# Voltage Tunable Spoof Surface Plasmon Polariton Waveguide Loaded with Ferroelectric Resonators
## 3 Results and Discussion
### Coupling Mechanism Between BCZT LSSP Resonator and SSPP Waveguide  
To reveal the formation of the Fano response of the LSSP resonator, the coupling mechanism of the near-field EM between the three Fano modes and the SSPP modes in  
Figure 6: Simulation and Fano fitting of (**a**) dipole peak, (**b**) quadrupole peak, and (**c**) hexapole peak of the LSSP resonator, where \(\varepsilon_{1}\) = 3.5.  
plasmonic waveguide is investigated in terms of temporal coupled mode theory (TCMT). The excited Fano modes should stem from the interference of the two transmission paths, as illustrated in Figure 7. An equivalent network is therefore employed to describe the EM field coupling characteristics between the two plasmonic structures, as illustrated in the inset of Figure 7.  
A dielectric loss as well as weak radiative loss of the LSSP resonator are considered azimuthally as a resistance R, where the amplitude damping factor in the LSSP is denoted as \(\gamma\). Regarding the distinction between \(\gamma\) and \(\Gamma\), \(\Gamma\) represents the resonance linewidth, which is a measure of the energy dissipation and directly determines the quality factor (Q) of the resonance. In contrast, \(\gamma\) describes the system's intrinsic damping characteristics, which contribute to the total loss, but are treated as a separate parameter in our model. The normalized EM power due to the interference of the two transmission paths can thus be depicted as follows:  
\[E_{2}=tE_{1}+jrE_{4} \tag{6}\]  
\[E_{3}=tE_{4}+jrE_{2} \tag{7}\]  
\[E_{4}=(1-\gamma)\text{exp}(j\theta)E_{3} \tag{8}\]  
Then, the transmission coefficient \(E_{2}/E_{1}\) can be deduced as follows:  
\[\frac{E_{2}}{E_{1}}=\frac{t+(1-\gamma)\text{exp}(j\theta)}{1+t(1-\gamma)\text{ exp}(j\theta)} \tag{9}\]  
where \(t\) represents the coupling through nonradiative damping channels, \(r\) describes the coupling process from the LSSP resonator back to the plasmonic waveguide, \(j\) stands for imaginary unit, and E stands for electric field strength. For the lossless coupling process between the SSPP waveguide and LSSP resonator, we assume \(t^{2}+r^{2}=1\). The rotated phase delay \(\theta\) distributed along the azimuthal direction of the LSSP resonator is in the range of 0 to \(2\pi\) for the dipole Fano mode. In terms of the low loss of the BCZT substrate in the LSSP resonator, we consider the weak value of R in the equivalent circuit. This parameter represents both the dielectric and radiative losses of the LSSP resonator, thereby influencing the system's damping characteristics. We set \(\gamma=0.05(\omega-\omega_{0})/\omega\)[54],  
Figure 7: When \(\epsilon_{1}\) is equal to 3.5, the calculated (blue line) and simulation results (red line) of the real part \(|E_{2}/E_{1}|\) are shown in terms of TCMT. The inset shows the equivalent network model revealing the near-field EM power transferring between the plasmonic waveguide and the LSSP resonator.  
where \(\omega_{0}\) represents the dipole resonant frequency, and \(t\) = 0.996 to fit the normalized transmission coefficient.  
This evidently indicates that the coupling effect between the SSPP waveguide and the LSSP resonator can deeply trap the power flow, appearing with a strong band notch in the transmission spectrum, as exhibited in Figure 7. Also, the theoretical calculated real part ratio \(|\mathrm{E_{2}/E_{1}}|\) in Equation (9), standing for transmission intensity ratio, matches quite well with the numerically simulated one. It can thus be validated that the asymmetric field fed by the SSPP waveguide could lead to a unidirectional EM coupling into the LSSP resonator, and therefore create an asymmetric resonant line shape, forming the Fano resonant frequencies.