# Voltage Tunable Spoof Surface Plasmon Polariton Waveguide Loaded with Ferroelectric Resonators
## 3 Results and Discussion
### Design of LSSP Resonator with BCZT Ferroelectric Compound Substrate  
The following design of the LSSP resonator starts with the design of a compound tunable ferroelectric substrate. As shown in Figure 3, a bilayer compound substrate was constructed using a laminated BCZT and PCB substrate. Then, the LSSP metal structure was printed on the surface of the compound substrate. Therefore, the substrate permittivity of the LSSP resonator can be flexibly adjusted by the applied voltage. According to the superposition principle of EM theory, the final effective permittivity of the compound substrate can be designed by controlling the relative thickness ratio of each layer, as depicted in Equations (1) and (2):  
\[\varepsilon_{i}=\sum_{i}(\varepsilon_{i}h_{i}P_{e,i}) \tag{1}\]  
where \(e_{i}\), \(h_{i}\), and \(P_{e,i}\) (\(i\) = 1, 2, 3) refer to the permittivity, laminated thickness, and filling coefficient of the \(i\)th layer in the LSSP resonator, respectively. As can be seen from Figure 3b, since the value of the dielectric permittivity of the whole substrate employed in the simulation is actually an equivalent average value, the equivalent dielectric permittivity can be adjusted by the \(P_{e,i}\) value, which ranges between 0 and 1 [58]. \[P_{e,i}=\frac{(1/2)\int\limits_{V}\varepsilon_{r,i}\overset{\rightarrow}{ \underset{V}{\overset{\rightarrow}{E}}}\times\overset{\rightarrow}{E}dV}{ \sum\limits_{i}\left((1/2)\int\limits_{V}\varepsilon_{r,i}\overset{\rightarrow }{\underset{E}{\overset{\rightarrow}{E}}}\times\overset{\rightarrow}{E}dV} \right)}=\frac{\int\limits_{S}\varepsilon_{r,i}E_{\theta}^{2}\times rdS}{\sum \limits_{i}\left(\int\limits_{S}\varepsilon_{r,i}E_{\theta}^{2}\times rdS \right)} \tag{2}\]  
where \(\varepsilon_{r,i}\) (\(i\) = 1, 2, 3) denotes the permittivity of the \(i\)th layer in the compound resonator, \(E^{\star}\) represents the complex conjugation of the electric field \(E\), while \(E_{\theta}\) depicts the component  
Figure 2: The dielectric permittivity and loss tangent (tan\(\delta\)) of the BCZT ferroelectric resonator in the 0.1 GHz–1 GHz frequency range. of the electric field oriented along the \(\theta\) direction. From the above equation, it can be revealed that the precise adjustment of the equivalent dielectric permittivity of the resonator can be achieved by judiciously designing the thickness and filling coefficient of each layer in the composite LSSP. For example, when the compound substrate is composed of two layers, where the lower layer is a PCB dielectric layer with a permittivity of 2.2 and thickness of h\({}_{1}\) = 0.5 mm, and the upper layer is a BCZT ferroelectric layer with a permittivity of 190 in the working frequency range and with a thickness of h\({}_{2}\) = 5.4 \(\upmu\)m, then the permittivity of the compound substrate can be calculated from Equations (1) and (2), and the result exhibits an equivalent permittivity value of 3.5. Then, as the applied voltage is loaded on BCZT layer, the permittivity will experience a 60% reduction, and the equivalent permittivity of the compound substrate will become 2.9. We numerically compared the ECS properties of the LSSP resonator with the compound substrate and with a single substrate of the equivalent permittivity, as illustrated in Figure 3b. The result reveals that the two LSSP resonators with the above two different substrates actually possess similar ECS properties, which indicates the equivalence of the two substrates and verifies the validity of the design strategy of the compound tunable ferroelectric substrate. Also, it is worth pointing out that although the regulation range employed in the LSSP ferroelectric resonator is from 3.5 to 2.9, which is similar to that of the nematic liquid crystals [38], the former actually possesses a larger regulation range than that of the latter, since the permittivity variation range of the former can be adjusted by the thickness ratio in the laminated composition.