# Near-Field Excitation of Bound States in the Continuum in All-Dielectric Metasurfaces through a Coupled Electric/Magnetic Dipole Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods: Coupled Electric/Magnetic Dipole Model for Point Dipole Excitation

## 3. Results and Discussion

#### 3.1. Si Nanosphere Metasurface

#### 3.2. Vertical Electric-Dipole BIC

#### 3.3. Vertical Magnetic-Dipole BIC

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Color maps of the reflectance as a function of wavelength and angle of incidence for both polarizations for a square array of dielectric spheres (radius $R=$100 nm) of constant dielectric permittivity $\u03f5=12.25$ ($n=3.5$, similar to that of Si in the visible), with lattice constants $a=b=4R=400$ nm, as shown in (

**a**), in the vicinity of two different bound states in the continuum (BICs) (at wavelengths indicated by black dashed horizontal lines): (

**b**,

**d**) TE and (

**c**,

**e**) TM polarizations. The symmetry-protected BICs emerge at: (

**b**) MD-BIC for TE polarization at $\lambda =709$ nm and (

**e**) ED-BIC for TM polarization at $\lambda =552$ nm. Dashed red lines in (

**d**,

**e**) delimit the diffractive regions (right); none in (

**b**,

**c**).

**Figure 2.**Color maps of the local density of states (LDOS) for the Si nanosphere metasurface shown in Figure 1a, at the wavelength of the symmetry-protected ED-BIC ($\lambda =552$ nm; see Figure 1e) within one unit cell at a distance of ${z}_{d}=a/3\approx 133$ nm, corresponding to the different electromagnetic dipolar contributions: (

**a**) ${p}_{x}$, (

**b**) ${p}_{y}$, (

**c**) ${p}_{z}$, (

**d**) ${m}_{x}$, (

**e**) ${m}_{y}$, (

**f**) ${m}_{z}$. The projection of the nanosphere cross section on the plane is denoted by a dashed circumference.

**Figure 3.**LDOS as a function of the distance from the metasurface plane ${z}_{d}$ (solid curves) for the Si nanosphere metasurface as in Figure 2, at the wavelength of the symmetry-protected ED-BIC ($\lambda =552$ nm), including separately the evanescent contributions (dashed curves): (

**a**) ${p}_{z}$ at ${x}_{d}=0$ (blue curves), ${x}_{d}=80$ nm (red curves), and ${x}_{d}=200$ nm (green curves); (

**b**) ${p}_{x}$ (blue curves) and ${m}_{y}$ (red curves) at ${x}_{d}=80$ nm; all for ${y}_{d}=0$.

**Figure 4.**Color maps of the normalized electric near-field on a plane at $z=a/3\approx 133$ nm scattered from a Si nanosphere metasurface as in Figure 1, when illuminated by a vertical point ED source at the wavelength of the symmetry-protected ED-BIC ($\lambda =552$ nm) located at $({x}_{d},{y}_{d},{z}_{d})=(0,0,a/3)$; thus, ${z}_{d}=z\approx 133$ nm. (

**a**) Total electric field amplitude, showing separately the contributions from the (

**b**) evanescent and (

**c**) propagating components, and the real (

**d**) and imaginary (

**e**) parts of the vertical electric field component.

**Figure 5.**Color maps of total electric near-fields as in Figure 4a at the wavelength of the symmetry-protected ED-BIC ($\lambda =532$ nm), but illuminating with various point-dipole sources located also at $({x}_{d},{y}_{d},{z}_{d})=(0,0,a/3)$: (

**a**) in-plane electric (${p}_{x}$); (

**b**) in-plane magnetic (${m}_{x}$); and (

**c**) vertical magnetic (${m}_{z}$). In all cases, the electric field is normalized for the sake of comparison to the maximum of the field when excited by a vertical electric dipole ${p}_{z}$.

**Figure 6.**LDOS for the Si nanosphere metasurface shown in Figure 1a as a function of $R/a$ for a fixed wavelength ($\lambda =552$ nm) at the unit cell center and at distance of ${z}_{d}=a/3\approx 133$ nm. All the different electromagnetic dipolar contributions are included: in-plane electric (${p}_{x},{p}_{y}$, blue) and magnetic (${m}_{x},{m}_{y}$, green), and vertical electric (${p}_{z}$, red) and magnetic (${m}_{z}$, purple), separating: (

**a**) total (solid curves) and evanescent (dashed curves) contributions; and (

**b**) propagating contributions. The maximum of the evanescent ${p}_{z}$ contribution at $R/a\sim 0.25$ corresponds to the wavelength of the symmetry-protected ED-BIC at the $\Gamma $ point in Figure 1e for $R=100$ nm.

**Figure 7.**Color maps of the LDOS as in Figure 2, but at the wavelength of the symmetry-protected MD-BIC ($\lambda =709$ nm and ${z}_{d}=a/3\approx 133\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}\sim 0.19\lambda $): (

**a**) ${p}_{x}$, (

**b**) ${p}_{y}$, (

**c**) ${p}_{z}$, (

**d**) ${m}_{x}$, (

**e**) ${m}_{y}$, (

**f**) ${m}_{z}$.

**Figure 8.**Color maps of near-fields as in Figure 4, but for the magnetic field at the wavelength of the symmetry-protected MD-BIC ($\lambda =709$ nm), illuminating with a vertical point MD source located at $({x}_{d},{y}_{d},{z}_{d})=(0,0,a/3)$; thus, ${z}_{d}=z\approx 133$ nm: (

**a**) Total magnetic field amplitude, showing separately the contributions from the (

**b**) evanescent and (

**c**) propagating components, and the real (

**d**) and imaginary (

**e**) parts of the vertical electric field component.

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**MDPI and ACS Style**

Abujetas, D.R.; Sánchez-Gil, J.A. Near-Field Excitation of Bound States in the Continuum in All-Dielectric Metasurfaces through a Coupled Electric/Magnetic Dipole Model. *Nanomaterials* **2021**, *11*, 998.
https://doi.org/10.3390/nano11040998

**AMA Style**

Abujetas DR, Sánchez-Gil JA. Near-Field Excitation of Bound States in the Continuum in All-Dielectric Metasurfaces through a Coupled Electric/Magnetic Dipole Model. *Nanomaterials*. 2021; 11(4):998.
https://doi.org/10.3390/nano11040998

**Chicago/Turabian Style**

Abujetas, Diego R., and José A. Sánchez-Gil. 2021. "Near-Field Excitation of Bound States in the Continuum in All-Dielectric Metasurfaces through a Coupled Electric/Magnetic Dipole Model" *Nanomaterials* 11, no. 4: 998.
https://doi.org/10.3390/nano11040998