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 resonance^{15.16} 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: *F*_{pl}(*d* = 0)/ *F*_{pl}(*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.

Considering only the first order of diffraction, since it is of interest in the frequency range up to 300 GHz, where Fano-type resonances have been observed, the dependence of the excited frequencies of the zero, first and second waveguide modes has been calculated. order with substrate thickness The calculation results are shown in Fig. 5 by solid lines. It is seen that the Fano resonance frequencies match the waveguide mode frequencies quite well, especially as the thickness of the substrate increases. This fact strongly supports the proposition that multiple Fano resonances appear as a consequence of the interaction of a broad plasmonic mode with narrow waveguide modes. There is some discrepancy in the calculated mode frequencies compared to the data obtained from the simulated spectra. The point is that the modes are calculated for a dielectric plate, both sides of which are surrounded by air. In the current situation, one side of the plate is covered by metallic SRRs, causing the appearance of plasmonic modes and Fano-like resonances in the transmission spectra. This obviously results in a change in the phase of the wave reflected from the metasurface which can influence the frequency of the excited waveguide mode.^{32}. It is evident that the difference between the results obtained from the modal approximation and from the spectral simulations decreases with increasing substrate thickness. This happens because the angle of incidence at the interface increases with increasing *d*and thus the additional phase shift due to the metallic lattice has a smaller and smaller contribution to the total phase shift accumulated by the beam traveling through the sample as a dielectric waveguide mode.

We measured the transmittance spectrum of the SRR arrays deposited on the PTFE substrate to confirm our theoretical consideration. The thickness of the substrate is 1 mm. The SSAIL technology described in Methods is applied. The experimental results along with the calculated spectrum are shown in Fig. 6. It is seen that three Fano-type resonances are theoretically predicted. The letters a, b and c label those resonances. Their calculated Q factors differ significantly. Very sharp (a) resonance has a Q factor of more than 200, while (b) and (c)—approximately 80 and 30, respectively. As can be deduced from the figure, the sharpest Fano resonance (a) is experimentally indistinguishable. The frequency of the measured Fano resonances (b) and (c) agree perfectly with the simulated data. However, the amplitudes of the resonances are lower than expected, as is often the case in the THz frequency domain.^{fifteen}. We find that the measured Fano peak amplitudes statistically scatter within 10%, due to their high sensitivity to manufacturing technology parameters.

Sharp resonances labeled with the letters d, e, and f have been theoretically predicted with Q factors ranging from about 300 (d) and (e) to 600 (f). Their characteristic frequencies approximately correspond to m = 0, 1 and 3 excited modes in the substrate due to second order diffraction on a 1200 µm period grating deposited on the surface of a 1 mm thick substrate (see Fig. 2 ). Resonances (d), (e), and (f) appear in the transmission spectrum as transmission minima at frequencies where the waveguide modes are excited. However, they are not resolved experimentally due to insufficient precision of sample processing.

In view of the obtained results, we have observed for the first time the appearance of multiple Fano-type resonances on metasurfaces with mirror-oriented resonators formed on sufficiently thick substrates due to the interaction of the waveguide modes with the plasmonic mode. The analysis of the surface currents at resonance (not shown) reveals that the physical reasons for the appearance of the Fano resonance in this article are practically the same as in our previous work, where more details are given on the dipole moment of the currents flowing in the SRR at maximum transmission. and minimum can be found^{fifteen}.