1. Introduction

Metamaterials attract significant research interest for their ability to generate electromagnetic (EM) responses that are normally unattainable in nature [1]. Despite recent advances, their wider utilization requires further improvements in many aspects. Enabling broadband, low-loss operation has been of utmost priority but so far its realization has been impeded by the resonator nature of metamaterials. Recently the concept of electromagnetically-induced transparency (EIT) was introduced to metamaterial design to overcome the loss problem [2]. EIT refers to the formation of a transmission window inside a medium’s absorption band upon the application of a secondary excitation and the destructive interference between the excitation routes. The strong dispersion and reduced group velocity that accompany EIT are potentially useful in many applications. Realizing conventional quantum EIT requires extensive experimental setups whose complexity has hindered a wide utilization of the effect. Since EIT stems from coupled resonances, analogous effects can be realized using classical systems such as spring-mass or RLC oscillators [3]. The metamaterial can also be configured to induce metamaterial-based EIT (MB-EIT). Several MB-EIT schemes have already been demonstrated at frequencies in RF [4–7], near-infrared [8], THz [9], and visible [10] regimes.

Common to all MB-EIT schemes is the exploitation of the dark mode, a hybridization of two resonance modes, either in electric field or current, that are equal in strength but opposite in phase [11]. Resonators supporting such modes are frequently referred to as dark resonators (DRs). Dark modes are inherently dipole-inactive and therefore won’t directly couple to EM waves for excitation or re-radiation. Their excitation requires special methods such as placing a dipole-active resonator whose resonance mode can easily be excited and transferred to a nearby DR through near-field coupling [6–10], making the DR structure asymmetric [4,5], or inducing heavily hybridized plasmon modes [12]. Once excited, the dark mode can induce MB-EIT by interacting with the excitation or other resonator modes, resulting in high transmission with strong dispersion within a limited span of frequency called an EIT window. The setups for MB-EIT are much simpler than those for their quantum counterparts. Moreover, the MB-EIT characteristics can be tuned by adjusting the metamaterial’s geometry, rather than material composition, which is a huge advantage in practical applications.

2. Bull’s eye metamaterial structure

Among many MB-EIT platforms, axially symmetric dual-rings [13] offer interesting potential. A dual-ring structure is shown in Fig. 1(a) as an example. Under excitation by normally incident, linearly polarized EM waves, charge motions or electrical currents are induced along the direction of polarization. Since the two rings are not identical, one ring will experience a stronger coupling to the incident optical excitation. The second ring, only weakly coupled to the incident optical field, will produce currents opposite to those in the first ring due to ring-to-ring inductive coupling. This concept of anti-parallel currents produced via inductive coupling is shown in Fig. 1(a). Such a current distribution occurs only over a very narrow range of frequency and serves as the dark mode which, in turn, induces EIT. The axial symmetry of the bulls-eye structure ensures that the whole process is totally insensitive to the excitation wave’s direction of polarization. Four-fold symmetric versions that are partially insensitive to the direction of polarization also exist [14,15].

 

Fig. 1 (a) A dual-ring as an example of symmetric multi-ring metamaterials. The opposite directions of induced currents (dotted lines) indicate the formation of a dark mode. (b) A schematic view of the bull’s eye metamaterial. The excitation wave’s directions of propagation and polarization are also specified.

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We note the space within the inner ring can accommodate more rings, as shown in Fig. 1(b), and hence support more MB-EIT windows at higher frequencies, which will greatly extend the MB-EIT bandwidth. In this Letter, we aim to confirm the feasibility of multiple MB-EIT from a single metamaterial in the form of multiple concentric rings, or a bull’s eye structure, and investigate its EIT characteristics. Bull’s eye structures have been studied as focusing elements [16,17] but not, as yet, in the light of MB-EIT.

3. Simulation method and results

The investigation is performed in the THz regime in which the metamaterial is often the only source of strong EM response. The excitation is a normally incident, linearly polarized EM wave whose frequency spans from 0.8 to 3.8 THz. The geometric parameters of the simulated bull’s eye structure are defined in Fig. 1. Bull’s eyes with two to four Au concentric rings with 400 nm thickness were simulated. R₄, d, and g were set to 12, 3, and 3 µm, respectively, so that the lowest frequency resonance occur at c_o /(2⋅π⋅r₁₂) ~1.7 THz, where r₁₂ = 28 µm is the average radius of the two outermost rings R₁ and R₂ and c_o is the vacuum speed of light.

Various materials with indices ranging from those of vacuum to GaAs (n > 3.2) are used as the substrate. The THz characteristic of Au is taken from Ref [18]. For numerical simulations, we used Comsol Multiphysics; a commercial finite element method solver. The transmission is measured by integrating the EM power passing through a circle with radius R₁ placed λ_o/2 beyond the bull’s eye metamaterial.

Figure 2a shows the calculated transmission spectra for three different bull’s eye configurations. The substrate index n_sub is set to 1.0. As expected, the single-ring resonance of R₁ induces strong absorption at the 1.7 THz frequency. R₂ alone induces strong absorption at 2.2 THz as well. In combination, R₁ and R₂ generate a dark mode, which leads to the formation of a transmission peak at 1.75 THz. The resultant transmission spectrum becomes different from the sum of the R₁ and R₂’s transmission spectra. Additional EIT windows appeared at higher frequencies as more rings are added. Since a dark mode requires a pair of rings, a quad-ring induce three EIT windows. The fact that the lower frequency transmission peaks maintain their original positions as new rings are added indicates that the new EIT windows do not affect the existing ones, which is desirable for expanding the operation bandwidth of metamaterials through MB-EIT. Dual-ring structures consisting of R₂-R₃ and R₃-R₄ have also been simulated. The resultant transmission spectra (not shown in Fig. 2a for clarity) are red-shifted versions of the R₁-R₂ dual-ring response with the peak positions coinciding with those of the overall bull’s eye transmission response. This means that the multi-peaked EIT-like transmission spectrum of the bull’s eye structure is not a simple sum of the dual-ring responses.

 

Fig. 2 (a) Calculated transmission spectra of different bull’s eye metamaterials. It is evident that the added rings brought in new EIT windows without affecting existing ones significantly. (b) The equivalent electric susceptibility χ_e retrieved from the phase response. The curve and its slope indicate a strong dispersion and reduction in velocity within the EIT windows.

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We also investigated the linear dispersion caused by the bull’s eye metamaterial. As was done in Ref [10], we retrieve the equivalent electric susceptibility χ_e from the phase responses. The results are plotted in Fig. 2(b). It is evident that χ_e undergoes rapid changes across the EIT windows. Those spectral ranges exhibiting positive slopes correspond to regimes of increased group index and slowed pulse propagation according to n_g = n_o + ω∙(dn/dω). At 1.8 THz around which the dn/dω slope is almost linear, the bull’s eye metamaterial’s group index is increased by 4.01 over the intrinsic value. Comparable slopes also exist within other EIT windows. Since the slow pulse propagation regimes occur within the EIT windows, the corresponding transmission can be high. The strong dispersion also ensures that the transmission peaks genuinely originate from MB-EIT and dark mode formation.

The dark mode nature of the bull’s eye metamaterial’s EIT behaviors can be further confirmed by simultaneously visualizing the induced currents and ambient EM power flows. Figure 3 shows the simulation results at frequencies (a)-(f) of Fig. 2(a) which correspond to 1.6, 1.7, 1.79, 2.0, 2.35, and 2.75 THz, respectively. The lines and ambient color represent the direction and intensity of EM power flow. The phase and strength of the currents induced along the direction of the excitation electric field (E in Fig. 1(b)) are also rendered on the ring surfaces. To better visualize the induced currents exhibiting varying levels of strength, different scales are used at each frequency. As shown in Fig. 3(a), before the onset of EIT, i₁ and i₂, i.e., the currents on R₁ and R₂, respectively, are already opposite in direction. But the color scale indicates that i₁ is much stronger than i₂, which leads to a dipole-active mode with high absorption. As the frequency is increased, the two currents become comparable in strength, forming a dark mode on the R₁−R₂ pair. The transmission maximum is reached near 1.79 THz. Further increase in frequency made i₂ much stronger than i₁, suppressing the dark mode and the transmission alike. At (d), an i₂-dominant current distribution appears on R₂-R₃ and the cycle repeats itself to form EIT windows at higher frequencies as shown in Figs. 3(e) and (f) as examples.

 

Fig. 3 (a) to (f) show the direction and intensity of EM power flow at four frequency points specified in Fig. 2(a). The phase and strength of the induced currents are also plotted on the ring surfaces. Note that different scales are used for induced currents at different frequencies. The formation of the dark mode is most prominent at (b).

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It is interesting to point out that the frequencies at which the dark modes’ current distributions become totally anti-symmetric always exhibit red-shifts with respect to the transmission maxima. For example, within the shaded EIT window of Fig. 2(a), the most anti-symmetric current distribution was formed at (b) while the transmission maximum occurred at (c). Similar red-shifts can also found in other multi-ring structures [14]. Examination of the field patterns and optical power distributions reveal that the additional transmission beyond what can be achievable through EIT-like transmission stems from the bull’s eye configuration’s ability to attract a small portion of its ambient optical power towards itself. Such a tendency is observed in both Figs. 3(c) and (e). As a result, near the transmission maxima at (c,e), the transmitted power concentration increases considerably, mimicking the annular slit-based subwavelength-scale focusing of EM waves [16,17]. Such a lens-like action and the resultant ability to focus EM waves below diffraction limit will find a number of new applications.

4. Impacts of structural parameters

We also investigate the impacts of the metamaterial’s structural parameters and dielectric environment on its EIT characteristics in light of their spectral tuning and practical use. For this, we simulate the previously used quad-ring bull’s eye on a semi-infinite substrate. As shown in Fig. 4(a) , the refractive index of the substrate n_sub affects the spectral response the most. Higher n_sub induces a significant red-shift to the overall transmission spectra as predicted from effective medium theory. On the other hand, the ring width d is most effective in fine-tuning the EIT spectrum. Figure 4(b) shows the shifts in the transmission spectra induced by varying d from 1 to 4 µm while adjusting g simultaneously to maintain d + g constant at 6 µm. For all cases, n_sub is maintained at 1.0. The consequential changes in average radii generate spectral shifts on the order of 0.1 THz. For both cases, the transmission spectra exhibit degradations in their depths of modulation since no further optimization was applied after adjusting n_sub or d. Scaling up the whole structure leads also to extension of the bull’s eye operation into the RF regime.

 

Fig. 4 Changes in transmission spectra due to variations in (a) the substrate index n_sub and (b) the ring width d

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5. Conclusion

We numerically investigate the EM response of the concentric multi-ring, or the bull’s eye, structure as an extension of the axially symmetric dual-ring metamaterial capable of supporting dark modes and MB-EIT. The calculated transmission spectra reveal that the addition of more concentric rings brings about new transmission windows while not affecting the existing ones, leading to an expansion in the operation bandwidth of the metamaterial. We confirm that the added transmission windows are caused by EIT-like effects based on the existence of strong dispersion, a signature of EIT. The patterns of induced currents on the metamaterial and the ambient EM power distribution also exhibit typical dark mode characteristics, further confirming their EIT nature. We also show that the MB-EIT’s spectral characteristics can be tuned by adjusting the metamaterial’s structural parameters and/or dielectric environment. With all the features in a single embodiment, the bull’s eye structure can be a useful building block in metamaterial technology.