1. Introduction

Optical resonators are the core components of modern optical devices. Typical examples of optical resonators include plasmonic nanoresonators, Fabry-Perot resonators, and ring resonators. However, the intrinsic Ohmic losses induced by conductive currents associated with metal materials may severely hinder the effective excitation of resonances with higher Q factors and at higher frequencies (such as the near-infrared and visible region). As a result, low-loss, high refractive index [1,2]and all-dielectric materials have emerged as a research hotspot in recent years. In general, various subwavelength nanophotonic structures made of high refractive index media are expected to support Fano resonances similar to nano-plasmonic resonances. While electric dipole (ED) and magnetic dipole (MD) Mie resonances can be readily excited in all-dielectric metamaterial structures [3–5], the excitation of toroidal dipole (TD) in these systems remains a challenge. The excitation of TD is achieved through the currents in the vicinity of the surface of the torus, which excite end-to-end magnetic dipoles inside the torus and eventually generate toroidal dipoles that oscillate along the central axis of the torus [6,7]. Toroidal multipoles are typically ignored in conventional multipole expansions. Due to their weak coupling with the incident electromagnetic wave, they are naturally masked by much stronger electromagnetic effects, making their detection also challenging. The moderate Q-factor of Mie resonance is always accompanied by moderate electric field confinement, which may limit light-matter coupling strength. Thus, the investigation of toroidal dipoles holds great significance. Recently, all-dielectric metasurfaces have been demonstrated to support optical bound states in the continuum (BICs) with infinite Q-factors [8–12]. Currently, symmetry-protected BICs are the most intensively studied because they can be easily found in the dielectric metasurface whose unit cell follows certain symmetry. By breaking unit cell’s symmetry, symmetry-protected BICs (SP-BICs) are converted into QBICs with high-Q factors [13–19]. There are currently a broad range of ways to break the symmetry of a structure, for instance metasurface detuning by breaking the geometric symmetry of the structure [20–22] or positional detuning by altering the position of the structure in the metasurface [23–26]. Among all these methods, the one based on surface lattice resonance (SLR) hybridization has attracted a wide attention in recent years. ED-BIC, MD-BIC, and EQ-BIC have been achieved by surface lattice resonance hybridization [26–29], and very high Q-factors have been obtained in experiments [26]. SLRs can emerge when the scattered field of a single nanoparticle is in phase with the emitted fields from the surrounding nanoparticles. By adjusting the interplay between Mie or localized surface plasmon resonance (LSPR) and Rayleigh anomaly (RA) enables to design ultra narrow resonances. However, it has been revealed that the contribution of TD in resonance is usually overlooked in the research of SLR hybridization. In fact, the influence of TD is of great significance for the analysis of BIC modes and the application of SLR hybridization. In this work, we investigated for the first time the Toroidal dipole bound states in the continuum (TD-BICs) based on the hybridization of Mie surface lattice resonances (SLRs) in an array. Firstly, the in-plane structural symmetry was broken by altering the relative positions of the composite nanorods in a periodic silicon dichotomy nanostrip array, so that the hybridization of Mie resonant SLR would lead to the excitation of the toroidal dipole(TD) BIC at $\Gamma$ points. The contribution of TD in resonances was verified through the discussion over the optical near-field capturing and far-field scattering characteristics of QBIC in detail. By leveraging the SLR characteristics, we demonstrate that this quasi-BIC can be conveniently tuned by varying the size of the nanorods or the lattice period. Meanwhile, through the study of the shape variation, we found that this quasi-BIC exhibits excellent robustness, whether in the case of two symmetric or asymmetric nanoscale structures. This will also provide large fabrication tolerance for the fabrication of devices. Our research results will improve the mode analysis of surface lattice resonance hybridization, and may find exciting applications in boosting light-matter interactions, including lasing, strong coupling, and nonlinear harmonic generation.

2. Toroidal dipole responses and analysis of induced SLR hybrid metasurfaces

As shown in Fig. 1(a), an all-dielectric metasurface structure consisting of two silicon nanostrips on a silica substrate was designed. Among all the parameters, the period $P_x=P_y=1000$ nm was set as the metasurface unit cell, the thickness of the nanorods h = 300 nm, the length and width of the nanostrip were designed as $l_y=500$ nm and $l_x=300$ nm, respectively, and the spacing between the two nanostrips was set as a=200 nm, as shown in Fig. 1(b). Under the plane wave of x-polarized, the in-plane symmetry of the structure was broken by moving the two nanostrips into opposite directions. The traveled distance traveled was marked as $d$ (the right nanostrip moved up when $d>0$ and moved downward when $d<0$), at the point of which the array could be regarded as the hybridization of the two nanostrip arrays (see Appendix 6.2 Fig. 7 for detailed information). Mie SLR hybridization was used for exciting a TD-BIC.(see Appendix 6.2 for detailed information) As shown in Fig. 1(c), two ring currents in opposite directions were observed when the structural symmetry was broken. At this moment, the coupling between the two nanostrips generated a magnetic field that was connected end-to-end, thus exciting the toroidal dipoles in the y direction. On top of that, the Mie SLR hybridization would also generate a strong ring response under array conditions.

 

Fig. 1. (a) The schematic of the unit cell of the proposed metasurface. (b) Top view of unit cells, where the nanostrip length $l_y=500$ nm, nanostrip width $l_x=300$ nm, nanostrip thickness $h=300$ nm, spacing a=200 nm, period $P_x=P_y=1000$ nm, and d stands for the distance that the nanostrip travels. (c) Schematic diagram of the resonance mechanism of toroidal dipoles, where the yellow arrow represents the displacement current j, the green arrow denotes the movement of magnetic dipoles, and the red arrow indicates the movement of excited toroidal dipoles.

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Figure 2(a) shows the transmission spectrum mapping diagram of the nanostrip moving from 0 to 100 nm. According to the figure, the symmetry of the in-plane structure was broken by adjusting the value of d, with the leakage channel coupled to the continuum opened and the BIC at a wavelength of about 1582 nm converted into a quasi-BIC suffering from a radiation loss. Apparently, the spectral line was broadened gradually as the value of d increased slowly. The inset presents the transmission line plot when $d=100$ nm and at a resonance wavelength of 1594.8 nm. The Fano fitting Q-factor was 1163.94 (see Appendix 6.3 for detailed information).

 

Fig. 2. (a) Transmission mapping versus wavelength wavelengt and parameter d, where the inset represents the transmission spectrum when $d=100$ nm. (b) Normalized scattered power of different multipoles. (c) Dependence of the Q factor and resonance wavelength of the quasi-BIC mode versus parameter d. (d) Scattered power of toroidal and electric dipole components along the $x$, $y$, and $z$ directions.

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In Fig. 2(c), the red line indicates the resonance wavelength under different values of parameter d. Since the symmetry of the structure was broken by moving the nanostrip without altering the volume of the structure, the overall effective refractive index of the structure hardly changed. This helped make the resonance wavelength obtained in the experiment relatively stable. In the meanwhile, the Q factor under different values of parameter d was also calculated, revealing that the Q factor decreased in a gradual manner as the of $|d|$ increased and the Q factor approached infinity when $d=0$. When d = 1 nm, the fano fitting Q-factor is 1054921 (see Appendix 6.3 for detailed information). To further comprehend the properties of this type of BIC, the multipolar decomposition method was used to reveal the physical mechanism of its formation in a multipolar mode. Accordingly, multipole decomposition of the resonance mode was performed in Cartesian coordinates (see Appendix 6.1 for detailed information). Figure 2(b) presents the calculation results when $d=100$ nm, indicating that the far-field radiation of the ring dipole at the resonant wavelength accounted for the majority of the contribution in spite of a weak magnetic quadrupole contribution, while the residual dipole contribution was strongly suppressed. According to Fig. 2(d), the major contribution of the toroidal dipole was mainly concentrated along the $y$ direction.

In Fig. 3(a) and 3(c), the electromagnetic field at the resonance wavelength was plotted with an aim to further verify that the resonance came as the result of the toroidal dipole resonance. This led to the finding that the electric field was mainly limited to the edge of the device with the magnetic field mainly concentrated inside the device. To better observe the field vector in the structure, the cross-sectional diagram of the $x-y$ plane and the $x-z$ plane was drawn, respectively. Figure 3(b) presents the distribution of the electric fields of the nanostrip in the $x-y$ plane. The figure clearly shows that the displacement currents circled in opposite directions in the two nanostrips, thus exhibiting excitation of opposite magnetic dipoles along the $z$-axis. As shown in Fig. 3(d) that takes incident light into consideration, a circular magnetic ring in the $x-z$ plane generated an intense ring-shaped dipole excitation along the $y$-axis. In the end, the collective oscillation of this kind of nanodisk array gave rise to an extremely strong ring response.

 

Fig. 3. (a),(c) Calculated distribution of electric and magnetic fields at the resonant wavelength, respectively. The arrows represent the directions of the fields, and the color scales reflect the intensity of the fields. (b),(d) Cross-sectional view of electric and magnetic fields in the $x-y$ and $x-z$ plane, respectively

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3. Tuning QBIC with structural parameters

In Fig. 4, to further analyze the interaction between TD-BIC and SLR, we change the resonance wavelength of TD-BIC by adjusting the structural parameters, which makes to change the coupling strength of QBIC and Rayleigh anomaly (RA). In the following analysis the resonance wavelength will be shifted because the effective refractive index of the metasurfaces will change due to the change of the structural parameters, and also the RA wavelength will be changed when the period is changed (See Eq. (1) for calculations).

 

Fig. 4. (a) Transmission mapping versus wavelength and parameter $l_x$. (b)wavelength and parameter $l_y$. (c)wavelength and parameter h. (d)wavelength and parameter period $P_x=P_y$

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As shown in Fig. 4(a) and Fig. 4(b), we adjust the $l_x$ or $l_y$ of the nanorods, here we find that when $l_x$ or $l_y$ decrease, the resonant wavelengths undergo a blueshift and eventually converge to (0, $\pm$1) or ($\pm$1,0) RA wavelengths. If we continue to decrease the $l_x$ or $l_y$ at this point, we find that the QBIC will not be excited at this point, i.e., no coupling will occur when the RA position is at the high energy of the local resonance. A similar phenomenon was observed when the height of the nanorods was adjusted. As shown in Fig. 4(c), where it is interesting to note that even if we increase the height to 500 nm, the resonance still exists and the spectral width of the resonance remains very narrow even at a large height due to the QBIC induced by the SLR. Similarly, we also focus on the variation of QBIC at different periods. As shown in Fig. 4(d), when the period becomes larger from 900 nm, the RA gradually approaches the QBIC and the coupling increases, and when the period reaches 1100 nm, a clear RA can be seen near 1595 nm (See Appendix Fig. 9(a) for details.).

4. Stability of QBIC

Finally, we discuss the effect of device shape on BIC. We compared four different nanostructures, each consisting of two identical nanodisks in every structural unit but with different shapes. Figure 5(a) shows that we classified these structures into four shape types on the x-y plane: Rectangle-Rectangle (R-R), Circular-Circular (C-C), Hexagon-Hexagon (H-H), and Ellipse-Ellipse (E-E). The R-R structure parameters were consistent with those described earlier. The C-C structure features a period size of 1000 nm, with a center-to-center distance of 500 nm between two identical nanodisks. Each nanodisk has a height of 300 nm and a disk diameter of 406 nm. The H-H structure, on the other hand, also has a period size of 1000 nm and a center-to-center distance of 500 nm between two identical nanodisks. Each nanodisk in the H-H structure has a height of 300 nm and a hexagonal side length of 224.3 nm. Finally, the E-E structure also has a period size of 1000 nm and a center-to-center distance of 500 nm between two identical nanodisks, with each nanodisk having a height of 300 nm. The E-E structure features elliptical nanodisks, with a half-major axis of 300 nm and a half-minor axis of 161 nm. The transmission spectrum mappings present the same BIC characteristics disappeared resonance peaks at d=0. When |d| is larger than zero, the BICs are converted into QBICs with finite Q-factors, and the resonance spectra are gradually broadened as |d| increases, suggesting reduced Q-factors. The four structures depicted in Fig. 5(b) introduce symmetry breaking within the structural plane and exhibit resonances (QBIC) in the range of 1585nm to 1595nm for d=100nm. As shown in Fig. 5(c), the resonance wavelength shift is less than 13 nm over the range of d from 0 to 100nm. Compared to the scenario of breaking the device structure, the resonant wavelengths of these four structures are considerably more stable. Figure 5(d) demonstrates that the Q factor of the structure retains the BIC characteristic even after undergoing a shape change. Notably, a slight variation in the Q factor is observed, with the E-E structure exhibiting the highest Q factor, and the H-H structure showing the lowest Q factor. This result can be attributed to the differences in coupling between the units caused by the variation in shape, leading to a minor change in the Q factor.

 

Fig. 5. (a) Top view ofunit cell ofthe (R-R) Rectangle-Rectangle , (C-C) Circular-Circular (H-H) Hexagon-Hexagon and (E-E) Ellipse-Ellipse nanodisk metasurfaces. (b) The transmission spectra of the four structures at d=100 nm. (c),(d) The resonance wavelengths and Q-factor as a function of the perturbation parameter d.

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However, using nanodisk structures with identical shapes did not fully reveal the superior robustness of our structure. Therefore, we recombined two nanodisks with different shapes using the four shapes mentioned above to demonstrate the ultra-strong robustness of our method to shape variations. Figure 6(a) illustrates that the four shapes of nanodisks were recombined to form four types of nanostructures with different shapes: Rectangle-Ellipse (R-E), Hexagon-Circular (H-C), Rectangle-Hexagon (R-H), and Ellipse-Circular (E-C), all of which were classified based on their x-y plane shape. The four structures share the same structural parameters as the previous design but differ in the shape of the two nanodisks within a unit. As shown in Fig. 6(b), the four structures exhibit quasi-bound states in the continuum (QBIC) within the range of 1585nm to 1595nm under the condition of d=100 nm. In Fig. 6(c), we also calculated the shift of the resonant wavelength at different d parameters. It can be found that the resonant wavelength shift is less than 13 nm, and the resonant wavelength is still very stable. At the same time, by calculating the Q factor, we found that the Q factor still exhibits the BIC characteristics even in the case of different shapes, and it is also relatively stable. As shown in Fig. 6(d), the E-C structure has the largest Q factor, while the R-E and R-H structures have the smallest Q factor. Therefore, such a BIC mode is excellent robust against the device shape and exhibits large fabrication tolerances. In addition, the QBIC is excited by changing the relative position of the nanodisks, which can be precisely controlled in the actual sample fabrication process, and provides an effective way for the realization of QBICs with high Q-factors.

 

Fig. 6. (a)Top view of unit cell of the (R-E) Rectangle-Ellipse, (H-C) Hexagon-Circular, (R-H) Rectangle-Hexagon, and (E-C) Ellipse-Circular nanodisk metasurfaces. (b) The transmission spectra of the four structures at d=100 nm. (c),(d) The resonance wavelengths and Q-factor as a function of the perturbation parameter d.

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5. Conclusion

In this work, we investigate a hybridization of TD-BIC based on surface lattice resonances (SLRs) in a periodic array of silicon nanostructures. The results demonstrate the existence of quasi-bound states in the continuum (QBIC) that exhibit both stable resonant wavelengths and remarkable robustness to shape variations, as supported by theoretical analysis. By moving the two nanostrips into opposite directions to break the in-plane symmetry of the structure, SLR hybridization was induced to generate TD-BIC. The mechanism of SLR hybridization for generation of TD-BIC was revealed through multipole decomposition and near-field analysis, confirming that TD resonances exhibit typical symmetry-protected BIC characteristics as well as un unlimited service life and can be converted to TD quasi-BIC resonances. Our results indicate that this quasi-BIC can be tuned by adjusting the size of the nanorods or the lattice period. Furthermore, we show that this type of BIC is shape-independent, exhibiting remarkable robustness even when the two nanodisks have different shapes. Our findings expand the mode analysis of SLRs hybridization, and the ultra-surface with high quality factor, strong robustness, and stable resonant wavelength provides a promising pathway for realizing high-Q resonators for nonlinear optical frequency conversion, low threshold laser, optical switching, and biosensors.

6. Appendix

6.1. RA wavelength calculation and multipolar decomposition

When incident light is incident perpendicularly, the RA position of the substrate can be expressed as:

(1)$$\lambda_{m_1,m_2}=\frac{n_{sub}}{((\frac{m_1}{P_x})^2+(\frac{m_2}{P_y})^2)^{1/2}},$$

In this paper, the base refractive index is $n_{sub}=1.45$, while the period $P_x=P_y=1000$ nm, hence $\lambda _{(0,\pm 1)}=\lambda _{(\pm 1,0)}=1450$ nm.

In order to deeply unveil the far-field radiation mechanism of the metasurface, we calculate multipole moments and their contributions for these Fano resonances using the multipole decomposition under the Cartesian coordinate. By integrating the current density within the unit cell of metasurface, all multipole moments can be defined as [24,30,31]: electric dipole moment:

(2)$$\vec P = \frac{1}{i\omega}\int {\vec j{d^3}r} ,$$

magnetic dipole moment:

(3)$$\vec M = \frac{1}{2c}\int {(\vec r \times \vec j){d^3}r} ,$$

toroidal dipole moment:

(4)$$\vec T = \frac{1}{10c}\int {[(\vec r \cdot \vec j)\vec r - 2{r^2}\vec j]{d^3}r} ,$$

electric quadrupole moment:

(5)$$Q_{\alpha \beta }^{(e)} = \frac{1}{2i\omega }\int {[{r_\alpha }{j_\beta } + {r_\beta }{j_\alpha } - \frac{2}{3}(\vec r \cdot \vec j){\delta _{\alpha ,\beta }}]{d^3}r} ,$$

magnetic quadrupole moment:

(6)$$Q_{\alpha \beta }^{(m)} = \frac{1}{3c}\int {[{{(\vec r \times \vec j)}_\alpha }{r_\beta } + ({{(\vec r \times \vec j)}_\beta }{r_\alpha })]{d^3}r} ,$$

where $c$ is the speed of light, $\omega$ is the angular frequency of light, and $\alpha, \beta =x, y, z$. $\vec {j}$ is the current density distribution within an unit cell:

(7)$$\vec{j}={-}i\omega \epsilon_{0} (n^2-1) \vec{E},$$

The far-field scattering intensity of these multipole moments can be calculated by the following formulas:

(8)$${I_P} = \frac{2{\omega ^4}}{3{c^3}}{\left| \vec P \right|^2},$$

(9)$${I_M} = \frac{2{\omega ^4}}{3{c^3}}{\left| \vec M \right|^2},$$

(10)$${I_T} = \frac{2{\omega ^6}}{3{c^5}}{\left| \vec T \right|^2},$$

(11)$${I_{Q^{(e)}} = \frac{\omega ^6}{5{c^5}}\sum {\left| {\vec Q _{\alpha \beta }^{(e)}} \right|}^2},$$

(12)$${I_{Q^{(m)}} = \frac{\omega ^6}{40{c^5}}\sum {\left| {\vec Q _{\alpha \beta }^{(m)}} \right|}^2}.$$

The total scattered power of the multipole moments can be given as

(13)$${I_{total}} = I_P+I_M+\frac{4{\omega ^5}}{3{c^4}}(\vec P \cdot \vec T)+I_T+I_{Q^{(e)}}+I_{Q^{(m)}}+O(\frac{1}{c^5}) ,$$

where $\frac {4{\omega ^5}}{3{c^4}}(\vec P \cdot \vec T)$ is the interference term of the electric and toroidal dipoles.

6.2. Surface lattice resonance hybridization

In Fig. 7 illustrates hybridization of arrays in composite lattices. The array can be seen as a hybridization of two single-nanostrip arrays, which are represented by different colors in the figure, where the green dotted line represents the composite lattice, the black dashed line represents the dark nanostrip lattice, and the red dotted line represents the light nanostrip lattice. When the lattice is not moving, the lattice is in isotope recombination, and when the nanostrip moves up and down, the two lattices achieve dislocation recombination, breaking the intraplane symmetry of the structural unit.

 

Fig. 7. Lattice hybridization (a) Lattice when not moving (b) After moving, the lattice moves down, and the dark color moves up.

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Here we calculated the optical response of two separate nanorods. In Fig. 8(a), there are one obvious resonant peaks near the wavelength 1490 nm, Also we can see that this resonance always appears at 1490 nm when the travel distance of the nanorods increases. The multipole expansion under the Cartesian coordinate system is performed to calculate multipole moments and their contributions to the far-field radiation [31,32]. As shown in Fig. 8(c), it is evident that in the selected wavelength range, the scattering cross section of the toroidal dipole (TD) dominates all other multipoles. This is consistent with Fig. 2(b) in the text. As shown in Fig. 8(b) and 8(d), we calculate the electric and magnetic fields at the resonant wavelengths, and we find that the electric and magnetic fields at this point are similar to the distributions in Fig. 3(b) and Fig. 3(d) in the paper. We can also see that the electric field is mainly localized at the edge of the device, which also allows the coupling of the near field between adjacent structures when forming the array to make the resonance shift.

 

Fig. 8. Optical response analysis of single nanorod dimer structure. (a) Scattering cross section. (b), (d) Electric and magnetic field distribution at the resonance wavelength. (c) Normalized scattered power of different multipoles.

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As shown in Fig. 9(a), when the structure becomes an array, by adjusting the period size, we can see that when the period decreases the adjacent structures are close to each other, the coupling strength is enhanced, and as the period becomes larger the RA line moves and will gradually become obvious, such as in the period of 1050 nm and 1100 nm, respectively, the resonance appears near 1522 nm and 1595 nm, and when the period becomes larger to a certain extent the RA and local resonance will no longer The resonance disappears at this point. As shown in Fig. 9(b), the electric field of 2$\times$2 at a period of 1000 nm can be seen as near-field coupling between adjacent structures, which is also consistent with the SLR mechanism.

 

Fig. 9. (a) Transmission line spectra at different periods. (b) Electric field distribution under array structure.

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6.3. Fano fitting of QBICs

The Fano resonaces can be fitted by the classical Fano formula [24] :

(14)$$T(\omega)=T_0+A_0\frac{[q+2(\omega-\omega_0)/\Gamma]^2)}{1+[2(\omega-\omega_0)/\Gamma]^2} ,$$

where $\omega _0$ is the resonant frequency, $\Gamma$ is the resonance linewidth, and $T_0$ is the transmission offset, $A_0$ is the continuum-discrete coupling constant, $q$ is the Breit-Wigner-Fano parameter determining asymmetry of the resonance profile. The Q-factor is evaluated by $\frac {\omega _0}{\Gamma }$. The fitting results of $d=100$ nm and $d=1$ nm are shown in Fig. 10. The Q-factors of $Q_{100 nm}$ and $Q_{1 nm}$ are 1163.94 and 1054921, respectively.

 

Fig. 10. The Fano fittings of $d=100$ nm and $d=1$ nm. The dashed curves are simulated results, and the solid curves are the Fano fittings. The marked Q-factor values are estimated by $\frac {\omega _0}{\Gamma }$.

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6.4. Electric field distribution of nanorods moving and not moving

As shown in Fig. 11(a), when the nanorods are not moving, the electric field is distributed between adjacent nanorods and the current distribution of both nanorods is the same, which is radiating free BICs. when the symmetry within the structural plane is broken by moving, the ideal BICs are transformed to leak modes. As shown in Fig. 11(b) and 11(c), the electric field is distributed at the edge of the nanorods and between the two nanorods. The in-plane symmetry is broken in the highly symmetric nanorods, so that the charge accumulation at the left and right edges of the nanorods is different from that at the center, thus generating a circular displacement current at each of the left and right sides, and the spins of the two are opposite, which excites two magnetic dipoles oscillating in opposite phases along the z direction, and the near-field coupling between the two nanorods can greatly suppress the outward radiation of the magnetic dipole oscillation along the z direction, forming a discrete dark mode. When the broad-spectrum radiation is coupled with the discrete dark mode, an asymmetric Fano characteristic QBIC is produced in the transmission spectrum.

 

Fig. 11. From left to right, the electric field distribution is shifted by 0 nm, 10 nm and 100 nm.

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6.5. Stability of BIC and field distribution under different shapes

As shown in Fig. 12(a), we calculated the transmission spectrum at d=0 for two nanorods with different shapes, i.e., without moving nanodiscs even if the shapes are different BIC modes are not converted to QBIC modes due to the asymmetry of the shapes. As shown in Fig. 12(b)-(e), we calculated the electric field distribution in the BIC mode, and we can find that the electric field distribution is almost the same even though the shapes are different.

 

Fig. 12. The transmission spectra and electric field distribution of the four structures at d=0.

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Also in the case of shape asymmetry we have calculated the electric and magnetic fields for $d = 100 nm$. As shown in Fig. 13, the distributions of electric and magnetic fields are almost the same for different shapes. However, we find that the electric field is more uniformly distributed at the edges of the structure when the structure is circular, because the circular structure has fewer tips compared to the other structures. The difference in shape makes a small difference in the field, but from the current distribution we can see that two toroidal currents are still formed around the opposite direction.

 

Fig. 13. The electric and magnetic fields distribution of the four structures at d=100 nm.

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Funding

Local Science and Technology Development Fund Project Guided by the Chinese Government (Grant No. QKZYD [2019] 4012); 2023 Scientific Research Project (Youth Project) of Ordinary Undergraduate Colleges and Universities of Guizhou Provincial Department of Education (Grant No. QJJ [2022] 279).

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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