Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials

Metamaterials have recently enabled coupling induced transparency due to interference effects in coupled subwavelength resonators. In this work, we present a three dimensional (3-D) metamaterial design with six-fold rotational symmetry that shows electromagnetically induced transparency with a strong polarization dependence to the incident electromagnetic wave due to the ultra-sharp resonance line width as a result of interaction between the constituent meta-atoms. However, when the six-fold rotationally symmetric unit cell design was re-arranged into a fourfold rotational symmetry, we observed the excitation of a polarization insensitive dual-band transparency. Thus, the 3-D split-ring resonators allow new schemes to observe single and multi-band classical analogues of electromagnetically induced transparencies that has huge potential applications in slowing down light, sensing modalities, and filtering functionalities either in the passive mode or the active mode where such effects could be tuned by integrating materials with dynamic properties.


The L-C resonant behavior of a single SRR
The Split-ring-resonator (SRR) is the most common resonant element of MMs, as shown in Fig. S1(a).
The SRR can be equivalent to inductor-capacitor (L-C) circuit, the metal loops function as inductors and the gap between the metal strips as capacitors of capacitance C. For the sub-wavelength MMs, the electric field penetrates the whole volume of the metallic structure, thereby resulting in equivalent dipolar response under the collective effect of induced currents. That is, the incident electric field (x-direction) perpendicular to the gap can excite loop currents (i) on the SRR and thereby induce magnetic dipole ( m ), as shown in Fig. S1(a). And because of the currents are not uniform distribution on the front and back strips, the oscillating currents can be equivalent to an oscillating electric dipole along the incident electric field direction. The surface electric currents, electric energy and magnetic energy distribution at the resonance are simulated and plotted in Fig. S1(b), where we observe the surface currents coincide with the description of Fig. S1(a). Moreover, the electric energy concentrates around the gap and the magnetic energy concentrates on the back metallic strip, which demonstrates the SRR is surely support inductor-capacitor (L-C) resonance.   For a single SRR1, the transmission curves show that the resonance is low-Q resonance to high-Q resonance and then non-resonance when the incident polarization angles change from 0° to 90°, as shown in Fig. S2(a) black curves. Similarly, a single SRR2 exhibits the lowest Q resonance at the 60° incidence, and the transmission curves are identical for (30°, 90°), (40°, 80°), and (50°, 70°), showing nice symmetry with respect to 60°. For a single SRR3, it exhibits a non-resonant transmission at 30° incidence, i.e. the incident electric field is parallel to the gap of SRR3. Moreover, the transmission curves show symmetric distribution with respect to the 30° incident polarization angle, and the transmission of SRR2 and SRR3 are identical for the 0° and 90° incidence. Fig. S2(a) shows that both high-Q and low-Q resonance can be tailored by rotating the orientation of the SRR. Despite showing different resonant frequency, two resonances with different Q-factor can still overlap and interfere destructively so that give rise to an interference-based transparent window, as shown in Fig. S2(b). In addition, two identical resonances with higher Q-factor overlap can form a lower Q-factor resonance, such as the resonant SRR combined by SRR2 and SRR3. Under 0° and 90° incidence, the combined SRR shows a higher-frequency lower-Q resonance (the red curves in Fig. S2(b)) compared to the resonance of SRR2 and SRR3 under 0° and 90° incidence (the red and blue curves in Fig. S2(a)). The decrease of Q-factor results from the combined SRR supports more bulky system so that increases the radiate loss. And compared to their single resonance, the resonant frequency of the SRR combined by SRR2 and SRR3 is blue-shift. The frequency shift origins from the fact that the combined SRR system can decrease the total capacitance and inductance. Because both of the capacitances and inductances are parallel connection by joining the SRRs together. Using the relation of

Evolution process of the electric/magnetic energy distribution
To clearly observe the evolution process of the near-field coupling between the SRRs, we simulated the electric/magnetic energy distributions at the transparent frequency with the incident polarization angles using steps of 10°, as shown in Fig S3. The frequencies for every polarization angle can be found in Table 1, as shown in fourth part. Figure S3. The evolution process of the electric energy and magnetic energy distribution by rotating the incident polarization angle from 0° to 90°.

The parameter fitting using coupled oscillator model
By fitting the transmission curves, we use the parameters as shown in Table 1, and the variation tendencies of every parameters versus different incident polarization angles are plotted in Fig. S4.
And the simulated, measured as well as the fitted results are plotted in Fig. S5, where it is clearly shown that the radiated loss of the bright mode (low-Q resonance) appears periodic property for the transparent modes. As we have illustrated, the bright mode will transit when from SRR1 to SRR2 when the incident polarization angle is rotated from 0° to 90°. For the parameters of coupling strength Ω and detuning δ, we can clearly observe that these two parameters were decreasing when we changed the incident polarization angle, which implies that the Q-factor of the transparent window will decrease as the coupling strength Ω and detuning δ decreasing. The transparent window will disappear when the coupling strength Ω deceases to zero. This makes sense because the coupling strength Ω characterize the interfering strength between bright mode and dark mode. Therefore, the transparent window will decease and disappear as the coupling strength deceasing from 1.268 GHz to 0. However, the detuning δ measure the frequency mismatch between bright mode and dark mode, which can be used to characterize the asymmetric degree of transparent window. For our proposed system, this property is clearly observed for the coupled oscillator mode, as shown in Figure S5 the fitting curves. 0.14 --0 0 1.5 Figure S4. The polarization-dependent behavior of fitting parameters. Figure S5. The simulated, measured as well as the fitted transmission spectra. The incident polarization vector angle is rotated with steps of 10°. Figure S6 demonstrates the effective parameters, i.e., the effective refractive index (neff), permittivity (εeff), and permeability (μeff) as a function of frequency. The effective parameters are retrieved through S-parameter retrieval method [1]. As shown in Figure S6, all of the retrieved parameters show strong peaks for the imaginary parts around the transmission dips, which imply strong absorptions and scatterings at these transmission dips. From Figure S6 (a) and (c), it is also observed that the imaginary parts of refractive index and effective permittivity close to zero at the transparent range, which mean the very low loss of the proposed metamaterial at the transparent range.