1. Introduction

Metallic split-ring resonators (SRRs) have recently gained much research interest as promising building blocks for the realization of negative permeability for negative refractive index materials [1–18]. In the microwave regime, negative refraction has been demonstrated with double SRRs (DSRRs) in combination with thin wires [2]. Negative permeabilities of such structures were achieved near the resonance of DSRRs. The role of the inner ring has been discussed as a reduction of the resonance frequency by increasing the gap capacitance of the outer ring [3]. Most investigations of SRRs in the optical wavelength range have focused on single SRRs [10–14]. Recently, it has been demonstrated that the resonances of a DSRR in the near infrared regime are a linear superposition of those of its constitute outer ring and inner ring [15]. In this paper, we analyze DSRRs as complex structured plasmonic systems from the viewpoint of resonance hybridization. The resonances in different coupling strength regimes are discussed with varied inner-ring sizes. This work underlines the need to consider coupling for the design and understanding of structured metamaterials. This can become critically important, especially as designs are increased in structural complexity, for example when extending into bulk regimes.

In near infrared and visible range, the resonances of SRRs excited by the external light field have been mostly modeled by a quasi-static LC circuit description [8–14]. Recently, the plasmon hybridization concept, which is an electromagnetic analogy of molecular orbital theory, has been successfully applied in describing the responses of complex nanostructures and stacked metamaterials [19–21]. We find that the response of a DSRR can be regarded as an interaction between the plasmons of the constitute inner ring and outer ring. This is similar to the optical response of nanoshells and nanorice [19, 20]. The concept is illustrated in Fig. 1 for a closely arranged DSRR. The resonances of the outer and inner rings alone are excited at the frequencies ω₁ and ω₂, which correspond to the plasmonic eigenmodes |ω₁〉 and |ω₂〉, respectively. The interaction between these plasmons results in a new coupled mode pair due to the plasmon hybridization. The modes at lower and higher energies correspond to an anti-symmetric plasmon mode |ω_-〉 and a symmetric plasmon mode |ω₊〉, respectively.

2. Numerical analysis

A schematic diagram of a DSRR is shown in Fig. 2. In order to investigate the interaction between the outer and inner rings in DSRRs, the outer-ring geometry is kept constant, whereas the geometry of the inner ring is varied. As the plasmonic eigenmode of an SRR depends strongly on its geometric length (L=l_x+2l_y-2w), the varied geometry of the inner ring leads to a resonant energy shift [17, 18]. A finite-integration time-domain algorithm (CST microwave studio) was used to simulate the resonant response of SRRs due to its ability to provide reliable results [11, 15]. The DSRRs are composed of gold and arranged in a square lattice on a quartz substrate (n=1.45). The permittivity of the gold is described by a Drude model [22, 23]. The incident light with the polarization parallel to the gap-bearing side of the SRR is used to excite the plasmon eigenmodes of SRRs.

 

Fig. 1. The energy diagram describes the plasmon hybridization in a DSRR. The interaction between the plasmonic modes (|ω₁〉 and |ω₂〉) of its constitute outer ring and inner rings results in a new coupled mode pair: an anti-symmetric mode |ω_-〉 and a symmetric mode |ω₊〉.

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Fig. 2. Schematic diagram of a DSRR. t denotes the thickness of the SRR, w the width of the wire, l_ox (l_ix) the length of the wire parallel to the gap bearing side in the outer (inner) ring, and l_oy (l_iy) the length of the wire perpendicular to the gap bearing side in the outer (inner) ring.

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2.1. Hybridization strength as a function of the inner-ring geometry

Figure 3(a)–(c) show the simulated reflectance spectra of DSRRs with various inner-ring geometries. The spectra of outer and inner rings alone are also given for comparison. In the spectra, only the resonances below 700 meV correspond to the fundamental plasmonic eigenmodes (namely, LC resonances) of SRRs, whereas the resonances above 700 meV correspond to the higher-order plasmonic modes of SRRs [17]. In order to investigate the LC resonance properties of DSRRs, we focus our discussion on the resonance modes below 700 meV. For explanation of the simulation results, we use the following notation for our discussion: the plasmonic modes of the outer and inner rings alone are denoted by |ω₁〉 and |ω₂〉, respectively. Note that |ω₁〉 is unchanged in each plot due to the fixed geometry, whereas |ω2〉 is shifted to higher energies with a decreased ring length. The new plasmonic eigenmodes of DSRRs at the lower and higher energies are denoted by |ω_-〉 and |ω₊〉, respectively.

 

Fig. 3. Simulated reflectance spectra for DSRR (dotted blue curves) compared to the outer SRR alone (dashed red curves) and the inner SRR alone (solid black curves) for various inner-ring sizes. |ω₁〉 and |ω₂〉 present the fundamental plasmonic eigenmodes of the outer and inner rings alone, respectively. For DSRRs, the lower energy anti-symmetric mode and higher energy symmetric modes are denoted as |ω_-〉 and |ω₊〉, respectively. In the simulations, the outer-ring size (l_ox=410 nm, l_oy=380 nm, w=50 nm, t=20 nm) and the array periods (p_x=p_y=550 nm) are kept constant. The inner-ring geometry is varied from (a) l_ix=270 nm and l_iy=290 nm; (b) l_ix=210 nm and l_iy=230 nm; and (c) l_ix=150 nm and l_iy=170 nm. The corresponding structure and plasmon hybridization diagrams are shown adjacent to the spectra. The individual spectra are shifted vertically for clarity.

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In Fig. 3(a), the outer and inner rings are closely arranged in the DSRR and the spacing between them is small (20 nm). Since the entire wire length difference between the outer and inner rings is 320 nm, the plasmonic eigenmodes of the outer and inner rings alone are closely spaced in the spectrum (at 123 meV distance). Due to the strong coupling between them, the plasmon hybridization induces a large plasmon energy shift of 83 meV (plasmon energy shift ΔE=‖ω ₊〉-|ω _-〉|-‖ω ₂〉-|ω ₁〉|). In this strong coupling regime, the strong interaction between the higher-order plasmonic modes of the outer and inner rings shifts the symmetric mode |ω₊〉 to an energy level lower than |ω₂〉. Fig. 3(b) shows the reflectance spectrum of a DSRR with reduced inner-ring size. The distance and entire wire length difference between the outer and inner rings are increased to 50 nm and 500 nm, respectively. In this case, the plasmonic eigenmodes of the outer and inner rings alone are more detuned from one another (at 209 meV distance). Due to the weaker coupling, only a slight plasmon energy shift of 53 meV is observed. Fig. 3(c) shows the results for an even further reduced inner-ring size. The distance and entire wire length difference between the outer and inner rings are increased to 80 nm and 680 nm, respectively. The resonances of the outer and inner rings alone are now strongly detuned (at 380 meV distance). Therefore, the coupling between them is strongly reduced and the induced plasmon energy shift is only 18 meV. The plasmonic modes of this DSRR can be regarded as a linear superposition of the plasmons of its constitute inner and outer rings alone. It is clear that the interaction strength between the plasmons is strongly influenced by the spatial distance and the resonant energy detuning between them. With the increased spacing and geometric length difference between the outer and inner rings, the plasmon energy shift is decreased. This corresponds to a decreased coupling strength.

The amplitudes of the DSRR resonances in the reflectance spectra are also influenced by the interaction between the outer and inner rings. For the symmetric mode |ω₊〉 of the DSRR, the induced electric dipoles in the outer and inner rings oscillate in phase; therefore the resonance amplitudes are increased. As for the anti-symmetric mode |ω_-〉, the induced electric dipoles in both rings oscillate out of phase, which leads to an electric field of the DSRR similar to an electric quadrupole rather than an electric dipole. As the coupling efficiency of the external light field to this quadrupole is much weaker, the resonance amplitude is not as pronounced as that of the out ring alone. This is consistent with the observation in Fig. 3(a) and (b), where the amplitude of resonance |ω_-〉 (|ω₊〉) decreases (increases) relative to that of |ω₁〉 (|ω₂〉) in the coupling case.

 

Fig. 4. Calculated reflectance spectra of the DSRRs with two different inner-ring orientations: the gap bearing side of the outer and inner rings are in the opposite direction (configuration A) and in the same direction (configuration B). Resonances at lower and higher energies in configuration A (B) are denoted by |ω_-〉 (|ω*_-〉) and |ω₊〉 (|ω*₊〉), respectively. The insets show the dipole oscillations at the corresponding resonances by the plasmon hybridization analysis. The individual spectra are shifted upwards for clarity.

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Fig. 5. Simulated electric maximum peak field distribution of E_z in a plane 30 nm above the structure for resonant modes of: (a) |ω_-〉, (b) |ω₊〉, (c) |ω*_-〉, (d) |ω*₊〉, corresponding to the peaks shown in Fig. 4. The amplitudes are normalized to that of the incident light field.

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Fig. 6. Simulated magnetic maximum peak field distributions of H_z in a plane 30nm above the structure for the resonant modes of (a) |ω_-〉 and (b) |ω*_-〉 corresponding to the lower energy peaks shown in Fig. 4. The amplitudes are normalized to those of incident field.

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2.2. Hybridization for different inner-ring orientations

The picture of the plasmon hybridization also explains intuitively the influence of the orientation of the inner ring with respect to the outer ring on the DSRR resonance. Fig. 4 compares the simulated reflectance spectra for two orientations of the inner-ring. The geometry of the DSRRs used in the simulations are the same as those of Fig. 3(a). We denote the previously used orientation by configuration “A” and the inverted orientation by configuration “B”. The resonances in configuration B at lower and higher energies are denoted by |ω*_-〉 and |ω*₊〉, respectively. In both cases, the introduction of an inner ring leads to two new eigenmodes when compared to the spectra of the outer and inner rings alone in Fig. 3(a). The calculated electric maximum peak field distributions of the resonances for both configurations are shown in Fig. 5. Out-of-phase charge oscillations are observed for both |ω_-〉 and |ω*_-〉, whereas for |ω₊〉 and |ω*₊〉, in-phase charge oscillation are observed. This is consistent with our analysis by the plasmon hybridization picture as indicated by the insets in Fig. 4.

Figure 6 shows the simulated magnetic maximum peak field distributions of resonance modes |ω_-〉 and |ω*_-〉, which correspond to the lower energy resonance peaks shown in Fig. 4. A strong magnetic field confinement is observed in both configurations. However, the magnetic field is confined in the center area of the DSRR in configuration A, but between the bottom arms of the inner and outer ring in configuration B. it is clear that the orientation of the inner ring with respect to the outer ring is not crucial for inducing the magnetic dipole in a DSRR in our discussion.

3. Conclusion

We have demonstrated that the optical response of a DSRR can be analyzed by a simple plasmon hybridization method. The interaction between the localized surface plasmons of the constitute inner and outer ring of a DSRR was discussed based on the variation of the inner-ring geometry. We have analyzed the influence of the inner-ring orientation on DSRR resonances based on the plasmon hybridization of the inner and outer ring. For the magnetic dipole excitation, the orientation of the inner ring is not crucial. The plasmon hybridization has been proven to be a well suitable model for describing the resonance behaviour of SRRs. It offers a simple and intuitive method for understanding the optical responses of coupled systems in metamaterials and provides insight for the design of specific resonance properties.