## 4 Performance and Discussion
### Frequency Response  
Here, the 3 dB-bandwidth (\(f_{\rm 3 dB}\)) and power consumption of the proposed DBOSW modulator with optimized geometric parameters are also confirmed. An equivalent circuit model is adopted to estimate the dynamic response of the DBOSW modulator, which is shown in Figure 13.  
In this model, the capacitance \(C_{\rm air}\) exists between two electrodes and the air, which is 12 fF [37]. The capacitance \(C\) that consists of the two graphene layers and SiO\({}_{2}\) isolation layer includes both the dielectric capacitance \(C_{\rm D}\) and the quantum capacitor \(C_{\rm Q}\). Assuming the graphene layer is undoped (\(n_{\rm s}=0\)), then we have \(C_{\rm Q}=0\). Hence, the capacitance \(C\) can be described by the following capacitor model:  
\[C=\varepsilon_{\rm r}\epsilon_{0}w_{\rm o1}L/d \tag{6}\]  
where \(L=20\) um and \(w_{\rm o1}=1385\) nm are the graphene modulation length and the overlap width of two graphene layers, respectively. \(\epsilon_{\rm r}\) and \(d\) are the permittivity and the thickness of SiO\({}_{2}\) isolation layer, respectively. \(R_{\rm total}\) is the total resistance, which includes the graphene sheet resistance \(R_{\rm s}\) of 100 \(\Omega/\square\)[38], and the metal electrode contact resistance to graphene layer \(R_{\rm c}\) of 150 \(\Omega\)-um [20]. According to Equation (6) below, \(R_{\rm total}\) is calculated to be about 33.85 \(\Omega\)[24; 30], where the effective graphene width \(w_{\rm g}\) is set to be 1.885 um. \[R_{\rm total}=2R_{\rm s}\times\frac{w_{\rm g}}{L}+\frac{2R_{\rm c}}{L} \tag{7}\]  
Figure 12: (a) Re(\(N_{\rm eff}\)) and Re(\(\Delta N_{\rm eff}\)), (b) Im(\(N_{\rm eff}\)) and MPA of the TM mode and complex mode versus the incident wavelength at \(\mu_{\rm c}=0.2\) and 0.6 eV, respectively. Figure 13: The equivalent circuit model. Based on Equation (8), the 3 dB-bandwidth is dominantly limited by \(RC\) delay. According to Equations (5)-(7), the DBOSW modulator shows a high \(f_{\rm 3 dB}\) of 101 GHz. For the proposed device structure, the silica as an isolation layer that is with a relatively low permittivity and a small thickness of 15 nm is favorable to compensate the disadvantages of the large effective width and overlapping area of graphene layers. In addition, the sheet resistance Rs and contact resistance \(R_{\rm c}\) are values adopted in related Ref. [17]. \[f_{\rm 3 dB}=\frac{1}{2\pi R_{\rm total}C_{\rm total}} \tag{8}\]  
The power consumption (\(E_{\rm bit}=C_{\rm total}(\Delta\rm U)^{2}/4\)) is estimated to be 0.271 pJ/bit at \(L=20~{}\upmu\)m, where the applied voltage change \(\Delta\)U is 4.834 V. This is in accordance with the change of chemical potential from 0.2 to 0.6 eV. To clarify the merits of proposed design, the theoretical performance of the reported polarization-insensitive graphene modulators is comprehensively compared with that of this work. As shown in Table 1, the proposed modulator has the largest optical bandwidth and the smallest Re(\(\Delta\)N\({}_{\rm eff}\)). Moderate MD and \(\Delta\)MD also can be obtained. The characteristics of \(f_{\rm 3 dB}\) and Ebit are also better than most reported works. The proposed DBOSW modulator can be implemented through the following fabrication process. The fabrication process starts with a commercial silica wafer. The E-beam evaporation deposits a silver layer with a thickness of 180 nm (\(h_{\rm b}\)) onto the silica surface, serving as a bottom layer. Subsequently, the plasma-enhanced chemical vapor deposition (PECVD) can be used to deposit the 100 nm-thick SiO\({}_{2}\) (\(h\)) middle layer. Next, the thermal evaporation is used to deposit a 200 nm-thick (\(h_{\rm t}\)) silver layer followed by electron-beam lithography (EBL) and inductively coupled plasma etching. The 125 nm-wide metal slot substructure (\(g\)) is then filled with SiO\({}_{2}\) by PECVD. The subsequent surface chemical mechanical polishing as well as the etching to three layers above can form a hybrid ridge with a width of 425 nm (\(w\times 2+g\)).