# Multipoles of Even/Odd Split-Ring Resonators

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multipolar Decomposition

**Figure 1.**(

**a**) Dimensions of the 1-SRR: L = 700 nm, W = 50 nm, T = 50 nm, and G = 200 nm (

**b**) Case 1: electric field in-plane and along the gap, magnetic field out-of-plane (

**c**) Case 2: electric field in-plane and normal to the gap, magnetic field out-of-plane (c) Case 3: electric and magnetic fields in-plane, along the gap and normal to it, respectively. All the gold SRRs are described using the Drude model with a plasma frequency (ω

_{p}= 1.367 × 10

^{16}rad/s) and collision frequency (ω

_{c}= 6.478 × 10

^{13}rad/s) based on values from [31].

**N**

_{nm}and

**M**

_{nm}[26,27]

_{nm}a scaling coefficient. Since the VSHs are orthogonal to each other, we can compute the expansion coefficients a

_{nm}and b

_{nm}by projection and appropriate normalization.

**p**, magnetic dipole,

**m**, and electric quadrupole,

**Q**). Finally, we can use these to decompose the scattering cross-section

_{x}, and, p

_{y}, magnetic dipole, m

_{z}, and electric quadrupole, Q

_{xy}.

## 3. From 1-SRR to 4-SRR

_{x}, cannot be excited in cases 1 and 3. As can be seen in Figure 2, in all three cases there are two resonances that are always excited (f

_{1}= 51 THz, f

_{3}= 151 THz) while in the second case there is also an additional resonance (f

_{2}= 106 THz). Resonance 2 is due to the excitation of the electric dipole, p

_{x}. Resonances 1 and 3 can be excited either by the electric field along the gap, or the magnetic field normal to the plane of SRR, or a combination of both, and yield both an electric dipole, p

_{y}, and a magnetic dipole, m

_{z}, response. In addition, resonance 3 shows an in-plane quadrupole, Q

_{xy}, response. Surprisingly, when both fields are present with the right polarization (case 1), their effects do cumulate for resonance 1 but not for resonance 3. This would seem to indicate that case 3 is the most favorable to multipole overlap but, in a planar configuration, the magnetic dipole, m

_{z}, does not radiate in the direction of propagation, thus making case 1 the preferred illumination scenario.

_{x}, cannot be excited in cases 1 and 3, the electric dipole, p

_{y}, cannot be excited in case 2, and the magnetic dipole, m

_{z}, and electric quadrupole, Q

_{xy}, cannot be excited in case 3. As a result, there can only be an overlap between the electric dipole, p

_{y}, and the magnetic dipole, m

_{z}, in case 1. However, as can be seen from Figure 3, both modes are only weakly excited in this case (f

_{1}= 98 THz). Unlike the 1-SRR, in case 2, the 2-SRR has a small overlap between the electric dipole, p

_{x}, and the magnetic dipole, m

_{z}, (f

_{1}= 98 THz, f

_{2}= 106 THz), which could prove useful in the design of novel metamaterials. This overlap is not due to symmetry but rather to the specific dimensions of the 2-SRR considered here and, hence, is accidental. For the 2-SRR, case 2 is the preferred illumination scenario.

**Figure 2.**Multipolar decomposition of the scattering cross-section, for a 1-SRR between 30 THz and 200 THz, into the lowest order multipoles; (

**a**) Case 1: electric field in-plane and along the gap, magnetic field out-of-plane; two resonances are observed at f

_{1}= 51 THz and f

_{3}= 151 THz (

**b**) Case 2: electric field in-plane and normal to the gap, magnetic field out-of-plane; three resonances are observed, those observed in case 1 plus another one at f

_{2}= 106 THz due to the excitation of the electric dipole, p

_{x}(

**c**) Case 3: electric and magnetic fields in-plane, along the gap and normal to it, respectively; two resonances are observed, just as in Case 1. The resonances excited in Case 1 and 3 yield both an electric dipole, p

_{y}, and a magnetic dipole, m

_{z}, response. Resonance 3 also shows an in-plane quadrupole, Q

_{xy}, response in all cases. Insets show the real part of E

_{z}at the corresponding resonance frequencies using a rainbow colormap (min in blue, zero in green, and max in red).

_{1}= 121 THz, f

_{2}= 219 THz) and, similar to the 1-SRR, when both fields are present with the right polarization (case 1), their effects do cumulate for resonance 1 but not for resonance 2. Furthermore, just as for the 1-SRR, the planar configuration of case 3 is not suitable for multipole overlap as it suffers from a lack of radiation along the direction of propagation. A notable difference for the 3-SRR is that resonance 2 has a much less prominent electric quadrupole, Q

_{xy}, than the 1-SRR. Overall, the 3-SRR is very similar to the 1-SRR, e.g., case 1 is the preferred illumination scenario, with the only important difference being that its resonances occur at higher frequencies.

**Figure 3.**Multipolar decomposition of the scattering cross-section, for a 2-SRR between 30 THz and 200 THz, into the lowest order multipoles; (

**a**) Case 1: electric field in-plane and along the gaps, magnetic field out-of-plane; one resonance is observed at f

_{1}= 98 THz. Weak excitation of electric dipole, p

_{y}, and the magnetic dipole, m

_{z}observed. (

**b**) Case 2: electric field in-plane and normal to the gaps, magnetic field out-of-plane; two resonances are observed, that observed in case 1 plus another one at f

_{2}= 106 THz. Small overlap between the electric dipole, p

_{x}, and the magnetic dipole, m

_{z}observed. (

**c**) Case 3: electric and magnetic fields in-plane, along the gaps and normal to them, respectively; no resonances are observed. Insets show the real part of E

_{z}at the corresponding resonance frequencies using a rainbow colormap (min in blue, zero in green, and max in red).

_{x}, cannot be excited in case 3, the electric dipole, p

_{y}, cannot be excited in case 2, the magnetic dipole, m

_{z}, and the electric quadrupole, Q

_{xy}, cannot be excited in case 3. As a result, there can only be an overlap between the electric dipole, p

_{x}, and the magnetic dipole, m

_{z}, in case 2. However, as can be seen from Figure 5, only the magnetic dipole, m

_{z}, is strongly excited in this case (f

_{1}=203 THz). For the 4-SRR, there is thus no preferred illumination scenario.

**Figure 4.**Multipolar decomposition of the scattering cross-section, for a 3-SRR between 30 THz and 350 THz, into the lowest order multipoles; (

**a**) Case 1: electric field in-plane and along the single gap, magnetic field out-of-plane; two resonances are observed at f

_{1}= 121 THz and f

_{2}= 219 THz (

**b**) Case 2: electric field in-plane and normal to the single gap, magnetic field out-of-plane; three resonances are observed, those observed in case 1 plus another one at f

_{3}= 297 THz (

**c**) Case 3: electric and magnetic fields in-plane, along the single gap and normal to it, respectively; two resonances are observed, just as in case 1. The resonances excited in Case 1 and 3 yield both an electric dipole, p

_{y}, and a magnetic dipole, m

_{z}, response similar to the 1-SRR. Insets show the real part of E

_{z}at the corresponding resonance frequencies using a rainbow colormap (min in blue, zero in green, and max in red).

**Figure 5.**Multipolar decomposition of the scattering cross-section, for a 4-SRR between 30 THz and 350 THz, into the lowest order multipoles; (

**a**) Case 2: electric field in-plane, magnetic field out-of-plane; one resonance is observed at f

_{1}= 203 THz. Strong magnetic dipole, m

_{z}, excitation is observed. (

**b**) Case 3: electric and magnetic fields in-plane; one resonance is observed at f

_{2}= 230 THz. Only electric dipole, p

_{y}, excitation is observed. Insets show the real part of E

_{z}at the corresponding resonance frequencies using a rainbow colormap (min in blue, zero in green, and max in red).

## 4. Conclusion

## Acknowledgments

## Author Contributions

## Conflict of Interest

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**MDPI and ACS Style**

Chen, A.; Kodigala, A.; Lepetit, T.; Kanté, B.
Multipoles of Even/Odd Split-Ring Resonators. *Photonics* **2015**, *2*, 883-892.
https://doi.org/10.3390/photonics2030883

**AMA Style**

Chen A, Kodigala A, Lepetit T, Kanté B.
Multipoles of Even/Odd Split-Ring Resonators. *Photonics*. 2015; 2(3):883-892.
https://doi.org/10.3390/photonics2030883

**Chicago/Turabian Style**

Chen, Andrew, Ashok Kodigala, Thomas Lepetit, and Boubacar Kanté.
2015. "Multipoles of Even/Odd Split-Ring Resonators" *Photonics* 2, no. 3: 883-892.
https://doi.org/10.3390/photonics2030883