1. Introduction

The plasmonic absorber has been researched owing to its potential applications in thermal emitters, solar cells, and photodetectors, and sensing [1–3]. At present, many absorbers based on precious metals have been proposed. In 2021, Lout et al. achieved multiple-band terahertz perfect light absorbers using multiple metallic bars [4]. In the same year, terahertz absorber based on holes array perforated into a metallic slab was proposed [5]. Unfortunately, the absorbers based on precious metals exist many drawbacks, such as static regulation and high ohmic loss. Graphene, due to its dynamic tunability, low loss and strong locality, has attracted widespread attention [6–8]. 2021 witnesses that Feng et al. proposed a broadband absorber in the terahertz region through patterned graphene [9]. Cai et al. realized a tunable multi-band absorber consisting of stacked and orthogonal elliptical graphene layers in the terahertz region [10]. Regrettably, monolayer graphene absorbs only 2.3% of light in the visible and near-infrared regions, which results in weak light-matter interaction. To enhance the interaction between light and matter, many mechanisms have been proposed, such as surface plasmon resonance [11], guided mode resonance (GMR) [12], and Fano resonance [13]. However, as a new way to enhance the interaction between light and matter, the bound states in the continuum (BIC) in absorbers is rarely mentioned.

In 1929, Von Neumann and Wigner first proposed the concept of BICs based on quantum theory, which refers to a particular localized state with eigenfrequencies inside the frequency band of an extended state [14]. In 1985, Friedrich and Wintgen discovered BICs in the model of multi-electron atoms [15]. Since then, BICs have aroused great interest among researchers, and have potential applications in optics [16], acoustics [17], and photonics [18]. In optical BICs, the ultra-high Q (theoretically reaching infinity) can effectively increase the interaction time between light and matter; therefore, they have potential applications in sensors [19], low-threshold lasers [20], low-loss optical fibres [21], and filters [22]. Moreover, different types of optical BICs have been proposed, including the symmetry-protected BICs, the Fabry-Perot BICs, and the Friedrich-Wintgen BICs. In April 2020, Wang et al. employed an asymmetric dielectric nanohole array to excite two symmetry-protected BICs by introducing symmetry breaking [23]. In March 2021, J. F. Algorri et al. realized a symmetry-protected BIC using a novel all-dielectric metamaterial composed of arrayed circular slots etched in a silicon layer [24]. In December 2018, S. I. Azzam et al. achieved a Friedrich-Wintgen BIC in a hybrid plasmonic-photonic structure [25]. In January 2020, a Friedrich-Wintgen BIC was achieved by different polarization states of incident waves in asymmetric terahertz metamaterials [26]. In November 2019, L. L. Doskolovich implemented a Fabry-Pérot BIC on the single-mode dielectric slab waveguide surface [27]. Therefore, it is great urgent to introduce BIC in the absorber to enhance the interaction between light and matter.

In this study, a grating-graphene-Bragg mirror lossy system, which can generate a dual-band perfect absorption contributed by guided-mode resonance (GMR) and Tamm plasmon polaritions (TPPs) modes, is proposed to form the Fabry-Pérot BICs. Research shows that the GMR mode and the Bragg reflection mirror can act as two identical resonators. Through the exquisite design, the phase difference between the two resonators meets the transverse resonance principle (TRP) to bound the light, forming the Fabry-Pérot BICs. The band structures and field distributions of the system are shown to further explain the mechanism of BIC. Besides, both the static (grating pitch P) and dynamic parameters (incident angle θ) can be modulated to generate the Fabry-Pérot BICs. We also explained the reason why the strong coupling between the GMR and TPPs modes does not produce the Friedrich-Wintgen BIC. Meanwhile, the plasmon-induced absorption (PIA) produced by strong coupling is calculated by the coupled mode theory (CMT). Interestingly, the optical switch can be achieved by switching the TE-TM wave. Therefore, the proposed structure can not only be applied to dual-band perfect absorbers and optical switches, but also guides the realization of Fabry-Pérot BICs in lossy systems.

2. Structural design and analysis

The 3D schematic diagram of a grating-graphene-Bragg mirror lossy system is shown in Fig. 1(a). A single-layer unpatterned graphene is sandwiched between the SiO₂ gratings and SiO₂ spacer which is placed on a thin silver slab coupled to a Bragg mirror. The mirror is composed of a silver substrate and 1D photonic crystal (PhC) with N = 6 pairs of alternately stacked SiO₂ and TiO₂ layers. Here, the refractive indices of SiO₂ and TiO₂ are 2.13 and 1.50, respectively. The thickness of graphene is 0.34 nm, and its surface conductivity is mainly composed of inter-band contribution σ_inter and intra-band contribution σ_intra, which can be expressed as [28,29]

 

Fig. 1. (a) 3D schematic diagram of a grating-graphene-Bragg mirror lossy system. (b) 2D side view of a grating-graphene-Bragg mirror lossy system. The parameters are as follows: d = 400 nm, w = 250 nm, P = 1160 nm, h = 1000 nm, l = 38 nm, a = 188 nm, b =276 nm. (c) Schematic diagram of transverse resonance principle. When the phase shift of wave oscillating between the resonator with GMR and PRM is an integer multiple of π, the electromagnetic waves are localized to form BICs inside the structure.

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Here, G(ξ) = sinh(ξ) / [cosh(E_f / k_BT) + coshξ], where ξ = ε / k_BT. ω is the angular frequency of incident light, k_B is the Boltzmann constant, ħ is the reduced Planck constant, τ = μ₀E_f / (ev_F²) [30–35] is the electron relaxation time, and σ_g is the graphene surface conductivity. In this work, the room temperature is T = 300 K, the Fermi level E_f of single-layer graphene is 0.1 eV, and the carrier mobility μ = 1.0 m²/(Vs). The complex permittivity of silver can be expressed as the Drude model ε(ω) = ε_∞ − ω²_p / (ω² + iωγ_p), where ε_∞ = 3.7 is the metal permittivity at infinite frequency, ω_p= 9.1 eV is the plasma frequency, and γ_p = 0.018 eV is the damping coefficient.

For a more direct illustration, a 2D side view of the proposed structure is shown in Fig. 1(b), and its specific structural parameters are illustrated in the caption in Fig. 1. Through reasonable design and parameter optimization, SiO₂ spacer guided-mode can be excited by grating to form a GMR mode [36]. As a lossy material, a single-layer graphene is introduced to interact with GMR mode to enhance the absorption of the system in the visible and near-infrared regions [37]. Moreover, a thin silver slab coupled to a Bragg mirror acts as a perfect reflection mirror (PRM) to block the transmission channel while exciting the TPPs mode [38], forming a two-channel system. In short, the grating-graphene-Bragg mirror lossy system can be composed of a resonator with GMR mode and PRM, as shown in Fig. 1 (c).

It is well known that two identical resonators can be total reflection mirrors for each other, confining the light near the resonance frequency. By modulating the distance between the two resonators or frequency of the wave, the phase shift of wave oscillating between the two resonators is changed to an integer multiple of π (transverse resonance principle, abbreviated as TRP), realizing the BIC [39]. Since a Fabry-Pérot cavity is constructed by the two resonators, the BIC is also called the Fabry-Pérot BIC. Of course, TRP is also applicable to the situation that a single resonator is close to the perfect reflection boundary, such as a lattice termination [40] and PhC with a band-gap [41]. Similarly, the system consisting of a resonator with GMR and PRM is also applicable, as shown in Fig. 1(c). It is worth mentioning that the substrate silver layer can also be used as a PRM to construct a Fabry-Pérot BIC. However, a thin silver slab coupled to a Bragg reflection mirror as a PRM can excite the TPPs mode to add an absorption channel, optimizing the absorption function of the system.

3. Simulation results and discussion

When the TE wave illuminates vertically the grating-graphene-Bragg mirror lossy system, a GMR mode with a 98.8% absorption rate is generated at a wavelength of 1594.6 nm due to the phase-matching between the incident light and SiO₂ spacer guided-mode, as shown in Fig. 2(a). Generally, the GMR mode is mainly localized in the dielectric spacer, and is consistent with the magnetic field distribution in Fig. 2(g). Meanwhile, a TPPs mode with a 99.9% absorption rate is excited at a wavelength of 1672.7 nm owing to the design of a thin silver slab coupled to a Bragg mirror [38]. In Fig. 2(h), the TPPs mode is localized between the silver slab and Bragg mirror, and its magnetic field distribution tends to attenuate from the silver slab to the mirror. To explore the role of graphene, the absorption spectrum and magnetic field distribution of the hybrid system without graphene are shown in Fig. 2(b) and (e). As we all know, single-layer graphene does not support plasmon resonance in the near-infrared region, and its light absorption rate is only 2.3%. When graphene does not exist in the hybrid system, the absorption rate of GMR mode is only 21.1%, and its magnetic field has a stronger ability to confine energy owing to the absence of loss material. Thus, the single layer graphene interacts with excited GMR to improve the absorption efficiency of the hybrid system. In Fig. 2(c), when the TM wave illuminates vertically the hybrid system, the GMR mode is not excited due to the phase mismatch. Thus, the optical switch can be realized by switching TE and TM waves, as shown in Fig. 2(d).

 

Fig. 2. Absorption spectra of the grating-graphene-Bragg mirror lossy system in the (a) TE and (c) TM wave. (b) Absorption spectra of the system without graphene in the TE wave. (d) The GMR mode can form an optical switch by switching TE and TM waves. (e-h) In the case of TE waves, the normalized magnetic field distributions of the GMR and TPPs modes at the whole system and the system without graphene.

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To investigate the mechanism of Fabry-Pérot BICs formation, the absorption spectra of different grating pitches P are plotted in Fig. 3(a). The phase-matching principle of the GMR mode can be represented by

 

Fig. 3. (a) The appearance and disappearance of the Fabry-Pérot BIC based on GMR at different grating pitches P. (b) Variation of the quality factor of GMR mode at different grating pitch P.

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Certainly, the formation of BICs can also be described by the bright-dark mode mechanism. The GMR mode that can be regarded as a bright mode is excited by the incident light, creating a Lorenz feature in the absorption. In comparison, a perfect bound state that is decoupled from far-field radiation has no the Lorenz feature. In the absorption spectra, we do indeed observe the Lorentz feature of bright mode disappears at P = 1310 nm, forming a BIC that can be regarded as a dark mode. Q factor vs grating pitch P for the Fabry-Pérot BIC is plotted in Fig. 3(b). Generally, the Q factor is mainly determined by material loss and incident light radiation loss. At the quasi-BIC (P = 1340 nm), since the incident light radiation loss is completely suppressed, the Q factor can be as high as 602. Compared with previous work [42–44], the Q factor of spectra response is the highest.

Literature [25] proposed a Friedrich-Wintgen BIC in hybrid plasmonic-photonic systems, which is the strong coupling between the photonic waveguide modes and the gap plasmons in the grating. The Friedrich-Wintgen BICs originates from the interference between two different essential resonance states. When a certain parameter becomes a fixed value, one of the modes completely disappears, forming a BIC with infinite Q. Similarly, in this work, the GMR and TPPs modes are excited by different mechanisms, and GMR mode completely disappears at P = 1310 nm. However, the strong coupling of the two modes, generating a plasmon-induced absorption (PIA), does not produce a Friedrich-Wintgen BIC, as shown in Fig. 4(a). Although the two modes are excited by different mechanisms, they are both surface plasmons in essence. It can also be seen from the electric field distributions that PIA is produced by the interaction between two bright modes. The electric field of the left absorption peak is mainly localized in the 1D PhC, which can be regarded as a TPPs-like mode. The electric field of the right absorption peak is chiefly bound in the grating-graphene-dielectric, which can be regarded as a GMR-like mode. The strong coupling of the two modes greatly increases the interaction between light and matter.

 

Fig. 4. (a) The destructive interference between the GMR mode and TPPs mode forms plasmon-induced absorption. The electric field distribution of the two modes also illustrates the strong coupling effect. (b) The coupled mode schematic diagram between the GMR mode (A) and TPPs mode (B).

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The coupling between the two modes can be explained by coupled mode theory [45], as shown in Fig. 4(b). The A and B are the GMR and TPPs modes, respectively, and their amplitudes, inter-loss, and extra-loss coefficient are a, b, γ_i1, γ_i2, γ_o1, and γ_o2, respectively. The total quality factor of the two modes can be obtained by Q₁₍₂₎ = λ₁₍₂₎ / Δλ₁₍₂₎, where λ₁₍₂₎ and Δλ₁₍₂₎ are the resonant wavelength and full width at half height of two modes, respectively. The total loss of the two modes are expressed as γ_A(B) = ω / 2Q₁₍₂₎. The coupling coefficients between them are μ₁₂ and μ₂₁, respectively. The superscript “+” and “–” indicate the states of electromagnetic waves propagating in the positive or negative direction, and the subscript “in” and “out” indicate the state of incidence or exit from the resonator. When light is incident perpendicularly, the coupling between the two modes can be expressed as

Here, γ₁₍₂₎= (iω – iω₁₍₂₎ − γ_i1(2) – γ_o1(2)), where γ_o1(2) = ω₁₍₂₎ / 2Q_o1(2), with ω₁₍₂₎ and Q_o1(2) being the resonant angular frequency and external-loss quality factor of the two modes, respectively. The internal-loss coefficient can be expressed as γ_i1(2)= γ_{A(B) −} γ_o1(2). The interaction between the GMR and TPPs modes satisfies the energy conservation,

The electromagnetic wave exits from the TPPs mode and then returns from the substrate silver (the phase difference of the electromagnetic wave propagating back and forth is 2ϕ), which can be expressed as

Thus,

From Eq. (10)–(11),

Finally, the reflection coefficient r of the system can be expressed as

Since the substrate silver blocks the transmission channel, the absorption rate of the proposed system can be expressed as A = 1 –r². In Fig. 4(a), the calculated total quality factor Q₁₍₂₎ of the two modes are 348.3 and 335.4, respectively. The phase difference φ and ϕ can be approximated as 0 due to the ratio of the distance to the velocity of light. The Fitting parameters Q_o1 and Q_o2 are 442.1 and 335.4. The calculation results are basically the same as the numerical simulation data, as shown in Fig. 4(a).

In practical applications, it is not feasible to realize the BIC by changing the structural parameters (static regulation). Researchers have been looking for effective methods such as temperature control [46], incident light angle control [47], and polarization angle control [26] to achieve dynamic control. Here, the appearance and disappearance of the Fabry-Pérot BIC can be observed in the reflection spectra by controlling the angle of incident light θ, as shown in Fig. 5(a). As the incident light angles increases, the GMR mode red-shifts and gradually disappears, and finally a BIC is formed at θ = 9.5°. When θ continues to increase, the BIC starts to disappear. In Fig. 5(b), the Q trend of the GMR mode at different incident light angles θ is plotted. When θ = 8°, the Q of the quasi-BIC can be as high as 598, which exceeds most plasmonic devices [48–55] based on graphene and metals.

 

Fig. 5. (a) The appearance and disappearance of the Fabry-Pérot BIC based on GMR at different incident light angles θ. (b) Variation of the GMR mode Q factor at different incident light angle θ.

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To explain the BIC phenomenon more clearly, we calculated the band structures of the proposed loss system and the corresponding Q-factors evolution, as shown in Fig. 6. Figure 6(a) shows the band structure when the grating period P = 1310 nm. To explore the location of the BIC, we also plot the Q-factors evolution of the band1. The maximum Q-factor corresponds to the BIC position of 1767nm for the band1, which is in perfect agreement with the BIC position in Fig. 3(a). However, since the structure is lossy, the BIC is an eigenmode with zero radiation loss, but can still have absorption loss. Therefore, to show the BIC, the electric field distribution of the BIC corresponding to kx = 0.01 is shown in Fig. 6(e). We can observe that the radiation loss in free space is 0, which proves that it is indeed a BIC. Similarly, the position of BIC in the band structure can be found when the grating period P = 1160 nm, as shown in Fig. 6(b). The BIC in the band1 does not exactly match the position of the BIC in Fig. 5(a) due to the effect of the angle of incidence. The relationship between the angle of incidence and the wave vector can be expressed by sin θ = 2kx, where x = P / 2π. The BIC appears when the angle of incidence θ = 9.5° in Fig. 5(a), and the solution is kx = 0.082, which is consistent with the position of the BIC in Fig. 6(c). The electric field distribution of the BIC corresponding to kx = 0.082 is shown in Fig. 6(f). The far field of the eigenmode decays rapidly to zero also proves the reality of BIC. It is worth mentioning that the BIC has a finite Q due to absorption loss. For a resonant mode with a complex frequency, the Q can be defined as 1/2 of the ration between the real and imaginary parts of the complex frequency. In summary, both static (grating pitch P) and dynamic parameters (incident angle θ) can be modulated to generate the BICs.

 

Fig. 6. The band structures of the proposed loss system when the grating period (a) P = 1310 nm and (c) P = 1160 nm. Simulated Q-factors evolution for the band1 when (b) P = 1310 nm and (d) P = 1160 nm. Electric field distributions of the BICs when (e) P = 1310 nm and (f) P = 1160 nm.

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4. Conclusions

In conclusion, we have demonstrated the formation of Fabry-Pérot BICs in a grating-graphene-Bragg mirror lossy system, which can produce a dual-band perfect absorption contributed by GMR and TPPs modes. In addition, the optical switch function can also be obtained by switching the TE-TM wave. Research has shown that the GMR mode and Bragg reflection mirror can act as two identical resonators. Through the exquisite design, the phase difference between the two resonators meets the TRP to limit light, forming the Fabry-Pérot BIC. Meanwhile, the band structures and field distributions further explain the mechanism of the BICs. Research has found both static parameter (grating pitch P) and dynamic parameter (incident light angle θ) can be modulated to generate the Fabry-Pérot BICs. Furthermore, we also explained the reason why the strong coupling between GMR and TPPs modes does not produce Friedrich-Wintgen BICs. To conclude, the proposed structure can not only be applied to dual-band perfect absorbers and optical switches, but also provides guidance for the realization of Fabry-Pérot BICs in lossy systems.