1. Introduction

The electromagnetic waves with the infrared frequencies that allocated between terahertz and near-infrared electromagnetic bands, can interact with two dimensional (2D) and three dimensional (3D) structured materials [1–17]. This interaction causes manipulation of the incident wave characteristics consisting of amplitude, polarization, and phase. Applications of the infrared metasurfaces have been developed in some categories, such as planar holographic [2–4], lensing [5–7], modulator [8,9], manipulating polarization [10–14], and cloaking [15–17]. For these applications, various bandwidth of responses are applied from ultra wideband regime to high Q-factor ones. The high Q-factor resonances can be achieved by some methods, such as using silver nanopan cavity [18], double-layer crystalline micro-disk resonator [19], photonic nano-cavity and nano-holes [20,21], and modes coupling [22,23]. Also, recently the physics of BIC has been applied widely for confining the electromagnetic waves. This idea was presented in an electronic system in the context of quantum mechanics [24]. BIC different from the conventional bound states, are special states that allocate in the continuum while remain localized and without radiation [25]. There are some approaches to create the BIC in metasurfaces and one can divide these approaches in two major categories. The first one can be appeared in destructive interference conditions with mode coupling regime [26], supercavities [22,27], and Fabry–Perot resonance [28]. The second one is well known as symmetry- protected BIC when leaking of the bound state is prevented by embedding a bound state with a given symmetry in a continuum with different symmetry. In this situation, the spectral linewidths are completely vanished and BIC is disappeared in the spectrum. By distorting the symmetry, quasi-BIC can be appeared in the spectrum with a specific linewidth and Q-factor [29–47]. This asymmetry in a metasurface can be achieved by some manners, such as, creating differences in horizontal dimensions of the meta-atoms relative to each other [29–35] or thickness of them [35,36], distinguishing the arrangement of the meta-atoms [30,37], adding or removing some small parts in different position into the meta-atoms [38–43], considering various materials for meta-atoms with different electromagnetic specifications [35], angular rotation of a meta-atoms set with respect to others [38,44–46], and assuming an incident angle to break the symmetry of the structure [47].

Here, graphene metasurfaces with tailored strong confinement of light based on symmetry-protected BICs in far-infrared regime are synthesized in far-infrared regime. The metasurfaces consist of rectangular graphene patches and breaking symmetry in the structure is controlled by manipulating these meta-atoms. Meta-atoms manipulation is pursued using three asymmetric parameters including length, rotation, and chemical potential. Changing the asymmetric parameters arises tailored coupling resonances with different linewidths and Q-factors. In contrast to previous works based on symmetry-protected BICs with graphene [31,39–42], the results are for spectral band of far-infrared, the proposed metasurfaces are feasible in terms of fabrication, the various methods of creating an asymmetric structure by applying length, angle, and chemical potential asymmetries are implemented in a specific structure, and the efficiency of these different approaches can be evaluated relative to each other. The efficiency not only depends on linewidth and Q-factor, but also depends on resonance intensity in the realistically modelled graphene metasurfaces with intrinsic losses. In this work, light confinement is quantitatively evaluated for three types of asymmetry methods by calculating the electric field enhancement.

2. Modeling a graphene BIC metasurface

To create a BIC structure, a metasurface consisting of rectangular graphene patches is considered, as shown in Fig. 1(a). Graphene is a two-dimensional material composed of a single layer of carbon atoms arranged in a hexagonal lattice. Graphene patches are typically fabricated using techniques such as chemical vapor deposition (CVD) or micromechanical cleavage from graphite masses. The size and shape of the rectangular patches can be controlled during fabrication by adjusting the growth conditions or cutting method. The thickness of the graphene layer is typically only a few atomic layers thick and it is assumed as h_g= 0.5 nm throughout this study. The electrical conductivity of graphene (σ_r + jσ_i) can be calculated by random-phase approximation in the temperature of T = 300 K [48]. The electric conductivity of graphene depends on the chemical potential of µ_c and this parameter can be changed by doping or by applying an external electric field. Doping refers to intentionally introducing impurities into the graphene lattice, which alters its electronic structure and therefore changes its chemical potential [49,50]. Also, for tuning the chemical potential with the external electric field, there is a relation between the gate voltage (V_g) and the charge carrier density (n_s) [51,52]:

The reflection of the metasurface, which consists of a rectangular graphene patch, can be obtained using the finite element method software (COMSOL Multiphysics), as shown in Fig. 1(b). The metasurface is defined by considering periodic boundary conditions for a unit cell at x- and y- directed boundaries exposed by a normal perpendicular plane wave with x and y polarizations. For the meta-atoms, dimensions of a = 400 nm and b = 100 nm are taken into account and the periodicities in x and y directions are P_x= 500 nm and P_y= 250 nm, respectively. The chemical potential of graphene is assumed to be µ_C= 500 meV and the dispersive permittivity of graphene is calculated by applying the relationship ɛ = ɛ₀ + j(σ_r + jσ_i)/ωh_g, where ɛ₀ is the permittivity of free space and ω is angular frequency. The metasurface substrate is considered as a silicon dioxide (SiO₂) slab with a thickness of L_s= 300 nm. Graphene patches imaged between dielectric media induce surface plasmon resonance (SPR) in the terahertz and infrared ranges [53–55].

 

Fig. 1. (a) Schematic view of the reflector consisting of rectangular graphene patches with periodicities Px and Py in the x and y directions, respectively, on a dielectric plate exposed to a normally polarized plane wave as an incident wave, (b) metasurface reflectance on top of a dielectric (SiO₂) slab with a thickness of L_s= 300 nm achieved by full-wave simulation with three surface plasmon resonances at 11.6 THz, and 28.2 THz, and 38.6 THz frequencies for x-polarization and other three resonances at 31.4 THz, and 34.8 THz, and 39.6 THz frequencies for y-polarization, the electric field distributions on the surface of the unit cell at the mentioned resonance frequencies for (c) x-polarization and (d) y-polarization.

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Figure 1(b) shows the reflectance spectra with the first three induced SPR modes induced due to the coupling between the structured graphene and the incident electric field for x and y polarizations. The first SPR modes for both polarizations have the largest amplitude. The electric field distributions in the xy cross section of the unit cell at the induced resonance frequencies of 11.6 THz, 28.2 THz, and 38.6 THz for x-polarization and 31.4 THz, 34.8 THz, and 39.4 THz for y-polarization are depicted in Fig. 1(c) and Fig. 1(d), respectively. The electric field profiles are normalized by the electric field intensity of the structure without the graphene patches (E₀). The confinement strength criteria can be taken into account by comparing in-plane wavelength λ_p with the corresponding wavelength of light in free space λ₀ [55]. The proposed structure addresses a strong confined light–matter interaction with a confinement factor of λ₀/λ_p≈64.

3. Quasi-BIC with three asymmetry approaches and efficiency analysis

In order to create the quasi-BIC in the structure mentioned, the symmetry of the metasurface can be broken using some approaches. As a straightforward manner, the length of a meta-atom in the x direction can be reduced by ΔL compared to the neighboring one. The asymmetry parameter α_L= ΔL/L can be defined for this metasurface. The second technique is to rotate a meta-atom by Δɵ degrees next to a fixed meta-atom and define an asymmetry parameter of α_ɵ= Δɵ/90. The third method is making a difference in the chemical potential for one of the couple meta-atoms in the unit cell of the metasurface with the asymmetry parameter of α_µ= Δµ_C/µ_C. Figure 2 shows schematically these different approaches. Each of these methods is then evaluated by checking the relevant reflection and electric field enhancement. The first excites the quasi-BIC resonance by reducing the length of one of the metaatom pairs in ΔL. The specifications of the metasurface are the same as the mentioned above and the values of ΔL are gradually decrease from 20 nm to 1 nm. Figure 3(a) depicts the reflectance spectra with the excited quasi-BIC resonance for different values of ΔL around the frequency of 11.4 THz. By reducing the value of ΔL, the amplitude of the excited resonance becomes smaller and, conversely, the Q-factor becomes larger. By reducing ΔL from 20 nm to 1 nm, the reflectance amplitude at the resonance frequency varies from 0.9 to 0.035 and the Q-factor increases from 38 to 493. The nature of graphene with intrinsic losses leads to a limitation of Q-factor and a damping intensity for small values of asymmetric parameters. The focused plot shows the quasi-BIC resonance with the highest Q-factor of 493 by the asymmetry parameter of α_L= 1/400. To evaluate the circumstances of field enhancement compared to the reflectance spectra in the asymmetry metasurface, Fig. 3(b) illustrates the normalized maximum values of the electric field as the red line with the reflectance for asymmetry parameter of ΔL = 5 nm. Breaking symmetry of the structure amplifies the enhancement of light at the quasi-BIC resonance even more than induced SPRs. This enhancement is more than 3-fold increase in maximum normalized values of the electric field. Inset plot of Fig. 3(b) shows the electric field distributions at the induced quasi-BIC resonance.

 

Fig. 2. Schematic representation of three methods for generating quasi-BIC on the metasurface of rectangular graphene patches. These methods involve making differences in Length, angle, and chemical potential of one of the two meta-atoms in a unit-cell.

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Fig. 3. Reflectance spectra with quasi-BIC resonance for different values of the asymmetry parameter in three asymmetry approaches of (a) length, (c) rotation, and (e) chemical potential with the inset plots of focused reflectance for the largest asymmetric parameters, the normalized maximum values of the electric field and reflectance spectra for (b) longitudinal asymmetry with ΔL = 5 nm, (d) rotational asymmetry with Δɵ=15°, (f) chemical potential based asymmetry with Δµ_C= 20 meV. The inset plots of (b), (d), and (f) show the electric field distributions at quasi-BIC resonances.

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The electric field profile shows two unequal electric dipole moments excited by the longitudinal asymmetric pair of rectangular graphene patches. Due to the non-zero net dipole moments of the structure, energy exchange with continuous radiation mode in free space can be realized and a quasi-BIC with such a high field enhancement can be achieved.

Another way to achieve quasi-BIC resonance is to break the symmetry of the lattice can be broken by rotating one of the pair of graphene patches by different angles. The specifications of the excited resonance depend on the value of rotation angle. Figure 3(c) depicts this dependency on the amplitude and Q-factor of the resonance in the reflectance spectra at the frequency of 11.4 THz. A significant difference between two meta-atoms with large angles of rotation causes big amplitude and the small value of Q-factor. On the other hand, a minor difference makes a resonance with short amplitude and big Q-factor. For example, rotation by Δɵ=20° produces an excited resonance with amplitude of 0.85 and the Q-factor of 45. While these values at Δɵ=5° are 0.01 and 945, respectively. Zoomed plot shows the quasi-BIC resonance with the highest Q-factor excited by a rotation of 5° with a reasonable and achievable asymmetry parameter of α_ɵ=5/90. The electric field enhancement and reflectance spectra with asymmetry parameter of Δɵ=15° are indicated by red and blue lines, respectively in Fig. 3(d). In complete similarity with the previous manner, the intensity of the normalized maximum value of the electric field at the quasi-BIC resonance is more than 3 times the intensity of the first SPR mode. Inset plot of Fig. 3(d) shows the electric field distribution at the quasi-BIC resonance and unbalanced electric dipole moments generated by rotational asymmetric pair of rectangular graphene patches.

In addition to the above-mentioned geometrically asymmetric meta-lattices, a manner based on differences in the electrical conductivities of the meta-atoms is proposed. Recently, graphene monolayers with two or more types of Fermi energy levels were presented by using non-uniform substrates [56–58]. As it is mentioned above, the electrical conductivity of the graphene depends on the chemical potential and this parameter, in turn, depends on other parameters such as carrier density and temperature. Figure 2 shows a pair of meta-atoms with two different chemical potentials of µ_C and µ_C+Δµ_C. I consider the specifications of the metasurface to be the same as those above, with the difference that one of the two meta-atoms has more chemical potential by Δµ_C. If one consider Δµ_C= 100 meV, a quasi-BIC resonance with low Q-factor of 33 and high amplitude of 0.91 can be achieved. As this chemical potential range gradually decreases, the Q factor increases and the intensity decreases, as shown in Fig. 3(e). When the value of Δµ_C is 2 meV, the Q-factor is 756 and the amplitude of the quasi-BIC resonance is 0.015, as emphasized in the inset plot of Fig. 3(e). Under these circumstances the asymmetry parameter is α_µ=2/500. Figure 3(f) represents the red and blue lines as the normalized maximum values of the electric field and the reflectance spectra for the asymmetry parameter of Δµ_C= 20 meV. The normalized maximum value of the electric field at the quasi-BIC is more than 3 times the intensity of the first SPR mode, quite similar to the previous ones. One can see the electric field distribution for the quasi-BIC resonance with different electric dipole moments derived from an asymmetric pair of rectangular graphene patches with various chemical potentials in inset plot of Fig. 3(f).

The above analysis shows that three methods for generating asymmetry features in the metasurface of graphene patches produce quasi-BIC resonances with various intensities and Q-factors that depend on the values of the asymmetry parameters. Figure 4 depicts the variation of the amplitude as the height of the quasi-BIC resonance and the Q-factor of the resonance versus the asymmetry parameters. At the first glance, creating asymmetry by rotation is the most effective of the three methods. I consider a benchmark at an asymmetry parameter of 10% as a vertical green dashed line.

 

Fig. 4. Q-factor and resonance amplitude as a function of the asymmetry parameter (α) for three different asymmetries of length, rotation, and chemical potential. The green dashed line indicates values for the specific asymmetry parameter of α=10%. It is clear that the rotation performance is much better than others and has a Q-factor of 151, while the length and chemical potential have Q-factors of 35 and 38, respectively.

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It means that by considering the asymmetric parameters of ΔL = 40 nm, Δɵ=9°, and Δµ_c= 50 meV one achieves resonances with Q-factors of 35, 151, and 38 respectively. While the highest value of the Q-factor (945) is belongs to the rotational asymmetry with 5° and the associated asymmetric parameter of α_ɵ=5.5%, two other methods provide Q factors smaller than 50 for this value of the asymmetric parameter. Trend of intensity values is the same for all three approaches and increasing the asymmetric parameter improves the corresponding amplitude.

4. Conclusion

In summary, the BIC metasurface composed of rectangular graphene patches is proposed in a far-infrared frequency band. This work begins with the realistic modeling of graphene with intrinsic losses and the numerical calculation of the reflectance of the metasurface. By breaking the symmetry with the asymmetric parameters of length, rotation and material characteristics of the chemical potential, a strong confinement of the reflectivity appears as a quasi-BIC resonance. For each asymmetry approach, the resulting resonances can be seen for different values of the symmetry parameters. In a general trend, by decreasing the symmetry parameter (α), the linewidth of the resonances and the Q factor increase and conversely the intensities decrease due to the intrinsic losses of graphene. Furthermore, the light enhancement is quantitatively evaluated and the highest light enhancement is observed in the quasi-BIC resonance with a single normalized maximum value of the electric field for three asymmetric structures. This enhancement is addressed by representing the unbalanced electric dipole moments at quasi-BIC resonance in the electric field distributions. Although the three mentioned asymmetric approaches induce strong confined quasi-BIC resonance with the same enhancement factor, there are some differences in the values of the corresponding asymmetric parameters to achieve an appropriate Q-factor. The results show the best performance of the rotational asymmetry form compared to other asymmetry methods. The rotational symmetry results in the Q factor of 945 with the asymmetry parameter of α=5.5%. Due to the results obtained in this work and the manufacturability of the rotational asymmetry approach, this method can be implemented in a wide range of optical and photonic applications.