# A Bound State in the Continuum Supported by a Trimeric Metallic Metasurface

^{*}

## Abstract

**:**

_{x}= p

_{y}= p = 18.5 mm. A gold metasurface was plated on a Poly tetra fluoroethylene (PTFE) substrate (thickness t = 2.4 mm, permittivity ε = 2.55 and loss tangent tan δ = 0.0002). Each unit has three slits with different lengths but the same width w = 2.2 mm (shown in Figure 1b). The properties of the resonance were studied by using an asymmetry index m = l/h, where l and h are the length of the left slit and the others, respectively. The distance of the adjacent slits is d = 3.7 mm. The period of the unit cell should stay smaller than the resonance wavelength in the simulation process. The simulation was carried out by using the COMSOL Multiphysics

^{®}software under a plane wave with the electric field

**E**

_{x}propagate to -z direction (the background material is air n = 1). Perfectly matched layers boundary conditions are used to absorb the electromagnetic waves scattering to the free space. A typical Fano resonance curve appears in the transmission spectra when m = 0.96 in Figure 1c, the amplitude changed rapidly at the resonance peak and dip.

^{4}(in Figure 2b) and tends to infinity at the BIC point. The state of the BIC point is no longer leaky resonance; it becomes eigenmodes that do not decay.

**J(r)**and

**E(r)**represent the current density and electric field intensity in the Cartesian coordinate system at the internal point

**r**= (x, y, z), ε

_{0}and n are the dielectric permittivity of the free-space and the complex refractive index from the material of the three stripes structure, respectively. The expressions of the moments and the scattered power corresponding to each multipole are shown in Table 1, where the δ and c denote the Dirac delta function and the speed of light, respectively. The subscripts α, β on behalf of the axis x, y, z in the Cartesian coordinates [23,24].

**H**

_{z}in the x-y plane, indicating the MQ. The z component of the electric field

**E**

_{z}is shown in Figure 3f. Clearly the field becomes perfectly confined in the structure at the BICs point and does not couple to radiation. The coupling of the multipole modes caused by the asymmetric structure leads to spatial scattering and the observable quasi-BIC, while the strong locality of the electromagnetic field in the symmetric structure can realize strong energy confinement (the character of BIC).

_{0})/γ is the reduced energy, E

_{0}and γ are the energy and width of the resonance, respectively. q = cot δ is the asymmetry parameter (δ represents the phase shift of modes), which is used to describe the degree of the asymmetry of the line shape. The curves of theoretical calculation from Fano’s formula and simulation are shown as the red dotted lines and black lines in Figure 4a. The retrieved fitted parameters are depicted in Figure 4b. It can be seen that the γ is smaller as the structure has minor asymmetry, indicating that the resonance linewidth is narrow when the structure slightly deviates from symmetry. The fitting parameter q is of the order of unity; in this case both the continuous state and discrete state are of the same strength, causing the asymmetrical Fano line shapes. q has a minor change with varying m from 0.96 to 1.02 except 1, indicating that the Fano lines have the same “peak and dip” shape no matter what m parameters change. While the curves calculated by Fano’s formula offset the simulation results (the green circles in Figure 4a) when m = 0.96 and m = 0.97, resulting in the fluctuations of the fitting parameters. Moreover, the fitting parameter curve can only reflect the changes of the degree of structural asymmetry, but the difference between structural symmetry and asymmetry is not obvious.

_{1}and Γ

_{2}are the radiation decay rate of eigen-frequencies of the two collective modes as mentioned above, and their magnitude affects the resonance bandwidth of the transmission spectrum. α and f

_{0}are the absorptive decay rate and the resonance frequency point, respectively. The radiation decay rate of the two modes decreases and the resonance bandwidth narrows when the structure turns to symmetry, shown in Figure 4d. Γ

_{1}approaches to 0 with the maximum value κ at m approaches to 1 (the red line shows the 0-label line), indicating that there is intensive coupling while no energy is radiating into free space. The symmetric structure results in the disappearance of the resonance bandwidth and the formation of an infinite Q-factor. Combined with the disappeared resonance bandwidth, the reason for the formation of BIC is illustrated. α keeps small throughout the fitting process. Clearly, the results calculated by the coupled-mode theory can not only be highly consistent with the simulation results at the resonant peaks and dips of all parameters, but also highlights the obvious transition between BIC and quasi-BIC.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of the trimeric metasurface and the electromagnetic field excitation, the red dashed box represents a unit of the metasurface shown in (

**b**). (

**c**) Normalized transmission spectra of the metasurface.

**Figure 2.**(

**a**) The normalized transmission spectra with different asymmetry parameter m. The red dot circle represents the position of the BIC. (

**b**) The quality factors as a function of asymmetry parameter m.

**Figure 3.**(

**a**,

**b**) Scattering power of logarithmic scale for multipoles moments with total scattering (pink line), ED (black line), MD (red line), TD (green line), EQ (deep blue line), and MQ (sky blue line) with m = 0.99 at around 8.24 GHz and m = 1, respectively. (

**c**–

**f**) are the nearfield profile of the metasurface, (

**c**) the color maps of the

**H**, the black arrows and red arrows (include red circles) stand for the magnetic and electric field vectors, and the green arrow is the direction of the TD. (

_{x}**d**) the deep blue arrows indicate the EQ, the three short black lines outline the three slits. (

**e**)

**H**in the x-y plane indicate the MQ, the arrows point in the direction of the magnetic field. (

_{z}**f**)

**E**in the y-z plane, the black box is the side view of the structure.

_{z}**Figure 4.**(

**a**,

**c**) the simulated (black dotted curves) and fitted (red dotted curves) transmission spectra with different m by Fano’s formula and coupled-mode theory (CMT), respectively, the green circles show the deviation from the resonant points. (

**b**,

**d**) fitted parameters retrieved from Fano’s formula and coupled-mode theory.

Multipole | Multipole Expression | Scattering Power |
---|---|---|

Electric dipole (ED) | $\mathit{P}=\frac{1}{i\omega}{\displaystyle \int \mathit{J}\mathit{(}\mathit{r}\mathit{)}{d}^{3}r}$ | ${I}_{P}=\frac{2{\omega}^{4}}{3{c}^{3}}{\left|\mathit{P}\right|}^{2}$ |

Magnetic dipole (MD) | $\mathit{M}=\frac{1}{ic}{\displaystyle \int \left(\mathit{r}\times \mathit{J}\mathit{(}\mathit{r}\mathit{)}\right){d}^{3}r}$ | ${I}_{M}=\frac{2{\omega}^{4}}{3{c}^{3}}{\left|\mathit{M}\right|}^{2}$ |

Toroidal dipole (TD) | $\mathit{T}=\frac{1}{10c}{\displaystyle \int \left[\left(\mathit{r}\u2022\mathit{J}\mathit{(}\mathit{r}\mathit{)}\right)\mathit{r}-2{\mathit{r}}^{2}\mathit{J}\mathit{(}\mathit{r}\mathit{)}\right]{\mathit{d}}^{3}r}$ | ${I}_{T}=\frac{2{\omega}^{6}}{3{c}^{5}}{\left|\mathit{T}\right|}^{2}$ |

Electric quadrupole (EQ) | ${Q}_{\alpha \beta}=\frac{1}{2i\omega}{\displaystyle \int \left[{\mathit{r}}_{\alpha}{\mathit{J}}_{\beta}\mathit{(}\mathit{r}\mathit{)}+{\mathit{r}}_{\beta}{\mathit{J}}_{\alpha}\mathit{(}\mathit{r}\mathit{)}-\frac{2}{3}{\delta}_{\alpha ,\beta}\left(\mathit{r}\u2022\mathit{J}\mathit{(}\mathit{r}\mathit{)}\right)\right]{d}^{3}r}$ | ${I}_{EQ}=\frac{{\omega}^{6}}{5{c}^{5}}{\displaystyle \sum _{}^{}{\left|{Q}_{\alpha \beta}\right|}^{2}}$ |

Magnetic quadrupole (MQ) | ${M}_{\alpha \beta}=\frac{1}{3c}{\displaystyle \int \left[{\left(\mathit{r}\times \mathit{J}\mathit{(}\mathit{r}\mathit{)}\right)}_{\alpha}{\mathit{r}}_{\beta}+{\left(\mathit{r}\times \mathit{J}\mathit{(}\mathit{r}\mathit{)}\right)}_{\beta}{\mathit{r}}_{\alpha}\right]{d}^{3}r}$ | ${I}_{MQ}=\frac{{\omega}^{6}}{20{c}^{6}}{\displaystyle \sum _{}^{}{\left|{M}_{\alpha \beta}\right|}^{2}}$ |

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**MDPI and ACS Style**

Fu, T.; Wang, Z.; Chen, Y.
A Bound State in the Continuum Supported by a Trimeric Metallic Metasurface. *Photonics* **2023**, *10*, 40.
https://doi.org/10.3390/photonics10010040

**AMA Style**

Fu T, Wang Z, Chen Y.
A Bound State in the Continuum Supported by a Trimeric Metallic Metasurface. *Photonics*. 2023; 10(1):40.
https://doi.org/10.3390/photonics10010040

**Chicago/Turabian Style**

Fu, Tao, Ziyan Wang, and Yonghe Chen.
2023. "A Bound State in the Continuum Supported by a Trimeric Metallic Metasurface" *Photonics* 10, no. 1: 40.
https://doi.org/10.3390/photonics10010040