Plexcitonic quasi-bound states in the continuum

Abstract

Enhancing light-matter interactions is fundamental to the advancement of nanophotonics and optoelectronics. Yet, light diffraction on dielectric platforms and energy loss on plasmonic metallic systems present an undesirable trade-off between coherent energy exchange and incoherent energy damping. Through judicious structural design, both light confinement and energy loss issues could be potentially and simultaneously addressed by creating bound states in the continuum (BICs) where light is ideally decoupled from the radiative continuum. Herein, we present a general framework based on the two-coupled resonances to first conceptualize and then numerically demonstrate a type of quasi-BICs that can be achieved through the interference between two bare resonance modes and is characterized by the considerably narrowed spectral line shape even on lossy metallic nanostructures. The ubiquity of the proposed framework further allows the paradigm to be extended for the realization of plexcitonic quasi-BICs on the same metallic systems. Owing to the topological nature, both plasmonic and plexcitonic quasi-BICs display strong mode robustness against parameters variation, thereby providing an attractive platform to unlock the potential of the coupled plasmon-exciton systems for manipulation of the photophysical properties of condensed phases.

Graphical Abstract

A general framework based on two-coupled resonances is presented to conceptualize a type of quasi-bound states in the continuum (quasi-BICs). Through judicious structural design, both plasmonic and plexcitonic quasi-BICs are numerically demonstrated on lossy metallic systems in the visible wavelength range. This study provides a promising platform for studying coupled plasmon-exciton systems for manipulating photophysical properties of condensed phases.

1. Introduction

Bound states in the continuum (BICs) are perfectly confined waves with an infinite lifetime that exist in the continuous spectrum of radiative waves but are completely decoupled from it.^1–3 Independent of the type of waves involved, being either photonic, phononic, or water waves, such a wave confinement paradigm is ubiquitous in nature and can be achieved by symmetry or separability and through parameter tuning for coupled resonances.⁴ Yet, ideal BICs are more of a theoretical construct than an experimental observation. That is because they require unbounded material systems with a vanishing energy loss rate or an epsilon-near-zero permittivity that do not naturally exist.⁵ Moreover, their communication with the continuum is forbidden by design, and thereby, they can neither be excited nor exhibit any spectral traces that are experimentally observable.

In practice, perturbation from the non-vanishing energy loss in a given physical system would downgrade ideal BICs to Fano resonance-like quasi-BICs.⁶ With a finite but long lifetime, quasi-BICs display characteristic peculiarities of narrow and sharp spectral line shapes that can be directly observed. Therefore, they provide an attractive platform for studying coherent light-matter interactions, and have been harnessed for the realization of BICs lasers,^7–9 supercavity modes,^10–13 BICs-enhanced spin-orbit strong coupling,¹⁴ BICs-enhanced second-harmonic generation,¹⁵ topologically enabled unidirectional guided resonances,¹⁶ and optical vortices.^17–18 These advancements, however, are made mostly on high-index dielectric metasurfaces while plasmonic metallic systems remain largely underexplored because of the strong energy loss caused by rapid plasmon dephasing and intrinsic loss in metals.

While the energy loss associated with surface plasmons (SPs) is detrimental to the creation of quasi-BICs, such an issue itself could also be potentially addressed by it. This is because the highly tunable resonance frequency of SPs,^19–22 which can be readily made by varying either the size, shape and/or materials of the nanoparticle, periodicity of patterned nanostructures or the dielectric environment, offers a vast parameter space for spectral tuning that could considerably suppress radiation to the continuum through destructive interference between the coupled resonances, generating quasi-BICs on the plasmonic metallic systems.⁶ Realization of plasmonic quasi-BICs with a prolonged lifetime not only offers a natural solution to the energy loss issue, it could also help significantly augment the role of SPs to facilitate coherent light-matter interactions, as exemplified by the interaction between SPs and excitons.

As an elementary excitation, excitons have rich relaxation dynamics and can transport energy through a series of coherent and incoherent processes.^23–25 Understanding and controlling their energy relaxation and migration is of fundamental importance in shaping the physiochemical properties of condensed phases. SPs, with its extraordinary spatial coherence, ultrafast dynamics, highly confined electromagnetic fields, and strong dipole moment, are uniquely positioned to modulate the excitonic photophysical and chemical properties via the near-field dipole-dipole interactions.^26–27 For weakly coupled plasmon-exciton systems, the electromagnetic energy is incoherently transferred from SPs to facilitate the desired excitonic energy relaxation dynamics and migration pathways, as demonstrated in plasmon-enhanced spectroscopies,^28–29 photocatalysis,^30–31 solar energy harvesting,^32–33 etc. For strongly coupled systems, Rabi splitting is activated that opens coherent energy exchange channels between SPs and excitons, ending up creating a pair of plasmon-exciton hybrid states, or plexcitons, which is the plasmonic counterpart of excitonic polaritons through the strong coupling of excitons and cavity photons.^{25, 34–35} The formed plexcitonic pair are not only spectrally shifted from the bare states and spectrally split by the Rabi frequency, but also tunable by the coupling strength, plasmon-exciton energy detuning, and energy dissipation rates. Essentially, plexcitons offer fundamentally new strategies for coherent manipulation of band structures, relaxation dynamics, and energy migration mechanisms, which are otherwise unachievable in the weak-coupling regime. The huge potential of plexcitons, as manifested by recent achievements and conceptualizations in Bose-Einstein condensation,³⁶ novel optoelectronic devices with new functionalities,^34–35 quantum information and sensing technologies,^{23, 37–38} has further galvanized efforts to create polaritons by strongly coupling excitons with quasi-BICs,^39–40 which, however, for the same energy loss reason, are exclusively demonstrated on high-index dielectric metasurfaces rather than plasmonic metallic systems.

To fill that knowledge gap and also to address the energy loss issues, we propose here a general framework to conceptualize a type of quasi-BICs that can be realized through the interference of two bare resonance modes, and is characterized by considerably narrowed spectral line shapes even on lossy metallic nanostructures. The two bare modes can be very general provided that they are spectrally and spatially in close proximity to each other so that the interference can occur. With that framework, we numerically demonstrate creation of not only the coveted plasmonic quasi-BICs, but also the plexcitonic counterpart of quasi-BICs. Both are achieved on the same lossy metallic systems and display strong mode robustness against parameters variation, highlighting the shared topological nature with ideal BICs.

2. Results and Discussion

2.1. Quasi-BICs from two coupled resonances

In a non-Hermitian system, the confined optical resonances are leaky owing to an irreversible energy exchange with the continuum. The associated complex frequency ω = ω₀ − iγ depicts a leaky mode with a resonance frequency ω₀ and a rate γ at which it loses energy to the continuum in the form of absorption and scattering (as well as the combination of the two, i.e. extinction), among other energy transfer channels. While selecting materials with a small energy loss rate γ can improve light confinement, it does not preclude energy exchange with the continuum. We aim to achieve quasi-BICs by harnessing Rayleigh anomaly^41–42, which is a surface-diffraction optical phenomenon ubiquitous on periodic nanostructures. For a hexagonally patterned periodic nanostructures with a periodicity a₀ sitting on an insulating substrate with a real refractive index n, under normal illumination, Rayleigh anomalies occurring on the substrate side with a grating order (s₁, s₂)_sub at the frequency ω₀ given by

$${\omega }_0=\frac{a_0}{\sqrt{\frac{4}{3}(s_1^2+s_1{s}_2+s_2^2)}}n$$ (1a)

For Rayleigh anomalies occurring on the air side with a grating order (s₁, s₂)_air, the frequency is only slightly modified by the refractive index change, which is given by

$${\omega }_0=\frac{a_0}{\sqrt{\frac{4}{3}(s_1^2+s_1{s}_2+s_2^2)}}$$ (1b)

The pure real frequency of Rayleigh anomalies suggests that they can be modelled as an optical mode with a vanishing energy loss rate. Such treatment of Rayleigh anomalies is grounded on its origin as a particular surface diffracted spectral order⁴³, which occurs at frequencies determined by the diffraction conditions (i.e. diffraction order and periodicity). Previous studies have established that Rayleigh anomalies can couple with surface plasmon polaritons^{41, 44–45} and localized surface plasmon resonance⁴⁶, just as typical optical modes do. Likewise, in the framework of leaky-wave theory^47–48, Rayleigh anomalies are treated as an impenetrable periodic surface reactance which stores but does not dissipate energy, consistent with our treatment of Rayleigh anomalies to be an optical mode that does not decay. Indeed, Rayleigh anomalies being surface diffractive phenomena manifest themselves as subtle spectral features, if any, that can be observed on periodic nanostructures. Notably, their mere dependence on the periodicity of metallic nanostructures and the surrounding environmental refractive indices (Eq. (1)) provides a facile, yet underexplored, route for broadband spectral tuning, which is independent of and unperturbed by the lossy metallic materials involved.

To start with, we consider two general resonances, which are in bare state before hybridization, with a phase difference ψ and a complex frequency ω_1,2 = ω_0,b − iγ_0,b interacting with each other through a coupling strength γ on the same periodic metallic nanostructure. Based on the temporal coupled-mode theory, the Hamiltonian for the coupled system in the absence of external driving source can be constructed as⁴

$$H=\left[\begin{array}{cc}{\omega }_0& \lambda \\ \lambda & {\omega }_b\end{array}\right]-i\left[\begin{array}{cc}{\gamma }_0& \sqrt{{\gamma }_0{\gamma }_b}{e}^{i\psi }\\ \sqrt{{\gamma }_0{\gamma }_b}{e}^{i\psi }& {\gamma }_b\end{array}\right]$$ (2)

By diagonalizing the Hamiltonian matrix in Eq. (2), the complex energy eigenvalues are obtained $E_{1,2}=\frac{1}{2}\left({\omega }_0+{\omega }_b\right)-i\frac{1}{2}\left({\gamma }_0+{\gamma }_b\right)\pm \frac{1}{2}\sqrt{{\left(\Delta -i4{\gamma }_{-}\right)}^2-4{\left(i\lambda +\sqrt{{\gamma }_0{\gamma }_b}{e}^{i\psi }\right)}^2}$, where the energy detuning is defined as Δ = ω₀ − ω_b and the reduced energy loss rate as γ₋ = (γ₀ − γ_b)/4. The two eigenvalues are the complex resonance frequency for a pair of newly formed hybrid modes. To create a quasi-BICs, the imaginary part of the complex frequency, i.e., the energy loss rate, for at least one of the two hybrid modes, needs to be made considerably small if not vanishing, so that the energy exchange with the continuum can be effectively suppressed. This can be realized by selecting appropriate bare states, as is discussed thereafter.

If we consider the first bare state ω₁ = ω₀ − iγ₀ to be a Rayleigh anomaly and suppose it has zero detuning from the second ω₂ = ω_b − iγ_b, the complex energy eigenvalues for the hybrid modes can be simplified as

$$E_1={\omega }_0-i\frac{{\lambda }^2}{\gamma }$$ (3a)

$$E_2={\omega }_0-i\left(\gamma -\frac{{\lambda }^2}{\gamma }\right)$$ (3b)

by setting γ₀ = 0, ω_b = ω₀, and γ_b = γ. Interestingly, the two new hybrid modes are degenerate in frequency and differ only in the energy loss rate. As the coupling strength is far smaller than the energy loss rate (λ ≪ γ) in the weak-coupling regime, the effective energy loss rate $\frac{{\lambda }^2}{\gamma }$ for the first eigenstate E₁ (Eq. (3a)) is significantly reduced albeit non-vanishing, which enables E₁ to become a quasi-BICs. Reduction of energy loss indicates that light can be confined for a prolonged time in the coupled system. This is evidenced as the considerably narrowed spectral line shape and sharpened peak, as shown in Fig. 1, where the two bare modes ω_1,2 are converted into two newly hybridized modes E_1,2. The newly formed hybrid mode E₁ thus possesses all the essential characters of a quasi-BICs mode. In comparison, the other newly formed hybrid mode E₂ has a complex resonance frequency largely unmodified from the second bare state (Eq. (3b)). Therefore, the spectrum for the hybrid mode E₂ remains almost the same as before hybridization, thereby mainly contributing to the background of the overall spectral profile (upper panel in Fig. 1).

Figure 1. Formation of quasi-BICs.

Two bare modes ω_1,2 = ω_0,b − iγ_0,b (lower panel) hybridize and produce two hybrid modes $E_1={\omega }_0-i\frac{{\lambda }^2}{\gamma }$ and $E_2={\omega }_0-i(\gamma -\frac{{\lambda }^2}{\gamma })$ (middle panel). The combined spectrum displays a significantly narrowed spectral line shape and sharpened peak characteristic of a quasi-BICs (upper panel). The spectra were plotted using a Lorentzian peak function based on Eq. (3) by setting ω_b = ω₀, γ₀ = 0 (Rayleigh anomaly line), γ_b = γ = 0.1ω₀, and γ = 0.3γ.

As the second bare mode ω₂ is not specifically defined in deriving Eq. (3), we reason that the results hold true for both plasmonic and plexcitonic states, among other optical modes, provided that they spectrally overlap with the Rayleigh anomaly. The generality of the second bare mode could open numerous opportunities that would allow us to perform light management by harnessing the interference between Rayleigh anomaly and SPs or plexcitons on periodic metallic nanostructures. Indeed, the generality of the coupled resonances approach has been similarly applied to study Fano resonances. Recent studies have unveiled that Fano resonances can become quasi-BICs when the Fano parameter (q = cotδ) diverges (i.e when there is a vanishing phase shift δ between the two coupled resonances with dramatically different damping rates).^49–52 In the following, we will take advantage of finite-difference time-domain (FDTD) simulations to launch a detailed study on the creation of both plasmonic and plexcitonic quasi-BICs. Both modes are found to be readily realized on the same platform and display strong mode robustness against parameters tuning.

2.2. Plasmonic quasi-BICs on Ag nanopyramid arrays

The studied Ag nanopyramid arrays are schematically shown in Fig. 2(a) with a periodicity a₀ sitting on a substrate with a refractive index n. Each pyramid has a height h and base radius r. Through tuning of these parameters, the plasmonic resonances of Ag nanopyramid arrays display strong dependence on the periodicity and the substrate refractive index, seen in Fig. 2(b–c). Evolving accordingly are Rayleigh anomalies with different grating orders, which are predicted based on Eq. (1). With a simple linear relationship with the periodicity and the substrate refractive index, they exhibit well discernible traces on the dispersion diagrams. At intersections between plasmonic resonances and the Rayleigh anomaly (1, 1)_sub for larger periodicities (e.g. a₀ > 1000 nm in Fig. 2(b)) or larger substrate refractive indices (e.g. n > 1.45 in Fig. 2(c)), the plasmonic spectral linewidths become significantly narrowed with the plasmonic peaks increasingly sharpened and collapsing into the Rayleigh anomaly line. We note that, although previously reported surface lattice resonances display similar sharp spectral line shapes on periodic nanostructures, they differ from our plasmonic quasi-BICs in that they need a homogeneous dielectric environment surrounding the studied metallic nanostructures in order to achieve constructive interference among scattered plasmonic resonances.^53–54 In contrast, in our study, it is in a non-symmetric dielectric environment and at the coalescence of plasmonic resonances and the Rayleigh anomaly that the spectral linewidth narrowing and peak sharpening are observed. This highlights the interference nature of two different types of optical modes that coalesce into each other, ending up forming two new hybrid modes that are degenerate in frequency but differ in the energy loss rate, as predicted in Eq. (3).

Figure 2. Plasmonic quasi-BICs.

(a) Scheme for the studied Ag nanopyramid arrays with a periodicity a₀ on a substrate with a refractive index n. Each nanopyramid has a base radius r and a height h. Incident direction and polarizations are specified as marked. Dispersion diagrams for reflection spectra under each polarization by (b) tuning the periodicity a₀ with n = 1.45, r = 100 nm, and h = 300 nm, and (c) tuning the substrate refractive index n with a₀ = 1000 nm, r = 100 nm, and h = 300 nm. Rayleigh anomalies with different grating orders (s₁, s₂) are traced by hollow circles and marked in parentheses with subscripts “sub” specifying the substrate side and “air” the air side. Two orthogonal polarizations are defined as $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$.

Representative reflection spectra under two orthogonal polarizations defined in Fig. 2(a) for Ag nanopyramid arrays are shown in Fig. 3(a), which display polarization independence despite the asymmetry of the structure on the studied polarization directions. We note that the simultaneously narrowed spectral linewidths and sharpened peaks are reminiscent of the predicted spectral line shapes in Fig. 1 (upper panel), which are characteristic of a quasi-BICs. Therefore, the resonance peak at around 725 nm in Fig. 3(a) can be determined to be a quasi-BICs that originates from the interference between the Rayleigh anomaly line (1,1)_sub and the plasmonic resonances of Ag nanopyramid arrays. While plasmonic quasi-BICs has been achieved due to symmetry in the middle-IR wavelength range⁵⁵, our study presents a general approach to realize visible plasmonic quasi-BIC by two coupled resonances, i.e. the interference between plasmonic resonances and Rayleigh anomalies. Given the predicted frequency degeneracy in Eq. (3) for the two hybrid modes but with considerably different energy loss rates, the combined spectral profile is thus featured by a narrow and sharp peak from the quasi-BICs and a spectral background largely from the bare plasmonic resonance. In the following discussions, we will no longer make distinctions between the two hybrid modes but will call them generically as a plasmonic quasi-BICs.

Figure 3. Electric field profiles of plasmonic quasi-BICs.

(a) Normalized reflection spectra under two orthogonal polarizations for Ag nanopyramid arrays with a₀=1000 nm, r=100 nm, and h=300 nm on a substrate with a refractive index n = 1.45. The Rayleigh anomaly line (1,1)_sub at 725 nm is marked by hollow circles (b). (c) Electric field profiles possessing the feature of trapped surface waves for the plasmonic quasi-BICs, in sharp contrast to those of the localized plasmon resonance modes seen in (d)-(f). The color-bars have arbitrary units.

In addition, more insights on the plasmonic quasi-BICs can be gained by looking at the electric field profiles that are obtained at different wavelengths. The plasmonic quasi-BICs behave like trapped surface wave phenomena seen in Fig. 3(b–c), in sharp contrast to the highly localized nature of plasmonic resonances seen in Fig. 3(d–f).

It is also worth mentioning that Rayleigh anomalies have been similarly harnessed to achieve BICs in previous studies, where it has been established based on the leaky-wave theory that the interaction between Rayleigh anomalies and leaky guided modes leads to the creation of embedded eigenstates (i.e. BICs) on periodic nanostructures.⁴⁸ Despite different approaches used, our realization of quasi-BICs by treating Rayleigh anomalies as a lossless optical mode based on the temporal coupled-mode theory is consistent with the newly unveiled BICs mechanism by treating Rayleigh anomalies as an impenetrable periodic surface reactance based on the leaky-wave theory, which highlights the distinct but underexplored feature of Rayleigh anomalies.

2.3. Robustness of plasmonic quasi-BICs against parameter tuning

Optical BICs have been found to be the vortex centers in the polarization direction of far-field radiation.^{18, 56} The conservation of the quantized topological charges, which are the winding number of the polarization vectors around the vortex center, protects optical BICs from perturbation in the parameter space. As the topological charges can only be generated or annihilated by drastic parameter changes, optical BICs display extraordinary robustness against parameters’ tuning that is highly desirable from a nanofabrication perspective.

To test the robustness of the plasmonic quasi-BICs on Ag nanopyramid arrays against parameters tuning, we have systematically investigated the spectral evolution by varying the structural dimensions. The resonance frequency for the plasmonic quasi-BICs is found to remain unchanged by tuning the pyramidal height from about 200 to 1000 nm, seen in Fig. 4(a–c). Similar mode robustness is also observed by tuning the pyramidal base radius from about 90 to 120 nm for different pyramidal heights, seen in Fig. 4(d–i). The fact that the quasi-BICs display robust topological attributes, just as BICs do, suggests that protected optical states do not necessarily rely on optical BICs that require complex structural design, among other stringent conditions. Instead, plasmonic quasi-BICs provide a facile route for the creation of robust optical states. Relaxation of parameter constraints further facilitates practical realization of these highly favorable plasmonic quasi-BICs.

Figure 4. Robustness of plasmonic quasi-BICs against parameter tuning.

(a)-(c) Height tuning with r=100 nm. Base radius tuning with (d)-(f) h=300 nm and (g)-(i) h=1000 nm. The periodicity is fixed as a₀=1000 nm and the substrate refractive index as n = 1.45. Studied in this case are Ag nanopyramid arrays under two orthogonal polarizations.

We note that the studied plasmonic quasi-BICs occur near the resonance frequency of Rayleigh anomaly. This implies that plasmonic quasi-BICs are dependent on and tunable by the periodicity and the substrate refractive index, which are the two key tuning parameters for Rayleigh anomaly. Such correlations suggest that the Rayleigh anomaly plays a dominant role in the creation of plasmonic quasi-BICs. At the same time, plasmonic resonances act as antennas that absorb and convert incident electromagnetic radiations into the confined energy in the form of trapped surface waves at the frequency of Rayleigh anomaly that are necessary for the formation of quasi-BICs. These observations are echoed in Eq. (3), which is derived with the first bare mode explicitly specified as a Rayleigh anomaly while the second bare mode not specifically defined. The generality of the second bare mode alludes to the ubiquity of quasi-BICs on periodic nanostructures, where the constituent elements are expected to play a lesser role than the resonances generated from them.

2.4. Ubiquity of plasmonic quasi-BICs

The ubiquity of plasmonic quasi-BICs is confirmed by studying the plasmonic resonance dispersions for different variants of the periodic metallic nanostructures. For Ag nanotriangle arrays with the same periodicity and base radius as Ag nanopyramid arrays, the plasmonic modes are found to collapse into the Rayleigh anomaly line (1,1)_sub at around 725 nm following almost the same behavior as those of the nanopyramid counterparts by height tuning, seen in Fig. 5(a). This demonstrates that it is indeed the plasmonic resonances generated from the shape of the constituent elements, rather than the shape of the constituent elements itself, that are crucial for the creation of plasmonic quasi-BICs.

Figure 5. Ubiquitous plasmonic quasi-BICs.

(a) Conversion of plasmonic modes into plasmonic quasi-BICs near the resonance frequency of the Rayleigh anomaly line (1,1)_sub by tuning the pyramidal height for both Ag nanotriangle and nanopyramid arrays (a₀=1000 nm, r=100 nm, and substrate refractive index n = 1.45). Plasmonic modes of Au nanopyramid arrays with the same parameters fail to be converted into a quasi-BICs owing to the spectral offset from the Rayleigh anomaly line (1,1)_sub. By tuning the periodicity (with the Rayleigh anomaly line tuned correspondingly), mode conversion is enabled on Au nanopyramid arrays that allows the formation of plasmonic quasi-BICs near the Rayleigh anomaly lines (b) (1,1)_sub for a₀=1083 nm and (c) (1,0)_sub for a₀=625 nm with the substrate refractive index fixed as n = 1.45 and r=100 nm. Hollow symbols represent corresponding data obtained under orthogonal polarizations.

However, as the plasmonic modes of Au nanopyramid arrays with the same dimensions as the Ag counterparts are spectrally offset from the Rayleigh anomaly line (1,1)_sub, they cannot be converted into plasmonic quasi-BICs. Nevertheless, such mode conversion can be realized by shifting the Rayleigh anomaly line to be spectrally overlapped with the plasmonic resonance through periodicity tuning. As demonstrated by Au nanopyramid arrays with a periodicity a₀ = 1083 nm, where the Rayleigh anomaly line (1,1)_sub is shifted to be about 785 nm in spectral overlap with the plasmonic resonances, the plasmonic modes from the structure with a height larger than 280 nm are converted into a plasmonic quasi-BICs, seen in Fig. 5(b). Likewise, the plasmonic modes of Au nanopyramid arrays with a periodicity a₀ = 625 nm can also be converted into quasi-BICs through hybridization with the Rayleigh anomaly line (1,0)_sub, seen in Fig. 5(c). The independence of mode conversion on either the shape of the constituent element or the metals involved further demonstrates the ubiquity of plasmonic quasi-BICs.

2.5. Strong coupling between plasmonic quasi-BICs and QEs

While a mechanistic understanding has been gained on the creation and ubiquity of quasi-BICs by harnessing the unique spectral features of Rayleigh anomalies on periodic metallic nanostructures, it remains unknown whether the mechanism for creation of plasmonic quasi-BICs can be extended to convert plexcitonic states into plexcitonic quasi-BICs. As the plexcitonic counterparts of plasmonic quasi-BICs, we envision that plexcitonic quasi-BICs shall display considerably narrowed spectral line shape and sharpened peak for at least one of the two plexcitonic branches. Efforts to create the envisioned plexcitonic quasi-BICs has been stimulated ever since the conceptualization of quantum plexcitonic biosensing paradigms by leveraging the distinct double-peak shift of a pair of plexcitonic modes.³⁷

Prior to studying the possibility of producing plexcitonic quasi-BICs based on the two coupled resonance model, we first propose a facile method to create plexcitonic states. Given the two-dimensional nature of Ag nanopyramid arrays, a layer of quantum emitters (QEs) can be readily coated on the surface with a thickness t, as shown in Fig. 6(a). Through near-field interaction, strong coupling between plasmonic resonances supported on Ag nanopyramid arrays and QEs can be achieved with the formation of plexcitons. To numerically understand the optical properties of the plexictonic states, we first proceed to model QEs as a Lorentz oscillator, which is given by⁵⁷

$$\epsilon \left(\Omega \right)={\epsilon }_{\infty }+\frac{f_0{\Omega }_0^2}{{\Omega }_0^2-{\Omega }^2-i{\Gamma }_0\Omega }$$ (4)

where the resonance frequency is Ω₀, the high-frequency permittivity, Lorentz permittivity, and damping rate are modelled as ε_∞ = 1.45, f₀ = 50 meV, and Γ₀ = 50 meV.

Figure 6. Strong coupling between plasmonic quasi-BICs and quantum emitters (QEs) under polarization $\stackrel{\rightharpoonup }{{\mathit{e}}_1}$ (defined in Fig. 2).

(a) Scheme for the studied Ag nanopyramid arrays (a₀ = 1000 nm, r = 100 nm) coated with a layer of QEs with a thickness t and transition energy Ω₀. (b) The absorption spectrum (inverted) for QEs with a transition energy Ω₀ = 725 nm in spectral overlap with the reflection spectrum for the plasmonic quasi-BICs with ω_b = 725 nm. (c) Observation of anti-crossing on the reflection spectra by tuning the QEs transition energy Ω₀ with the plasmonic quasi-BICs fixed as ω_b = 725 nm. A pair or plexcitonic states, i.e. upper polariton (UP) and lower polariton (LP), are clearly seen and marked using hollow circles. (d) Observation of spectral splitting on the reflection spectra by tuning the QEs thickness t with the resonance frequency for QEs and plasmonic quasi-BICs fixed as ω_b = Ω₀ = 725 nm. Marked by the double arrow is the Rabi frequency Ω_R that quantifies the plasmon-QEs coupling strength λ with the relation Ω_R = 2λ. (e) Select reflection spectra from (d) for t = 0, 2, 20, 40 nm, suggesting that the spectral splitting starts at t = 2 nm. (f) Plot of the calculated Rabi frequency Ω_R for QEs with different thicknesses t.

By setting the QEs to be in resonance with the plasmonic quasi-BICs, i.e. Ω₀ = ω_b = 725 nm, the absorption spectrum (inverted) for QEs is spectrally overlapped with the reflection spectrum for Ag nanopyramid arrays, seen in Fig. 6(b). Through tuning of the QEs transition frequency with the thickness fixed as t = 20 nm, a pair of anti-crossing plexcitonic states that are featured as two branches on both sides of the QEs transition line are clearly seen in the reflection spectra in Fig. 6(c). By tuning the QEs thickness at the resonance condition, it is found that the plexcitonic pair can be created with a mere thickness of t = 2 nm, seen in Fig. 6(d–e). The Rabi frequency Ω_R, which quantifies the amount of energy splitting between the plexcitonic pair, increases gradually with the QEs thickness and reach about 180 meV for the QEs with a thickness of 50 nm, seen in Fig. 6(f). Given that the coupling strength $\lambda =\frac{1}{2}{\Omega }_R\gg {\Gamma }_0, {\gamma }_b$ with γ_b estimated to be ~20 meV and Γ₀ = 50 meV, the coupled plasmonic quasi-BICs and QEs are well into the strong coupling regime.

It is important to note that, in addition to the Rabi doublet that is manifested as a pair of plexcitonic states, also observed is a third branch that originates from the QEs excitonic transition. Unambiguous demonstration of the triple-branch polaritonic states in the coupled plasmon-QEs system underscores the “trapped” surface wave nature of the plasmonic quasi-BICs that are tightly confined at the interface between the dielectric substrate and the periodic nanostructures. Therefore, the quasi-BICs enable considerable Rabi splitting even with a layer of QEs of merely 2 nm in thickness. Absence of the QEs excitonic transition peak in Fig. 6(e) (t = 2 nm) implies that all the excitons are involved in the strong coupling with the plasmonic modes and thus get completely converted into the plexcitons. However, further increasing the QEs thickness beyond 20 nm does not significantly improve Rabi splitting, as shown in Fig. 6(f), as only the excitons closer to the interface can effectively contribute to the strong coupling. Similar observations in Fig. S2 are also confirmed by the results obtained under an orthogonal polarization $\stackrel{\rightharpoonup }{{\mathit{e}}_2}$ as defined in Fig. 2.

Also noted are the broadened spectral linewidths for the plexcitonic pairs with a full width at half maximum (FWHM) of about 70 meV in Fig. 6(e) as compared to the plasmonic (FWHM ~ 20 meV) and excitonic (FWHM ~ 50 meV) bare modes in Fig. 6(b). Such unfavorable linewidth broadening could be potentially reversed by the very creation of plexcitonic quasi-BICs.

2.6. Creation and robustness of plexcitonic quasi-BICs

Owing to Rabi splitting, the plexcitonic pair is always spectrally offset from the Rayleigh anomaly line (1,1)_sub when the plasmonic quasi-BICs is in resonance with the QEs excitonic transition. To facilitate the interference between plexcitonic states and the Rayleigh anomaly, it is necessary to create the plexcitonic pair at off-resonance condition so that the plexcitonic states can be tuned to be in resonance with the Rayleigh anomaly line (1,1)_sub. The off-resonance condition can be established by varying the periodicity of Ag nanopyramid arrays with the Rayleigh anomaly line (1,1)_sub shifted accordingly based on Eq. (1). Other parameters remain unchanged, as schematically shown in Fig. 7(a).

Figure 7. Creation and robustness of plexcitonic quasi-BICs under polarization $\stackrel{\rightharpoonup }{{\mathit{e}}_1}$ (defined in Fig. 2).

(a) Scheme for the studied Ag nanopyramid arrays (r=100 nm) by tuning the periodicity a₀. The coated QEs layer has a thickness t=20 nm with a transition frequency ω₀=725 nm. (b) Observation of spectral narrowing at the intersection between plexcitonic branchs (UP and LP) and the Rayleigh anomaly line (1,1)_sub on the reflection spectra, suggesting the formation of plexcitonic quasi-BICs, which are also seen on (c) select reflection spectra for a₀=960, 1000, and 1070 nm from (b). To test the robustness of the plexcitonic quasi-BICs, the QEs transition frequency is fixed as ω₀=725 nm while performing height tuning (d-f) and base radius tuning (g-i). The formed plexcitonic quasi-BICs at both branches display strong robustness against height and base radius tuning.

The obtained dispersion diagram in the form of reflection spectra is shown in Fig. 7(b), where the Rayleigh anomaly line (1,1)_sub is traced by hollow circles. It is notable that at the intersection between the Rayleigh anomaly line (1,1)_sub and either plexcitonic branch, the spectral line shape becomes narrower and sharper, which can also be clearly seen in Fig. 7(c) for representative reflection spectra obtained with the structural periodicity at a₀ = 960, 1000, 1070 nm. Such phenomena of spectral linewidth narrowing and peak sharpening are the distinct characters of quasi-BICs, as has been demonstrated in the study of the plasmonic quasi-BICs, thereby, demonstrating the creation of plexcitonic quasi-BICs for both branches. Observations obtained by tuning the pyramidal height and base radius in Fig. 7(d–i), along with similar observations in Fig. S3 under an orthogonal polarization $\stackrel{\rightharpoonup }{{\mathit{e}}_2}$, collectively prove the robustness of the formed plexcitonic quasi-BICs on both branches against parameter variation (including polarization variation), which highlights the topological nature of quasi-BICs. Similarity in creating plasmonic and plexcitonic quasi-BICs also underscores the generality of creating quasi-BICs from two coupled resonances.

3. Conclusion

In this study, we have developed a general model based on the two-coupled resonances to conceptualize the creation of quasi-BICs on lossy metallic systems, which was demonstrated through rigorous numerical simulations. By modelling Rayleigh anomaly as a bare optical mode with vanishing energy loss in resonance with the second non-specifically defined bare mode, we have obtained two new hybrid states that are degenerate in frequency but differ in the energy loss rate. Spectrally, they display a narrow line shape and a sharp peak, which is characteristic of quasi-BICs. With Ag nanopyramid arrays as a model system, the quasi-BICs have been unambiguously observed at the intersection between plasmon resonances and the Rayleigh anomaly line. Owing to the generality of the second bare mode, such quasi-BICs are independent of the constituent elements. Consequentially, not only have other variants of the metallic systems, such as Ag nanotriangle arrays and Au nanopyramid arrays, been demonstrated to host plasmonic quasi-BICs, we also show that plexcitons can be successfully converted into plexcitonic quasi-BICs through the interaction with the Rayleigh anomaly. Both plasmonic and plexcitonic quasi-BICs are also found to possess topological attributes and display strong robustness against parameter variation. Collectively, the presented findings highlight the huge potential of quasi-BICs in the fields of optoelectronics, nanophotonics, and quantum optics.

Additionally, we note that the surface-diffractive wave nature of Rayleigh anomaly requires long-range ordered periodic nanostructures. It remains an open question whether similar quasi-BICs could be achieved on short-range ordered nanostructures or if Rayleigh anomaly could be replaced with other possible optical phenomena which can be modelled as an optical mode with vanishing energy loss. For the creation of plexcitonic quasi-BICs, the QEs layer needs to be made as thin as 20~50 nm, as a thicker layer would not further significantly improve the coupling strength but could compromise the optical measurements. Given the unique layered structure of two-dimensional transition metal dichalcogenides and other materials such as graphene, they provide an attractive testbed for future experimental studies of the coupled plasmon-QEs system.