# Voltage Tunable Spoof Surface Plasmon Polariton Waveguide Loaded with Ferroelectric Resonators
## 3 Results and Discussion
### Design of LSSP Resonator with BCZT Ferroelectric Compound Substrate
Figure 3: **(a)** The variation in the simulated ECS spectra with the permittivity \(\varepsilon_{1}\) of the substrate. **(b)** The comparison of the ECS spectra of the ferroelectric composite dielectric resonator vs. the effective single layer LSSP res-nator. The inset is the schematic diagram of the composite substrate, which consists of two layers. The upper layer is the BCZT ferroelectric layer with a permittivity of 190 and lower layer is the PCB dielectric layer with a permittivity of 2.2. The spoke electrodes of the LSSP resonator is on the surface. The resonator has a radius of 6.25 mm and a thickness of 0.65 mm. **(c)** The electric field distribution of the first three resonant peaks of the proposed resonator when the dielectric constant \(\varepsilon_{1}\) = 3.5. **(d)** The magnetic field distribution of the first three resonant peaks of proposed resonator when the dielectric constant \(\varepsilon_{1}\) = 3.5. **(e)** The electric field distribution of the first three resonant peaks of proposed resonator when the dielectric constant \(\varepsilon_{1}\) = 2.9. **(f)** The magnetic field distribution of the first three resonant peaks of proposed resonator when the dielectric constant \(\varepsilon_{1}\) = 2.9. **(g)** The excitation direction of the plane wave. Bearing the above substrate equivalence in mind, the LSSP can be constructed on a single equivalent substrate instead of the compound substrate for convenience in the following calculation model. As depicted in panels a and b of Figure 4, the configuration of the LSSP resonator consists of three parts; the upper layer is composed of two disjoint electrodes, the middle layer is the equivalent substrate with a permittivity of \(\varepsilon_{1}\) = 3.5 and thickness of t = 0.65 mm, and the bottom layer is composed of a metal disk with radius r\({}_{1}\) and six evenly distributed metal spokes. Figure 4c shows the unit structure diagram of the LSSP resonator, where the geometric parameters are listed in Table 1. The top electrode is composed of two parallel and nonintersecting rectangular metals, in which each metal electrode has a width of 0.06 mm and a gap of 0.7 mm. The permittivity of \(\varepsilon_{1}\) can be regulated by applying voltage at both ends of the electrode. The metal electrode is annealed copper with a thickness of 0.018 mm. The resonant mode of the LSSP can be essentially understood as a standing wave mode propagating in a waveguide with circular arranged periodic units. When the disk circumference L of the LSSP resonator is equal to an integer n multiple the effective wavelength of the propagating mode, the standing wave resonance will be aroused [53]. Thus, the dipole mode, quadrupole mode, and hexapole mode can be excited, where the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure** & **Symbol** & **Size (mm)** \\ \hline the lower surface metal strip length & l\({}_{1}\) & 5.25 \\ the upper metal strip length & l\({}_{2}\) & 4.71 \\ the gap between upper center circle and metal strip & w\({}_{1}\) & 0.1 \\ the upper surface metal strip width & w\({}_{2}\) & 0.08 \\ the upper and lower center metal circle radius & r\({}_{1}\) & 1 \\ the upper metal ring width & w & 0.44 \\ the lower surface metal strip width & a & 0.7 \\ the periodic of the circle slots & p & 6.545 \\ the thickness of the dielectric substrate & t & 0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The denotation and physical dimensions of each part of the designed SSPP waveguide loaded with the LSSP resonator. Figure 4: (a) Schematic diagram of the top of the resonator, with the parameters listed in Table 1. (b) Schematic diagram of the bottom of the resonator with the LSSP structure of periodic straight spokes. (c) Local unit structure diagram of the LSSP resonator with the parameters listed in Table 1. (d) Dispersion curve of the LSSP resonator with a different substrate permittivity \(\varepsilon_{1}\). The substrate thickness is 0.5 mm.