## 3 Results and Discussion  
As previously mentioned, normal-incidence transmittance measurements were performed on both direct and complementary SRRs in the range of 0.5 to 4 THz at different  
Figure 2: Experimental transmittance spectra for direct (**a**) and complementary (**b**) SRR arrays at different temperatures. In both cases, the fitting curves (empty symbols) at 6 K and 110 K are reported, too. The insets show a sketch of the optical measurement configuration for direct (**a**) and complementary (**b**) SRR arrays with the corresponding electric field polarization: perpendicular with respect to the SRR gap (**a**), and parallel with respect to the complementary SRR gap (**b**). Yellow areas indicate YBCO, and light-blue areas indicate Al\({}_{2}\)O\({}_{3}\) substrate. \(I_{0}(v)\) indicates the incoming radiation, whereas \(I_{T}(v)\) indicates the transmitted one. temperatures, ranging from 6 K to 110 K and crossing \(T_{c}\). For each temperature, the transmittance was normalized with respect to that of the bare Al\({}_{2}\)O\({}_{3}\) substrate, to isolate the SRR contribution. Figure 2 shows the normalized transmittance for both direct and complementary devices at four different temperatures. The fitting curves (empty symbols) at 6 K and 110 K are shown as well. More specifically, Figure 2a reports the transmittance spectra for the direct device, with the electric field polarized perpendicular to the SRR gap, and Figure 2b for the complementary one, with the electric field polarization rotated by 90\({}^{\circ}\), as shown in the insets and in agreement with the Babinet principle [39; 40]. Indeed, the latter implies that the transmittance of the two SRR arrays is complementary given a 90\({}^{\circ}\) rotation of the polarization between the two structures [37; 38]. When the electric field is polarized perpendicular to the SRR gap (for the direct case), two main resonances appear. The one at lower frequency (located at about 1 THz) is called LC resonance [41], since the electric field couples with the geometric capacitance and inductance of the SRR, generating a closed resonant circulating current inside the metallic ring. This resonance is also defined as \"magnetic\" because the time-varying current creates a magnetic dipole moment perpendicular to the ring plane. The second resonance, instead, appears at higher frequency (around 2.7 THz) and is called dipolar because it is characterized by the excitation of an electric dipole along the SRR arm opposite the gap. These features appear as dips in the direct device and as peaks in the complementary one, and become broader and weaker for increasing temperature. The transmittance \(T_{dir}(\
u,T)\) of the direct array (complementary, \(T_{comp}(\
u,T)=1-T_{dir}(\
u,T)\)) can be expressed through the following equation, already used in Reference [42] and here properly modified to take into account LC and dipolar resonance contributions simultaneously:  
\[T_{dir}(\
u,T)=T_{0}\frac{1+(1+\frac{\beta_{LC}}{2})^{2}Q_{LLC}^{2}(\frac{\
u} {v_{LC}}-\frac{v_{LC}}{\
u})^{2}}{1+Q_{L,LC}^{2}(\frac{\
u}{v_{LC}}-\frac{v_{ LC}}{\
u})^{2}}\frac{1+(1+\frac{\beta_{Dij}}{2})^{2}Q_{L,Dip}^{2}(\frac{\
u}{v_{Dip}}- \frac{v_{Dip}}{\
u})^{2}}{1+Q_{L,Dip}^{2}(\frac{\
u}{v_{Dip}}-\frac{v_{Dip}}{ \
u})^{2}} \tag{1}\]  
where \(T_{0}\) is the transmittance level far from the resonances, \(\
u_{res}\) (\(res=\) LC, Dip) is the resonance frequency, \(Q_{L,res}=\frac{v_{res}}{\Gamma_{res}}\) is the loaded quality factor, \(\Gamma_{res}\) being the resonance width, and \(\beta_{res}\) is the coupling coefficient among different resonators. Since the equation is written in a product form, it is still valid when one of the two resonances is absent, with its corresponding term in the product going to 1. This equation, usually used in the microwaves region [42], takes into account both the resonating behavior of the single split-rings and their mutual interaction through the \(\beta\) coefficients.