# Meta-Atoms with Toroidal Topology for Strongly Resonant Responses

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Metasurface Full-Wave Simulations and Multipole Expansion

#### 2.2. Meta-Atom Fabrication with 3D Printing

#### 2.3. Electromagnetic Characterization with Rectangular Waveguide Setup

## 3. Results

#### 3.1. Free-Space Metasurface with Controllable Strongly Resonant Response

#### 3.2. Meta-Atoms in a Rectangular Waveguide Setup—Experimental Verification

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Periodic metasurface illuminated with a normally incident plane wave of ${E}_{y}$ polarization. (

**b**) Two unit cells fitted in a rectangular metallic waveguide. (

**c**) 3D-printed unit cells using a PLA filament. (

**d**) Structure after coating with a conductive silver paste. (

**e**) Test fitting of the structure under study on the flange of the waveguide-to-coax adapter. The unit cells are lightweight and can be simply glued on a piece of paper and inserted at any junction between waveguide segments. (

**f**) Measurement setup including a vector network analyzer allowing to measure reflection (S${}_{11}$) and transmission (S${}_{21}$) coefficients.

**Figure 2.**(

**a**) Meta-atom geometry. When ${R}_{\mathrm{out}}>a/2$, the neighboring meta-atoms are touching; in this case, no substrate is necessary. (

**b**) R/T/A power coefficients for plane-wave scattering by the periodic metasurface under normal incidence (${E}_{y}$ polarization). The dimensions are $a=15$ mm, ${R}_{\mathrm{out}}=7$ mm, ${R}_{\mathrm{inn}}=5$ mm ($w=2$ mm), ${g}_{1}=0.5$ mm and ${g}_{2}=0.5$ mm. The thickness of the conductive meta-atom is $h=1$ mm. (

**c**) Supported resonance associated with the spectral feature in panel (

**b**). The color corresponds to the ${E}_{y}$ component (real part) and the arrows correspond to the magnetic field distribution. The eigenmode is characterized by a residual electric dipole moment due to counteracting contributions from the inner gap vs. the outer gaps. The magnetic field circulation gives rise to a strong toroidal dipole moment, ${T}_{y}$. (

**d**) Power scattered from each multipole. The toroidal dipole and magnetic quadrupole moments are dominant at the resonant frequency. However, their scattered fields cancel out, since ${E}_{\mathrm{sca}}^{{T}_{y}}=-{E}_{\mathrm{sca}}^{{Q}_{xz}^{m}}$ exactly. Thus, the response is dictated by the residual electric dipole moment, ${p}_{y}$. (

**e**) Comparison of the reflection coefficient calculated from the full-wave simulation with those reconstructed from the multipole moments. The response can be described fairly accurately when only the electric dipole moment is considered.

**Figure 3.**(

**a**) Tuning the central gap (${g}_{1}$), while keeping other gaps (${g}_{2}$) constant at 0.5 mm. This controls the residual electric dipole moment and consequently the radiative strength of the resonance. (

**b**) Varying the outer radius for ${R}_{\mathrm{inn}}={R}_{\mathrm{out}}-w$ with $w=2$ mm. As the radius increases, the non-resonant (background) electric dipole moment changes and with it the Fano lineshape. For ${R}_{\mathrm{out}}$ values of 7.6, 7.7, and 7.8 mm, the meta-atoms are touching; this modifies the residual electric dipole moment, as the outer gaps of each meta-atom begin to merge with those of the neighboring one.

**Figure 4.**Simulations of the 3D-printed conductive unit cells within the WR-187 rectangular waveguide. The reflection ($R=|{S}_{11}{|}^{2}$), transmission ($T=|{S}_{21}{|}^{2}$) and absorption ($A=1-|{S}_{11}{|}^{2}-{\left|{S}_{21}\right|}^{2}$) power coefficients are plotted for different conductivities: (

**a**) $\sigma ={10}^{5}$ S/m, (

**b**) $\sigma =5\times {10}^{4}$ S/m, (

**c**) $\sigma ={10}^{4}$ S/m and (

**d**) $\sigma =5\times {10}^{3}$ S/m. As the conductivity is decreased, the spectral feature becomes broader and the absorption on resonance increases up to a maximum of 0.5 before starting to decrease again.

**Figure 5.**Measurement of the 3D-printed conductive unit cells within the WR-187 rectangular waveguide. The reflection ($R=|{S}_{11}{|}^{2}$) and transmission ($T=|{S}_{21}{|}^{2}$) power coefficients are plotted in the frequency range of 4.5–5.5 GHz.

**Table 1.**Quality factors for the cases depicted in Figure 3a. ${g}_{2}$ is held constant at 0.5 mm. They have been calculated using the complex eigenfrequency through $Q={\omega}^{\prime}/\left(2{\omega}^{\u2033}\right)$.

Central Gap ${\mathit{g}}_{1}$ (mm) | Resonant Frequency (GHz) | ${\mathit{Q}}_{\mathbf{tot}}$ |
---|---|---|

0.3 | 8.02 | 446 |

0.4 | 8.37 | 14,000 |

0.5 | 8.71 | 2,114 |

0.6 | 8.94 | 517 |

0.7 | 9.14 | 260 |

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## Share and Cite

**MDPI and ACS Style**

Tsilipakos, O.; Viskadourakis, Z.; Tasolamprou, A.C.; Zografopoulos, D.C.; Kafesaki, M.; Kenanakis, G.; Economou, E.N. Meta-Atoms with Toroidal Topology for Strongly Resonant Responses. *Micromachines* **2023**, *14*, 468.
https://doi.org/10.3390/mi14020468

**AMA Style**

Tsilipakos O, Viskadourakis Z, Tasolamprou AC, Zografopoulos DC, Kafesaki M, Kenanakis G, Economou EN. Meta-Atoms with Toroidal Topology for Strongly Resonant Responses. *Micromachines*. 2023; 14(2):468.
https://doi.org/10.3390/mi14020468

**Chicago/Turabian Style**

Tsilipakos, Odysseas, Zacharias Viskadourakis, Anna C. Tasolamprou, Dimitrios C. Zografopoulos, Maria Kafesaki, George Kenanakis, and Eleftherios N. Economou. 2023. "Meta-Atoms with Toroidal Topology for Strongly Resonant Responses" *Micromachines* 14, no. 2: 468.
https://doi.org/10.3390/mi14020468