Flat-band Friedrich-Wintgen bound states in the continuum based on borophene metamaterials

Yan-Xi Zhang; Qi Lin; Qi Lin; Xiao-Qiang Yan; Ling-Ling Wang; Gui-Dong Liu; Gui-Dong Liu

doi:10.1364/OE.515152

1. Introduction

For many different applications, such as solar thermophotovoltaic devices [1], optical modulators [2], and optical sensors [3], the ability of materials to absorb electromagnetic radiation is critical. The absorption of the system is generally positively correlated with the near-field enhancement, which can reach a maximum of 0.5 for a single-port input when the critical coupling condition is met, if there are not degenerate modes [4,5]. By optimizing the structural parameters, one can achieve a balance between radiation and non-radiation loss rates, resulting in critical coupling. Under the critical coupling condition, Salusbury design [6,7] and Dellenbach type [8] achieve perfect absorption by adding a certain thickness of the traditional noble metal plate to block the transmission of the system. It suffers from some drawbacks, such as cannot exhibit coherent control of absorption through beam modulation.

The other method is coherent perfect absorption (CPA), which is the interferometric all-optical control of absorption under the illumination of two counter-propagating input beams by controlling their intensities and relative phases [9,10]. The ability to flexibly control the absorption of light makes CPA for a very wide range of applications. For example, it provides a way towards the realization of linear optical switches, modulators and logical gates [11–13]. CPA has also been proposed for pulse restoration and coherence filtering of optical signals [14]. Studies have shown that by using materials with tunable conductivity, such as graphene or black phosphorus, as the absorbing medium based on CPA, a deep subwavelength plasmonic absorber with tunable operating frequency and absorption rate can be achieved [15,16]. Unfortunately, the plasmonic frequencies of these 2D materials are limited to the terahertz or infrared regions due to their low carrier densities, making it difficult to support plasmon resonances in commercially important communication bands. Recently, monolayer borophene was successfully synthesized on metallic surfaces by molecular beam epitaxy [17,18]. Borophene, an emerging 2D material, has unique optical properties and an electronic structure that gives it excellent potential for optical absorption. Due to the higher carrier density (∼10¹⁹m⁻²), borophene is capable of supporting surface plasmon polaritons (SPPs) in the near-infrared region [19,20], making it a promising candidate for optical communication band applications.

Controlling and matching the radiative loss channel to the inherent loss of the system is essential to achieve CPA. Bound states in the continuum (BICs) have gained considerable attention due to their inability to couple with the far-field, resulting in infinite radiative Q-factors and intensified electromagnetic near fields [21,22]. In practice, small parameter adjustments (geometric or excitation conditions) may disturb the BIC, resulting in the emergence of quasi-BICs that are reachable from the far field [23,24]. Meanwhile, the radiative Q-factors become finite and proportional to the magnitude of the disturbance. The introduction of loss materials into this system can be very convenient to achieve critical coupling under one-port source excitation, leading to coherent perfect absorption under double coherent light source excitation [25]. For example, the research groups of Shuyuan Xiao and Feng Wu recently presented a general method to successfully for tailoring the absorption bandwidth of graphene via critical coupling in the near-infrared region [26]. Hongjian Li and Zhimin Liu et al. investigated optical BIC in a grating-graphene-Bragg mirror structure and achieved a dual-band perfect absorber [27]. To control light in an efficient and coherent way, we proposed gap-perturbed dimerized gratings based on bulk WS₂ for flexible control of self-hybridization of a quasi-BIC and excitons, and achieved polaritonic coherent perfect absorption under strong critical coupling condition [28]. However, in these studies, most of the electromagnetic energy is loosely localized in the all-dielectric resonators, i.e., its mode volume is large, which is not conducive to device integration. Guangtao Cao et al. proposed a new strategy based on Friedrich-Wintgen bound states in the continuum (F-W BICs) to realize a tunable perfect absorber with a large dynamic modulation range by employing graphene metamaterials [29]. In the vicinity of the communication band, Andreas Tittl et al. proposed a novel approach for plasmonic perfect absorbers by combining mirror-coupled resonances with the unique loss engineering capability of plasmonic quasi-BICs. Based on the above studies, it is theoretically possible to realize a borophene-supported plasmonic BIC to satisfy the strong critical coupling condition and achieve CPA, and then eventually obtain a deep subwavelength plasmonic absorber with tunable frequency and absorption strength near commercially important communication bands.

In this work, we propose F-W BIC based on borophene metamaterials to realize coherent perfect absorption with a dual-band absorption peak in commercially important communication bands. The metamaterials consist of borophene gratings and a borophene sheet which can simultaneously support a Fabry-Perot plasmon resonance and a guided plasmon mode. It is found that the formation and dynamic modulation of the F-W BIC can be achieved by adjusting the width or carrier density of the borophene grating, while the strong coupling leads to the anti-crossover behavior of the absorption spectrum. In the case of oblique incidence, due to the weak angular dispersion originating from the intrinsic flat-band characteristic of the deep sub-wavelength periodic structure, we found that the system has wide angle (within 70 degrees) and high absorption. In addition, we employ the temporal coupled-mode theory including near- and far-field coupling to obtain strong critical coupling, and successfully achieve coherent perfect absorption. It is found that by changing the phase difference between two coherent beams, we can switch between complete absorption and complete transparency to achieve absorption switching.

2. Design and simulations

As shown in Fig. 1(a), we propose an absorption system consisting of borophene gratings and a borophene sheet, separated by a silica layer in the middle. The period is P, the width of the borophene grating is w, the thickness of the silica layer is d, and the carrier densities of the borophene grating and the borophene sheet are n and n₀. The structural parameters are set as follows: P = 168 nm, w = 108 nm, d = 10 nm, n = n₀ = 4.3 × 10¹⁹m⁻², and the relative refractive index of the silica layer is 1.6. Figure 1(b) shows the xz cross-sectional view of the system in one period, which allows the structure to be observed more clearly. The finite-difference time-domain (FDTD) method is used to investigate the electromagnetic properties of the structure. In the simulation, the polarization direction of the plane wave is along the x-direction and is incident along the z-direction. The simulation temperature and time are set to 300 K and 20000 fs, respectively. Periodic boundary conditions and perfectly matched layers (PMLs) are set in the x- and z-directions, respectively. The grid accuracy is set to Δx = 1 nm and Δz = 0.5 nm.

Fig. 1. (a) Structure diagram of the absorption system. Inset: the optical axis direction of the χ₃ phase borophene. (b) The xz cross-sectional view in one period.

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The material of choice is the anisotropic borophene, and its electrical conductivity is given by the Drude formula [30]:

(1)$${\sigma _{jj}} = \frac{{i{D_j}}}{{\pi (\omega + i{\tau ^{ - 1}})}}\textrm{ },\textrm{ }{D_j} = \pi {e^2}\frac{n}{{{m_j}}}\textrm{, }$$

where j represents either the x or y optical axis of the borophene crystal, ω is the angular frequency of the incident wave, τ is the electron relaxation time set to 65 fs, n and e are the carrier density and electron charge, respectively. D_j and m_j stand for the Drude weight and the effective electron mass, here m_x = 1.4 m₀, m_y = 3.4 m₀, m₀ represents the standard electron rest mass. It should be noted that the edge of a two-dimensional material can have an influence on the bands of the 2D electronic structure and the plasmon resonance it supports [31,32]. However, the effect on plasmon resonance can be ignored when the size is greater than 20 nm [33,34]. Considering that the width of the borophene in this paper is greater than 20 nm, we can ignore the influence of the edge of borophene on plasmon resonance.

3. Results and discussion

Under suitable phase matching conditions, it is well-established that the destructive interference between different modes can form F-W BIC. Therefore, we first investigate the mechanism of the two modes before analyzing the physical mechanism of F-W BIC. The absorption spectrum of the system is shown in Fig. 2(a), when the incident wave is irradiated vertically on the system, two absorption peaks with 0.5 appear near 1399 nm and 1864nm, where the absorption is defined by the equation A = 1 - R - T. Here, the n = n₀ = 4.3 × 10¹⁹m⁻². Figures 2(b) and (c) show the z-component of electric field distributions at the positions for the two resonance peaks. It can be seen that these are the guided mode resonance (GMR) and the Fabry-Perot resonance (FPR). The dotted line represents the position of the borophene grating and the borophene sheet. Then, we analyze the generation conditions of the two modes. With the borophene grating, enabling the incident wave to achieve the phase-matching with the guided modes in the SiO₂ plate, this phenomenon is called GMR [35,36]. The tangential component of the wave vector in the system can be denoted by k_x = k₀sinθ, where k₀ = ω / c is the wave vector in free space, θ is the angle of incidence, ω is the angular frequency, and c is the speed of light in air. The GMR mode condition can be approximated as follows:

(2)$${\textrm{Re}} (\beta (\omega )) = |{{k_x} + m{G_x}} |\textrm{ ,}$$

here m is the order of the GMR mode, we consider the case of m = 0. G_x = 2π / P represents the reciprocal lattice vector caused by the borophene grating. β(ω) is the propagation constant of the guided mode, which can be obtained by solving the dispersion relation of the guided mode [37]:

(3)$$\frac{{{n_v}^2}}{{\sqrt {\beta {{(\omega )}^2} - {n_v}^2{k_0}^2} }} + \frac{{{n_s}^2}}{{\sqrt {\beta {{(\omega )}^2} - {n_s}^2{k_0}^2} }} ={-} \frac{{i{\sigma _{jj}}}}{{c{k_0}{\varepsilon _0}}},$$

where n_v and n_s are the refractive index in vacuum and the relative refractive index of the silica, respectively.

Fig. 2. (a) FDTD simulations of the absorption spectrum and results of the theoretical TCMT calculation. (b) and (c) are the z-component of electric field distributions for the two modes.

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Due to the impedance at the two ends is mismatched in the regions with and without borophene, the borophene SPPs are reflected back and forth at the two boundaries, forming a standing wave. The standing wave is called lateral FPR, and the frequency of the FPR mode can be expressed as [38]:

(4)$${\omega _{FPR}} = \frac{{Kc}}{{2{n_M}w}}\textrm{ },$$

where K is the order of the FPR mode, c is the speed of light, n_M is the effective modal index of the standing wave, and w is the width of the borophene grating.

To further explore the strong coupling between the GMR mode and the FPR mode, we employ temporal coupled-mode theory (TCMT) for analysis. The dynamical equations of the two resonant modes can be expressed as [39]:

(5)$$\frac{{d{a_\textrm{1}}}}{{dt}} = (i{\omega _1} - {\gamma _{e1}} - {\gamma _{i1}}){a_1} + i\kappa {a_2} + i\sqrt {{\gamma _{e1}}} {s^ + } + i\sqrt {{\gamma _{e1}}} (i\sqrt {{\gamma _{e2}}} {e^{ - i\varphi }}{a_2})\textrm{ ,}$$

(6)$$\frac{{d{a_2}}}{{dt}} = (i{\omega _2} - {\gamma _{e2}} - {\gamma _{i2}}){a_2} + i\kappa {a_1} + i\sqrt {{\gamma _{e2}}} {s^ + } + i\sqrt {{\gamma _{e2}}} (i\sqrt {{\gamma _{e1}}} {e^{ - i\varphi }}{a_1})\textrm{ ,}$$

where a₁ and a₂ represent the amplitudes of the two modes, ω₁₍₂₎, γ_e1(2) and γ_i1(2) represent the eigenfrequency, external loss and internal loss of the two modes, respectively, κ is the coupling coefficient between the two modes, s⁺ is the incident wave, φ is the phase difference between the two modes. By solution of Eq. (5) and Eq. (6), the amplitudes of the two modes can be expressed as:

(7)$${a_1} = \frac{{\{i(\omega - {\omega _2}) + {\gamma _{e2}} + {\gamma _{i2}}\} i\sqrt {{\gamma _{e1}}} + (i\kappa - \sqrt {{\gamma _{e1}}{\gamma _{e2}}} {e^{ - i\varphi }})i\sqrt {{\gamma _{e2}}} }}{{\{ i(\omega - {\omega _1}) + {\gamma _{e1}} + {\gamma _{i1}}\} \{ i(\omega - {\omega _2}) + {\gamma _{e2}} + {\gamma _{i2}}\} - {{(\sqrt {{\gamma _{e1}}{\gamma _{e2}}} {e^{ - i\varphi }} - i\kappa )}^2}}}{s^ + }\textrm{ ,}$$

(8)$${a_2} = \frac{{\{i(\omega - {\omega _1}) + {\gamma _{e1}} + {\gamma _{i1}}\} i\sqrt {{\gamma _{e2}}} + (i\kappa - \sqrt {{\gamma _{e1}}{\gamma _{e2}}} {e^{ - i\varphi }})i\sqrt {{\gamma _{e1}}} }}{{\{ i(\omega - {\omega _1}) + {\gamma _{e1}} + {\gamma _{i1}}\} \{ i(\omega - {\omega _2}) + {\gamma _{e2}} + {\gamma _{i2}}\} - {{(\sqrt {{\gamma _{e1}}{\gamma _{e2}}} {e^{ - i\varphi }} - i\kappa )}^2}}}{s^ + }\textrm{ ,}$$

and the absorption of the system is given by the formula [40]:

(9)$$\textrm{A} = \textrm{2}{\gamma _{i1}}{\left|{\frac{{{a_1}}}{{{s^ + }}}} \right|^2} + \textrm{2}{\gamma _{i2}}{\left|{\frac{{{a_2}}}{{{s^ + }}}} \right|^2}\textrm{ ,}$$

The phase difference φ can be approximately to zero, primarily due to the significant order of magnitude difference between the propagation velocity of the incident wave and the distance. In the fitting results, the absorption formula derived from TCMT fits well with the numerical simulation results obtained from FDTD, as illustrated in Fig. 2(a). External loss and internal loss values of the two modes are consistent, i.e., both modes achieve a balance between the radiation loss rate and the non-radiation loss rate, satisfying the critical coupling condition. In the following content, we will introduce each part of the work, and use FDTD software to construct the model for numerical simulation.

The investigation focuses on the strong coupling between the GMR mode and FPR mode, which leads to the formation of the F-W BIC. Figure 3(a) shows the variation trend of the absorption spectrum with the width of the borophene grating. The figure illustrates anti-crossover behavior in the absorption spectrum, with the formation of F-W BIC near the frequency intersection. Figure 3(b) shows the absorption spectra under different grating widths to explore the interaction between the two modes. As the width of the borophene grating decreases from 96 nm to 76 nm, the resonance frequencies of the two modes gradually become equal, resulting in the quasi-BIC evolving into a BIC. F-W BIC is formed near 1465 nm. Figures 3(c)-(f) illustrate z-component of electric field distributions for the two resonance modes before and after the BIC point, it can be seen that the two modes exchange. Consequently, we can infer that the strong coupling between the two modes makes the absorption spectrum have anti-crossover behavior, and mutual interference between the modes leads to the disappearance of the line width, forming F-W BIC. Next, we calculate the Q-factor under the coupling of the two modes using the formula Q = f₀ / Δf, where f₀ represents the center frequency of the absorption peak and Δf represents the full width at half maximum (FWHM). The Q-factor diverges at w = 76 nm due to the vanishing absorption linewidth, which is consistent with the physical properties of the BIC point, as shown in Fig. 3(g).

Fig. 3. The process of strong coupling of the GMR and FPR modes to form the F-W BIC. (a) Absorption spectra at different borophene grating width, with n = n₀= 4.3 × 10¹⁹m⁻². (b) Absorption spectra at different borophene grating width, where destructive interference of the GMR and FPR modes avoids crossing and linewidth vanishing. (c)-(f) The z-component of electric field distributions for the two resonance modes at w = 96 nm and w = 66 nm. (g) Variation of the Q-factor with the width of the borophene grating.

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In the previous discussion, we achieved F-W BIC by adjusting the width of the borophene grating. However, once devices are manufactured, their structure cannot be changed, resulting in the absorption and working frequency being unable to be dynamically tuned. In theory, the gate voltage can be used to adjust the carrier density of borophene [41,42], which can change the resonance wavelength of borophene SPPs. Under this condition (w = 66 nm), we successfully verified that the F-W BIC can be dynamically tuned by designing the carrier density of the borophene grating, as shown in Fig. 4(a). As the carrier density decreases, both GMR and FPR modes produce a redshift. At n = 2.6 × 10¹⁹m⁻², the F-W BIC is formed due to the destructive interference of these two modes. Figures 4(b)-(e) show the z-component of electric field distributions for the two modes at n = 3.5 × 10¹⁹m⁻² and n = 2.0 × 10¹⁹m⁻². The figures demonstrate that two modes exchange energy before and after strong coupling, confirming that the F-W BIC can be dynamically modulated by designing the carrier density of the borophene grating.

Fig. 4. (a) Demonstrating the dynamic modulation process of the F-W BIC (w = 66 nm) by designing the carrier density of the borophene grating. (b)-(e) At the carrier density of the borophene grating is n = 3.5 × 10¹⁹m⁻² and n = 2.0 × 10¹⁹m⁻², the z-component of electric field distribution of the two resonance modes.

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In addition, from a practical standpoint, evaluating the impact of oblique incidence on the absorber is crucial for assessing its performance. Figure 5(a) illustrates the schematic of oblique incidence. If the incident wave is a transverse magnetic (TM) polarized wave, i.e., the polarization perpendicular to the grating or has a polarization angle of 0°, high absorption can be maintained even at an incidence angle of up to 70°, as depicted in Fig. 5(b). It can be attributed to the weak angular dispersion originating from the intrinsic flat-band characteristic of the deep sub-wavelength periodic structure. When the incident wave is polarized as transverse electric (TE), meaning the polarization is parallel to the grating or the polarization angle is 90°, the effective excitation of borophene SPPs is hindered. It results in weak energy localization, which is why the absorption is almost zero, as shown in Fig. 5(c). Figure 5(d) shows the absorption spectrum of the incident wave at different polarization angles. It maintains high absorption when the polarization angle is less than 30°. In a large range of angles, the absorption of system is almost insensitive to the incident angle. This means that it is capable of achieving wide-angle absorption due to its high absorption rate at a specific polarization angle.

Fig. 5. (a) Schematic of oblique incidence. (b) Absorption spectra at different incident angles. (c) Absorption spectra of TM and TE polarized waves under normal incidence. (d) Absorption spectra at different polarization angles.

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In order to enhance the absorption of the system, coherent perfect absorption is explored under the strong critical coupling condition, as shown in Fig. 6(a). The scattering matrix describes the relationship between the input beam and the output beam:

(10)$$\left( {\begin{array}{c} {{s_1}^ + }\\ {{s_2}^ + } \end{array}} \right) = \left( {\begin{array}{cc} {{r_1}}&{{t_2}}\\ {{t_1}}&{{r_2}} \end{array}} \right)\left( {\begin{array}{c} {{s_1}^ - }\\ {{s_2}^ - } \end{array}} \right)$$

CPA occurs when every output beam component vanishes, i.e., the determinant of the scattering matrix equals zero. Two coherent beams with input intensity |s₁⁺|² = |s₂⁺|² = 1 and a phase difference of zero are chosen to excite the system such that the output intensity |s₁⁻|² + |s₂⁻|² ≈ 0 near 1399 nm and 1864nm, respectively, as shown in Fig. 6(b). The output intensities I₁ and I₂ are obtained from two monitors placed behind the light source. Simulation results show that the total output intensity at the resonance wavelength is almost zero, achieving coherent perfect absorption. The absorption reaches almost 1, as illustrated in Fig. 6(c). The variation trend of the system absorption as the phase difference of the two coherent beams changes from 0 to 4π is shown in Fig. 6(d). System absorption can reach 1 when the phase differences of the two coherent beams are even multiples of π, and it is practically zero when the phase differences are odd multiples of π. The result shows that the absorption system can be dynamically tuned between complete absorption and complete transparency to achieve the switching effect. That is, another beam can be flexibly adjusted by an input beam. This provides a new idea for realizing linear optical switch and modulator in communication band.

Fig. 6. (a) Schematic diagram of coherent perfect absorption. (b) The output intensities I₁ and I₂ of the system under two-port source excitation. (c) Absorption spectrum of the system under one-port and two-port source excitation. (d) The change of absorption spectra with the phase difference of two incident waves.

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4. Conclusion

In conclusion, we propose F-W BIC based on borophene metamaterials to realize coherent perfect absorption (CPA) in commercially important communication bands. The metamaterials consist of borophene gratings and a borophene sheet, which can simultaneously support a Fabry-Perot plasmon resonance and a guided plasmon mode. The formation and dynamic modulation of the F-W BIC can be achieved by adjusting the width or carrier density of the borophene grating, while the strong coupling leads to the anti-crossover behavior of the absorption spectrum. In the case of oblique incidence, we find that the system has the characteristics of wide angle (within 70 degrees) and high absorption. In addition, based on the strong critical coupling, we realize coherent perfect absorption and absorption switching by the phase difference between two coherent beams. Our research results can provide theoretical support for the design of absorbers and all-optical tuning, and provide new ideas for the application of optical absorption.

Funding

Scientific Research Foundation of Hunan Provincial Education Department (22B0105); Natural Science Foundation of Hunan Province (2020JJ5551, 2021JJ40523); National Natural Science Foundation of China (11947062, 62105276, 62205278).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Flat-band Friedrich-Wintgen bound states in the continuum based on borophene metamaterials

Abstract

1. Introduction

2. Design and simulations

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (10)

Optics Express