
of the fit confirms the results of our analytical model anticipating
variations of the imaginary eigenvalues like $k^{4}$. The parameters are:
$\Lambda=1\ \mu m,\ h=0.25\ \mu m,\ L=0.3\ \mu m$ and $D=0.3\ \mu m$.

Close to the $\Gamma$ point (small $k$), it is possible to derive the complex
dispersion characteristic of this double transverse BIC using Eqs. (4)- (6)
where $\eta=1$ and $\psi(k=0)\equiv\psi_{0}=\pi$ (mod $2\pi)$. The expanded
real part $\omega_{R}$ of the complex dispersion characteristic close to the
$\Gamma$ point is given by Eq. (78) of appendix C.2:

$\omega_{R}(k)=\omega_{R_{0}}+C_{R}\frac{k^{2}}{2},$ (9)

where

$\omega_{R_{0}}=\omega_{R}(k=0)=\omega_{0}+\kappa_{d}+K+\kappa_{c},$ (10)

and $C_{R}$ is the curvature of the real dispersion characteristic at the
$\Gamma$ point (i.e. the second derivative of $\omega_{R}(k)$). The curvature
$C_{R}$ can be written:

$C_{R}=\frac{\tau_{wg}C_{wg}+\tau_{rad}C_{rad}}{\tau_{wg}+\tau_{rad}},$ (11)

where:

$\displaystyle\tau_{wg}$ $\displaystyle=\frac{1}{2G}$ (12a)
$\displaystyle\tau_{rad}$ $\displaystyle=\frac{L_{opt}}{c}$ (12b)

and:

$\displaystyle C_{wg}$ $\displaystyle=\frac{v^{2}}{K+\kappa_{c}}$ (13a)
$\displaystyle C_{rad}$
$\displaystyle=\frac{\left(c/n\right)^{2}}{\omega_{R_{0}}}.$ (13b)

$\tau_{wg}$ is the average lifetime of photons in the wave-guided state before
being emitted into the continuum and $\tau_{rad}$ is the average lifetime of
photons in the radiated state during one way trip between the two gratings.
$C_{wg}$ and $C_{rad}$ are the guided and radiated curvature respectively,
their contribution to the total curvature is weighted by the relative time
spent by photons in the wave-guided state, i.e.
$\tau_{wg}/(\tau_{wg}+\tau_{rad})$, and radiated state, i.e.
$\tau_{rad}/(\tau_{wg}+\tau_{rad})$, respectively. The expression of the
curvature is an eloquent illustration of the hybrid character of this double
transverse BIC, which is altogether guided and radiated [45].

Real dispersion characteristic of double transverse BIC can be made very flat
if the condition $C_{R}=0$ is met. This is shown to be achievable in the
supplemental appendix (section C.5), provided that $L_{opt}$ is significantly
larger than $\lambda$, that is for relatively thick structures. As shown in
Fig. 7, RCWA simulation demonstrates indeed that very flat transverse BIC with
zero curvature can be achieved provided that the thickness of the structure
exceeds a few times $\lambda$. The double transverse BIC with flat band (zero
curvature at the $\Gamma$ point ) is observed for a rather thick structure:
the gratings are separated by a $13\lambda$ thick spacer.

In that case, as $L_{opt}$ varies rapidly with the wavelength, solving
equations (4) and (7) for the sole even modes gives rise to more than 2
solutions, as this is confirmed by simulations in Fig. 7. All these modes are
quasi-Fabry-Pérot modes where the photons share their life time between the
guided state in the corrugated membranes (propagation in the lateral
direction, see Eq. (12a)) which act as resonant reflectors, and the radiated
state in the spacer (propagation in the transverse direction, see Eq. (12b)).

Figure 7: (a) Angle-resolved reflectivity spectra obtained by RCWA
simulations, showing a flat double BIC with zero curvature of the real
eigenvalue at the $\Gamma$ point (red arrow). (b) FEM simulations of the real
eigenvalues of the three branches A, B, C. The real curvature changes from
positive value (A) to negative value (C) and vanishes for the intermediate
plot (B). Common origin of the energy has been chosen at $\Gamma$ point for
the three bands to improve readability of the curvature changes. (c) The real
part of the eigenvalue of the flat double BIC is zoomed around the $\Gamma$
point: it shows $k^{4}$ variation, thus confirming a zero curvature. The
optical thickness of the spacer $D$ is about 13$\lambda$.The other parameters
are $\Lambda=1\ \mu m,\ h=0.25\ \mu m,\ L=0.3\ \mu m$ and $D=16.4423\ \mu m$.

As recalled in the introduction, achieving flat band (or zero band curvature)
conditions of the real dispersion characteristic provides slow light of zero
group velocity with high density of state (DOS) for a broad range of the
Brillouin zone, which is an attractive feature for a variety of applications.
We propose here a specific approach for that purpose, based on the interplay
between radiated and guided hybridized components of a double transverse BIC.

The imaginary part $\omega_{I}$ of the complex dispersion characteristics of
the transverse BIC close to the $\Gamma$ point can also be obtained from
expansion of Eqs. (4)-(7) where $\eta=1$. The general expression of the
imaginary part $C_{I}$ of the complex curvature given by equation 80 of
appendix C.2, for the even bright branch, indicates that $C_{I}$ is null when
$\psi_{0}=\pi$, that is the case of a double transverse BIC. Expansion of
$\omega_{I}$ is therefore required to be extended up to the order 4. This
means that $\omega_{I}$, which is null at the $\Gamma$ point, increases like
$k^{4}$. In other word the imaginary part of the complex dispersion
characteristics of the double transverse BIC (belonging to the bright branch)
is flat close to the $\Gamma$ point. This is confirmed by FEM simulations (see
Fig. 6), showing the quality factor of the double transverse BIC to vary like
$k^{-4}$ in the vicinity of the $\Gamma$ point.

This result is consistent with our interpretation of the transverse double BIC
at the $\Gamma$ point as resulting from the merging of two transverse BICs
occurring off the $\Gamma$ point at $k=\pm k_{BIC}$, when $k_{BIC}$ tends to
zero momentum. We remind that the imaginary part of the dispersion
characteristics varies like $(k-k_{BIC})^{2}$ for the two transverse BICs with
$\pm k_{BIC}$ momentum. For the merging BIC design, that is $k_{BIC}=0$, the
imaginary part varies like $(k-k_{BIC})^{2}(k-k_{BIC})^{2}\sim k^{4}$ (see
appendix C.3 and equation (87) for complementary analysis). This transverse
BIC merging scenario, derived from analytical model, is in line with recent
interpretation of radiative losses of BICs based on their topological nature
[46, 40, 18].

####  III.3.3  Triple BIC

At the $\Gamma$ point of the dark branch, the mode is systematically dark
since it is protected by the lateral symmetry. If, in addition, conditions are
met for the formation of a double transverse BIC, the eigenmode turns to be
doubly protected. We call triple BIC the mode generated in these particular
conditions. This specific accidental circumstance corresponds to the merging
of two ordinary transverse BICs belonging to the dark branch and occurring at
$\pm k_{BIC}$ vectors, when $k_{BIC}$ approaches zero, with the lateral BIC
occurring systematically at the $\Gamma$ point. Close to the $\Gamma$ point
(small $k$), it is also possible to derive the complex dispersion
characteristic of the triple BIC from expansion of Eqs. (4)-(7).

The expanded real part $\omega_{R}$ of the complex dispersion characteristics
close to the $\Gamma$ point is given by Eqs. (81) and (82) of the appendix
C.2, for $\psi(k=0)\equiv\psi_{0}=\pi$ (mod $2\pi)$:

$\omega_{R}(k)=\omega_{R_{0}}-\frac{v^{2}k^{2}}{2(K+\kappa_{c})},$ (14)

where, for the dark branch,
$\omega_{R_{0}}=\omega_{0}+\kappa_{d}-K-\kappa_{c}$.

We may note the fully wave-guiding nature of the real dispersion
characteristics, which expresses the efficient protection of the triple BIC
from the surrounding continuum. This is clearly unlike the case of the double
transverse BIC issued from the bright branch, which hybrid character, both
wave-guided and radiated, has been pointed out before in Eq. (11) and (95).

The imaginary part $\omega_{I}$ of the complex dispersion characteristics of
the triple BIC close to the $\Gamma$ point can also be obtained from expansion
of Eqs. (4)-(7). Equation 83 of the appendix C.2 indicates that the second
derivative $d^{2}\omega_{I}/dk^{2}$ is null when $\psi_{0}=\pi$ (mod $2\pi$).
It can be shown that expansion of $\omega_{I}$ is required to be extended up
to the order 6 (see appendix C.3 for complementary analysis). This means that
$\omega_{I}$, which is null at the $\Gamma$ point, increases like $k^{6}$. In
other word the imaginary part of the complex dispersion characteristics of the
triple BIC is ultra-flat close to the $\Gamma$ point. This also means that the
optical losses of a structure provided with a triple BIC, which are strictly
null at the $\Gamma$ point, remain very low off the $\Gamma$ point, up to
large lateral $k$ momentum. These predictions of the model are confirmed by
RCWA and FEM simulations, illustrated in Fig. 8. They indicate that, when a
triple BIC is generated, the quality factor of the resonance retains very
large values (around $10^{4}$) up to large momentum angle of the $\Gamma$
point ($\pm 10^{\circ})$. FEM data show the quality factor of the triple BIC
structure to vary like $k^{-6}$ in the vicinity of the $\Gamma$ point.

Figure 8:  (a) Angle-resolved reflectivity spectra obtained by RCWA
simulations, showing the occurrence of a triple BIC at the $\Gamma$ point
(black arrow) of the dark branch (here dielectric). (b) log-log plot of the
quality factor of the same branch versus $k$ vector in the vicinity of the
$\Gamma$ point. Black squares are numerical results obtained by FEM
simulations. Red line is a polynomial fit. The slope $-5.93$ of the fit
confirms the results of our analytical model anticipating variations of the
imaginary eigenvalues like $k^{6}$. Q factors larger than $10^{12}$ should not
be taken into account because they lies beyond the FEM accuracy. The
parameters are $\Lambda=0.55\ \mu m,\ h=0.279\ \mu m,\ L=0.275\ \mu m$ and
$D=0.$

This result is consistent with our interpretation of the triple BIC at the
$\Gamma$ point as resulting from the merging of two transverse BICs occurring
off the $\Gamma$ point at $k=\pm k_{BIC}$ momentum, with a lateral BIC at the
$\Gamma$ point, when $k_{BIC}$ tends to zero momentum. For the triple BIC
design, that is $k_{BIC}=0$, the imaginary part varies like
$(k-k_{BIC})^{2}(k-k_{BIC})^{2}k^{2}\sim k^{6}$, given that the lateral BIC
imaginary part varies like $k^{2}$ (see complementary analysis in the appendix
section C.3 and Eq. (88)).

####  III.3.4  Dirac point at triple BIC

The Dirac point at triple BIC is the central and most outstanding character
among all distinctive photonic species described in previous sections, which
combines all the remarkable characteristics at a time. We again concentrate on
the two even modes, given that conclusions are qualitatively similar for odd
modes. We remind that one of the two even modes is systematically dark at the
$\Gamma$ point, being lateral symmetry protected (lateral BIC), while the
other is bright in general. Let us start with the double exceptional point
(see subsection III.3.1) which is further described in the supplemental
appendix (section C.4). We remind that a double exceptional point is formed
when conditions for full degeneracy of the complex eigenvalues of the dark and
bright branches is achieved [27]. It is shown in section C.4 of the appendix
(Eqs. (91) and (92)) that the double exceptional points show up in the
dispersion characteristics at the wave vector $k=k_{EP}=\pm
G(1+\cos\psi(k_{EP}))/v$, with the additional condition
$K+\kappa_{c}+G\sin\psi(k_{EP})=0$. If $K+\kappa_{c}$ turns to be null,
$\psi(k_{EP})=\pi$ (mod $2\pi$) (for the even mode) and $k=k_{EP}=0$. The two
exceptional points merge at the $\Gamma$ point. It also results obviously that
$\psi_{0}=\psi(k=0)=\pi$ (mod $2\pi$). Consequently, the bright mode at the
$\Gamma$ point turns into a double transverse BIC, which therefore degenerate
with a lateral BIC. The double transverse BIC and the lateral BIC having the
same real eigenfrequency, we may conclude that the lateral BIC is also a
triple BIC, as described in the previous section, since it benefits from both
lateral and transverse protections (the condition $\psi_{0}=\pi$ (mod $2\pi$)
holds also for it). While the double transverse BIC and the triple BIC are
degenerated at the $\Gamma$ point, the regime of strong coupling between the
two eigenmodes applies off the $\Gamma$ point, as soon as the lateral $k$
momentum differs from 0 222This is in contrast with the behaviour of a double
exceptional point, where the weak coupling regime occurs first for $k$
momentum below $k_{EP}$, followed by the strong coupling regime, for $k$
momentum exceeding $k_{EP}$, where the real dispersion characteristics of the
dark and bright branches are linear ($\pm vk$) and where both branches equally
share the loss rate (see appendix C.4). As a result, the real dispersion
characteristics follow a Dirac like linear variation. We call therefore Dirac
point at triple BIC this specific type of mode, since it features zero-index
behaviour at the $\Gamma$ point, with very weak losses, owing to a fine
interplay between transverse and lateral BICs. Dirac point at triple BIC
characteristics are precisely accounted for by our analytical model, as
further detailed below.

The expanded real part $\omega_{R}$ of the complex dispersion characteristics
close to the $\Gamma$ point is derived from Eqs. (4)-(7), with
$\psi=\psi_{0}=\pi$ (mod $2\pi$) at the $\Gamma$ point and $K+\kappa_{c}=0$.
It yields the linear relation below:

$\omega_{R}(k)=\omega_{0}+\kappa_{d}\pm v_{D}k,$ (15)

where:

$v_{D}=\frac{v}{\sqrt{1+\frac{\tau_{rad}}{\tau_{wg}}}}.$ (16)

We remind that $\tau_{wg}$ and $\tau_{rad}$, given by Eqs. (12a) and (12b)
respectively, are the average lifetime of photons in the wave-guided state
before being emitted into the continuum, and the radiated state during a one
way trip between the two gratings. The dispersion characteristics include an
upper Dirac branch ($-v_{D}k$ when $k<0$ and $+v_{D}k$ when $k>0$) and a lower
Dirac branch ($+v_{D}k$ when $k<0$ and $-v_{D}k$ when $k>0$); both branches
degenerate at the $\Gamma$ point, where the curvature is infinite.

The expanded imaginary part $\omega_{I}$ of the complex dispersion
characteristics close to the $\Gamma$ point can be again derived from Eqs.
(4)-(7), with $\psi=\psi_{0}=\pi$ (mod $2\pi$) at the $\Gamma$ point and
$K+\kappa_{c}=0$. If $\tau_{wg}\gg\tau_{rad}$, it can be shown that for both
Dirac branches, which share the same optical loss rate in strong coupling
regime:

$\omega_{I}(k)=-\frac{1}{2}G\left(\frac{L_{opt}}{n_{G}}\right)^{2}k^{2},$ (17)

where $n_{G}$ is the group index of the non-corrugated grating waveguide. This
relation expresses that the wave-guided photons retain the protecting umbrella
of the triple BIC, within a domain of the reciprocal space extending over
$\Delta k\sim n_{G}/L_{opt}$, around the $\Gamma$ point. This is another
manifestation of the interplay between radiated and wave-guided propagations.

In summary, while the real part of dispersion characteristics for the Dirac
point at triple BIC is a linear function of $k$, the imaginary part is a
quadratic function of $k$, which means that complex Dirac point at triple BIC
characteristics retain low optical losses, for $k$ momentum around the
$\Gamma$ point. Eq. (17) shows that optical losses are minimized, if the
optical losses of each individual grating and if the transit path of photons
in free continuum are kept low. These predictions of the model are confirmed
by RCWA and FEM simulations, as illustrated in Fig. 9.

Figure 9:  (a) Angle-resolved reflectivity spectra obtained by RCWA
simulations, showing the occurrence of a Dirac point at triple BIC at the
$\Gamma$ point (black arrow). (b) log-log plot of the quality factor of the
two branches A and B versus $k$ vector in the vicinity of the $\Gamma$ point.
The blue squares and red circles are numerical results obtained by FEM
simulations. The black line is a polynomial fit. The slope $-2$ of the fit
confirms the results of our analytical model anticipating variations of the
imaginary eigenvalues like $k^{2}$. The parameters are $\Lambda=1\ \mu m,\
h=0.3593\ \mu m,\ L=0.2735\ \mu m$ and $D=0$.

The practical design rules of a Dirac point at a triple BIC are given in the
appendix C.6.

Figure 10: Synthetic overview of the variety of BIC configurations, which are
achievable using ordinary fully symmetrical coupled 1D PC grating structures,
and the scenarii for their generation. Here *B are lateral BICs, *B are
transverse BICs, *B are triple BIC at Dirac point, *B are EPs, *B indicate
opening/closing of real gaps, *B indicate merging/splitting of tranverse BICs.

###  III.4  Synthetic Summary

The diagrams of Fig. 10 thereafter summarize the principal results and
conclusions of this section and of the related appendix C. The basic BIC
building blocks are respectively the lateral BICs (represented by the dark
dots) and the transverse BICs (yellow dots). The lateral BICs originate from
the lateral symmetry protection, and occur systematically at the $\Gamma$
point on the dark dispersion branch (denoted as “Dark”, in the figure, against
“Bright”, which stands for the bright branch, free from lateral BICs). The
transverse BICs are “accidental” and may occur anywhere on both the dark and
bright branches: their specific occurrences are set by specific opto-
geometrical characteristics of the structure. The green dots represent the
exceptional points, where the complex eigenvalues are degenerated. The
successive steps of the different scenarii are indicated by the double arrows.
The orange double arrows are related to merging-splitting processes of the
BICs. The yellow double dots stand for double BICs, originating from the
merging at the $\Gamma$ point of two transverse BICs on the bright branch.
Triple BICs (yellow-black-yellow triple dots) may also occur from the merging
at the $\Gamma$ point of two transverse BICs with a lateral BIC on the dark
branch: these triple BICs benefit from both lateral and transverse
protections, which help them retaining very low optical losses away from the
$\Gamma$ point. The blue double arrows point upon band inversion phenomena,
where a dark branch transmute into a bright branch and vice versa.
Transmutation event manifests itself by an intermediate state where bands
coalesce and show flat dispersion characteristics. Finally, the Dirac
characteristic can be viewed as the central character of the plot: it is a
particular double exceptional point, where the two exceptional points merge at
the $\Gamma$ point and convert into two transverse BICs, which merge in their
turn at the $\Gamma$ point with a lateral BIC. In addition, the bright and
dark branches degenerate at the $\Gamma$ point. In brief, the Dirac point
coincides with two degenerated triple BICs, represented by three red dots.
Strong coupling is triggered between the two branches of the Dirac
characteristic, right at the $\Gamma$ point, as soon as the lateral $k$ vector
deviates from zero, resulting in linear dispersion characteristics. At last,
we may recall that the double transverse BIC at the $\Gamma$ point of the
bright branch can be made flat (cancellation of the curvature); this is also
the case of the lateral BIC at the $\Gamma$ point of the dark branch, in the
configuration leading to the band inversion.

##  IV  LATERAL SYMMETRY BREAKING: ALIGNED IDENTICAL GRATINGS WITH LATERAL
BROKEN SYMMETRY

We propose now to analyse in detail the case of a coupled grating structure
which is symmetrical along transverse direction and which presents a lateral
symmetry breaking (See Fig. 11). We concentrate on the sole effect on
dispersion characteristics of breaking the lateral symmetry of the unit cell.

###  IV.1  Analytical expression of Hamiltonian

The general Hamiltonian is similar to that derived for the fully symmetrical
structure, except for the phases
$\varphi_{1+},\varphi_{2+},\varphi_{1-},\varphi_{2-}$ related to the coupling
step of forward (+) and backward (-) guided waves to the radiation continuum
in gratings. The gratings being identical, we have
$\varphi_{1\pm}=\varphi_{2\pm}=\varphi_{\pm}$, but the lateral symmetry
breaking of the unit cell leads to $\varphi_{+}\neq\varphi_{-}$. As a result,
the Hamiltonian can be written along the following modified version, with
respect to the fully symmetrical case:

$H(k)=\left(\omega_{0}-iG\right)\mathbb{1}_{4}+\left(\begin{matrix}v_{0}k&K-iGe^{-i\varphi}&\kappa_{d}-iGe^{i\psi}&\kappa_{c}-iGe^{i\left(\psi-\varphi\right)}\\\
K-iGe^{i\varphi}&-v_{0}k&\kappa_{c}-iGe^{i\left(\psi+\varphi\right)}&\kappa_{d}-iGe^{i\psi}\\\
\kappa_{d}-iGe^{i\psi}&\kappa_{c}-iGe^{i\left(\psi-\varphi\right)}&v_{0}k&K-iGe^{-i\varphi}\\\
\kappa_{c}-iGe^{i\left(\psi+\varphi\right)}&\kappa_{d}-iGe^{i\psi}&K-iGe^{i\varphi}&-v_{0}k\\\
\end{matrix}\right),$ (18)

where $\varphi=\varphi_{+}-\varphi_{-}$. All other parameters are the same as
in the fully symmetrical case.

From diagonalization of the Hamiltonian, we obtain four complex eigenvalues
whose expressions are also given by Eq. (4) (See appendix D for more details
and physics), where the "complex splitting" $S\left(k\right)$ is now written:

$S(k)=2\sqrt{v_{0}^{2}k^{2}+\left(K+\eta\kappa_{c}-iG_{\psi}e^{-i\varphi}\right)\left(K+\eta\kappa_{c}-iG_{\psi}e^{i\varphi}\right)}.$
(19)

As in the fully symmetrical case, the even and odd branches of the complex
eigenvalues ignore each other, since the corresponding modes have opposite
parity along the transverse direction. Therefore, possible crossing of these
branches is not avoided. The analytical expression of the phase $\psi$ given
by Eq. (7) remains valid.

Figure 11: Schematic view of the simplest coupled grating structure with
broken lateral symmetry. The vertical symmetry is preserved.

###  IV.2  General features of the complex eigenvalues

In the following, we restrict the discussion to the couple of even modes
($\eta=1$ in Eq. (19)), given that the features of the eigenvalues are
qualitatively similar for odd modes ($\eta=-1$). The expressions of the
eigenvalues are similar to that applying for fully symmetrical structures,
except for the factors $e^{\pm i\varphi}$, whose major impact lies in the
disappearance of the lateral BICs, thus confirming the lack of lateral
symmetry protection. It is observed, in general, two bright modes for each
transverse opposite parities. $\varphi=0$ (mod $2\pi$) or $\varphi=\pi$ (mod
$2\pi$) stand for gratings with symmetrical unit cell along the lateral
direction. Note that the eigenvalues are identical if $K+\kappa_{c}\geq 0$
with $\varphi=0$, or if $K+\kappa_{c}\leq 0$ with $\varphi=\pi$. Therefore, in
general, all situations are accounted for, if one chooses either
$K+\kappa_{c}\geq 0$ with $\varphi$ ranging in the interval $[0,\pi]$, or
$K+\kappa_{c}$ spanning positive as well as negative values, with $\varphi$
limited to the interval $[0,\frac{\pi}{2}]$.

When $\varphi\neq 0$ (or $\varphi\neq\pi$), room is solely left to the
category of accidental transverse BICs. It is indeed possible to generate a
transverse BIC for each of the two eigenvalues for a given transverse parity.
For example, in the case of even transverse parity ($\eta=1$), this is
achieved when the condition $\psi=\pi$ (mod $2\pi$) is met, for any of the two
eigenvalues and for two distinct structures having specific opto-geometrical
parameters (e.g. the grating thickness). The condition $\psi=\pi$ (mod $2\pi$)
can be realized for any $k$ value as well as for any phase $\varphi$.

Figure 12:  Angle-resolved reflectivity spectra obtained by RCWA simulations,
showing (a) the generation of transverse BICs (red arrow) with on demand
angular resolution achievable in lateral symmetry broken structures
($e=0.02$), where they are solely observed, lateral BICs at the $\Gamma$ point
being forbidden ; (b) in structures with preserved lateral symmetry ($e=0$),
transverse BICs in oblique (right black arrow) have to cohabit with
unavoidable lateral BICs at the gamma point (left black arrow). (c) log-log
plot of the quality factor of the corresponding branches versus $k$ vector,
confirming the sole presence of a transverse BIC in the structure with broken
lateral symmetry, while, in the structure with preserved lateral symmetry,
both the transverse BIC and lateral BIC coexist. Here the red and black
squares are numerical results obtained from FEM simulations. The others
parameters are $\Lambda=1\ \mu m,\ h=0.25\ \mu m,\ L_{1}=0.1\ \mu m,\
L_{e}2=0.2\ \mu m$ and $D=0.25\ \mu m$.

The dispersion characteristics around the $\Gamma$ point (small $k$ values) of
the two double transverse BICs are obtained from expansion of Eq. (4), where
$S(k)$ is now given by Eq. (19). The two double transverse BICs are referred
to as BIC+ and BIC-, below. The mathematical process is the same as in the
case of fully symmetrical structures, and leads to the real dispersion
characteristics, in the case of even modes for example ($\eta=1$).

$\omega_{R}^{\pm}(k)\approx\omega_{0}+\kappa_{d}\pm
K+\kappa_{c}+C_{R}^{\pm}\times\frac{k^{2}}{2},$ (20)

where:

$C_{R}^{\pm}=\frac{\pm\tau_{wg}^{\pm}(\varphi)C_{wg}+\tau_{rad}C_{rad}}{\tau_{wg}^{\pm}(\varphi)+\tau_{rad}},$
(21)

$\tau_{wg}^{\pm}(\varphi)=\frac{1}{G\left(1{\pm}\cos\varphi\right)},$ (22)

and where $\tau_{rad}$, $C_{wg}$ and $C_{rad}$ are given by Eqs. (12b), (13a)
and (13b) respectively.

In the case of fully symmetrical structure ($\varphi=0$), we find, as
expected, that the BIC+ coincides with the double transverse BIC issued from
the bright branch mode given by Eq. (9) and that the BIC- turns into a triple
BIC given by Eq. (14). As to the imaginary eigenvalue $\omega_{I}$, which is
null at the $\Gamma$ point (transverse BIC), it can be shown easily that it
increases like $k^{4}$, for both the double transverse BIC+ and the double
transverse BIC- (if $\varphi\neq 0$).

###  IV.3  Principal asset of structures with lateral symmetry breaking:
angular resolved BICs

Breaking some of the chains imposed upon fully symmetrical structures by
symmetry rules is the main asset of structures with broken lateral symmetry.
For example, it is possible to get full resolution of the angular
characteristics of BICs which can be formed in broken lateral symmetry
structures. This is particularly the case at the $\Gamma$ point where the
presence of a BIC can be decided or avoided on demand, unlike in fully
symmetrical structures, where lateral BIC is systematically present. This
particular aspect, accounted for by our analytical model, is illustrated in
RCWA simulations showed in Fig. 12 in the case of a lateral symmetry broken
structure where $\Lambda=1\mu m$, $h=0.25\mu m$, $D=0.25\mu m$, $L_{1}=0.1\mu
m$, $L_{2}=0.2\mu m$ and $e=0.02$.

##  V  Transverse symmetry breaking: aligned and different symmetrical
gratings

In this section, we analyse the impact of breaking the transverse symmetry of
1D PC wave-guiding structures on the complex dispersion characteristics. Fig.
13 shows the schematic view of a coupled grating structure, where the two
gratings are aligned and different: this is the general configuration allowing
for the analysis of the sole effect on dispersion characteristics of breaking
the transverse symmetry.

###  V.1  Analytical expression of Hamiltonian

With regard to the general Hamiltonian presented in section II.1 and in
appendix B.2, the only specific simplified characteristics of the 4×4
Hamiltonian are: $\phi=0$, since the two gratings are aligned, and
$\varphi_{1,2+}=\varphi_{1,2-}$, since they are formed with laterally
symmetric unit cells (no lateral symmetry breaking) and since the two gratings
are different, $\varphi=\varphi_{1}-\varphi_{2}\neq 0$.

Writing and diagonalizing the Hamiltonian in the base formed by base vectors
$(a_{1+},a_{1-},a_{2+},a_{2-})$ results in heavy mathematical wording, which
is detrimental to the physical readability. The latter is considerably
improved if the Hamiltonian is written in the new base
$(a_{1+}+a_{1-},a_{2+}+a_{2-},a_{1+}-a_{1-},a_{2+}-a_{2-})$, as below:

$H=\left(\begin{matrix}\omega_{1}+K_{12}-iG_{12}&K_{+}-i\sqrt{G_{12}G_{21}}e^{i(\psi-\varphi)}&v_{1}k&0\\\
K_{+}^{*}-i\sqrt{G_{12}G_{21}}e^{i(\psi+\varphi)}&\omega_{2}+K_{21}-iG_{21}&0&v_{2}k\\\
v_{1}k&0&\omega_{1}-K_{12}&K_{-}^{*}\\\
0&v_{2}k&K_{-}&\omega_{2}-K_{21}\end{matrix}\right),$ (23)

where $K_{ij}=\kappa_{i}+\beta_{i}\kappa_{j}$,
$G_{ij}=(\sqrt{\gamma_{i}}+\sqrt{\beta_{i}\gamma_{j}})^{2}$ and
$K_{\pm}=\kappa_{d}\pm\kappa_{c}$.

###  V.2  General features of the complex eigenvalues

As a result of broken transverse symmetry, it is no more possible to separate
the four eigenvalues into two couples of eigenvalues of opposite parity along
the transverse direction, which would ignore each other and whose crossing
would be allowed, as in the case of symmetric structures. The 4 eigenvalues
can be now classified into two couples of eigenvalues with quasi-even and
quasi-odd parities at $k=0$, respectively. This is schematically illustrated
in Fig. 14.

Figure 13:  Schematic view of the general coupled grating structure, with
broken transverse symmetry. The lateral symmetry is preserved. Figure 14:
Schematic representation of real dispersion characteristics of structures with
preserved transverse symmetry (a) and with broken transverse symmetry (b). For
the latter, crossing of branches with opposite quasi-parities is avoided and
modes are fully hybridized in the anti-crossing areas

###  V.3  Specific properties of eigenvalues at the $\Gamma$ point

At the $\Gamma$ point, the Hamiltonian is a diagonal block matrix, and the 4
eigenvalues can be easily determined through the diagonalization of the two
2×2 matrices being on the diagonal, resulting respectively in two eigenvalues
which are systematically complex for the first and purely real for the second.
The latter correspond to 2 lateral BICs which are systematically observed, as
a result of the lateral symmetry, while the former correspond to bright modes
which cannot turn into transverse BICs, owing to the broken transverse
symmetry. It can be easily deduced that the non-existence of transverse BICs
results from the finite phase $\varphi$, the two aligned gratings being
different. This fact is confirmed by results of numerical simulations, which
do not reveal the presence of transverse BICs, unlike the case of fully
symmetrical structures where $\varphi=0$, and where transverse BICs are
present for $\psi=0$ (mod $\pi$). Complementary physical discussion concerning
the generation of quasi transverse BICs at the $\Gamma$ point is given in the
appendix E.

###  V.4  Principal asset of structures with transverse symmetry breaking:
flat BIC

Lateral BICs can be made flat under specific conditions at the $\Gamma$ point:
this category of flat BIC has not the same nature as the flat transverse
double BICs observed in structures with preserved transverse symmetry and
presented in sections III and IV. For the latter, flat BICs result from the
interplay between the radiated and wave-guided components of transverse BICs.
In structures with broken transverse symmetry, flat lateral BICs may result
from the interplay between eigenmodes with quasi-even and quasi-odd
symmetries. This is unlike the case of structures with preserved transverse
symmetry, where such an interplay does not exist. It is noteworthy that this
category of flat lateral BIC is the counterpart of the loss-less flat band
characteristics exhibited in structures with broken transverse symmetry
operating under the protection of the light cone, around the first Brillouin
zone boundary [28]. This category of flat BIC is also present in the “fish-
bone” structures, as exposed in details in subsection VI.4.

###  V.5  Practical implementation of transverse symmetry breaking

A practical implementation of structures with broken transverse symmetry is
schematically represented in Fig. 15.

Figure 15:  Schematic views of a “Comb” structure. The right view shows a more
standard configuration, where the two superimposed “gratings” are not
separated

It consists in coupling a symmetrical grating with a plain membrane, which can
be considered as a non-corrugated grating. The right view is representative of
a more standard configuration, where the two parts of the structure are not
separated. It consists of an asymmetric grating, which is viewed as two
superimposed non separated symmetric gratings, the second “grating" being non
corrugated. A straightforward “joy-stick" to tune the vertical symmetry
breaking is the etch depth ratio $\epsilon$ (the etch depth is equal to
$\epsilon h$, where $h$ is the total thickness of the grating), which may span
the range 0 to 1. This configuration is very little demanding in terms of
technological constraints, yet featuring properties quite similar to those
presented in this section. Also, the mathematical treatment of the analytical
mode is significantly simplified, since such parameters as $\kappa_{2}$,
$\gamma_{2}$, in addition to the phase $\phi$, vanish.

##  VI  Transverse and lateral symmetry breaking: misaligned gratings

When the two coupled gratings are misaligned, both the lateral and transverse
symmetry are broken and a very wide variety of configurations can be
considered and handled using the general Hamiltonian presented in the
theoretical approach in section II. However, as noted in the introduction, the
number of parameters involved for full generality is rather large and it is
appropriate to implement simplified versions of the general Hamiltonian,
encompassing a wide variety of specific practical cases. In this section we
concentrate on a simple case of misaligned structure where the two gratings
are identical and symmetrical: we call this structure a “fish-bone” structure.
In this particular configuration, both the lateral and the transverse symmetry
are broken. Fig. 16 shows a schematic view of a “fish-bone” structure.

###  VI.1  Analytical expression of Hamiltonian

The general $4\times 4$ Hamiltonian can be written along the following
simplified version:

$H=\left(\begin{matrix}\omega_{0}+vk-i\sqrt{GG^{*}}&K^{*}-iG^{*}&\kappa_{d}-ie^{i\psi}G^{*}&\kappa_{c}-ie^{i\psi}\sqrt{GG^{*}}\\\
K-iG&\omega_{0}-vk-i\sqrt{GG^{*}}&\kappa_{c}-ie^{i\psi}\sqrt{GG^{*}}&\kappa_{d}-ie^{i\psi}G\\\
\kappa_{d}-ie^{i\psi}G&\kappa_{c}-ie^{i\psi}\sqrt{GG^{*}}&\omega_{0}+vk-i\sqrt{GG^{*}}&K-iG\\\
\kappa_{c}-ie^{i\psi}\sqrt{GG^{*}}&\kappa_{d}-ie^{i\psi}G^{*}&K^{*}-iG^{*}&\omega_{0}-vk-i\sqrt{GG^{*}}\par\end{matrix}\right).$
(24)

Figure 16:  Schematic view of a “fish-bone” structure. It consists in two
identical gratings, super-imposed and misaligned along the lateral direction.
The misalignment is given by the lateral off-set $\pm\delta/2\times\Lambda$
with $\delta\in[-0.5,0.5]$ of the upper and lower grating respectively. This
corresponds to a relative displacement $\delta\times\Lambda$ between the two
gratings.

The Hamiltonian is very similar to the case of a fully symmetrical structure:
the only difference lies in the phase parameter $\phi$, related to the
respective x-coordinate off-set $\pm\delta/2\times\Lambda$ of the two gratings
(see Fig. 16). One may show that this phase parameter is given by:

$\phi=2\pi\delta\,(mod\,2\pi)$ (25)

It results that the coupling parameters $G$ and $K$ are complex in general:

$\sqrt{G}=\sqrt{\gamma}\left(e^{i\phi/2}+\sqrt{\beta}e^{-i\phi/2}\right),$
(26)

$K=\kappa\left(e^{i\phi}+\beta e^{-i\phi}\right).$ (27)

The misalignment, characterized by the relative displacement
$\delta\times\Lambda$ of the two gratings results in breaking both the
transverse and lateral symmetry of the structure, except for two cases:
$\delta=0$ (perfect alignment) and $\delta=\pm 0.5$ (half period
misalignment). As discussed in details in the appendix section A, one may show
that both lateral and transverse symmetries of eigenmodes are preserved in
these two cases (the lateral symmetry requires in plus an operation at the
$\Gamma$ point). Thus the aligned gratings and half period misalignment are
equivalent in term of symmetry for the complex eigenmodes. Therefore
transverse BICs and lateral BICs can be obtained in both configurations.

The equivalence of the two cases $\delta=0$ and $\delta=\pm 0.5$ is also found
in the resolution of complex eigenvalues. Indeed, these two cases correspond
to the value of phase parameter $\phi=0$ (mod $2\pi$) and $\phi=\pi$ (mod
$2\pi$) respectively. This leads to real values of coupling parameters $G$ and
$K$ given by Eqs. (26),(27). Thus, the Hamiltonian given by Eq. (24) is
simplified into the one given by Eq. (2). In other words, the two
configurations share the same Hamiltonian description of aligned gratings.
Also, as noted in the appendix section F.1 [eq. (99)], the formal compact
expressions of the Hamiltonian for aligned gratings and half period
misalignment are identical. This means that for these two limit cases, the
properties of the eigenvalues are formally identical. For example, when
$\phi=\pi$ (mod $2\pi$), in the same way as when $\phi=0$ (mod $2\pi$), the
four eigenvalues consists also in two couples of fundamental and excited
branches, which ignore each other, respectively, and whose crossing is not
avoided (see appendix section F.1, Eqs (100),(101)). Consequently, the general
as well as specific features of the fully symmetrical structures described in
sections III.2 and III.3, apply to the case of structures with half period
lateral off-set.

In structures with arbitrary misalignment, physical readability and
exploitation of the Hamiltonian is significantly improved if one choose the
alternative base ($a_{1+}+a_{2-},a_{2+}+a_{1-},a_{1+}-a_{2-},a_{2+}-a_{1-}$).
In this base, the Hamiltonian can be written as below:

$H=\left(\begin{matrix}\omega_{0}+\kappa_{c}-i\sqrt{GG^{*}}(1+e^{i\psi})&\kappa_{d}+K^{*}-iG^{*}(1+e^{i\psi})&vk&0\\\
\kappa_{d}+K-iG(1+e^{i\psi})&\omega_{0}+\kappa_{c}-i\sqrt{GG^{*}}(1+e^{i\psi})&0&vk\\\
vk&0&\omega_{0}-\kappa_{c}-i\sqrt{GG^{*}}(1-e^{i\psi})&\kappa_{d}-K^{*}-iG^{*}(1-e^{i\psi})\\\
0&vk&\kappa_{d}-K-iG(1-e^{i\psi})&\omega_{0}-\kappa_{c}-i\sqrt{GG^{*}}(1-e^{i\psi})\end{matrix}\right).$
(28)

###  VI.2  General features of the complex eigenvalues

As a result of broken transverse symmetry, it is no more possible to separate
the four eigenvalues into two couples of eigenvalues of opposite parity along
the transverse direction, which would ignore each other and whose crossing
would be allowed, as in the case of symmetric structures. The 4 eigenvalues
can be now classified into two couples of eigenvalues with quasi-even and
quasi-odd parities, respectively. This is schematically illustrated in Fig.
17.

Figure 17:  Schematic representation of real dispersion characteristics of
aligned grating structures, i.e. where $\phi=0$ [(a), (c)], and misaligned
grating structures, i.e. where $\phi\neq 0$ [(b), (d)]. Blue (red) curves
represent respectively odd (even) branches for the former and quasi-odd
(quasi-even) branches for the latter. For the misaligned structures, the
eigenvalues are fully hybridized in the area of dispersion characteristics
[(b), (d)] coloured in purple. There are two possible configurations depending
on whether $K_{0}=\kappa(1+\beta)=K(\phi=0)$ is smaller [(a), (b)] or larger
[(c), (d)] than $|\kappa_{d}|$ (remind that $\kappa_{d}<0$). The numbering 1
to 4 of eigenvalues at the $\Gamma$ point corresponds to the numbering of Eqs.
(29) to (32).

The two possible configurations shown in Fig. 17, depending on whether
$K_{0}=\kappa(1+\beta)=K(\phi=0)$ is smaller [(a) and (b)] or larger [(c) and
(d)] than $|\kappa_{d}|$, will be discussed further, later on in this paper,
in connection with the topic of flat BICs at the $\Gamma$ point.

###  VI.3  Specific properties of eigenvalues at the $\Gamma$ point

At the $\Gamma$ point, the Hamiltonian is a diagonal block matrix, and the 4
eigenvalues can be easily determined through the diagonalization of the 2×2
Hamiltonians, resulting in 4 eigenvalues, numbered 1 to 4 below:

$\omega_{1}=\omega_{0}-\kappa_{c}-i\sqrt{GG^{*}}(1-e^{i\psi})+\sqrt{[\kappa_{d}-K+iG(1-e^{i\psi})][\kappa_{d}-K^{*}+iG^{*}(1-e^{i\psi})]}),$
(29)

$\omega_{2}=\omega_{0}-\kappa_{c}-i\sqrt{GG^{*}}(1-e^{i\psi})-\sqrt{[\kappa_{d}-K+iG(1-e^{i\psi})][\kappa_{d}-K^{*}+iG^{*}(1-e^{i\psi})]}),$
(30)

$\omega_{3}=\omega_{0}+\kappa_{c}-i\sqrt{GG^{*}}(1+e^{i\psi})+\sqrt{[\kappa_{d}+K-iG(1+e^{i\psi})][\kappa_{d}+K^{*}-iG^{*}(1+e^{i\psi})]}),$
(31)

$\omega_{4}=\omega_{0}+\kappa_{c}-i\sqrt{GG^{*}}(1+e^{i\psi})-\sqrt{[\kappa_{d}+K-iG(1+e^{i\psi})][\kappa_{d}+K^{*}-iG^{*}(1+e^{i\psi})]}).$
(32)

In general, the 4 eigenvalues are complex and correspond to bright modes; in
absence of lateral symmetry protection, lateral BICs are forbidden. On the
other hand, for $\psi=0$ or $\psi=\pi$, that is for opto-geometrical
parameters of the structures (e.g. the grating thickness and/or separation
distance) specific to each of the 4 considered eigenvalues, 4 transverse BICs
can be formed (one single transverse BIC at a time, for a given coupled
grating structure):

$\omega_{1,2}(\psi=0)=\omega_{R\pm}(\psi=0)=\omega_{0}-\kappa_{c}\pm|\kappa_{d}-K|,$
(33)

$\omega_{3,4}(\psi=\pi)=\omega_{R\pm}(\psi=\pi)=\omega_{0}+\kappa_{c}\pm|\kappa_{d}+K|.$
(34)

Note that for $\phi=0$ mod($2\pi$) or $\phi=\pi$ mod($2\pi$) (structure with
preserved lateral symmetry), two of these transverse BICs coincide with
lateral BICs, the two other being transverse double BICs.

Eqs. (33) and (34) show that the eigenfrequency values of the BICs depend on
the parameter $\phi$, that is on the lateral off-set between the two gratings.
Hence, if $\phi$ is varied, the effective transverse optical distance
$L_{opt}$ between the grating wave-guided resonances has to be adjusted to
meet the condition required for the formation of a transverse BIC ($\psi=0$ or
$\psi=\pi$). In summary, a fine interplay of the lateral and transverse shifts
between the two gratings may be implemented to control the strength of the
transverse eigenresonance.

At this stage, it must be emphasized that the analytical model does not
anticipate the presence of transverse BICs in “fish-bone” structures off the
$\Gamma$ point in such a formal and straightforward way as in the case of
aligned or half period shifted gratings. This is confirmed by RCWA
simulations, which show the presence of transverse BICs solely at the $\Gamma$
point, when $\phi\neq 0,\pi$ mod($2\pi$) . Interestingly, since the transverse
symmetry is broken, this transverse BIC is not inherent to the transverse
symmetry but another symmetry called “reversal symmetry” which is preserved
for any misalignment but requires operation at the $\Gamma$ point. This
symmetry is fully described in the Appendix section A. Here the accidental BIC
at the $\Gamma$ point goes by the name transverse symmetry only because its
formation relies on the value of the transverse phase shift $\psi$. As a
result, BICs numbered 1 to 4 are single transverse BICs and not double
transverse BICs as in aligned grating structures or with half-period offset:
they do not result, indeed, from the merging of two transverse oblique BICs.
In summary, fish-bone structures of arbitrary misalignment can accommodate in
general only single transverse BICs at the $\Gamma$ point, with lateral BICs
at the $\Gamma$ point and transverse BICs off the $\Gamma$ point being
excluded by lateral symmetry breaking and transverse symmetry breaking
respectively.

As a final comment about the behaviour of eigenvalues at the $\Gamma$ point,
we draw the attention of the reader to the remarks below:

  * •

Since, in general, that is for $\phi\neq 0,\pi$ mod($2\pi$) , one single BIC
belonging to the sole category of transverse BICs can be formed, it results
that the generation of triple BIC is prohibited, in absence of lateral BICs.
We remind indeed that they may occur only in aligned ($\phi=0$ mod($2\pi$) or
half-period shifted ($\phi=\pi$ mod($2\pi$) gratings, where lateral and
transverse (or reversal) symmetry protection is made possible.

  * •

The generation of Dirac point at triple BICs is also prohibited, in general.
They can be formed only in aligned [$\phi=0$ mod($2\pi$)] or half-period
shifted [$\phi=\pi$ mod($2\pi$)] gratings, when degeneracy of a double
transverse BIC and a triple BIC is achieved.

Figure 18:  (a) Angle-resolved reflectivity spectra obtained by RCWA
simulations, showing a flat BIC for a lateral offset $\delta=0.133$, with zero
curvature of the real eigenvalue at the $\Gamma$ point. The protocol adopted
for the generation of this flat BIC is based on the hybridization of
eigenmodes of opposite parities, as a result of transverse symmetry breaking.
The curvature of the dispersion characteristic is efficiently controlled by
the lateral off-set $\delta=\phi/2\pi$ of the two gratings: it vanishes for
$K_{0}\cos\phi\approx|\kappa_{d}|$, it is positive for
$K_{0}\cos\phi>|\kappa_{d}|$ and negative for $K_{0}\cos\phi<|\kappa_{d}|$.
(b) FEM simulations of real eigenvalues for several values of lateral offset
$\delta$. The others parameters are $\Lambda=1\ \mu m,\ h=0.25\ \mu m,\ L=0.5\
\mu m$ and $D=0$.

###  VI.4  Flat BIC at the $\Gamma$ point

Apart from the $\Gamma$ point, the analytical resolution of equations is, in
practice, either a bit heavy in its close vicinity (small $k$ momentum), or
impossible further away in the $k$ momentum space. For the latter, numerical
resolution of the analytical equations is mandatory. We concentrate in the
present work on analysis in the vicinity of the $\Gamma$ point. See
complementary information in appendix section F.2.

For small $k$ momentum, it is possible to derive the analytical expression of
the dispersion characteristics by expanding Eqs. (102) and (7). We concentrate
specifically on the study of flat BICs, which can be considered as being among
the most attractive features of “fish-bone” structures. We therefore
concentrate on the analysis of the curvature or second derivative $C_{R}$ of
dispersion characteristics at the $\Gamma$ point. In subsection III.3.2, we
have shown that it is possible to design flat double transverse BICs in fully
symmetrical structures, by engineering a fine interplay between the radiated
and wave-guided components of the hybrid double transverse BIC. We apply here
a quite different protocol to design flat BICs at the $\Gamma$ point: it is
based on the hybridization of eigenmodes of opposite parities, as a result of
transverse symmetry breaking. Therefore, the relevant eigenmodes lending
themselves to efficient hybridization, are related to eigenvalues numbered 3
and 4, given to their spectral proximity. The general expression of $C_{R}$ is
a bit heavy. We give below a version of the expression of $C_{R}$ whose
physical significance is made fully readable; it is in line with the
expression of $C_{R}$ obtained for double transverse BICs, in fully
symmetrical structures (see Eq. (11)):

$C_{R}=\frac{\tau_{wg}C_{wg}+\tau_{rad}C_{rad}}{\tau_{wg}+\tau_{rad}},$ (35)

where $\tau_{rad}$ and $C_{rad}$ are given by Eq. (12b) and (13b)
respectively, and:

$\tau_{wg}=\frac{1}{(1+\alpha)|G|},$ (36)

$C_{wg}=\frac{v^{2}}{K_{eff}}.$ (37)

The expression of the parameters $\alpha$ and $1/K_{eff}$ are given in the
appendix F.2 (Eqs. (105) and (106)). The module of $\alpha$ is smaller than 1.
The parameter $K_{eff}$ includes all the diffractive processes in action
within the structure. In the limit case of aligned gratings ($\phi=0$), we
have $\alpha=1$ and $K_{eff}=K+\kappa_{c}$. We remind that $\tau_{wg}$ and
$\tau_{rad}$ are respectively the average lifetime of photons in the wave-
guided state before being emitted into the continuum and in the radiated state
during a one way trip between the two gratings, and that $C_{wg}$ and
$C_{rad}$ are the guided and radiated curvatures respectively.

The similarity of the formal expressions of transverse BIC curvatures in
aligned ($\phi=0$) and misaligned ($\phi\neq 0$) structures should not hide
the major difference introduced by the lateral off-set $\delta$ between the
two gratings. This difference lies in the guided curvature $C_{wg}$, which is
heavily dependent of the parameter $\phi=2\pi\delta$ (mod $2\pi$). $C_{wg}$
can be controlled, ad libitum and continuously, from negative to positive
values by varying the lateral offset $\delta$. A detailed analysis of the
parameter $1/K_{eff}$ (see Eq. (106)) indicates that this is indeed possible
provided that $K_{0}=\kappa(1+\beta)=K(\phi=0)$ is larger than |$\kappa_{d}$|.
This condition corresponds to the dispersion characteristic configuration
shown in Fig. 17, (c) and (d). In the limit of negligible losses (small $G$),
it can be shown, from the expression of $1/K_{eff}$ , that the condition for
the production of a flat BIC is reduced to the simple relation:
$K_{0}\cos\phi\approx|\kappa_{d}|$.

In summary the lateral offset $\delta$ offers a very efficient joystick for
the production of flat transverse BICs in coupled grating structures, with no
request for the thickness of the coupled grating structure to be large and to
exceed a few times $\lambda$, as this is the case in the design scenario
specific to aligned gratings (see section III.3.2). These conclusions are
faithfully confirmed by the results of RCWA numerical simulations presented in
Fig. 18.

##  VII  Experimental Demonstration

Although the present paper is essentially devoted to theoretical work, we
chose however to include a proof of concept of the Dirac point at triple BIC
in the current experimental section, since it may be considered as the most
achieved photonic specie based on the combination and interaction of BICs.

The fabrication of the sample includes the different steps described
thereafter, starting from a commercial quartz substrate on which a 590 nm-
thick amorphous silicon film is deposited by plasma-enhanced chemical vapor
deposition (PECVD) using SiH4 as a precursor and helium as the plasma gas. The
substrate temperature is kept at 300 ∘C and the pressure in the chamber is 2
Torr. The plasma is set by an RF signal at 25W. A 100 nm-thick hydrogen
silsesquioxane (HSQ) resist is spun on the sample and baked at 80 ∘C for 4
minutes. The resist is then exposed by electron-beam lithography and developed
with a solution of TMAH. The patterns are subsequently transferred to the a-Si
by inductively-coupled reactive ion etching (ICP-RIE) using a mixture of Cl2
and O2. To finalize the device, 828 nm of PMMA is spun directly on top of the
patterns (see Fig. 19(a)).

Devices are characterized using a home-made setup that measures the angle-
resolved reflectivity. A broadband white light source (Halogen) is focused on
the sample through a microscope objective (NA=0.42) and the reflectivity of
the device is collected via the same objective. The back-focal plane of the
objective is imaged using a lens, focused at the entrance slit of a
spectrograph and then collected on an InGaAs camera sensor. With this
configuration, we obtain a direct measurement of the experimental energy- and
momentum-resolved dispersion of samples. A polarizer is placed between the
Fourier lens and the focusing lens to select the measured polarization.

Figure. 19(b) depict the result of angle-resolved reflectivity measurement.
The Dirac point at triple BIC dispersion is experimentally demonstrated around
a wavelength of 1.5 $\mu m$ in TE-polarization.

Figure 19: (a) Sketch of the fabricated structure, (b) Experimental angle-
resolved reflectivity spectra of the sample with a close-up view of the Dirac
point at triple BIC. Parameters are $\Lambda=0.83\ \mu m,\ h=0.3\ \mu m,\
L=0.23\ \mu m$ and $D=0$.

##  VIII  Prospects

The theoretical approach for analytical modelling of complex dispersion
characteristics of optical modes operating in PCs proposed in this work is,
all at the same time, extremely concise and very generic. It shows an
unprecedented degree of completeness and generality, owing to its capability
to provide a faith-full description of the photonic properties of an unlimited
number of photonic structures. It reveals in particular that apparently simple
photonic structures, featuring plain fully symmetrical 1D PCs, may offer an
unexpected wealth of remarkable complex dispersion characteristics. It proves,
in addition, to be a powerful enabler of the design tool-box in order to
predict, control and assess the impact of breaking the transverse and lateral
symmetry of grating structures, on their complex dispersion characteristics.
Among the variety of conceptual and practical outcomes provided by our
approach, we wish to emphasize that all remarkable photonic species being
generated in 1D PCs rely fully or, at least to an essential extent, on two
basic building blocks so-named lateral and transverse (accidental) BICs in
this paper, and explicitly unveiled by our modelling approach. It results that
this theoretical analytical approach provides very efficient support to
numerical simulations which, taken alone, may not be able to predict the
underlying physics, when it comes to design photonic structures with
remarkable desired properties. Along this design track, we selected, for
experimental demonstration, such a remarkable structure as the so-named “Dirac
point at triple BIC”, which results from the merging / degeneracy of two
couples of transverse BICs and one lateral BIC. This work opens a unique
playground for both exotic Dirac and flatband physics. In particular, it will
be possible to study the light delocalization when a dispersion is gradually
transformed from flat dispersion (exceptional localized states and ultra-
sensitive to disorders [48, 49, 50, 51]) to Dirac cones (very robust versus
disorder effect of Anderson localization [52]). This gradual transformation
may be naturally implemented in moiré lattices, where, for example, two
grating structures with slightly different periods [53] or two identical
gratings twisted around a common axis [54] would be superimposed and coupled
to each-other with an adjustable coupling rate, thus resulting in a fish-bone
like moiré.

More generally, the conceptual and design approach proposed in the present
work offers a promising practical route for on demand tailoring of the local
density of optical states and processing of light at the nanometer scale [55],
which are paramount for applications in optoelectronic devices. At this stage,
plenty of room is left for an extra wide variety of configurations, with,
among other prospects, the extension to 2D PC slab, which naturally lend
themselves to additional degrees of freedom in 3D manipulation of light.
Indeed, extending these concept to 2D PC slab would benefit richer in-plane
symmetry (reflections and rotations) as well as the possibility to harness
diffractive couplings between different polarization guided modes. In terms of
non-Hermitian features, one may expect more complicated scenario for
merging/splitting BICs, engineering exceptional rings/lines instead of EPs,
studying high-order EPs.

## Acknowledgements

This work was partly funded by the the IDEXLYON from Université de Lyon,
Scientific Breakthrough project TORE within the Programme Investissements
d’Avenir (ANR-19-IDEX-0005).

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##  Appendix A Symmetry properties of bilayer structures made of two identical
symmetric gratings

The lateral and transverse symmetries are at the heart of the Hamiltonian
construction and the properties of its complex eigenmodes. In this section, we
will discuss in details the symmetry operators and their interplay with the
$4\times 4$ Hamiltonian of two identical symmetric gratings. The two gratings
can be eventually misaligned to make a “fishbone” structure as discussed in
section VI of the main text, with a lateral offset
$\pm\frac{\delta}{2}\times\Lambda$ for the upper and lower grating
respectively (see Fig. 16). Note that the case of aligned symmetric gratings
of section III is a particular case of “fishbone” structure in which
$\delta=0$.

####  A.0.1 Lateral symmetry of “fishbone”

Since each grating is symmetric, the lateral symmetry is dictated by the
x-reflection operator $\sigma_{x}$ that swaps the forwards and backwards
modes:

$\displaystyle a_{1\pm}$ $\displaystyle\xrightarrow{\sigma_{x}}a_{1\mp}$ (38a)
$\displaystyle a_{2\pm}$ $\displaystyle\xrightarrow{\sigma_{x}}a_{2\mp}$ (38b)

In the basis formed by $(a_{1+},a_{1-},a_{2+},a_{2-})$, the corresponding
matrix of the operator $\sigma_{x}$ is therefore given by:

$M_{x}=\left(\begin{matrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{matrix}\right).$ (39)

Moreover, the same operator would inverse the wavevector and the lateral off-
set:

$(k,\delta)\xrightarrow{\sigma_{x}}(-k,-\delta),$ (40)

Thus the Hamiltonian $H(k,\delta)$ needs to satisfy the symmetry condition:

$H(k,\delta)=M_{x}.H(-k,-\delta).M_{x}$ (41)

It is straightforward to demonstrate that the Hamiltonian in Eq. (24) verifies
nicely the condition above.

We now look for the sets of $(k,\delta)$ for which the eigenmodes of the
system preserve the lateral symmetry. These “high symmetry points” require the
commutation relationship $[H(k,\delta),M_{x}]=0$. From Eq. (41), this
requirement is equivalent to $H(k,\delta)=H(-k,-\delta)$. Therefore a trivial
configuration is with $k=0$ and $\delta=0$. In other words, the lateral
symmetry is preserved at the $\Gamma$ point with perfectly aligned gratings.

Interestingly, the Hamiltonian in Eq. (24) shows that the requirement
$H(k,\delta)=H(-k,-\delta)$ is also fulfilled when $k=0$ and $\delta=\pm 0.5$,
i.e. half period misalignment. This is due to the fact that the phase
parameters $\phi=\pi$ (mod $2\pi$) and $\phi=-\pi$ (mod $2\pi$) are strictly
equivalent. As a consequence, the lateral symmetry is also preserved at the
$\Gamma$ point with gratings of half period misalignment.

####  A.0.2 Transverse symmetry of “fishbone”

Since the two gratings are identical, the transverse symmetry is dictated by
the z-reflection operator $\sigma_{z}$ that swaps the co-propagating modes
from different layers:

$\displaystyle a_{1\pm}$ $\displaystyle\xrightarrow{\sigma_{z}}a_{2\pm}$ (42a)
$\displaystyle a_{2\pm}$ $\displaystyle\xrightarrow{\sigma_{z}}a_{1\pm}$ (42b)

In the basis formed by $(a_{1+},a_{1-},a_{2+},a_{2-})$, the corresponding
matrix of the operator $\sigma_{z}$ is therefore given by:

$M_{z}=\left(\begin{matrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\\\
\end{matrix}\right).$ (43)

Moreover, the same operator would inverse the lateral off-set but keeping the
wavevector unchanged:

$(k,\delta)\xrightarrow{\sigma_{z}}(k,-\delta),$ (44)

Thus the Hamiltonian $H(k,\delta)$ needs to satisfy the symmetry condition:

$H(k,\delta)=M_{z}.H(k,-\delta).M_{z}$ (45)

Again, one may easily show that the Hamiltonian in Eq. (24) verifies nicely
the condition above.

In the same fashion as the lateral symmetry in previous section, we now look
for the sets of $(k,\delta)$ for which the eigenmodes of the system preserve
the transverse symmetry. These “high symmetry points” require the commutation
relationship $[H(k,\delta),M_{z}]=0$. From Eq. (45), this requirement is
equivalent to $H(k,\delta)=H(k,-\delta)$. Following the discussion from the
lateral symmetry, we deduce that there are two configurations: the first one
is the trivial case is with $\delta=0$ (perfectly aligned gratings), and the
second one is with $\delta=\pm 0.5$ (half period misalignment). Unlike the
“high symmetry points” of the lateral symmetry, here both configurations are
valid for any value of the wavevector $k$.

####  A.0.3 Reversal symmetry of “fishbone”

Other than lateral and transverse symmetry, the group symmetry of “fishbone”
structure also exhibits another symmetry inherited from the two previous ones.
This symmetry is dictated by the operator $\sigma_{r}$ that is defined by
executing successively the z-reflection operator $\sigma_{z}$ and x-reflection
operator $\sigma_{x}$ or vice versa. Such operator, that is called here
“reversal symmetry”, swaps the counter-propagating modes from different
layers:

$\displaystyle a_{1\pm}$ $\displaystyle\xrightarrow{\sigma_{r}}a_{2\mp}$ (46a)
$\displaystyle a_{2\pm}$ $\displaystyle\xrightarrow{\sigma_{r}}a_{1\mp}$ (46b)

In the basis formed by $(a_{1+},a_{1-},a_{2+},a_{2-})$, the corresponding
matrix of the operator $\sigma_{r}$ is therefore given by:

$M_{r}=\left(\begin{matrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
\end{matrix}\right).$ (47)

One may verify that $M_{r}=M_{x}.M_{z}=M_{z}.M_{x}$.

Interestingly, since $\sigma_{x}$ inverses both the wavevector and the lateral
off-set while $\sigma_{z}$ only inverses the lateral off-set, $\sigma_{r}$
would only inverse the wavevector but keeping the lateral off-set unchanged:

$(k,\delta)\xrightarrow{\sigma_{r}}(-k,\delta),$ (48)

This action corresponds to the time reversal symmetry, thus explains the name
“reversal symmetry”.

As a consequence, the Hamiltonian $H(k,\delta)$ needs to satisfy the symmetry
condition:

$H(k,\delta)=M_{r}.H(-k,\delta).M_{r}$ (49)

Again, one may easily show that the Hamiltonian in Eq. (24) verifies nicely
the condition above.

The reversal symmetry is preserved in eigenmodes of the system if
$H(k,\delta)=H(-k,\delta)$. That limits operation at the $\Gamma$ point in a
similar way as the lateral symmetry. However, unlike the lateral and
transverse symmetry which require $\delta=0$ or $\pm 0.5$, the reversal
symmetry is preserved for all lateral off-set. Thus this symmetry is the only
one that is preserved for any arbitrary misalignment.

####  A.0.4 Symmetry and dispersion of eigenmodes in “fishbone” structures

As discussed in previous subsections, the lateral and transverse symmetry are
preserved in eigenmodes only for the cases of perfect alignment or half period
misalignment. Moreover, the lateral symmetry requires an operation at $\Gamma$
point. When both symmetries are preserved, the four operator $H$, $M_{x}$,
$M_{z}$ and $M_{r}$ commute one to each other and having the same four
eigenmodes, given by:

$\displaystyle\Psi_{1}$
$\displaystyle=\frac{a_{1+}-a_{1-}}{2}+\frac{a_{2+}-a_{2-}}{2}=\left(\begin{matrix}1\\\
-1\\\ 1\\\ -1\\\ \end{matrix}\right)$ (50a) $\displaystyle\Psi_{2}$
$\displaystyle=\frac{a_{1+}-a_{1-}}{2}-\frac{a_{2+}-a_{2-}}{2}=\left(\begin{matrix}1\\\
-1\\\ -1\\\ 1\\\ \end{matrix}\right)$ (50b) $\displaystyle\Psi_{3}$
$\displaystyle=\frac{a_{1+}+a_{1-}}{2}+\frac{a_{2+}+a_{2-}}{2}=\left(\begin{matrix}1\\\
1\\\ 1\\\ 1\\\ \end{matrix}\right)$ (50c) $\displaystyle\Psi_{4}$
$\displaystyle=\frac{a_{1+}+a_{1-}}{2}-\frac{a_{2+}+a_{2-}}{2}=\left(\begin{matrix}1\\\
1\\\ -1\\\ -1\\\ \end{matrix}\right)$ (50d)

Here $\Psi_{1,2}$ and $\Psi_{3,4}$ are eigenvectors of $M_{x}$ with
corresponding eigenvalues $1$ (even mode) and $-1$ (odd mode) respectively.
Therefore $\Psi_{3,4}$ are lateral BICs which are protected by the lateral
symmetry. The photonic bands having $\Psi_{1,2}$ and $\Psi_{3,4}$ at the
$\Gamma$ points are one called bright and dark branches respectively in the
section III.2.

Moreover, $\Psi_{1,3}$ and $\Psi_{2,4}$ are eigenvectors of $M_{z}$ with
corresponding eigenvalues $1$ (even mode) and $-1$ (odd mode) respectively.
Since the transverse symmetry is preserved even at out of the $\Gamma$ point,
the photonic modes $\tilde{\Psi}_{n}(k)$ ($n=1..4$) that are $\Psi_{n}$ at the
$\Gamma$ points preserve the transverse parity of $\Psi_{n}$. The eigenmodes
are divided into two groups of opposite transverse symmetry and the
dispersions of two modes from different groups do not avoid crossings (see
Fig. 14(a)). Most importantly, transverse BICs can take place at any value of
wavevector $k$ as already discussed in section III.2.

Finally, for an arbitrary misalignment having $\delta\neq 0,\pm 0.5$, both
lateral and and transverse symmetry are broken. However, the reversal symmetry
is still preserved as long as we stay at the $\Gamma$ point. Therefore, it is
still possible to obtain accidental BIC at $\Gamma$ point if the transverse
phase shift $\psi$ is equal to 0 or $\pi$ (mod 2$\pi$) (see discussions in
section VI.2). Interestingly, although this accidental BIC is inherent to the
reversal symmetry and the transverse symmetry is broken, its formation is
still dictacted by the transverse phase shift. Therefore we still call it by
the name of transverse BIC.

##  Appendix B Complemental information on the theoretical approach

###  B.1 Discussion on the approximations of the model

####  B.1.1 Complements on the rationale and phenomenology of the model

Figure 20:  rationale of the H4x4 versus the H2x2 approach

As recalled in section II.1, leaky optical resonances in PC slabs are
generally described as complex eigenvalues of 2×2 non-Hermitian Hamiltonian.
However this H2×2 approach, although attractive in terms of simplified
mathematical resolution, provides a rather partial description of the
structures. In particular, it ignores explicitly the transverse dimension of
the structures, as illustrated in Fig. 20(a): along the H2×2 approach, a 1D
grating is viewed as a flat zero thickness periodic structure, able to
accommodate wave-guided leaky resonances, which are, by design, restricted to
be mono-modal in the transverse direction; also the H2×2 approach cannot
account for events encountered by the leaky light when travelling across the
grating structure along the transverse direction.

Let us now proceed one step further by considering the case of two identical
and aligned h thick gratings, with are apart by the distance D, as shown in
Fig. 20(b) left. They may communicate via the near field, as well as via the
radiated field. In Fig. 20(b) right, it is proposed a simplified view of this
coupled grating structure as two coupled flat zero thickness periodic
structures, which are considered as being strictly mono-modal. Coupling of the
two single mode grating structures results in the occurrence of additional
modes of opposite (even and odd) parity along the transverse direction, which
are the photonic expression of the transverse dimension of the structure. It
is then required to move to the H4×4 formalism proposed in this work for a
proper analytical description of the real grating structure. The parameter
$L_{opt}$ is the effective optical transverse distance between the wave-guided
resonances accommodated by each of the two periodic structures. $L_{opt}$
encompasses transverse optical paths across the two gratings as well as across
the spacer and may be written along the following phenomenological relation:
$L_{opt}=nh+n_{0}D$, where $n$($n_{0}$) is the refractive index of the
effective (surrounding) medium. Our modelling approach applies to the case of
two gratings in contact ($D=0$, Fig. 20(c)), at least for a physical
trustworthy description of the physics, although the small perturbation
approximation of the coupled mode formalism may not strictly apply. For
example this modelling approach provides a faithful account of the variety of
transverse BICs occurring in real grating structures, as fully demonstrated in
the present work.

One must also emphasize that, unlike in the H2x2 approach, our model can
account for the impact of breaking the lateral as well transverse symmetry of
the coupled grating structure, as illustrated in Fig. 20(d) (different
gratings of same period with arbitrary lateral offsets), and as thoroughly
developed in the main text of this article.

As explained in the main text (section II.5), in order to get a tractable
Hamiltonian, two main approximations are made. These assumptions will be
discussed in the following two subsections and exemplified in the case of two
identical gratings.

####  B.1.2 Neglecting the imaginary part of $\psi$

In the case of two identical gratings, the implicit equation for the complex
eigenvalue $\omega$ at $\Gamma$-point for the bright branch can be written
(see Eq. (4) and following):

$\omega=\omega_{R}+i\omega_{i}=\omega_{b}-i\gamma_{b}e^{i\psi},$ (51)

where $\omega_{b}$ ($\gamma_{b}$) is a complex (real) constant. Rigorously
speaking $\psi=\omega\frac{L_{opt}}{c}$, whereas we have assumed
$\psi=\omega_{R}\frac{L_{opt}}{c}$. This leads to an error which can be
quantified on the term:

$|\omega-\omega_{b}|=\gamma_{b}e^{-\psi_{i}}.$ (52)

Neglecting $\psi_{i}$ results in a relative error:

$1-e^{-\psi_{i}}=1-e^{-\omega_{i}\frac{L_{opt}}{c}},$ (53)

which is small if:

$\omega_{i}\frac{L_{opt}}{c}=\frac{\tau_{rad}}{\tau}<<1,$ (54)

where $\tau$ is the lifetime of the eigenmode. In this wortk we always
consider high quality factor modes, which valids this approximation.

####  B.1.3 Neglecting the reflectivity of the interfaces

From first principle Coupled Mode Theory [56] and taking into account the
effect of the interfaces on the transverse propagation of photons, a rigorous
Hamiltonian can be derived in the case of two identical gratings. Following
this approach, new implicit equations are obtained for the eigenvalues. For
example Eq. (4) in the main text becomes (for $\eta$=+1):

$\omega=\omega_{R}+i\omega_{i}=\omega_{0}+\kappa_{d}-iG(1+\mu)\pm\sqrt{v^{2}k^{2}+\left(K+\kappa_{c}-iG(1+\mu)\right)^{2}},$
(55)

where:

$\mu=\frac{r_{m}+t_{m}}{1-r_{m}e^{i\theta}}e^{i\theta},$ (56)

with $r_{m}$ and $t_{m}$ the complex reflection and transmission coefficients
of one grating at frequencies far from resonance. These coefficients can be
easily expressed by considering the propagation across a plain (non patterned)
slab. $\theta$ is the phase acquired by photons during their travel in the
spacer layer between the 2 gratings. The term $\mu$ accounts for the
interferential process experimented by photons in the transverse stack.
Obviously, if $r_{m}=0$ (which is our approximation):

$\mu=e^{i(\theta+\theta_{m})}=e^{i\psi},$ (57)

where $\theta_{m}$ is the phase corresponding to the transmission across the
plain slab ($t_{m}=|t_{m}|e^{i\theta_{m}}$). In other words, the Hamiltonian
in the main text is strictly valid if the slab thickness is half-wavelength
(for the eigenfrequency). On the contrary, the error is maximum if the slab
thickness is quarter-wavelength. A numerical solving of Eq. (55), for usual
double gratings, shows that this error can be significant for the imaginary
part of the eigenfrequencies, while being negligible for the real one. More
importantly, Eq. (55) leads to the same qualitative results which are exposed
in the main text (especially $k^{n}$ dependencies of the dispersion). As an
illustration, in the case of the Dirac point at triple BIC (section III.3.4),
simple algebra allows to get the dispersion in the vicinity of the
$\gamma$-point:

$\frac{d\omega_{R}}{dk}=\frac{\pm
v}{\sqrt{1+\frac{2G}{1-|r_{m}|^{2}}\frac{\partial\psi}{\partial\omega_{R}}}}.$
(58)

Out of resonance, the stack can be consider as a simple Fabry-Pérot cavity
closed by mirrors with reflection coefficient $|r_{m}|^{2}$. The lifetime,
$\tau_{rad}$, inside this cavity is therefore (see e.g. [57]):

$\tau_{rad}=\frac{1}{1-|r_{m}|^{2}}\frac{\partial\psi}{\partial\omega_{R}},$
(59)

and the dispersion obtained in the main text (Eqs. (15) and (16)) is found
again:

$\frac{d\omega_{R}}{dk}=\frac{\pm v}{\sqrt{1+\frac{\tau_{rad}}{\tau_{wg}}}}.$
(60)

###  B.2 Derivation of the Hamiltonian matrix elements

As explained in section II (Theoretical approach), the general 4$\times$4
Hamiltonian is written:

$H=\left(\begin{matrix}H_{11}&H_{21}\\\ H_{12}&H_{22}\end{matrix}\right),$
(61)

where $H_{ij}$ are 2$\times$2 matrices which describe the optical interactions
intra each of the two gratings $(H_{ij=i})$, and inter the 2 gratings
$(H_{ij\neq i})$, with $i=1,2$.

The general expressions of $H_{ij}$ matrices are given below:

$H_{11}=\left(\begin{matrix}\omega_{1}+v_{1}k-i\sqrt{G_{1}G_{1}^{*}}&K_{1}^{*}-iG_{1}^{*}e^{-i(\varphi_{1+}-\varphi_{1-})}\\\
K_{1}-iG_{1}e^{i(\varphi_{1+}-\varphi_{1-})}&\omega_{1}-v_{1}k-i\sqrt{G_{1}G_{1}^{*}}\end{matrix}\right),$
(62)
$H_{22}=\left(\begin{matrix}\omega_{2}+v_{2}k-i\sqrt{G_{2}G_{2}^{*}}&K_{2}^{*}-iG_{2}^{*}e^{-i(\varphi_{2+}-\varphi_{2-})}\\\
K_{2}-iG_{2}e^{i(\varphi_{2+}-\varphi_{2-})}&\omega_{1}-v_{2}k-i\sqrt{G_{2}G_{2}^{*}}\end{matrix}\right),$
(63)
$H_{12}=\left(\begin{matrix}\kappa_{d}-ie^{i\psi}\sqrt{G_{1}G_{2}^{*}}e^{i(\varphi_{1+}-\varphi_{2+})}&\kappa_{c}^{*}-ie^{i\psi}\sqrt{G_{1}^{*}G_{2}^{*}}e^{i(\varphi_{1-}-\varphi_{2+}})\\\
\kappa_{c}-ie^{i\psi}\sqrt{G_{1}G_{2}}e^{i(\varphi_{1+}-\varphi_{2-})}&\kappa_{d}-ie^{i\psi}\sqrt{G_{1}^{*}G_{2}}e^{i(\varphi_{1-}-\varphi_{2-})}\end{matrix}\right),$
(64)
$H_{21}=\left(\begin{matrix}\kappa_{d}^{*}-ie^{i\psi}\sqrt{G_{1}^{*}G_{2}}e^{-i(\varphi_{1+}-\varphi_{2+})}&\kappa_{c}^{*}-ie^{i\psi}\sqrt{G_{1}^{*}G_{2}^{*}}e^{-i(\varphi_{1+}-\varphi_{2-})}\\\
\kappa_{c}-ie^{i\psi}\sqrt{G_{1}G_{2}}e^{-i(\varphi_{1-}-\varphi_{2+})}&\kappa_{d}^{*}-ie^{i\psi}\sqrt{G_{1}G_{2}^{*}}e^{-i(\varphi_{1-}-\varphi_{2-})}\end{matrix}\right),$
(65)

where:

$\sqrt{G_{1}}=\sqrt{\gamma_{1}}exp^{i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}exp^{-i\frac{\phi}{2}},$
(66)
$\sqrt{G_{2}}=\sqrt{\gamma_{2}}exp^{-i\frac{\phi}{2}}+\sqrt{\beta_{2}\gamma_{1}}exp^{i\frac{\phi}{2}},$
(67) $K_{1}=\kappa_{1}e^{i\phi}+\beta_{1}\kappa_{2}e^{-i\phi},$ (68)
$K_{2}=\kappa_{2}e^{-i\phi}+\beta_{2}\kappa_{1}e^{i\phi}.$ (69)

An approximate expression of $\kappa_{c}$ may be derived from classical
coupling rate evaluation based on the spatial overlap of the field
distribution within the two coupled gratings.

$\kappa_{c}=\sqrt{\beta_{1}}\kappa_{2}e^{-i\phi}+\sqrt{\beta_{2}}\kappa_{1}e^{i\phi}.$
(70)

This relation indicates that $\kappa_{c}$ is real when the two gratings are
aligned ($\phi=0$), or for any lateral off-set if they are identical. We give
below the clues for establishing the expressions of the Hamiltonian matrix
elements through a few illustrative exemplifying cases. Let us concentrate on
the elements of the first column of $H$ (first columns of $H_{11}$ and
$H_{12}$), which express the optical field transfers supplied by the forward
wave of grating 1 $a_{1+}$, to all waves $a_{i\pm}$ (i=1,2).

$\left(\begin{matrix}\omega_{1}+v_{1}k-i\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\sqrt{\gamma_{1}}e^{i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}e^{-i\frac{\phi}{2}})e^{i\varphi_{1+}}(\sqrt{\gamma_{1}}e^{-i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}e^{i\frac{\phi}{2}})e^{-i\varphi_{1+}}\\\
\kappa_{1}e^{i\phi}+\beta_{1}\kappa_{2}e^{-i\phi}-i(\sqrt{\gamma_{1}}e^{i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}e^{-i\frac{\phi}{2}})e^{i\varphi_{1+}}(\sqrt{\gamma_{1}}e^{i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}e^{-i\frac{\phi}{2}})e^{-i\varphi_{1-}}\\\
\kappa_{d}-ie^{i\psi}(\sqrt{\gamma_{1}}e^{i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}e^{-i\frac{\phi}{2}})e^{i\varphi_{1+}}(\sqrt{\beta_{2}\gamma_{1}}e^{-i\frac{\phi}{2}}+\sqrt{\gamma_{2}}e^{i\frac{\phi}{2}})e^{-i\varphi_{2+}}\\\
\kappa_{c}-ie^{i\psi}(\sqrt{\gamma_{1}}e^{i\frac{\phi}{2}}+\sqrt{\beta_{1}\gamma_{2}}e^{-i\frac{\phi}{2}})e^{i\varphi_{1+}}(\sqrt{\beta_{2}\gamma_{1}}e^{i\frac{\phi}{2}}+\sqrt{\gamma_{2}}e^{-i\frac{\phi}{2}})e^{-i\varphi_{2-}}\end{matrix}\right).$
(71)

The matrix element in red expresses the optical field transfer from $a_{1+}$
to $a_{1+}$, mediated by the $1^{st}$ order diffraction processes at the
$\Gamma$ point occurring between the guided wave $a_{1+}$ and the radiation
continuum. This transfer takes place in two successive steps, as illustrated
in the figure below:

Figure 21: $a_{1+}/a_{1+}$ radiative coupling

Diffraction of $a_{1+}$ is induced by the corrugation of grating 1 (factor
$\sqrt{\gamma_{1}})$ and by the corrugation of grating 2, the action of the
latter being limited to the evanescent tail of the guided wave $a_{1+}$
(factor $\gamma_{2}$, weighted by the parameter $\beta_{1}$). The coefficients
$e^{\pm i\frac{\phi}{2}}$, with $\phi=2\pi\delta$, correspond to the $1^{st}$
order diffraction phase-shift at $\Gamma$ point between the two gratings due
to the lateral offset $\delta$. The coefficients $e^{\pm i\varphi_{1+}}$ and
$e^{\pm i\varphi_{2+}}$ account for the phase shifts resulting from the
diffraction of $a_{1+}$ to (+) or from (-) the radiation continuum, induced by
gratings 1 and 2 respectively. If the grating unit cell is laterally
symmetric, $\varphi_{i+}=\varphi_{i-}$. If the two gratings are identical,
$\varphi_{1\pm}=\varphi_{2\pm}$. When the two gratings are far apart, this