Interaction of plasmonic bound states in the continuum

Fengzhao Cao; Mimi Zhou; Chang-Wei Cheng; Haojie Li; Qianwen Jia; Anwen Jiang; Bokun Lyu; Dahe Liu; Dezhuan Han; Shangjr Gwo; Shangjr Gwo; Shangjr Gwo; Jinwei Shi; Jinwei Shi

doi:10.1364/PRJ.480968

1. INTRODUCTION

Surface plasmons can confine light into deep subwavelength scale [1–3], which leads to various applications such as near-field enhancement and the Purcell effect [4,5], high-sensitivity sensors [6–8], lasers [9–11], modulators [12,13], logic gates [14,15], nonlinear optics [16–18], and strong light–matter interaction [3,19,20]. However, the intrinsic dissipative and radiative losses of plasmonic material also limit its application, especially the linewidth of the plasmonic metasurface. Plasmonic lattices support surface lattice resonances (SLRs), which can increase the quality factor ( $Q$ factor) efficiently [21–23]. However, the $Q$ factor of SLRs in the visible range is still limited by the radiation loss.

Another option to realize high- $Q$ resonances in optical regime is based on the bound state in the continuum (BIC). The BIC is considered to be a confined state, which is located in a continuous band and coexists with the extended wave, but without any radiation channels [24]. It can be considered to be an extreme case of SLR without radiation loss. The concept of BICs was first proposed by von Neumann and Wigner in 1921 [25], and it has not been realized in quantum mechanics. Generally, BICs can be classified in two main categories: the symmetry-protected BIC and Friedrich–Wintgen (FW) BIC [24]. The latter was also known as “accidental BIC” in some literatures. The symmetry-protected BIC is usually located at the high symmetry point of the structure with reflection or rotational symmetry, such as Γ point. It is caused by perfect destructive interference of symmetrical modes in the same channel. The FW BIC is usually formed by the destructive interference of two or more different modes [26]. The ideal optical BIC is a mathematical concept, which only exists in completely lossless or infinite structures. Due to structural absorption, finite size, and fabrication error, BICs with infinite $Q$ factor will be transformed into quasi-BIC with a still large $Q$ factor. Over the past decades, structures supporting BICs and their applications have been explored in many fields [27–38]. But most of them work at long wavelengths, such as the microwave and terahertz (THz) regimes, where the ohmic losses of metals are sufficiently small. The ones working in the visible regime are still limited [19,39–44], and many of them are hybrid plasmonic–photonic systems. Especially, there is no experimental report on metal–insulator–metal (MIM) structure, which is most commonly used for plasmonic near-field confinement. Also, the interactions of BICs are not studied.

In this work, we design and fabricate an MIM metasurface and realize one-dimensional plasmonic BICs. This structure has two kinds of pure plasmonic modes, the localized surface plasmon resonance (LSPR) mode and lattice surface plasmon polariton (lattice SPP, LSPP). The LSPR modes with different orders have different parity. The even-order LSPR modes have even parity. It is found that, because of the radiation property of even-order resonance, the symmetry-protected BIC can be realized when light is normally incident on the structure. The band structure can be tuned by both the LSPR and LSPP. Interestingly, the hybrid modes are also BICs. Meanwhile, there is an isolated bright mode located in the center of the $k$ -space. In other words, although BICs cannot interact with far field, we successfully demonstrate BIC splitting through far-field excitation. By tuning the pitch of the MIM grating, the FW BIC is also observed. The best $Q$ factor achieved by simulation is over 106, and that measured experimentally is over 66. Finally, based on these studies, we propose and demonstrate preliminarily an ultrathin bandpass spatial filter. These findings provide a new MIM platform to study plasmonic BICs. Because of the excellent field confinement, plasmonic BICs of the MIM metasurface can further boost the applications of plasmonics in many fields.

2. MODEL

The proposed MIM metasurface is shown in Fig. 1(a). It is composed of a layer of silver, a thin layer of aluminum oxide ( ${Al}_{2} O_{3}$ ) as spacer, and a silver grating. The mode can be tuned by the width of the grating ridge, the pitch of the grating, and the thickness of the spacer. There are two types of LSPR in the MIM structure, even- and odd-order modes. For even-order LSPR, the mode constrained in the spacer forms a special spatial pattern. Figure 1(b) shows the schematic diagrams of the even-order mode and odd-order mode within the dielectric spacer. As will be discussed later, the even mode shown here corresponds to electric dipoles along $z$ ( $E D_{z}$ ) and along $x$ ( $E D_{x}$ ) directions, and the odd mode corresponds to electric toroidal dipole along the $y$ direction ( $E T D_{y}$ ). Under normal incidence, the even mode has even parity and cannot interact with the far field, making it a symmetry-protected BIC. From another point of view, it corresponds to a one-dimensional topologic defect [45]. It is worth mentioning that these $E D_{z}$ and $E D_{x}$ components in this one-dimensional structure correspond to a magnetic toroidal dipole along $z$ direction ( $M T D_{z}$ ) in a two-dimensional structure.

Fig. 1. Metasurface supporting plasmonic BIC modes. (a) Schematic of the structure composed of a lattice of MIM structure. (b) Schematic diagrams of the even mode and odd mode within the dielectric spacer. (c) Even-mode BIC and strong coupling with LSPPs in momentum space.

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Figure 1(c) shows schematically the strong coupling between LSPR and LSPPs in far-field momentum space. The left part of Fig. 1(c) shows that, when the LSPR and LSPPs are highly detuned, an even mode at $Γ$ point of momentum space can be considered as a symmetry-protected BIC. The right part of Fig. 1(c) shows that, when the detuning between the LSPR and LSPPs is small, the new hybrid modes also inherit the BIC feature from the original modes. From the scheme, an isolated spot is located right at the center of the far-field angle-resolved spectrum. In the following, we will discuss these phenomena in detail.

3. RESULTS AND DISCUSSION

We start from the case in which the LSPR and LSPP modes are highly detuned and no obvious strong coupling is shown. The angle-resolved reflection spectra of the metasurface are calculated by the finite-difference time-domain method (FDTD). The incident light is TM polarized [along the $x$ direction in Fig. 1(a)]. The calculated angle-resolved spectrum, i.e., the far-field measured band structure, of the sample with a pitch of 300 nm is shown in Fig. 2(a). The incident angle and the parallel component of the wave vector $k$ are linked by the photon energy $ω$ and the index of SPP $n_{SPP}$ ,

- \begin{matrix} k_{∥} = 2 π \sin (θ) / λ = 2 π \sin (θ) (n_{SPP} ω / c) . \end{matrix}

Fig. 2. Angle-resolved reflection spectra of the MIM metasurface with the pitch of 300 nm. (a) Full-wave simulation of the angle-resolved reflection spectra of the metasurface. The dashed ellipse denotes the even mode BIC at $Γ$ point. (b) Reflectance spectra at different incident angles extracted from (a). (c) Experimental results of the metasurface corresponding to (a). (d) Reflectance spectra at different incident angles extracted from (c). The inset in (b) shows the polarization vortex in momentum space at normal incidence.

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We can see that, because of the small pitch, the band structure around 590 nm measured from the far field is an LSPR-like flatband, which is not much affected by the propagating mode. At $Γ$ point, this LSPR-like mode cannot be excited by the far field. Figure 2(b) is the reflectance spectra at different incident angles, showing clear bandwidth shrinking and mode weakening with the decrease of the incident angle, which is the typical phenomenon of symmetry-protected BICs. We can see that the reflectance is not unitary even at normal incidence because the plasmonic system has significant ohmic loss; therefore, the plasmonic BICs discussed in this paper are all quasi-BICs. The inset in Fig. 2(b) shows the directions of the polarization vector field, revealing a vortex with topological charges of $+ 1$ . This is one of the key evidences of BICs.

We also perform the multipole expansion of the metasurface under different incident angles. The near-field distributions of $E_{x}$ and $E_{z}$ components of a unit cell are shown in Figs. 3(a) and 3(b) (normal incidence, 0°) and Figs. 3(e) and 3(f) (oblique incidence, 20° in $x - z$ plane). Based on them, the multipole expansion spectra are obtained and shown in Fig. 3(c) (0°) and Fig. 3(g) (20°). The contributions from different components of $E D$ and $E T D$ are shown in Fig. 3(d) (0°) and Fig. 3(h) (20°). Obviously, with the decrease of the incident angle, the dominant moment around this even-order LSPR (585 nm) changes from $E D$ to $E T D$ . And the dominant component changes from $E D_{z}$ (accompanied by a small amount of $E D_{x}$ and $E T D_{y}$ ) to $E T D_{y}$ . Note that the value below $10^{- 40}$ is actually simulation noise. At normal incidence, the disappearance of $E D_{z}$ and minimization of $E D_{x}$ correspond to the generation of a symmetry-protected BIC, as pointed out above. The residual $E D_{x}$ at normal incidence has no peak and is caused by material loss and other odd modes, which also clearly explains why the plasmonic BIC is always a quasi-BIC [see Fig. 2(b)]. At normal incidence, the $E T D_{y}$ component is the source of near-perfect reflectance. This component exists at any odd modes and non-resonant region, and it has no contribution to the quasi-BIC.

Fig. 3. Near-field distribution and multipole expansion under different incident angles. (a) and (b) Electric near-field distribution of $E_{x}$ $E_{x}$ and $E_{z}$ components of a unit cell under normal incidence (0°). (c) Multipole expansion spectra under normal incidence. Around resonance (585 nm), the dominant moment is $E T D$ . (d) The contribution from different components of $E D$ and $E T D$ . The dominant component is $E T D_{y}$ . (e) and (f) Electric near-field distribution of $E_{x}$ and $E_{z}$ components of a unit cell under oblique incidence (20° in $x - z$ plane). (g) Multipole expansion spectra under oblique incidence. Around resonance (585 nm), the dominant moment is $E D$ . (h) The contribution from different components of $E D$ and $E T D$ . The dominant component is $E D_{z}$ , accompanied by a small amount of $E D_{x}$ and $E T D_{y}$ . The color bar is fixed for (a), (b) and (e), (f).

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We also fabricate the metasurface. The proposed MIM metasurface preparation requires the following three steps. Step 1: First, a silver epitaxial film with surface roughness less than 5 nm was grown on 5.12-cm-diameter, c-plane sapphire (0001) wafer by a thermal evaporator of silver slug (99.99%). Prior to growth, the sapphire substrate was subjected to a battery of treatments. The sapphire substrate needed to be heat-cleaned at 750°C for 3 h. In order to prevent dewetting of the deposited silver film, the 50-nm-thick titanium (Ti) layer was deposited at a rate of 2 Å/s on the sapphire substrate as an adhesive between the substrate and the silver film. Then, the 300-nm-thick silver film was deposited at a rate of 10 Å/s under a high vacuum condition ( $10^{- 7} T o r r$ ). The silver film was annealed at 550°C for 90 min in the chamber to improve the quality of the film. Step 2: A 20-nm-thick layer of aluminum oxide ( ${Al}_{2} O_{3}$ ) was deposited on the silver film by the atomic layer deposition technique. Step 3: A polymethyl methacrylate (PMMA) layer was spin-coated on top of the ${Al}_{2} O_{3}$ film. Electron beam lithography and electron beam evaporation techniques were used to fabricate the silver gratings with different pitches on the dielectric layer.

The angle-resolved reflection spectra of the metasurface were measured by a homemade angle-resolved spectrometer [19]. A white halogen lamp (HL-2000-CAL, Ocean Optics, Inc.) was used as the light source. The polarization of the incident light was set to be perpendicular to the grating lines by a polarizer. The Fourier plane of reflections was collected with an objective lens with $N.A. = 0.83$ and was then projected on the slit of the spectrometer with a charge-coupled device (CCD). The experimental result of the metasurface with a grating pitch of 300 nm is shown in Figs. 2(c) and 2(d). Obviously, it is consistent with the simulation results.

Under normal incidence, the mode around 585 nm by simulation disappears in the far-field spectrum, which is the typical evidence of a symmetry-protected BIC. However, in the experiment, even under normal incidence, the peak does not completely disappear. This is because the experimental sample has limited size and bigger intrinsic (absorption) and extrinsic loss due to imperfect fabrication, which slightly breaks the symmetry of the even mode, as will be shown in Fig. 4.

To gain insight into the BICs formed in MIM structures, we further calculate the angle-resolved reflection spectra of metasurfaces with different pitches. The results are shown in Fig. 4, where the pitches are 400–700 nm and the width and height of the ridge are fixed at 170 and 50 nm. For pitch 400 nm, the first-order LSPP mode starts to appear. However, the center is still highly detuned, and the mode at $Γ$ point is still an even-mode BIC. When the pitch is increased to 500 nm, strong coupling between the two modes significantly modifies the band structure. Several BIC-like modes are observed. The BIC at $Γ$ point is pushed to the red side, and the band is not flat anymore.

Fig. 4. Simulation (top row) and experimental (bottom row) results of angle-resolved reflectance spectra of the MIM metasurface with the pitch of (a), (e) 400 nm, (b), (f) 500 nm, (c), (g) 600 nm, and (d), (h) 700 nm. The white circles denote the symmetry-protected BIC, the purple circles highlight the FW BIC, and the dashed white ellipse in (g) indicates another set of modes induced by fabrication defects.

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Besides, two more BICs appear at large wave vector (marked by purple ellipses). According to the definition, these are FW BICs caused by destructive interference between the two modes [24]:

\begin{matrix} H = (\begin{matrix} E_{1} & g \\ g & E_{2} \end{matrix}) - i (\begin{matrix} γ_{1} & \sqrt{γ_{1} γ_{2}} \\ \sqrt{γ_{1} γ_{2}} & γ_{2} \end{matrix}), \end{matrix}

where

E_{1}

and

E_{2}

are the resonant frequencies of the two modes,

g

is the coupling strength,

γ_{1}

and

γ_{2}

are the radiation loss rate, and

\sqrt{γ_{1} γ_{2}}

is the radiation coupling term. Note that the intrinsic absorption loss is not included. The corresponding eigenvalues are

\begin{matrix} E_{a, b} = \frac{E_{1} + E_{2}}{2} - i \frac{γ_{1} + γ_{2}}{2} \pm \sqrt{{(\frac{E_{1} - E_{2}}{2} - i \frac{γ_{1} - γ_{2}}{2})}^{2} + {(g - i \sqrt{γ_{1} γ_{2}})}^{2}} . \end{matrix}

Considering the FW condition [26],

\begin{matrix} (E_{1} - E_{2}) \cdot \sqrt{γ_{1} γ_{2}} = g (γ_{1} - γ_{2}) . \end{matrix}

Then

\begin{matrix} E_{a} = \frac{E_{1} + E_{2}}{2} + \frac{g}{\sqrt{γ_{1} γ_{2}}} \cdot \frac{γ_{1} + γ_{2}}{2} - i (γ_{1} + γ_{2}), \end{matrix}

\begin{matrix} E_{b} = \frac{E_{1} + E_{2}}{2} - \frac{g}{\sqrt{γ_{1} γ_{2}}} \cdot \frac{γ_{1} + γ_{2}}{2} . \end{matrix}

Obviously, one of the two hybrid states becomes more lossy, and another state has no radiation loss, corresponding to a BIC. We also performed the multimode expansion of the FW BIC in Fig. 4(d). The results are shown in Fig. 5, where the incident angle of 15° corresponds to the FW BIC at $\sim 725 nm$ . Obviously, near BIC [Figs. 5(b) and 5(e)], the dominant component is also $E T D_{y}$ , which corresponds to the specular reflectance. Both $E D_{x}$ and $E D_{z}$ components reach minimum here, while at the other angles, they reach local maxima. The non-zero minimum stems from the other bright modes, which shows that the plasmonic FW BIC is also a quasi-BIC.

Fig. 5. Multipole expansion of FW BIC in Fig. 4(d) under different incident angles. (a)–(c) Multimode expansion results of FW BIC, where (a) corresponds to the incident angle of 2°, (b) corresponds to the FW BIC at $\sim 725 nm$ , and (c) represents the incident angle of 30°. (d)–(e) Dominant components of $E D$ and $E T D$ at the three incident angles. All the results are shown in logarithmic coordinate, and the other noise components are out of the range.

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Fig. 6. SEM image showing the top view of the fabricated MIM structure with the pitch of 600 nm. The fabrication errors are quite obvious.

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By further increasing the pitch, the phenomena are clearer. For pitch 700 nm, we can see that there are four BICs in the angle-resolved spectrum. Two of them are symmetry-protected BICs (white circles), and the other two are FW BICs (purple ellipses). At $Γ$ point, the symmetry-protected BICs are generated between the near-resonance three-mode coupling [46],

\begin{matrix} H = (\begin{matrix} E_{1} & g_{str} & g_{str} \\ g_{str} & E_{2} - i γ_{2} & g_{bg} + i \sqrt{γ_{2} γ_{3}} \\ g_{str} & g_{bg} + i \sqrt{γ_{2} γ_{3}} & E_{3} - i γ_{3} \end{matrix}), \end{matrix}

where

g_{str}

represents the coupling strength between the even-parity LSPR and the LSPP modes, and

g_{bg}

is the coupling strength between the LSPP modes.

E_{1}

denotes the resonant frequency of the even-parity mode

| ψ_{1} ⟩

at

Γ

point, and

E_{2, 3}

represent the resonant frequencies of the two LSPP modes

| ψ_{2, 3} ⟩

, which always have the same resonant energy at

Γ

point,

E_{2} = E_{3}

and

γ_{2} = γ_{3}

. The radiation loss of the even-parity mode is

γ_{1} = 0

. It should be noticed that the imaginary part of the coupling strength between two LSPP modes is

+ i \sqrt{γ_{2} γ_{3}}

, rather than

- i \sqrt{γ_{2} γ_{3}}

, which was frequently used in the literature [24]. According to Ref. [46], for two-mode coupling, both of them are correct. However, when the third mode is considered and

g_{str}

is defined, then only one of them is correct. If the LSPR mode

E_{1}

is a bright mode, then

- i \sqrt{γ_{2} γ_{3}}

should be used; however, here

E_{1}

is a BIC mode, and then

+ i \sqrt{γ_{2} γ_{3}}

should be taken, which means that the two LSPP modes have opposite phase when coupling to the same channel.

The new eigenvalues are

\begin{matrix} E_{a} = E_{2} - g_{bg} - 2 i γ_{2}, \end{matrix}

E_{b} = \frac{E_{1} + E_{2} + g_{bg} - \sqrt{{(E_{2} - E_{1} + g_{bg})}^{2} + 8 {(g_{str})}^{2}}}{2},

E_{c} = \frac{E_{1} + E_{2} + g_{bg} + \sqrt{{(E_{2} - E_{1} + g_{bg})}^{2} + 8 {(g_{str})}^{2}}}{2} .

The corresponding eigenstates are

\begin{matrix} | ψ_{a} ⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 0 \\ - 1 \\ 1 \end{matrix}), \end{matrix}

\begin{matrix} | ψ_{b} ⟩ \propto (\begin{matrix} \frac{E_{1} - E_{2} - g_{bg} - \sqrt{{(E_{2} - E_{1} + g_{bg})}^{2} + 8 {(g_{str})}^{2}}}{2 g} \\ 1 \\ 1 \end{matrix}), \end{matrix}

\begin{matrix} | ψ_{c} ⟩ \propto (\begin{matrix} \frac{E_{1} - E_{2} - g_{bg} + \sqrt{{(E_{2} - E_{1} + g_{bg})}^{2} + 8 {(g_{str})}^{2}}}{2 g} \\ 1 \\ 1 \end{matrix}) . \end{matrix}

When the LSPR is also on resonance with LSPP modes (triple resonant), $E_{1} = E_{2} + g_{bg} = E_{3} + g_{bg}$ , then

\begin{matrix} E_{b} = E_{1} - \sqrt{2} g_{str}, \end{matrix}

\begin{matrix} E_{c} = E_{1} + \sqrt{2} g_{str}, \end{matrix}

\begin{matrix} | ψ_{b} ⟩ = \frac{1}{2} (\begin{matrix} - \sqrt{2} \\ 1 \\ 1 \end{matrix}), \end{matrix}

\begin{matrix} | ψ_{c} ⟩ = \frac{1}{2} (\begin{matrix} \sqrt{2} \\ 1 \\ 1 \end{matrix}) . \end{matrix}

The Rabi splitting on resonance is similar to that of the Jaynes–Cummings mode [20]. However, since the even mode $| ψ_{1} ⟩$ is a BIC at $Γ$ point, $γ_{1} = 0$ , obviously $| ψ_{b} ⟩$ and $| ψ_{c} ⟩$ are both BICs. This is also evident in Eqs. (8)–(10), where $E_{b}$ and $E_{c}$ have no radiation loss. Therefore, in spite of the fact that a BIC mode cannot interact with the far field, we have demonstrated the strong coupling and mode splitting of BIC by the far-field spectrum. To our knowledge, this is the first time that the splitting of a BIC is reported.

Interestingly, there is always a spot at the center of the far-field spectrum. It can be understood by Eq. (7) and by the parity selection rule. Based on parity conservation, only modes with the same parity can have strong coupling with each other. From Eq. (7), $| ψ_{a} ⟩$ is the odd-parity bright state left at the center of the far-field angle-resolved spectrum, which cannot interact with the even-parity mode, and is more lossy than the original modes. Therefore, we also realize a super lossy isolated mode in the normal direction.

The experimental results of the metasurface with different pitches are shown in Figs. 4(e)–4(h). The fabrication and measurement process are the same as the above. We can see that the experimental results basically reproduce all the content in the simulation results, which proves that both the symmetry-protected BIC and FW BIC can be realized in this MIM metasurface.

There is another set of diffraction modes in the experimental results, as marked by the dashed circle in Fig. 4(g). These modes do not seem to interact with even-parity mode, and they disappear completely in the simulation results. They also block the up-branch BIC in Fig. 4(g). By careful inspection, we notice that the fabricated metasurface has a rough edge (10–40 nm error), as can be seen in Fig. 6. These edge defects support unexpected LSPR, which can act as the scattering source of the second set of diffraction modes. They are also one of the possible reasons why the experimental results are not as good as simulation (Fig. 2). In the future, they can be eliminated by improving the fabrication process.

The $Q$ factors [47] of the corresponding even-parity BIC and FW BIC in the metasurface with pitch 700 nm are shown in Fig. 7. Figure 7(a) shows the $Q$ factor of the mode at 832 nm and different angles. As shown in the figure, the $Q$ factor increases significantly with the decrease of the angle, and the maximum $Q$ factor measured in the simulation is $\sim 106$ . The $Q$ factor obtained in the experiment is about 66 [Fig. 7(b)]. Figures 7(c) and 7(d) show the $Q$ factors of the FW BIC extracted from simulation and experimental results. The $Q$ factor of FW BIC obtained in the simulation is about 26, while that obtained in the experiment is about 14. It should be noticed that the $Q$ factor value measured by reflectance is limited by the background reflection and is not the real maximal value of the metasurface.

Fig. 7. $Q$ factors of even-mode BIC and FW BIC extracted from simulation and experimental results of pitch 700 nm. (a) and (b) are simulation and experimental results of even-mode BIC. (c) and (d) are simulation and experimental results of FW BIC.

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The application of the BIC as a high $Q$ cavity, such as a laser and sensor, has been studied before. Here we propose another potential application, the optical spatial filter. As can be seen from Figs. 2 and 4, the MIM metasurface shows very rich far-field band structure, which could be used as all kinds of narrow frequency band spatial filters. Figure 2 is actually a low-pass filter, which can be used for light beam collimation. In Fig. 8, we demonstrate a bandpass/band-stop filter by both simulation and experiment. The pitch is 1000 nm. The working wavelength is 691 nm. We can see that, because of the strong modal resonance and the Ohmic loss, the metasurface can absorb incident light with specific wavelength and spatial frequency. The contrast in Figs. 8(a)–8(d) can be further improved by adding absorbing material into the spacer of the MIM metasurface.

Fig. 8. Optical spatial frequency filter. The pitch of the metasurface is 1000 nm. (a) Far-field spatial frequency spectrum of different wavelengths obtained by simulation. (b) Cut of (a) at 699 nm. (c) Far-field spatial frequency spectrum of different wavelengths obtained by experiment. (d) Cut of (c) at 691 nm.

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4. CONCLUSION

In conclusion, we realize a one-dimensional plasmonic BIC by designing an MIM metasurface. The symmetry-protected BIC is observed both in simulation and experiment when light is normally incident on the structure. By tuning the even-order LSPR and LSPP modes, the splitting of symmetry-protected BIC is achieved by far-field excitation, which is counterintuitive because usually the BIC does not interact with the far field. The super-radiative mode and FW BIC mode are also observed in the momentum space. The best $Q$ factor obtained by simulation is over 106, and that measured in experiment is over 66. Finally, we propose and demonstrate preliminarily an ultrathin bandpass spatial filter. These findings provide a new platform to study optical TD BICs and can have applications in many fields such as nano lasers, ultrasensitive sensors, filters, nonlinearity enhancement, and quantum optics.

Funding

National Natural Science Foundation of China (91950108, 12174031, 12074049, 11774035); National Science and Technology Council (111-2123-M-007-002).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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Interaction of plasmonic bound states in the continuum

Abstract

1. INTRODUCTION

2. MODEL

3. RESULTS AND DISCUSSION

4. CONCLUSION

Funding

Disclosures

Data Availability

REFERENCES

Data Availability

Cited By

Figures (8)

Equations (17)

Photonics Research