# Voltage Tunable Spoof Surface Plasmon Polariton Waveguide Loaded with Ferroelectric Resonators
## 3 Results and Discussion
### Tunability of SSPP Waveguide Loaded with BCZT LSSP Resonator  
According to the above, this investigation is conducive to the higher tunability of \(\varepsilon_{\text{r}}\) in the BCZT compound resonator, which, in turn, will be beneficial for the following tunability of the plasmonic waveguide. Figure 5 presents the plasmonic waveguide with the BCZT compound resonator loaded symmetrically along the center of SSPP waveguide. The transmitted power distribution was numerically monitored in the x-y plane 0.5 mm above the coupled LSSP resonator at the resonant frequencies. The plasmonic waveguide is settled on a Rogers RT5880 substrate, with a substrate thickness of 0.5 mm, a permittivity of 2.2, and a loss tangent of 0.0009.  
Figure 5: **(a,b)** The schematic structure of the plasmonic waveguide. **(c)** The simulated transmission spectra loaded with the BCZT compound resonator. **(d,e)** The internal electric and magnetic field distribution diagram at three notched points, 1, 2, and 3, respectively, where \(\varepsilon_{1}\) = 3.5. (**f,g**) The internal electric and magnetic field distribution diagram at three notched points 1\({}^{\ast}\), 2\({}^{\ast}\), and 3\({}^{\ast}\), respectively, where \(\varepsilon_{1}\) = 2.9.  
After loading the LSSP resonator, we conducted a numerical simulation on the S parameter of the filter, and the results are shown in Figure 5. The power flow distribution forms counterclockwise closed loops in the LSSP and the first three excited resonant modes are dipole, quadrupole, and hexapole resonant modes, as depicted in Figure 5c. Also, when the dielectric permittivity of the resonator varies from 3.5 to 2.9, the entire resonance peak position undergoes a blue shift, where the resonant frequency of the dipole mode shifts from 3.44 GHz to 3.65 GHz, with a blue shift of 210 MHz, or a 6.1% controllable range. Meanwhile, the resonant frequency of the quadrupole mode shifts from 5.39 GHz to 5.71 GHz, with a blue shift of 320 MHz, or a regulation range of 5.9%, and the hexapole mode shifts from 6.07 GHz to 6.44 GHz, with a tuning range of 6.1%, which is consistent with the variation pattern in the ECS spectra of the LSSP resonator illustrated in Figure 3a. Also, the transmission properties show that the excited dipole, the quadrupoles, as well as the hexapole of the loaded LSSP resonator are all Fano resonance frequencies because of the asymmetrically distributed resonant line shape. Thus, in the spectral domain, the interference between these plasmonic modes can be expressed with an analytical Fano interference model to fit the spectral lines as follows:  
\[T(\omega)=T_{0}+A_{0}\frac{\left[q+2(\omega-\omega_{0})/\varGamma\right]^{2}}{ 1+\ \left[2(\omega-\omega_{0})/\varGamma\right]^{2}} \tag{4}\]  
where \(\omega_{0}\) represents the Fano resonant frequencies, \(\omega\) is the working angular frequency, \(\varGamma\) is the resonance line width, \(T_{0}\) characterizes the background scattering parameter, \(A_{0}\) is the coupling parameter between the background and the Fano resonances, and \(q\) depicts the Breit-Wigner-Fano parameter, which determines the asymmetry of the resonance spectral line.  
The fitting spectral line utilizing the analytical Fano format as well as the simulated spectra are shown in the red and black line, respectively, in Figure 6. As can be seen, a well-matched spectra within the three oscillations (n = 1, 2, 3) can be obtained for the LSSP structure. The resonant frequencies of the dipole, quadrupole, and the hexapole modes center at 3.43 GHz, 5.39 GHz, and 6.06 GHz, respectively, and the corresponding line widths of the modes are 0.0125 GHz, 0.078 GHz, and 0.045 GHz, respectively. Thus, the Q value of the resonator can be calculated using the following formula:  
\[Q=\frac{\omega_{0}}{\varGamma} \tag{5}\]  
The results show that the \(Q\) values of the dipole, quadrupole, and hexapole modes of the LSSP resonator can be calculated as 275, 80, and 135, respectively.