Enhanced toroidal localized spoof surface plasmons in homolateral double-split ring resonators

Bo Sun; Yingying Yu; Wenxing Yang

doi:10.1364/OE.395068

1. Introduction

The toroidal dipole was first proposed by Zel’dovich in 1957 [1]. In contrast to the common dipole, toroidal dipoles and higher toroidal multipoles constitute a third independent family of elementary electromagnetic sources [2]. The toroidal dipole is created by induced magnetic dipoles connecting in a head-to-tail manner [3]. It shows growing interest due to their unusual electromagnetic properties. Many successful designs were demonstrated in microwave frequency range [4–7]. A variety of appealing applications are realized through toroidal dipoles including lasing spasers [8], sensing [9] and absorbers [10].

Metamaterials, which have become a growing interest in recent years, have the advantage of being artificially adjusted by tailoring the geometry of the unit cells [11–14]. Metamaterial structures can also be used to induce a toroidal dipole. In 2007, Marinov et al., first theoretically proposed toroidal metamaterials [15]. In 2010, toroidal metamaterials based on the arranging four three-dimensional split rings was first experimentally demonstrated in microwave region [16]. Subsequently, many toroidal metamaterial structures were proposed to excite toroidal resonance, such as a split ring resonator [17,18], double-bar [19], oligomer nanocavities [20] and other structures [21–23].

Surface plasmons (SPs) have been a hot spot due to the features of subwavelength field confinement and field enhancement at optical frequencies. SPs can be categorized into two types: localized surface plasmons (LSPs) and surface plasmon polaritons (SPPs). SPPs are electromagnetic modes that arise from the coupling of light with collective oscillations of electrons propagating along an interface between a dielectric and a metal. LSPs are resonance modes supported by subwavelength metal particles. Different from SPPs, LSPs are non-propagating excitations, which arise from the scattering effect of subwavelength metal particle in an oscillating electromagnetic field. To transfer the capabilities of LSPs and SPPs to low frequencies, the concept of localized spoof surface plasmons (LSSPs) and spoof surface plasmon polaritons (spoof-SPPs) were developed. The spatial confinements and the dispersion properties of LSSPs and spoof-SPPs are similar to those of LSPs and SPPs. In the terahertz and microwave bands, plasmonic metamaterials have been proposed to produce the LSSPs or spoof-SPPs [24–28]. Apart from conventional LSSPs, many novel phenomena have been proposed and experimentally verified for LSSPs, such as the unusual magnetic and electric LSSPs resonance in metamaterials. As a part of the family of magnetic and electric LSSPs resonance, toroidal LSSPs can also be excited in metamaterials [29–31]. The toroidal metamaterial is usually based on a single-split ring resonator, which hinders the further enhancement of toroidal resonance, especially in toroidal LSSPs. In traditional toroidal LSSPs, the toroidal dipole is imperfect due to the source (the toroidal dipole shows a gap in the place of the source). Consequently enhanced and perfect toroidal LSSPs are worthy of further in-depth investigation.

For this purpose, a novel homolateral double-split ring resonator metamaterial is proposed in this paper. We show enhanced and perfect toroidal LSSPs. By introducing a new split in the single-split ring resonator, the magnetic field in the resonator is locally modified, leading to the enhancement of the magnetic field. The rest of this paper is organized as follows: first, the physical model of the toroidal metamaterial is given, and the reflection spectra are discussed. Second, the comparison between single- and double-split ring resonators is discussed. We qualitatively understand the advantage of the proposed structure. Third, to analyze the toroidal dipole resonance quantitatively, radiated powers are calculated, and the results show that the toroidal dipole can be excited in the metamaterial. Finally, we experimentally demonstrate that enhanced toroidal LSSPs are sensitive to the background medium, and the sensitivity is higher than the metamaterial based on the single-split ring resonators. The potential of this metamaterial as plasmonic sensing has been evaluated.

2. Model structure design

As shown in Fig. 1, the geometrical configuration of the proposed structure is exhibited. The structure can be characterized by twelve homolateral double-split ring resonators with a height H=3mm and w=1.6mm. The inner radius R₁=6mm, the outer radius R₂=15mm, the length L=2.6mm, and the width t=1.2mm. By using a standard printed circuit board (PCB) fabrication, the structure of copper (yellow region) with a thickness of 0.035mm and conductivity of 5.7×10⁷S m⁻¹ is printed on a 3mm thick FR4 dielectric substrate with a relative permittivity of 4.3×(1 + 0.025i). The top-layer structure is connected to the bottom-layer structure by through-hole, the diameter of through-hole is 0.8mm. Number simulations are performed by using a commercial software package CST Microwave Studio. In the simulation, a discrete port is introduced at one gap of one resonator to excite the resonance. Open boundary conditions are applied to the proposed structure along the X, Y, and Z directions.

Fig. 1. (a) Geometrical configuration of the proposed structure. (b) Fabricated sample of the proposed structure.

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3. Results and analysis

Figure 2(b) shows the simulated and measured reflection spectral curves for the proposed structure. In the measurement, we performed all spectrum measurements using vector network analyzer (R&S ZVB8), as described in Fig. 2(a). One monopole antenna is placed at the inner gap of SRR to excite the toroidal dipole. The antenna is capacitively coupled to the resonator allowing us to excite the electric field strength in vicinity of the gap of SRR. The monopole antenna is connected with the vector network analyzer, which measures the reflection from the structure. From Fig. 2(b), we can find that the structure shows two resonances, which are located at f₁=4.04 and f₂=5.44 GHz, respectively. Compared to the simulated resonance peaks located at 3.95 and 5.51 GHz, the difference between measurement and simulation results is observed. It is due to the limitation of fabrication process and our measurement setup. To reveal the nature of LSSPs in the proposed structure, the dispersion relation of spoof SPPs on the single and double-split ring resonators structure are shown in Fig. 2(c). It can be seen that they have asymptotic frequencies, at which the group velocity becomes zero. Moreover, the high cutoff frequency of the double-split ring resonators shifts to a lower value. This shift is due to the influence of the field distribution due to the mixed coupling.

Fig. 2. (a) Measurement setups of the structure. (b) Simulated and measured reflection spectral curves for the proposed structure. (c) Dispersion curves of the spoof SPPs on the single- and double-split ring resonators structure.

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In order to understand the mechanism of the two resonances, Figs. 3(a) and 3(b) show the cross-profile of the simulated magnetic field on an x-y plane in the middle of the structure. The magnetic field is only excited at the resonator of source at f₁. However, the magnetic field generated by the surface current loop on double-split ring resonators are connected in a head-to-tail way at f₂, which is an important signature of a toroidal dipole, demonstrating the existence of a toroidal dipole T in z direction. Figures 3(c) and 3(d) show the electric field distribution on the plane 1 mm above the structure at 3.95 and 5.51 GHz. From Fig. 3(d), we can find that the electric field varies between positive and negative, which looks like a dipole electromagnetic field radiation pattern. Moreover, the dipole locates in the center of the structure, which is an obvious feature of the toroidal LSSPs resonance.

Fig. 3. Magnetic field distributions at (a) f₁=3.95 and (b) f₂=5.51 GHz. Electric field distributions at (c) f₁=3.95 and (d) f₂=5.51 GHz.

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To further demonstrate the advantages of the structure, the structure of the single-split ring resonator is used for comparison. Figure 4 shows the simulated reflection spectral curves for the single- and double-split ring resonators structure. As can be seen from Fig. 4, the toroidal dipole can be excited for single-split ring resonators at f₃=4.268 GHz. The toroidal dipole cannot be excited at 5.676 GHz. Compared with the double-split ring resonators, the red shift is due to the increase of physical length with the decrease in the number of splits. Furthermore, the Q factor is defined as Q = f_r/Δf, where f_r is the resonance frequency and Δf is the full width at half maxima. The Q factors are 19.45 and 18.69 for double- and single-split ring resonators, respectively. The FoM defined as the product of the Q factor and resonance intensity (δI), are 131.75 and 48.67, respectively. The measured FoMs are 84.6 and 51.7 for double- and single-split ring resonators [29], respectively. Although the Q factor does not significantly increase, it shows a stronger resonance intensity and results in a higher FoM (near 3 and 1.6 times higher than that of the single-split ring resonators in the simulation and experiment, respectively). Figure 5(a) shows the cross-profile of simulated magnetic field on x-y plane in the middle of the single-split ring resonators structure at f₃=4.268 GHz. From the figure, we can find that the toroidal dipole shows a gap in the place of the source. Meanwhile, compared with Fig. 3(b), the magnetic field intensity shows a significant decrease in the single-split ring resonators structure. To compare the magnetic field intensity between the single- and double-split ring resonators, the magnetic field intensity curves on a different plane in the middle of the structure are obtained. As shown in Fig. 5, the magnetic field strength of the double-split ring resonators is enhanced with varying degrees, and results in the enhancement of the toroidal dipole.

Fig. 4. Simulated reflection spectral curves for the single- and double-split ring resonators structure.

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Fig. 5. (a) Magnetic field distribution of single-split ring resonators at f₃=4.268 GHz. (b-d) The magnetic field intensity curves on a different plane in the middle of the single- and double-split ring resonators structure.

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To understand and clarify the enhanced magnetic field phenomena, we analyze the role of the double-split ring resonator. To excite the toroidal LSSPs, a monopole antenna is introduced at one gap of one resonator to excite the resonance. Figures 6(a) and 6(b) show the equivalent circuits of single- and double-split ring resonator, respectively. From Fig. 6(a), we can find that the resonance cannot be excited when the source is placed at the gap. It can be considered as the connection of capacitance. From Fig. 5(a), we can find that there is a gap where the source is placed in the toroidal LSSPs. In the single-split ring resonator structure, the resonator of source serves as a short electric antenna. The antenna is coupled to the other resonators allowing us to excite the electromagnetic field.

Fig. 6. Equivalent circuits of (a) single-split ring resonator and (b) double-split ring resonator. (c) Simulated reflection spectral curves for the single- and double- split ring resonator.

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But in the double-split ring resonator as shown in Fig. 6(b), another capacitance can be used to excite the resonance. The double-split ring resonator can be considered as LC resonance mode. The resonance frequency is defined as f_LC= (LC)^−1/2/2π, where C and L are the capacitance and inductance, respectively. The C is determined by the geometry of the gap, C≈8pF. The L is determined by the size of the ring, L≈167 pH. The resonance frequency is calculated, f_LC≈4.356 GHz. Therefore, the strong resonance also can be excited at the resonator of source. From Fig. 5(b), we can find that the magnetic field shows a significant increase in the source. To further prove the increase, the simulated reflection spectral curves for the single- and double-split ring resonator are shown in Fig. 6(c). The resonance can only be excited in the double-split ring resonator. It shows one resonance, which is located at 4.402 GHz. Compared to the calculated resonance frequency, there is a little deviation due to the approximation.

The double-split ring resonator can be considered as LC resonance mode with single capacitance, the capacitance is the series of two splits. The proposed structure can be considered as a resonator at source and the other resonators as shown in Fig. 7, where C and L are the self-capacitance and self-inductance, respectively. C_m and L_mrepresent the mutual capacitance and mutual inductance. From Fig. 3(a), the power cannot realize the energy transfer (consider as open circuit) in the structure. However, from Fig. 3(b), the power can realize the energy transfer (consider as short circuit) in the structure. The resonance frequency can be written as [32]

(1)$${f_1} = \frac{1}{{2\pi \sqrt {(L + {L_m}/a)(L + {C_m}/a)} }}$$

(2)$${f_2} = \frac{1}{{2\pi \sqrt {(L - {L_m})(L - {C_m})} }}$$

From Fig. 3(a), we can find that half of the resonators in the right side play the role in blocking energy transfer. Therefore, in the resonant frequency f₁, we introduce the coefficient a=2. The C_m and L_m can be calculated with those given resonance frequencies as C_m=2.29 pF, L_m=20.97pF or C_m=1 pF, L_m=47.78pF. The mixed coupling coefficient k can be found to be

(3)$$k = {k_m} + {k_e} = \frac{{{L_m}}}{L} + \frac{{{C_m}}}{C}$$

The mixed coupling coefficient results from the superposition of the electric and magnetic coupling. It can be calculated as k=0.412.

Fig. 7. Equivalent circuits of double-split ring resonators.

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In the single-split ring resonator structure, we can find that the toroidal dipole is excited at resonance frequency f₃=4.268 GHz from Fig. 4, which is very close to the resonance frequency of double-split ring when the source is placed at one gap (4.402 GHz as shown in Fig. 6(c)). Therefore, the toroidal dipole results from the magnetic inductive coupling [33,34] that like the traditional propagation of magnetoinductive waves in arrays of split ring resonators [35]. The mixed coupling cannot be excited and results in the decrease of magnetic field strength.

In order to analyze the toroidal dipole resonance quantitatively, the radiated powers of the toroidal dipole and electric- and magnetic-multipoles are calculated. The radiation power magnitude of these multipoles can be described as follows [36,37]:

(4)$$I = \frac{{2{\omega ^4}}}{{3{c^3}}}{\left| P \right|^2} + \frac{{2{\omega ^4}}}{{3{c^3}}}{\left| M \right|^2} + \frac{{4{\omega ^5}}}{{3{c^4}}}{\left| {P \cdot T} \right|^2} + \frac{{2{\omega ^6}}}{{3{c^5}}}{\left| T \right|^2} + \frac{{{\omega ^6}}}{{5{c^5}}}\sum {{{\left| {{Q_{a\beta }}} \right|}^2}} + \frac{{{\omega ^6}}}{{40{c^5}}}\sum {{{\left| {{M_{a\beta }}} \right|}^2}} + O\left( {\frac{1}{{{c^5}}}} \right)$$

where P, M, T, Q_αβ, and M_αβ correspond to the electric, magnetic, toroidal dipole, electric and magnetic quadrupole. ω is the angular frequency, c is the speed of light, and α, β=x, y, z. In Fig. 8(a), the magnitudes of these multipole moments are calculated from surface current extracted from simulation. These multipoles can be described as follows [36,37]:

(5)$$\hbox{electric dipole}:\qquad \qquad \qquad P = \frac{1}{{i\omega }}\int {J{d^3}r}$$

(6)$$\hbox{magnetic dipole}:\qquad \qquad \qquad M = \frac{1}{{2c}}\int {(r \times j){d^3}r}$$

(7)$$\hbox{toroidal dipole}:\qquad \qquad \qquad T = \frac{1}{{10c}}\int {[(r \cdot j)r - 2{r^2}j]{d^3}r}$$

(8)$$\hbox{electric quadrupole}:\qquad \quad {Q_{\alpha \beta }} = \frac{1}{{i\omega }}\int {[{r_a}{j_\beta } + {r_\beta }{j_a} - \frac{2}{3}(r \cdot j)]{d^3}r}$$

(9)$$\hbox{magnetic quadrupole}:\qquad \quad {M_{\alpha \beta }} = \frac{1}{{3c }}\int {[({r}\times{j})_{\alpha}r_\beta + ({r}\times{j})_{\beta}r_\alpha]{d^3}r}$$

We need to point out that if there are any other multipoles, contributed to the reflection spectra, other than the toroidal dipole. From Fig. 8(a), we can find that at f₂, the toroidal dipole T_z has the largest intensity, which is an order larger than that of the electric dipole P_x. The x component of electric dipole comes from the asymmetry of the toroidal dipole, the surface current of right side is larger than that of left side due to the source located in the right side, hence a net vector resultant along x direction is produced causing P_x, its intensity is about 10 times weaker than the predominant toroidal dipole T_z. For the magnetic dipole M_y, the distribution of closed loop makes the scattering field cancel each other, and results in the weak far-field radiation. Its intensity is about 10² times weaker than the predominant toroidal dipole T_z. Meantime, the electric quadrupole Q_e and magnetic quadrupole Q_m also have no obvious contribution to the far-field radiation. From the figure, one can see that the intensity of the toroidal dipole reaches the maximum value at 5.65 GHz, and the intensity of electric dipole shows a decrease because of the weakened of the asymmetry of the toroidal dipole. Figure 8(b) shows the cross-profile of the simulated magnetic field on x-y plane in the middle of the structure at 5.65 GHz. From the figure, one can see that compared with Fig. 3(b), although the resonance intensity shows a decrease, the higher symmetry of the toroidal dipole also shows a higher intensity of the toroidal dipole. Meantime, the higher symmetry decreases the intensity of P_x (shown in Fig. 8(a)).

Next, the effect of location of source on the toroidal dipole is analyzed. Figure 2(b) shows the reflection spectra with the source located at the inner gap. When the source is located at the outer gap, the reflection spectra is shown in Fig. 9(a). The resonances are located at 3.9 GHz and 5.588 GHz, respectively, and the intensity of resonance shows a significant decrease. The Q factor and FoM are 21.56 and 95.38, respectively. The Q factor and FoM are larger than that of single-split ring resonators. Therefore, no matter where the location of source is, the magnetic field can be significantly enhanced in the double-split ring resonators, and it shows the higher the Q factor and FoM. Meantime, compared with Fig. 2(b), when the source is placed at the outer gap, the FoM shows a significant decrease. The weakness of resonance is due to the increase of relative distance between split ring resonators.

Fig. 8. (a) Dispersion of radiation power for various multipole moments include in the metamaterial. The black shadow is for the eye guide. (b) Magnetic field distribution of double-split ring resonators metamaterial at 5.65 GHz.

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Fig. 9. (a) When the source is put at the outer gap, simulated and measured reflection spectral curves for the proposed structure. (b) Simulated results of the reflection spectral curves with different D of the structure.

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We also study the impact of the structure to the toroidal LSSPs resonance. Figure 9(b) shows the simulated reflection spectra as a function of spacing (D) between two splits. One can see that the toroidal LSSPs shows a blue shift with increasing D of the structure. When the D=2 mm, the toroidal LSSPs resonance first appears at the frequency of 5.292 GHz. When the D=5 mm, the toroidal LSSPs resonance is blueshifted by 0.328 GHz. Moreover, the Q factor and FoM show a significant increase. Increasing the D of the structure from 2 mm to 5 mm, the Q factor is increased from 20.66 to 26.48, the FoM is increased from 68 to 452.51. Therefore, the toroidal LSSPs resonance can be further enhanced by increasing D.

4. Application

The toroidal LSSPs is applicable for practical application including sensor. We have found that the toroidal dipole of double-split ring resonators is sensitive to the background medium. Here a dielectric pad (polylactic acid (PLA) n=1.65) is put on the surface of the structure. Changing the refractive index of the background medium is demonstrated experimentally. When the structure is covered by air and PLA, the measured response spectra are shown in Fig. 10. As the structure is covered with the PLA, the resonance frequency f₁ shifts from 4.04 GHz to 3.75 GHz, and the resonance frequency f₂ shifts from 5.44 GHz to 4.82 GHz. It means that we can obtain a 0.29 GHz shift for the resonance frequency f₁ and a 0.62 GHz shift for the resonance frequency f₂. The sensitivity is defined as S=Δf/Δn. Here the sensitivities for the resonance frequency f₁ and f₂ are 0.45 GHz/RIU and 0.95 GHz/RIU, respectively. One can see that the resonance frequency f₂ shows a higher sensitivity due to the local magnetic field enhancement phenomenon occurring in the double-split ring resonators. Compared with the toroidal LSSPs resonance based on the single-split ring resonator [29], the sensitivity shows a significant increase because the magnetic field is enhanced.

Fig. 10. The measured reflection response of the proposed structure covered by different dielectric pads.

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5. Conclusion

In this paper, we have proposed a toroidal metamaterial based on the homolateral double-split ring resonator. The simulation and experiment results demonstrate that a toroidal LSSPs can be excited. In order to understand the mechanism of the proposed structure, the magnetic field distribution has been discussed. The comparison between single- and double-split ring resonators were discussed. The results show that the intensity of magnetic field is enhanced by introducing a new split. To understand and clarify the enhanced magnetic field phenomena, we outline a simple theoretical analysis. Analysis shows the mixed coupling can be introduced by adding a new split. The radiated powers were calculated to analyze the toroidal dipole resonance quantitatively. In addition, the effect of location of source and spacing between two splits on the resonance intensity were also discussed. The toroidal LSSPs resonance can be further enhanced by increasing D. The FoM can be increased by nearly 10 times by increasing D. The designed toroidal metamaterial structure can be used as plasmonic sensors and find potential applications in the terahertz and microwave frequencies.

Funding

National Natural Science Foundation of China (11774054, 11775190); Hubei Provincial Department of Education (Q20181302); Natural Science Foundation of Jiangsu Province (BK20161410).

Disclosures

The authors declare no conflicts of interest

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Enhanced toroidal localized spoof surface plasmons in homolateral double-split ring resonators

Abstract

1. Introduction

2. Model structure design

3. Results and analysis

4. Application

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (10)

Equations (9)

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