# Taming Friedrich-Wintgen interference in resonant metasurface: vortex laser
emitting at on-demand tilted-angle

Raphael Mermet-Lyaudoz  Univ Lyon, Ecole Centrale de Lyon, INSA Lyon,
Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, Ecully 69130,
France  Clémentine Symond  Univ Lyon, Université Claude Bernard Lyon 1, CNRS,
Institut Lumière Matière, F-69622, Lyon, France  Florian Berry  Univ Lyon,
Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon,
CNRS, INL, UMR5270, Ecully 69130, France  Emmanuel Drouard  Univ Lyon, Ecole
Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS,
INL, UMR5270, Ecully 69130, France  Céline Chevalier  Univ Lyon, Ecole
Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS,
INL, UMR5270, Ecully 69130, France  Gaëlle Trippé-Allard  Lumière, Matière et
Interfaces (LuMIn) Laboratory, Université Paris-Saclay, ENS Paris-Saclay,
CNRS, CentraleSupélec, 91190 Gif-sur-Yvette, France  Emmanuelle Deleporte
Lumière, Matière et Interfaces (LuMIn) Laboratory, Université Paris-Saclay,
ENS Paris-Saclay, CNRS, CentraleSupélec, 91190 Gif-sur-Yvette, France  Joel
Bellessa  Univ Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière
Matière, F-69622, Lyon, France  Christian Seassal  Univ Lyon, Ecole Centrale
de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL,
UMR5270, Ecully 69130, France  Hai Son Nguyen<EMAIL_ADDRESS>Univ
Lyon, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CPE
Lyon, CNRS, INL, UMR5270, Ecully 69130, France

###### Abstract

Friedrich-Wintgen (FW) interference is an atypical coupling mechanism that
grants loss exchange between leaky resonances in non-Hermitian classical and
quantum systems. Intriguingly, such an mechanism makes it possible for
destructive interference scenario in which a radiating wave becomes a bound
state in the continuum (BIC) by giving away all of its losses. Here we propose
and demonstrate experimentally an original concept to tailor FW-BICs as
polarization singularity at on-demand wavevectors in optical metasurface. As a
proof-of-concept, using hybrid organic-inorganic halide perovskite as active
material, we empower this novel polarization singularity to obtain lasing
emission exhibiting both highly directional emission at oblique angles and
polarization vortex in momentum space. Our results pave the way to steerable
coherent emission with tailored polarization pattern for applications in
optical communication/manipulation in free-space, high-resolution imaging
/focusing and data storage.

###### keywords:

Friedrich-Wintgen inteference, vector vortex beam, bound states in a
continuum, halide perovskites, metasurfaces, micro-laser, nanophotonics

Institut Universitaire de France (IUF), 75231 Paris, France

##  1 Introduction

Vector vortex light are optical beams having non-uniform polarization pattern
that rotates around a singularity point and exhibiting intensity cross section
in form of a doughnut shape 1, 2, 3. Over the last years, these exotic light
sources have aroused great interests due to tremendous applications in quantum
optics 4, 5, optical manipulation/trapping 6, 7, light tight-focusing 8,
optical communication in free-space9, 10 and data storage 11. Notably, the
generation of vector vortex beams often relies on bulky and/or complex setups:
transformation of a Gaussian beam in free space or guided light via optical
elements 12, 13, 14, 15, integrated circuit including lasers with integrated
polarization or phase modulation components 16, combining external frequency
feed-back and anisotropic media 17. Thus making on-chip vector vortex laser
that gets rid of all external components on one hand, and having advantages in
terms of robustness, efficiency and reliability on the other, is an appealing
research topic of contemporary photonics.

An elegant strategy to make on-chip vector vortex laser is to use optical
bound states in the continuum (BICs) in photonic lattice 18, 19, 20. They are
peculiar photonic resonances forbidden to radiate into free space despite
lying in a continuum of propagating waves. As consequence, the radiation
pattern of isolated BICs exhibits polarization vortex in momentum space 21, 22
with the vorticity dictated by the topological nature of the resonances 23,
24, 25. When combined with high gain materials, one may obtain lasing action
of which the emission is a vector vortex beam pinned at the BIC location in
momentum space. Up to now, vector vortex lasers based on BIC have been
reported for different lattice geometries and gain materials 26, 26, 27, 28,
29, 30, 31. Remarkably, the polarization pattern or the lasing wavelength of
BIC vector beams could be dynamically tuned by harnessing non-linearity
originated from free-carriers injections 28, phase change material 29 or
exciton-polariton strong coupling regime 30. However, to our knowledge, all of
vector vortex microlasers in the literature are pinned at the $\Gamma$ point
of the momentum space. Thus they are not ready for many applications such as
light tracking, manipulation or steering without external optical elements.

Seminal work of Friedrich and Wintgen (FW) showed that BICs can be obtained by
harnessing destructive interference mechanism between two leaky resonances32.
In such a paradigm, one resonance gives away all of its losses to the other
one and becomes a FW-BIC. Interestingly, FW-BICs usually take place in the
vicinity of the avoid crossing point between the two resonances 20. Thus
tailoring the avoid crossing point in momentum space is an effective route to
design FW-BIC at arbitrary oblique wavevector. This promises a salient
configuration to make vector vortex microlasers pointing at high-tilted angle.

In this work, we propose a novel proof-of-concept for on-chip vector vortex
laser emitting at high oblique angle. The lasing resonance is an original FW-
BIC that is tailored by engineering destructive interference out of a
Hermitian triple degeneracy of Bloch resonances in subwavelength metasurface.
This concept allows isolating the FW-BIC in both momentum space directions.
Moreover, the location of this FW-BIC can be designed on-demand by tuning the
in-plane anisotropy of the structure. We demonstrate experimentally lasing
action at 550nm from a halide perovskite metasurface. Angle-resolved combined
with polarization resolved measurements reveal a coherent vortex emission,
pointing at 21 degrees. All experimental results are perfectly explained by an
analytical model based on FW interference mechanism, and nicely matched with
numerical simulations. Our demonstrated vector vortex on-chip would open the
way for steerable coherent emission with tailored polarization patterns.

##  2 Results and discussion

###  2.1 Nanophotonic concept and design

The photonic resonance of our vector vortex laser is designed to exhibit
farfield polarization singularity that is located at a photonic band edge of
the energy-momentum dispersion $E(k_{x},k_{y})$. These singularities,
corresponding to FW-BICs, is tailored at arbitrary point of the momentum space
by engineering FW hybridization between Bloch resonances from subwavelength
metasurface. Indeed, a suppression of radiative losses, accompanied by an
avoid crossing, results from the interplay between different coupling
mechanisms (direct couplings via nearfield effects and radiative couplings via
the radiative continuum) in the vincinity of the crossing point for uncoupled
resonances when sweeping parameters such as wavevectors, geometrical sizes,
refractive index… Here the set of Bloch resonances is chosen so that the
crossing point takes places in momentum space with both $k_{x}$ and $k_{y}$
wavevectors as sweeping parameters. Thus the destructive interference
configuration leads to an isolated FW-BIC. This isolated FW-BIC corresponds to
a farfield radiation-singularity that is pinned at a photonic band edge with
infinite quality factor.

Figure 1: (a) Scheme of the perovskite based metasurface and the corresponding
unit cell, and Brillouin zone. The unit cell is a rectangular lattice having
$\eta$ as the anisotropy factor of period mismatch along $x$ and $y$-axis. The
Brillouin zone is also a rectangular cell having $2\pi/\eta a$ and $2\pi/a$ as
size along $k_{x}$ and $k_{y}$ respectively. (b) Angle-resolved absorption
simulations along $k_{x}$ direction, when $k_{y}=0$. The orange rectangular
corresponds to a zoom in to highlight the FW-BIC. (c) Zoom in of the angle-
resolved absorption simulation along $k_{y}$ direction, when $k_{x}=k_{BIC}$,
to highlight the FW-BIC. In both (b) and (c), the FW-BIC is indicated by the
red arrow; black lines are fittings using effective theory of FW couplings
between the three guided resonances $(-1,\pm 1)$ and $(+1,0)$; white dashed
lines indicate the uncoupled modes. (d,e) Results from numerical simulations
of the quality factor (d) and polarization texture (e) in the vicinity of the
FW-BIC. (f) Location of the FW-BIC in $k_{x}$ direction as a function of the
anisotropy factor $\eta$. Red circles are results from numerical simulations.
Black line is the analytical prediction.

Hybrid organic-inorganic halide perovskite has been chosen as active material
for our proof-of-concept due to their maturity and excellent properties for
solution-processable lasers (direct bandgap, high gain, low trap density) 33,
34, with lasing effect in perovskite metasurface previously reported by
different groups 35, 36, 37, 38, 39, 28, 29. Our photonic metasurface,
depicted in Fig. 1a, is made of a CH3NH3PbBr3 perovskite slab of thickness
$t=270$ nm. The perovskite slab is partially corrugated by a rectangular
lattice of air holes, with patterning depth $h=80$ nm and diameter $d=200$ nm.
The lattice period along $y$-axis of $a=340$ nm and the one along $x$-axis of
$\eta\times a$ are slightly mismatched, with the anisotropy factor $\eta=0.9$.
The anisotropy of the unit cell in real space correspond to the same
anisotropy of the Brillouin zone in momentum space, having a $C_{2}$ symmetry
$\Gamma$, X and Y as high symmetry points (see Fig. 1a). This anisotropy plays
a key role to tune the farfield location of the targeted FW-BIC, as explained
in the following.

Figure. 1b shows the simulated photonic band structure of the metasurface
along $\Gamma$X direction (i.e. sweeping $k_{x}$, with $k_{y}=0$) obtained by
absorption calculation using the Rigorous Coupled Wave Analysis (RCWA) (see
Methods section). We pay particular attention to three photonic bands shown in
the zoom of the simulation results (orange rectangle in Fig. 1b). These modes
are results of the hybridization of three Bloch resonances $(+1,0)$, $(-1,+1)$
and $(-1,-1)$ that are all originated from the fundamental transverse electric
guided mode of the perovskite slab. Here the indexes $n$ and $m$ in $(n,m)$
are diffraction orders along $x$ and $y$ direction, corresponding to the
propagation vector $\bm{\beta}_{n,m}(k_{x},k_{y})=\left(\frac{2n\pi}{\eta
a}+k_{x}\right)\bm{u_{x}}+\left(\frac{2m\pi}{a}+k_{y}\right)\bm{u_{y}}$ of the
original guided mode. We note that $(-1,+1)$ and $(-1,-1)$ are always
degenerated along $\Gamma X$ direction ($k_{y}=0$), a triple degeneracy point
occurs at $k_{x}=\frac{\pi}{2a}\eta$ and $k_{y}=0$. Most importantly, the
coupling between these three resonances gives rise to an avoid crossing effect
that is accompanied by the formation of a FW-BIC at the lowest band (see
Methods and the Supplemental Information for details of our analytical
theory). Indeed, in the zoom of Fig. 1b, we distinguish clearly an avoid
crossing of the resulted resonances, and a local vanishing of the lowest
resonance at its band-edge
($k_{x}=k_{BIC}$=$4.16\text{\,}\mathrm{\SIUnitSymbolMicro m}$-1) that is the
hall-mark of a FW-BIC 20. Figure. 1c depicts simulated photonic band
structures along $k_{y}$ when fixing $k_{x}=k_{BIC}$ in the same spectral
range as the zoom in Fig. 1b, showing the three hybridized resonances
previously discussed. We distinguish again a local vanishing of the lowest
resonance at its band-edge, here at $k_{y}=0$. As consequence, our photonic
structure exhibits a FW-BIC that is localized at $k_{x}=k_{BIC}$ and
$k_{y}=0$. From Figs. 1b,c, we note also an excellent agreement between the
band structure obtained from numerical simulations and the theoretical one
obtained by our analytical model based on FW theory.

The quality factor (Q-factor) of the lowest band when scanning $k_{x}$ and
$k_{y}$ is numerically obtained from RCWA simulation, and reported in Fig. 1d.
A Q-factor enhancement at the FW-BIC is clearly evidenced, with $Q\approx
10^{5}$ at $k_{x}=k_{BIC}$ and $k_{y}=0$. A finite numerical value of the
Q-factor, unlike an infinite value predicted by the analytical model detailed
in the Supplementary Information, can be explained by two reasons: i)
parasitic absorption of the perovskite material bellow its band-gap; ii)
imperfect destructive interference of radiative losses due to the presence of
other guided mode resonances that are not taken into account in the analytical
model. Thus our FW-BIC is of quasi-BIC nature, which may still possess some
radiative and non-radiative losses. However, these losses are negligible for
lasing applications based-on halide perovskites that only require quality
factors in order of $10^{3}$.35

Figure 1e presents the polarization texture of the lowest hybrid band,
extracted from RCWA calculations. A vortex texture of linear polarization
light rotating around the FW-BIC is clearly evidenced. This vortex is
associated to a topological charge 23 of
$q=\frac{1}{2\pi}\oint_{\mathcal{C}}d\bm{k}\nabla_{\bm{k}}\phi=-1$ where
$\phi(\bm{k})$ is the orientation of the farfield polarization vector. We note
that the same topological charge exists at $(k_{x}=-k_{BIC},k_{y}=0)$ and at
the same energy due to the $C_{2}$ symmetry of our structure. This twin
singularity is a FW-BIC resulting from the coupling between $(-1,0)$,
$(+1,+1)$ and $(+1,-1)$ guided resonances. Therefore our design suggests an
original engineering of slow-light singularities at arbitrary oblique angle.

The location of the FW-BIC in the momentum space can be tailored on-demand via
the anisotropy factor $\eta$. Indeed, as shown in Fig. 1b, $k_{BIC}$ almost
coincides to the momentum of the triple degeneracy point between $(+1,0)$ and
$(-1,\pm 1)$. Therefore the FW-BIC position is given by:

$k_{BIC}\approx\frac{\pi}{2a}\eta.$ (1)

Such a simple dependence is nicely confirmed when compared to the value of
$k_{BIC}$ extracted from RCWA simulations for different values of $\eta$ (see
Fig. 1f). In the following, for the experimental demonstration, we chose
$\eta=0.9$, so that the FW-BIC corresponds to an angle of
$\theta_{BIC}=21.3^{o}$ which is a high-tilted angle but still inside the
numerical aperture of our experimental setup ($\theta_{max}=26.7^{o}$)..

###  2.2 Experimental results:

The designed metasurface is fabricated by direct thermal nano-imprint
lithography on solution-processed CH3NH3PbBr3 perovskite thin film (see
fabrication details in Methods). The perovskite layer is first deposited via
spincoating onto a substrate of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of
SiO2 on silicon. It is then followed by a nano-imprinting process using a
patterned silicon mold. Figure. 2a depicts Scanning Electron Microscopy (SEM)
image of the fabricated structure.

Figure 2: (a) SEM top view of the perovskite metasurface. (b) Angle-resolved
reflectivity measurement along $k_{x}$ and a zoom in the vicinity of the FW-
BIC that is indicated by the red arrow. (c) Angle-integrated spectra of
photoluminescence at different pump powers. (d) Spectral and angle-integrated
intensity as the function of the pump power. The lasing threshold $p_{th}$ is
indicated by the black arrow.

We first characterize the photonic modes of the perovskite metasurface via
angle-resolved reflectivity measurements obtained from a house-built setup.
Figure. 2b depicts the reflectivity spectra along $\Gamma$X direction. The
dispersion of Bloch resonances shown in these spectra is in good agreement
with the one predicted by the theory and numerical simulations in Fig. 1b. In
particular, the avoid crossings of three hybrid modes, and the formation of
FW-BIC in the lower branch are clearly evidenced.

To investigate lasing action of the fabricated sample, the perovskite
metasurface is optically pumped (100 Hz, 5 ns pulsed laser at 450 nm, spot-
size diameter of $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$). Since the
lifetime of polycristalline perovskite is about few ns40, only a fraction of
the 5-ns pulse laser is used for carrier injection while most of the pulse
energy is attributed for heating effect. Thus we adopt the average power
instead of energy per pulse for evaluating the pump intensity. The angle-
integrated photoluminescence spectra, and spectrally integrated
photoluminescence intensity for different pump powers are shown in Fig. 2c and
Fig. 2d respectively. These results show a lasing action at the threshold
$p_{th}\approx$11\text{\,}\mathrm{\SIUnitSymbolMicro W}$$: i) Below threshold,
the photoluminescence spectra corresponds to a broad spontaneous emission
spanning over 40 nm (from 530 to 570 nm). ii) At threshold, a sharp peak of
0.49nm linewidth corresponding to stimulated emission starts emerging from the
spontaneous emission background. The gain and loss are balanced, thus we can
use the measured linewidth to estimate the quality factor of the lasing mode
that amounts to 1130. iii) Above threshold, the emission intensity exhibits a
jump and increases drastically. The emission spectrum is completely dominated
by the stimulated emission.

Figure 3: (a-f) Angle-resolved photoluminescence spectra along $k_{x}$
direction at $k_{y}=0$ (a-c), and along $k_{y}$ direction at $k_{x}=k_{BIC}$
(d-f) for pump power below threshold (a,d), at threshold (b,e) and above
threshold (c,f). The emission is filtered along $\mathrel{\updownarrow}$ for
measurements along $k_{x}$ and $\mathrel{\leftrightarrow}$ for measurements
along $k_{y}$. (g,h) Angle-resolved absorption spectra along $k_{x}$ direcion
at $k_{y}=0$ (g), and along $k_{y}$ direction at $k_{x}=k_{BIC}$ (h).

The origin of this lasing action is revealed by monitoring the angle-resolved
photoluminescence spectra when increasing the pump power across the lasing
threshold. Figure. 3 depicts the experimental results of these measurements
below threshold [Figs. 3a,d], at threshold [Figs. 3b,e] and above threshold
[Figs. 3c,f] for angle-resolved emission along $k_{x}$ with $k_{y}=0$ [Figs.
3a-c], and along $k_{y}$ with $k_{x}=k_{BIC}$ [Figs. 3d-f].For the sake of
clarity, here we only analyse the emission in the polarization of the FW-BIC
($\mathrel{\updownarrow}$ for $k_{x}$ dispersions and
$\mathrel{\leftrightarrow}$ for $k_{y}$ dispersions). As references, the
corresponding angle-resolved absorption spectra are also simulated and shown
in Figs. 3g,h. The FW-BIC is clearly evidenced as the lasing mode: i) Below
threshold, the spontaneous emission populates the whole photonic dispersion in
our spectral window [Figs. 3a,d] and these emission spectra are nicely matched
with the simulated absorption ones [Figs. 3g,h]. ii) At threshold, most of the
emission is located spectrally and momentumly at the FW-BIC [Figs. 3b,e]. This
localized signal is the stimulated emission that is nurtured by long lifetime
photons of FW-BIC. However, one may still distinguish a weak spontaneous
emission background from other photonic modes. iii) Above threshold, only
emission in the vicinity of the FW-BIC mode is observed. Indeed, similar to
lasing action at $\Gamma$ point of symmetry-protected BIC, our laser beam in
momentum space exhibits a doughnut shape centred at the BIC 26, 41. This
doughnut shape is projected as two lobes on both sides of the BIC in the
energy-momentum dispersion measurements. The doughnut centre imposes the
pointing angle and the doughnut diameter (i.e. distance between the two lasing
lobes in energy-momentum dispersion measurements) dictates the beam
divergence. This corresponds to a directional emission pointing at $k_{BIC}$
and localized in the reciprocal space within $\Delta
k_{x}=$$0.37\text{\,}\mathrm{\SIUnitSymbolMicro m}$-1. Light emission thus
occurs at 21-degree oblique angle with a divergence less that 2 degrees.
Similar lasing action also takes place at the twin FW-BIC and corresponds to a
lasing emission at -21 degrees.

Finally we study the polarization properties of the lasing emission. Its
farfield is analysed in four different polarizations
$\mathrel{\leftrightarrow}$, $\mathrel{\updownarrow}$,
$\mathrel{\text{$\nearrow$\hbox to0.0pt{\hss$\swarrow$}}}$ and
$\mathrel{\text{$\nwarrow$\hbox to0.0pt{\hss$\searrow$}}}$, then is directly
imaged onto the camera sensor without any spectral selection [see Fig. 4a,b)].
It is shown that only two lobes of the doughnut beam are selected for each
polarization. To reconstruct the polarization texture, the Stokes parameters
$S_{1}$ and $S_{2}$ are extracted from the four intensity mappings:
$S_{1}=\left(I_{\mathrel{\leftrightarrow}}-I_{\mathrel{\updownarrow}}\right)/\left(I_{\mathrel{\leftrightarrow}}+I_{\mathrel{\updownarrow}}\right)$
and $S_{2}=\left(I_{\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}}-I_{\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}}}\right)/\left(I_{\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}}+I_{\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}}}\right)$. Then the polarization orientation angle
$\phi$ is given by $\tan{2\phi}=S_{1}/S_{2}$. The experimental results of
polarization texture are depicted in Fig. 4c (left panel), and are in perfect
agreement with the analytical results from the effective theory [Fig. 4c
(middle panel)], and numerical results from RCWA simulations [Fig. 4c (right
panel)]. It shows that the polarization angle rotates a full 2$\pi$ clock-wise
when encircling the singularity point pinned at the FW-BIC. Therefore our
lasing emission is a vector vortex light in momentum space with topological
charge $q=-1$. The same topological charge is also observed for the FW-BIC
located at $k_{x}=-k_{BIC}$ (see supplemental material). Therefore our device
exhibit two oblique vector vortex lasing emissions as illustrated in Fig. 4d.
We note that while high-angle emitting micro-laser using BIC concepts have
been recently demonstrated42, 31, the reported lasing actions do not take
place at singularity point and exhibits only gaussian beam. Our observation is
then the first demonstration of oblique vector vortex lasing action.

Figure 4: (a) Farfield image of the lasing emission, analysed in
$\mathrel{\updownarrow}$ polarization. (b) Zoom in of the farfield emission in
the vicinity of the FW-BIC in four different polarizations
$\mathrel{\updownarrow}$, $\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}$, $\mathrel{\leftrightarrow}$, and
$\mathrel{\text{$\nwarrow$\hbox to0.0pt{\hss$\searrow$}}}$. (c) Polarization
texture of the lasing emission extracted from the experimental measurements
(left panel), calculated by the FW theory (middle panel), and provided by RCWA
simulations (right panel). (d) Illustration of the vortex lasing emissions
demonstrated in this work.

##  3 Conclusion

In conclusion, we have proposed and demonstrated experimentally for the first
time a micro-laser emitting vector vortex beam at high tilted-angle. The
photonic concept is based on an original engineering of FW interference
between three guided mode resonances, leading to the formation of polarization
singularity at off-$\Gamma$ band-edge. The lasing action is achieved thanks to
the implementation of high gain material (i.e. halide perovskite) at the FW-
BIC. Moreover, our work provides a unique scheme to tailor vortex beam at
arbitrary oblique angle via the use of anistrotropic lattice. Since our proof-
of-concept employs perovskite metasurface fabricated by solution-based method
and nano-imprint structuration, a beam steering mechanisms can be
straightforwardly implemented via the use of flexible or piezoelectric
substrates. Therefore, our work would pave the way for dynamical control of
vortex beam lasing for applications in optical manipulations, trapping or
communication in free space.

##  4 Methods

##### Mold fabrication:

A 100 nm-thick hydrogen silsesquioxane (HSQ) resist is spun on a silicon
substrate, then baked at 80 ∘C for 4 minutes. The resist is exposed by
electron-beam lithography to define negative patterns (i.e. pillar lattices)
and developed with a solution of TMAH. The patterns are subsequently
transferred to the Si substrate by inductively-coupled reactive ion etching
(ICP-RIE) using a mixture of Cl2 and O2.

##### Perovskite preparation and deposition:

The perovskite solution is prepared by mixing CH3NH3PbBr and PbBr2 (1.2: 1) in
N,N-dimethylformamide (DMF), then stirred for 1 h at room temperature. Then
$80\text{\,}\mathrm{\SIUnitSymbolMicro l}$ of the precursor solution is spin-
coated at 5000 rpm for 30 s. In the aim to obtain homogeneous and uniform
active layer $60\text{\,}\mathrm{\SIUnitSymbolMicro l}$ of Chlorobenzene
antisolvent is dropped at 5s after launching the spin coating. The sample is
then annealed by baking at 70 ∘C for 5 min and 1 hour for the layer followed
by nanoprinting and without nanoprinting, respectively.

##### Perovskite nano-imprint:

Before imprinting, the mold is functionalized by a silanization process to
avoid sticking with perovskite. The imprinting process is operated in ambient
atmosphere. The nano-imprint is conducted with the use of a Rondol manual
press at 3 kN and 100 ∘C for 15 minutes. After 15 minutes, the heat is turned
off and the pressure is released when the temperature reaches 35 ∘C.

##### Experimental setup:

Perovskite metasurfaces are characterized using a house-made setup that
measures the angle-resolved reflectivity and photoluminescece. Excitation
(white light source for reflectivity and laser for photoluminescence) is
focused onto the sample via a microscope objective (NA=0.45) and the signal is
collected via the same objective in a confocal geometry. The back-focal plane
of the objective is imaged onto the entrance slit of a spectrometer and the is
dispersed onto a CCD sensor at the output of the spectrometer. With this
configuration, the dispersion is measured along the direction that is imposed
by the spectrometer slit.

##### Analytical model:

The couplings between the three modes (1,0), (-1,1) and (-1,-1) are described
by the FW Hamiltonian (see details in the Supplemental Material):

$H_{FW}(\bm{k})=\begin{pmatrix}\omega_{1}(\bm{k})&U_{1}&U_{1}\\\
U_{1}&\omega_{2}(\bm{k})&U_{2}\\\
U_{1}&U_{2}&\omega_{3}(\bm{k})\end{pmatrix}+i\begin{pmatrix}\gamma_{1}&\Gamma_{12}(\bm{k})&\Gamma_{13}(\bm{k})\\\
\Gamma_{12}(\bm{k})&\gamma_{2}&\Gamma_{23}(\bm{k})\\\
\Gamma_{13}(\bm{k})&\Gamma_{23}(\bm{k})&\gamma_{2}\end{pmatrix}$ (2)

In the Hermitian part (first term), $\omega_{j=1..3}(\bm{k})$ is the
dispersion of uncoupled modes [white dashed line in Fig 1(b,c)]; $U_{1}$ is
the diffractive coupling strength between (1,0) and (-1,$\pm$1); $U_{2}$ is
the diffractive coupling strength between (-1,1) and (-1,-1). In the non-
Hermitian part (second term), $\gamma_{1}$ and $\gamma_{2}$ are the radiative
losses of (1,0) and (-1,$\pm$1) respectively. The coupling terms
$\Gamma_{12(\bm{k})}$, $\Gamma_{13(\bm{k})}$ and $\Gamma_{23(\bm{k})}$
correspond to the radiative couplings (i.e. losses exchange mechanism) between
the uncoupled modes. The analytical expressions of $\omega_{j=1..3}(\bm{k})$
and $\Gamma_{ij}(\bm{k})$ are given in the supplemental material. The real and
imaginary part of the complex eigenvalues of (2) correspond to the energy and
losses of the hybrid modes respectively. Moreover, the eigenvectors of (2)
provide the near/far field patterns and the polarization texture of the hybrid
modes.

##### Numerical simulationss:

The RCWA simulations have been performed with the S4 package provided by the
Fan Group at the Stanford Electrical Engineering Department.43 The refractive
index and extinction coefficient of perovskite, used in RCWA simulations, is
obtained from ellipsometry measurement. The ellipsometry experiment has been
performed on perovskite layer that is pressed by a flat mold.

The authors thank Xavier Letartre and Pierre Viktorovitch for fruitful
discussions. This work benefited from the facilities of the Nanolyon
technological platform, member of the Renatech+ and CARAT networks for all
nanofabrication processes. Simulations were carried out using the facilities
of the PMCS2I (Ecole Centrale de Lyon). This work is partly supported by the
French National Research Agency (ANR) under the projects POPEYE
(ANR-17-CE24-0020) and EMIPERO (ANR-18-CE24-0016). It is also supported by the
Auvergne-Rhône-Alpes region in the framework of PAI2020 and the Vingroup
Innovation Foundation (VINIF) annual research grant program under Project Code
VINIF.2021.DA00169

In the Supplemental Material document, a full derivation of the FW model and
the formation of FW-BIC provided. We also include additional results on the
perovskite deposition and nanoimprint, as well as photoluminescence images in
real and momentum space of of the structure.

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— SUPPLEMENTAL MATERIAL —

Figure S1: Pristine vs Pressed perovskite layer. SEM images of perovskite
layer obtained by (left-panel): spincoating + anti-solvent dripping; (right-
panel): spincoating + anti-solvent dripping + pressing with flat mold under
high pressure and high temperature. Figure S2: Silicon mold vs Imprinted
perovskite layer. SEM images of the silicon mold and imprinted perovskite
layer that is obtained by using the same mold. Figure S3: Farfield imaging in
real space. Experimental image in real space of the photoluminescece signal
(without any spectral filter and polarization elements) when pumping just
below and just above threshold. The dashed line indicates the metasurface
region where perovskite has been patterned. We observe clearly a drastic
enhancement of the photoluminesce signal and the appearance of speckles,
indication of coherent emission, for the lasing emission. Figure S4: Farfield
imaging in momentum space. Experimental image in momentum space of the
photoluminescece signal when pumping below and above threshold. The dashed
line indicates the numerical aperture of the microscope objective. The
polarization is selected along y-axis. There is no spectral filter. Below
threshold, the photoluminesce takes place in a broad region of momentum space.
Above threshold, we only observe a lasing emission that is concentrated in the
vicinity of the twin FW-BICs.

##  5 Theoretical model for Friedrich-Wintgen couplings

In this section, we provide the detailed derivation of the effective
Hamiltonian representing leaky Bloch modes in a general photonic lattice of
period $a_{x}=\eta\times a$ along x-axis, and $a_{y}=a$ along y-axis with
$\eta\approx 1$. We limit our consideration for unit cells that exhibit
lateral mirror symmetries $x\rightarrow-x$ and $y\rightarrow-y$.

In a perturbative approach, the Bloch resonances of the photonic lattice can
be described in the basis of propagating in-plan guided modes in a dielectric
slab of effective medium without corrugation. The Bloch resonances are results
of the couplings within these guided modes and also between these guided modes
with the radiative continuum.

###  5.1 Basis made of folded guides modes

For the sake of simplicity, we suppose that the dielectric slab of effective
medium only exhibits a single guided mode in the spectral range of interest.
The folding of this guided mode from the $(n,m)^{th}$ Brillouin zone given by
the Bloch vector
$\bm{B}_{n,m}=\frac{2\pi}{a_{x}}n\bm{u_{x}}+\frac{2\pi}{a_{y}}m\bm{u_{y}}$ to
the first Brillouin zone correspond to the folded guided mode $\lvert
n,m\rangle$ of wavevector $\bm{k}=k_{x}\bm{u_{x}}+k_{y}\bm{u_{x}}$ with
$k_{x,y}<2\pi/a_{x,y}$.

The ensemble of $\lvert n,m\rangle$ with $n,m\in\mathbb{Z}$ is the basis of
our theory. We note that for a general case of multi-mode operation, our
theory can be easily extended by considering the basis made of $\lvert
n,m\rangle_{p}$ where $p$ is the index of the unfolded guided mode.

Due to the folding, the propagation vector of the folded guided mode $\lvert
n,m\rangle$ is given by:

$\bm{\beta}_{n,m}(\bm{k})=\frac{2\pi}{\eta
a}n\bm{u_{x}}+\frac{2\pi}{a}m\bm{u_{y}}+\bm{k}$ (S1)

####  5.1.1 Dispersion characteristic

Assuming that the dispersion characteristic in the spectral range of interest
of the unfolded guided mode is linear with the slope given by the effective
group index $n_{g}$, one may show that the dispersion characteristic of
$\lvert n,m\rangle$ is given by:

$\displaystyle\begin{split}\omega_{n,m}(\bm{k})&=\omega_{0}+\frac{c}{n_{g}}\left(\lvert\bm{\beta}_{n,m}(\bm{k})\rvert-\frac{2\pi}{a}\right)\\\
&=\omega_{0}+\frac{2\pi
c}{n_{g}a}\left[\sqrt{\left(\frac{n}{\eta}+\frac{k_{x}a}{2\pi}\right)^{2}+\left(m+\frac{k_{y}a}{2\pi}\right)^{2}}-1\right]\end{split}$
(S2)

where $c$ is the speed of light and $\omega_{0}$ is the pulsation of the
unfolded guided mode at $\lvert\bm{\beta}\rvert=2\pi/a$.

To simplify the notations, it is preferable to work with dimensionless
quantities. We thus redefine the wavevector and pulsation as:

$\displaystyle\bm{q}$ $\displaystyle=\bm{k}a/2\pi$ (S3)

and

$\hat{\omega}_{n,m}(\bm{q})=\frac{\left[\omega_{n,m}(\bm{q})-\omega_{0}\right]n_{g}a}{2\pi
c}.$ (S4)

Using these quantities, the propagation vector (S1) and the dispersion
characteristic (S2) can be rewritten as:

$\bm{\beta}_{n,m}(\bm{q})=\frac{2\pi}{a}\left[\left(\frac{n}{\eta}+q_{x}\right)\bm{u_{x}}+\left(m+q_{y}\right)\bm{u_{x}}\right]$
(S5)

and

$\hat{\omega}_{n,m}(\bm{q})=\sqrt{\left(\frac{n}{\eta}+q_{x}\right)^{2}+\left(m+q_{y}\right)^{2}}-1$
(S6)

where $q_{x}=k_{x}a/2\pi$ and $q_{y}=k_{y}a/2\pi$.

From (S6), we deduce that the lowest bands in the folded dispersion band
structure are $\lvert\pm 1,0\rangle$, $\lvert 0,\pm 1\rangle$, $\lvert\pm
1,1\rangle$ and $\lvert\pm 1,-1\rangle$.

####  5.1.2 Polarization vectors

Polarizations of guided modes are classified into TE and TM modes (i.e. the
electric field $\bm{E}$ and magnetic field $\bm{H}$ are in-plan and
perpendicular to the propagation vector respectively). Here we are only
interested in the in-plane components of the electric field, given by in-plane
polarization vector $\bm{u}_{n,m}^{TE}(\bm{q})$ and
$\bm{u}_{n,m}^{TM}(\bm{q})$ for TE and TM guided modes respectively. Since
$\bm{u}_{n,m}^{TE}(\bm{q}).\bm{\beta}_{n,m}(\bm{q})=0$,
$\bm{u}_{n,m}^{TM}(\bm{q})||\bm{\beta}_{n,m}(\bm{q})$ and
$\lvert\bm{u}_{n,m}^{TE}(\bm{q})\rvert=\lvert\bm{u}_{n,m}^{TM}(\bm{q})\rvert=1$,
from the propagation vector (S5), we obtain:

$\bm{u}_{n,m}^{TE}(\bm{q})=\frac{1}{\sqrt{\left(\frac{n}{\eta}+q_{x}\right)^{2}+\left(m+q_{y}\right)^{2}}}\left(\begin{matrix}-(m+q_{y})\\\
\frac{n}{\eta}+q_{x}\end{matrix}\right)$ (S7)

and

$\bm{u}_{n,m}^{TM}(\bm{q})=\frac{1}{\sqrt{\left(\frac{n}{\eta}+q_{x}\right)^{2}+\left(m+q_{y}\right)^{2}}}\left(\begin{matrix}\frac{n}{\eta}+q_{x}\\\
m+q_{y}\end{matrix}\right).$ (S8)

In the following, we assume that the single unfolded mode is the fundamental
TE mode, thus the TE and TM index will be omitted and the polarization vector
$\bm{u}_{n,m}(\bm{q})$ is the one of TE modes $\bm{u}_{n,m}^{TE}(\bm{q})$.

###  5.2 Coupling mechanism of folded guided modes

The coupling between folded guided modes are possible if the phase matching
conditions are fulfilled. There are two phase matching conditions:

  * •

Symmetry matching: they are originated from unfolded guided modes of the same
vertical symmetry parity if the structure exhibits a vertical symmetry mirror.
Here we only considered folded guided modes from the same unfolded guided
modes, thus this condition is automatically satisfied.

  * •

Energy and Wavevector matchings: the coupling is only effective in the
vicinity of the crossing points of the dispersions.

####  5.2.1 Diffractive coupling

Thanks to the periodic corrugation, the folded guided modes can be coupled via
near-field effect as soon as the phase matching conditions mentioned above are
fulfilled. A diffractive coupling of strength $U_{ij}$ between two folded
guided modes $\lvert i\rangle=\lvert n_{i},m_{i}\rangle$ and $\lvert
j\rangle=\lvert n_{j},m_{j}\rangle$ would lead to an avoid crossing of band
gap $\approx 2U_{ij}$ from a doubly degenerated crossing point between $\lvert
i\rangle$ and $\lvert j\rangle$.

####  5.2.2 Radiative losses and radiative couplings

Due to the bandfolding, the dispersion of $\lvert n,m\rangle$ can locate
inside the light cone. Thus this folded guided mode can couple to the
radiative continuum and exhibiting radiative losses of normalized rate
$\gamma_{n,m}$ in the unit of $\frac{2\pi c}{n_{g}a}$.

A radiative coupling between two folded guided modes $\lvert i\rangle=\lvert
n_{i},m_{i}\rangle$ and $\lvert j\rangle=\lvert n_{j},m_{j}\rangle$ is
possible via the radiative continuum. This coupling, discovered by Friedrich
and Wintgen in a general scheme, is the inteference between the two folded
guided modes via the farfield radiation. One may describe the coupling
strength $\Gamma_{ij}$ by a simple expression:

$\Gamma_{ij}=\sqrt{\gamma_{i}\gamma_{j}}.\cos\alpha_{ij}$ (S9)

where $\gamma_{i}$ and $\gamma_{j}$ are simplified notations of
$\gamma_{n_{i},m_{i}}$ and $\gamma_{n_{j},m_{j}}$ respectively. The
coefficient $\cos\alpha_{ij}=\bm{u}_{n_{i},m_{i}}.\bm{u}_{n_{j},m_{j}}$ is due
to the fact that only the same electric field component can interfere.

####  5.2.3 Friedrich-Wintgen Hamiltonian

In the vicinity of a $N$ times degenerated crossing point $\bm{q_{c}}$ between
$N$ folded guided modes $\lvert 1\rangle$, $\lvert 2\rangle$, …,$\lvert
N\rangle$, all of these modes can be coupled one to each other via diffractive
and radiative couplings. In the basis of uncoupled modes $B=\left\\{\lvert
1\rangle,\lvert 2\rangle,...\lvert N\rangle\right\\}$, the system is depicted
by an non-Hermitian Hamiltonian, given by:

$H_{FW}(\bm{q})=H_{0}(\bm{q})+U(\bm{q})+i\Gamma(\bm{q})$ (S10)

The construction of this Hamiltonian is explained as the following:

  * •

The first term $H_{0}$ is the Hamiltonian of uncoupled modes and only
containing diagonal elements, given by $H_{0(i=j)}=\hat{\omega}_{i}(\bm{q})$.
Here $\hat{\omega}_{i}(\bm{q})$ is the simplified notation of
$\hat{\omega}_{n_{i},m_{i}}(\bm{q})$ given by (S6).

  * •

The second term $U$ corresponds to diffractive couplings and only containing
non-diagonal elements, given by $U_{i\neq j}=U_{ij}$. Here $U_{ij}$ is the
diffractive coupling strength between $\lvert i\rangle$ and $\lvert j\rangle$.
Of course $U_{ij}=U_{ji}$.

  * •

The last term $\Gamma$ corresponds to the radiative losses and radiative
coupling. Its diagonal elements are the radiative losses of each mode, given
by $\Gamma_{i=j}=\gamma_{i}$. Its non-diagonal elements are given by
$\Gamma_{i\neq j}=\sqrt{\gamma_{i}\gamma_{j}}\cos\alpha_{ij}$ as (S9). Of
course $\Gamma_{ij}=\Gamma_{ji}$.

####  5.2.4 Eigenvalues of $H_{FW}$: radiative losses and FW conditions

The diagonalization of $H_{FW}$ provides $N$ eigenmodes
$\lvert\Omega_{1}\rangle$, $\lvert\Omega_{2}\rangle$…
$\lvert\Omega_{N}\rangle$ with corresponding eigenvalues $\Omega_{1}$,
$\Omega_{2}$, …,$\Omega_{N}$. These eigenvalues are complex numbers of which
the real parts are pulsation and the imaginary part are radiative losses.

An eigenmode $\lvert\Omega_{BIC}\rangle$ is a Friedrich-Wintgen bound state in
the continuum (FW-BIC) if the imaginary part of its eigenvalue is zero, i.e.
$\text{Im}\left(\Omega_{BIC}\right)=0$. Such a destructive interference
configuration requires special relations between the elements of $H_{0}$, $U$
and $\Gamma$. These relations are called Friedrich-Wintgen conditions (FW
conditions). For example, in the case of a $2\times 2$ $H_{FW}$, given by
$H_{FW}=\begin{pmatrix}\hat{\omega}_{1}&0\\\
0&\hat{\omega}_{2}\end{pmatrix}+\begin{pmatrix}0&U\\\
U&0\end{pmatrix}+i\begin{pmatrix}\gamma_{1}&\sqrt{\gamma_{1}\gamma_{2}}\cos\alpha\\\
\sqrt{\gamma_{1}\gamma_{2}}\cos\alpha&\gamma_{2}\end{pmatrix}$, the FW
conditions are $\sin\alpha=0$ and
$\hat{\omega}_{1}-\hat{\omega}_{2}=(\gamma_{1}-\gamma_{2})U/\sqrt{\gamma_{1}\gamma_{2}}\cos\alpha$.

####  5.2.5 Eigenmodes of $H_{FW}$: eigenvectors, nearfield and farfield
radiation

The nearfield and farfield pattern of a given eigenmode $\lvert\Omega\rangle$
can be calculated from the its eigenvector
$\bm{v}_{\Omega}=\begin{pmatrix}C_{1}\\\ C_{2}\\\ ...\\\ C_{N}\end{pmatrix}$
obtain from $H_{FW}$. Indeed, the decomposition of $\lvert\Omega\rangle$ in
the basis of folded guided modes is given by:

$\lvert\Omega\rangle=\sum_{j=1}^{N}{C_{j}\lvert j\rangle}$ (S11)

Thus the nearfield and farfield pattern of $\lvert\Omega\rangle$ is simply
given by linear combination of the ones of the folded guided modes.

The nearfield of the folded guided mode $\lvert j\rangle$ is a plane wave
propagating in-plane (x,y) with wavevector $\bm{\beta}_{n_{j},m_{j}}(\bm{q})$
and vertical confinement profile $f(z)$. Thus it is given by:

$\displaystyle\begin{split}\bm{\mathcal{E}}_{\lvert
j\rangle}^{\text{nearfield}}&=\mathcal{E}_{0}f(z)\exp{\left[i\bm{\beta}_{n_{j},m_{j}}(\bm{q}).\bm{r}_{||}\right]}\bm{u_{n_{j},m_{j}}}\\\
&=\mathcal{E}_{0}f(z)\exp{\left(i\frac{2\pi\bm{q}.\bm{r}_{||}}{a}\right)}\exp{\left(i\bm{B}_{n_{j},m_{j}}.\bm{r}_{||}\right)}\bm{u_{n_{j},m_{j}}}\end{split}$
(S12)

where $\bm{r}_{||}=x\bm{u_{x}}+y\bm{u_{y}}$ is the in-plane coordinates;
$\bm{B}_{n_{j},m_{j}}=\frac{2\pi}{a_{x}}n_{j}\bm{u_{x}}+\frac{2\pi}{a_{y}}m_{j}\bm{u_{y}}$
is the Bloch vector of the band folding; $\bm{u_{n_{j},m_{j}}}$ is the
polarization vector from (S7) for TE modes, and the constant $\mathcal{E}_{0}$
is a the field amplitude.

The Bragg scattering mechanism due to periodic corrugation folds the guided
mode of propagation vector $\bm{\beta}_{n,m}(\bm{q})$ to the same wavevector
$\frac{2\pi\bm{q}}{a}$ for any couples $(n,m)$. As a consequence, the folded
guided mode $\lvert j\rangle$ radiates to the free space as plane wave having
$\frac{2\pi\bm{q}}{a}$ as the in-plane wavevector component. In the vicinity
of the normal emission and of the crossing point, these radiating waves are
approximately described as plane waves of the same wavevector
$\frac{\omega_{c}}{c}\bm{u_{z}}$ while preserving the polarization
$\bm{u}_{n_{j},m_{j}}$ of the unfolded modes (here $\omega_{c}$ is the
pulsation of the crossing point). Moreover, the amplitude of the radiating
wave is dictated by the radiative losses, thus propotional to
$\sqrt{\gamma_{j}}$. Therefore the farfield of the folded guided mode $\lvert
j\rangle$ is given by

$\bm{\mathcal{E}}_{\lvert
j\rangle}^{\text{farfield}}\propto\mathcal{E}_{0}\sqrt{\gamma_{j}}\exp\left(\frac{i\omega_{c}z}{c}\right)\bm{u}_{n_{j},m_{j}}(\bm{q})$
(S13)

Therefore,the nearfield and farfield pattern of the eigenmodes
$\lvert\Omega\rangle$ are given by:

$\displaystyle\bm{\mathcal{E}}_{\lvert\Omega\rangle}^{\text{nearfield}}$
$\displaystyle=\mathcal{E}_{0}f(z)\exp{\left(i\frac{2\pi\bm{q}.\bm{r}_{||}}{a}\right)}\sum_{j=1}^{N}{C_{j}\exp{\left(i\bm{B}_{n_{j},m_{j}}.\bm{r}_{||}\right)}\bm{u_{n_{j},m_{j}}}}$
(S14) $\displaystyle\bm{\mathcal{E}}_{\lvert\Omega\rangle}^{\text{farfield}}$
$\displaystyle\propto\mathcal{E}_{0}\exp\left(\frac{i\omega_{c}z}{c}\right)\sum_{j=1}^{N}{C_{j}\sqrt{\gamma_{j}}\bm{u}_{n_{j},m_{j}}(\bm{q})}$
(S15)

Figure S5: An arbitrary polarized light is determined by two angles $\phi$ and
$\chi$ defining the orientation and ellipticity of the polarization.

From the distribution in momentum space of the complex electric field
$\bm{\mathcal{E}}_{\lvert\Omega\rangle}^{\text{farfield}}$, we can retrieve
the its polarization texture given by the orientation angle $\phi$ and
ellitpicity angle $\chi$ (see Fig. S5):

$\displaystyle\tan 2\phi$
$\displaystyle=\frac{2\text{Re}(\mathcal{E}_{x}^{*}.\mathcal{E}_{y})}{|\mathcal{E}_{x}|^{2}-|\mathcal{E}_{y}|^{2}}$
(S16) $\displaystyle\sin 2\chi$
$\displaystyle=\frac{2\text{Im}(\mathcal{E}_{x}^{*}.\mathcal{E}_{y})}{|\mathcal{E}_{x}|^{2}+|\mathcal{E}_{y}|^{2}}$
(S17)

where
$\mathcal{E}_{x}=\bm{\mathcal{E}}_{\lvert\Omega\rangle}^{\text{farfield}}.\bm{u_{x}}$
and
$\mathcal{E}_{y}=\bm{\mathcal{E}}_{\lvert\Omega\rangle}^{\text{farfield}}.\bm{u_{y}}$.

Importantly, if the FW conditions are fulfilled and $\lvert\Omega\rangle$ is a
FW-BIC, one may show that
$\sum_{j=1}^{N}{C_{j}\sqrt{\gamma_{j}}\bm{u}_{n_{j},m_{j}}}=0$. Therefore the
farfield radiation vanishes at FW-BIC. Moreover, if the FW-BICs are located at
isolates position in the momentum space, they correspond to singularities of
polarization vortex.

###  5.3 Friedrich-Wintgen coupling between $\lvert 1,0\rangle$,
$\lvert-1,1\rangle$ and $\lvert-1,-1\rangle$

####  5.3.1 Dispersion characteristics and polarization of uncoupled modes

In previous sections, we have developped a general theory for Friedrich-
Wintgen inteference between folded guided modes in rectangular lattice. In the
following, we will focus on the coupling between the three TE folded guided
modes given by:

$\displaystyle\begin{split}\lvert 1\rangle&\equiv\lvert n=1,m=0\rangle\\\
\lvert 2\rangle&\equiv\lvert n=-1,m=1\rangle\\\ \lvert 3\rangle&\equiv\lvert
n=-1,m=-1\rangle\end{split}$ (S18)

From (S6), the dispersion of these modes are given by:

$\displaystyle\begin{split}\hat{\omega}_{1}(\bm{q})&=\hat{\omega}_{1,0}(\bm{q})=\sqrt{\left(\frac{1}{\eta}+q_{x}\right)^{2}+q_{y}^{2}}-1\\\
\hat{\omega}_{2}(\bm{q})&=\hat{\omega}_{-1,1}(\bm{q})=\sqrt{\left(-\frac{1}{\eta}+q_{x}\right)^{2}+\left(1+q_{y}\right)^{2}}-1\\\
\hat{\omega}_{3}(\bm{q})&=\hat{\omega}_{-1,-1}(\bm{q})=\sqrt{\left(-\frac{1}{\eta}+q_{x}\right)^{2}+\left(-1+q_{y}\right)^{2}}-1\end{split}$
(S19)

It can be easily demonstrated that the three mode degenerate triply at a
crossing point of wavevector $\bm{q}_{c}=\begin{pmatrix}q_{c}=\eta/4\\\
0\end{pmatrix}$ and normalized pulsation
$\hat{\omega}_{c}=\frac{1}{\eta}+\frac{\eta}{4}-1$. In the vicinity of the
triple crossing point, the dispersions are given by:

$\displaystyle\begin{split}\hat{\omega}_{1}(\bm{q})&\approx\hat{\omega_{c}}+\left(q_{x}-q_{c}\right)+\frac{4+\eta^{2}}{8\eta}q_{y}^{2}\\\
\hat{\omega}_{2}(\bm{q})&\approx\hat{\omega_{c}}-\frac{4-\eta^{2}}{4+\eta^{2}}\left(q_{x}-q_{c}\right)+\frac{4\eta}{4+\eta^{2}}q_{y}\\\
\hat{\omega}_{3}(\bm{q})&\approx\hat{\omega_{c}}-\frac{4-\eta^{2}}{4+\eta^{2}}\left(q_{x}-q_{c}\right)-\frac{4\eta}{4+\eta^{2}}q_{y}\end{split}$
(S20)

From these results, we deduce that in the vicinity of the crossing point, i)
along $q_{x}$ with $q_{y}=0$: the three modes exhibit linear dispersion and
the two modes $\lvert 2\rangle$ and $\lvert 3\rangle$ are degenerated; ii)
along $q_{y}$ with $q_{x}=q_{c}$: $\lvert 1\rangle$ exhibits parabolic
dispersion while $\lvert 2\rangle$ and $\lvert 3\rangle$ exhibit linear
dispersions of opposite group velocity. The dispersion surfaces and different
cuts along $q_{x}$ and $q_{y}$ are depicted in Fig. S6.

Figure S6: Dispersion characteristics of the three folded guided modes $\lvert
1\rangle\equiv\lvert 1,0\rangle$, $\lvert 2\rangle\equiv\lvert-1,1\rangle$ and
$\lvert 3\rangle\equiv\lvert-1,-1\rangle$. (a) The dispersion surfaces in
$(q_{x},q_{y})$ plane of . (b,c) Dispersion curves along $q_{x}$ with
$q_{y}=0$. (c) Dispersion curves along $q_{y}$ with $q_{x}=q_{c}$

From (S7), the polarization of these three modes are given by:

$\displaystyle\begin{split}\bm{u}_{1}(\bm{q})&=\frac{1}{\sqrt{\left(\frac{1}{\eta}+q_{x}\right)^{2}+q_{y}^{2}}}\left(\begin{matrix}-q_{y}\\\
\frac{1}{\eta}+q_{x}\end{matrix}\right)\\\
\bm{u}_{2,3}(\bm{q})&=\frac{1}{\sqrt{\left(\frac{1}{\eta}-q_{x}\right)^{2}+\left(1\pm
q_{y}\right)^{2}}}\left(\begin{matrix}\mp 1-q_{y}\\\
-\frac{1}{\eta}+q_{x}\end{matrix}\right)\end{split}$ (S21)

####  5.3.2 Friedrich-Wintgen Hamiltonian

Due to the mirror symmetry $y\rightarrow-y$, the diffractive coupling strength
$U_{12}$ and $U_{13}$ are equal. For the same reason, $\lvert 2\rangle$ and
$\lvert 3\rangle$ would have the same radiative losses (i.e.
$\gamma_{2}=\gamma_{3}$). To simplify the notations, in the following we
denote $U_{1}\equiv U_{13}=U_{12}$ and $U_{2}\equiv U_{23}$.

Therefore in our case, the Friedrich-Wintgen Hamiltonian from (S10) is reduced
to:

$H_{FW}(\bm{q})=\begin{pmatrix}\hat{\omega}_{1}(\bm{q})&U_{1}&U_{1}\\\
U_{1}&\hat{\omega}_{2}(\bm{q})&U_{2}\\\
U_{1}&U_{2}&\hat{\omega}_{3}(\bm{q})\end{pmatrix}+i\begin{pmatrix}\gamma_{1}&\Gamma_{12}(\bm{q})&\Gamma_{13}(\bm{q})\\\
\Gamma_{12}(\bm{q})&\gamma_{2}&\Gamma_{23}(\bm{q})\\\
\Gamma_{13}(\bm{q})&\Gamma_{23}(\bm{q})&\gamma_{2}\end{pmatrix}$ (S22)

where

$\displaystyle\begin{split}\Gamma_{12}(\bm{q})&=\bm{u}_{1}(\bm{q}).\bm{u}_{2}(\bm{q})\sqrt{\gamma_{1}\gamma_{2}}\\\
\Gamma_{13}(\bm{q})&=\bm{u}_{1}(\bm{q}).\bm{u}_{3}(\bm{q})\sqrt{\gamma_{1}\gamma_{2}}\\\
\Gamma_{23}(\bm{q})&=\bm{u}_{2}(\bm{q}).\bm{u}_{3}(\bm{q})\gamma_{2}\end{split}$
(S23)

####  5.3.3 Formation of Friedrich-Wintgen BIC

Using (S20) and (S21), one may demonstrate that the FW conditions are
established at $\bm{q}_{BIC}=\begin{pmatrix}q_{BIC}\\\ 0\end{pmatrix}$ with
$q_{BIC}\approx q_{c}$ ; and the three corresponding complex eignevalues given
by:

$\displaystyle\Omega_{1}(\bm{q}_{BIC})$
$\displaystyle\approx\frac{U_{1}}{\sqrt{2}}\left(\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}-\sqrt{\frac{\gamma_{1}}{\gamma_{2}}}\right)-U_{1}^{2}\left(\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}+\sqrt{\frac{\gamma_{1}}{\gamma_{2}}}\right)^{2}$
(S24) $\displaystyle\Omega_{2}(\bm{q}_{BIC})$
$\displaystyle\approx\frac{U_{1}}{\sqrt{2}}\left(\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}-\sqrt{\frac{\gamma_{1}}{\gamma_{2}}}\right)+U_{1}^{2}\left(\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}+\sqrt{\frac{\gamma_{1}}{\gamma_{2}}}\right)^{2}+i(\gamma_{1}+\gamma_{2})$
(S25) $\displaystyle\Omega_{3}(\bm{q}_{BIC})$
$\displaystyle\approx\sqrt{2}U_{1}\left(\sqrt{\frac{\gamma_{2}}{\gamma_{1}}}-\sqrt{\frac{\gamma_{1}}{\gamma_{2}}}\right)-2U_{2}+i\gamma_{2}$
(S26)

Here the BIC state corresponds to $\lvert\Omega_{1}(\bm{q}_{BIC})\rangle$ that
exhibits zero losses. Figure S7 represent the dispersion characteristics of
the eignmodes $\lvert\Omega_{1}\rangle$, $\lvert\Omega_{2}\rangle$ and
$\lvert\Omega_{3}\rangle$ of the FW Hamiltonian. The formation of FW-BIC in
the vicinity of the crossing point is clearly evidenced.

Figure S7: Dispersion characteristics of eigenmodes $\lvert\Omega_{1}\rangle$,
$\lvert\Omega_{2}\rangle$ and $\lvert\Omega_{3}\rangle$ resulted from the FW
Hamiltonian (S22). (a) The dispersion surfaces of $\text{Re}(\Omega_{1,2,3})$
in $(q_{x},q_{y})$ plane. (b,c) Dispersion curves along $q_{x}$ with $q_{y}=0$
of (b) $\text{Re}(\Omega_{1,2,3})$ and (c) $\text{Im}(\Omega_{1,2,3})$. (d,e)
Dispersion curves along $q_{y}$ with $q_{x}=q_{c}$ of (d)
$\text{Re}(\Omega_{1,2,3})$ and (e) $\text{Im}(\Omega_{1,2,3})$. The
calculation has been done with $\eta=0.9$, $U_{1}=5.7\times 10^{-4}$,
$U_{2}=U_{1}/1.5$, $\gamma_{1}=0.4\lvert U_{1}\rvert$ and
$\gamma_{2}=0.5\lvert U_{1}\rvert$. Figure S8: Dispersion characteristics:
RCWA vs FW model. (a,b) The dispersions along (a) $k_{x}$ and (b) $k_{y}$
obtained from RCWA simulations (pseudo colormap of resonances) and the real
part of the eigenmodes obtained from the FW theory (black solid lines).
Uncoupled folded modes are also presented by white dashed lines. (c,d) Quality
factor mappings of the eigenmode $\lvert\Omega_{1}\rangle$ that is extracted
from (a) RCWA simulations and (b) FW theory. The FW model is calculated with
$n_{g}=5.5$, $\eta=0.9$, $U_{1}=5.7\times 10^{-4}$, $U_{2}=U_{1}/1.5$,
$\gamma_{1}=0.4\lvert U_{1}\rvert$ and $\gamma_{2}=0.5\lvert U_{1}\rvert$.

We now confront the results of our theoretical model with the ones obtained
from numerical simulations using RCWA methods. Figures S8(a,b) represent the
dispersion along $k_{x}$ and $k_{y}$ obtained from RCWA simulations of angle-
resolved abosrption (pseudo colormap of resonances) and the real part of the
eigenmodes obtained from the FW theory (black solid lines), as well as the
uncoupled folded modes (white dashed lines). It shows that the resonant
energies are perfectly reproduced by the FW theory. Figures S8(c) and (d)
represent the quality factor mappings of $\lvert\Omega_{1}\rangle$ obtained
from RCWA simulations and the FW theory respectively. The two mappings both
exhibit maximum of quality factor at $\bm{k}_{BIC}$ and are in good agreement
for the variation of quality factor when going out from $\bm{k}_{BIC}$.
However, while the quality factor from FW theory diverges at $\bm{k}_{BIC}$,
the one from RCWA simulation is bounded at $10^{5}$. Therefore, the FW-BIC in
becomes a quasi-BIC for the real structure. The residual losses, although very
small, is due to parasitic absorption of the perovskite material bellow its
band-gap and the coupling with other modes that are not considered in the
$3\times 3$ model.

We note that a twin FW-BIC is expected at $-\bm{k}_{BIC}$ due to the mirror
symmetry $x\rightarrow-x$. This twin FW-BIC is resulted from the coupling
between $\lvert\tilde{1}\rangle=\lvert-1,0\rangle$,
$\lvert\tilde{2}\rangle=\lvert 1,1\rangle$ and $\lvert\tilde{3}\rangle=\lvert
1,-1\rangle$.

####  5.3.4 Polarization texture of the FW-BIC twins

Using (S15), one can obtain the farfield patterns from the eigenvectors of
(S22). Then the polarization textures can be calculated using (S16). Figures
S9(a-d) depict the polarization texture, obtained by RCWA simulations and FW
theory, in the vicinity of the two FW-BICs. The results show a perfect
agreement between the numerical simulations and the FW model. Most
importantly, we evidence polarization singularity pinned at the FW-BIC
location.

Figure S9: Polarization textures: RCWA vs FW model. Mapping of the
polarization orientation angle $\phi$ in the vicinity of the FW-BIC located at
(a,b)$\bm{k}_{BIC}$, (c,d)$-\bm{k}_{BIC}$. These textures are obtained from
(a,c) RCWA simulations, (b,d) the FW theory. The parameters for the FW theory
are the same as the one used in Fig. S7.