## 3 Metasurface Design  
The metasurface that is going to be analyzed, and where the plasma discharge will be inserted, is the known Sievenpiper mushroom [16]. This structure consists of a metallic patch, normally etched on a dielectric substrate, grounded with a metallic via. When used in a periodic lattice configuration, the existing gap between two adjacent patches behaves as a capacitor and the grounded via as an inductor, which together generate a resonant circuit and therefore induce a high-impedance surface at the correspondent resonant frequency. This condition avoids the propagation of surface waves within a band called stop-band or band-gap. Figure 2 shows a description of the mushroom-like Sievenpiper metasurface and its equivalent electric circuit for two adjacent mushrooms. From this circuit, we can have the equivalent surface impedance \(Z_{s}\) for the structure, as exposed in Equation (5) [1]. However, with this analysis, we cannot obtain properly the stop-band generated by this lattice:  
\[Z_{s}=\frac{j\omega L}{1-\omega^{2}LC} \tag{5}\]  
Figure 1: Plasma permittivity as a function of the \(\omega_{p}/\omega\) and \(v_{p}/\omega\) ratios. (**a**) Real part; (**b**) Imaginary part.  
To evaluate the stop-band generated by this structure, we proceed to do an irreducible Brillouin zone analysis, from which we can retrieve its dispersion diagram and therefore the frequency range where there are no propagating modes [16; 35]. This analysis is done using the software ANSYS HFSS [36], where a unit cell of the mushroom structure is implemented and analyzed. The unit cell is shown in Figure 3 and the obtained dispersion diagram is shown in Figure 4. For this simulation, all the conductors are considered as perfect electric conductors (PEC), while the substrate used is a PTFE composite whose electrical characteristics correspond to a ROGERS 5870, with a relative permittivity \(\varepsilon_{r}=2.33\). To notice that we have added a top plate over the mushroom separated with an air gap, to have a structure similar to the one that will be used on the final simulations. The dimensions used for the unit cell are the following: \(p=4.55\) mm, \(gap=3.5\) mm, \(h_{sub}=3.175\) mm, \(l_{m}=w_{m}=2.8\) mm, \(g=3.5\) mm and \(\phi=0.36\) mm.  
The resulting dispersion diagram shows a stop-band generated between the first and second modes that can propagate, which has a low cutoff frequency of 7.7 GHz and a high cutoff frequency of 12.8 GHz. However, and in order to implement the metasurface, we will assess its behavior in a full-wave wideband simulation. For that, we will design a \(8\times 5\) mushroom lattice to be inserted into a parallel plate configuration of lateral dimensions \(W_{sub}=L_{sub}=50\) mm to be simulated. In Figure 5, the designed mushroom lattice is shown in the simulation configuration used to assess the transmission losses, and therefore the frequencies where the stop-band is generated.  
Figure 4: Dispersion diagram of the three first modes for the proposed mushroom metasurface.  
Figure 3: Proposed Sievenpiper mushroom metasurface. **Left**: cross-view; **Right** top-view.  
Figure 2: Sievenpiper mushroom-like metasurface and its equivalent electric circuit.  
Figure 6 shows the transmission coefficient \(|S_{21}|\) as a function of the frequency for the parallel plate waveguide simulation with the metasurface inserted, compared to the same parallel plate structure without the metasurface. We can see that, for this structure, there is a good agreement between the cutoff frequencies of the stop-band obtained on the eigenmode analysis and in the full wave simulation of the structure.