# Experimental demonstration of multiple Fano resonances in a reflected array of split-ring resonators on a thick substrate

In Fig. 2, we present an example of modes that can be excited on a 1000 µm thick substrate with a 1200 µm period grating deposited on its surface. Solid lines show the angle of incidence at the dielectric-air interface of excited waveguide modes in a dielectric plate surrounded by air. The angle of incidence θ at the lowest frequency for any mode is equal to the critical angle, which is indicated by the horizontal dashed line in the figure. The dashed lines show the angles at which the beam is deflected by diffraction from a periodic structure deposited on the surface of the plate. Obviously, the intersection of the dashed and solid curves shows the modes that can be excited on a plate with a periodic metasurface. In this particular case, all six modes can be excited. Two of them (m = 0, 1) appear due to first-order diffraction and four due to second-order diffraction (m = 0–3). Their frequencies and angles. θ are shown in Fig. 2.

The calculated frequency dependencies of the transmittance of the arrays of mirrors formed on different thicknesses of a substrate are shown in Fig. 3. For clarity, the curves are offset in the ordinate axis relative to each other. As can be seen in the figure, the previously observed Fano resonance15.16 in arrays formed on a relatively thin substrate, it shifts towards lower frequencies with increasing substrate thickness. First-order plasmon resonance behaves in a similar way. However, as the thickness of the substrate increases, instead of one Fano-type resonance, two, and in samples on the thicker substrate, even three clear Fano resonances can be distinguished.

Since we have studied metasurfaces formed on thick substrates, they should also exhibit Fabry-Perot resonances. However, due to the low dielectric permittivity of the substrate, the depth of the bandwidth modulation caused by these resonances is not large. This is confirmed by the computational results shown in Fig. 4. In addition to the spectrum calculated for a metasurface formed on a 1.2 mm thick substrate, the Fabry–Perot resonance-mediated bandwidth modulation is shown. . It is of the order of 10%, and its influence on the metasurface transmittance is not significant in the frequency range where sharp Fano resonances are observed.

From the calculated spectra, shown in Fig. 3, we determined the frequency dependence of the Fano resonance and the first plasmon resonance on the thickness of the substrate. The symbols in Fig. 5 show these results. As Fig. 5 shows, the plasmonic resonance frequency decreases with increasing thickness of the dielectric substrate until the thickness reaches about 100 μm. A further increase in thickness does not influence the plasmonic frequency. It can be assumed that for d > 100 μm, the effective dielectric permittivity of the interface can be expressed as the average of the dielectric permittivities on both sides of the interface. $${\varepsilon}^{*}=(\varepsilon +1)/2,$$ where ε is the permittivity of the dielectric substrate, and unity corresponds to the relative dielectric permittivity of free space. Considering the plasmon resonance as the resonance of the LC circuit, it is clear that by increasing d we are changing the capacitance of the equivalent circuit, while the inductance remains unchanged. Therefore, formally the dependence of the plasmon resonance frequency of ε* can be expressed $${f}_{pl}=1/\left(2\pi \sqrt{LC}\right)\sim \frac{1}{\sqrt{{\varepsilon }^{*}}}$$. Taking into account that ε= 2, the resonance frequency can be obtained to decrease by a factor of 1.225 when d is increasing. Surprisingly, this is precisely the same as the relationship obtained from the simulation results: Fpl(d = 0)/ Fpl(d> 100 μm) = 82/67 = 1.224.

As seen in Fig. 5, compiled from the spectral dependencies of transmittance (ref. to Fig. 3), the Fano resonance frequencies demonstrate a much stronger dependence on substrate thickness than the first plasmon resonance. . Consequently, it can hardly be explained by the variation of the effective dielectric permittivity. However, a periodic array of SRRs deposited on the dielectric leads to diffraction of electromagnetic radiation, and diffracted rays falling on the substrate at an angle greater than the total internal reflection angle cannot escape from the substrate. Therefore, the waveguide modes are excited in the dielectric, which interacts with the plasmon resonance (north= 3) giving rise to the Fano resonances mentioned above.