# Exploring the route from leaky Berreman modes to bound states in continuum

Ghanasyam Remesh1, Pravin Vaity2, Venu Gopal Achanta2 and Subhasish Dutta
Gupta 1,3 111Corresponding author, e-mail<EMAIL_ADDRESS>1School of
Physics, University of Hyderabad, Hyderabad-500046, India.
2Department of Condensed Matter Physics and Material Science, Tata Institute
of Fundamental Research, Mumbai 400005, India.
3Tata Centre for Interdisciplinary Sciences, TIFRH, Hyderabad 500107, India.

###### Abstract

We study coupling of leaky Berreman modes in polar dielectric films (SiO2)
through a thin metallic layer (gold) and show the familiar signatures of
normal mode splitting. Due to very large negative real part of the dielectric
function of gold, the splitting shows up only for extremely thin coupling
layers. In contrast, coupling of Berreman modes through a dielectric spacer
layer reveals novel possibilities of having bound states in continuum, albeit
in the limit of vanishing losses. It is shown that the corresponding
dispersion branches of the symmetric and antisymmetric modes can cross. BIC is
shown to occur on one of these branches which is characterized by lower loss.
In fact the BIC corresponds to the point where the radiative losses are
minimized. For thicker layers (both spacer and the polar dielectric) BIC is
shown to occur on the higher order dispersion branches. The origin of BIC is
traced to the Fabry-Perot type mechanism due of the excitation of the leaky
guided modes in the central layer.

##  I Introduction

In recent years there has been a great deal of interest in the bound states in
continuum (BIC) DouglasStone2016 ; Koshelev:20 ; azzam2021 ; Monticone2018 ;
Li2016 ; Sadreev2020 ; Sadreev2018 . The interest in BICs is motivated by the
complete confinement of the associated fields leading to small mode volume
with no leakage, resulting in extremely large quality factor and local field
enhancements. Recall that all these features facilitate efficient light-matter
interaction. Moreover, these discrete bound states, also referred to as
embedded eigenstates, coexist with the continuum. It is not surprising that
the generic nature of BIC’s irrespective of the specialized area have found
many practical applications. Recent reviews DouglasStone2016 ; azzam2021
summarize some of the areas where BIC’s have been discovered starting from the
pioneering works of von Neumann and Wigner, Stillinger and Herrick, Friedrich
and Wintgen vonNeuman1929 ; Stillinger1975 ; Friedrich1985 . Perhaps the
simplest possible structure that can lead to BIC exploits the Fabry-Perot (FP)
type systems supporting coupled resonances Borisov2008 ; Fan2003 . This could
be a generic symmetric coupled waveguide/plasmonic system. Ocurrence of BIC in
such structures is closely linked to the epsilon near zero (ENZ) behavior of
the films that support them. The literature on the ENZ materials and their
various posibilities and applications is truly vast Shalaev2019 ; Engheta2017
; Reshe2019 ; Alu2006 . BIC has been reported in a single ENZ slab Li2017  and
also in spherical core-shell or layered spherical structures with ENZ
constituents silveirinha2014 ; monticone2014 . Very recent studies report the
possibility of multiple BICs in two identical ENZ films separated by a
dielectric Li2016 ; sakotic2020 . The ENZ behavior was modeled with a Drude
model ignoring the intrinsic losses. Since the BIC can be considered as a
limiting case with zero leakage and null line width with diverging quality
factor, it is pertinent to investigate the route from standard leaky modes
(resonances with finite width) to the BIC (resonance with null width).

In this context we pick a polar dielectric film on a metal or dielectric
substrate, which can support the Berreman modes. Since the pioneering work of
Berreman Berreman1963 , these modes have been studied in detail both
theoretically and experimentally Shaykhutdinov2017 ; Vassant2012a ;
Harbecke1985 ; Bichri1997 ; Bichri1993 ; Newman2015 . Several studies have
been carried out to understand the non-standard properties of the leaky modes
Marcuvitz1956 ; Alu2015 ; Hu2009 . Existence of the Berreman modes is also
connected to the epsilon near zero (ENZ) behavior. There are several reports
of Berreman modes in ENZ films Vassant2012a ; Passler2019 ; Chen2013 ;
Caligiuri . A detailed analysis of the dispersion and the field profiles of
these modes in an SiO2 film on a gold substrate has been presented by Vassant
et al. Vassant2012a . Thereafter, other studies highlight the local field
enhancement and associated applications in sensing and nonlinear optics
Passler2019 ; Bello2017 . Only very recently the Berreman modes have been
linked to the ocurrence of BIC in the context of a Drude model sakotic2020 .
In the same reference a polar dielectric (SiC) was considered briefly to show
the feasibility of high-Q quasi-BIC in presence of losses. In this paper we
study coupled Berreman modes where the coupling is mediated by (a) a metal
(Au) film and (b) a dielectric (air) layer. It is clear that (a) leads to an
evanescent coupling with usual normal mode splittings and avoided crossings as
with coupled plasmons of a thin metallic film sdgbook . Our major interest is
in scheme (b) where the polar dielectric films near the epsilon near zero
(ENZ) condition act like ideal mirrors of the FP cavity, leading thereby to
the possibility of BIC Li2016 ; sakotic2020 . Our main target is to explore
the route from the leaky Berreman modes to the BIC. In particular, in order to
establish the connection of the leaky symmetric or the antisymmetric mode
(hereafter symmetry will relate to the symmetry of the magnetic field
distribution) to the BIC we ignore the intrinsic losses in the SiO2 film. We
show that, depending on the symmetry of the mode (symmetric or antisymmetric)
a continuous tuning of the operating point moves the leaky mode to the BIC. In
other words the BIC can be recognised as the limiting case of the one of the
eigen modes which can exhibit null radiation leakage. The results obtained are
explained by a generic Hamiltonian as in Haus1983 . We also studied the leaky
guided modes of the spacer layer with larger thickness to show analogous
effects with the symmetric and antisymmetric branches of the dispersion. The
results are also validated by direct calculation of the reflectivity and
superposing the dispersion data on it.

The structure of the paper is as follows. In Section II, we present the
formulation of the problem. Section III presents the numerical results for
both the schemes (a) and (b) and discusses the results. In Conclusion we
summarize the main findings of the paper.

##  II Formulation of the problem

Consider the symmetric structure shown in Fig. 1 where two thin films of polar
dielectric (e.g. SiO2) are separated by a spacer layer of metal (Au) or
dielectric (e.g. air). Let the dielectric function of Au and SiO2 be given by
Vassant2012a ; Meyer2006

Figure 1: Schematics of the symmetric structure with Silica-Metal/Dielectric-
Silica layers. The medium outside is assumed to be air.

$\begin{split}\epsilon_{Au}(\lambda)&=\epsilon_{Au,\infty}-\frac{1}{\lambda_{p}^{2}(1/\lambda^{2}+i/\gamma_{p}\lambda)}\\\
&+\sum_{i=1,2}\frac{A_{i}}{\lambda_{i}}\left[\frac{e^{i\Phi_{i}}}{(1/\lambda_{i}-1/\lambda-i/\gamma_{i})}+\frac{e^{-i\Phi_{i}}}{(1/\lambda_{i}+1/\lambda+i/\gamma_{i})}\right],\end{split}$
(1)

where $\epsilon_{Au,\infty}=1.53$, $\lambda_{p}=0.145\mu$m ,
$\gamma_{p}=17.0\mu$m, $A_{1}=0.94$, $\phi_{1}=-\pi/4$,
$\lambda_{1}=0.468\mu$m, $\gamma_{1}=2.3\mu$m , $A_{2}=1.36$,
$\phi_{2}=-\pi/4$, $\lambda_{2}=0.331\mu$m, $\gamma_{2}=0.94\mu$m, and

$\epsilon_{SiO2}=\epsilon_{\infty}\frac{\omega^{2}-\omega_{L}^{2}+i\omega\Gamma}{\omega^{2}-\omega_{T}^{2}+i\omega\Gamma},$
(2)

where $\epsilon_{\infty}=2.095$, $\omega_{L}=2.30\times 10^{14}$ rad/s
($\lambda_{L}=8.19\,\mu m$), $\omega_{T}=1.98\times 10^{14}$ rad/s
($\lambda_{T}=9.51\,\mu m$) and $\Gamma=1.35\times 10^{13}$ rad/s
($\lambda_{\tau}=139.53\,\mu m$).

We have carefully studied the applicability of models given by Eqs.(1) and (2)
over the range of frequencies in our study. Eq.(1) for gold was designed to
fit the experimental data of Johnson and Christie Johnson1972 , which does not
cover our frequency range. However, a later study by Babar and Weaver
Weaver2015  covers a much wider range including our spectral domain. A
comparison of the results for the real and imaginary parts of the dielectric
function of gold from these experimental sources with the model in their
respective ranges of wavelength reveals that over the range of wavelengths
considered in Johnson1972  the fit is truly good, while in the extended range
of Weaver2015  it is qualitative retaining the essential features of gold at
larger wavelengths (not shown). As will be shown below in section III.1, the
important feature is the large and negative value of the real part of
dielectric function of gold, which is adequately captured by Eq.(1), yielding
the correct physics.

The model for SiO2 (Eq.(2)) with the parameters mentioned in the text was
checked against the experimental data of Philipp Philipp1985  and as can be
seen from the comparison in Fig. 2, is an acceptable model. What is important
is that the ENZ behavior (Real($\epsilon_{SiO_{2}})=0$ near $\lambda\sim
8.12\,\mu$m corresponding to $\omega=\omega_{L}$) occurs nearly at the same
location in Fig. 2. Recall that for BIC we need to look at a lossless system
with $\Gamma=0$ in Eq.(2). The corresponding $\epsilon_{SiO_{2}}$ is shown by
the dashed line in Fig. 2(a).

The generic features of a polar dielectric (like SiO2) given by Eq.(2) show
that close to the frequency $\omega_{L}$, one has an epsilon near zero
behavior, while a resonant behavior is exhibited at $\omega=\omega_{T}$ (see
Fig. 2(a)). The ENZ behavior can lead to interesting electromagnetic
phenomena, which can be easily seen if one considers an interface between a
conventional dielectric (say, with $\epsilon_{1}$) and a polar dielectric
(with $\epsilon_{2}$). Due to the boundary conditions for the normal component
of the electric field, one can expect a large enhancement of the field
($E_{2}^{\perp}=\epsilon_{1}/\epsilon_{2}E_{1}^{\perp}$ as $\epsilon_{2}$
tends to zero). Further, as shown in Vassant2012a , one can expect the
presence of two modes (inside and outside the lightcone) near the ENZ
condition $\epsilon_{\text{SiO}_{2}}(\omega)=0$. The first mode is referred to
as the Berreman mode (which is leaky in nature), and it is responsible for the
large absorption in thin polar dielectric films first demonstrated by Berreman
in his seminal paper Berreman1963 . The other mode (evanescent in character),
present outside the lightcone, was labeled as the ENZ mode due to its
proximity to the $\omega=\omega_{L}$ line. It was shown that the ENZ mode was
related to the long-range surface modes, and exhibits a nearly dispersionless
behavior for thin films. It is clear from the evanescent nature of this mode
that one cannot couple incident plane waves to the evanescent ENZ mode via
oblique incidence due to the momentum mismatch, unless one uses a high index
prism or a grating.

We use the analytical expressions of the dielectric functions of both metal
and polar dielectric so as to enable us to calculate the roots of the
dispersion relations in order to make connection with the existing results and
validate our codes. We first consider the manifestations of the coupling
through the gold film retaining the full losses in the SiO2 layer. We focus
only on the coupled Berreman modes. As will be shown below, the coupling leads
to the familiar normal mode splitting.

Figure 2: (a) Real and (b) imaginary parts of the permittivity of SiO2 as
functions of wavelength. Blue solid line is from Eq.(2), and the red line with
dots are the experimental data taken from Philipp1985 . The ENZ behavior due
to $\omega=\omega_{L}$ can be seen near $\lambda\sim 8.12\,\mu m$. The
resonant behavior at $\omega_{T}$ is near $\lambda\sim 9.52\,\mu m$. The
dashed blue line is the value of $\epsilon_{SiO_{2}}$ without damping
($\Gamma=0$).

It is pertinent to mention here that we used a standard characteristic matrix
method sdgbook  to calculate the reflection/transmission characteristics of
the structure for a p-polarized plane wave incidence at an angle $\theta_{i}$.
The corresponding symmetric and antisymmetric dispersion relation for the
structure (shown in Fig. 1) is given by sdgbook

$\displaystyle D_{sym}(k_{x},\omega)$ $\displaystyle=$ $\displaystyle
m_{21}+m_{22}p_{tz}=0$ (3) $\displaystyle D_{antisym}(k_{x},\omega)$
$\displaystyle=$ $\displaystyle m_{11}+m_{12}p_{tz}=0$ (4)

where $p_{jz}={k_{jz}}/{k_{0}\epsilon_{rj}}$ is the normalized transverse wave
vector, given by $k_{jz}=\pm\sqrt{\epsilon_{rj}k_{0}^{2}-k_{x}^{2}}$, and
$\epsilon_{rj}$ is the relative dielectric permittivity of the $j^{th}$
medium. The correct branch of $k_{jz}$ is chosen such that
$\text{real}(k_{jz})+\text{imag}(k_{jz})>0$ Vassant2012a . $m_{ij}$ are the
elements of the characteristic matrix for the upper half of the structure
$M_{T}=M_{0}(d_{0}/2)M_{1}(d_{1})$, with $M_{j}$ being the characteristic
matrix of the $j$th layer. In Eqs.(3) and (4) $k_{x}$ is the surface component
of the wave vector which is assumed to be real as in Vassant2012a . We
investigated the roots (complex $\omega$) of the dispersion relation by using
standard root finding algorithm. The results were validated by additional
Lumerical FDTD simulations of the same structure.

As mentioned in the Introduction our aim is to establish a link between the
Berreman modes and possible BIC’s in such structures. Keeping in view of the
fact that pure BIC in quantum systems occurs in absence of any losses, we look
at the coupled system with dielectric spacer layer in the limit of vanishing
material losses ($\Gamma=0$). We show below that the BIC occurs on one of the
branches of the coupled system. Note that BIC in analogous systems of two
coupled metal films with null losses has been reported using ENZ behavior of
metals at the plasma frequency Li2016 ; sakotic2020 . The details of our
investigation and the results are presented in the next section.

##  III Numerical results and discussions

In order to validate our code we first recovered the results presented by
Vassant et al. Vassant2012a  for a silica film on a gold substrate (not
shown). Since our target is to understand the link between the leaky modes and
BIC, we focus our attention only to the Berreman modes occurring on the left
of the light line ($k_{x}<\sqrt{\epsilon_{i}}k_{0}$) close to the ENZ
condition $\omega\sim\omega_{L}$. The results for the middle gold and air film
are presented separately in the following subsections.

###  III.1 Coupling via evanescent waves

It is well known that at lower frequencies ($\omega<<\omega_{p}$), the real
part of the dielectric function of gold is large and negative indicating a
very small skin depth. Thus coupling of the Berreman modes is possible only
for extremely low thickness of the coupling layer. The coupling manifests
itself as normal mode splittings into the symmetric and the antisymmetric
modes. As mentioned earlier symmetry will be defined by the symmetry of the
$p-$polarized magnetic fields. The results for the normal mode splittings for
the real and imaginary parts of the roots of the dispersion relations Eqs.(3)
and (4) are shown in Fig. 3. We have plotted the real (imaginary) parts of the
roots in the left (right) column. The top row shows these roots as functions
of $d_{0}$ (Figs. 3(a),(b)) for fixed $k_{x}=0.7~{}\mu m^{-1}$. The middle
(bottom) row depicts the same as functions $k_{x}$ for fixed $d_{0}=0.02~{}\mu
m$ in Figs. 3(c),(d) ($d_{0}=0.005~{}\mu m$ in Figs. 3(e),(f)). The results
are analogous to surface plasmon coupling in a thin metal film. For the
parameters of SiO2 and for large gold thickness $\sim 15$ nm, there was no
experimental evidence of coupling between the Berreman modes Bichri1997 . The
splitting shows up for gold film thickness of about $d_{0}\lesssim 0.005~{}\mu
m$. As can be seen from these figures, the coupling induced splittings are
more prominent for lower thickness of the gold film, since the thin metal film
allows for better coupling between the modes. These features are all expected,
though they were not well studied in the context of Berreman modes. As will be
shown below, novel features appear when the spacer layer between the two polar
dielectric films is a dielectric. In order to avoid complications due to
material dispersion, we assumed the spacer layer to be air and the
communication between the Berreman modes is established by propagating waves.

Figure 3: (a) Real and (b) imaginary parts of the normalized frequency
$\omega/\omega_{L}$ for symmetric (solid red) and antisymmetric (dashed blue)
modes as functions of the gold thickness, $d_{0}=d_{\text{Au}}$ for fixed
surface component of wave vector $k_{x}=0.7~{}\mu m^{-1}$. (c) Real and (d)
imaginary part of $\omega/\omega_{L}$ as functions of of $k_{x}$ for
$d_{1}=d_{\text{SiO}_{2}}=0.02~{}\mu m$ and $d_{0}=0.02~{}\mu m$. (e),(f) same
as in (c),(d) except for $d_{0}=0.005~{}\mu m$. The black dashed line
represent the lightline $k_{x}=\displaystyle\omega/c$.

###  III.2 Coupling via propagating waves

The results of the previous subsection were obtained retaining the full losses
both in SiO2 and the Au layers. The pure BIC can exist only in systems without
losses. Keeping this in view we consider a theoretical model where we study
the same structure albeit with vanishing losses ($\Gamma=0$). In short, we
study Fabry-Perot type BIC mediated by the modes of the symmetric structure
where two thin SiO2 films with vanishing losses are separated by an air gap.
The coupling via the propagating waves leads to the symmetric and the
antisymmetric modes. These modes are calculated exploiting the symmetry of the
structure as in sdgbook .

Figure 4: (a) Real and (b) imaginary parts of the normalized frequency
$\omega/\omega_{L}$ as functions of $k_{x}$ for
$d_{1}=d_{\text{SiO}_{2}}=0.02~{}\mu m$ and $d_{0}=d_{\text{Air}}=5.6~{}\mu
m$. The BIC is marked by a black point P1 in (a). (c),(d) Same as in (a),(b),
but retaining full losses in the SiO2 layers. The red solid (blue dashed) line
depicts the symmetric (antisymmetric) mode. The black dashed (dotted) line
represents the lightline (ENZ line $\omega/\omega_{L}=1$). (e) Shows the
intensity reflection $R$ as a function of angle of incidence and frequency.
The dispersion curves for symmetric (white solid) and antisymmetric (dashed
white) are superimposed. (f) Intensity reflection as a function of the
frequency at a fixed angle of incidence ($\theta_{i}\simeq 42.96^{0}$ at which
BIC occurs).

.

We now pay attention to the central result of this study, namely, the
occurrence of the BIC on either of the dispersion branches having the least
radiation leakage. For very thin SiO2 layers the two dispersion branches for
the real part can cross. Crossing in the imaginary part albeit at a different
value of $k_{x}$ is also observed. The possibility of crossing of the real or
imaginary parts of the dispersion curves has been discussed in detail
Rotter2003 . In a different context of exceptional points in non-Hermitian
systems crossing of the dispersion branches for the real part has been pointed
out Alu2019 . Such crossing in the real part is associated with the repulsion
of the branches for the imaginary part. Another important aspect is the
overlap of the crossing of the dispersion curves with the ENZ line,
Re$(\omega)=\omega_{L}$ in the $\omega$–$k_{x}$ plane. Note that when the ENZ
condition is met, the polar dielectric layers act as ideal mirrors with unity
reflectivity, thus meeting the requirement for the Fabry-Perot type BIC. The
results for the real and imaginary parts of the roots of the dispersion
relation for complex $\omega$ (normalized to $\omega_{L}$) for
$d_{0}=5.6~{}\mu m$, $d_{1}=0.02~{}\mu m$ are shown in Figs. 4(a) and 4(b),
respectively. Recall that the presence of the finite imaginary part of the
root implies losses of the modes in a system where intrinsic material losses
were ignored. Here losses owe their origin to the leaky character of the
modes. It is interesting to note that for Berreman modes of very thin films, a
fine tuning of the system parameters can make the crossing occurs on the ENZ
line. The BIC is then realized on either of the symmetric or the antisymmetric
branch, depending on which mode can exhibit the least leakage (null loss). It
is clear from Fig. 4(a) that for the said system parameters, the BIC occurs on
the symmetric mode at $k_{x}\simeq 0.5225~{}\mu m^{-1}$ having null leakage
(see Fig. 4(b). This is validated by direct calculation of the reflectivity
(for incidence of a plane p-polarized light at an angle $\theta_{i}$) as a
function of frequency and angle of incidence and superposing the real part of
the dispersion curves on it. While superposing the real root on the
reflectivity data, we have moved from $k_{x}$ to angle of incidence
$\theta_{i}$ for better clarity. BIC can be noted at the crossing of the white
solid and dashed lines (dispersion data), where the blue region shrinks to
null width (see Fig. 4(e)). Fig. 4(f) shows the intensity reflection profile
for a fixed angle of incidence $\theta_{i}\simeq 42.96^{0}$. It is thus clear
that for the said system parameters the symmetric Berreman mode has vanishing
leakage at $k_{x}\simeq 0.5225~{}\mu m^{-1}$, and it is the one which has
evolved to BIC coexisting with the leaky antisymmetric mode. Note that because
of the crossing of the dispersion branches, the same value of $k_{x}$, namely
$k_{x}\simeq 0.5225~{}\mu m^{-1}$, corresponds to the momentum matching for
the excitation of the leaky antisymmetric mode.

In order to confirm the total confinement of the fields at BIC, we have
plotted the eigen mode spatial profiles at and away from the BIC. The results
for upper half of the structure are shown in Fig. 5 for the tangential
intensity profiles $\mu_{0}|H_{y}|^{2}$ and $\epsilon_{0}|E_{x}|^{2}$ as
functions of distance from the center. The inset in Fig. 5(f) shows an
expanded portion just to verify the continuity of the tangential component of
$|E_{x}|$. It is clear that away from BIC one excites the leaky modes while
the BIC corresponds to total confinement with null leakage.

Figure 5: Spatial distribution of the tangential field intensities (a)
$\mu_{0}|H_{y}|^{2}$ and (b) $\epsilon_{0}|E_{x}|^{2}$ of the symmetric
eigenmodes for the structure studied in Fig. 4. (a),(b) before BIC (at
$k_{x}=0.3~{}\mu m^{-1}$). (c),(d) at BIC (P1 at $k_{x}\simeq 0.5225~{}\mu
m^{-1}$) and (e),(f) after BIC (at $k_{x}=0.7~{}\mu m^{-1}$). The dashed
vertical lines represent the boundaries between two different layers. The
inset in (f) shows an expanded part near the boundary to reflect the
continuity of the fields. The other parameters are as in Fig. 4.

In order to understand what happens in a realistic system, we repeated the
above calculations retaining the losses ($\Gamma=1.35\times 10^{13}~{}rad/s$).
The results are shown in Figs 4(c) and 4(d). A comparison of Fig. 4(c) with
Fig. 4(a) reveals that, despite the persistence of the crossing on the ENZ
line both the split modes have losses ruling out the possibility of BIC or a
quasi-BIC with high Q-factor. In fact, the highest quality factor for the
symmetric Berreman mode for a realistic SiO2 film of thickness $0.02$ $\mu m$
with a gap of $5.6\,\mu m$ between them was estimated to be $\sim 17$.
Moreover, the minimum of the absorption for the symmetric mode occurs away
from the crossing point. This is also verified by direct reflectivity
calculations (not shown). These results, along with lack of experimental
evidence of normal mode splittings may suggest that SiO2 may not be the ideal
material for practical applications due to its large intrinsic losses.
However, for other polar materials like AIN, a quality factor of $\sim 81$ was
reported experimentally (see Fig. 2 in Passler2019 ). The feasibility of
quasi-BIC with quality factors as high as $10^{3}$ have been reported in other
structures sakotic2020 ; Sadrieva2017 .

We now show that it is possible to have the BIC occurring on both the
antisymmetric and symmetric branches. To this end we double the central air
layer thickness accomodating the possibility of higher order symmetric and
antisymmetric modes. In particular we take the following system parameters:
$d_{1}=d_{\text{SiO}_{2}}=0.02~{}\mu m$, $d_{0}=d_{\text{Air}}=11.2~{}\mu m$.
The results are shown in Fig. 6(a) and 6(b). The corresponding reflectivity
spectra and the reflectivity profiles are shown in Figs. 6(c) and 6(d),
respectively. As can be seen from these figures that now the BIC occurs at two
points. While the first occurs on the antisymmetric Berreman branch at
$k_{x}\simeq 0.5225~{}\mu m^{-1}$, the second shows up on the symmetric branch
at values of $k_{x}\simeq 0.7135~{}\mu m^{-1}$ corresponding to vanishing
leakage for these modes. Note also that the symmetric Berreman branch can not
lead to BIC at the first crossing at $k_{x}\simeq 0.5225~{}\mu m^{-1}$ due
finite leakage. The remaining features are the same as in Fig. 4. For
completeness we have presented the eigen mode spatial profiles at and away
from BIC in Fig. 7.

The possibility of having BIC with both symmetric and antisymmetric Berreman
modes for thin SiO2 layers is due the fact that both the dispersion branches
oscillate near the ENZ line and they can cross on the same. The situation is
different for thicker SiO2 layers, when the dispersion branches can move away
from the ENZ line with larger radiative leakage. Nevertheless, some of these
dispersion branches can cross the ENZ line leading to BIC, though there is no
crossing of the dispersion branches. The results for a system with
$d_{1}=1~{}\mu m$ with other parameters as in Fig. 6 are shown in Fig. 8. The
corresponding eigen function plots are shown in Fig. 9 for both the BIC’s in
Fig. 8.

Results presented in Fig. 4, Fig. 6 and Fig. 8 can easily be understood in
terms of a generic Hamiltonian given by the following DouglasStone2016 ;
Haus1983

$\displaystyle H=\begin{pmatrix}\omega_{0}&g\\\
g&\omega_{0}\end{pmatrix}-i\gamma\begin{pmatrix}1&e^{ik_{z}d_{0}}\\\
e^{ik_{z}d_{0}}&1\end{pmatrix},$ (5)

where $\omega_{0}$ is the resonance frequency of the uncoupled resonators, $g$
is the coupling constant between them, $\gamma$ is the radiation rate of the
individual resonances and $k_{z}d_{0}$ is the propagation phase shift between
the two resonators with $k_{z}$ being the component of the wave vector along
the $z$-axis. The two eigenvalues of $H$ are

$\displaystyle\omega_{\pm}=\omega_{0}\pm g-i\gamma(1\pm e^{ik_{z}d_{0}}).$ (6)

Here, ‘+’ (‘-’) represents the symmetric (antisymmetric) eigenmodes. When
$k_{z}d_{0}$ is an integer multiple of $\pi$ one of the two eigenmodes becomes
more lossy with twice the original decay rate, and the other eigenmode becomes
a BIC with a purely real eigenfrequency. Odd (even) integer multiples of $\pi$
correspond to symmetric (antisymmetric) BICs. It is thus clear how when we
change the order of the mode, BICs shift from the symmetric to the
antisymmetric branch. For example for $d_{0}=5.6~{}\mu m$, $k_{z}d_{0}=\pi$ at
$k_{x}\simeq 0.5225~{}\mu m^{-1}$ leading to the BIC on the symmetric branch
with $Im(\omega_{+})=0$ (see Fig. 4). In contrast, for $d_{0}=11.2~{}\mu m$,
$k_{z}d_{0}=2\pi$ at $k_{x}\simeq 0.5225~{}\mu m^{-1}$, the BIC occurs on the
antisymmetric branch with $Im(\omega_{-})=0$. It is clear that 1st order BIC
takes place again on the symmetric branch for $k_{z}d_{0}=\pi$ at $k_{x}\simeq
0.7135~{}\mu m^{-1}$ (see Figs. 6 and 8).

Indeed for a valid ENZ condition $\epsilon_{SiO_{2}}=0$ (or
$\omega=\omega_{L}$) ), the solution to the dispersion relation (Eqs.(3) and
(4)) can be solved easily for the values of $k_{x}=k_{0}\sin\theta$ recovering
the above results. For example, Eq.(3) for the symmetric mode can be rewritten
in the form

$\displaystyle-i\frac{p_{0z}}{p_{1z}}\cos(k_{1z}d_{1})\sin\left(\frac{k_{z}d_{0}}{2}\right)-i\sin(k_{1z}d_{1})\cos\left(\frac{k_{z}d_{0}}{2}\right)$
$\displaystyle+\frac{p_{0z}}{p_{1z}}\cos(k_{1z}d_{1})\cos\left(\frac{k_{z}d_{0}}{2}\right)-\frac{p_{0z}^{2}}{p_{1z}^{2}}\sin(k_{1z}d_{1})\sin\left(\frac{k_{z}d_{0}}{2}\right)=0.$
(7)

In Eq.(7) we have replaced $k_{0z}$ by $k_{z}$. For
$\epsilon_{1}=\epsilon_{SiO2}\rightarrow 0$, the terms with
$1/p_{1z}=\frac{k_{0}\epsilon_{1}}{\sqrt{k_{0}^{2}\epsilon_{1}-k_{x}^{2}}}=\frac{\epsilon_{1}}{i\sin(\theta)}$
vanish, leading to the FP BIC condition

$k_{z}d_{0}={k_{0}d_{0}\cos\theta}=\pi$

for the symmetric mode. Analogous procedure for the antisymmetric mode leads
to the other condition.

$k_{z}d_{0}=k_{0}d_{0}\cos\theta=2\pi$

All through this study we have not discussed the trivial BIC occurring at
normal incidence, as expected in any symmetric structure. These are not
interesting since we cannot control their position unlike the BIC’s emerging
from the Berreman or other higher order modes.

Figure 6: (a),(b),(c) The same as in Fig. 4(a),(b),(e), respectively, except
for $d_{0}=d_{\text{Air}}=11.2~{}\mu m$. Note the 2nd (1st) order BIC on the
antisymmetric (symmetric) branch, marked by P2 (P1). (d) The red solid (blue
dashed) line shows the intensity reflection as a function of the frequency at
a fixed angle of incidence $\theta_{i}\simeq 68.52^{0}$ ($\theta_{i}\simeq
42.96^{0}$) corresponding to the 1st (2nd) order BIC. Figure 7: Spatial
distribution of the tangential field intensities of (a) $\mu_{0}|H_{y}|^{2}$
and (b) $\epsilon_{0}|E_{x}|^{2}$ of the symmetric eigenmodes for the
structure studied in Fig. 6(a), (a),(b) at $1^{st}$ order BIC ( P1 at
$k_{x}\simeq 0.7135~{}\mu m^{-1}$), (c),(d) after the 1st order BIC (at
$k_{x}=0.75~{}\mu m^{-1}$). (e),(f) Antisymmetric eigenmode intensity
distributions at $2^{nd}$ order BIC (at $k_{x}\simeq 0.5225~{}\mu m^{-1}$) and
(g),(h) after $2^{nd}$ order BIC (at $k_{x}=0.75~{}\mu m^{-1}$). The inset in
(b) shows an expanded part near the boundary to reflect the continuity of the
fields.The other parameters are as in Fig. 6. Figure 8: (a),(b),(c) The same
as in Fig. 6(a),(b),(c), but for $d_{\text{SiO}_{2}}=1.0~{}\mu m$ and
$d_{\text{Air}}=11.2~{}\mu m$. The dotted red (blue) line represents the
coupled symmetric (antisymmetric) Berreman mode. The red solid (blue dashed)
line represents symmetric (antisymmetric) higher order leaky guided mode. The
2nd (1st) order BIC lies on the antisymmetric (symmetric) branch, marked by P2
(P1). (d) The red solid (blue dashed) lines gives the intensity reflection as
a function of frequency at a fixed angle of incidence $\theta_{i}\simeq
68.52^{0}$ ($\theta_{i}\simeq 42.96^{0}$) corresponding to the 1st (2nd) order
BIC. Figure 9: Spatial distribution of the tangential field intensities of the
eigen modes (a) $\mu_{0}|H_{y}|^{2}$ and (b) $\epsilon_{0}|E_{x}|^{2}$ of the
symmetric eigenmodes for the structure studied in Fig. 8(a). (a),(b) at
$1^{st}$ order BIC (P1 at $k_{x}\simeq 0.7135~{}\mu m^{-1}$), (c),(d) after
the 1st order BIC (at $k_{x}=0.75~{}\mu m^{-1}$). (e),(f) Antisymmetric
eigenmode intensity distributions at $2^{nd}$ order BIC (P2 at $k_{x}\simeq
0.5225~{}\mu m^{-1}$) and (g),(h) after $2^{nd}$ order BIC (at
$k_{x}=0.7~{}\mu m^{-1}$). The other parameters are as in Fig. 8.

##  IV Conclusions

We have studied a symmetric layered structure involving two polar dielectric
(SiO2) films on both sides of a metallic (Au) or dielectric (air) film. The
structure can support coupled Berreman modes. For evanescent coupling through
a thin metallic film, the standard features of avoided crossing and normal
mode mode splittings are reported. For the structure with the dielectric
spacer layer, BIC’s are shown to exist in the limit of vanishing intrinsic
material losses. A thorough analysis of the dispersion branches of the
symmetric and the antisymmetric modes revealed that the BIC’s always occur on
one of these dispersion branches which has the least radiative loss. It is
also shown that for very thin SiO2 films the symmetric and antisymmetric
branches can cross. Moreover, BIC’s are shown to exist near the ENZ condition
of SiO2, when these layers can act like ideal mirrors for the central air gap
justifying the mechanism to be of Fabry-Perot type. Thus a continuous tuning
of the operating point along one of the dispersion branches of the leaky modes
eventually leads to the BIC. These results are validated by direct
calculations of the reflectivity spectra, along with spatial eigen mode
profiles depicting complete confinement. The results can be useful for having
a deeper understanding of the BIC in such structures as the limiting case of
the leaky modes for very rigid set of system parameters.

Acknowledgment
One of the authors (SDG) would like to thank Rishav Banduri for help in
calculations.

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# Reply to “Comment on ‘Nonstandard superconductivity or no superconductivity
in hydrides under high pressure’ ”

J. E. Hirscha and F. Marsigliob aDepartment of Physics, University of
California, San Diego, La Jolla, CA 92093-0319
bDepartment of Physics, University of Alberta, Edmonton, Alberta, Canada T6G
2E1

###### Abstract

In Ref. nonstandard  we surveyed the known hydride superconductors, and
compared their resistive behavior to that of typical known superconductors,
including conventional (e.g. NbN and MgB2) and unconventional (e.g. YBCO)
superconductors, and concluded that the behavior of the hydrides was
indicative of nonstandard or no superconductivity. In the preceding comment,
Talantsev, Minkov, Balakirev and Eremets talantsev_comment  (arXiv:2311.07865)
claim that we presented a “flawed analysis and a selective and inaccurate
report of published data.” Here we show that this claim is wrong.

##  I Introduction

In their Comment, Talantsev, Minkov, Balakirev and Eremets talantsev_comment
claim (first sentence in their abstract, and elsewhere in the Comment) that in
our paper nonstandard  we “assert that hydrogen-rich compounds do not exhibit
superconductivity”. That claim is non-factual: our paper claimed instead that
“Our results indicate that either these materials are unconventional
superconductors of a novel kind, which we term “nonstandard superconductors,”
or alternatively, that they are not superconductors”, and that “If the second
is the case, which we believe is more likely, we suggest that the signals
interpreted as superconductivity are either experimental artifacts or they
signal other interesting physics but not superconductivity.” We will assume in
what follows that their statement that our claim “relies on a flawed analysis
and a selective and inaccurate report of published data” refers to our true
claim, not an imagined one. To refute that claim, we follow the sequence of
their sections II-VI, their titles given in italics in the listing below, and
discuss their figures which purport to support their claim.

##  II detailed analysis

1) The transition width definition.

Our observation of nonstandard behavior in the hydrides compared to that in
known superconductors is based on observation of qualitative trends. That such
an approach is required is exemplified by the myriad examples of resistivity
vs. temperature curves with knees, tails, kinks, non-monotonic behavior, etc,
for which an arbitrary rigid definition of width such as given in Eq. (1) of
the Comment is not appropriate. That is, choosing $\Delta T_{0.1-0.9}$ as in
Eq. (1) of the Comment, versus $\Delta T_{0.15-0.85}$ or $\Delta
T_{0.05-0.95}$, would yield qualitatively different answers. For all the
examples discussed in our paper nonstandard , both standard superconductors
and hydride materials, the curves of resistance versus temperature for varying
magnetic fields from which we extracted the widths shown in our Fig. 11 are
shown in our paper also, enabling readers to easily perform their own analysis
with their favorite definition of transition width such as Eq. (1) in the
Comment, and compare with our estimates given in Fig. 11 of nonstandard .
Surprisingly, this was not done in the Comment.

2) Type I superconductors.

We do not believe that the hydrides are type I superconductors, nor does
anyone else as far as we know, so we consider this discussion in the Comment
to be irrelevant. Note that if these hydrides really were type I
superconductors with such high $T_{c}$’s, that don’t expel magnetic fields and
trap magnetic fields according to what is reported by the authors of the
Comment in other publications e2021p ; etrappedp , then indeed they would be
$very$ nonstandard superconductors.

3) Type II superconductors (non-hydrides)

Various examples of resistivity widths for non-hydride type II superconductors
have been provided in the Comment. Shown in their Fig. (3a) are results for
$2.9$ nm thin films of MgB2, with fields applied both perpendicular and
parallel to the plane of the thin film, reported in Ref. mgb2thin . There is
an initial decrease of transition width as a function of applied field,
followed by the standard increase with magnetic field. These films have
thickness of only a few unit cells, smaller than the superconducting coherence
length and much smaller than the London penetration depth. Clearly the
anomalous initial decrease of transition width with magnetic field highlighted
in the Comment is a property connected to ultrathin films. The hydride samples
have thicknesses that are larger by three orders of magnitude and surely
resemble the polycrystalline samples used in the study referred to in our
paper and many other such studies more than they resemble highly oriented
ultrathin films.

Fig. (3b) in the Comment are results reported in Ref. wang2010  for
polycrystalline MgB2 samples doped by SiC, that also show an upturn for low
fields. The authors of Ref. wang2010  discuss this anomalous feature of their
data in great detail, and conclude that “ the origin of the low field $\Delta
T$ upturn is due to multiple superconducting transitions resulting from the
replacement of B and/or Mg atoms by other elements.” We cannot rule out that
multiple transitions in the hydrides could also conceivably give rise to such
an upturn. We are not aware however of such upturns in the hydrides, instead
our paper Ref. nonstandard  highlighted the fact that in many cases the
transition widths are approximately constant as function of field.

In Fig. (3c) in the Comment, a figure is presented for a pnictide,
NaFe1-xCoxAs. In this case, referring to the Comment’s Ref. 27, which we cite
as Ref. choi2017 , the authors “cherry-picked” one composition ($x=0.01$) for
an applied field parallel to the c-axis, from Fig. (4a) of Ref. choi2017 .
However, as subsequent panels in that same Fig. 4 of Ref. choi2017  show,
reproduced here in Fig. (1), all higher compositions ($x=0.03$ and $x=0.07$)
clearly show the normal broadening as a function of applied field. This
compound at this stoichiometry is a poor choice, because, as the authors of
choi2017  indicate, the compound with $x=0.01$ is actually also
antiferromagnetic, with a Néel temperature of 33 K, not too far above the
superconducting transition temperature. The authors also point out that the
superconducting transition at $x=0.01$ is extremely broad for this reason
(compare with panels (c) and (e) in Fig. (1)), so one can imagine that a
reduction in the width with applied field could well reflect the impact of
that field on the underlying antiferromagnetic state. Since no
antiferromagnetic transitions are expected in the hydride materials this is
hardly a suitable choice of material with which to critique our results,
especially leaving out the higher dopant values.

Figure 1: Fig. 4 from Ref. choi2017 . Note the variation of the temperature
scales for panels (a) and (b) ($x=0.01$), compared to (c) and (d) ($x=0.03$)
and to (e) and (f) ($x=0.07$). At $x=0.01$ the material has entered an
antiferromagnetic state, so application of a magnetic field also affects the
underlying magnetic state.

In Fig. (3d) in the Comment, an example of a cuprate material, Bi2Sr2CaCu2O8,
with applied pressure of $9$ GPa is given, using data obtained from the
Comment’s Ref. (28), cited here as Ref. guo2020 . Again, this is a poor choice
of material with which to critique our work, as it was used by the authors of
Ref. guo2020  to observe “a pressure-induced crossover from two- to three-
dimensional (2D to 2D) superconducting states,” resulting in “Berezinskii-
Kosterlitz-Thouless-like behavior”. Therefore a significant amount of
nonstandard physics is occurring in this material, that also exhibits multiple
transitions. We show Fig. (1) from that paper here as Fig. (2). The displayed
resistance versus temperature curves clearly indicate abnormal behavior at low
applied fields, along with the expected broadening with field at higher
magnetic fields, where the resistivity curves lack the abnormality present at
lower fields.

Figure 2: Fig. 1 from Ref. guo2020 . It is a strikingly poor material with
which to conclude anything about transition widths, particularly at low
applied field where a pronounced knee is present in the resistivity data
(panels (b) and (d)). Note the clear indication of two transitions at 9 GPa
(in panel (a)) so the comments from sintered MgB2 paper (Wang et al. wang2010
) about the “abnormal” decrease in transition width could well apply.

4) Highly compressed scandium.

These measurements are at high pressure, and only under the highest pressure
measured (262 GPa) is there an indication from Fig. (4) in the Comment that
the width decreases with increasing fields at very low fields. But this is
precisely where a significant knee occurs (see Fig. 8d in the supplementary
material of the Comment), and so concerns about the interpretation of this
initial decrease echo those above.

5) Hydride superconductors.

Figure 5 of the Comment shows resistance width versus applied field
constructed by the authors based on their Eq. (1), but applied to very
complicated resistance versus temperature measurements. It is not clear what
the purpose of this Figure is. In Fig. 3 we show a number of resistance versus
temperature profiles used by the authors of the Comment in their Fig. 5. The
upper left panel shows the data from Mozaffari et al. mozaffari2019  on H3S
for a variety of applied magnetic fields whose width is shown in Fig. 5a of
the Comment. It is evident that there are difficulties with interpretation, as
a significant kink exists in the low field (including zero field) data.
Mozaffari et al. mozaffari2019  attribute this to inhomogeneities, but do not
have an explanation for why application of a sufficiently high magnetic field
seemingly removes these inhomogeneities, or why there was a time dependence
for these resistance measurements. Whatever the explanation, at the very
least, their resistance versus temperature measurements constitute strong
evidence for nonstandard superconductivity and, in our opinion, it is
inprudent to try to analyze the transition width as a function of applied
magnetic field for such data.

To construct the result in their Fig. (5b), Talantsev et al. use resistance
versus temperature data from Troyan et al. troyan2021  for YH6. We show this
data in the upper right panel of Fig. (3); this figure clearly shows anomalous
characteristics. So once again, if we cannot qualitatively understand the
resistance versus temperature measurements, it seems ill-advised to apply a
formula such as the Comment’s Eq. (1) to analyse their widths.

Figure 3: Some of the resistance versus temperature profiles used in Fig. 5 of
the Comment, and one additional figure (lower right panel). A detailed
description is given in the text. Briefly, all of these results show anomalous
behavior over some temperature and/or magnetic field range, and determining a
transition width using Eq. (1) of Ref. talantsev_comment  is clearly
problematic. In the lower left panel we have included (red) parallel lines to
indicate that the widths of these transitions are not changing at low applied
magnetic field. In the lower right panel we show recent results for $LaH_{23}$
from Ref. recent .

.

In panel (c) of Fig. 5 of the Comment, a number of compounds are included. The
(La,Ce)H9 data (red diamonds) are very similar to the ones we presented for
LaH10 (black squares) and therefore support our claim of very little or
decreasing broadening with applied field. The plotted results for SnH4 are
again misleading as a perusal of the resistance versus temperature data
indicate. These data, from Troyan et al. troyan2023 , are included in the
lower left panel of our Fig. 3, together with four parallel red lines that
represent the slopes of the resistance drops for the three lowest magnetic
fields quite well. The fact that they are parallel indicates that the width is
not changing at all with increasing field, contrary to the claim made in the
Comment, where an increasing slope is indicated as a function of magnetic
field. As the field increases further the resistance drops begin to broaden,
but largely because of an increasing tail at the low temperature end. This is
most apparent for very high applied fields, where the normal state resistance
takes a semiconductor-like upward turn and eventually the near-zero resistance
limit is never achieved. To attribute this abnormal behavior simply to
transition width broadening is very misleading. Finally, the HfHx results
shown in panel (c) of Fig. 5 of the Comment show both a very small width for
zero field and a very small increase as a function of applied magnetic field,
for fields as large as $B_{appl}/B_{c2}=0.3$. This is qualitatively different
from the broadenings seen in standard superconductors shown in Fig. 11a of our
paper Ref. nonstandard , hence we consider these results to be consistent with
the message in our paper.

To conclude, we show in the lower right panel of Fig. 3 recent results for
$La_{4}H_{23}$ from Ref. recent , showing anomalous narrowing of the
resistance drop with increasing field. The authors of Ref. recent  comment
that “The sample demonstrates the absence (and even negative value) of
broadening of superconducting transitions in a magnetic field, as previously
observed for yttrium ($YH_{6}$) and lanthanum-yttrium ($(La,Y)H_{10}$)
hydrides”, confirming the theme of our paper Ref. nonstandard  that the
anomalous behavior in resistance width that we pointed out is common in so-
called hydride superconductors.

##  III Summary and conclusion

To conclude this Reply, we address each point of the “Detailed explanation of
the primary message of the Comment” in its Sect. I, where the authors claim to
“demonstrate the following”:

“1. There is no universal in-field transition width dependence that can serve
as the criterion for determining the existence of the superconductivity in any
given material”

There is a near-universal broadening of the resistive transition versus field
in standard type II superconductors, whose origin is well understood
nonstandard ; blatter . As shown in this Reply, for all examples of
“anomalous” behavior for standard superconductors presented in the Comment we
have shown that there are clear reasons for the anomalous behavior that are
not expected to be present in the hydrides. Absence of such broadening in many
hydride materials suggests that the resistance drops seen in hydrides are not
due to superconductivity but due to other reasons hmnature ; dc ; enormous ;
vdp .

“2. Hydrides at high pressures exhibit a similar in-field transition width
dependence to other superconductors, such as MgB2”

The typical behavior of $MgB_{2}$ samples is very standard, the few samples
that show anomalous behavior do it for reasons that don’t apply to the
hydrides.

“3. The authors employed undisclosed definitions/methods for extracting: the
transition temperature and the transition width from raw experimental
resistance data”

We showed figures for the resistance curves for all cases discussed in our
paper, the qualitative behavior that we highlighted is very clear from those
figures. The authors of the Comment have not shown that their criteria for
transition width gives qualitatively different behavior than what we showed in
Fig. 11 of our paper Ref. nonstandard .

“4. The authors[1] selectively presented datasets that support their claims
while excluding datasets that contradict them”

We presented a representative set of examples available at that time. Since
then other materials have been measured, and several of them show nonstandard
behavior as discussed in the previous section.

“5. The authors[1] misinterpreted the magnetic phase diagrams of
superconductors by overlooking the existence of the intermediate state in
type-I superconductors.”

The intermediate state in type-I superconductors is irrelevant to the hydrides
claimed to be high temperature type II superconductors with strong pinning
centers etrappedp .

In conclusion, we argue that the Comment talantsev_comment  has no merit. The
points that we made in our paper Ref. nonstandard  are valid and important, an
early warning that the resistance drops observed for hydrides under pressure
may not be due to superconductivity. Since we wrote Ref. nonstandard ,
substantial additional evidence has emerged that the hydrides are either
“nonstandard superconductors” or, more likely, not superconductors nsr ;
hysteresis ; hmtrapped2 . In addition, we point out that two papers csh ; luh
claiming high temperature superconductivity in hydrides that showed anomalous
behavior in the resistance data hmnature ; enormous  have been retracted
cshretr ; luhretr  and a third one yttrium  is under investigation, and other
claims hemleycsh ; hemleyluh ; hemley550  have not been reproduced.

###### Acknowledgements.

FM was supported in part by the Natural Sciences and Engineering Research
Council of Canada (NSERC).

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