High-Q Resonances Induced by Toroidal Dipole Bound States in the Continuum in Terahertz Metasurfaces

Abstract

The radiation mode of the interaction between electromagnetic waves and materials has always been a research hotspot in nanophotonics, and bound states in the continuum (BICs) belong to one of the nonradiative modes. Owing to their high-quality factor characteristics, BICs are extensively employed in nonlinear harmonic generators and sensors. Here, the influence of structural parameters on radiation modes has been systematically analyzed using band theory; the mechanisms of quasi-BIC mode and BIC mode were also analyzed through multipole decomposition of scattered power and near-field distribution. Notably, this study presents the discovery that the toroidal dipole-BIC (TD-BIC) arises from the interference and cancellation of electric and toroidal dipoles. The research results indicate that the structure, which supports symmetry-protected BICs, is sensitive to variations in the concentration of NaCl solution in its surroundings, making it applicable for liquid detection in miniaturized metal sensors. The proposed scheme broadens the applicability of BIC-based sensors and provides a prospective platform for biological and chemical sensing.

1. Introduction

Nonradiative states have been extensively studied in quantum mechanics, including nonradiative motion, extended electron models, radiation processes, and the Aharonov–Bohm phenomenon [1,2,3,4]. Nonradiative states, arising from oscillating charge–current distributions, are electromagnetic states that do not emit energy into the far-field region, instead confining fields to a localized space [5]. Recent advancements in metasurface photonics have led to a growing recognition of the significant potential of nonradiative optical modes for various applications. These include the enhancement of localized electric fields, control of far-field scattering in metasurfaces, and amplification of nonlinear effects [6]. Consequently, research on nonradiative electromagnetic modes has emerged as a prominent topic within the realms of classical electrodynamics and nanophotonics. In Mie-resonant dielectric structures, the coupling and complex interaction of multipole modes can give rise to nonradiative electromagnetic states [7]. These states are characterized by the absence of energy leakage and spatially localized profiles, which are known as BICs [8,9,10]. When a system has reflection or rotational symmetries, bound states and the continuum can belong to separate symmetry classes, which prohibits coupling if the symmetry remains intact [11,12,13,14,15]. When external disturbances affect the structure, such as the breaking of symmetry or material radiation and ohmic losses, symmetry-protected BICs will transition from dark modes to bright modes, becoming quasi-BICs. At this point, the energy confined within the structure is released, leading to Fano resonance with the external environment [16]. The characteristics of BICs enable their application in a wide range of technologies, including ultra-narrowband filters [12,17,18,19,20], laser action [21,22], highly sensitive sensing [23,24,25], enhanced nonlinear effects [26,27,28,29,30], and active devices [31,32,33,34,35].

Regardless of the context of nuclear, atomic, and molecular physics, solid state physics, classical electrodynamics, and photonics, the systems with toroidal topology structures need to consider the existence of toroidal multipole moments [36,37,38]. The existence of toroidal multipole moments was not proposed in Mie theory [38,39,40], but they have been proven to have significant effects in many applications, such as anapole states [41,42,43], lasing spasers [44,45], optical forces [46,47,48], and chirality [49]. Recently, several studies have been presented, including a theoretical analysis of TD-BICs [36], a polarization-insensitive TD-BIC implemented in a permittivity-asymmetric all-dielectric metasurface [50], a symmetry analysis of TD resonance in all-dielectric metasurfaces [51], and a study on high-Q factor-resonant dielectric metasurfaces [52]. Nevertheless, research on toroidal dipole BIC-driven high-Q resonances with a metallic terahertz metasurface is rare, especially when the BIC is mainly composed of interference cancellation between electric dipoles and toroidal dipoles; another mode that supports this special situation is the anapole mode.

In this study, we propose a terahertz metallic metasurface, which supports the BIC mode driven by a TD dipole. The metasurface structure consists of a periodically arranged aluminum double split-rectangular resonator (DSRR) array on a SiO₂ substrate. We first demonstrate the BIC characteristics of resonances and choose the combination of an aluminum double split-rectangular as the asymmetric structure. To further explore the radiation modes of the structure, we conduct an in-depth examination of its dipole composition and orientation through multipole decomposition. Finally, perturbation theory is employed to theoretically analyze the sensing performance of the proposed structure. For example, our work proposes a sensor designed to detect the concentration of NaCl solution. Refractive index sensing reveals that the sensitivity of our structure reaches 335 GHz/RIU, higher than that of earlier reported metasurface THz sensors [53,54,55,56,57,58,59]. Our findings facilitate applications in optics and photonics, such as low-threshold lasers and sensing.

2. Theory and Model Design

The simulations are performed using the finite element method with COMSOL Multiphysics 6.2. A unit cell is simulated with periodic boundary conditions along the in-plane x- and y-axes and perfectly matched layers (PML) along the propagation direction (z-axis). Terahertz (THz) plane waves are incident perpendicularly on the metasurface along the z-axis, with the electric field linearly polarized along the x-axis. The electromagnetic and scattering properties of the proposed metasurface are analyzed by decomposing the fields within the metal elements into Cartesian multipole moments.

2.1. Metal Metasurface

The proposed metasurface structure consists of a periodically arranged aluminum double split-rectangular resonator (DSRR) array on a SiO₂ substrate, as shown in Figure 1a. Figure 1b presents the top view (x-y plane) of the DSRR: the periodic constants in the x and y directions are Px = Py = 100 μm; the side lengths of the outer and inner rectangular shapes of the DSRR are a₁ = 20 μm and a₂ = 30 μm, respectively; the widths of the upper and lower gaps of the rectangular body in the DSRR are g₂ = 4–10 μm and g₁ = 4 μm, respectively; the distance between the two rectangular bodies is t = 6 μm and the height of the rectangular bodies is h = 8 μm; here, the asymmetry parameter α is defined as $\mathsf{\alpha }=|{\mathit{g}}_2-{\mathit{g}}_1|/({\mathit{g}}_2+{\mathit{g}}_1)$; the DSRR is built on a SiO₂ substrate with a thickness of 100 μm and a refractive index of n = 1.54 [60,61,62]. The DSRR is made of aluminum (Al) and the conductivity of Al is 3.56 × 10⁷ $\mathrm{S}/\mathrm{m}$ [23].

2.2. Theoretical Formula

To identify the radiation effects between electromagnetic waves and the material, as well as the corresponding dipole composition and orientation, we conducted a multipole expansion analysis of the DSRR based on the following equations. The formulas for multipole decomposition are derived from the current. We first need to calculate the current based on the electric field at the corresponding position [40,63]:

$$\mathit{j}=-\mathbf{i}\omega [\epsilon \left(\mathrm{r}\right)-1]\mathit{E}(\mathit{r})$$

(1)

Here, j is the current vector, $\omega$ is the frequency of the incident wave, $\epsilon$ is the dielectric constant of the material, and E(r) is the electric field at the corresponding position. Then, the electric dipole moment ED, magnetic dipole moment MD, toroidal dipole moment TD, electric quadrupole moment EQ, and magnetic quadrupole moment MQ can be derived as follows [40,63]:

$$\mathit{E}\mathit{D}={\displaystyle \frac{1}{i\omega }}{{\displaystyle \int }}_{\mathrm{V}}\mathit{j}(\mathit{r})dr$$

(2)

$$\mathit{M}\mathit{D}={\displaystyle \frac{1}{2c}}{{\displaystyle \int }}_{\mathrm{V}}\mathit{r}\times \mathit{j}(\mathit{r})dr$$

(3)

$$\mathit{T}\mathit{D}={\displaystyle \frac{1}{10c}}{{\displaystyle \int }}_{\mathrm{V}}\{[\mathit{r}\cdot \mathit{j}(\mathit{r})]\mathit{r}-2(\mathit{r}·\mathit{r})\mathit{j}(\mathit{r})\}\mathrm{d}r$$

(4)

$$E{Q}_{\alpha ,\beta }={\displaystyle \frac{1}{2i\omega }}{{\displaystyle \int }}_{\mathrm{V}}[r_{\mathsf{\alpha }}{j}_{\mathsf{\beta }}(\mathit{r})+r_{\mathsf{\beta }}{j}_{\mathsf{\alpha }}(\mathit{r})-{\displaystyle \frac{2}{3}}(\mathit{r}\cdot \mathit{j}(\mathit{r}{))\mathsf{\delta }}_{\mathsf{\alpha },\mathsf{\beta }}]\mathrm{d}r$$

(5)

$$M{Q}_{\alpha ,\beta }={\displaystyle \frac{1}{3c}}{{\displaystyle \int }}_{\mathrm{V}}\{[\mathit{r}\times \mathit{j}(\mathit{r}{)]}_{\mathsf{\alpha }}{r}_{\mathsf{\beta }}+r_{\mathsf{\alpha }}{[\mathit{r}\times \mathit{j}(\mathit{r})]}_{\mathsf{\beta }}\}\mathrm{d}r$$

(6)

Finally, by using the following equation, the corresponding scattering cross-section $C_{sca}^{sum}$ can be obtained [64]:

$${{\displaystyle C}}_{sca}^{sum}\simeq {\displaystyle \frac{{k_0}^4}{6{{\mathsf{\pi }\mathsf{\epsilon }}_0}^2{\left|E_{\mathrm{inc}}\right|}^2}}{\left|\mathit{E}\mathit{D}+i{k}_0\mathit{T}\mathit{D}\right|}^2+{\displaystyle \frac{{k_0}^4{\epsilon }_d{\mu }_0}{6{\mathsf{\pi }\mathsf{\epsilon }}_0{\left|E_{\mathrm{inc}}\right|}^2}}{\left|\mathit{M}\mathit{D}\right|}^2+{\displaystyle \frac{{k_0}^6{\epsilon }_d}{720{{\mathsf{\pi }\mathsf{\epsilon }}_0}^2{\left|E_{\mathrm{inc}}\right|}^2}}{\left|\mathit{E}\mathit{Q}\right|}^2+{\displaystyle \frac{{k_0}^6{{\epsilon }_d}^2{\mu }_0}{80{\mathsf{\pi }\mathsf{\epsilon }}_0{\left|E_{\mathrm{inc}}\right|}^2}}{\left|\mathit{M}\mathit{Q}\right|}^2$$

(7)

k₀ is the wave number in vacuum; $E_{\mathrm{inc}}$ is the magnitude of the incident electric field, which is 1 V/m; ${\mu }_0$ is the magnetic permeability in vacuum; ${\epsilon }_d$ is the dielectric constant of the corresponding material.

3. Results and Discussion

Figure 2a depicts the simulated spectra of the DSRR composed of a perfect electric conductor (PEC) and a realistic metal (aluminum) with different g₂; the simulation of the PEC represents the ideal simulation of a metal with no loss in THz waves. As shown in

Table 1. The Fano resonance FWHM of BICs and quasi-BICs under different g₂ values.

Material Simulation	FWHM (THz)
	The Value of g2 (μm)
	4	6	8	10
Aluminum	0	0.01162	0.02411	0.02996
PEC	0	0.00359	0.01132	0.01966

and Figure 2a, when g₂ is varied from 4 to 10 μm, it can be observed that the Fano resonance gradually strengthens within the structure, and the transmission curve shows a typical Fano resonance curve, and the absorption of the DSRR gradually increases; when g₂ = 4 μm, the Fano resonance half-peak full width (FWHM) of the DSRR structure at 2.11–2.15 THz disappears, and the corresponding absorption also tends to a smooth curve, it means a transform in radiation dipole modes in the DSRR. The mode at 2.1 THz is a weakly coupled mode of Fano resonance [8]. Figure 2b shows the two-dimensional transmission spectrum in different g₂, and it can be easily observed that Fano resonance FWHM gradually decreases and disappears at 2.112 THz (marked by the green star).

In order to further investigate the properties of BIC mode, a perfect electric conductor (PEC) was used for the DSRR in simulations to calculate characteristic frequency and quality factors. The Q-factor is defined as Q = ω₀/2γ, where ω₀ represents the resonant characteristic frequency and γ represents the resonance damping [12]. Figure 2c shows the characteristic frequency and radiation Q-factor under the influence of the periodic boundary conditions of the structure, where a represents the period size of the structure. The position of the wave vector is from the M-point to the highly symmetric center point Γ-point and then to the X-point, which mainly indicates whether the disturbance of periodic boundary conditions affects the mirror symmetry and rotational symmetry (C_2v symmetry) of the structure. From Figure 2c, the characteristic frequency at the highly symmetrical center point is 2.112 THz, and the radiation Q-factor at this position is at its maximum and close to infinity. When gradually moving away from the highly symmetrical center point, the radiation Q factor sharply decreases. This indicates that the DSRR maintains mirror symmetry and rotational symmetry at the highly symmetrical center Γ-point. Combining Figure 2a,b, it can be observed that when g₂ = 4 μm, the structure maintains C_2V symmetry. When the symmetry of the structure is disrupted, either by moving away from the highly symmetrical center Γ-point or when the structural parameter g₂ exceeds 4 μm, the energy originally confined within the structure is gradually released. It leads to mutual coupling between the structure and the extended wave, resulting in increasingly pronounced Fano resonance. Figure 2d presents a logarithmic plot of the radiation Q-factor as a function of asymmetry α, illustrating the relationship between asymmetry α and the radiation Q-factor as ${\mathrm{Q}}^{\propto }{\mathsf{\alpha }}^{-2}$. According to the definition of symmetric protected BIC, two conditions must be met: first, when the structure maintains C_2v symmetry, it has an infinitely high Q-factor in an infinitely narrow frequency spectrum [6,8]; second, the relationship between the asymmetric parameter α of the structure and the Q-factor is ${\mathrm{Q}}^{\propto }{\mathsf{\alpha }}^{-2}$ [11,12]. From Figure 2a–d, it can be inferred that the mode at 2.112 THz is a symmetric protected BIC.

Figure 2 clearly demonstrates that when the structure is in a BIC mode, electromagnetic waves within the structure do not couple with extended waves; the electromagnetic wave absorption within the structure tends toward 0. In contrast, in a quasi-BIC mode, these waves do couple, accompanied by a noticeable Fano resonance, and the electromagnetic wave absorption within the structure gradually increases with the increase of asymmetry. This indicates that in structures with mirror or inversion symmetry, BICs prevent coupling between different modes.

To investigate the reason for the increase in absorption within the DSRR, we performed a multipole decomposition in the Cartesian coordinate system. For specific calculations, please refer to the theoretical discussion mentioned earlier. Figure 3a depicts the total scattering power in different g₂. It is worth noting that the minimum total scattering rate is at the BIC mode position (at 2.112 THz), while the maximum total scattering rate is at the quasi-BIC position. Figure 3b illustrates the composition of dipoles at the positions of the extremum, corresponding to Figure 3a. For BICs, it can be observed clearly that the contribution of the ED (at 2.112 THz) is dominant, indicating that BICs are primarily induced by the ED. For quasi-BICs, the scattering is mainly contributed by the superposition of other dipole powers. By decomposing the x and y components of the ED and TD scattering powers in Figure 3a, we find that the x component of the scattering power from the ED and TD predominates and their values are nearly equal. The y and z components of scattering power from the ED and TD approach zero when compared to the x component. This indicates that the ED and TD are aligned along the x direction when the metasurface is illuminated by x-polarized light. For the BIC mode, Figure 3a shows that the total scattering power at 2.112 THz reaches a minimum value, indicating the presence of interference and the cancellation behavior of dipoles. On the contrary, for the quasi-BIC mode, the total scattering power is at a maximum, indicating the presence of dipole superposition behavior. To verify the proposed interference cancellation and superposition mechanisms, we plotted the scattering patterns between ED_x and TD_x, as well as the scattering power maps of ED-ik₀TD at g₂ = 4 μm and g₂ = 10 μm, respectively, as shown in Figure 3c,d. The blue and green lines represent ED_x and −ik₀TD_x, respectively. When these two lines intersect, it indicates that the angular difference between their directions is π, reflecting opposite orientations and the cancellation of dipole interference. When these two lines do not intersect and differ by an angle of π, it indicates that the directional difference between them is 0, meaning they are aligned in the same direction, which leads to dipole superposition behavior. The red dashed line represents the total radiated power between ED_x and TD_x. When there is a minimum value of the scattering power, it indicates the interference and cancellation behavior between the ED_x and TD_x. Conversely, when there is a maximum value of the scattering power, it signifies the superposition behavior of ED_x and TD_x. This observation is consistent with the solid blue and green lines. For the BIC mode (at g₂ = 4 μm), the blue and green lines intersect at the 2.112 THz, while the red dashed line shows a minimum value at 2.112 THz (see Figure 3c). This clearly indicates that the direction of ED_x and TD_x are opposite, demonstrating that ED and TD undergo interference cancellation behavior. Thus, the generation of the BIC mode at this location arises from the interference cancellation behavior of the ED and TD. For the quasi-BIC mode (at g₂ = 10 μm), the difference in value between the blue and green lines at 2.242 THz is π, while the red dashed line exhibits a maximum value at 2.242 THz (see Figure 3d). This clearly indicates that the direction of ED_x and TD_x are aligned, demonstrating that the ED and TD undergo superposition behavior. Thus, the generation of the quasi-BIC mode at this location arises from the superposition behavior of the ED and TD.

To further verify the directions of the ED and TD in the BIC and quasi-BIC modes, we plotted the electromagnetic field diagrams for g₂ = 4 μm and g₂ = 10 μm, as shown in Figure 4a–d. The direction of the ED is defined as the path from the negative charge to the positive charge, indicating that the opposite direction of the electric field vector corresponds to the direction of the ED. The generation of the TD arises from the MD, which in turn is derived from the surface current. The directions of the TD and MD can be determined using the right-hand screw rule. According to Figure 4a,c, the electric field vector of the DSRR is oriented from left to right in the central gap, while the direction of the ED is from right to left. In the BIC mode, the dipole direction at the upper and lower open gaps is from right to left; in contrast, for the quasi-BIC mode, the dipole direction at these gaps is from left to right. As shown in Figure 4b, the surface current in the upper cube flows in a left-handed circular direction, indicating that the direction of the MD is oriented upward. In the lower cube, the surface current flows in a right-handed circular direction, resulting in a downward orientation of the MD. Therefore, the direction of the TD is to the right. In contrast, as shown in Figure 4c, the direction of the TD is oriented to the left. Figure 4a–d confirm the observations presented in Figure 3c,d, indicating that the BIC mode induced by the DSRR structure results from destructive interference between the ED and TD, while the quasi-BIC mode arises from the constructive interference between dipoles.

The increase in the scattering power of the TD significantly enhances the absorption of the structure, which is also reflected in the transmission dips observed in Figure 2a,b. In the BIC mode, due to the destructive interference between the ED and TD, the interaction between the THz electromagnetic waves and the structure does not exhibit a pronounced Fano resonance, with the primary contribution to radiation arising from the ED mode. On the other hand, in the quasi-BIC mode, the ED and TD are in a state of constructive interference, and the relatively strong scattering power of the TD significantly influences the interaction between the THz electromagnetic waves and the structure. This results in a pronounced Fano resonance, with the primary contributions to radiation arising from the ED, TD, and MQ modes.

The Fano resonance characteristics of metasurfaces are influenced by the surrounding background medium, making them promising for liquid sensing applications [65]. We designed a metasurface liquid sensor featuring Al cuboid dimer clusters, a SiO₂ substrate, and an integrated microfluidic channel, as shown in Figure 5a. The sensitivity (S) is defined as the resonant frequency shift per unit refractive index change, using the formula $\mathrm{S}=\Delta \mathrm{f}/\Delta \mathrm{n}$ [66], where $\Delta \mathrm{f}$ represents the resonance peak frequency shift and $\Delta \mathrm{n}$ is the refractive index change of the analyte liquid. The figure of merit (FOM) is defined as FOM = S/FWHM, where FWHM is the full width at half maximum. As shown in Figure 5b,c, increasing the refractive index of liquid from 1.3333 to 1.351 results in a redshift of the resonant frequency. When g₂ is 12, 10, and 8 μm, the maximum absorption rates of the DSRR in quasi-BIC mode are 47%, 44%, and 19%, respectively. The overall structure of the DSRR is immersed in water at room temperature and pressure, and the thermal radiation generated by THz electromagnetic waves can be ignored [67]. The refractive index corresponding to NaCl solutions at concentrations of 0%, 1%, 5%, and 10% are 1.3333, 1.3352, 1.3424, and 1.351, respectively [67]. When g₂ = 12 μm, the Fano resonance dips across different background media occur between 2.06 and 2.07 THz, with an average FWHM of 64.53 RIU⁻¹. When g₂ = 10 μm, the Fano resonance dips occur between 2.054 and 2.064 THz, with an average FWHM of 99.04 RIU⁻¹. When g₂ = 8 μm, the Fano resonance dips occur between 2.047 and 2.057 THz, with an average FWHM of 159.52 RIU⁻¹. As shown in Figure 5c,d, when g₂ = 12, 10, 8 μm, the average values of refractive index sensitivity are S = 332 ± 3, 272 ± 2, and 230 ± 2 GHz/RIU⁻¹, respectively. Notably, when g₂ = 12 μm, the refractive index sensitivity better linearity compared to other g₂. Additionally, we compared our work with others, showing better performance (as shown in

Table 2. Comparing THz sensing performance achieved in different works.

Types of Resonance	Materials	Sensitivity (GHz/RIU)	FOM (RIU−1)	References
EIT	Silicon	231	64.7	[53]
BIC	Gold	105	7.501	[54]
Fano	Aluminum	139.2	-	[55]
TD	Aluminum	186	-	[56]
TD	Aluminum	124.3	-	[57]
EIT	Graphene	177.7	59.3	[58]
BIC	Silicon	77	11.1	[59]
TD-BIC	Aluminum	335	64.53	This work

EIT, electromagnetically induced transparency.

). This enhanced linearity ensures more accurate and consistent detection, making the metasurface sensor highly suitable for liquid sensing applications, where precise measurement of refractive index changes is crucial.

4. Conclusions

In summary, a THz metal liquid sensor capable of generating toroidal dipole BIC-driven has been proposed, and the influence of structural parameters on radiation modes has been systematically analyzed using band theory through the relationship between structural characteristic frequency and radiation Q-factor. The mechanisms of the quasi-BIC mode and BIC mode were also analyzed through multipole decomposition of scattered power and near-field distribution. With sensitivities of 335 GHz/RIU and an FOM of 64.53 RIU⁻¹, the quasi-BIC resonant sensor outperforms previously reported designs, demonstrating promising liquid sensing capabilities for precision sensing applications.