# Coexistence of Surface Lattice Resonances and Bound states In the Continuum
in a Plasmonic Lattice

Quoc Trung Trinh1 Sy Khiem Nguyen1 Dinh Hai Nguyen1 Gia Khanh Tran1 Viet Hoang
Le1 Hai-Son Nguyen2,3 Quynh Le-Van1<EMAIL_ADDRESS>1College of
Engineering and Computer Science, VinUniversity, Gia Lam district, Hanoi
14000, Vietnam  2Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA Lyon,
Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, 69130 Ecully,
France  3Institut Universitaire de France (IUF), Paris, France

###### Abstract

We present a numerical study on a 2D array of plasmonic structures covered by
a subwavelength film. We explain the origin of surface lattice resonances
(SLRs) using coupled dipole approximation and show that the diffraction-
assisted plasmonic resonances and formation of bound states in the continuum
(BICs) can be controlled by altering the optical environment. Our study shows
that when the refractive index contrast $\Delta$n ¡ -0.1, the SLR cannot be
excited, while a significant contrast ($\Delta$n ¿ 0.3) not only sustains
plasmonic-induced resonances but also forms both symmetry-protected and
accidental BICs. The results can aid the streamlined design of plasmonic
lattices in studies on light–matter interactions and applications in
biosensors and optoelectronic devices.

Surface lattice resonances (SLRs) are delocalized modes emerging as localized
surface plasmon resonances (LSPRs) of individual plasmonic particles that are
radiatively coupled to the diffraction orders of the incident light waves
induced by the lattices [1, 2]. The lattice mode is also studied in lossless
systems [3]. The coupling is a mechanism to mitigate the ohmic losses from the
LSPRs to the diffracted waves. As a result, SLRs are characterized by an
increased quality factor over LSPRs and by a sharper Fano resonance in the
scattering spectra [4, 5, 6]. These remarkable properties have attracted
significant interest for potential applications in lasers [7, 8, 9], strong
light-matter interactions [10, 11], nonlinear optics [12], and biosensors
[13]. Another peculiar aspect of Fano resonances is their disappearance in the
scattering spectra. The collapse of Fano resonances is a signature of the
bound state in the continuum (BIC) [14, 15]. This disappearance leads to an
infinite Q-factor and has been experimentally proven in dielectric systems
[16, 17, 18]. Recently, several groups have theoretically proposed [19, 20,
21, 22, 23, 24] and reported the BICs in lossy media [25, 26, 27].
Interestingly, both SLRs and BICs can be realized in periodic structures
despite having opposite features in terms of the Fano resonances. To date,
there has been no report on any plasmonic system that supports both SLRs and
BICs. Here, we propose a simple plasmonic structure made of a silver nanodisks
array that can host both SLRs and BICs simultaneously. We elaborate on the
physics of the system through numerical studies under varied conditions of the
excitation angle, periodicity, and refractive index of the covered layer. We
show that three distinct regions with different physics can be obtained for
the given system.

We begin our study by addressing the origin of the SLRs in a 2D square lattice
at normal incidence. Our system comprises an array of silver nanodisks (radius
$r$ = 40 nm and height $h$ = 40 nm) with periodicity $p$, as illustrated in
Fig. 1a. They are placed on a glass substrate ($n_{sub}$ = 1.46) and covered
by a 200 nm thick superstrate with refractive index $n_{sup}$. We used the
coupled dipole approximation (CDA) to calculate the spectral positions of
LSPRs, Rayleigh anomalies, and SLRs [28]. In this framework, when a periodic
array of identical subwavelength nanoparticles is impinged by an external
plane wave, each particle can be characterized as an electric dipole with
polarizability $\alpha_{E}$111
$1/\alpha_{E}=\frac{3}{V}\left(L+\frac{\epsilon_{m}}{\epsilon-\epsilon_{m}}\right)$
where V is the volume of the particle, L is the geometric factor that varies
with different shapes [41], $\epsilon_{m}$ is the relative permittivity of the
medium and $\epsilon(\omega)=1-\frac{\omega_{p}^{2}}{\omega(\omega+i\gamma)}$
with $\omega_{p}$ being the plasmon frequency and $\gamma$ is the linewidth of
the LSPR. and subjected to the total applied fields, including the incident
waves and retarded fields by the neighboring particles. Thus the effective
polarizability $\alpha$ of each particle is given by
$\alpha^{-1}=\alpha_{E}^{-1}-S$
222$S\approx\left(\frac{4\sqrt{2}\pi^{2}}{\sqrt{\beta}}-118\right)\frac{1}{p^{3}}+2i\left(\frac{\pi
k}{p^{2}}-\frac{k^{3}}{3}+\frac{\pi^{2}}{(\pi N)^{3/2}\beta^{2}p^{3}}\right)$
where $k=(2\pi/\lambda)\sqrt{\epsilon_{r}}$ is the amplitude of wavevector in
the medium with relative permittivity $\epsilon_{r}$, $N$ is the number of the
particle in the array, and $\beta=\frac{2\pi}{kp}-1$[42]. Here $S$ is the
lattice sum describing the dipole coupling mechanisms between a particle and
its neighbors. Note that $\alpha_{E}$ and $S$ are complex values due to the
presence of Ohmic and radiative losses. In addition to dipole interactions,
the diffraction of the periodic array is another pronounced effect in response
to the incident waves. The diffracted waves can radiate in the regions above,
possibly below, and in the array. An interesting physical phenomena occur as
the wave diffracts in the plane of the array, known as Rayleigh anomaly. The
corresponding condition for such an effect is the zero-propagation constant in
the propagation direction. In other words, the condition for Rayleigh
anomalies in the Cartesian system is given by
$\epsilon_{r}=\left(\sin\theta\cos\Phi+m\frac{\lambda}{p}\right)^{2}+\left(\sin\theta\sin\Phi+n\frac{\lambda}{p}\right)^{2}$
where the pair $(m,n)$ corresponds to the diffraction orders, and $\theta$ and
$\Phi$ are the incident and azimuthal angle, respectively (see Fig. 1a). The
far-field reflectance for normal incidence can be expressed as $R=|\frac{2\pi
k}{p^{2}}\alpha|^{2}$[28, 31], leading to photonic resonances at the zeros of
the real part of $\alpha^{-1}$. Figure 1b shows the calculated
$\Re(\alpha_{E}^{-1})$ and real values of the lattice sum $S_{1}$ and $S_{2}$
corresponding to two periods $p_{1}$ = 325 nm and $p_{2}$ = 425 nm. The
$\Re(S_{1})$ value is low, indicating a modest contribution of the lattice in
the spectral range. In contrast, the $\Re(S_{2})$ shows an abrupt drop to
zero-value at 2.0 eV. The intersection of $\Re(\alpha_{E}^{-1})$ with
$\Re(S_{1})$ is evidence of an LSPR, whereas it crosses $\Re(S_{2})$ at three
distinct values.

Figure 1: The origin of SLR. a) Sketch of the plasmonic array. A square
lattice of silver nanodisks on a substrate ($n_{sub}$ = 1.46) is covered by a
thin layer. b) The calculated inverse polarizability (1/$\alpha$) of the
nanodisk ($r=h=$ 40 nm) and lattice sums $S_{1}$ of $p_{1}=$ 325 nm and
$S_{2}$ of $p_{2}=$ 425 nm at $\theta=0^{\circ}$. c) Simulated reflectances at
$\theta=0^{\circ}$ of the structures with $n_{sup}$ = 1.5. Three peaks in
$\mathbb{c)}$ agree well with the crossing points in $\mathbb{b)}$. The
electric fields of peaks: A (d-e), B (f-g) and C (h-i) for a unit cell. The
orange dash lines indicate the interfaces between layers.

First, the intersection with the sharp drop (2.0 eV) corresponds to the
diffracted waves or Rayleigh anomaly. The crossing at 1.95 eV (635.9 nm) and
2.33 eV (532.1 nm) are signatures of the SLR and LSPR, respectively. To verify
the predictions, we performed a numerical simulation of the scattering spectra
for such geometries using Comsol Multiphysics with TE polarization and Floquet
boundary conditions. Figure 1c shows the simulated reflectance of the nanodisk
lattices as shown in Fig. 1b. The narrowest resonance at 1.95 eV (peak A) only
emerges for the lattice spacing $p_{2}$ = 425 nm, and a kink at $E$ = 2.0 eV
is the position of the Rayleigh anomaly. Figures 1d and e confirm the
characteristics of the SLR, whose electric field is expanding to the edges of
the unit cell. The broad resonant peak at 2.15 eV (peak C) and 2.33 eV (peak
B) correspond to the LSPRs of the particles as $p_{1}$ = 325 nm and $p_{2}$ =
425 nm, respectively. The electric fields of LSPRs are illustrated in Figs.
1f-g and Figs. 1h-i. The peaks are in good agreement with the CDA predictions
shown in Fig. 1b.

Figure 2: Excitations of the SLR with varying refractive index contrast. a)
Evolution of the resonances in the index contrast $\Delta n=-0.46-0.94$. Three
regions (I, II, and III) are observed. The numbers in parenthesizes indicate
the diffracted orders of the lattice. b) Truncated spectra at specific values
in Fig. 2a. c) Calculated electric field in y-direction for three resonances
as a function of z position for $\Delta n=0.34$. d), e) Electric field of GR.

We investigated the role of the superstrate for the nanodisk array with a
period $p$ = 425 nm in sustaining both LSPRs and SLRs. $n_{sup}$ is varied
from 1.0 (air) to 2.4 (TiO2 [32] or perovskite film [33, 34]). This
corresponds to a variation in the index contrast, defined by $\Delta
n=n_{sup}-n_{sub}$, from -0.46 to 0.94. Although SLRs have been studied in
inhomogeneous media [35], only low index contrast (¡9$\%$) was considered.
Here, the contrast range was extended up to 64$\%$ ($\Delta n=0.94$). Figure
2a shows three regions with distinct features: the region I, $\Delta n<-0.1$,
which only supports the LSPR because the waves propagate in the substrate
owing to its higher refractive index. Region II supports both LSPR and SLR as
the index contrast is kept at a minimum ($-0.1<\Delta n<0.3$). Notably, an
additional sharp peak appears in the region III ($\Delta n>0.3$) between the
LSPR and SLR. This sharp resonance is attributed to a guided resonance (GR)
[36, 37, 38] which is a dielectric guided mode confined within the superstrate
and is diffracted into the radiative continuum owing to the lattice effect.
Naturally, this GR is less exposed to the nonradiative losses of the metallic
particles than the LSPR and SLR, thus, it would exhibit much higher Q-factors.
To verify the nature of the resonances, we truncated the spectra at three
representative values for each region (see Fig. 2b). The spectrum when $\Delta
n=-0.3$ shows a single broad LSPR at 2.54 eV while the one of $\Delta n=0.04$
exhibits two distinct peaks, (LSPR and SLR) as shown in Fig. 1c. Lastly, the
spectrum for $\Delta n=0.34$ reveals three peaks: SLR (1.72 eV), GR (1.92 eV),
and LSPR (2.02 eV). Figure 2c presents the calculated electric field in
y-direction for the resonances. As expected, the SLR field is maximum in the
lattice plane, whereas the GR field is maximum at the boundary between the
superstrate and the air. However, the GR field also exhibits a local maximum
near the plasmonic particle. This suggests that this GR is resulted from the
hybridization between pure guided mode and LSPR. Figures 2d and e confirm this
conclusion, which shows that the electric field is mainly confined within in
the superstrate and on the particle. Hybridization is also reflected in th
Q-factors of the resonances. When $\Delta n$ increases from 0.04 $\rightarrow$
0.34, $Q_{SLR}$ decreases from 60 $\rightarrow$ 52, while $Q_{LSPR}$ increases
from 17 $\rightarrow$ 28, with $Q_{GR}$ = 232 at $\Delta n=0.34$. The increase
in $Q_{LSPR}$ is attributed to its hybrid with GR, in which a loss exchange
occurs between two resonances.

Figure 3: The reflectance of the system (at $\theta=0^{\circ}$) with tuned
periodicity for two refractive index contrasts. a) $\Delta n=0.04$ and b)
$\Delta n=0.34$. The dash curves with numbers in parenthesizes are diffraction
orders of the lattice.

We investigated the influence of the spatial distance between the particles on
the optical responses of the system. We focused on two representative cases:
low ($\Delta n=0.04$) and high ($\Delta n=0.34$) refractive index contrasts,
which are presented in Figs. 3a and 3b, respectively. As shown in Fig. 3a, the
LSPR frequency decreases with increasing periodicity. This contradicts the
common belief that the frequency is solely determined by the materials and the
optical environment. The change in the frequency of the LSPR results from the
reshaping of the band diagram as the periodicity is tuned. We also observe
that the SLR becomes sharper as the periodicity approaches the diffraction
line. This can be attributed to the enhanced coupling between the LSPR and
Rayleigh anomalies. Notably, under such a large index contrast $\Delta
n=0.34$, the GR is not always excited. Instead, the GR is excited when it
couples with the LSPRs. As a result, the losses of the LSPR are alleviated via
hybridization with the GR, and hence, the Q-factors of these LSPRs
significantly increase. Conversely, the Q-factors of the SLRs deteriorate
owing to reduced coupling with the in-plane diffractions. We analyzed the
angle dependence of the SLRs when excited in a quasi-homogeneous environment,
$\Delta n=0.04$. As they are probed in wavevector $k_{x}$ (Fig. 4a), their
optical responses are shaped by the coupling between the LSPR and the
diffraction orders ($\pm$1,0). The LSPR in Figs. 4a and b exhibits relatively
constant energy when excited from oblique angles. In contrast, the SLRs
emerging from the coupling between the LSPR and the diffraction order
($\pm$1,0) show strong angular dependence. Two SLR branches can be observed
along $k_{x}$, both exhibiting a linear dispersion as they approach the
diffraction order ($\pm$1,0). The upper SLR branch fades closer to the
$\Gamma$ point and is not visible at normal incidence. In comparison, the
lower SLR branch has a pronounced resonance signal at the $\Gamma$ point and
fades at oblique angles. When probed in $k_{y}$, only the lower SLR branch is
visible with a quadratic dispersion, closely following the characteristics of
the diffraction order (0,$\pm$1). We note that changing polarization and
fixing the probe angle will produce the same results as a fixed polarization
and changing probing directions as in this work. Figure 4c presents the
Q-factors of the different SLR branches plotted as a function of the
wavevector. As $k_{x}$ deviates from the $\Gamma$ point, the Q-factor
increases rapidly for the SLR branch (-1,0) and reaches a plateau (Q = 300)
when fading to the Rayleigh anomalies ($k_{x}>1\mu m^{-1}$).

Figure 4: Angle-resolved reflectance of the silver nanodisk array for $\Delta
n$ = 0.04 and $p$ = 425 nm. The reflectance is probed a) in $k_{x}$ and b) in
$k_{y}$. c) Q-factor of different bands of SLRs. d) Spectra truncated in a)
and b) at $\theta=0^{\circ}$ and $\theta=6^{\circ}$. The solid and dashed
lines are associated with the spectra probed in $k_{x}$ and $k_{y}$,
respectively.

Finally, we investigated the angle-resolved response of the structure with a
high-index contrast $\Delta n$ = 0.34. As shown in Fig. 5, we find strikingly
different responses compared with the low-index contrast case, as shown in
Fig. 4. First, the LSPR and lower SLR bands are reshaped by the contraction of
the band diagram owing to the change in the refractive index. This leads to a
quadratic dispersion of the LSPR bands. More importantly, three GR-like bands
with a high curvature emerged between the LSPR and lower SLR bands. The
highest energy band corresponds to the one denoted as $GR$ in our previous
studies (Figs. 2 and 3), resulting from the hybridization between a pure GR
and the LSPR. Here, the angular-resolved reflectance of this band reveals that
it gets sharper when moving away from the G point and eventually reaches its
highest quality factor at $k_{x}=0.78\,\mu m^{-1}$ and $E$ = 1.925 eV
($\Lambda$ = 675.3 nm) when its scattering resonance vanishes locally in the
momentum space, see Fig. 5b. Such behaviors are the hallmarks of an accidental
BIC (a-BIC) that occurs when the hybridization mechanism leads to a
destructive interference configuration [39].

Figure 5: Angle-resolved reflectance when $\Delta n$ = 0.34: a) Reflectance
spectra are probed along $k_{x}$. Besides LSPR and SLR, the structures form an
accident BIC (a-BIC) at $k_{x}=0.78~{}\mu m^{-1}$ and two quasi symmetry-
protected BICs (s-BICs) at $\Gamma$ point. b) Total Q-factors of the BICs
marked in a). c) Magnified reflectance around quasi-BIC points indicated in
a). The color scale is unchanged compared to that in a). d) Side view of the
electric fields of points D, E, and F in c). They confirm that the a-BIC is an
hybrid LSPR-GR state while the two s-BICs are both only a GR nature.

On the other hand, the spectral range and curvature of the second and third
GR-like bands suggest that they are hybridizations between a pure GR and the
upper SLR branch depicted in Fig. 5a. Moreover, because the upper SLR branch
does not exist in the vicinity of the $\Gamma$ points (i.e. $|k_{x}|<0.1\,\mu
m^{-1}$, Fig. 5a), these two bands are of a GR nature solely in this range.
Interestingly, their scattering resonances vanish at the $\Gamma$ point when
energies $E$ = 1.845 eV ($\Lambda$ = 672 nm) and 1.836 eV ($\Lambda$ = 675.3
nm), while becoming broader as soon as they are under oblique excitation.
Therefore, they are symmetry-protected BICs (s-BICs), in which the coupling to
the radiative continuum is strictly forbidden owing to the $C_{4}$ symmetry
mismatch between their in-plane pattern and the radiating plane waves
[39].Magnified views of the a-BIC and the two s-BICs are presented in the
upper and lower panels of Fig. 5c. Figure 5d depicts the electric field
distributions of the three BIC states, confirming that the a-BIC is indeed of
a hybrid GR–LSPR nature, while the two s-BICs are solely of a GR nature. We
used the Q-factor to compare these BICs with others reported for plasmonic
systems. The Q-factors in our system reached remarkably high values ($\sim
10^{4}$) in the BIC states, as shown in Fig. 5b. We note that in plasmonic
systems, the Q-factor is bounded by the nonradiative losses from the metallic
component, and most of the reported BICs from plasmonic systems designed for
infared and visible frequencies exhibit Q-factors only in the range of
$10^{2}-10^{3}$ [20, 26, 27]. Our results show that by using GR-like modes,
the lossy flaw in plasmonic systems can be mitigated, thus providing ultra-
high-quality factors even with metallic components. A similar result was
recently reported for 1D plasmonic gratings [40].

In conclusion, we demonstrated that plasmonic structures can enable a
variation in the resonance as well as in the trapped light in the form of
BICs. The key constraint to unlock this interesting phenomenon is the large
refractive index contrast surrounding the plasmonic particles. Our study
demonstrated a new application of guided resonance as an effective mechanism
to mitigate the absorptive losses of the plasmonic particles and revealed that
the waveguide effect can lead to the formation of BICs in a lossy medium. The
findings could be implemented in plasmonic lattice covered by a thin film of
perovskite or semiconductor nanocrystals for applications in sensing and
light–emitting devices.

Funding Information This work was funded by Vingroup Big Data Institute,
Vingroup VINIF.2021.DA00169.

Acknowledgement. Q. L-V would like to thank Bojorn Maes for his critical
assistance with simulation models.

Disclosures. The authors declare no conflicts of interest.

## References

  * Zou  _et al._ [2004] S. Zou, N. Janel, and G. C. Schatz, Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes, The Journal of Chemical Physics 120, 10871 (2004), https://doi.org/10.1063/1.1760740 .
  * Markel [2005] V. A. Markel, Divergence of dipole sums and the nature of non-lorentzian exponentially narrow resonances in one-dimensional periodic arrays of nanospheres, Journal of Physics B: Atomic, Molecular and Optical Physics 38, L115 (2005).
  * Babicheva and Evlyukhin [2017] V. E. Babicheva and A. B. Evlyukhin, Resonant lattice kerker effect in metasurfaces with electric and magnetic optical responses, Laser & Photonics Reviews 11, 1700132 (2017), https://onlinelibrary.wiley.com/doi/pdf/10.1002/lpor.201700132 .
  * Miroshnichenko  _et al._ [2010] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys. 82, 2257 (2010).
  * Le-Van  _et al._ [2019] Q. Le-Van, E. Zoethout, E.-J. Geluk, M. Ramezani, M. Berghuis, and J. Gómez Rivas, Enhanced quality factors of surface lattice resonances in plasmonic arrays of nanoparticles, Advanced Optical Materials 7, 1801451 (2019), https://onlinelibrary.wiley.com/doi/pdf/10.1002/adom.201801451 .
  * Bin-Alam  _et al._ [2021] M. S. Bin-Alam, O. Reshef, Y. Mamchur, M. Z. Alam, G. Carlow, J. Upham, B. T. Sullivan, J.-M. Ménard, M. J. Huttunen, R. W. Boyd, and K. Dolgaleva, Ultra-high-q resonances in plasmonic metasurfaces, Nature Communications 12, 974 (2021).
  * Zhou  _et al._ [2013] W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, Lasing action in strongly coupled plasmonic nanocavity arrays, Nature Nanotechnology 8, 506 (2013).
  * Yang  _et al._ [2015] A. Yang, T. B. Hoang, M. Dridi, C. Deeb, M. H. Mikkelsen, G. C. Schatz, and T. W. Odom, Real-time tunable lasing from plasmonic nanocavity arrays, Nature Communications 6, 6939 (2015).
  * Ramezani  _et al._ [2018] M. Ramezani, Q. Le-Van, A. Halpin, and J. Gómez Rivas, Nonlinear emission of molecular ensembles strongly coupled to plasmonic lattices with structural imperfections, Phys. Rev. Lett. 121, 243904 (2018).
  * Hakala  _et al._ [2018] T. K. Hakala, A. J. Moilanen, A. I. Väkeväinen, R. Guo, J.-P. Martikainen, K. S. Daskalakis, H. T. Rekola, A. Julku, and P. Törmä, Bose-einstein condensation in a plasmonic lattice, Nature Physics 14, 739 (2018).
  * Wang  _et al._ [2019] S. Wang, Q. Le-Van, F. Vaianella, B. Maes, S. E. Barker, R. H. Godiksen, A. G. Curto, and J. Gomez Rivas, Limits to strong coupling of excitons in multilayer ws2 with collective plasmonic resonances, ACS Photon. 6, 286 (2019).
  * Czaplicki  _et al._ [2016] R. Czaplicki, A. Kiviniemi, J. Laukkanen, J. Lehtolahti, M. Kuittinen, and M. Kauranen, Surface lattice resonances in second-harmonic generation from metasurfaces, Opt. Lett. 41, 2684 (2016).
  * Li  _et al._ [2021] N. Li, T. D. Canady, Q. Huang, X. Wang, G. A. Fried, and B. T. Cunningham, Photonic resonator interferometric scattering microscopy, Nature Communications 12, 1744 (2021).
  * von Neumann and Wigner [1929] J. von Neumann and E. Wigner, Physik. Z. 30, 467 (1929).
  * Friedrich and Wintgen [1985] H. Friedrich and D. Wintgen, Interfering resonances and bound states in the continuum, Phys. Rev. A 32, 3231 (1985).
  * Hsu  _et al._ [2013] C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, Observation of trapped light within the radiation continuum, Nature 499, 188 (2013).
  * Kodigala  _et al._ [2017] A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, Lasing action from photonic bound states in continuum, Nature 541, 196 (2017).
  * Ha  _et al._ [2018] S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Domínguez, and A. I. Kuznetsov, Directional lasing in resonant semiconductor nanoantenna arrays, Nature Nanotechnology 13, 1042 (2018).
  * Marinica  _et al._ [2008] D. C. Marinica, A. G. Borisov, and S. V. Shabanov, Bound states in the continuum in photonics, Phys. Rev. Lett. 100, 183902 (2008).
  * Azzam  _et al._ [2018] S. I. Azzam, V. M. Shalaev, A. Boltasseva, and A. V. Kildishev, Formation of bound states in the continuum in hybrid plasmonic-photonic systems, Phys. Rev. Lett. 121, 253901 (2018).
  * Mermet-Lyaudoz  _et al._ [2019] R. Mermet-Lyaudoz, F. Dubois, N.-V. Hoang, E. Drouard, L. Berguiga, C. Seassal, X. Letartre, P. Viktorovitch, and H. S. Nguyen, Realization of bound state in the continuum induced by vertical symmetry breaking in photonic lattice, arXiv , 1905.03868 (2019).
  * Contractor  _et al._ [2020] R. Contractor, W. Noh, Q. Le-Van, and B. Kanté, Doping-induced plateau of strong electromagnetic confinement in the momentum space, Opt. Lett. 45, 3653 (2020).
  * Hsu  _et al._ [2021] L. Hsu, F. I. Baida, and A. Ndao, Local field enhancement using a photonic-plasmonic nanostructure, Opt. Express 29, 1102 (2021).
  * Wang  _et al._ [2020] X. Wang, J. Duan, W. Chen, C. Zhou, T. Liu, and S. Xiao, Controlling light absorption of graphene at critical coupling through magnetic dipole quasi-bound states in the continuum resonance, Phys. Rev. B 102, 155432 (2020).
  * Doeleman  _et al._ [2018] H. M. Doeleman, F. Monticone, W. den Hollander, A. Alù, and A. F. Koenderink, Experimental observation of a polarization vortex at an optical bound state in the continuum, Nat. Photon. 12, 397 (2018).
  * Liang  _et al._ [2020] Y. Liang, K. Koshelev, F. Zhang, H. Lin, S. Lin, J. Wu, B. Jia, and Y. Kivshar, Bound states in the continuum in anisotropic plasmonic metasurfaces, Nano Lett. 20, 6351 (2020).
  * Seo  _et al._ [2020] I. C. Seo, S. Kim, B. H. Woo, I.-S. Chung, and Y. C. Jun, Fourier-plane investigation of plasmonic bound states in the continuum and molecular emission coupling, Nanophotonics 9, 4565 (2020).
  * García de Abajo [2007] F. J. García de Abajo, Colloquium: Light scattering by particle and hole arrays, Rev. Mod. Phys. 79, 1267 (2007).
  * Note [1] $1/\alpha_{E}=\frac{3}{V}\left(L+\frac{\epsilon_{m}}{\epsilon-\epsilon_{m}}\right)$ where V is the volume of the particle, L is the geometric factor that varies with different shapes [41], $\epsilon_{m}$ is the relative permittivity of the medium and $\epsilon(\omega)=1-\frac{\omega_{p}^{2}}{\omega(\omega+i\gamma)}$ with $\omega_{p}$ being the plasmon frequency and $\gamma$ is the linewidth of the LSPR.
  * Note [2] $S\approx\left(\frac{4\sqrt{2}\pi^{2}}{\sqrt{\beta}}-118\right)\frac{1}{p^{3}}+2i\left(\frac{\pi k}{p^{2}}-\frac{k^{3}}{3}+\frac{\pi^{2}}{(\pi N)^{3/2}\beta^{2}p^{3}}\right)$ where $k=(2\pi/\lambda)\sqrt{\epsilon_{r}}$ is the amplitude of wavevector in the medium with relative permittivity $\epsilon_{r}$, $N$ is the number of the particle in the array, and $\beta=\frac{2\pi}{kp}-1$[42].
  * Teperik and Degiron [2012] T. V. Teperik and A. Degiron, Design strategies to tailor the narrow plasmon-photonic resonances in arrays of metallic nanoparticles, Phys. Rev. B 86, 245425 (2012).
  * Sun  _et al._ [2017] S. Sun, Z. Zhou, C. Zhang, Y. Gao, Z. Duan, S. Xiao, and Q. Song, All-Dielectric Full-Color Printing with TiO2 Metasurfaces, ACS Nano 11, 4445 (2017).
  * Lu  _et al._ [2020] L. Lu, Q. Le-Van, L. Ferrier, E. Drouard, C. Seassal, and H. S. Nguyen, Engineering a light-matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points, Photon. Res. 8, A91 (2020).
  * Dang  _et al._ [2020] N. H. M. Dang, D. Gerace, E. Drouard, G. Trippé-Allard, F. Lédée, R. Mazurczyk, E. Deleporte, C. Seassal, and H. S. Nguyen, Tailoring dispersion of room-temperature exciton-polaritons with perovskite-based subwavelength metasurfaces, Nano Lett. 20, 2113 (2020).
  * Auguié  _et al._ [2010] B. Auguié, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, Diffractive arrays of gold nanoparticles near an interface: Critical role of the substrate, Phys. Rev. B 82, 155447 (2010).
  * Zentgraf  _et al._ [2009] T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems, Phys. Rev. B 80, 195415 (2009).
  * Monticone and Alù [2017] F. Monticone and A. Alù, Bound states within the radiation continuum in diffraction gratings and the role of leaky modes, New Journal of Physics 19, 093011 (2017).
  * Bian  _et al._ [2018] Y. Bian, Q. Ren, L. Kang, T. Yue, P. L. Werner, and D. H. Werner, Deep-subwavelength light transmission in hybrid nanowire-loaded silicon nano-rib waveguides, Photon. Res. 6, 37 (2018).
  * Hsu  _et al._ [2016] C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Bound states in the continuum, Nature Reviews Materials 1, 16048 (2016).
  * Deng  _et al._ [2021] Z.-L. Deng, F.-J. Li, H. Li, X. Li, and A. Alù, Perfect diffraction metagratings supporting bound states in the continuum and exceptional points, arXiv , 2109.10533 (2021).
  * Barnes [2016] W. L. Barnes, Particle plasmons: Why shape matters, American Journal of Physics 84, 593 (2016).
  * Špačková and Homola [2013] B. Špačková and J. Homola, Sensing properties of lattice resonances of 2d metal nanoparticle arrays: An analytical model, Opt. Express 21, 27490 (2013).